T Hudson
email: hudsont@cermics.enpc.fr

UPSCALING A MODEL FOR THE THERMALLYDRIVEN MOTION OF SCREW DISLOCATIONS

Keywords: July 7, 2016 Screw dislocations, antiplane shear, lattice models, Kinetic Monte Carlo, Large Deviations

We formulate and study a stochastic model for the thermallydriven motion of interacting straight screw dislocations in a cylindrical domain with a convex polygonal crosssection. Motion is modelled as a Markov jump process, where waiting times for transitions from state to state are assumed to be exponentially distributed with rates expressed in terms of the potential energy barrier between the states. Assuming the energy of the system is described by a discrete lattice model, a precise asymptotic description of the energy barriers between states is obtained. Through scaling of the various physical constants, two dimensionless parameters are identied which govern the behaviour of the resulting stochastic evolution. In an asymptotic regime where these parameters remain xed, the process is found to satisfy a Large Deviations Principle. A suciently explicit description of the corresponding rate functional is obtained such that the most probable path of the dislocation conguration may be described as the solution of Discrete Dislocation Dynamics with an explicit anisotropic mobility which depends on the underlying lattice structure.

Introduction

Dislocations are topological line defects whose motion is a key factor in the plastic behaviour of crystalline solids. After their existence was hypothesised in order to explain a discrepancy between predicted and observed yield stress in metals [START_REF] Orowan | Zur Kristallplastizität[END_REF][START_REF] Polanyi | Über eine Art Gitterstörung, die einen Kristall plastisch machen könnte[END_REF][START_REF] Taylor | The mechanism of plastic deformation of crystals. Part I. Theoretical[END_REF], they were subsequently experimentally identied in the 1950s via electron microscopy [START_REF] Hirsch | LXVIII. Direct observations of the arrangement and motion of dislocations in aluminium[END_REF][START_REF] Bollmann | Interference eects in the electron microscopy of thin crystal foils[END_REF]. Dislocations are typically described by a curve in the crystal, called the dislocation line, which is where the resulting distortion is most concentrated, and their Burgers vector, which reects the mismatch in the lattice they induce [START_REF] Price | Theory of Dislocations[END_REF].

Although the discovery of dislocations is now over 80 years distant, the study of these objects remains of signicant interest to Materials Scientists and Engineers today. In particular, a cubic centimetre of a metallic solid may contain between 10 5 and 10 9 m of dislocation lines [START_REF] Hull | Introduction to dislocations[END_REF], leading to a dense networked geometry, and inducing complex stress elds in the material which are relatively poorly understood. Accurately modelling the behaviour of dislocations therefore remains a major hurdle to obtaining predictive models of plasticity on a single crystal scale.

In this work, we propose and study a discrete stochastic model for the thermallydriven motion of interacting straight screw dislocations in a cylindrical crystal of nite diameter. The basic assumptions of this model are that all screw dislocations are aligned with the axis of the cylinder, and that the motion of dislocations proceeds by random jumps between `adjacent' equilibria, with the rate of jumps being governed by the temperature and the energy barrier between states: this is the minimal additional potential energy which must be gained in order to pass to from one state to another. To describe the system, we prescribe a lattice energy functional, variants of which have been extensively studied in recent literature [START_REF] Ponsiglione | Elastic energy stored in a crystal induced by screw dislocations: from discrete to continuous[END_REF][START_REF] Hudson | Existence and stability of a screw dislocation under anti-plane deformation[END_REF][START_REF] Alicandro | Metastability and dynamics of discrete topological singularities in two dimensions: a Γ-convergence approach[END_REF][START_REF] Hudson | Analysis of stable screw dislocation congurations in an antiplane lattice model[END_REF][START_REF] Alicandro | Dynamics of discrete screw dislocations on glide directions[END_REF].

By rescaling the model in space and time, we identify two dimensionless parameters, and with a specic family of scalings corresponding to a regime in which dislocations are dilute relative to the lattice spacing, the time over which the system is observed is long and the system temperature is low, we nd we may apply the theory of Large Deviations described in [START_REF] Feng | Large deviations for stochastic processes[END_REF] to obtain a mesoscopic evolution law for the most probable trajectory of a dislocation conguration.

The major novelties of this work are the demonstration of uniqueness (up to symmetries of the model) of equilibria containing dislocations, a precise asymptotic characterisation of the energy barriers between dislocation congurations, and the rigorous identication of both a parameter regime in which the twodimensional Discrete Dislocation Dynamics framework [START_REF] Rj Amodeo | Dislocation dynamics. i. a proposed methodology for deformation micromechanics[END_REF][START_REF] Van Der Giessen | Discrete dislocation plasticity: a simple planar model[END_REF][START_REF] Cai | Mobility laws in dislocation dynamics simulations[END_REF][START_REF] Bulatov | Computer Simulations of Dislocations, volume 3 of Oxford Series on Materials Modelling[END_REF] is valid, as well as a new set of explicit nonlinear anisotropic mobilities which depend upon the underlying lattice structure. The nonlinearity and anisotropy of the mobilities obtained is in contrast to the linear isotropic mobility often assumed in Discrete Dislocation Dynamics simulations.

1.1. Kinetic Monte Carlo models. The stochastic model we formulate is based on the observation that at low temperatures, thermallydriven particle systems spend long periods of time close to local equilibria, or metastable states, before transitioning to adjacent states, and repeating the same process. It is a classical assertion that such transitions are approximately exponentially distributed at low temperatures, with a rate which depends upon the temperature and energy barrier which must be overcome to pass into a new state; the transition rate from state µ to state ν, R(µ → ν), is given approximately by the formula R(µ → ν) = A(µ → ν) e -βB(µ→ν) , (1.1) where

• β := (k B T ) -1 is the inverse of the thermodynamic temperature of the system, with k B being Boltzmann's constant and T being the absolute temperature; • B(µ → ν) is the energy barrier, that is, the additional potential energy relative to the energy at state µ that the system must acquire in order to pass to the state ν; and • A(µ → ν) is the entropic prefactor which is related to the `width' of the pathway by which the system may pass from the state µ to the state ν with minimal potential energy.

The discovery and renement of the rate formula (1.1) is ascribed to Arrhenius [START_REF] Arrhenius | Über die reaktionsgeschwindigkeit bei der inversion von rohrzucker durch säuren[END_REF], Eyring [START_REF] Eyring | The activated complex in chemical reactions[END_REF], and Kramers [START_REF] Kramers | Brownian motion in a eld of force and the diusion model of chemical reactions[END_REF], and a review of the physics literature on this subject may be found in [START_REF] Hänggi | Reaction-rate theory: fty after kramers[END_REF]. For Itô SDEs with small noise (the usual mathematical interpretation of the correct lowtemperature dynamics of a particle system) (1.1) has recently been rigorously validated in the mathematical literature: for a review of recent progress on this subject, we refer the reader to [START_REF] Berglund | Kramers' law: validity, derivations and generalisations[END_REF].

We may use the observation above to generate a simple coarsegrained model for the thermally driven evolution of a particle system. Begin by labelling the local equilibria of the system, µ, and prescribe a set of neighbouring equilibria N µ which may be accessed from µ, along with the transition rates R(µ → ν), for ν ∈ N µ . Given that the system is in a state µ at time 0, we model a transition from µ to a new state ν ∈ N µ as a jump at a random time τ , where

τ ∼ min ν∈Nµ Exp R(µ → ν) = Exp ν∈Nµ R(µ → ν) and P[µ → ν | t = τ ] = R(µ → ν ) ν∈Nµ R(µ → ν)
.

This denes a Markov jump process on the set of all states: such processes are sometimes called Kinetic Monte Carlo (KMC) models, and are highly computationally ecient for certain problems in Materials Science [START_REF] Arthur | Introduction to the kinetic monte carlo method[END_REF]. As an example of their use, KMC models have recently been particularly successful in the study of pattern formation during epitaxial growth [START_REF] Boateng | Approximating o-lattice kinetic Monte Carlo[END_REF][START_REF] Schulze | Coupling kinetic Monte-Carlo and continuum models with application to epitaxial growth[END_REF]. Due to the ease with which samples from exponential random variables may be computed, KMC models allow attainment of signicantly longer timescales than Molecular Dynamics simulations of a particle system, with the tradeo being that ne detail on the precise mechanisms by which phenomena occur may be lost.

A major hurdle in the prescription of a computational KMC model is the denition of the rates R(µ → ν). In practice, these must be derived or precomputed by some means, normally via a costly ab initio or Molecular Dynamics computation run on the underlying particle system to be approximated. Likewise, a large part of the analysis we undertake here is devoted to rigorously deriving an asymptotic expression for energy barrier B(µ → ν), which then informs our choice of R(µ → ν) using formula (1.1).

1.2. Modeling screw dislocations. In order to use the KMC framework described above to model the motion of dislocations, we must give an energetic description of the system which allows us to dene both corresponding metastable states µ and the energy barriers B(µ → ν). In several recent works [START_REF] Ponsiglione | Elastic energy stored in a crystal induced by screw dislocations: from discrete to continuous[END_REF][START_REF] Hudson | Existence and stability of a screw dislocation under anti-plane deformation[END_REF][START_REF] Alicandro | Metastability and dynamics of discrete topological singularities in two dimensions: a Γ-convergence approach[END_REF][START_REF] Hudson | Analysis of stable screw dislocation congurations in an antiplane lattice model[END_REF][START_REF] Alicandro | Dynamics of discrete screw dislocations on glide directions[END_REF], variants of an antiplane lattice model have been studied in which the notion of the energy of a conguration of straight screw dislocations can be made mathematically precise, and in which screw dislocations may be identied using the topological framework described in [4].

Here, we will follow [START_REF] Hudson | Existence and stability of a screw dislocation under anti-plane deformation[END_REF][START_REF] Hudson | Analysis of stable screw dislocation congurations in an antiplane lattice model[END_REF] in considering the energy dierence E n (y; ỹ) := e∈D n,1 ψ(dy(e)) -ψ(dỹ(e)) ,

which compares the energy of deformations y and ỹ of a long cylindrical crystal with crosssection nD: the scaled crosssection nD ⊂ R 2 is a convex lattice polygon in either the square, triangular or hexagonal lattice, D n,1 denotes a set of pairs of interacting columns of atoms, d is a nite dierence operator, y and ỹ are antiplane displacement elds, and ψ is a periodic intercolumn interaction potential, here taken to be ψ(s) := 1 2 λ dist(s, Z) 2 . We dene a locally stable equilibrium to be a displacement y such that u = 0 minimises E n (y +u; y) among all perturbations which are suciently small in the energy norm Congurations containing dislocations are identied by considering bondlength 1forms associated with dy, the denition of which is recalled in 2.5. In analogy with the procedure described in 1.3 of [START_REF] Price | Theory of Dislocations[END_REF], this construction allows us to dene the Burgers vector in a region of the crystal subject to the deformation y as the integral of the bondlength 1form around the boundary of the region. This denes a eld µ, which we call the dislocation conguration, and we say that the displacement eld y contains the dislocations µ.

The results of [START_REF] Hudson | Analysis of stable screw dislocation congurations in an antiplane lattice model[END_REF][START_REF] Alicandro | Metastability and dynamics of discrete topological singularities in two dimensions: a Γ-convergence approach[END_REF][START_REF] Alicandro | Dynamics of discrete screw dislocations on glide directions[END_REF] demonstrate that there are a large number of locally stable equilibria in this model which contain dislocations for a range of underlying lattice structures. Nevertheless, since these existence results are ultimately all based upon compactness methods, they do not provide a ne description of the equilibria, nor a guarantee of uniqueness up to lattice symmetries, for a xed choice of the dislocation conguration µ. The rst achievement of this work is therefore Theorem 3.1, which provides a novel construction of the equilibria corresponding to dislocation congurations in the particular case where ψ(s) := 1 2 λ dist(s, Z) 2 . This construction uses a form of lattice duality to show that these minima may be characterised as the `discrete harmonic conjugate' (interpreted in an appropriate sense) of lattice Green's functions satisfying Dirichlet boundary conditions on a nite lattice domain. In particular, this representation enables us to show that, given a dislocation conguration, there exist corresponding equilibria which are unique up to lattice symmetries. E n (γ(t); u µ ),

where u µ , u ν are locally stable equilibria containing dislocation congurations µ and ν respectively, and Γ n (µ → ν) is the space of continuous paths connecting these equilibria. The second major achievement of this work is Theorem 3.2, which gives a precise asymptotic formula for B n (µ → ν) as the domain and dislocation conguration are scaled, in terms of the gradient of the renormalised energy [START_REF] Cermelli | Renormalized energy and forces on dislocations[END_REF][START_REF] Sandier | From the Ginzburg-Landau model to vortex lattice problems[END_REF][START_REF] Alicandro | Metastability and dynamics of discrete topological singularities in two dimensions: a Γ-convergence approach[END_REF]. In the course of proving this result, in 5 we constructively demonstrate the existence of transition states u ↑ , such that

E n (u ↑ ; u µ ) = B n (µ → ν).
The construction of u ↑ again uses the form of lattice duality we describe and lattice Green's functions on the nite domain. Moreover, the properties of Green's functions allow us to compute B n (µ → ν)

explicitly in terms of a single nite dierence of the dual lattice Green's function. In Theorem 4.6, we obtain a precise asymptotic description of this nite dierence in terms of the gradient of the continuum renormalised energy as the domain is rescaled, and hence to provide an asymptotic formula for B n (µ → ν). Our strategy for proving Theorem 4.6 is t develop a theory akin to the classical gradient estimates for solutions of Poisson's equation (see 3.4 of [START_REF] Gilbarg | Elliptic partial dierential equations of second order[END_REF]) in a discrete setting.

1.4. Upscaling via a Large Deviations Principle. Once we have obtained the asymptotic representation of B n (µ → ν) given in Theorem 3.2, we apply formula (1.1) to dene the rates R n (µ → ν)

and hence the stochastic model considered. We then seek to understand the behaviour of this model in the regime where the distance between dislocations is signicantly larger than the lattice spacing. Scaling the various physical constants inherent in the model enables us to identify two nondimensional constants which govern the evolution.

Fixing these constants leads us to consider the asymptotic regime in which the temperature is low, the diameter of the cylindrical domain and the spacing between dislocations is large relative with the lattice spacing, and the time over which the process is observed is long. In this regime, we nd that the processes satisfy a Large Deviations Principle, which provides a means of describing the asymptotic probability of rare events in random processes. A general theoretical framework for proving such results has been developed over the last 50 years, and major treatises on the subject describing a variety of approaches include [START_REF] Freidlin | Random perturbations of dynamical systems[END_REF][START_REF] Ellis | Entropy, large deviations, and statistical mechanics[END_REF][START_REF] Dembo | Large deviations techniques and applications[END_REF][START_REF] Feng | Large deviations for stochastic processes[END_REF].

More precisely, a sequence of random variables X n taking values on a metric space M is said to satisfy a Large Deviations Principle if there exists a lower semicontinuous functional

I : M → [0, +∞] such that for any open set A ⊆ M , lim inf n→∞ 1 n log P[X n ∈ A] ≥ -inf x∈A I(x),
and for any closed set B ⊆ M , we have that lim sup

n→∞ 1 n log P[X n ∈ B] ≤ -inf x∈B I(x).
The function I is called the rate function of the Large Deviations Principle, and is called good if each of the sublevel sets {x | I(x) ≤ a} for a ∈ R is compact in M (a property normally referred to as coercivity in the Calculus of Variations literature). The existence of a Large Deviations Principle may be interpreted as saying that, for any Borel set A,

P[X n ∈ A] exp -n inf x∈A I(x) , as n → ∞,
i.e. the probability of observing events disjoint from I -1 (0) becomes exponentially small as n → ∞.

In the setting considered here, the random variables X n correspond to trajectories of the dislocation conguration through an appropriate state space. In order to prove a Large Deviations Principle, we apply the theory developed in [START_REF] Feng | Large deviations for stochastic processes[END_REF] and summarise the main results of this treatise in a form suited to our application in Theorem 3.3. The existence of a Large Deviations Principle is then asserted in Theorem 3.4, which also gives an explicit description of the `most probable' trajectory of the system.

This trajectory corresponds to a solution of the equations usually simulated in the study of Discrete Dislocation Dynamics [START_REF] Rj Amodeo | Dislocation dynamics. i. a proposed methodology for deformation micromechanics[END_REF][START_REF] Cai | Mobility laws in dislocation dynamics simulations[END_REF][START_REF] Bulatov | Computer Simulations of Dislocations, volume 3 of Oxford Series on Materials Modelling[END_REF], with an explicit anisotropic mobility function M L A,B which depends upon the underlying lattice structure.

We conclude our study by discussing the interpretation of this result, and show that the additional regimes identied in [START_REF] Bonaschi | Quadratic and rate-independent limits for a large-deviations functional[END_REF] also apply here: in particular, we show it is possible to recover the linear gradient ow structure normally used in Discrete Dislocation Dynamics simulations [START_REF] Rj Amodeo | Dislocation dynamics. i. a proposed methodology for deformation micromechanics[END_REF][START_REF] Cai | Mobility laws in dislocation dynamics simulations[END_REF][START_REF] Bulatov | Computer Simulations of Dislocations, volume 3 of Oxford Series on Materials Modelling[END_REF] in a further parametric limit, but we argue that in the appropriate parameter regimes, a stochastic evolution problem may be more appropriate to model dislocation evolution. 1.5. Structure and notation. In order to give a precise statement of our main results, 2 is devoted to describing the geometric framework which is both used to describe the Burgers vector of a lattice deformation in our model and the notion of duality which we use in the subsequent analysis.

In 3, we state and discuss our main results. These are Theorem 3.1, which characterises equilibria containing dislocations, Theorem 3.2, which provides a precise asymptotic formula for the energy barrier between equilibria, and Theorem 3.4, which asserts the existence of a Large Deviations Principle for the Markov processes and asymptotic regime we consider. The proofs of these results are given in 4, 5 and 6 respectively.

Since we introduce signicant amounts of notation in order to concisely state our results, Table 1 is provided for convenience.

Preliminaries

As stated in the introduction, the construction of the local minima corresponding to dislocation congurations we give below relies upon a particular dual construction which corresponds in some sense to the construction of a `discrete harmonic conjugate'. This construction is most conveniently expressed using a discrete theory of dierential forms, which also provides the basis for a denition of the Burgers vector of a deformation. The reader already familiar with this theory may wish to refer to Table 1 for our choice of notation and skip to 2.3, where the particular examples necessary for the subsequent analysis are given. 2.1. Lattice complex. We begin by recalling some facts about lattice complexes, which provide the correct tools to study dislocations in the model we consider. Lattice complexes are a particular class of CW complex, which are objects usually studied in algebraic topology, and were dened with a particular view to applications in the modelling of dislocations in crystals in [4]: we follow the same basic denitions and terminology here. For further details on the denitions below, we refer the reader to Section 2 of [4], and for background on such constructions in a general setting, see either the Appendix of [START_REF] Hatcher | Algebraic topology[END_REF], or [START_REF] Munkres | Elements of algebraic topology[END_REF].

To provide some intuition to those less familiar with the notions described here, we remark at the outset that a lattice complex may be thought of as a `skeleton' of sets of increasing dimension which is built on the lattice points and lls R d . The elements of this skeleton are pcells, where p refers to the `dimension' of the particular element. The key idea behind the denition of a lattice complex is that it provides a means by which to make rigorous sense of • the boundaries of sets;

• operators analogous to the gradient, divergence, and curl, and • versions of the Divergence and Stokes' theorems which relate the above notions.

Since these are likely to be familiar, we will point out some analogies with these more familiar calculus concepts along the way. The reader is invited to refer to Figure 1 for an illustration of the particular lattice complexes used in the subsequent analysis.

2.1.1. Construction of a lattice complex. Given a Hausdor topological space S, a 0cell is simply a member of some xed subset of points in S. Higherdimensional cells are then dened iteratively: for p ≥ 1, a pdimensional cell (or pcell) is e ⊂ S for which there exists a homeomorphism mapping the interior of the pdimensional closed ball in R p onto e, and mapping the boundary of the ball onto a nite union of cells of dimension less than p.

A CW complex is a Hausdor topological space along with a collection of cells as dened above, such that S is the disjoint union of all cells. The CW complex is ddimensional if the maximum dimension of any cell is d, and S is referred to as the underlying space of the complex: S p will denote the set of all pcells in the complex.

Each pcell may be assigned an orientation consistent with the usual notion for set in R d , and we write -e to mean the pcell with opposite orientation to that of e. We may dene an operator ∂, called the boundary operator, which maps oriented pcells to consistently oriented (p -1)cells, which intuitively are `the boundary' of the original cell. Similarly, the coboundary operator δ may be dened, mapping an oriented pcell, e, to all consistently oriented p + 1cells which have e as part of their boundary.

We now recall from [4] that a lattice complex is a CW complex such that:

• the underlying space is all of R d , • the set of 0cells forms an ddimensional lattice, and • the cell set is translation and symmetry invariant. Throughout, we will denote such a lattice complex L, and the set of pcells of the corresponding complex will be L p . Due to the translation invariance of L, it will be particularly convenient to consider translations of lattice pcells, so for e ∈ L p and a vector a ∈ R d , we dene e + a := x ∈ R d x = y + a, y ∈ e .

Q r evaluated at e ∈ Ext(Q r 0 ) G µ * Solution to ∆ * G µ * = µ * in W 0 (D * n,0 ) G y Continuum Green's function, solving -∆G y = 1 V δ y in D, G y = 0 on ∂D M n Set of `wellseparated' dislocation positions
We will always assume that we have chosen coordinates such that {0} ∈ L 0 and, abusing notation, we will write 0 to refer to this 0cell.

A second convenient notational convention we will occasionally use is the representation of a 1cell

through its boundary; we write e = [e 0 , e 1 ] to mean e ∈ L 1 such that ∂e = e 1 ∪ -e 0 .

2.1.2. Spaces of pforms and calculus on lattices. For the application considered here, we wish to describe deformations of a crystal. These are appropriately described in the lattice complex framework as pforms, which are realvalued functions on pcells which change sign if the orientation of the cell on which they are evaluated is reversed.

We dene W (L p ) to be the space of all pforms, that is

W (L p ) := f : L p → R f (e) = -f (-e),
for any e ∈ L p .

It is straightforward to check that this is a vector space under pointwise addition. We also dene the set of compactlysupported pforms,

W c (L p ) := f ∈ W (L p ) {e | f (e) = 0} is compact in R d ,
where here and throughout, A denotes the closure of A ⊂ R d . Let A ⊂ L p be nite; then for f ∈ W (L p ), we dene the integral For a 0form on a lattice complex, the dierential is simply the nite dierence operator dened for a pair of nearest neighbours, and in a continuous setting the same operator is the gradient. Similary, δ acting on 1forms is either (the negative of ) the discrete or continuum divergence operator. In a threedimensional complex, both d acting on 1forms and δ acting on 2forms may be thought of as the curl operator.

A f := e∈A f (e).
The bilinear form

(f, g) := Lp f g is welldened whenever f ∈ W c (L p ) or g ∈ W c (L p ). Moreover, if f ∈ W c (L p ) and g ∈ W c (L p+1 ), we
have the integration by parts formula (df, g) = (f, δg);

(2.1) this statement should be compared with that of the Divergence Theorem and variants, using the vector calculus interpretation of d and δ given above. Furthermore, by dening the space L 2 (L p ) := f ∈ W (L p ) (f, f ) < +∞ , this bilinear form denes an inner product. It is straightforward to show that this is then a Hilbert space with the induced norm, which we denote u 2 := (u, u) 1/2 . We recall the denition of the Hodge Laplacian as the operator

∆ : W (L p ) → W (L p ) with ∆f := (δd + dδ)f (2.2)
when p = 0 and p = m, and in the cases where p = 0 and p = m, ∆ = δd and ∆ = dδ respectively. Note that, in a continuum setting, this denition of the Laplacian agrees with the interpretation of d as the gradient on 0forms and δ as the negative of the divergence on 1forms. Any function satisfying ∆f = 0 on A ⊂ L p is said to be harmonic on A.

Finally, 1 e will always denote the pform 1 e (e ) := ±1 e = ±e, 0 otherwise. 2.2. Dual complex. The common notion of duality which occurs in algebraic topology relating to CW complexes is that of the cohomology. This is usually presented as an abstract algebraic structure, since it is only this structure which is needed to deduce topological information about a CW complex.

In some cases it may also be given a more concrete identication, which will be particularly important for the subsequent analysis.

Given an mdimensional lattice complex, when possible, we dene the dual complex as follows:

• For any e ∈ L m , let e * := e x dx, the barycentre of set e in R d , and let

L * 0 := {e * | e ∈ L m }.
• For a collection of elementary mcells A ∈ L m , let

A * := e∈A e * .
(2.3)

• Now, iterate over p = m -1, m -2, . . . , 0: for each p, let e ∈ L p , and consider δe ∈ L p+1 as a sum of elementary pcells. Find the corresponding cells in L * m-p-1 . Dene e * ∈ L * m-p to be the convex hull of (δe) * with (δe) * removed, assigning e * the same orientation as e. For A, a sum of elementary pcells, we again dene A * via (2.3).

We dene boundary and coboundary operators on the dual lattice complex, ∂ * and δ * , so that ∂ * e * = (δe) * , and δ * e * = (∂e) * .

(2.4)

By construction, * : L p → L * m-p denes an isomorphism of the additive group structure usually dened on lattice complexes (see 2.2 of [4]). The equalities stated in (2.4) may then be interpreted as the statement of the Poincaré duality theorem (see for example Section 3.3 of [START_REF] Hatcher | Algebraic topology[END_REF]), and the construction described above is succinctly represented in the following commutation diagram.

L p+1 L p L p-1 L * m-p-1 L * m-p L * m-p+1 * * * ∂ ∂ δ * δ * δ δ ∂ * ∂ *
Since the dierential and codierential operators inherit features from the structure of the CW complex on which pforms are dened, we now show that similar duality properties hold for the dierential complexes on L and L * . For any f ∈ W (L p ), we dene

f * ∈ W (L * m-p ) via f * (e * ) := f (e).
Again, it may be checked that * : Again, this relationship is concisely expressed in the following diagram.

W (L p ) → W (L * m-p ) is
W (L p+1 ) W (L p ) W (L p-1 ) W (L * m-p-1 ) W (L * m-p ) W (L * m-p+1 ) * * * d d δ * δ * δ δ d * d * 2.
3. Examples: the square, triangular and hexagonal lattices. In the analysis which follows, we focus exclusively on 2dimensional lattice complexes, and in particular the triangular, square and hexagonal lattices denoted Tr, Sq and Hx respectively. Let R 4 and R 6 be the rotation matrices

R 4 := 0 -1 1 0 and R 6 := 1 2 - √ 3 2 √ 3 2 1 2
.

For convenience, we dene e 1 := a 1 := (1, 0) T , and e i := R i-1 4 e 1 for i ∈ {1, 2, 3, 4}, and a j := R j-1 6 a 1 for j ∈ {1, . . . , 6}.

The triangular, square and hexagonal lattices are dened to be

Tr := [a 1 , a 2 ] • Z 2 , Sq := Z 2 ,
Tr * = √ 3 3 R 4 Hx + 1 3 (a 2 + a 3 ), Sq * = Sq + 1 2 (e 1 + e 2 )
, and Hx * =

√ 3R 4 Tr + √ 3
3 (a 1 + a 2 ). Figure 1 illustrates the three lattices and the duality mapping between L and L * . At this point, we give the denitions of some latticedependent constants which will arise during our analysis:

K :=    3 if L = Hx, 4 if L = Sq, 6 if L = Tr. and V :=    2 if L = Hx, 4 if L = Sq, 6 if L = Tr, (2.5)
For convenience, we will write V * and K * to mean the relevant constants for the dual lattice. Note that K is the number of nearest neighbours in the lattice.

2.4. Finite lattice subcomplexes. For the particular application we will consider, we will make use of nite subcomplexes of the full lattice complex, and so we now make precise the notation we use as well as the particular assumptions made throughout our analysis. The reader may nd it useful to refer to Figure 2, which illustrates the construction in a couple of simple cases.

2.4.1. Induced subcomplexes. Given a nite subset A 0 ⊂ L 0 , we dene the induced lattice subcomplex by inductively dening

A p := e ∈ L p ∂e ⊂ A p-1 .
This is a welldened CW complex when the corresponding boundary ∂ A and coboundary δ A operators are dened by restriction, i.e.

∂ A e := ∂e ∩ A p-1 , and δ A e = (δe) ∩ A p+1 for all e ∈ A p .

The induced dierential and codierential operators d A and δ A are then dened in the same way as d and δ, using ∂ A and δ A in place of ∂ and δ, and we may dene the spaces W (A p ) and L 2 (A p ).

It will be convenient to distinguish what we term the exterior and interior pcells of the CW complex A, respectively dened to be

Ext(A p ) := {e ∈ A p | δe = δ A e}, and Int(A p ) := A p \ Ext(A p ).
The former set may be thought of as the `edge' of the lattice subcomplex, and the latter as the `interior' of the lattice subcomplex.

We now dene a subcomplex of the dual lattice complex which we call the dual subcomplex induced by

A 0 . Let A * m := {e * ∈ L * m | e ∈ A 0 }, and inductively dene A * m-p := e * ∈ L * m-p e * ∈ ∂ * a * for some a * ∈ A * m-p+1 } for p ≥ 1.
We remark that this denition is not equivalent to dening sets of sets of dual pcells by directly taking the dual of the primal pcells; however, we do have the inclusion From now on, it will always be clear from the context whether we are referring to the relevant operators on L and L * , or on A and A * , so for the sake of concision, we will suppress A from our notation.

A p * ⊆ A * m-p for each p,
2.4.2. Subcomplexes induced by a domain. We will say that an induced lattice subcomplex is path connected if for any e, e ∈ A 0 , there exists γ ⊂ A 1 such that ∂γ = e ∪ -e , and call such γ ⊂ A 1 a path which connects e and e . We will say a lattice subcomplex is simply connected if for any γ ⊂ A 1 such that ∂γ = ∅, γ = ∂A for some A ∈ A 2 . Throughout our analysis, D will always denote a closed convex lattice polygon, i.e. a nonempty compact convex subset of R 2 which has corners c l ∈ L and internal angles ϕ l where l = 1, . . . , L indexes the corners, following [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF]. We consider the scaled domains nD, for n ∈ N, noting that nD remains a lattice polygon, and denote D n to be the largest induced lattice subcomplex with respect to inclusion such that • D n,p ⊂ nD for all p, • D * n,p ⊂ nD for all p, • D n and D * n are both path connected and simply connected.

It can be shown that such a complex always exists as long as n is suciently large, since D is convex:

we give an example on the righthand side of Figure 2. As for the space of forms dened on L, we dene the inner product and induced norm

(u, v) := Dn,p u v, and u 2 := (u, u) 1/2 .
Since D n,p is nite, these are always welldened; we will also make occasional use of the norm

u ∞ := max e∈Dn,p |u(e)|.
We denote the subspace of pforms vanishing on Ext(D n,p )

W 0 (D n,p ) := u ∈ W (D n,p ) u = 0 on Ext(D n,p ) ,
which is clearly a vector space, and the bilinear form

((u, v)) := D n,1
du dv is a welldened inner product on W 0 (D n,0 ). W 0 (D n,0 ) is thus a Hilbert space with the corresponding norm, denoted u 1,2 := ((u, u)) 1/2 . We now demonstrate positivedeniteness of the inner product, since we will use the resulting version of Poincaré inequality below.

Since D n is pathconnected, for any e ∈ Int(D n,0 ), there exists γ ⊂ D n,1 such that ∂γ = e ∪ -e , with e ∈ Ext(D n,0 ) and #γ ≤ C L 0 n diam(D). For any u ∈ W 0 (D n,0 ), we then have u(e) = γ du, so applying the CauchySchwarz inequality, we have

|u(e)| 2 = γ du 2 ≤ #γ γ |du| 2 ≤ #γ D n,1 |du| 2 .
Integrating over D n,0 , and noting that there exists a constant C L 1 > 0 which depends only on the underlying lattice

L such that #D n,0 ≤ C L 1 n 2 diam(D) 2 , we have D n,0 |u| 2 ≤ C L 2 n 3 diam(D) 3 D n,1 |du| 2 , (2.6)
where

C L 2 = C L 0 C L 1 .
We note that the same inequality also holds for u ∈ W 0 (D * n,0 ) by a similar argument.

2.4.5. Duality for pforms on lattice subcomplexes. We dene the duality mapping * :

W (nD n,p ) → W 0 (D * n,2-p ) as follows: u * (a) = u(e) a = e * ∈ Int(D * n,2-p ), 0 a ∈ Ext(D * n,2-p
). We note that this mapping is welldened since as noted in 2.4.1, a ∈ Ext(D * n,2-p ) if and only if there exists no e ∈ D p with a = e * . This duality mapping denes an isomorphism from W (D n,p ) to W 0 (D * n-p ) as vector spaces; as, in addition (2.7) 2.5. Dislocation congurations. We now recall some denitions from [START_REF] Hudson | Analysis of stable screw dislocation congurations in an antiplane lattice model[END_REF] which will permit us to give a kinematic description of screw dislocations in the setting of our model. Given u ∈ W (D n,0 ), we dene the associated set of bondlength 1forms

(u, u) = nDn,p |u| 2 = Int(nD * n,p ) |u * | 2 = nD * n,p |u * | 2 = (u * , u * ),
[du] := α ∈ W (D n,1 ) α ∞ ≤ 1 2 , α -du ∈ Z . A dislocation core is any positivelyoriented 2cell e ∈ D 2 such that dα(e) = ∂e α = 0. Let µ ∈ W (D 2 ), with µ : D 2 → {-1, 0, +1}. We will say that u is a deformation containing the dislocation conguration µ if ∃ α ∈ [du] such that dα = µ.
The 2form µ represents the Burgers vectors of the dislocations in the conguration, which are the topological `charge' of dislocations; see [START_REF] Price | Theory of Dislocations[END_REF][START_REF] Hull | Introduction to dislocations[END_REF] for general discussion of the notion of the Burgers vector and its importance in the study of dislocations, and [4,[START_REF] Hudson | Analysis of stable screw dislocation congurations in an antiplane lattice model[END_REF] for further discussion of the physical interpretation of this specic denition.

For the purposes of our analysis, we dene sets of admissible dislocation congurations. For > 0, n ∈ N, and b i ∈ {±1} for i = 1, . . . , m, we dene the set

M n (b 1 , . . . , b m ) of 2forms M n (b 1 , . . . , b m ) := µ = m i=1 b i 1 e i e i ∈ D 2 positively oriented, dist(e i , Ext(D n,0 )) ≥ n ,
dist(e i , e j ) ≥ n, for all i, j ∈ {1, . . . , m}, i = j .

Main Results

3.1. Energy and equilibria. As stated in the introduction, we follow [4,[START_REF] Ponsiglione | Elastic energy stored in a crystal induced by screw dislocations: from discrete to continuous[END_REF][START_REF] Alicandro | Metastability and dynamics of discrete topological singularities in two dimensions: a Γ-convergence approach[END_REF][START_REF] Hudson | Existence and stability of a screw dislocation under anti-plane deformation[END_REF][START_REF] Alicandro | Dynamics of discrete screw dislocations on glide directions[END_REF][START_REF] Hudson | Analysis of stable screw dislocation congurations in an antiplane lattice model[END_REF] and consider a nearestneighbour antiplane lattice model for the cylinder of crystal. Let ψ : R → R be given by ψ(x) := 1 2 λdist(x, Z) 2 ; we consider the energy dierence functional

E n (y; ỹ) := D n,1 ψ(dy) -ψ(dỹ) .
This functional is a model for potential energy per unit length of a long cylindrical crystal, and points D n,0 correspond to columns of atoms which are assumed to be periodic in the direction perpendicular to the plane considered. For further motivation of this model, we refer the reader to 1 of [START_REF] Alicandro | Dynamics of discrete screw dislocations on glide directions[END_REF].

Following Denition 1 of [START_REF] Hudson | Analysis of stable screw dislocation congurations in an antiplane lattice model[END_REF], we will say that y ∈ W (D n,0 ) is a locally stable equilibrium if there exists > 0 such that E n (y + u; y) ≥ 0 whenever u 1,2 ≤ .

Due to the periodicity of ψ, we note that any locally stable equilibrium generates an entire family of equilibria: letting z ∈ W (D n,0 ) taking values in H + Z for some H ∈ R, if y is a locally stable equilibrium, then so is y + z. These equilibria are physically indistinguishable, since they correspond to a vertical `shifts' of columns by an integer number of lattice spacings, and a rigid vertical translation of the entire crystal by H. We therefore dene the equivalence relation

u ∼ v if and only if u = v + z, where z : D n,0 → Z + H for some H ∈ R, (3.1)
and denote the equivalence classes of this relation as y .

We recall that Theorem 3.3 in [START_REF] Hudson | Analysis of stable screw dislocation congurations in an antiplane lattice model[END_REF] gives sucient conditions such that locally stable equilibra containing dislocations exist in the case of a more general choice of ψ than that chosen here. Our rst main result is similar, but in addition provides a very precise representation of the corresponding bondlength 1form in the case considered here, and asserts the uniqueness (up to lattice symmetries) of local equilibria containing a given dislocation conguration.

Theorem 3.1. Fix > 0 and D a convex lattice polygon; then for all n suciently large, the following statements hold:

(1) For every 2form µ ∈ M n , there exists a corresponding locally stable equilibrium u µ which contains the dislocation conguration µ;

(2) Each such equilibrium u µ is unique up to the equivalence relation dened in (3.1); and

(3) For any u ∈ u µ , there is a unique bondlength

1form α ∈ [du] satisfying α * = d * G µ * ,
where µ * is the 0form dual to µ, and

G µ * ∈ W 0 (D n,0 ) is the solution to ∆ * G µ * = µ * in Int(D * n,0 ), with G µ * = 0 on Ext(D * n,0 ). (3.2)
Strategy of proof. The proof of this theorem is the main focus of 4. We begin by showing that if u is a locally stable equilibrium containing dislocations µ, then α ∈ [du] must necessarily satisfy

α ∞ < 1 2 , dα = µ on D 2 ,
and δα = 0 on D n,0 .

(

We show that these conditions are satised by at most one α ∈ W (D n,1 ), and using the duality transformation described in 2.

Γ n (µ → ν) := γ ∈ C [0, 1]; W (D n,0 ) γ(0) ∈ u µ , γ(1) ∈ u ν , ∀t ∈ [0, 1], α ∈ [dγ(t)] implies dα = µ or dα = ν .
In the case where we will apply this denition, i.e. where ν -µ = b i [1 q -1 p ] with q * = p * +a * for some nearestneighbour direction a * in the dual lattice, corresponding to a single dislocation `hopping' to an adjacent site, the nal condition on the paths in the above denition ensures that the Burgers vectors of the congurations along the path vary only on the 2cells p and q. In other words, we make the modelling assumption that dislocations move strictly from one site to an adjacent site, and not via a more complicated route.

We dene the energy barrier for the transition from µ to ν for µ, ν ∈ M n to be

B n (µ → ν) := min γ∈Γn(µ→ν) max t∈[0,1]
E n (γ(t); u µ ).

(3.4)

Our second main result concerns an asymptotic representation of this quantity.

Theorem 3.2. Suppose that µ, ν ∈ M n are 2forms such that ν -µ = b i [1 q -1 p ] for some i, where

q * = p * + a * for some nearest neighbour direction a * in L * . For i = 1, . . . , m, let x i ∈ D be such that dist(x i , 1 n e * i ) ≤ 1 n .
Then there exist a constant c 0 which depends only on the underlying lattice complex L such that

B n (µ → ν) = λc 0 + 1 2 λn -1 b 2 i ∇ȳ j (x j ) • a * + i | i =j b j b i ∇G x i (x j ) • a * + o(n -1 ),
where [START_REF] Alicandro | Dynamics of discrete screw dislocations on glide directions[END_REF] λ is given in the denition of ψ, (2) ȳj solves the boundary value problem

∆ȳ j = 0 in D, ȳj (•) = 1 Vπ log(| • -x j |) on ∂D, (3)
G y is the solution to

∆G y = V 2 δ y in D,
with G y = 0 on ∂D, where we recall the denition of V from (2.5), and ( 4) o(n -1 ) satises no(n -1 ) → 0 as n → ∞, uniformly for all µ ∈ M n .

Strategy of proof. The proof of this result is the main focus of 5. Our main task is the explicit construction of a transition state, i.e. u ↓ ∈ W (D n,0 ) such that

E n (u ↓ ; u µ ) = min γ∈Γn(µ→ν) max t∈[0,1] E n γ(t); u µ .
This may be seen as a generalisation of the notion of a critical point, but is not a true critical point, since E n is not dierentiable at u ↓ . Nevertheless, we show that α ∈ [du ↓ ] has a dual which is closely related to the interpolation of d * G µ * and d * G ν * which are solutions of (3.2). This dual representation, combined with the precise asymptotics obtained for d * G µ * in order to prove Theorem 3.1, allow us to derive the expression of B n (µ → ν).

3.3. Remarks on the model. Here, we collect a few remarks concerning the choice of model, the notion of duality we use, and some further links between the results above and the way in which dislocations are modelled in continuum elastoplasticity.

More general potentials. The derivation of the energy we consider as given in 2.2 of [START_REF] Alicandro | Metastability and dynamics of discrete topological singularities in two dimensions: a Γ-convergence approach[END_REF] suggests that potential ψ should be chosen to be smooth, in keeping with the usual assumptions on interatomic potentials. On the other hand, our results rely heavily on the denition of ψ, since the structure of the potential chosen permits us both to prove the characterisation and uniqueness of α given in Theorem 3.1, and to be precise about the set on which B(µ → ν) is attained. This ultimately provides us with a means by which to prove Theorem 3.2.

In spite of this, a result similar to Theorem 3.2 may hold in cases where ψ is more general, but is suciently `close' to the choice made here (see for example the structural assumptions made in 5 of [START_REF] Alicandro | Metastability and dynamics of discrete topological singularities in two dimensions: a Γ-convergence approach[END_REF]). Since the interatomic distances rapidly approach those predicted by linear elasticity as one moves away from a dislocation core (see Theorem 3.5 in [START_REF] Ehrlacher | Analysis of boundary conditions for crystal defect atomistic simulations[END_REF]), and much of the potential energy is carried by the elastic eld at signicant distances from the dislocation core where a harmonic approximation of the energy is valid, heuristically one might expect that the energy barrier should be similar to that given in Theorem 3.2. However, due to the complexity of possible transitions in a more general case, such a result does not seem tractable without very strong assumptions on the potential, and signicant additional technicalities: we therefore do not pursue such results here.

Dynamics in the innite lattice. We remark that a signicant amount of our analysis is devoted to verifying the rst condition in (3.3) holds. This aspect of the proof of Theorem 3.1 would be signicantly simplied if we were to consider the problem in an innite domain, since in this case integral representations of the lattice Green's function are available via Fourieranalysis. Nevertheless, we pursue the evolution on a nite domain here, both because this is a case of physical relevance, and because we are able to demonstrate that the boundary aects the evolution of the conguration in exactly the manner described in 2.1 of [START_REF] Van Der Giessen | Discrete dislocation plasticity: a simple planar model[END_REF].

Equilibrium conditions and geometry. Finally, we remark that the two latter conditions in (3.3) are analogous to the requirement that a continuum strain eld ε satises curl(ε) = µ and div(C : ε) = 0.

These are the conditions usually prescribed on a strain eld ε which contain dislocations described by a measure µ in a linear elastic setting (see for example (1.1) in [START_REF] Cermelli | Renormalized energy and forces on dislocations[END_REF]).

We also note that the precise notion of duality which we use is specic to twodimensional modelling of dislocations, as it is only in this case that L 1 and L * 1 are related by duality. The fact that dual 1cells are orthogonal segments suggest that one should view the construction of α by duality as a version of the CauchyRiemann equations for harmonic conjugate functions.

3.4. KMC model for dislocation motion. With the asymptotic expression for B n (µ → ν) given by Theorem 3.2, we are now in a position to apply (1.1) and formulate the KMC model for dislocation motion we wish to study. In doing so, we make several modeling assumptions, which we now discuss in detail.

Our rst assumption is that the only possible transitions are from µ ∈ M n to ν ∈ M n satisfying ν -µ = b i [1 q -1 p ] for some i ∈ {1, . . . , m}, with p * = q * + a * for some dual lattice nearestneighbour direction a * . This requirement prevents the following possible situations from arising:

(1) Multiple dislocations cannot move together in a coherent way: it seems reasonable to dismiss this possibility since we consider a regime where dislocations are far apart.

(2) Single dislocations cannot make successive correlated jumps over several lattice sites. Since we consider a low temperature regime, we expect the probability of multiple correlated jumps to be negligible.

(3) Dislocations cannot be spontaneously generated in the material during the course of the evolution. In this case, we expect the energy barrier for dipole creation to be higher than that for the motion of single dislocations, so once again, we expect such events to be of very small probability and we therefore neglect them.

We therefore assume that the transition time for a dislocation µ to ν is exponentially distributed

with rate R n (µ → ν) := A n (µ → ν) exp -βB n (µ → ν) , (1.1)
where:

(1) B n (µ → ν) is the energy barrier for the transition from µ to ν dened by (3.4), (2) β = (k B T ) -1 is the inverse thermodynamic temperature, and (3) A n (µ → ν) is the preexponential rate factor which is related to the entropic `width' of the pathways connecting µ and ν, and hence also depends on the inverse temperature β.

Formula (1.1) may be interpreted as follows: the exponential factor encodes the probability that thermal uctuations will result in the system achieving the potential energy necessary for a transition to happen. The prefactor then determines how often such energy levels will lead to a transition: if the passage between states in the energy landscape is very `narrow', then even if the system achieves sucient energy to exit, it may only rarely nd the the pathway to achieve such a transition.

Our second main assumption will be that A n (µ → ν) = A 0 + o(1), as β → ∞ and n → ∞, where A 0 is independent of µ and ν. In the case of a nitedimensional system with a smooth potential energy V , having local minima at x and y, and a saddle point at z with a single unstable direction where the minimal energy barrier between x and y is achieved, the form of the prefactor is (see formula [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF] in [START_REF] Kramers | Brownian motion in a eld of force and the diusion model of chemical reactions[END_REF] for the original onedimensional derivation, or [START_REF] Hänggi | Reaction-rate theory: fty after kramers[END_REF] for an overview of variants derived in a variety of situations)

A(µ → ν) = γ 2 + 4|λ 1 (z)| -γ 2π det ∇ 2 V (x) | det ∇ 2 V (z)| + o(1). (3.5)
Here γ is a friction coecient, with units of time -1 , and λ 1 (z) is the eigenvalue of the Hessian at z which corresponds to the unstable direction. The rate can be reduced if either the eigenvalues of ∇ 2 V (x) are made smaller, reducing its determinant, or if the positive eigenvalues of ∇ 2 V (z) are increased. The former means the potential energy `basin' around x is wider, and the latter means that the `mountain pass' in the energy landscape through which the system can travel most easily to arrive at state ν is narrower. This coecient therefore encodes entropic eects related to the shape of the energy landscape.

In our model, we have shown that there is a discontinuity in the rst derivative at the energy barrier between states, so the exact expression (3.5) cannot be valid; however, in directions for which second derivatives exist, the Hessian of the energy at the transition state and at equilibria are identical, motivating the assumption that A n is constant as n → ∞ and β → ∞. We remark that it is usual in practice (except in symmetric situations where multiple transition pathways with the same energy barrier exist) to choose a constant prefactor in KMC simulations, since eigenvalue decompositions of the Hessian of the energy are often be unavailable, and transition events may be too rare to obtain a suciently accurate numerical estimate of the rate. In order to describe the limit, we dene the set of admissible (macroscale) dislocation positions to be

M ∞ := (x 1 , . . . , x m ) ∈ D m x i ∈ D, |x i -x j | ≥ , dist(x i , ∂D) ≥ , ∀i, j with i = j ,
and identify M n with a subset of this space by the embedding

ι n : M n → M ∞ , where ι n m i=1 b i 1 e i = 1 n e * 1 , . . . , 1 n e * m . (3.6)
It is clear that this map is welldened, and by endowing M n with the metric ι n is an isometric embedding. It is straightforward to see that each of these spaces is compact. Given a dierentiable function f : M ∞ → R, we will write ∂ i f (x) to mean the R 2 valued function such that

∂ i f (x) • a = f (x 1 , . . . , x i + a, . . . , x m ) -f (x 1 , . . . , x m ) + o(|a|) for all a ∈ R 2 .
Let D([0, T ]; M n ) denote the Skorokhod space of càdlàg maps from [0, T ] ⊂ R with values in M n , and denote the space of continuous realvalued functions dened on M n to be C(M n ; R): this is in fact the space of all realvalued functions on M n , since the metric r n induces the discrete topology.

Dene

N µ := ν ∈ M n r n (µ, ν) = d L , where d L =    √ 3 3 L = Tr, 1 L = Sq, √ 3 L = Hx.
Since we expect our modelling assumptions to break down as dislocations either approach one another or the domain boundary, we stop the evolution in such an event. We therefore denote what we term the boundary of M n , dened to be

∂M n := µ = m i=1 b i 1 e i ∈ M n ∃ν / ∈ M n such that r n (µ, ν) = d L .
We consider the sequence of Markov processes Y n ∈ D [0, T ]; M n which are killed on the boundary ∂M n , having innitesimal generator

Ω n : C M n ; R → C M n ; R where [Ω n f ](µ) :=    ν∈Nµ T n R n (µ → ν)[f (ν) -f (µ)], µ ∈ M n \ ∂M n , 0 µ ∈ ∂M n ,
and R n (µ → ν) is dened in (1.1). Since R n (µ → ν) is strictly positive and bounded for all µ, ν ∈ M n and n ∈ N, Ω n is a bounded linear operator. Dening X n t := ι n (Y n t ), it follows that X n t is a Markov process on the space M ∞ .

3.5. The FengKurtz approach to Large Deviations Principles. The last of our main results will be to show that in a specic asymptotic regime, the Markov processes X n satisfy a Large Deviations Principle. To do so, we apply the general theory developed in [START_REF] Feng | Large deviations for stochastic processes[END_REF], which provides an approach to proving such results by demonstrating the convergence of a sequence of nonlinear semigroups. For convenience, we provide the following theorem as a synthesis of the results of Theorem 6.14 and Corollary 8.29 in [START_REF] Feng | Large deviations for stochastic processes[END_REF], adapted to our application.

Theorem 3.3. Suppose that the following conditions hold:

(1) M is a compact subset of R N , viewed a metric space with the usual metric induced by the Euclidean norm. (2) For all n ∈ N, (M n , r n ) is a complete separable metric space and there exists a sequence ι n : M n → M of Borel measurable maps such that for any x ∈ M , there exists

z n ∈ M n satisfying ι n (z n ) → x. (3) For each n ∈ N, Ω n : C(M n ; R) → C(M n ; R) is the innitesimal generator of a Markov
process on M n . Suppose the martingale problem is wellposed, i.e. for any initial distribution µ 0 on M n , the distribution of the Markov process at all later times is uniquely determined, and the mapping from y ∈ M n to trajectories with initial distribution δ y is Borel measurable under the weak topology on the space of probability measures dened on D([0, +∞); M n ). (4) For any n ∈ N, and any f ∈ C(M n ; R), dene the nonlinear generator

H n f (x) := 1 n e -nf (x)
Ω n e nf (x).

(3.7)

Let H be an operator mapping C 1 (M ; R) to the space of bounded measurable functions on M , which is represented as

Hf (x) = H x, ∇f (x) ,
where H : M × R N → R satises the following conditions:

• H is uniformly continuous on the interior of M × B r (0) for all r > 0,

• H is dierentiable in p on the interior of M × R N ,

• H(x, p) = 0 for all p ∈ R N when x ∈ ∂M , and

• For all x ∈ M , p → H(x, p) is a convex function. For each pair (f, g) such that g = Hf , there exists a sequence (f n , g n ) such that g n = H n f n , f • ι n -f n → 0,
g n is uniformly bounded, and for any sequence

z n ∈ M n satisfying ι n (z n ) → x, we have g l (x) ≤ lim inf n→∞ g n (z n ) ≤ lim sup n→∞ g n (z n ) ≤ g u (x),

g(y).

(5) There exists L :

M × R N → [0, +∞] such that L(x, ξ) = sup p∈R N ξ • p -H(x, p) , lim |ξ|→∞ L(x, ξ) |ξ| = +∞ for all x ∈ M and ξ ∈ R N , (3.9)
and for each x 0 ∈ M , there exists

x ∈ W 1,1 ([0, T ]; R N ) satisfying x(0) = x 0 and T 0 L x(t), ẋ(t) dt = 0. (3.10)
Then the sequence of M valued processes

X n := ι n (Y n ) with X n (0) = ι n (y n )
, where y n ∈ M n and ι n (y n ) → x 0 as n → ∞, satisfy a Large Deviations Principle with rate functional

J (x) :=    ∞ 0 L(x, ẋ) dt x ∈ W 1,1 [0, +∞); R N with x(0) = x 0 , +∞ otherwise. (3.11)
6 contains the proof of this result, which amounts to checking that the assumptions above correspond to a series of conditions in [START_REF] Feng | Large deviations for stochastic processes[END_REF].

3.6. Asymptotics for the KMC model. An important condition of Theorem 3.3 is the verication of the convergence of the nonlinear generator, H n . It will be this which motivates our particular choice of regime after we have nondimensionalised the model. Since we are interested in the physically relevant case of observing a large system over a long timescale, we let T n 1 be the timescale of observation, which will be taken relative to the typical timescale on which a dislocation conguration changes. We then multiply all rates by this timescale, which we view as corresponding to observing the process over a long timescale. Now, recalling the denition of the nonlinear generator given in (3.7), suppose that f ∈ C 1 (M ∞ ; R), and and let x n = ( 1 n e * 1 , . . . , 1 n e * m ). By Taylor expanding f , we nd that

H n (f • ι n )(x n ) = m i=1 K * j=1 T n R n (µ → ν) n exp ∂ i f (x n ) • s i,j + o(1) -1 as n → ∞,
where s i,j are the nearest neighbour directions in L * at e * i , and K * is the number of nearest neighbours in L * . Now, by applying Theorem 3.2 and the assumption that

A n (µ → ν) = A 0 + o(1), we have that T n R n (µ → ν) n = T n A 0 e -βλc 0 n exp - βλ 2n ∂ i E(x n ) • s i,j + o Tn n ,
where

E(x) := m i=j b 2 j ȳj (x j ) - m i,j=1 i<j 1 2 b i b j G x i (x j ).
Here, following [START_REF] Alicandro | Metastability and dynamics of discrete topological singularities in two dimensions: a Γ-convergence approach[END_REF] we have dened the renormalised energy, E. -∂ i E(x) is the PeachKöhler force on the dislocation at x i , and hence the gradient ow dynamics of E corresponds to Discrete Dislocation Dynamics. We identify two parameters in this expression,

A := T n A 0 e -βλc 0 n and B := βλ 2n , which are dimensionless, upon recalling that:

(1) T n has units of time, (2) n is the diameter of the domain relative to a xed reference domain, and hence is dimensionless,

(3) β = (k B T ) -1 is the inverse thermodynamic temperature of the system per particle, (4) λ has units of energy per particle, and (5) A 0 is the rate of successful exits from µ → ν, and has units of time -1 . We may think of A 0 e -βλc 0 as being the number of times a dislocation hops a single spacing in the full lattice per unit time, when subject to zero stress. Dividing by n and multiplying by T n , this becomes the proportion of the domain crossed per proportion of time over which the system is observed. The product βλ is the ratio between the potential energy required to allow transitions to occur relative to the available thermal energy; dividing by n gives this quantity relative to the ratio between the lattice spacing and the domain diameter.

We therefore consider the asymptotic regime where n → ∞ with A and B are held constant: assuming that λ and A 0 remain constant as n, β and T n vary, this entails that β and T n tend to innity, and hence we consider a regime in which a large system is observed at low temperature for a long time. In this regime, we obtain the following result, which is proved in 6 as an application of Theorem 3.3. It corresponds to a rigorous validation of the equations of twodimensional Discrete Dislocation Dynamics [START_REF] Rj Amodeo | Dislocation dynamics. i. a proposed methodology for deformation micromechanics[END_REF][START_REF] Bulatov | Computer Simulations of Dislocations, volume 3 of Oxford Series on Materials Modelling[END_REF][START_REF] Cai | Mobility laws in dislocation dynamics simulations[END_REF] for screw dislocations in the given physical parameter regime. Theorem 3.4. Suppose that L = Hx, L = Sq or L = Tr and X n 0 = ι n (x n ) where x n → x 0 ∈ M ∞ as n → ∞. Then the sequence of processes X n t satises a Large Deviation Principle with a good rate function as n → ∞ with A and B xed.

Moreover, in each case, the rate function is minimised by the unique solution of the ODE

ẋ = M L A,B -∇E(x) , with x(0) = x 0 , (3.12)
where E : M ∞ → R is the renormalised energy, and M L A,B is the mobility function

M L A,B [ξ] :=                      m i=1 6 j=1 A sinh(Bξ i • a j )a j L = Hx, m i=1 4 j=1 A sinh(Bξ i • e j )e j L = Sq, m i=1 6 j=1 A sinh(Bξ i • a j )a j 3 j=1 2 cosh(Bξ i • 1 3 a 2j + a 2j-1 L = Tr, (3.13)
where a j and e j are as dened in 2.3.

3.7. Generalised gradient ows and mobility functions. As has been noted in [START_REF] Mielke | On the relation between gradient ows and the large-deviation principle, with applications to Markov chains and diusion[END_REF][START_REF] Bonaschi | Quadratic and rate-independent limits for a large-deviations functional[END_REF], there is a close link between minimisers of Large Deviations rate functionals and gradient ows: we also observe this phenomenon here in the cases where L = Hx and L = Sq. In those cases, it is shown in 6.2 that the rate functional takes the form

J L A,B (x) = B T 0 Φ L A,B ( ẋ) + Ψ L A,B (-∇E x) + ∇E(x), ẋ dt
where M L A,B = ∇Ψ L A,B , and Φ L A,B is the LegendreFenchel transform of Ψ L A,B . This entails that the minimiser of the rate functional is a solution of a generalised gradient ow in the sense described in [START_REF] Mielke | On evolutionary Γ-convergence for gradient systems[END_REF]. Furthermore, as in Theorem 3.1 of [START_REF] Bonaschi | Quadratic and rate-independent limits for a large-deviations functional[END_REF], we may recover a quadratic dissipation in the limit where A → ∞ and B → 0. Proposition 3.5. Suppose that A → ∞ and B → 0 with AB → ω. Then

M L A,B [ξ] → 1 2
ωV * ξ uniformly on compact subsets of R 2m , where V * is the constant V for L * . Consequently, for sucently small T , solutions x : [0, T ] → D m of (3.12) converge uniformly converge to the solution of ẋ = -1 2 ωV * ∇E(x), with x(0) = x 0 as A → +∞ and B → 0 with AB → ω.

The proof of this result follows directly from representing M L A,B via series expansion, and we omit it. Recalling the interpretation of A and B given in 3.6, this could be viewed as suggesting a Large Deviations Principle in the regime where the thermal energy is much larger than the energy barrier to dislocation motion, but where the proportion of the cylinder crossed by a dislocation during the observed time is small. However, recalling the denition of A and B from 3.4, we note that AB = T n n 2 A 0 βλ exp c 0 βλ .

If β, λ and A 0 are xed as n → ∞, choosing AB → ω corresponds to a diusive scaling of the Markov process. We would therefore expect that randomness would persists on a macroscopic scale in such an asymptotic regime, a connection which should be explored in future work.

Finally, we remark that is also possible to consider the other scaling regime analysed in [START_REF] Bonaschi | Quadratic and rate-independent limits for a large-deviations functional[END_REF], in which B → ∞ with log(A) = -c 1 B for some c 1 . In terms of the parameters described in 3.4, this entails that log

T n A 0 n = βλ c 0 - c 1 2n .
Assuming that λ and A 0 remain xed, the only way in which this scaling regime can be attained is if n remains small and xed, with β → ∞ and T n → ∞. Since our analysis relies upon the fact that n → ∞ to ensure that lowerorder terms vanish in R n (µ → ν), we cannot be certain that this limit corresponds to a physicallyrelevant limit, and thus we do not study it here.

4. Proof of Theorem 3.1

In this section, we develop discrete elliptic estimates which will allow us to prove Theorem 3.1; many of the tools used are analogous to those used in the regularity theory of scalar elliptic partial dierential equations. To motivate our approach, and to provide the reader with some intuition, we recall the following result, proved in 3.4 of [START_REF] Gilbarg | Elliptic partial dierential equations of second order[END_REF]:

given Q = {x ∈ R 2 | |x • e 1 |, |x • e 2 | ≤ d}, f ∈ C(Q) and u ∈ C 2 (Q) ∩ C(Q) satisfying ∆u = f , then ∇u(0) • e i ≤ 2 d sup ∂Q |u| + d 2 sup Q |f |.
Our approach will be to apply the discrete analogue of the techniques used to prove this bound, i.e. the maximum principle and elementary potential theory. The application of these techniques in combination with ne residual estimates, will then allow us to conclude the proof. Proof. We employ a discrete version of the Dirichlet principle: extend g to a 0form by dening g(e) := 0 for all e ∈ Int(D n,0 ), and let I : W 0 (D n,0 ) → R be given by

I(v) := 1 2 (d(v + g), d(v + g)) - Int(D n,0 ) f v.
It is straightforward to verify that this functional is twice Gateaudierentiable, with DI(v), u = (d(v + g), du) -

Int(D n,0 )
f u, and D 2 I(v)u, u = ((u, u)).

It follows that I is strictly convex, so has a unique minimiser. By setting u = 1 e for any e ∈ Int(D n,0 ),

this minimiser v satises ∆(v + g) = f in Int(D n,0 ),
and v + g = g on Ext(D n,0 ) by denition.

Our next auxiliary result is to prove the following discrete maximum principle. u(e).

Proof. We prove only the former statement, the proof of the latter being almost identical. Suppose that u satises ∆u ≥ 0 on Int(D n,0 ), and that there exists e ∈ Int(D n,0 ) such that u(e) = min e ∈D n,0

u(e ).

Either e ∈ Ext(D n,0 ), so there is nothing to prove, or else e ∈ Int(D n,0 ). Since ∆u(e) = e ∈D n,0 dist(e ,e)=1

[u(e) -u(e )] ≥ 0, it follows that u(e) = u(e ) for all e with dist(e, e ) = 1. Iterating, and using the fact that D n,0 is nite, we nd that min e∈D n,0 u(e) = min e∈Ext(D n,0 ) u(e), as required. 4.2. Green's functions in the full lattice. We next assert the following lemma, concerning the existence of a full lattice Green's function G L . Lemma 4.3.

Suppose that L = Sq, Tr, or Hx. Then there exists a lattice Green's function

G L ∈ W (L 0 ) such that G L (0) = 0, ∆G L = 1 0 .
In addition:

(1) G L is invariant under the group of lattice point symmetries, i.e. if R : R 2 → R 2 is an orthogonal linear transformation such that RL 0 = L 0 , then

G L (Re) = G L (e).
(2) sup

e∈L 1 |dG L (e)| = K -1 , where K is dened in (2.

5).

( The usual method of constructing G L is via the Fourier transform, and the existence of such a Green's function in the case of Bravais lattices is a classical assertion, as is the symmetry asserted in [START_REF] Alicandro | Dynamics of discrete screw dislocations on glide directions[END_REF]. The bounds (4.1) and (4.2) are proved in Theorem 3.5 of [START_REF] Ehrlacher | Analysis of boundary conditions for crystal defect atomistic simulations[END_REF] for all Bravais lattices, thus covering the cases where L = Sq and L = Tr. It therefore remains to prove (2) and the other results in the L = Hx case: the main observation used here is that Hx may be viewed as a subset of Tr.

Proof. We rst prove (2) for L = Sq and L = Tr. Fix a to be a nearest neighbour direction in the lattice. By the symmetry of G L from (1), we have

1 = ∆G L (0) = KG L (0) -KG L (0 + a) = -KG L (0 + a). Hence dG L ([0, 0 + a]) = K -1 . Now consider v ∈ W (L 0 ) dened to be v(e) := G L (e + a) -G L (e).
It follows that ∆v = 1 0-a -1 0 . Applying Lemma 4.2 on the lattice subcomplex induced by the set B r := e ∈ L 0 dist(e, 0) ≤ r, e = 0, 0 -a , we note that the maximum and minimum of v are attained on Ext(B r,0 ), since G L is harmonic on Int(B r,0 ). Now, applying (3) and letting r tend to innity implies the desired result, noting that v(0) = -v(0 -a) = K -1 .

It remains to prove the theorem for the case where L = Hx. Recall from 2.3 that Hx may be written as

Hx = √ 3R 4 Tr ∪ √ 3R 4 Tr + e 1 ,
and dene G Hx ∈ W (Hx 0 ) to be Using the denition of v Tr , and then applying statement (3) in the case L = Tr as well as (4.3) gives the result.

G Hx (e) :=      3 G Tr (R T 4 e/ √ 3) e ∈ √ 3R 4
4.3. The harmonic measure and interior dierential estimates. We now dene the harmonic measure, which allows us to express functions which are harmonic in a region in terms of their boundary values. In order to do so, we introduce Q r , which should be thought of as `balls of radius r' in the lattice, and are dened to be: on Ext(Q r 0 ).

Q r := [-r, r] 2 ∩ L L = Sq, x ∈ R 2 x • (a 1 + a 2 ) , |x • (a 2 + a 3 )|, |x • (a 3 + a 4 )| ≤ 1 2 r ∩ L L = Tr, Hx.
In addition, ω r e satises the following properties:

(

1) If ∈ W (Q r 0 ) is harmonic in Q r 0 , then for any e ∈ D n,0 , u(e ) = e∈Ext(Q r 0 )
ω r e (e )u(e).

(2) There exists a constant C L > 0 depending only on L such that |dω r e ([0, 0 + a])| ≤ C L log(r)r -2 (4.5)

for any nearestneighbour direction a.

The function ω r e is called the harmonic measure, and enjoys a variety of interpretations, both probabilistic and and analytic: for further detail, we refer the reader to [START_REF] Garnett | of New Mathematical Monographs[END_REF]. Its principal use will be as a tool by which we can estimate the eect of the boundary conditions on the solution in the domain interior.

The existence of ω r e and statement (1) follow directly from Lemma 4.1. In the case where L = Sq, a proof of (4.5) with the improved upper bound C L r -2 is given in Lemma 3 of [START_REF] Alamirew | Harmonic Functions On Square Lattices: Uniqueness Sets and Growth Properties[END_REF] using an explicit construction of ω r e . Further results on the harmonic measure in the square lattice may also be found in Chapter 8 of [START_REF] Lawler | Random walk: a modern introduction[END_REF].

Proof. It remains to prove [START_REF] Alicandro | Metastability and dynamics of discrete topological singularities in two dimensions: a Γ-convergence approach[END_REF]. We use the discrete analogue of Green's formula:

Int(Q r 0 ) u∆v -v∆u = Ext(Q r 0 ) u∆v -v∆u = e∈Ext(Q r 0 )
u(e) |g(e )|.

4.4. Asymptotics for Green's functions on nite subcomplexes. We have now collected the necessary analytical tools with which we will prove Theorem 3.1: our nal auxiliary result is the following precise description of the dierential of solutions to (3.2).

Theorem 4.6. Suppose that µ = m i=1 b i 1 e i ∈ M n , and let G µ * ∈ W (D * n,0 ) be the solution to (3.2). Let e ∈ D * n,0 with [e, e + a] ∈ D * 1 , and let x ∈ R 2 correspond to the dual 0cell e; then we have

d * G µ * ([e, e + a]) = b i d * G L * ([e -x i , e + a -x i ]) + n -1 b i ∇ȳ i ( 1 n x) • a + j =i b j ∇G x j ( 1 n x) • a + O n -1-δ log(n) ,
where:

(1) G L * is the full lattice Green's function for L * , whose existence was asserted in Theorem 4.3, (2) e * i minimises dist(x, e * i ) over all i = 1, . . . , m, (3) for each i,

x i ∈ D satises dist(x i , 1 n e * i ) ≤ 1 n , ( 4
) G y is the continuum Dirichlet Green's function on D corresponding to the point y, i.e. the solution to

-∆G y (•) = V 2 δ(• -y) in D, with G y = 0 on ∂D, (5) ȳi solves -∆ȳ i = 0 in D, with ȳi (s) = 1 Vπ log(|s -x i |) on ∂D, (6)
δ > 0 is an exponent which depends only on ϕ l , the interior angles at the corners of the lattice polygon D, and ( 7) O(n -1-δ log(n)) denotes an error term which is uniform for all µ ∈ M n .

The proof of this result is technical, so we rst outline the main strategy, which is similar in spirit to the approach taken in the proof of Theorem 3.3 in [START_REF] Hudson | Analysis of stable screw dislocation congurations in an antiplane lattice model[END_REF]. We decompose G µ * as a sum of (1) full lattice Green's functions restricted to D * n,0 ,

(2) continuum boundary correctors ȳi , and

(3) discrete correctors.

Each of these components are treated separately, applying Lemma 4.3, the regularity theory of [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF],

the maximum principle proved in Lemma 4.2 and the interior estimate of Lemma 4.5 to analyse each piece. Since the entire proof takes place in the dual complex D * n , for brevity we drop * from our notation throughout.

Decomposition of G µ . For i = 1, . . . , m, let x i ∈ R 2 be the vector corresponding to the point e i . We begin by decomposing

G µ (e) = m i=1 b i G L (e -x i ) + m i=1 ȳn i (e) + u(e) + v(e), (4.7)
where:

(1) G L is the full (dual) lattice Green's function, Regularity of ȳi . We now recall some facts concerning the regularity of ȳj from [25]. Applying Theorem 6.4.2.6 in [START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF], there exists σ ∈ (0, 1) such that ȳi lies in the space

X := C 4,σ (D) + span S l,m m ∈ N, 0 < m < (4 + σ)ϕ l /π ,
where S l,m is given in polar coordinates (r l , θ l ) about the c l , the lth corner of D as S l,m (r l , θ l ) := r mπ/ϕ l l sin mπ ϕ l θ l η(r l ), mπ/ϕ l / ∈ N, r mπ/ϕ l l log(r l ) sin mπ ϕ l θ l + θ l cos mπ ϕ l θ l η(r l ), mπ/ϕ l ∈ N. We recall that ϕ l is the interior angle at c l , and we set η ∈ C ∞ 0 (R) to be a cuto function so that η(x) = 1 for |x| suciently small, and supp{S l,m } ∩ supp{S l ,n } = ∅ for any n, m ∈ N and any l = l .

We note that ȳi only fails to be C 4,σ at the corners of the domain D, and since D is convex, π/ϕ l > 1, which implies that S l,m ∈ C 1,δ (D). Hence ȳi ∈ C 1,δ (D) with δ := min l {π/ϕ l -1} ∈ (0, 1 2 ]. X is a Banach space when endowed with the norm Estimating dG L and dȳ n j . Applying (4.2) for any e j which is not the closest point to e in the support of µ, we have that dG L (e -x j , e + ax j ) -

v + 0<m<(4+σ)ϕ l /π C l,m S l,m X := v C 4,σ (D) + 0<m<(4+σ)ϕ l /π |C l,m |.
1 0 1 Vπ e + ta -x j |e + ta -x j | 2 • a dt log |dist(e -x j , 0)| dist(e -x j , 0) 2 = O n -2 log(n) ,
which holds uniformly for µ ∈ M n since dist(e, e j ) ≥ 1 2 dist(e i , e j ) ≥ 1 2 n. Furthermore, using the homogeneity and regularity of (x, y) → x-y |x-y| 2 to Taylor expand under the integral, we have

dG L ([e -x j , e + a -x j ]) = n -1 1 Kπ 1 n e -x j | 1 n e -x j | 2 • a + O(n -2 log(n)). (4.9)
Using the representation of ȳj ∈ X and Taylor expanding, we have By applying (4.1) and again invoking the denition and regularity of ȳn j to Taylor expand near the boundary, we have that

dȳ n j ([e, e + a]) -n -1 ∇ȳ j ( 1 n e) • a ≤ n -2 L l=1 |dist( 1 n e, c l )| π/ϕ l -2 ȳj X .
g ∞ (Ext(D n,0 )) n -1 ȳ X + O n -1 log(n) ,
where the term is uniform in n for xed . Lemma 4.5 now implies that |du(e)| n -1 log(n) dist e, Ext(D n,0 ) -1 .

(4.12)

Estimating ∆ȳ n j . For the purpose of estimating dv, we rst obtain bounds on ∆ȳ n j . Let e ∈ D n,0 \ Ext(D n,0 ), and x ∈ D be the corresponding vector. We use the regularity of ȳj to Taylor expand, obtaining

∆ȳ n j (e) = m j=1 K i=1 1 0 ∇ȳ j ( 1 n (x + ts i )) • 1 n s i dt, = m j=1 K i=1 1 0 1 2 n -3 ∇ 3 ȳj ( 1 n x)[s i , s i , s i ] + 1 6 n -4 (1 -t) 3 ∇ 4 ȳj ( 1 n (x + ts i ))[s i , s i , s i , s i ] dt, (4.13)
where s i are nearest neighbour directions in the dual lattice, and the terms involving ∇ȳ j and ∇ 2 ȳj cancel respectively by lattice symmetry and the fact that ȳj is harmonic. If the dual lattice is Sq or Tr, then the terms involving ∇ 3 ȳj also cancel, which entails that

∆ȳ n j (e) ≤ 1 6 n -4 m i=1 1 0 (1 -t) 3 ∇ 4 ȳj 1 n (x + ts i ) dt.
By using the description of ȳj as a sum of v ∈ C 4,σ (D) and S j,m , it can be seen that each of the integrands in the estimate above is bounded any e ∈ D n,0 and s i , and moreover

∆ȳ n j (e) ≤ 1 6 Kn -4 ȳ X l dist 1 n e, c l π/ϕ l -4 . (4.14)
Returning to the case where the dual lattice is Hx, we rst Taylor expand to thirdorder to obtain that then for all e ∈ A, we have

|∆ȳ n j (e)| ≤ 1 2 n -3 l dist( 1 n e, c l ) π/ϕ l -3 ȳ X . (4.15)
1 2 n -3 3 i=1 ∇ 3 ȳj ( 1 n e)[a 2i , a 2i , a 2i ] -∇ 3 ȳj 1 n (e + e 1 ) [a 2i , a 2i , a 2i ] = 1 2 n -4 3 i=1 1 0 ∇ 4 ȳj ( 1 n (e + te 1 ))[e 1 , a i , a i , a i ] , ≤ 3 2 n -4 l dist( 1 n e, c l ) π/ϕ l -4 ȳ X .
Using this estimate, and the argument used above in the case where the dual lattice was Tr, for any e ∈ A, we deduce that ∆ȳ n j (e) + ∆ȳ n j (e + e 1 ) ≤ 2n -4 l dist( 1 n e, c l ) π/ϕ l -4 ȳ X .

(4.17)

Estimating dv. It remains to bound dv. We proceed by constructing upper and lower bounds on v by using estimates (4.14), (4.15) and (4.17

G L (• -x) + 1 Vπ log |n diam(D)| ≥ 0 on Ext(D n,0 ).
Next, we dene neighbourhoods of each corner of the domain

B l, := e ∈ Int(D n,0 ) dist( 1 n e, c l ) ≤ . Recalling that δ := min{ π ϕ l -1} ∈ (0, 1 2 ], estimate (4.14) that |∆ȳ n j (e)| n -4 δ-3 ȳj X on Int(D n,0 ) \ l B l, . (4.18)
We now dene v ± (e) :=e ∈D n,0

∆ȳ n j (e ) G L (e -e ) ± C n ,
where C n is a small constant depending upon n that we will choose later. We note that ∆[v-v ± ] = 0, so choosing C n such that v + ≥ 0 and v -≤ 0 on Ext(D n,0 ), Lemma 4.2 would imply that v -(e) ≤ v(e) ≤ v + (e) for all e ∈ Int(D n,0 ).

When the dual lattice is either Tr or Sq, applying estimate (4.18), and summing,

|v ± (e)| ȳ X e ∈Int(D n,0 ) e / ∈ l B l, n -4 δ-3 G L e -e + n -1-δ e∈ l B l, dist(e , nc l ) δ-3 G L e -e + C n n 2 .
Treating each sum separately, we see that

e ∈Int(D n,0 ) e / ∈ l B l, G L e -e e ∈Int(D n,0 ) e / ∈ l B l, log dist(e, e ) + 1 n 2 log(n), e ∈ l B l, |dist(e , nc l )| δ-3 G L e -e log(n) e∈ l B l,r |dist(e , nc l )| δ-3 log(n), recalling that statement (3) of Theorem 4.3 implies that |G L (e)| log |dist(e, 0)|, diam(nD) = O(n),
and the sum on the second line converges since δ ≤ 1 2 < 1.

These estimates imply that

|v ± (e)| ȳ X n -1-δ log(n) + C n n 2 , so choosing C n = O(n -3-δ log(n)) gives |v(e)| = O n -1-δ log(n) , and hence |dv(e)| = O n -1-δ log(n) (4.19)
for all e ∈ D n,1 .

When the dual lattice is Hx, recall the denition of A from (4.16), and set

A := {e ∈ Int(D n,0 ) | e -e 1 / ∈ Int(D n,0 )}.
For any e ∈ A let x ∈ D be the corresponding vector. We apply (4.15), (4.17), and the conclusions of Theorem 4.3 to deduce that

G Hx (e -x )∆ȳ n j (e) + G Hx (e + e 1 -x )∆ȳ n j (e + e 1 ) ≤ |G Hx (e -x )|n -4 |dist( 1 n e, c l )| π/ϕ l -4 + |dG Hx (e -x , e + e 1 -x )|n -3 |dist( 1 n e, c l )| π/ϕ l -3 ≤ log |dist(e, x )|n -4 |dist( 1 n e, c l )| π/ϕ l -4 + log |dist(e, x )| dist(e, x ) n -3 |dist( 1 n e, c l )| π/ϕ l -3 .
By summing over e ∈ A, we obtain e ∈A G Hx (e-x )∆ȳ n j (e )+G Hx (e+e 1 -x )∆ȳ n j (e +e 1 ) ȳj X log(n)n -2 δ-3 +n -1-δ log(n) . e ∈A e / ∈ l B l,

n -3 δ-2 + n -1-δ e ∈A e ∈ l B l, dist(e , nc l ) δ-2 , ȳj X δ-2 log(n)n -2 + n -1-δ log(n) . (4.21)
Putting (4.20) and (4.21) together, and applying similar arguments to that for the other cases above, we deduce that (4.19) also holds in the case where the dual lattice is Hx.

Conclusion.

Combining (4.9), (4.10), (4.12) and (4. [START_REF] Evans | Measure theory and ne properties of functions[END_REF]) and noting that

∇ȳ j (x) + 1 Vπ x -x j |x -x j | 2 = ∇G x j (x),
we have proved Theorem 4.6. Theorem 4.6 implies the following corollary. Corollary 4.7. Given > 0 and a convex lattice polygon D ⊂ R 2 , for all n suciently large,

sup e∈D * n,1 |d * G µ * (e)| < 1 2 for any µ ∈ M n . Proof. Let e * ∈ D * n,
G x i (x) = 1 Vπ log(|x -x i |) + ȳi (x)
, we obtain the estimate

|dG µ * (e)| ≤ sup e∈L * 1 |d * G L * (e)| + n -1 (m-1) Vπ + m i=1 ȳi X + O n -1-δ log(n) ,
where we recall the denition of the norm . X from (4.8). Further, from 4.4 we have that ȳi X is uniformly bounded for x i ∈ x ∈ D dist(x, ∂D) ≥ , and so applying statement (2) of Theorem 4.3, we have the result.

4.5. Proof of Theorem 3.1. We now complete the proof of Theorem 3.1 using the results above.

Our rst step is to verify the necessity of the equilibrium conditions given in (

Let u be a locally stable equilibrium containing the dislocation conguration µ ∈ M n . By inspect- ing the proof of Lemma 5.1 in [START_REF] Alicandro | Metastability and dynamics of discrete topological singularities in two dimensions: a Γ-convergence approach[END_REF], it follows that if du(e) ∈ 1 2 + Z for some e ∈ D n,1 , then there exist lower energy states arbitrarily close to u, and so any α ∈ [du] has α ∞ < 1

2

. By denition, we have that dα = µ. Finally, let v ∈ W (D n,0 ); then for t suciently small, α + tdv ∞ < 1

2

, hence

E n (u + tv; u) = D n,1 ψ(α + tdv) -ψ(α) = D n,1 λt α dv + 1 2 λt 2 |dv| 2 .
It follows that (α, dv) = 0 for any v ∈ W (D n,0 ), hence δα = 0.

Next, we show that if α satises the equilibrium conditions (3.3), then it is unique. Suppose that α and α satisfy (3.3). We dene β = α -α , and note that β * ∈ W 0 (D * n,1 ) satises d * β * = 0 and δ * β * = 0. Since nD * is simply connected, the former condition implies that β * = d * w for some w ∈ W 0 (D * n,0 ), which must satisfy ∆ * w = 0: by the uniqueness of the solution proved in Lemma 4.1, it follows that w = 0, hence β = 0, and thus α = α .

Since * is a bijection between W (D n,1 ) and W 0 (D * n,1 ), there exists α ∈ W (D n,1 ) such that α * = d * G µ * . Furthermore, by using (2.7), we have that Finally, we note that α ∞ = dG µ * ∞ , hence applying Corollary 4.7, it follows that α satises (3.3) if n is suciently large.

To demonstrate that α ∈ [du µ ] for some u µ ∈ W (D n,0 ), x e ∈ D n,0 , and dene u µ (e ) = 0. Using the fact that D n is pathconnected, let γ e be the path such that ∂γ e = e ∪ -e, and dene u µ (e) := γ e α. Letting b = [e 0 , e 1 ] ∈ D n,1 , we nd that

du(b) = γ e 1 α - γ e 0 α, = γ e 1 ∪-γ e 0 ∪-b α + b α.
Noting that ∂(γ e 1 ∪ -γ e 0 ∪ -b) = ∅, we apply the that D n is simply connected to assert that γ e 1 -γ e 0 -b = ∂A, for some A ∈ D 2 , hence

du(b) = α(b) + ∂A α = α(b) + A µ ∈ α(b) + Z.
It follows that α ∈ [du µ ]. To prove that u µ is unique up to the equivalence (3.1), we note that if α ∈ [du] and α ∈ [dv], then by the denition of a bondlength 1form (see 2.5), it follows that du(e) = dv(e) + Z(e) for all e ∈ D n,1 , with Z : D n,1 → Z.

Moreover, dZ = 0, so Z = dz, and it is straightforward to check that z : D n,0 → H + Z for some H ∈ R, completing the proof of Theorem 3.1.

Proof of Theorem 3.2

This section is devoted to the proof of Theorem 3.2, and we proceed in several steps. We rst demonstrate that there exists u which `solves' the minmax problem used to dene B n (µ → ν) via a compactness method. We then identify necessary conditions for such a solution, and show that these necessary conditions identify a pair of bondlength 1forms. The required bondlength 1forms are then constructed via duality using an interpolation of dual Green's functions, and we verify that the necessary conditions are satised to conclude. 5.1. The minmax problem. To establish existence of a solution, we transform the problem via taking the quotient of the space of deformations with respect to the equivalence relation dened in (3.1); in other words, we identify deformations `up to lattice symmetries'. This space turns out to be compact, hence the existence of a critical point follows directly by a compactness argument. 5.1.1. Quotient space. Recall from (3.1) that ∼ is the equivalence relation on u ∈ W (D n,0 ) u ∼ v whenever u = v + z + C for some z : D n,0 → Z and some C ∈ R.

Dene the quotient space Q := W (D n,0 )/ ∼ of equivalence classes u ; we claim that this is a metric space when endowed with the metric

d Q ( u , v ) = α 2 ,
where α ∈ [du -dv], for any u ∈ u and v ∈ v . The triangle inequality follows, and hence the metric is welldened. Moreover, the space is complete and totally bounded, so the HeineBorel theorem applies, and Q is compact. We recall that the mapping u → u is the natural embedding of W (D n,0 ) in Q.

If u ∼ v, then du ∈ dv + Z, and hence [du] = [dv]. Symmetry is immediate, and 0 ∈ [du -dv] implies that u -v ∼ 0, hence d Q (u, v) = 0 implies that u ∼ v.
u, u , v ∈ W (D n,0 ) such that u ∼ u , E n (u, v) = E n (u , v). It follows that the mapping E n : Q → R, E n ( u ) := E n (u, v) for some u ∈ u
is welldened. Suppose that u ∈ u , and u ∈ u , and that α ∈

[du -du ]. Then E n ( u ) -E n ( u ) = D n,1 ψ(du + α) -ψ(du ) C D n,1 |α| α 2 = d Q u , u ,
where we use the fact that ψ is uniformly Lipschitz, and then apply the CauchySchwarz inequality. It follows that E n is uniformly Lipschitz on Q.

; Q) continuous functions from [0, 1] to Q, with the usual metric d ∞ Q (γ, γ ) := sup t∈[0,1] d Q γ(t), γ (t) .
The mapping γ → max t∈[0,1] E n (γ(t)) is continuous with respect to this metric, since E n is uniformly continuous on Q.

We suppose that n is large enough such that the conclusion of Theorem 3.1 holds, and write u µ to mean the equivalence class containing u µ , which is the set of all locally stable equilibria corresponding to the dislocation positions µ ∈ M n . Dene the sets of paths

Γ n (µ → ν) := γ ∈ C([0, 1]; Q) γ(0) = u µ , γ(1) = u ν , α ∈ [dγ(t)] has dα ∈ {µ, ν}, ∀t ∈ [0, 1] ;
this should be thought of as the set of paths through phase space which move dislocations from µ to ν without visiting any intermediate states.

5.1.4. Existence. We recall that the energy barrier was dened to be

B n (µ → ν) = inf γ∈Γn(µ,ν) sup t∈[0,1]
E n (γ(t); u µ ).

The following lemma now demonstrates the existence of a transition state.

Lemma 5.1. If n is suciently large, for any µ, ν ∈ M n such that Γ n (µ → ν) is nonempty, there

exists u ↑ ∈ W (D n,0 ) such that E n (u ↑ ; u µ ) = B n (µ → ν).
We will call u ↑ a transition state for the transition from µ to ν.

Proof. We rst note that since E n γ(t) .

Q is compact, C([0, 1]; Q) is compact. By assumption, Γ n (µ → ν)
Since γ → max t∈[0,1] E n (γ(t)) is continuous, there exists a minimiser γ ∈ argmin max

t∈[0,1] E n (γ(t)) γ ∈ Γ n (µ → ν) , where Γ n (µ → ν) denotes the closure of Γ n (µ → ν) in Q. As t → E n γ(t) is also continuous, it follows that there exists u ↑ ∈ u ↑ with u ↑ = γ(t * ) ∈ Γ n (µ → ν) for some t * ∈ [0, 1], which satises E n (u ↑ ; u µ ) = min γ∈ Γn(µ→ν) max t∈[0,1] E n γ(t) = min γ∈Γn(µ→ν) max t∈[0,1]
E γ(t); u µ .

5.2. Necessary conditions. We now identify necessary conditions on the transition states identied in Lemma 5.1. We remark that the proof of the following lemma relies crucially on the particular choice of potential ψ.

Lemma 5.2. Suppose that u ↑ ∈ W (D n,0 ) is a transition state for the transition from µ to ν, where ν -µ = b i [1 q -1 p ] and q * = p * + a for some nearestneighbour direction in the dual lattice, a. Then

u ↑ ∈ u ∈ W (D n,0 ) α ∈ [du] has α(l) = ± 1 2
, where l * = [p * , q * ], and moreover there exist exactly two α ↑ , α ↓ ∈ [du ↑ ], satisfying [START_REF] Alicandro | Dynamics of discrete screw dislocations on glide directions[END_REF] dα ↑ = µ, dα ↓ = ν, (2) δα ↑ (a) = δα ↓ (a) = 0 for all a± / ∈ ∂l, (3) δα ↑ (e 0 ) + δα ↑ (e 1 ) = 0 and δα ↓ (e 0 ) + δα ↓ (e 1 ) = 0 for e 0 and e 1 such that l = [e 0 , e 1 ], and (4) -α ↑ (l) = α ↓ (l) = 1 2 b i . and in particular, ∆ * G t (p * ) = b j (1 -t) and ∆ * G t (q * ) = b j t.

As in Lemma 5.2, set l ∈ D n,1 with l * = [p * , q * ]. Since Lemma 5.2 entails that the the transition state must have α ↓

(l) = 1 2 b i , we choose t ∈ [0, 1] such that ∆ * G t (p * ) + d * G t (l * ) = 1 2 b j , ⇔ (1 -t)b j + d * G t (l * ) = 1 2 b j , (5.1)
and

∆ * G t (q * ) -d * G t (l * ) = 1 2 b j , ⇔ tb j -d * G t (l * ) = 1 2 b j .
(5.2)

Solving, we nd that

t = 1 2 b j + d * G µ * (l * ) b j + d * G µ * (l * ) -d * G ν * (l * )
.

(5.3) Again, by duality we also have

Noting that d * G µ * (l * ) = d * G L * ([0, 0+a])+o(1) =
d[α ↑ -α µ ](p) = ∆ * G t (p * ) + d * G t (l * ) + 1 2 b j -∆ * G µ * (p * ) = 0, d[α ↑ -α µ ](q) = ∆ * G t (q * ) -d * G t (l * ) -1 2 b j -∆ * G µ * (p * ) = 0. Similarly, d[α ↓ -α ν ] = 0. It follows therefore that there exist v ↑ and v ↓ such that α ↑ ∈ [du µ + dv ↑ ], and α ↑ ∈ [du µ + dv ↓ ].
We also note that if a / ∈ ±∂l and l = [e 0 , e 1 ], then

δ[α ↑ -α µ ](a) = (d * ) 2 (G t -G µ * )(a * ) = 0, δ[α ↑ -α µ ](e 0 ) + δ[α ↑ -α µ ](e 1 ) = (d * ) 2 [G t -G µ * ](e * 0 ) + (d * ) 2 [G t -G µ * ](e * 1 ) = 0.
It follows that α ↑ and α ↓ satisfy conditions (1)(4) of Lemma 5.2, and hence we have constructed the bondlength one forms corresponding to the transition state.

Finally, we dene γ ∈ Γ n (µ → ν) via γ(t)

:= u µ + 2tv ↑ t ∈ [0, 1 2 ], u µ + v ↑ + (2t -1)v ↓ t ∈ ( 1 2 , 1],
which demonstrates that Γ n (µ → ν) is nonempty, and hence B n (µ → ν) exists.

We recall that a comparison principle is the statement that viscosity sub and supersolutions of (6.1) are globally ordered. When x lies on the boundary of M , H vanishes, hence F δ (x, r, p) = r-h(x) for all x ∈ ∂M Thus any subsolution f and supersolution f must satisfy f (x) ≤ f (x) for all x ∈ ∂M.

Theorem 3.3 in [START_REF] Michael | User's guide to viscosity solutions of second order partial dierential equations[END_REF] asserts that F δ satises a comparison principle on the interior of M if (1) There exists γ > 0 such that γ(r -s) ≤ F δ (x, r, p) -F δ (x, s, p)

for all x in the interior of M , r, s ∈ R and p ∈ R N ; and (2) There exists a function ω : [0, +∞) → [0, +∞) with lim t→0 ω(t) = 0, such that F δ x, r, α(x -y) -F δ y, r, α(x -y) ≤ ω α|x -y| 2 + |x -y| for all x and y in the interior of M , γ ∈ R and r ∈ R.

It is straightforward to verify that the former condition holds with γ = 1 for F δ as dened in (6.1); since we have assumed uniform continuity and dierentiability of H on the interior of M × R N , the second condition is also straightforward to verify, since M × {α(x -y) | x, y ∈ M } is compact in M × R N . Thus, a comparison principle holds on the entirety of M , and it follows that the conclusion of Theorem 6.14 holds, i.e. the sequence of Markov processes satises a Large Deviations Principle.

To conclude that the rate function takes a variational form, we will rst apply Corrolary 8.29. This requires us to check the conditions of Theorem 8.27. In the case considered here, the operators H † = H ‡ = H, thus we need only check that Conditions 8.9, 8.10 and 8.11 hold.

To verify Condition 8.9, we note the following, which demonstrate that each of the subconditions (1)(5) are satised.

(1) In our case, Af (x, u) = u • ∇f (x), (6.2) which is welldened on C 1 (M ; R); this set separates points, so Condition 8.9.1 is veried.

(2) Here, Γ := M × R N : for any x 0 ∈ M , dene x(t) = x 0 for all t ∈ [0, +∞) and λ(ds × du) = δ 0 (du) × ds to verify Condition 8.9.2.

(3) Condition 8.9.3 is satised by assumption [START_REF] Mp Ariza | Discrete dislocations in graphene[END_REF].

(4) Condition 8.9.4 is trivially satised by taking K = M , since M is compact.

(5) Condition 8.9.5 is satised due to our assumption that L satises the growth condition (3.9).

Condition 8.10 is satised upon taking λ(ds × du) = δ ẋ(s) (u)ds × du, where x is the function whose existence was asserted in (3.10).

Finally, to verify Condition 8.11, we follow the LegendreFenchel transform approach described in 8.3.6.2. Dene

q f (x) := ∂ p H x, ∇f (x) x ∈ M \ ∂M, 0 x ∈ ∂M.
This is welldened, is continuous on the interior of M , and there exists a solution to the ODE ẋ = q f (x) with x(0) = x 0 and x(t) ∈ M for all t ∈ [0, +∞), for any initial condition x 0 ∈ M . Therefore, Condition 8.11 holds upon choosing x to be this solution, and λ(du×ds) = δ q f (x(s)) (du)×ds.

We have thus veried the assumptions of Corollary 8.29, which allows us to conclude that the rate functional has a variational representation as a control problem, given in (8.18) in [START_REF] Feng | Large deviations for stochastic processes[END_REF]. To conclude that the rate function takes the precise form we have here, where the solution to the minimisation problem over admissible controls is stated explicitly, we may apply an identical proof to that given for Theorem 10.22, noting that, under our assumptions, I 0 (x) is 0 if x = x 0 , and +∞ otherwise. We have therefore proved Theorem 3.3.

6.2. Proof of Theorem 3.4: the cases L = Hx and L = Sq. When the lattice is Hx or Sq, the respective dual lattices are isomorphic to Tr and Sq, and hence the set of nearest neighbour directions in the dual is always the same; on the other hand, since Tr * is isomorphic to Hx, which is a multi-lattice, dierent techniques are required, and we therefore treat this case separately in the following section.

Take f ∈ C 1 (M ∞ ; R); as M ∞ is compact, there exists a uniform modulus of continuity ω f : [0, +∞) → [0, +∞) with lim r→0 ω f (r) = 0, such that for x = (x 1 , . . . , x m ) ∈ M ∞ and y ∈ M ∞ . Thus, for all x ∈ M ∞ , n ∈ N and s ∈ R 2 such that x i + 1 n s ∈ M ∞ , we have

f (x 1 , . . . , x j + 1 n s, . . . , x m ) -f (x) -1 n ∂ j f (x) • s ≤ |s| n ω f |x -y| .
As n → ∞ in the parameter regime we prescribed in 3.6, with x in the interior of M ∞ , and a sequence x n ∈ M n with dist ι n (x n ), x → 0 as n → ∞, we have

H n f • ι n (x n ) = Hf (x) + O ω f dist(ι n (x n ), x) , setting Hf (x) := m i=1 K * j=1 A exp ∂ j f (x) • s j -1 exp -B ∇ȳ i (x i ) + k =j b i b k ∇G x j (x i ) • s j ,
where s j are the nearest neighbour directions in L

* . If x n ∈ ∂M n , then H n f • ι n (x n ) = 0, so we dene Hf (x) := 0 for x ∈ ∂M ∞ .
Recall the denition of ȳi as the solution to

-∆ȳ i = 0 in D, and ȳi (x) = 1 Vπ log(|x -x i |) on ∂D.
Following the approach of [START_REF] Alicandro | Metastability and dynamics of discrete topological singularities in two dimensions: a Γ-convergence approach[END_REF] and [START_REF] Alicandro | Dynamics of discrete screw dislocations on glide directions[END_REF], we dene the renormalised energy for x ∈ M ∞ to be

E(x) := - m i,j=1 i =j b i b j 2Vπ log |x i -x j | + m i,j=1
b i b j ȳi (x j ).

Recalling the denition of G y from Theorem 3.2, we have that

∂ i E(x) = ∇ȳ i (x i ) + j =i b i b j ∇G x j (x i ).
For x in the interior of M ∞ , this allows us to write

Hf (x) = m i=1 K * j=1 A cosh ∂ i f (x) -B∂ i E(x) • a j -cosh -B∂ i E(x) • a j .
We dene the Hamiltonian,

H L A,B : M ∞ × R 2m → R, as H L A,B (x, p) :=      m i=1 K * j=1 A cosh p i -B∂ i E(x) • s j -cosh -B∂ i E(x) • s j x ∈ M ∞ \ ∂M ∞ , 0 x ∈ ∂M ∞ ,
where p = (p 1 , . . . , p m ) with p i ∈ R 2 for each i. The Lagrangian is the LegendreFenchel transform (for further details on this topic, see 26 in [START_REF] Rockafellar | Convex analysis[END_REF]) of the Hamiltonian of H L A,B with respect to its second argument, i.e.

L L

A,B (x, ξ) := sup

p∈R 2m ξ, p -H L A,B (x, p) ,
where •, • is the inner product on R 2m given by ξ, p := m i=1 ξ i • p i . We now follow [START_REF] Bonaschi | Quadratic and rate-independent limits for a large-deviations functional[END_REF] in dening

Ψ Hx A,B , Ψ Sq A,B : R 2m → R via Hx A,B (f ) := m i=1 6 j=1 A B [cosh(Bf i • a j ) -1], Ψ Sq A,B (f ) := m i=1 4 j=1 A B [cosh(Bf i • e j ) -1],
which permits us to write

H L A,B (x, p) = B Ψ L A,B 1
B p -∇E(x) -Ψ L A,B -∇E(x) . Ψ L
A,B is (strictly) convex, and hence has a convex dual, given by its LegendreFenchel transform, denoted Φ L A,B . Moreover, by properties of the LegendreFenchel transform, we have that

∇Φ L A,B (ξ) = 1 B p -∇E(x) ⇔ ξ = ∇Ψ L A,B 1
B p -∇E(x) ⇔ p ∈ argmax p { ξ, p -H L A,B (x, p )}.
Using this fact, we have that

L L A,B (x, ξ) = B∇Φ L A,B (ξ) + B∇E(x), ξ -H L A,B x, B∇Φ L A,B (ξ) + B∇E(x) = B∇Φ L A,B (ξ) + B∇E(x), ξ -BΨ L A,B ∇Φ L A,B (ξ) + BΨ L A,B -∇E(x) Using the property that u, v = Ψ L A,B (u) + Φ L A,B (v), we then have L L A,B (x, ξ) = BΦ L A,B (ξ) + BΨ L A,B -∇E(x) + B ∇E(x), ξ ,
which leads us to dene the rate functional J

L A,B : D([0, T ]; M ∞ ) → R with J L A,B (x) :=    ∞ 0 L L A,B x, ẋ dt x ∈ W 1,1 [0, +∞); M ∞ , +∞ otherwise, and L L A,B (x, q) :=    Φ L A,B (q) + Ψ L A,B -∇E(x) + ∇E(x), q x ∈ M ∞ \ ∂M ∞ , 0
x ∈ ∂M ∞ and q = 0, +∞ x ∈ ∂M ∞ and q = 0.

We may now state the following result, asserting a Large Deviation Principle for the sequence of processes in this case. Lemma 6.1. Suppose that L = Hx or L = Sq, and that X n 0 = ι n (x n ) where x n → x 0 ∈ M n as n → ∞ with δ > . Then the processes X n t satisfy a Large Deviations principle with good rate function J L A,B .

This result is very similar to those obtained in Chapter 5, 2 of [START_REF] Freidlin | Random perturbations of dynamical systems[END_REF], or 10.3 of [START_REF] Feng | Large deviations for stochastic processes[END_REF], the main dierence being that there is no diusive part of the process. We also refer the reader to [START_REF] Bovier | Sample path large deviations for a class of markov chains related to disordered mean eld models[END_REF] for related results concerning a discretetime model on a lattice.

Proof. As stated, we wish to apply Theorem 3.3 to prove Lemma 6.1. The main conditions we are required to check are (3), ( 4) and ( 5), since conditions (1) and ( 2) are straightforward to check when Verifying Condition (3). M n is a nite state space and is endowed with a topology which is equivalent to the discrete topology. Therefore, by the Lebesgue Decomposition Theorem (see for example Theorem 3 of 1.6 in [START_REF] Evans | Measure theory and ne properties of functions[END_REF]) all measures µ on M n may be expressed as µ(dx)

M n = M n , M = M ∞ ,
y∈M n f (y)δ y (dx),
where the density f : M n → R is continuous (as are all realvalued functions on M n ). Next, x a probability measure µ y (dx, 0) = δ y (dx) on M n , and dene

µ y (dx, t) = z∈M n f y (z, t)δ z (dx), where f y : M n × [0, +∞) → R solves ∂ t f y (x, t) = Ω T n f (x, t)
with Ω T n being the adjoint of Ω n , and f y (x, 0) = δ y (x). It is straightforward to see that f y exists and is unique, since Ω T n is a bounded linear operator, and therefore the ODE system ∂ t f y = Ω T n f y has a unique solution for all time, so the martingale problem is wellposed. Moreover, the mapping from y to µ y is trivially measurable, since the topology on M n is the discrete topology. Verifying Condition (4). H L A,B clearly satises the regularity conditions required, since E is harmonic on the interior of M , and cosh is smooth, and hence H L A,B is smooth on the interior of M × R 2m . The third condition holds by denition, and since

x → cosh(Bx • a) is a convex function on R 2 for any xed a ∈ R 2 , H L A,B is convex in p for any x ∈ M ∞ . We take g = Hf for f ∈ C 1 (M ∞ ; R) and f n = f • ι n , which trivially satises the required convergence condition. Since f ∈ C 1 (M ∞ ; R), it is straightforward to check that g n = H n f n is uniformly bounded, since
∇f is uniformly continuous, and H L A,B is smooth, so the convergence statement made at the beginning of 6.2 holds uniformly for sequences ι n (x n ) which approximate points in the interior of M ∞ . When (x, p) is in the interior of M ∞ × R 2m , H L A,B is continuous, so verication of (3.8) follows from the same arguments. When x ∈ ∂M ∞ , there are two possible limits, either 0 or the limiting value for lim y→x H L A,B (y, p) for sequences of points y lying in the interior of M ∞ . Supposing ι n (x n ) → x ∈ ∂M , we therefore have

min lim y→x H L A,B (y, p), 0 ≤ lim inf n→∞ H n f n (x n ) ≤ lim sup n→∞ H n f n (x n ) ≤ max lim y→x H L A,B (
y, p), 0 , which veries the statement (3.8).

Verifying Condition [START_REF] Mp Ariza | Discrete dislocations in graphene[END_REF]. For x ∈ ∂M ∞ , we have that

L L A,B (x, ξ) = 0 ξ = 0, +∞ ξ = 0 , so (3
.9) is trivially satised. Next, using hyperbolic trigonometric identities and the fact that

| sinh(x)| ≤ cosh(x) for all x ∈ R, we obtain cosh [p -∇E(x)] • a -cosh [-∇E(x)] • a ≤ 2 cosh(|p|) cosh(|∇E(x)|).
Applying this estimate to the denition of H L A,B , we nd that for some M suciently large,

H L A,B (x, p) ≤ M m i=1 cosh(|p i |).
Dene ψ(y) := M cosh(|y|) for y ∈ R 2 ; it may be veried that the LegendreFenchel transform of this function, ψ * , is

ψ * (y) = |y| sinh -1 |y| M -1 + |y| 2 M 2 .
By the ordering properties of the LegendreFenchel transform, we therefore have

L L A,B (x, ξ) ≥ m i=1 ψ * (ξ i ),
and since sinh -1 (r) → ∞ as r → ∞, we have that ψ(ξ i )

|ξ i | → ∞ as |ξ i | → ∞, thus L L A,B satises (3.9).
Next, recalling that Ψ L A,B and Φ L A,B conjugate functions, we have that for any α, β ∈ R 2m , Ψ L A,B (α) + Φ L A,B (β) ≥ α, β , where equality holds if and only if β = ∇Ψ L A,B (α). This implies that L L A,B (x, ẋ) ≥ 0 for all (x, ẋ) ∈ M ∞ × R 2m , and L L A,B (x, ẋ) = 0 when x lies in the interior of M ∞ if and only if ẋ = ∇Ψ L A,B -∇E(x) . Given that the function on the righthand side is uniformly Lipschitz for x in the interior of M ∞ , it follows that there exists a solution x ∈ C([0, T ]; M ) to the ODE ẋ(t) = ∇Ψ L A,B -∇E(x(t)) , x(0) = x 0 , where T is chosen such that x(T ) ∈ ∂M ∞ , and x 0 lies in the interior of M ∞ . Then, setting x(t) = x(T ) for all t > T , we have that ẋ(t) = 0 for all t > T , and thus

∞ 0 L L A,B x(t), ẋ(t) dt = 0;
we have therefore veried condition (5) of Theorem 3.3, so applying its conclusion, we have proved the result.

To conclude the proof of Theorem 3.4, we note that, by properties of the LegendreFenchel trans- requires an additional step in this case. The technique which we use to prove convergence bears signicant similarities to the use of a periodic `corrector' as used in the theory of homogenisation for dierential operators with rapidly oscillating coecients, and our approach may be viewed as the discrete analogue of the strategy used in Example 1.10 in [START_REF] Feng | Large deviations for stochastic processes[END_REF].

form, for x ∈ M ∞ \ ∂M ∞ , L L A,B (x, q) ≥ 0, L L A,B (x, q) = 0 if and only if q = ∇Ψ L A,B -∇E(x)
For clarity, we rst x some notation which we use throughout the proof: recall from 2.3 the denition of a i , and the fact that Tr * is the union of 2 translated copies of √ 3

3 Tr. It will therefore be convenient to dene We see that the generator oscillates in value depending upon whether each e * i ∈ Tr * + or e * i ∈ Tr * -. To obtain a Large Deviations Principle, we must show that the nonlinear generator converges in the sense of condition (4) in Theorem 3.3. We suppose that f ∈ C 1 M ∞ ; R), and dene a sequence + o(n -1 ).

a * i = 1 3 (a 2i + a 2i-1 ) for i = 1,
f n (µ) = f • ι n (µ) + 1 n h f (ι n (µ); µ), where h f : M ∞ × (Tr * ) m →
Our aim is now to dene h f such that for some g ∈ C(M ∞ ; R),

sup µ∈M n H n (f • ι n + 1 n h f )(µ) -g • ι n (µ) → 0 as n → ∞.
As long as h f (x, µ) is uniformly bounded for (x, µ) ∈ M ∞ × (Tr * ) m , this will imply the convergence condition required in Theorem 3.3. We make the ansatz that A exp B∂ i E(x) -

∂ i f (x) • a * j + h + f,i (x) -h - f,i (x) -exp B∂ i E(x) • a * j . (6.3)
Equating terms which contain ±[h + f,i (x) -h - f,i (x)] and solving, we set

h ± f,i (x) = ± 1 2 log (γ + i -γ - i ) + (γ + i -γ - i ) 2 + 4δ + i δ - i 2δ + i
+ i + γ - i ) 2 + δ + i δ - i -γ + i γ - i -1 2 (γ + i + γ - i ),
where

γ ± i = 3 j=1 exp ∓ B∂ i E(x) • a * j and δ ± i = 3 j=1 exp ± ∂ i f (x) -B∂ i E(x) • a * j .
By the convexity of the exponential function and the fact that a * 1 + a * 2 + a * 3 = 0, we have γ ± i , δ ± i ≥ 3;

in addition, (γ + i -γ - i ) 2 + 4δ + i δ - i + γ + i -γ - i ≥ 0, so h ± f,i (x) is welldened for all x. Since γ ± i and δ ± i are continuous functions of x ∈ M ∞ , we also have that h ± f,i (x) depend continuously on x, and is thus uniformly bounded for x ∈ M ∞ . We now state the following theorem, which asserts the existence of a Large Deviations Principle for the model for dislocation motion for the case L = Tr.

Lemma 6.2. Suppose that L = Tr, and that X n 0 = ι n (x n ) where x n → x 0 ∈ M ∞ as n → ∞. Then the processes X n t satisfy a Large Deviations Principle with good rate function J Tr A,B .

Once more, we prove this result by checking the conditions of Theorem 3.3.

Proof. As in the proof of Lemma 6.1, conditions (1) and ( 2) are straightforward to verify with M n = M n and M = M ∞ , and condition (3) holds by an identical argument.

Verifying Condition (4). It is clear from the arguments of the previous section that H Tr A,B (x, p) satises the necessary regularity conditions, and by denition the H Tr A,B vanishes for x ∈ ∂M ∞ ; the convexity condition is also evident for x ∈ ∂M ∞ . Next, let x lie in the interior of M ∞ : then the second derivative of H Tr A,B (x, p) with respect to p i is

∂ 2 p i H Tr A,B (x, p) = 1 2B 2 ∇ 2 Ψ Tr A,B [ 1 B p i -∂ i E(x)] Υ A,B [∂ i E(x)] 2 + Ψ Tr A,B [ 1 B p i -∂ i E(x)] -Ψ Tr A,B [-∂ i E(x)] 1/2 - 1 4B 2 ∇Ψ Tr A,B [ 1 B p i -∂ i E(x)] ⊗ ∇Ψ Tr A,B [ 1 B p i -∂ i E(x)] Υ A,B [∂ i E(x)] 2 + Ψ Tr A,B [ 1 B p i -∂ i E(x)] -Ψ Tr A,B [-∂ i E(x)]
3/2 .

To verify convexity of H Tr

A,B , we check that this matrix is positive denite. This reduces to verifying that, as symmetric matrices, Premultiplying by v T and postmultiplying the matrices in the above expression by v for some v ∈ R

A 4 B 2 sinh[Bξ • a j ] sinh[Bξ • a j ](v • a j )(v • a k ).
Using the identity (v

• a j )(v • a k ) = 1 2 [v • (a j + a k )] 2 -1 2 (v • a j ) 2 -1 2 (v • a k ) 2
and the symmetry of the vectors a j , we have

1 4 6 j,k=1 sinh[Bξ • a j ] sinh[Bξ • a k ](v • a j )(v • a k ) = 1 8 6 j,k=1 sinh[Bξ • a j ] sinh[Bξ • a k ][v • (a j + a k )] 2 -1 4 6 j,k=1 sinh[Bξ • a j ] sinh[Bξ • a k ](v • a j ) 2 . (6.4)
The latter sum vanishes, since sinh is an odd function and v •a k = -v •a k+3 for k = 1, 2 or 3. Splitting the sum, interchanging indices j and k and then using convexity and the fact that cosh is postive, we have (6.5)

1 2 6 j,k=1 cosh[Bξ • a j ] cosh[Bξ • a k ](v • a j ) 2 = 1 2 6 j,k=1 cosh[Bξ • a j ] cosh[Bξ • a k ] 1 2 (v • a j ) 2 + 1 2 (v • a k ) 2
Combining (6.4) and (6.5), and using the addition formula for hyperbolic cosine, then bounding cosh below by 1 and dropping all terms except for those where j = k, we nd that

1 2 Ψ Tr A,B [ξ] ∇ 2 Ψ Tr A,B [ξ] : [v, v] -1 4 v • ∇Ψ[ξ] 2 ≥ 1 8 A 4 B 2 6 j,k=1 cosh Bξ • (a j + a k ) [v • (a j + a k )] 2 , ≥ 1 2 A 4 B 2 6 j=1 (v • a j ) =
Υ A,B [ζ] 2 -Ψ Tr A,B [ζ] = 1 4 (γ + i + γ - i ) 2 -γ + i γ - i = 1 4 (γ + i -γ - i ) 2 = 6 j=1 A sinh[Bξ • a j ] 2 ≥ 0, (6.7)
and using the positivedeniteness of ∇ 2 Ψ Tr A,B [ξ]. Estimates (6.6) and (6.7) entail that ∂ 2 p i H Tr A,B (x, p) is strictly positive denite for all x in the interior of M ∞ , and therefore H Tr A,B satises the convexity condition.

To verify that the convergence requirement of Condition ( 4) is satised, we dene h n : M n → R to be h n (µ) := h ι n (µ), µ). Then as h f (x, e * 1 , . . . , e * m ) is uniformly bounded for all x ∈ M ∞ and e * i ∈ Tr * , so

f • ι n + 1 n h n -f • ι n ≤ cn -1 → 0 as n → ∞.
Since ∇f , ∂ i E and x → h f (x, µ) are uniformly continuous on M ∞ , and H Tr A,B is smooth and hence uniformly continuous on the interior of M ∞ ×B r (0) for any r > 0, we have that x → H Tr A,B x, ∇f (x) is continuous. Using the fact that h f was chosen to satisfy (6.3), it is now straightforward to check that

H n f • ι n + 1 n h n (µ) -Hf • ι n → 0 as n → ∞,
and so convergence is veried.

Verifying Condition [START_REF] Mp Ariza | Discrete dislocations in graphene[END_REF]. Given that H Tr A,B is a signicantly more complex function than the Hamilto- nians obtained in the previous cases, we do not have as explicit an expression for L Tr To verify the growth condition (3.9), we estimate H Tr A,B (x, p) above. Using the elementary inequality √ a + b ≤ √ a + √ b for any a, b ≥ 0, the AMGM inequality, and the property that γ ± i ≥ 3, we nd

1 4 (γ + i -γ - i ) 2 + δ + i δ - i -1 2 (γ + i + γ - i ) ≤ 1 2 |γ + i -γ - i | + δ + i δ - i -1 2 (γ + i + γ - i ) ≤ 1 2 (δ + i + δ - i ) -min{γ + i , γ - i } ≤ 1 2 (δ + i + δ - i ). (6.8)
Noting that cosh(v • a j ) ≤ cosh( A similar argument to that used in the proof of Theorem 6.1 now allows us to conclude that (3.9) also holds in this case.

Next, we note that 0 = ∂ ξ L with x(0) = x 0 , (6.9) where x 0 is in the interior of M ∞ , then (3.10) is veried. As Ψ Tr A,B and Υ A,B are smooth, Υ A,B is bounded below, and ∂E(x) is bounded on M ∞ , an identical argument to that given in the proof of Theorem 6.1 entails that this condition is satised.

Having now veried all conditions of Theorem 3.3, its application implies Lemma 6.2.

Finally, upon noting that L Tr A,B is minimised when (6.9) is satised, and setting

M Tr

A,B (ξ) =

∇Ψ Tr A,B [ξ] 2 Υ A,B [ξ]
,

we have proved Theorem 3.4.

1. 3 .

 3 Energy barriers. For two dislocation congurations µ and ν, we dene the energy barrier for the transition from µ to ν as B n (µ → ν) := min γ∈Γn(µ→ν) max t∈[0,1]

  Characteristic timescale of observationD([0, T ]; M )Skorokhod space of càdlàg functions from [0, T ] to a metric space M Ω n , H n Innitesimal and nonlinear generators of the KMC processH L A,B , L L A,BThe Hamiltonian and Lagrangian for KMC process E

  The dierential and codierential are respectively the linear operators d : W (L p ) → W (L p-1 ) and δ : W (L p ) → W (L p+1 ), dened to be df (e) :=

  an isomorphism; in fact, * denes an isometry of the spaces L 2 (L p ) and L 2 (L * m-p ). The dierential, denoted d * : W (L * p ) → W (L * p-1 (L * )), and codierential, denoted δ * : W (L * p-1 ) → W (L * p ), are then d * f * (e * ) := ∂ * e * f * = δe f = δf (e), and δ * f * (e * ) := δ * e * f * = ∂e f = df (e).

Figure 1 .

 1 Figure 1. The square, triangular and hexagonal lattices respectively, and their duals. Primal lattices are shown in grey, dual lattices in red; 2cells are left uncoloured. Particular primal cells are highlighted in black, and their respective dual cells are given in blue.

  where equality always holds when p = m by denition. The other inclusions follow by induction on p: note that e ∈ A p with p ≥ 1 implies that e ∈ δ A a for some a∈ A p-1 , δ A a ⊆ δa,and hence e * ∈ ∂ * a * for some a * ∈ A * m-p+1 . As before, we may dene ∂ A * and δ A * by restriction, which in turn leads us to dene operators d A * and δ A * analogously. Similarly, let Ext(A * p ) := e * ∈ A * p δ * e * = δ D * e * , and Int(A * p ) := A * p \ Ext(A * p ). By construction, Ext(A * 2 ) = ∅, and e ∈ Ext(A * n-p ) if and only if there exists no a ∈ A p with e = a * (see Figure 2 for an illustration).

Figure 2 .

 2 Figure2. On the left, an example of primal (in black) and dual (in red) induced subcomplexes for a general subset of the triangular lattice: A 0 is the set of black points. On the right, a lattice polygon D in the square lattice, and the corresponding primal and dual subcomplexes, which are both path and simplyconnected.

  it follows that * denes an isometry of the spaces L 2 (nD n,p ) to L 2 (D * n,2-p ). Moreover, for any e ∈ nD n,p , we verify that du(e) = ∂e u = (∂e) * u * = δ * e * u * = δ * u * (e * ), and δu(e) = δe u = (δe) * u * = ∂ * e * u * = d * u * (e * ).

  Each 2form in this set represents a collection of m dislocations with respective Burgers vectors b 1 , . . . , b m and cores e 1 , . . . , e m : these dislocations are separated from each other and from the boundary by a distance of at least n. Since we will assume that the number of dislocations m, and the Burgers vectors b 1 , . . . , b m are xed throughout, we will suppress the dependence on (b 1 , . . . , b m ) from now on.

r n (µ, ν) = m i=1 1 nb i 1 e i and ν = m i=1 b i 1

 11 dist e * i , (e i ) * where µ = m i=1 e i , and M ∞ with the metric r ∞ (µ, ν) = m i=1 dist(x i , x i ) where µ = (x 1 , . . . , x m ) and ν = (x 1 , . . . , x m ),

( 3 . 8 )

 38 where g l and g u are respectively the lower and uppersemicontinuous regularizations of g, g l (x) := lim r→0 inf y∈Br(x) g(y) and g u (x) := lim r→0 sup y∈Br(x)

4. 1 .

 1 The discrete Poisson boundary value problem. We begin by proving existence of solutions to the Poisson boundary value problem in a general pathconnected subcomplex D n,0 . Lemma 4.1. Suppose D n,0 is a pathconnected lattice subcomplex; let g : Ext(D n,0 ) → R, and f : Int(D n,0 ) → R then there exists a unique solution u ∈ W (D n,0 ) to the problem ∆u = f in Int(D n,0 ) with u = g on Ext(D n,0 ).

Lemma 4 . 2 .

 42 Suppose that u ∈ W (D n,0 ). Then ∆u ≥ 0 on Int(D n,0 ) implies min e∈D n,0 u(e) = min e∈Ext(D n,0 ) u(e), and ∆u ≤ 0 on Int(D n,0 ) implies max e∈D n,0 u(e) = max e∈Ext(D n,0 )

  ) There exists a constant C L ∈ R such that if u(e) := G L (e) + C L + 1 Vπ log |dist(0, e)| for e ∈ L \ {0}, then |u(e)| |dist(e, 0)| -1 log |dist(e, 0)| (4.1) and |du(e)| |dist(e, 0)| -2 log |dist(e, 0)|.

4 4 = 6 G 3 G√ 3R 4 4 4 1 2π

 44634441 Tr, e |dist(e ,e)=1 G Hx (e ) e ∈ √ 3R 4 Tr + e 1 , where G Tr is the lattice Green's function for L = Tr. We note that ∆G Hx (e) = 0 by denition for e ∈ √ 3R Tr + e 1 , and for e ∈ √ 3R Tr, ∆G Hx (e) = 9 G Tr (e/ √ 3)e |dist(e ,e)=1 e | dist(e ,e )=1 G Hx (e ) , Tr (e/ √ 3)e |dist(e ,e)= √ Tr (e / √ 3), = 1 0 (e). Moreover G Hx (0) = G Tr (0) = 0, and the symmetry of G Tr also implies (1) for G Hx . Let C Tr be the constant in statement (3) for the case where L = Tr, and for e ∈ Hx \ {0}, dene u Hx (e) := G Hx (e) + 1 2π log dist e/ √ 3, 0 + 3C Tr = G Hx (e) + 1 2π log |dist(e, 0)| + 3C Tr -1 4π log(3). we see that for e ∈ Tr, u Hx satises (4.1) by the assertion for the case where L = Tr. For e ∈ √ 3R Tr \ {0}, dene v Tr (e) := G Tr (e/ √ 3) + 1 6π log dist e/ √ 3, 0 ; then for e ∈ √ 3R Tr + e 1 , we have u Hx (e) = 1 2π log dist e/ √ 3, 0 + e |dist(e,e )=1 v Tr (e ) -1 6π log dist e / √ 3, 0 .Since log |x| is harmonic away from 0, Taylor expanding to thirdorder about the point e and using the symmetry of Hx implies that

(4. 3 )√ 3R 4 4 3 -

 3443 Applying this estimate and (4.1) for L = Tr, we obtain that |u(e)| |dist(e, 0)| -1 log |dist(e, 0)| for all e ∈ Hx \ {0}. To demonstrate (4.2), suppose without loss of generality that e ∈ Tr and e + a ∈ √ 3R Tr + e 1 for some nearest neighbour direction a. Recalling the denition of a i from 2.3, for some i, we have |du Hx ([e, e + a])| = v Tr e + √ 3a i+1 + v Tr e + √ 3a i -2v Tr (e) + 1 2π log dist(e + a, 0 / √ e |dist(e+a,e )=1 1 6π log dist(e , 0)/ √ 3 .

(4. 4 )

 4 Lemma 4.4. Let Q r be as dened in(4.4). Then for each e ∈ Ext(Q r 0 ), there exists ω r e ∈ W (Q r 0 )) satisfying∆ω r e = 0 in Int(Q r 0 ), with ω r e = 1 e#Ext(Q r 0 )

  δe dv -v(e) δe dv , which follows by applying (2.1) to the extension of u, v, du and dv by 0 to the full lattice complex. Now, consider v which is the solution to ∆v = 0, v(e) = G L (e + a) -G L (e) on Ext(Q r 0 ). Such v clearly exists by Lemma 4.1, and by applying Lemma 4.2 and then Lemma 4.3, we obtain that sup e ∈Q r 1 |dv(e )| ≤ 2 sup e ∈Q r 0 |v(e )| ≤ 2 sup e ∈Ext(Q r 0 )|v(e )| ≤ C L r -1 log(r).

(4. 6 ).

 6 Dening u ∈ W (Q r 0 ) to be u(e) := G L (e + a) -G L (e) -v(e), u vanishes on Ext(Q r 0 ), and we have that ∆u = 1 e+a -1 e , and hence dω r e ([0, 0 + a]) = Now, applying (4.2) and (4.6), we obtain |∆u(e)| ≤ C L r -2 log(r) which completes the proof. The harmonic measure now allows us to obtain the following interior bound on the dierential of a harmonic lattice form u. Lemma 4.5. Suppose that u ∈ W (D n,0 ) satises ∆u = 0 and and u = g on Ext(D n,0 ) with g : Ext(D n,0 ) → R. Then there exists a constant C L > 0 depending only on L such that |du(e)| ≤ C L log(dist(e, Ext(D n,0 )) dist(e, Ext(D n,0 )) sup e ∈Ext(D n,0 ) |g(e )| for any e ∈ D n,1 .Proof. Suppose that e = [e 0 , e 1 ] ∈ D n,1 , and let x ∈ D be the vector corresponding to e 0 . Let Q r be as dened in(4.4), where r = dist(e 0 , Ext(D n,0 )) ; then x + Q r 0 ⊂ D n,0 , and statement (1) in Lemma 4.4 implies that w(e) = e ∈Ext(x+Q r 0 ) ω r e (e)w(e ), so dw(e) = e ∈Ext(a+Q r 0 )dω r e (e)w(e ). Applying statement of Lemma 4.4 and Lemma 4.2, it follows that |dw(e)| ≤ sup e ∈Ext(x+Q r 0 ) |w(e )| e ∈Ext(x+Q r 0 ) |dω r e (e)| ≤ C L log(r) r sup e ∈Ext(D n,0 )

( 2 )

 2 ȳn i (x) := ȳi ( 1 n x), where ȳi solves ∆ȳ i = 0 on D with boundary values ȳi (x) = 1 Vπ b i log(|x -x i |), (3) u is the solution to the discrete Poisson problem ∆u = 0 with u(e) = -m i=1 b i G L (e -x i ) + ȳn i (e) on Ext(D n,0 ), and (4) v the solution to the discrete Poisson problem ∆v = -e) = 0 on Ext(D n,0 ).In combination, Lemma 4.3, the theory of boundary value problems on polygons in[START_REF] Grisvard | Elliptic problems in nonsmooth domains[END_REF], and Lemma 4.1 allow us to conclude that each of the terms in this decomposition is welldened. Furthermore, equality follows since solutions to the Poisson problem are unique by Lemma 4.1.

( 4 . 8 )

 48 Furthermore, it can be checked that the mapping S : x ∈ D | dist(x, ∂D) ≥ /2 → X where S(x i ) := ȳi is continuous, and is hence bounded, since the domain of S is compact.

(4. 10 )

 10 Estimating du. We now use Lemma 4.5 to estimate du. Dening g : Ext(D n,0 ) → R to be g(e) := m j=1 b j G L (e -x j ) + ȳn j (e).

DeneA

  := e ∈ Int(D n,0 ) e, e + e 1 ∈ Int(D n,0 ) ;(4.16)

  ) and the full lattice Green's function. Recalling the result of Lemma 4.3, for any x ∈ D, we note that ∆ G L (• -x) + 1 Vπ log |n diam(D)| = 1 e in Int(D n,0 ), and

(4. 20 )

 20 Next, we sum (4.15) over A , noting that #A = O(n), to obtain e ∈A |G L (e -x )||∆ȳ n j (e )| ȳj X log(n)

  dα(e) = δ * d * G µ * (e * ) = µ * (e * ) = µ(e), for e ∈ D 2 , and δα(e) = (d * ) 2 G µ * (e * ) = 0 for e ∈ D n,0 .

  Finally, for the triangle inequality, by checking cases it may be shown that β ∈ [du], β ∈ [dv] and α ∈ [du + dv] imply that |α(e)| ≤ |β(e)| + |β (e)| for all e ∈ D n,1 .

5. 1 . 3 .

 13 Space of continuous paths. Dene the metric space C([0, 1]

E

  n (γ(t); u µ ) = max t∈[0,1]

  α * ↑ (e * ) := d * G t (e * ) e * = ±l * , ∓ 1 2 b j e * = ±l * , and α * ↓ (e * ) := d * G t (e * ) e * = ±l * , ∓ 1 2 b j e * = ±l * , where t is given by (5.3). Letting α µ ∈ [du µ ], for any e ∈ D 2 with e = p, q, by duality we have d[α ↑ -α µ ](e) = ∆ * G t (e * ) -∆ * G µ * (e * ) = t ∆ * G ν * (e * ) -∆ * G µ * (e * ) = 0.

  and ι n is as dened in (3.6).

A,B . 6 . 3 .

 63 , and by denition, M LA,B = ∇Ψ L Proof of Theorem 3.4: the case L = Tr. The case where L = Tr is more complicated than the cases treated above, since Tr * is isomorphic to Hx, which is a multilattice rather than a simple Bravais lattice. The value of Ω n f therefore oscillates depending upon the specic sublattice on which each dislocation lies, and so the verication of the convergence condition in Theorem 3.3

H 3 j=1exp- 1 + A i|e * i ∈Tr * - 3 j=1exp

 313 n f (µ) = 1 n e -nf (µ) Ω n e nf (µ) B∂ i E(x) • a * j exp ∂ i f (x) • a * j + h f (x, e 1 , T i a * j µ) -h f (x, µ) -B∂ i E(x) • a * j exp -∂ i f (x) • a * j + h f (x, T i -a * j µ) -h f (x, µ) -1

h 3 j=1A

 3 f (x; e * 1 , . . . , e * m ) = m i=1 h f,i (x; e * i ), where h f,i (x; e * i ) = h + f,i (x) e * i ∈ Tr * + , h - f,i (x) e * i ∈ Tr * -; thus, each h f,i : M ∞ × Tr * → Rdepends only on whether e * i ∈ Tr * + or e * i ∈ Tr * -. In order that H n (f • ι n + 1 n h f ) -g • ι n tends to zero independently of the choice of sublattice for each e * i , we then choose h ± f,i (x) to satisfy the `corrector problem' exp ∂i f (x) -B∂ i E(x) • a * j + h - f,i (x) -h + f,i (x) -exp -B∂ i E(x) • a * j

1 B 6 j=1cosh

 16 By now expressing g(x) in terms of hyperbolic trigonometric functions, we dene Hf (x) := H Tr A,B x, ∇f (x) , where the limiting Hamiltonian H Tr A,B (x, p) is dened to beH Tr A,B (x, p) p i -∂ i E(x) -Ψ Tr A,B -∂ i E(x) -Υ A,B ∂ i E(x) , for x ∈ M ∞ \ ∂M ∞ , and H Tr A,B (x, p) := 0 for x ∈ ∂M ∞ ,whereΨ Tr A,B [ξ] := A 2 Bξ i • a j , and Υ A,B [ξ] := A 3 j=1 cosh[Bξ i • a * j ].We dene the conjugate functionL Tr A,B : M ∞ × R 2m → R ∪ {+∞} to be L Tr A,B (x, ξ) := sup p∈R 2m ξ • p -H Tr A,B (x, p) ,and the corresponding rate functional J : D([0, +∞); M ∞ ) → R ∪ {+∞} to be J Tr A,B (x) := x, ẋ dt x ∈ W 1,1 [0, +∞); R 2m , +∞ otherwise.

1 2 Υ

 2 A,B [ζ] 2 -Ψ Tr A,B [ζ] + Ψ Tr A,B [ξ] ∇ 2 Ψ Tr A,B [ξ] -1 4 ∇Ψ Tr A,B [ξ] ⊗ ∇Ψ Tr A,B [ξ] ≥ 0 for all ξ, ζ ∈ R 2 .

  • a j ] cosh[Bξ • a k ] v • (a j + a k ) 2 .

A,B

  as we obtained in the cases where L = Sq and L = Hx. We therefore verify Condition[START_REF] Mp Ariza | Discrete dislocations in graphene[END_REF] indirectly using properties of the LegendreFenchel transform. First, we verify that L Tr A,B (x, ξ) ≥ 0. We note that since H Tr A,B (x, p) is smooth and strictly convex in p, L Tr A,B (x, ξ) is also smooth and strictly convex, and H Tr A,B (x, p) = sup ξ∈R 2m p • ξ -L Tr A,B (x, ξ) . It follows that 0 = H Tr A,B (x, 0) = sup ξ∈R 2m -L Tr A,B (x, ξ) = -inf ξ∈R 2m L Tr A,B (x, ξ).

√ 3 3 3 j=1A 3 3

 333 |v|), formula (6.8), along with the denition of H Tr A,B , the convexity of cosh and the fact that ∂ i E(x) is uniformly bounded for all x ∈ M ∞ , implies that there exists a constant C > 0 independent of x such thatH Tr A,B (x, p) ≤ m i=1 cosh [p i -B∂ i E(x)] • a * j ≤ |p i | + C.

Table 1 .

 1 Notation conventions. Set of pcells in the primal and dual lattice subcomplexes induced by nD Ext(D n,p ) Set of pcells in D n,p at the `edge' of the complex induced by nD Int(D n,p ) Set of pcells in D n,p which lie `away from the edge of nD' e A pcell [e 0 , e 1 ] 1cell e such that ∂e = e 1 ∪ -e 0 . e + a pcell obtained by translating e by the vector a ∂, δ Dierential and codierential on forms dened on the lattice complex ∆ Hodge Laplacian on forms W (L p ),W (D n,p ) Set of pforms on L and D L 2 (L p ) Hilbert space of squareintegrable pforms on L W 0 (D n,p ) Set of pforms on D which vanish on Ext(D n,p ) (•) *

	Symbol	Description
	L	mdimensional multilattice identied with a lattice complex
	Tr, Sq, Hx	Triangular, square and hexagonal lattices in R 2
	K, K * , V, V *	Constants depending on L
	D	Convex lattice polygon
	c l , ϕ l	Position and interior angle of corner l of D
	L p ,L * p	Set of pcells in the primal and dual lattice complexes induced by L
	D n,p ,D * n,p
		Boundary and coboundary operators
	d, δ
		Duality mapping on pcells and pforms
	E n (y; ỹ)	Energy dierence between deformations y and	ỹ
	ψ	Potential giving energy per unit length of interaction between columns of atoms
	[du]	Set of bondlength 1forms corresponding to du
	u µ	Locally stable equilibrium containing dislocations µ
	R n (µ → ν)	Exponential transition rate to pass from µ to ν
	A n (µ → ν)	Entropic prefactor for transition from µ to ν
	B n (µ → ν)	Potential energy barrier to transition from µ to ν
	Γ n (µ → ν)	Space of paths in deformation space connecting u µ and u ν
	u ↑	Transition state, i.e. deformation where B n (µ → ν) is attained
	α ↑ ,

α ↓ Bondlength 1forms corresponding to the transition state 1 e pform which is ±1 on ±e and 0 otherwise

	G L	Green's function for the full lattice L
	Q r	Polygonal set of radius r in the lattice
	ω r e	Harmonic measure for

  Tr + e 1 ;the nearest neighbour directions in Sq are therefore e i , and a i in Tr or Hx.

	and Hx :=	√	3 R 4 Tr ∪	√ 3 R 4 We may dene lattice
	complexes based on these sets (see 2.3.2 and 2.3.3 of [4] and [5]), and moreover

  2, we verify that α ∈ W (D n,1 ) satisfying α * = d * G µ * veries the latter two conditions. Showing that α ∞ = d * G µ * ∞ < 1 2 is the most technical aspect of the proof, and requires us to develop a theory which is analogous to obtaining interior estimates for solutions of a boundary value problem for Poisson's equation in the continuum setting. To conclude, we obtain the class u µ by `integrating' α.3.2. Energy barriers. Let C [0, 1]; W (D n,0 ) denote the space of continuous paths from [0, 1] to W (D n,0 ). For µ and ν ∈ M n , we dene the set of continuous paths which move any local equilibrium in u µ to any other local equilibrium in u ν to be

  1 , and let e * i ∈ argmin dist(e * , e * i ) e * i ∈ supp{µ} . Applying Theorem 4.6, and splitting

  5.3. Construction of the transition state. In Theorem 3.1, we found the bond length 1 forms corresponding to local equilibria containing dislocations are related to dual Green's functions.By considering this relationship, it is natural to consider strains dual to interpolations of these Green's functions as possible candidates for the transition state u. We therefore dene G t := (1-t)G µ * +tG ν * , where t ∈ [0, 1], G µ * and G ν * solve (3.2). We note that for any e * ∈ D * n,0 , ∆ * G t (e * ) = (1 -t)∆ * G µ * (e * ) + t∆ * G ν * (e * ) = (1 -t)µ * (e * ) + tν * (e

* );

  1 K * b j +o(1) and similarly, d * G ν * (l * ) = -1 K * b j +o(1), as n → ∞ by applying Theorem 4.6 and statement (2) of Lemma 4.3, we see that t ∈ [0, 1]; indeed, t → 1 2 as n → ∞. We now dene α ↑ and α ↓ via

  2, 3, Tr * + := Tr + a * 1 , and Tr * -:= Tra * 2 . By denition, we have that Tr * = Tr *

	directions in Tr * are	+ ∪Tr * -; the subscripts refer to the fact that the nearest neighbour
	a * 1 , a * 2 , a * 3	for e * ∈ Tr * +	and	-a * 1 , -a * 2 , -a * 3	for e * ∈ Tr * -.
	With this notation, if µ * = (e * 1 , . . . , e * m ) with r ∞ (µ * , x) = O(n -1 ), we have
		3
	Ω n f (µ) =	nA exp -B∂ i E(x) • a * j + o(1) f (e * 1 , . . . , e * i + a * j , . . . , e * m ) -f (µ)
	i|e * i ∈Tr * +	j=1
		+
		i|e * i ∈Tr *

-3 j=1 nA exp B∂ i E(x) • a * j + o(1) f (e * 1 , . . . , e * i -a * j , . . . , e * m ) -f (µ) .

  R will be dened shortly. For convenience, we also dene T i s µ := (e 1 , . . , e i + s, . . . , e m ), and calculate

  It is immediate that ∇ 2 Ψ Tr A,B [ξ] : [v, v] ≥ 0 for all v ∈ R 2, since cosh is bounded below by 1, and the vectors a j span R 2 . Next, we note that1 2 Ψ Tr [ξ] ∇ 2 Ψ Tr A,B [ξ] : [v, v] -1 4 v • ∇Ψ Tr A,B [ξ] 4 B 2 cosh[Bξ • a j ] cosh[Bξ • a k ](v • a j ) 2

			2
	=	1 2	6 j,k=1
			-	1 4	6 j,k=1

2 , we have

∇ 2 Ψ Tr A,B [ξ] : [v, v] = 6 j=1 A 2 B 2 cosh[Bξ • a j ](v • a j ) 2 , and v • ∇Ψ Tr A,B [ξ] 2 = 6 i=1 A 2 B sinh[Bξ • a j ](v • a j ) 2 .

A

  3 2 A 4 B 2 |v| 2 (6.6) It remains to verify that Υ A,B [ζ] 2 -Ψ Tr A,B [ζ] 1 2 ∇ 2 Ψ Tr A,B [ξ] ≥ 0 for all ξ, ζ ∈ R 2 . This is immediate upon noting that

  Tr A,B (x, ξ) if and only if ξ = ∂ p H Tr A,B (x, 0). Tr A,B , we nd that if x solves ẋi = ∇Ψ Tr A,B [-∂ i E(x)] 2Υ A,B [∂ i E(x)]

Computing ∂ p H

Acknowledgements

Thanks: The author would like to thank Giovanni Bonaschi and Giacomo Di Gesú for informative discussions on Large Deviations Principles while carrying out this work, and the two anonymous referees for helpful suggestions of a variety of improvements to this paper.

Funding: This study was funded by a public grant overseen by French National Research Agency (ANR) as part of the Investissements d'Avenir program (reference: ANR-10-LABX-0098).

Conict of interest: The author declares that there is no conict of interest regarding this work.

Proof. We begin by proving that all transition states lie in the set B := u W (D n,0 ) α ∈ [du] has α(e) = ± 1 2 for some e ∈ D n,1 .

We remark that any γ ∈ Γ n (µ → ν) must pass through B, since it is only on this set that we may have α, α ∈ [dγ(t)] with dα(p) = b i , dα(q) = 0, and dα (p) = 0, dα (q) = b i .

Suppose that γ ∈ Γ n (µ → ν) solves the minimisation problem (3.4), and attains a transition state u ↑ = γ(t * ) at t = t * . Suppose further that u ↑ / ∈ B.

This is a valid competitor for the minimum problem, and moreover by using strict convexity of ψ(x)

which is a contradiction.

Suppose once more that γ is a minimal path, and max t∈[0,1] E[γ(t); u µ ] attaining a transition state at t = t * . Suppose also that α ∈ [dγ(t * )] has α(e) = ± 1 2 for some e = ±l. Let a ∈ ∂e such that e / ∈ ±∂l. Then by considering γ(t * ) + s1 a , and following the strategy of proof of Lemma 5.1 in [START_REF] Alicandro | Metastability and dynamics of discrete topological singularities in two dimensions: a Γ-convergence approach[END_REF], it may be checked that there exists δ > 0 such that for all s ∈ [0, δ) or for all s ∈ (-δ, 0],

By redening γ to pass through γ(t * ) + s1 a in a neighbourhood of t * , it follows that γ(t * ) cannot be a transition state, and hence if u is a transition state with α ∈ [du], α(e) = ± 1 2 if and only if e = ±l. By considering paths β which have β(t * ) = γ(t * ) + s1 a with a / ∈ ∂l, we obtain that

for all s suciently small. If α ∈ [dγ(t * )], we have that

, where l = [e 0 , e 1 ], we obtain that δα(e 0 ) + δα(e 1 ) = 0, hence we have proved that a transition state must satisfy conditions (1)(4).

Next, we prove that dα ↑ = µ, α ↑ (l) = 1 2 and conditions (2) and (3) dene a unique 1form, which is an elastic strain at the transition state. Suppose that α ↑ and α ↑ satisfy these conditions. Dening θ := α ↑ -α ↑ , we have that dθ = 0, hence θ = dv for some 0form v since D n is simply connected. Furthermore, dv(l) = θ(l) = 0, ∆v(b) = s and ∆v(c) = -s for some s ∈ R. Then we have

implying that θ = 0, and hence α ↑ is unique. It may be similarly veried that α ↓ exists and is unique,

where we use (2.1) and the denition of G µ * and G ν * as Green's functions; to arrive at the penultimate line, we factorise and use (5.2), and use the denition of G t to obtain the nal line. As a consequence of Theorem 4.6, we have

and

, it follows that t =

).

(5.8) Substituting (5.5), (5.7) and (5.8) into (5.4), we obtain

Finally, setting c 0 := 1 8 + 1 4 dG L * ([0, 0 + a]) completes the proof of Theorem 3.2.

Proofs of Large Deviations results

This section is devoted to the proofs of the Large Deviations Principles. 6.1 veries Theorem 3.3 by using the results of [START_REF] Feng | Large deviations for stochastic processes[END_REF]. Theorem 3.4 is then split into the cases where L * is a Bravais lattice, i.e. L = Hx or L = Sq, and where L * is a multilattice, i.e. where L = Tr. These separate cases are covered by Lemma 6.1 and Lemma 6.2, and the proofs of these results constitute the remainder of the section.

6.1. Proof of Theorem 3.3. Where not otherwise stated, all references given in this section are to results in [START_REF] Feng | Large deviations for stochastic processes[END_REF].

Conditions (1)(4) assumed in Theorem 3.3 are particular cases of assumptions of Theorem 6.14.

The only additional conditions we need to verify to apply this theorem are rst, that the equation F δ x, f (x), ∇f (x) := f (x) -δH x, ∇f (x) -h(x) = 0, (

where F δ : E × R × R N → R, satises a comparison principle for all δ > 0 suciently small, and second, that the domain of H is dense in C(E; R). The second condition is immediate, since H is dened on C 1 (E; R).