Bertrand Duplantier
email: bertrand.duplantier@cea.fr.

R Émi Rhodes

Scott Sheffield
email: sheffield@math.mit.edu.

Vincent Vargas
email: vincent.vargas@ens.fr.

Log-correlated Gaussian fields: an overview

We survey the properties of the log-correlated Gaussian field (LGF), which is a centered Gaussian random distribution (generalized function) h on R d , defined up to a global additive constant. Its law is determined by the covariance formula

which holds for mean-zero test functions φ 1 , φ 2 . The LGF belongs to the larger family of fractional Gaussian fields obtained by applying fractional powers of the Laplacian to a white noise W on R d . It takes the form h = (-∆) -d/4 W . By comparison, the Gaussian free field (GFF) takes the form (-∆) -1/2 W in any dimension. The LGFs with d ∈ {2, 1} coincide with the 2D GFF and its restriction to a line. These objects arise in the study of conformal field theory and SLE, random surfaces, random matrices, Liouville quantum gravity, and (when d = 1) finance. Higher dimensional LGFs appear in models of turbulence and early-universe cosmology. LGFs are closely related to cascade models and Gaussian branching random walks. We review LGF approximation schemes, restriction properties, Markov properties, conformal symmetries, and multiplicative chaos applications.

Introduction 1.Overview

The log-correlated Gaussian free field (LGF) is a beautiful and canonical Gaussian random generalized function (a tempered distribution) that can be defined (modulo a global additive constant) on R d for any d ≥ 1.We present a series of equivalent definitions of the LGF in Section 1.4, but the simplest is that it is a centered real-valued Gaussian random tempered distribution h on R d , defined modulo a global additive constant, whose law is determined by the covariance formula Cov (h, φ 1 ), (h, φ 2 ) =

R d ×R d -log |y -z|φ 1 (y)φ 2 (z)dydz, (1)
which holds for mean-zero test functions φ 1 , φ 2 . Here "mean-zero" means φ i (z)dz = 0, which ensures that (h, φ i ) is well-defined even though h is only defined modulo a global additive constant. The statement that h is centered means that E[(h, φ)] = 0 for each mean-zero test function φ. This note provides a brief overview of the LGF and is somewhat analogous to the survey of the Gaussian free field (GFF) presented in [START_REF] Sheffield | Gaussian free fields for mathematicians[END_REF]. It should be accessible to any reader familiar with a few standard notions from real analysis (such as generalized functions, Green's function, and Gaussian Hilbert spaces). The reader who is unfamiliar with Gaussian Hilbert spaces and Wiener chaos (as used in the construction of Brownian motion or the Gaussian free field) might wish to consult [START_REF] Janson | Gaussian Hilbert Spaces[END_REF] or [START_REF] Sheffield | Gaussian free fields for mathematicians[END_REF] for a discussion of these issues with more detail than we include here. Also, as we will explain in Section 1.2, the LGF belongs to the more general family of fractional Gaussian fields (FGFs) surveyed in [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF] (which is co-authored by one of the current authors). This article can be viewed as a companion piece to [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF]. We will cite [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF] for results that hold for general FGFs and emphasize here the results and perspectives that are specific to (or particularly natural for) the LGF.

The LGF in dimension d = 1 has been proposed as a model of (the log of) financial market volatility [START_REF] Bacry | Log-Normal continuous cascades: aggregation properties and estimation. Application to financial time-series[END_REF][START_REF] Duchon | Forecasting volatility with the multifractal random walk model[END_REF]. When d = 2, the LGF coincides with the 2D Gaussian free field (GFF), which has an enormous range of applications to mathematical physics [START_REF] Sheffield | Gaussian free fields for mathematicians[END_REF]. When d = 3, the LGF plays an important role in early universe cosmology, where it approximately describes the gravitational potential function of the universe at a fixed time shortly after the big bang. 1 When d = 4 the LGF is the continuum analog of the so-called Gaussian membrane model, 2 which is a Gibbs measure whose defining energy is the L 2 norm of the discrete Laplacian (c.f. the discrete Gaussian free field, whose defining energy function is the L 2 norm of the discrete gradient) [START_REF] Kurt | Maximum and entropic repulsion for a Gaussian membrane model in the critical dimension[END_REF][START_REF] Sakagawa | On the free energy of a Gaussian membrane model with external potentials[END_REF][START_REF] Cipriani | High points for the membrane model in the critical dimension[END_REF]. The literature on these subjects is large and complex, and we will not attempt to survey it further here.

We will see that the LGF has conformal invariance symmetries in any dimension. It is also closely related, in any dimension, to additive cascade models and branching random walks. We will see that when d is even the LGF has an interesting type of Markov property: namely, the conditional law of h in a spherical domain D, given the behavior of h outside of D, depends only on the given values of h on ∂D and the first d/2 -1 normal derivatives of 1 The Laplacian of the LGF is a random distribution ψ that is believed to approximately describe perturbations of mass/energy density from uniformity. This field follows the Harrison-Zel'dovich spectrum, which means that

E[ ψ(k) * ψ(k )] = δ(k -k )P (k),
where δ is the Dirac delta function, the power spectrum P (k) is given by |k| ns , and the spectral index n s is 1. (In the notation of Section 1.2, ψ is an FGF -ns/2 .) An overview of this story appears in the reference text [START_REF] Dodelson | Modern Cosmology[END_REF]. This formula for P has been empirically observed to approximately hold for a range of k values spanning many orders of magnitude. However, some of the most recent experimental data, including data from the Planck Observatory [PAA + 13], suggest that while the assumptions of Gaussianity, translation invariance, and rotational invariance are consistent with the data (and any "non-Gaussianianity" that exists must be limited), there are statistically significant differences between the empirically calculated power spectrum and the Harrison-Zel'dovich spectrum. Section 2 of [PAA + 13] provides a historical overview of this issue and many additional references, and Section 4 explains the recent observations. Another analysis combining this data with more recent BICEP2 data finds n s between .95 and .98 [START_REF] Wu | Cosmological parameter fittings with the BICEP2 data[END_REF].

2 To be precise, if we are given a finite Λ ⊂ Z d and a boundary function h δ : (Z d \ Λ) → R, then the membrane model on Λ is a probability measure on the finite-dimensional space of functions from Z d to R that agree with h δ outside of Λ. The probability density function is e -H(h)/2 (times a normalizing constant), where

H(h) = v∈Λ (∆h(v)) 2
and ∆ is the discrete Laplacian and Λ is the union of Λ and set of vertices adjacent to a point in Λ. The membrane model on all of Z d can be defined as a Gibbs measure on functions h from Z d to R in the usual way (which, depending on d, may be defined modulo global additive constants or modulo discrete harmonic polynomials of some degree): conditioned on the h values outside of Λ, the conditional law within Λ is as described above. Note that this conditional law depends on and is determined by the given values on those vertices of Z d \ Λ whose graph distance from Λ is 1 or 2. By contrast, in the case of the discrete Gaussian free field, the conditional law would be determined by the values on the boundary vertices, i.e., those vertices in Z d \ Λ whose graph distance from Λ is 1.

h on ∂D. We will also show for general d that the restriction of an LGF on R d to a lower dimensional subspace is an LGF on that subspace. As mentioned above, in two dimensions, the GFF and the LGF coincide. Recent years have seen an explosion of interesting results about the 2D GFF, and it is interesting to note that while many of these results can be naturally generalized to other dimensions, the generalizations often apply to general LGFs, not general GFFs. This is true, for example, of several recent results about multiplicative chaos and the so-called KPZ formula [DS11, RV11, DRSV12, DRSV14, CJ14]. In some cases, the generalizations are unsolved problems. For example, it was shown in [START_REF] Schramm | A contour line of the continuum Gaussian free field[END_REF] that even though the 2D GFF is a distribution, not a continuous function, it is possible to define continuous zero-height "contour lines" of the GFF using so-called SLE 4 curves (i.e., Schramm-Loewner evolutions with parameter κ = 4).

It remains an open question whether the level surfaces of the three (or higher) dimensional

LGF can be canonically defined in a similar way (although it is possible to draw level surfaces of continuous approximations to the LGF, as Figure 1

illustrates).

Figure 1: Level surfaces of simulated fine-mesh approximations of the 3D LGF. Since the restriction of a 3D LGF to the plane is a 2D GFF, we expect (based on the GFF results in [START_REF] Schramm | A contour line of the continuum Gaussian free field[END_REF]) that the intersection of one of these surfaces with a plane will be comprised of loops that look locally like SLE 4 .

Relation to other fractional Gaussian fields

The LGF belongs to a larger one-parameter family of Gaussian random functions (or generalized functions) on R d : namely, the family of Gaussian fields obtained as

h = (-∆) -s/2 W (2)
where s ∈ R, and ∆ is the Laplacian, and W is a white noise on R d . Following the survey article [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF], we will refer to a field of the form (-∆) -s/2 W as a fractional Gaussian field and denote it by FGF s . (The parameter s is related to the so-called Hurst parameter, as we will explain below.) Making sense of (2) requires us to make sense of the operator (-∆) s/2 in this context. Fractional powers of the Laplacian can be rigorously constructed in various ways and are the subject of a large and rapidly growing literature (see [START_REF] Caffarelli | An extension problem related to the fractional Laplacian[END_REF][START_REF] Luis A Caffarelli | Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian[END_REF][START_REF] Serra | The Dirichlet problem for the fractional Laplacian: regularity up to the boundary[END_REF][START_REF]Fractional Laplacian Wiki Page[END_REF] and the references therein, as well as our discussion in Section 1.3) but they are not generally well-defined for all tempered distributions, so there is still some thought required to make sense of the h in (2). As we explain below the basic idea is to define (-∆) s/2 to be the operator that multiplies the Fourier transform of its input by the function | • | s .

It turns out that the LGF, as defined by (1), is equivalent (up to a multiplicative constant) to (2) when s = d/2. We will explain how to make sense of (2) when s = d/2 in Section 1.4. The survey [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF] contains more detail about general FGF s processes, and much of that analysis applies to the LGF as a special case. For example, [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF] explicitly derives the constant relating (1) and (2), which we will not do here.

The family of fractional Gaussian fields includes white noise itself (s = 0) and the Gaussian free field (s = 1). The field (-∆) -s/2 W can also be understood as a standard Gaussian in the Hilbert space with inner product

(-∆) s/2 f, (-∆) s/2 g = (-∆) s f, g , (3)
as we will explain (in the LGF context) in Section 1.4. The Hurst parameter of an FGF s is the quantity

H = s -d/2.
The Hurst parameter describes the scaling relation that an instance h of the FGF s possesses: if a > 0 is a constant, then h(az) has the same law as |a| H h(z). Note that for the LGF we have H = 0 and s = d/2. The fact that H = 0 is related to the fact that, as a random generalized function, the LGF is exactly invariant under conformal transformations of the domain (without rescaling the range) in any dimension, which is not true of FGF s when s = d/2.3 When d = 1 and s ∈ (1/2, 3/2), we have H ∈ (0, 1) and the FGF s is (up to multiplicative constant; see [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF]) the fractional Brownian motion with Hurst parameter H. It is a Gaussian random function h : R → R, except that instead of requiring h(0) = 0 (which would break the translation invariance of the process) we generally consider h to be defined only modulo a global additive constant. This means that while the quantity h(t) is not a well-defined random variable when t is fixed, the quantity h(t 1 ) -h(t 2 ) is a well-defined random variable when t 1 and t 2 are given. The law of this process, the fractional Brownian motion with Hurst parameter H, is then determined by the variance formula

Var h(t 1 ) -h(t 2 ) = |t 1 -t 2 | 2H .
(Covariances of the form Cov h(t 1 ) -h(t 2 ), h(t 3 ) -h(t 4 ) can be derived from this.) More general FGF s correlation formulas, which apply when H ∈ (0, 1), are explained in [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF].

When d = 1, the LGF corresponds to s = 1/2 and is thus in some sense the limit of the fractional Brownian motion processes as H decreases to 0.4 For each d ≥ 1, it turns out that when s ∈ (d/2, d/2+1) the FGF s is an a.s. continuous random function and has the property that its restriction to any one-dimensional line has the law of a fractional Brownian motion (up to a global additive constant) [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF]. 5 In this range, FGF s is often called a fractional Brownian field and has been studied in a variety of contexts (see e.g. [Lin93, BG99, ZS02]). If s = (d + 1)/2 then the restriction of FGF s to any one-dimensional line is a Brownian motion (up to a global additive constant), and the FGF s itself is known as Lévy's Brownian motion [START_REF] Lévy | Le mouvement brownien[END_REF][START_REF] Mckean | Brownian motion with a several-dimensional time[END_REF][START_REF] Lévy | Processus stochastiques et mouvement brownien[END_REF][START_REF] Ciesielski | On Lévy's Brownian motion with several-dimensional time[END_REF]. The LGF in any dimension d ≥ 1 can be understood as the limit of these (continuous) fractional Brownian fields obtained when s decreases to d/2.

When s ≤ d/2 the FGF s can only be defined as a random distribution, and not as a continuous function. In this sense, the value s = d/2 corresponding to the LGF is critical: it divides FGF s fields that (like Brownian motion) can be defined as random a.s. continuous functions from those that (like the LGF and the GFF) can only be defined as random generalized functions [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF].

Basic definitions

We recall a few basic definitions and facts that can be found in many textbooks on Fourier analysis (see, e.g., Chapter 11 of [START_REF] Hunter | Applied analysis[END_REF]). Fix d ≥ 0. The Schwartz space S is the space of smooth complex-valued functions on R d whose partial derivatives all decay superpolynomially. In other words,

S = {φ ∈ C ∞ (R d ) : φ α,β < ∞ ∀α, β},
where α and β are multi-indices (i.e., d-tuples of non-negative integers) and

||φ|| α,β = sup x∈R d |x α D β φ(x)|,
where x α and D β respectively mean product of multi-order α of the d coordinates of x, and partial derivative of multi-order β with respect to those coordinates. One can define a countable family of semi-norms by

φ → sup x∈R d |x α D β φ(x)|,
for multi-indices α and β. These semi-norms are actually norms when restricted to S, and they induce a locally convex topology on S, w.r.t. which S is metrizable, complete, and separable [START_REF] Pietsch | Nuclear locally convex spaces[END_REF]. A tempered distribution h is a continuous linear map from the Schwartz space to C. For φ ∈ S, we write (h, φ) for the value of this map applied to φ. Let T denote the space of tempered distributions. Derivatives and integrals of tempered distributions are defined by integration by parts

(D β h, φ) = (-1) |β| (h, D β φ),
where |β| is the sum of the indices of β, and the product of h with a function like x α can be defined by (x α h, φ) = (h, x α φ).

At first glance, T appears to be a fairly large and unwieldy space. However, the Schwartz representation theorem states that for any u ∈ T , there is a finite collection

u αβ : R d → C of bounded continuous functions, |α| + |β| ≤ k, for some k < ∞, such that u = |α|+|β|≤k x β D α u αβ .
In other words, tempered distributions are just (finite sums of) products of polynomials and derivatives of bounded continuous functions [START_REF] Melrose | Introduction to microlocal analysis[END_REF]. Another way to say this is that T is the smallest linear space that contains all bounded continuous functions and is closed under differentiation and monomial multiplication. (Equivalently, it is the smallest linear space that includes the bounded continuous functions and is closed under differentiation and the Fourier transform -see the Fourier transform discussion below.) It is often natural to equip T with the weak- * topology (i.e., the weakest topology for which the maps h → (h, φ) are continuous for each φ ∈ S). In this topology h 1 , h 2 , . . . converge to h if and only if (h 1 , φ), (h 2 , φ), . . . converge to (h, φ) for all φ ∈ S.

We also use (•, •) to denote the standard L 2 (R d ) inner product defined by (f, g) = f (x)ḡ(x)dx, where z denotes the complex conjugate of z. (One can define this inner product for elements of S and extend it to all of L 2 (R d ) by noting that L 2 (R d ) is the Hilbert space completion of S w.r.t. this norm.) Throughout this paper, we define the Fourier transform using the normalization that makes it a unitary transformation on the complex function space L 2 (R d ), namely:

f (ω) := (2π) -d/2 R d f (x)e -iω•x dx, so that f (x) = (2π) -d/2 R d f (ω)e iω•x dω.
The Fourier transform is a continuous one-to-one operator on Schwartz space that changes differentiation D α to multiplication by x α , and vice versa. The Fourier transform also preserves the L 2 norm. Since S is dense in L 2 (R d ), the operator can be continuously extended to a map from L 2 (R d ) to L 2 (R d ). The Fourier transform defined on S also induces a one-to-one operator on the space T of tempered distributions: if h is a tempered distribution, then the definition of ĥ is fixed by the identity for φ ∈ S ( ĥ, φ) = (h, φ).

(4)

Note that if h ∈ L 2 (R d ) then the map φ → (h, φ) defined via the L 2 (R d ) inner product can also be understood as a tempered distribution. When h ∈ L 2 (R d ) ⊂ T , the two definitions of the Fourier transform given above (for L 2 (R d ) and for T ) coincide. We stress again that the Fourier transform fixes each of the three spaces S ⊂ L 2 (R d ) ⊂ T .

A tempered distribution modulo additive constant can be understood in two ways: as an equivalence class of tempered distributions (with two tempered distributions considered equivalent if their difference is a constant function), or as a continuous linear functional defined only on the subspace S 0 of Schwartz space consisting of functions φ with φ(z)dz = 0. It is not hard to see that these two notions are equivalent. A continuous linear map φ → (h, φ) on S is determined by its restriction to S 0 together with the value of (h, φ) for one fixed φ ∈ S \ S 0 (and knowing this value is equivalent to knowing the global additive constant). Let T 0 denote the space of tempered distributions modulo additive constant.

The Fourier transform of an element of T 0 , defined via (4) as above, is a continuous linear functional on the Fourier transform of S 0 , i.e., on the subspace Ŝ0 of Schwartz space consisting of functions that vanish at zero. Accordingly, denote by T0 the set of continuous linear functionals on Ŝ0 , so that T0 is the Fourier transform of T 0 .

A standard example of a random distribution is white noise: recall that the white noise W on R d is the so-called "standard Gaussian" random variable associated with the L 2 inner product (•, •) on R d . This means that if f 1 , f 2 , . . . are an orthonormal basis for L 2 (R d ), then an instance of white noise can be written as W = i α i f i , where the α i are i.i.d. N (0, 1) (i.e., normal with mean zero, variance one). The sum i α i f i almost surely diverges pointwise, but for each ρ ∈ L 2 (R d ), we can write

(W, ρ) := i (α i f i , ρ),
a sum that converges almost surely. The random variables (W, ρ), for ρ ∈ L 2 (R d ) form a Hilbert space of centered (i.e., mean-zero) Gaussian random variables with covariances given by Cov (W, ρ 1 ), (W, ρ 2 ) = (ρ 1 , ρ 2 ).

(5) for all test functions ρ 1 and ρ 2 in L 2 . Hilbert spaces of centered Gaussian random variables (where the covariance is the inner product) are discussed in much more detail in the reference text [START_REF] Janson | Gaussian Hilbert Spaces[END_REF]. Note that, in particular, if ρ 1 and ρ 2 have disjoint support then (W, ρ 1 ) and (W, ρ 2 ) are independent. Also, although there a.s. exist exceptional elements ρ ∈ L 2 (R d ) for which i (α i f i , ρ) does not converge, the sum does a.s. converge for all elements of S, and this allows W to be a.s. defined as an element of T . This fact is explained (with more references and detail) in [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF]. White noise can be defined as the unique centered Gaussian random tempered distribution for which (5) applies.

A complex white noise takes the form W 1 + iW 2 where W 1 and W 2 are independent and each is a (real) white noise as defined above. The Fourier transform of a complex white noise is itself a complex white noise.

Denote by ∆ the Laplacian operator. Note that ∆ is a continuous operator on S. If f ∈ T then we can define -∆f via the integration by parts formula (-∆f, φ) := (f, -∆φ), for φ ∈ S. Thus the Laplacian also makes sense as a map from T to itself. Next, observe that if ρ = (-∆)f then ρ(ω) = f (ω)|ω| 2 . That is, the operation -∆ corresponds to multiplication of the Fourier transform by the function ω → |ω| 2 , in the sense that if φ ∈ S then

(-∆)f, φ = (| • | 2 f , φ) := ( f , | • | 2 φ).
We can now define the powers (-∆) s/2 of the Laplacian, where s ∈ R, as the operators that multiply a function's Fourier transform by

| • | s . (-∆) s/2 f, φ := (| • | s f , φ) = ( f , | • | s φ). (6)
However, we observe that multiplication by | • | s describes a continuous function from S to itself only when s/2 is a non-negative integer: for other values of s, the RHS of (6) only formally makes sense if all derivatives of φ vanish at zero. Thus, in the most straightforward sense, our definition of Laplacian powers only applies for positive integer values of s/2. Another way to say this is to note that the middle expression in (6) is not necessarily defined, because the product of a tempered distribution and a not-necessarily-smooth function like | • | s is not necessarily a tempered distribution. However, we can still consider on a case-by-case basis whether we can make sense of | • | s f as a tempered distribution. If not, we can always at least make sense of the RHS of (6) for some subspace of test functions φ. (For example, the RHS of (6) makes sense at least whenever φ and all of its derivatives vanish at the origin.) We take (6) to be the definition of (-∆) s/2 f , with the understanding that this restriction to a subspace of test functions is sometimes necessary.

A more in depth discussion of the domain and range of the operator (-∆) s/2 appears in the FGF survey [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF], which also contains additional references to the functional analysis literature. What will be important for us is that (-∆) s/2 f is well-defined as a tempered distribution modulo a global additive constant when f is a complex white noise (in which case f is also a complex white noise) and s = -d/2. This fact is derived in [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF] as a consequence of the Bochner-Minlos theorem. In this case, | • | s f is a complex white noise times the deterministic real valued function | • | s . This implies that if φ is a real test function in the Schwartz space then the real part of (| • | s f , φ) has variance given by |ω| s φ(ω)

2 dω = |ω| -d φ(ω)
2 dω, which is easily seen to be finite if and only if φ(0) = 0.

(Similar statements hold for the imaginary part.) In this case, considering (-∆) s/2 f only as a tempered distribution modulo additive constant (i.e., as an element of T 0 ) corresponds to restricting to the subspace S 0 of test functions φ for which φ(0) = 0.

LGF definitions

We will generally define the LGF to be a real-valued generalized function h (so that (h, φ) is real when φ is real). However, we remark that it is often natural to consider a complex analog of the LGF by writing h 1 + ih 2 , where each h i is an independent real LGF. We will give several equivalent definitions of the LGF on R d below, each of which has a real and a complex analog. In the definitions involving Fourier transforms, note that requiring h to be real is formally equivalent to imposing the symmetry ĥ(z) = ĥ(-z). We will explain some of the equivalences directly in the statement of Proposition 1.1 below.

Proposition 1.1. Each of the LGF definitions listed below describes a random element h of T 0 . The corresponding probability measures on T 0 are equivalent (up to multiplying h by a deterministic constant).

1. A random h ∈ T 0 (i.e., a random distribution modulo global additive constant) whose law is the centered Gaussian determined by the covariance formula

Cov (h, φ 1 ), (h, φ 2 ) = R d ×R d -log |y -z|φ 1 (y)φ 2 (z)dydz, (7)
for all test functions φ 1 and φ 2 in S 0 .

2. A "standard Gaussian" in a particular Hilbert space that we view as a subspace of T 0 .

To construct that Hilbert space, first note that each element of S can be interpreted as an element of T 0 (by considering its equivalence class modulo global additive constant).

Then consider the Hilbert space closure of S ⊂ T 0 w.r.t. the inner product

(f, g) d := | • | d f , ĝ = f , | • | d ĝ = R d |z| d f (z)ĝ(z)dz. ( 8
)
The statement that h is a "standard Gaussian" means that we can write

h = i α i f i (9)
where α i are i.i.d. centered normal random variables and the f i are an orthonormal basis for the Hilbert space described above. The sum converges a.s. within the space T 0 .

3. A standard Gaussian in a particular Hilbert space: namely, the Hilbert space closure of S ⊂ T 0 w.r.t. the inner product

(f, g) d := (-∆) d/2 f, g = f, (-∆) d/2 g = (-∆) d/4 f, (-∆) d/4 g , (10)
where (•, •) denotes the standard L 2 inner product and ∆ is the Laplacian. This definition is formally equivalent to Definition 2 once one posits the identity

(-∆) a f = | • | 2a f and the fact that f → f is an isometry of L 2 (R d ).
This definition is especially natural when d is a multiple of 4, since in this case (-∆) d/4 is a local operator on S, and (f, f ) d is simply the square of the L 2 norm of (-∆) d/4 f . Note that (-∆) d/2 is still a local operator if d is merely a multiple of 2, which gives us an easy way to define the middle expressions of (10). For odd values of d, one has to define (-∆) d/4 via Fourier transforms (which amounts to returning to Definition 2).

4. A standard Gaussian in a particular Hilbert space: namely, the Hilbert space closure of S ⊂ T 0 w.r.t. the inner product

(f, g) d := (-∆) (d-2)/2 f, g ∇ = f, (-∆) (d-2)/2 g ∇ = (-∆) (d-2)/4 f, (-∆) (d-2)/4 g ∇ ,
where (•, •) ∇ denotes the standard Dirichlet inner product (f, g) ∇ := ∇f (x)•∇g(x)dx. This is equivalent to Definition 3 by the identity

(f, g) ∇ = (-∆f, g) = (f, -∆g).
Nonetheless, this definition is especially natural when d -2 is a multiple of 4, since in this case (-∆) (d-2)/4 is a local operator and (f, f ) d is simply the Dirichlet energy of (-∆) (d-2)/4 f . 5. (-∆) -d/4 W where W is a real white noise. (Since W is by definition the standard Gaussian in the Hilbert space L 2 (R d ), where the inner product is (f, g), this definition is equivalent to Definition 3.)

6. The real part of the Fourier transform of |z| -d/2 W (z), where W is complex white noise on R d . (This is formally equivalent to Definition 2, since |z| -d/2 W (z) is the standard Gaussian in the Hilbert space with inner product given by the RHS of (8).)

Proof. The Hilbert spaces described are all equivalent by definition. The fact that the sums converge in T 0 is shown in [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF]. By Definition 1 of Proposition 1.1, this implies the invariance of the law of the LGF under the inversion map φ. A similar calculation implies invariance when φ is a translation or a dilation. The proposition now follows from the well known fact (Liouville theorem) that for d > 2 all conformal automorphisms of R d are compositions of inversions, translations and dilations.

Proposition 1.2. If φ is a conformal automorphism of R d ,
Next, we would like to say that, in some sense, applying the operator "multiplication by | • | -d/2 " (which fails to be defined on the entire space of tempered distributions) to a function's Fourier transform corresponds to applying the operator "convolution with | • | -d/2 " (which also fails to be defined on the entire space of tempered distributions) to the function itself. The first definition in the proposition below is inspired by this idea.

Proposition 1.3. The following are equivalent to the definitions given in Proposition 1.1.

1. A formal convolution of a real white noise W and the function |z| -d/2 . Formally, this means that h(a) = W (z)|a -z| -d/2 dz, but this integral is almost surely undefined for any given a (which makes sense, since the h we seek to define this way is a distribution, not a function, and cannot be defined pointwise). However, one can define h ε to be the convolution of W with

ψ ε (z) := |z| -d/2 1 ε<|z|<1/ε .
This object can be defined pointwise, and we can construct h by taking the limit in law as ε → 0.

The formal integral

h(a) = C(a) W (x, y) dxdy, (11)
where C(a) := {(x, y) : x ∈ R d and 0 ≤ y ≤ |x -a| -d } and W is a real white noise on R d ×[0, ∞). The h defined this way is a distribution, not a function, and can be defined precisely by first replacing C(a) with the subset C ε (a) := C(a) ∩ {(x, y) : ε < y < ε -1 } (so that the the approximation h ε thus defined is a random continuous function), and we can construct h by taking the limit in law as ε → 0.

Proof. We will sketch the proof here. In each of the two cases above, we can compute the variance of (h ε , ρ) explicitly (assuming ρ is Schwartz with mean zero) and check that as ε → 0 it converges to the variance of (h, ρ) when h an instance of the LGF. Since the h ε are all Gaussian, it follows from this that the h ε converge in law to the LGF as ε → 0. For any fixed ρ, the convergence also holds in L 2 , which implies almost sure convergence of (h ε , ρ) to a limit. A similar argument shows that for any countable dense collection of ρ i , the restriction of (h ε , •) converges almost surely to a limit with the law of the LGF restricted to these test functions. Definition 2 in Proposition 1.3 involves integrating white noise over an extra dimension of space: intuitively one has a different white noise for each y (a convolution of white noise with the indicator function of a ball) and h is the integral of these processes over y ∈ [0, ∞). This construction will in fact turn out to be closely related to another construction of the LGF as a continuum analog of the additive Mandelbrot cascade, which we will describe in Section 2. This construction, explained in more detail in Section 2, involves averaging a continuum of rescaled instances of a stationary bounded-variance Gaussian random function on R d . Section 4 will describe a variety of ways to obtain the LGF as an integral of this sort, following the method of Kahane.

One can find in [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF] more detail on how to enlarge the space of test functions beyond the Schwartz space. We cannot make sense of (h, ρ) when ρ is a δ-function (which would correspond to h being defined pointwise) but we can sometimes make sense of (h, ρ) when ρ represents a measure supported on a set of dimension strictly between 0 and d, so that, for example, (h, ρ) represents the mean value of h on a line-segment. We will see that (7) still holds for the random variables (h, ρ) defined this way. This allows one to make precise the following so-called restriction property of the LGF (which is a special case of a more general result explained in [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF]): Proposition 1.4. If h is an LGF on R d , and a lower-dimensional subspace Λ of R d is fixed, then the restriction of h to Λ is an LGF on Λ. Proposition 1.4 is actually immediate from (7) once we know that (h, ρ) is well-defined for test functions ρ supported on the lower dimensional subspace. The reader familiar with the 2D GFF may find this property intriguing. It implies that no matter what d is, the restriction of h to any fixed two-dimensional slice of R d is just an ordinary 2D GFF. The higher dimensional LGF can thus be understood as a way to couple together all of these Gaussian free fields (one for each two-dimensional plane). All of the structure that exists for the 2D GFF can be found on a two-dimensional slice of an LGF in R d . For example, as mentioned earlier, it was shown in [START_REF] Schramm | A contour line of the continuum Gaussian free field[END_REF] that even though the 2D GFF is a distribution, not a continuous function, it is possible to define continuous zero-height "contour lines" of the GFF, which are forms of the Schramm-Loewner evolution (SLE). Although we are not able to do this, it seems natural to try to rigorously construct a level surface of the 3D LGF (as in Figure 1) by joining together the contour lines defined on each plane in a family of parallel planar slices of R 3 .

Sections 3 and 4 will establish Markov properties of h, decompositions of h into independent sums, joint laws for the averages of h on circles and spheres, and various schemes for approximating h.

Acknowledgements

Scott Sheffield learned much about this subject from personal conversations with Linan Chen and Dmitry Jakobson, whose recent work [START_REF] Chen | Gaussian Free Fields and KPZ Relation in R 4[END_REF] considers random distributions similar to the LGF on R 4 , along with variants defined on compact and possibly curved manifolds. They also consider analogs of the Liouville quantum gravity measures for these objects and establish a general expected box counting version of the KPZ formula for these measures. Vincent Vargas would like to thank Vincent Millot for useful discussions on fractional Laplacians. Scott Sheffield thanks the astronomer John M. Kovac for conversations about early universe cosmology and for the corresponding references. Kovac is the leader of the BICEP2 team, which has been collecting and analyzing cosmic microwave background radiation data at the South Pole. Their recent detection of so-called B-modes in this data has been widely discussed and analyzed this year [AAB + 14], and interpretation efforts are ongoing. Sheffield also thanks Asad Lodhia, Xin Sun, and Samuel Watson, his co-authors on the FGF survey [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF], for a broad range of useful insights.

Comparison to cascades

If one types "log-correlated Gaussian field" into an online search engine, one finds that there has a been a good deal of research on random fields that have approximately logarithmic correlations (as opposed to the exactly logarithmic correlations enjoyed by the LGF). 6 In practice, many of the tools that are useful for studying the LGF can be applied equally well to Gaussian random functions with approximately logarithmic correlations.

To motivate the definition of the LGF and build some intuition, we will first describe one of these closely related families of random generalized functions: the so-called additive cascades constructed from branching random walks. In their simplest form, additive cascades are defined as follows. Let Y 0 be a random function from R d to R that assigns an independent mean-zero, unit-variance Gaussian random variable to each unit cube of Z di.e., each cube obtained by translating [0, 1] d by an element of Z d . (The value assigned to the measure-zero boundaries between cubes does not matter.) For each integer k, let Y k be an independent random function on R d such that

(a) Y 1 (b) Y 2 (c) Y 3 (d) Y 4
Y k (•) = LAW Y 0 (2 k •). (12)
Thus, for each k the function Y k is a.s. piecewise constant, assigning a different random value to each cube obtained by translating [0, 2 -k ] d by an element of 2 -k Z d , as illustrated in Figure 2. We mention without proof the following facts. (These are not hard to verify, but it would be a distraction to give detailed proofs here.)

1. The infinite sum ∞ k=0 Y k a.s. converges in the space of distributions to a limiting random distribution (generalized function) Y + whose law is invariant under translations of R d by integer vectors (elements of Z d ). In other words, there is a.s. a distribution Y + such that for each smooth, compactly supported test function φ, we have

(Y + , φ) = ∞ k=0 (Y k , φ).

The infinite sum

Y = Y 0 + ∞ k=1 (Y k + Y -k ) (13)
also converges to a limit in the space of distributions understood modulo additive constant (which amounts to limiting the space of test functions φ to those whose global mean is zero) if we restrict our attention to [0, ∞) d ⊂ R d . This follows from the convergence of ∞ k=0 Y k and the fact that if φ is compactly supported in [0, ∞) d , then the function Y -k is a.s. constant on the support of φ for all sufficiently large k. However, the discontinuity of Z n := Y 0 + n k=1 (Y k + Y -k ) at the origin becomes arbitrarily large as n → ∞, so the limit does not converge on all of R d .

3. Suppose x 1 , x 2 are distinct points in [0, ∞) d with irrational coordinates. Let L(x 1 , x 2 ) be the smallest k such that x 1 and x 2 lie in different cubes of 2 -k Z d . Observe that

Y k (x 1 ) = Y k (x 2 ) a.s. when k < L(x 1 , x 2 ) and Y k (x 1 ) and Y k (x 2 ) are independent when k ≥ L(x 1 , x 2 ). This implies that Cov Z n (x 1 ), Z n (x 2 ) = n + L(x 1 , x 2 ) ( 14
)
whenever n is larger than |L(x 1 , x 2 )|. When we compute covariances of differences (and n > |L(x 1 , x 2 )|) the n terms on the right side of (14) cancel out:

Cov Z n (x 1 ) -Z n (x 2 ), Z n (x 3 ) -Z n (x 4 ) = L(x 1 , x 3 ) -L(x 1 , x 4 ) -L(x 2 , x 3 ) + L(x 2 , x 4 ).
Taking the n → ∞ limit, we obtain that for any smooth mean-zero test functions φ 1 and

φ 2 compactly supported on [0, ∞) d , Cov (Y, φ 1 ), (Y, φ 2 ) = [0,∞) d ×[0,∞) d L(y, z)φ 1 (y)φ 2 (z)dydz. ( 15
)
The LGF is a continuum version of the random distribution Y defined above, satisfying (7) in place of (15), which amounts to replacing L(x, y) by -log |x -y|. (Note that the function L(x, y) is in some sense a discrete analog of the function -log 2 |x -y|.) However, unlike Y , the LGF is defined on all of R d and it has a law that is invariant under arbitrary translations, rotations, and dilations (and not only dilations by powers of 2). Nonetheless, we will see in Section 4 that the LGF can still be defined via (12) and (13) provided we replace Y 0 with an appropriate stationary bounded-variance Gaussian random function on R d (indeed there are many choices for Y 0 that suffice). This Y 0 can take the form of a white noise convolved with a bump function, so that Y 0 (x 1 ) and Y 0 (x 2 ) are independent when |x 1 -x 2 | is large enough. Alternatively, we can let Y k denote the portion of the integral in (11) corresponding to 2 dk < y < 2 d(k+1) . (It is easy to see that (12) holds for the Y k defined this way.) Thus, much of the intuition suggested by the cascade model (where each k represents a "scale" and there is an independent constant-order-variance noise at each scale) is valid for the LGF as well.

Remark 2.1. A more general class of random distributions on [0, ∞) d (modulo additive constant) can be constructed by replacing k∈Z Y k with k∈Z α k Y k . The latter are invariant under transformations that dilate the domain [0, ∞) d by a factor of 2 and the range R by a factor of α. Just as k∈Z Y k is analogous to the LGF, the more general family k∈Z α k Y k is analogous to the general family of FGFs.

Remark 2.2. If in the construction of Y above we made Y 0 constant on integer translations of [-1/2, 1/2] d , instead of integer translations of [0, 1] d , then the corresponding random distribution Y could be defined on all of R d , instead of just on [0, ∞) d , since the support of any compactly supported test function lies in 2 k [-1/2, 1/2] d when k is large enough.

Orthogonal decompositions and Markov properties

Conditional expectation on a bounded domain

Consider a bounded subdomain D ⊂ R d . Let HARM D be the (f, g) LGF Hilbert-space closure of the space of smooth functions f for which (-∆) d/2 f = 0 on D. Let SUPP D be the Hilbertspace closure of the space of smooth, compactly supported functions on D. The following is a relatively simple observation (see [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF] for a proof of a generalization of this statement that applies to general d and s; an analogous result for the GFF appears in [START_REF] Sheffield | Gaussian free fields for mathematicians[END_REF]):

Proposition 3.1. The spaces HARM D and SUPP D are orthogonal subspaces, and indeed they form an orthogonal decomposition of the entire LGF Hilbert space described by (3).

The projection of an instance h of the LGF onto HARM D gives the conditional expectation of h in D given the values of h outside of D. Similarly, we can define the zero-boundary LGF on D to be the projection of h onto SUPP D (a construction explained more generally in [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF]). When d is even, the operator (-∆) d/2 is simply a positive integer power of the ordinary Laplacian, and the functions in HARM D satisfy ∆ d/2 f = 0. Thus, to understand conditional expectations of h on D (given the observed field outside of D) one needs to understand what functions in HARM D are like.

Radially symmetric functions

In light of the previous section, it will be useful to us to better understand functions g that satisfy (-∆) d/2 g = 0. We begin with spherically symmetric functions. The following is a straightforward calculation of the Laplacian of powers of the d-dimensional norm function x → |x|:

g(x) = |x| k =⇒ ∆g(x) = k(k + d -2)|x| k-2 , (16)
and

g(x) = log |x| =⇒ ∆g(x) = (d -2)|x| -2 . ( 17
)
Note that ∆g(x) = 0 away from the origin if d = 2 and g(x) = log |x|; and similarly, ∆g(x) = 0 away from the origin if d = 2 and either g(x) = |x| 2-d (in which case g(x) is a multiple of Green's function) or g(x) = |x| 0 . This implies the following:

Proposition 3.2. Suppose d ≥ 2 is even. We have (-∆) d/2 g(x) = 0 for x ∈ R \ {0} whenever g(x) = log |x|. If g(x) = |x| k then we have (-∆) d/2 g(x) = 0 for x ∈ R \ {0} if and only if k is even and 2 -d ≤ k ≤ d -2.
Readers familiar with the Gaussian free field in dimension d = 2 will recall that the radially symmetric harmonic functions, namely the linear span of the function 1 and log | • |, play an important role. Within this linear span, we can find a function that takes any specified value on the circle ∂B r (0) and any other specified value on ∂B s (0), where 0 < r < s. The function can be interpreted as the expectation of a free field on the annulus {z : r < |z| < s} with these boundary conditions. Proposition 3.2 suggests that in a general even dimension d, the analogous space is spanned by d spherically harmonic functions. Within the linear span of these functions, we can find a function such that it and its first d/2 -1 radial derivatives take specified values on ∂B r (0) and ∂B s (0). We interpret this function as the expectation of the LGF with the specified values and derivative values on the annulus boundary.

Other spherical harmonics

We follow here the spherical coordinate decomposition given for the GFF in [START_REF] Jerison | Internal DLA and the Gaussian free field[END_REF]. We write the Laplacian in spherical coordinates as

∆ = r 1-d ∂ ∂r r d-1 ∂ ∂r + r -2 ∆ S d-1 . (18)
A polynomial ψ ∈ R[x 1 , . . . , x d ] is called harmonic if ∆ψ is the zero polynomial. (It is called bi-harmonic, tri-harmonic, etc. if ∆ 2 ψ = 0, ∆ 3 ψ = 0, etc.) Suppose that ψ is harmonic and homogeneous of degree k. Letting f = ψ| S d-1 , we have ψ(ru) = f (u)r k for all u ∈ S d-1 and r ≥ 0. Setting (18) to zero at r = 1 yields

∆ S d-1 f = -k(k + d -2)f,
i.e., f is an eigenfunction of ∆ S d-1 with eigenvalue -k(k + d -2). Note that the expression -k(k + d -2) is unchanged when the nonnegative integer k is replaced with the negative integer k := -(d -2) -k. Thus f (u)r k is also harmonic on R d \ {0}. We can also use (18) to precisely derive the bi-harmonic, tri-harmonic, etc. functions that are given by f (u)g(r) in spherical coordinates where f is spherical harmonic. We mention a few basic results about spherical harmonics that appear, e.g., in [START_REF] Elias | Introduction to Fourier Analysis on Euclidean Spaces (PMS-32)[END_REF]. Assume d ≥ 2. Let A k be the set of homogenous degree k harmonic polynomials on R d and let H k be the space of functions on the unit sphere S d-1 ⊂ R d obtained by restriction from A k . Then the dimension of H k is given by

d + k -1 d -1 - d + k -3 d -1 ,
which is finite. An important property is that the spaces H k are pairwise orthogonal (for the standard inner product of L 2 (S d-1 )) and the sum is dense in L 2 (S d-1 ). Now we will discuss how to describe the LGF in spherical coordinates, which will give us a natural way to describe a higher dimensional Gaussian field with a countable collection of one-dimensional Gaussian fields. (A more general version of this story appears in [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF].) For each k, let {ψ k,j } be an orthonormal basis of H k , where 1 ≤ j ≤ dim H k . Then let H k,j denote the space of functions on all of R d that have the form ψ(ur) = ψ k,j (u)f (r), whenever u is in the unit sphere and r ∈ [0, ∞), with f : [0, ∞) → R being a continuous and sufficiently smooth function. More precisely, let H k,j be the Hilbert space closure, with respect to the inner product (•, •) d , of the set of smooth functions of this type.

Note that the LGF is a standard Gaussian on the Hilbert space spanned by the orthogonal subspaces H k,j . Thus, we can write an instance h of the LGF as a sum

h = ∞ k=0 dimH k j=1 h k,j ,
where the h k,j are independent standard Gaussians on the space H k,j .

Note that by the definition of the spaces H k,j , we can write:

h k,j (ru) = hk,j (r)ψ k,j (u),
for u ∈ S d-1 and r ∈ [0, ∞), where hk,j is a one-dimensional Gaussian field from [0, ∞) to R.

In order to describe hk,j , it is useful to recall the spherical coordinate Laplacian (18). Indeed, we have

∆ψ(ur) = r 1-d ∂ ∂r r d-1 f (r) ψ k,j (u) -k(k + d -2)r -2 ψ(ur) (19) = r -1 (d -1)f (r) + f (r) -k(k + d -2)r -2 f (r) ψ k,j (u). (20) (21)
This gives us a definition of the Laplacian restricted to the space H k,j , i.e.

L k,d (r, f )(r) := r -1 (d -1)f (r) + f (r) -k(k + d -2)r -2 f (r).
This restriction can now be viewed as a one-dimensional object: i.e, an operator that maps a function on [0, ∞) to another function on [0, ∞). Therefore, hk,j is the standard Gaussian on the Hilbert space defined by the scalar product

∞ 0 r d-1 (-L k,d ) d/2 (r, f )(r)g(r)dr.
When d is a multiple of four, we can write this as

∞ 0 r d-1 (-L k,d ) d/4 (r, f )(r)(-L k,d ) d/4 (r, g)(r)dr.
When d is two more than a multiple of 4, we can use the identity

(f, g) d = (-∆) (d-2)/4 f, (-∆) (d-2)/4 g ∇ .
In each case, we find that the corresponding norm obtained on functions f : R + → R is an integral over r of a quantity depending on r, f (r), f (r), f (r), . . . , f (d/2+1) (r). It turns out that the h k,j are continuous functions when d ≥ 2 and are d/2 -1 times differentiable for general even d > 2 [START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF]. From this and the above discussion it follows that if we are given the values and first d/2 -1 derivatives of hk,j at a fixed value s, then conditionally the restriction of hk,j to [0, s) is independent of the restriction of hk,j to (s, ∞].

By combining the results for each of the H k,j spaces, we obtain the entire conditional law of the LGF inside of a sphere of radius s, given the values of the LGF outside. We find in particular that this law only depends on the values and first d/2 -1 derivatives of each of the hk,j at s. This is the Markov property mentioned in the introduction. Since the LGF is conformally invariant, these observations also determine the conditional law of the LGF on one side of a d -1 dimensional hyperplane given its values on the other side.

Finally, we remark that one could in principle also write r = e t for t ∈ R, and write the Laplacian in terms of the t parameter. In this way, one would obtain a map from functions on all of R to functions on all of R. The (small) advantage to this approach is that the conformal symmetry with respect to the change r → 1/r then reduces to a symmetry with respect to the change t → -t. In t coordinates, one has both translation and reflection invariance of the Laplacian operator.

Cutoffs and approximations

Kahane decomposition

We discuss different kinds of "cutoffs" which are essentially ways to write the LGF (or analogs on unbounded or bounded domains) as a sum of two independent pieces, one of which looks like the LGF on small scales (but looks approximately constant on large scales) and one of which looks like the LGF on large scales (and is approximately constant on small scales).

The most straightforward way to do this is to recall from Proposition 1.1 that when h is a complex LGF, we can write its Fourier transform ĥ(z) = |z| -d/2 W , where W is white noise. One can write ĥ as a sum of two terms as follows: ĥ

(z) = |z| -d/2 1 |z|≤1 W (z) + |z| -d/2 1 |z|>1 W (z),
and applying the inverse Fourier transform to these pieces gives a way to write h as a sum of two terms. Then, one can decompose further and write h = ∞ j=-∞ h j where ĥj (z) = |z| -d/2 1 2 j ≤z<2 j+1 W (z).

Since h j and h j+1 agree in law (up to a scaling of space by a constant factor) this construction is rather analogous to the additive cascade model in Section 2.

Whole plane approximations

We will now discuss a decomposition scheme (in the spirit of the multiplicative chaos constructions of Kahane [START_REF] Kahane | Sur le chaos multiplicatif[END_REF][START_REF] Allez | Lognormal -scale invariant random measures[END_REF]) that allows us to decompose the LGF into a continuum of independent pieces, each corresponding to a different scale, in a translation and scale invariant way. See [START_REF] Duplantier | Critical Gaussian Multiplicative Chaos: Convergence of the Derivative Martingale[END_REF][START_REF] Duplantier | Renormalization of Critical Gaussian Multiplicative Chaos and KPZ Relation[END_REF] for more details and references. We consider a family of centered stationary Gaussian processes ((X t (x)) x∈R d ) t≥0 where, for each t ≥ 0, the process (X t (x)) x∈R d has covariance given by:

K t (x) = E[X t (0)X t (x)] = e t 1 k(ux) u du, (22)
for some covariance kernel k satisfying k(0) = 1, of class C 1 and vanishing outside a compact set. It is natural to let k be the covariance kernel describing the convolution of white noise on R d with a compactly supported test function. A convenient construction of X is the following. Write k as the Fourier transform of a smooth nonnegative function g ∈ L 2 (R d ) and define f (x) = (2π) -d/4 √ g in order to get k = f * f . Then we consider a white noise W on R d × R + and we define

X t (x) = R d ×[1,e t ]
f (y -ux)u -1/2 W (dy, du).

It can be checked that E[X t (x)X t (x )] = e min(t,t )

1 k(u(x -x )) u du,
in such a way that ( 22) is valid and the process (X t (x) -X s (x)) x∈R d is independent of the processes (X u (x)) x∈R d u≤s for all s < t. Put in other words, the mapping t → X t (•) has independent increments. This allows to stack independent innovations so as to recover, as t → ∞, a Gaussian random distribution X with covariance kernel given by

E[X(x)X(x )] = ∞ 1 k(u(x -x )) u du.
The important thing is that in this Kahane approach, when one integrates log u from 0 to ∞, one obtains (modulo additive constant) the whole plane LGF. The object one obtains by just integrating between 0 and t (as we do to obtain X t ) can be understood as the expectation of the LGF given the information in this range.

Kahane's cutoff on bounded domains

Kahane originally considered centered random Gaussian distributions X the covariance kernels of which can be rewritten as K(x, y) = n≥1 k n (x, y) for some sequence (k n ) n of continuous covariance kernels. Thus there exists a sequence (Y n ) n of independent centered Gaussian processes such that E[Y n (x)Y n (y)] = k n (x, y).

A sequence of cutoffs/approximations of X is then given by

∀x ∈ D ⊂ R d , X n (x) = n k=1 Y k (x).
From now on, we stick to the notations of Subsection 3.1. As suggested in [START_REF] Duplantier | Liouville quantum gravity and KPZ[END_REF] in the case of the two-dimensional GFF, a natural choice of cutoffs/approximations of the LGF can be achieved by choosing k n (x, y) = (-λ n ) -d 2 e n (x)e n (y), corresponding to Y n (x) = (-λ n ) -d 4 β n e n (x), where λ n and e n denote the eigenvalues and eigenfunctions of the Dirichlet Laplacian on D, and β n i.i.d. centered normal random variables. The important point here is that the approximating sequence (X n ) n can be almost surely defined as a function of the whole LGF distribution X in terms of projections onto an orthonormal basis of the Dirichlet Laplacian.

In dimension 2, a construction of the GFF with Dirichlet boundary conditions, close to that in Subsection 4.1.1, has been suggested in [START_REF] Rhodes | KPZ formula for log-infinitely divisible multifractal random measures[END_REF]. Consider a bounded open domain D of R 2 . The Green function of the Laplacian can then be rewritten as One can check that E[X t (x)X t (x )] = ∞ e -2t p D (s, x, x ) ds, and that the process (X t (x) -X s (x)) x∈D is independent of the processes (X u (x)) x∈D u≤s for all s < t, allowing the possibility of dealing with independent innovations. For finite t, X t is a cutoff approximation of the GFF that is obtained by taking t = ∞.

G D (x, y) =

LGF in dimension 1: financial volatility and cutoffs

We briefly recall a cutoff-based construction of the LGF that appeared in work on the volatility of financial markets involving one of the current authors [START_REF] Duchon | Forecasting volatility with the multifractal random walk model[END_REF]. It is proposed in [START_REF] Bacry | Log-Normal continuous cascades: aggregation properties and estimation. Application to financial time-series[END_REF], based on some empirical data, that the log volatility of many assets (e.g., stocks, currencies, indices) can be modeled by a Gaussian random distribution h T living in T and with the following covariance structure (for test functions φ 1 , φ 2 ):

Cov (h T , φ 1 ), (h T , φ 2 ) = R×R log + T |y -z| φ 1 (y)φ 2 (z)dydz,

where log + (x) = max(log(x), 0) and T is the so-called correlation length. When calibrated on real data, one sometimes finds a T which is larger than the calibration window (see [START_REF] Bacry | Log-Normal continuous cascades: aggregation properties and estimation. Application to financial time-series[END_REF]). If T is larger than the window of time being considered, then it makes sense to replace T by ∞ in (23) and consider the log volatility as an element of T 0 . This was the framework considered in [START_REF] Duchon | Forecasting volatility with the multifractal random walk model[END_REF] to forecast volatility. This gives another construction of the LGF in dimension 1, namely as the limit when T → ∞ of h T in the space T 0 .

  then the law of h • φ agrees with the law of h. Proof. Consider the inversion map φ(z) = z/|z| 2 and suppose that µ and ν are finite signed measures on R d , each with total mass zero. Then either direct calculation or geometrical reasoning gives |z||w||z 1 |z| 2 -w 1 |w| 2 | = |z |w| |z| -w |z| |w| | = |z -w| and hence log |z -w|dµ(z)dν(w) = log |φ(z) -φ(w)|dµ(z)dν(w).

Figure 2 :

 2 Figure 2: Possible instances of the independent random functions Y 1 , Y 2 , Y 3 , and Y 4 , restricted to the unit square [0, 1] 2 (with function values indicated by grey scale). The sum ∞ k=0 Y k is a well-defined random distribution on R 2 . The sum ∞ k=-∞ Y k is well-defined as a random distribution, modulo global additive constant, on [0, ∞) 2 .

∞ 0 p

 0 D (t, x, y)dt, where p D is the (sub-Markovian) semi-group of a Brownian motion B killed upon touching the boundary of D, namely for a Borel set A ⊂ D, A p D (t, x, y) dy = P x (B t ∈ A, T D > t), with T D = inf{t ≥ 0, B t ∈ D}. Note that, for each t > 0, p D (t, x, y) is a continuous symmetric and positive definite kernel on D. Therefore, by considering a white noise W distributed on D × R + , we define X t (x) = D×[e -2t ,∞] p D ( s 2 , x, y)W (dy, ds).

If Φ is a conformal automorphism of R d , then we define the pullback h • Φ -1 of h to be the distribution defined by (h • Φ -1 , φ) = (h, φ) whenever φ is a test function and φ = |Φ | -d φ • Φ -1 . (Here |Φ | d is the Jacobian-since Φ is conformal, this means that |Φ (z)| is the factor by which Φ stretches distances of pairs of points near z. And (h, φ) is the value of the distribution h integrated against φ.) Note that if h is a continuous function (viewed as a distribution via the map φ → φ(z)h(z)dz), then h • Φ -1 is the continuous function given by the ordinary composition of h and Φ -1 (with this function again being interpreted as a distribution).

When d = 1 the derivative of an FGF s process is always an FGF s-1 process, and hence all FGF s process are obtained by starting with either a fractional Brownian motion or an LGF, and then integrating or differentiating some integer number of times. From this, it is clear that if an FGF s , for s ∈ (1/2, 3/2], is defined modulo additive constant, then the distributional derivatives FGF s-1 , FGF s-2 , etc. are defined without an additive constant. Thus FGF s is defined as a random distribution without an additive constant when s ≤ 1/2. Since the FGF s , for s ∈ (1/2, 3/2], is defined modulo additive constant, the indefinite integrals FGF s+1 , FGF s+2 , etc. are respectively defined modulo linear polynomials, quadratic polynomials, etc.[START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF]

It is shown more generally in[START_REF] Lodhia | Fractional Gaussian fields: a survey[END_REF] that if s > 0, d ≥ 1, and k ≥ 1 then the FGF s in dimension d can in some sense be obtained as the restriction of a (d + k) dimensional FGF s+k/2 to a d dimensional plane.

In contrast to our specific usage here, the phrase "log-correlated Gaussian field" is sometimes used more broadly to describe approximately logarithmic fields.

Partially supported by grant ANR-11-JCJC. Partially supported by NSF grants DMS 064558 and DMS 1209044. Partially supported by grant ANR-11-JCJC.


Nermine Chaari

Emmanuel Clouet

David Rodney

First order pyramidal slip of 1/3 1 210 screw dislocations in zirconium *

Keywords: Zirconium, Cross-slip, screw dislocations, etc

Atomistic simulations, based either on an empirical interatomic potential or on ab initio calculations, are used to study the pyramidal glide of a 1/3 1 210 screw dislocation in hexagonal close-packed zirconium. Generalized stacking fault calculations reveal a metastable stacking fault in the first order pyramidal {10 11} plane, which corresponds to an elementary pyramidal twin. This fault is at the origin of a metastable configuration of the screw dislocation in zirconium, which spontaneously appears when the dislocation glides in the pyramidal plane.

Introduction

Hexagonal close-packed (hcp) Zirconium is an important material for the nuclear industry where it is used as structural component in nuclear reactors. In particular the cladding of nuclear fuel is made of zirconium alloys. Like most crystalline material, the mechanical behavior is mainly driven by dislocations motion.

In α-zirconium, dislocations with Burgers vector a = 1/3 [1 210], named a dislocations, are the most frequently observed with transmission electron microscopy [START_REF] Rapperport | Room temperature deformation processes in zirconium[END_REF][START_REF] Caillard | Thermally Activated Mechanisms in Crystal Plasticity[END_REF][START_REF] Ferrer | Étude des Mécanismes de Déformation du Zirconium entre 25C et 400C. Influence d'une Faible Teneur en Soufre[END_REF]. These dislocations glide principally in prismatic {10 10} planes [START_REF] Caillard | Thermally Activated Mechanisms in Crystal Plasticity[END_REF][START_REF] Vitek | Non-planar dislocation cores: A ubiquitous phenomenon affecting mechanical properties of crystalline materials[END_REF][START_REF] Kubin | Dislocations, mesoscale simulations and plastic flow[END_REF] due to a lower critical resolved shear stress than in the basal and pyramidal planes [START_REF] Rapperport | Room temperature deformation processes in zirconium[END_REF][START_REF] Ferrer | Étude des Mécanismes de Déformation du Zirconium entre 25C et 400C. Influence d'une Faible Teneur en Soufre[END_REF][START_REF] Akhtar | Plastic deformation of zirconium single crystals[END_REF][START_REF] Akhtar | Prismatic slip in zirconium single crystals at elevated temperatures[END_REF][START_REF] Tyson | Basal and prismatic slip in h.c.p. crystals[END_REF][START_REF] Akhtar | Schmid's law and prismatic slip of zirconium[END_REF]. At low temperature, screw components of a dislocations can be distinguished as long rectilinear segments while mixed and edge components are observed in their equilibrium state as curved lines. This is because screw dislocations have a larger lattice friction opposing their motion and making them less mobile compared to mixed and edge dislocations [START_REF] Rapperport | Room temperature deformation processes in zirconium[END_REF][START_REF] Caillard | Thermally Activated Mechanisms in Crystal Plasticity[END_REF][START_REF] Ferrer | Étude des Mécanismes de Déformation du Zirconium entre 25C et 400C. Influence d'une Faible Teneur en Soufre[END_REF]. For this reason, screw dislocations with Burgers vector a control the material plasticity at low temperature and are mostly considered in the literature. In addition, experiments show that the ease of glide of a screw dislocations in the prismatic planes is strongly temperature dependent and also decreases when the amount of impurities such as oxygen, sulfur and carbon, increases in the material [START_REF] Ferrer | Étude des Mécanismes de Déformation du Zirconium entre 25C et 400C. Influence d'une Faible Teneur en Soufre[END_REF][START_REF] Kubin | Dislocations, mesoscale simulations and plastic flow[END_REF][START_REF] Baldwin | Some effects of oxygen on tensile deformation of polycrystalline zirconium[END_REF][START_REF] Sastry | An evaluation of rate-controlling obstacles for low-temperature deformation of zirconium[END_REF][START_REF] Mills | The plastic deformation of zirconium-oxygen alloy single crystals in the range 77 to 950 k[END_REF].

At higher temperatures and strain levels, secondary slip systems are activated such as 1/3 1 213 first order pyramidal slip [START_REF] Numakura | 11 23 {1011} slip in zirconium[END_REF], which has been evidenced at room temperature as an important glide system to accommodate the crystal deformation along the 0001 direction. Thermal activation also enhances a dislocation cross-slip. Experimental evidence shows that above 300 K (573 • C), screw dislocations with Burgers vector a = 1/3 [1 210] initially gliding in the prismatic planes, may leave their habit plane to glide in a first order pyramidal {10 11} plane (π 1 ) [START_REF] Ferrer | Étude des Mécanismes de Déformation du Zirconium entre 25C et 400C. Influence d'une Faible Teneur en Soufre[END_REF][START_REF] Baldwin | Some effects of oxygen on tensile deformation of polycrystalline zirconium[END_REF][START_REF] Rautenberg | Microstructural characterization of creep anisotropy at 673K in the M5(R) alloy[END_REF], or less frequently in a basal {0001} plane [START_REF] Akhtar | Plastic deformation of zirconium single crystals[END_REF][START_REF] Akhtar | Basal slip in zirconium[END_REF][START_REF] Yapici | Plastic flow anisotropy of pure zirconium after severe plastic deformation at room temperature[END_REF][START_REF] Knezevic | Anomalous basal slip activity in zirconium under high-strain deformation[END_REF]. Screw a dislocations cross slip is more frequently observed with increasing impurity content, especially with oxygen, while the hardening effect due to impurities manifests itself on the prismatic glide [START_REF] Ferrer | Étude des Mécanismes de Déformation du Zirconium entre 25C et 400C. Influence d'une Faible Teneur en Soufre[END_REF][START_REF] Kubin | Dislocations, mesoscale simulations and plastic flow[END_REF][START_REF] Baldwin | Some effects of oxygen on tensile deformation of polycrystalline zirconium[END_REF][START_REF] Sastry | An evaluation of rate-controlling obstacles for low-temperature deformation of zirconium[END_REF], as mentioned above. The same dislocation behavior has also been evidenced in titanium [START_REF] Churchman | The slip modes of titanium and the effect of purity on their occurrence during tensile deformation of single crystals[END_REF][START_REF] Shechtman | Orientation dependent slip in polycrystalline titanium[END_REF][START_REF] Rosi | Mechanism of plastic flow in titanium -determination of slip and twinning elements[END_REF][START_REF] Farenc | An in situ study of prismatic glide in α titanium at low temperatures[END_REF][START_REF] Naka | Étude des mécanismes de déformation plastique à basse température de monocristaux de titane α[END_REF][START_REF] Naka | The low-temperature plastic deformation of α titanium and the core structure of a-type screw dislocations[END_REF], a transition metal with similar properties to zirconium.

In agreement with the experiments, atomistic simulations have established that, in pure zirconium, a screw dislocation with Burgers vector a dissociates spontaneously in the prismatic plane into two partial dislocations with Burgers vector a/2 [START_REF] Clouet | Screw dislocation in zirconium: An ab initio study[END_REF][START_REF] Bacon | Atomic-scale modeling of dislocations and related properties in the hexagonal-close-packed metals[END_REF]. The dissociation is explained by a stable stacking fault with low energy in the prismatic plane [START_REF] Clouet | Screw dislocation in zirconium: An ab initio study[END_REF][START_REF] Domain | Atomic-scale ab initio study of the Zr-H system: II. interaction of H with plane defects and mechanical properties[END_REF][START_REF] Poty | Classification of the critical resolved shear stress in the hexagonal-closepacked materials by atomic simulation: Application to α-zirconium and α-titanium[END_REF]. In turn, dissociation leads to a low lattice friction that makes screw dislocations glide easily in the prismatic planes [START_REF] Clouet | Screw dislocation in zirconium: An ab initio study[END_REF][START_REF] Khater | Dislocation core structure and dynamics in two atomic models of alpha-zirconium[END_REF]. Considering pyramidal and basal slip of a dislocations in pure zirconium, it has been shown in a previous article that both slip systems share the same thermally-activated mechanism involving a metastable core structure, where the screw dislocation with Burgers vector a partially spreads in a pyramidal plane [START_REF] Chaari | First principles study of secondary slip in zirconium[END_REF]. The aim of the present paper is to study in more details pyramidal slip and to justify the glide mechanism suggested in Ref. [START_REF] Chaari | First principles study of secondary slip in zirconium[END_REF].

As dislocation mobility results from their core structure, accurate atomic-scale study of the dislocation core is required to understand their mobility. Besides, it has been shown that in hcp transition metals, the relative ease of dislocation glide is directly related to the stacking fault energies in the glide planes, which are in turn controlled by the electronic structure of the metal [START_REF] Legrand | Relations entre la structure électronique et la facilité de glissement dans les métaux hexagonaux compacts[END_REF]. Atomistic simulations incorporating a full description of the electronic structure are therefore necessary to model dislocations in zirconium. In the present work, we used both ab initio calculations and an empirical potential to calculate the energy associated with shearing the hcp lattice in a π 1 pyramidal plane, both homogeneously to determine generalized stacking faults, and inhomogeneously when a a dislocation, initially dissociated in a prismatic plane glides along a π 1 pyramidal plane.

Methods

As described in previous papers [START_REF] Clouet | Screw dislocation in zirconium: An ab initio study[END_REF][START_REF] Chaari | First principles study of secondary slip in zirconium[END_REF], we performed ab initio calculations based on the density functional theory using the PWSCF code [START_REF] Giannozzi | QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials[END_REF]. Atomistic simulations with the embedded atom method (EAM) potential developed by Mendelev and Ackland (potential #3) [START_REF] Mendelev | Development of an interatomic potential for the simulation of phase transformations in zirconium[END_REF] were also performed to study the effect of simulation cell size. A comparison between results obtained with both energy models was established to assess the ability of this empirical potential to describe pyramidal slip.

Dislocation core structure and glide are directly related to the stacking fault energy in the glide plane. To characterize the shearing of the crystal in the pyramidal plane, we calculated the generalized stacking fault [START_REF] Vitek | Non-planar dislocation cores: A ubiquitous phenomenon affecting mechanical properties of crystalline materials[END_REF] in this plane using fully periodic boundary conditions. Only one fault is introduced in the simulation cell, and the corresponding shift is applied to the periodicity vector perpendicular to the fault plane. Atom relaxation is only allowed in the direction perpendicular to the fault plane, but we also performed calculations with full atomic relaxation to identify stable stacking faults.

To model dislocations, we used a fully periodic arrangement of dislocation dipole described as an S arrangement in Ref. [START_REF] Clouet | Screw dislocation in zirconium: An ab initio study[END_REF]. The dimensions of the simulation cells are n × √ 3 × a in the [10 10] direction, m × c in the [0001] direction and a in the [1 210] direction along the dislocation line (n and m are two integers). Atoms are relaxed until all components of the atomic forces are smaller than 10 meV/ Å. We checked for some configurations that relaxation at 2 meV/ Å does not change any of our results. We employed the Nudged elastic Band (NEB) method [START_REF] Henkelman | Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points[END_REF] to determine the energy barrier against dislocation glide in the first order pyramidal plane. This method gives the minimum energy path between two stable states. All energy barriers are calculated while moving both dislocations composing the dipole in the same direction and the path is relaxed with a tolerance on atomic forces of 20 meV/ Å.

Stacking fault energy in the pyramidal plane

Since dislocation glide is directly related to the stacking fault energy in the corresponding plane, we started by investigating stacking fault energies in the first order pyramidal plane. In the hcp structure, pyramidal planes are corrugated. As a consequence, there are two different ways to shear the crystal along a pyramidal plane. The crystal might be sheared either inside a corrugated pyramidal plane, which we call a dense plane π 1D , or between two pyramidal corrugated planes, i.e. inside a loose plane π 1L (Fig. 1). The generalized stacking fault is calculated for both types of pyramidal planes using both ab initio calculations and the EAM potential.

We used a simulation cell with a height h = q × ζ, where ζ ∼ 5.9 Å is the height of the elementary cell and q = 4 is the number of the atomic planes separating two faults. The The results obtained with ab initio calculations and with the EAM potential in both pyramidal π 1D and π 1L planes are in good agreement. We show in Fig. 2 the γ-surfaces obtained with ab initio calculations in both pyramidal planes. From a general point of view, the γ-surfaces show that shearing the crystal in a π 1D plane costs higher energy than in the π 1L plane. This is a consequence of the fact that atoms are close to each other in this plane, and the shearing may bring them closer, which strongly increases the energy. But this high energy landscape is explored only when a [10 12] fault component is involved. Focusing now on the [1 210] direction, which is the relevant direction for a dislocation glide, the γ-surfaces calculated in both π 1L and π 1D plane show a valley of low energy along this direction.

0 1 a/3 [12 - 10] 0 1 2 a/2 [101 - 2] 0 0.5 1 E (J/m 2 ) (a) π 1L 0 1 a/3 [12 - 10] 0 1 2 a/2 [101 - 2] 0 2.5 5 E (J/m 2 ) (b) π 1D
To compare both pyramidal planes, we plot in Fig. 3 the generalized stacking fault energy only along the [1 210] direction for both planes. The energy obtained with ab initio calculations is higher than with the EAM potential in the case of the π 1L plane, while it is nearly the same for the π 1D plane. According to the EAM potential, the energy cost to shear the crystal along the [1 210] direction in a pyramidal plane is almost the same for both π 1L and π 1D planes. Ab initio calculations, however, predict that it is easier to shear in a π 1D plane than in a π 1L plane. Our work therefore shows that both π 1L and π 1D pyramidal planes may need to be considered when studying stacking faults in the first order pyramidal plane. This contrasts with the previous stacking fault calculations in hcp materials where only the π 1L plane was considered [START_REF] Poty | Classification of the critical resolved shear stress in the hexagonal-closepacked materials by atomic simulation: Application to α-zirconium and α-titanium[END_REF][START_REF] Ghazisaeidi | Core structure of a screw dislocation in Ti from density functional theory and classical potentials[END_REF][START_REF] Shin | Orbital-free density functional theory simulations of dislocations in magnesium[END_REF].

Considering both Figs. 2 and3, no energy minimum is found along the [1 210] direction, or in its immediate vicinity. This is true for the ab initio calculations and the EAM potential. One should notice that the present γ-surfaces were obtained by relaxing the atoms only perpendicularly to the fault plane. A full atomic relaxation, however, allows for some atomic shuffling and reveals an energy minimum that corresponds to a metastable stacking fault in the π 1D plane. The corresponding fault vector sketched by red arrow in Fig. 2(b), is f = 1/2 a + b e , where a = 1/3 [1 210] and b e is a component orthogonal to a to be detailed below. This minimum is obtained with both ab initio calculations and the EAM potential. On the other hand, no relevant minimum could be found for the π 1L plane, even with full atomic relaxations.

The atomic structure of the metastable stacking fault is shown in Fig. 4. The displacement map shows that in the shearing direction [1 210], the atoms below the shearing plane S have their color switched, which means that they have been displaced by a/2. The shearing is thus perfectly localized in the S plane. Perpendicularly to the shearing direction, the blue arrows show a displacement of all the atoms below the S plane following one same vector b e , which corresponds to the orthogonal component of the fault vector, as well as a shuffling of the atoms which is extended to several planes at both sides of the S plane. This shuffling explains why the metastable fault did not appear on the γ-surfaces of Fig. 2, where only atomic relaxation perpendicularly to the fault plane was allowed.

Analysis of this metastable core structure reveals that it corresponds to an elemen- tary two-layer pyramidal {10 11} twin [START_REF] Chaari | First principles study of secondary slip in zirconium[END_REF] bordered by the M and M' mirror planes, as illustrated by the broken line on Fig. 4. We also looked at the positions of the first nearest neighbors for each atom and compared the obtained pattern with the ones existing in a perfect hcp structure, both for the parent and the twinned lattices. We thus managed to characterize whether an atom belongs to the parent or the twinned hcp lattice. On figure 4, atoms plotted with diamonds corresponds to the atoms belonging to the twin layer.

The twin is produced by the glide of a two-layer disconnection with a Burgers vector that corresponds to the fault vector f = 1/2 a + b e where the edge component of the disconnection is defined by f e = a(4γ 2 -9)/2 3 + 4γ 2 (γ is the c/a ratio) [START_REF] Serra | Computer simulation of the structure and mobility of twinning disclocations in h.c.p. metals[END_REF][START_REF] Wang | Twinning dislocations on { 1011} and { 1013} planes in hexagonal close-packed crystals[END_REF]. Two layer disconnections are well-known to be stable on pyramidal twins [START_REF] Serra | Computer simulation of the structure and mobility of twinning disclocations in h.c.p. metals[END_REF][START_REF] Wang | Twinning dislocations on { 1011} and { 1013} planes in hexagonal close-packed crystals[END_REF][START_REF] Leclercq | Atomic-scale comparison between { 1101} and { 1102} twin growth mechanisms in magnesium[END_REF][START_REF] Bursill | {1 101} twin dislocation structures in evaporated titanium thin films[END_REF][START_REF] Li | Zonal dislocations mediating twinning in magnesium[END_REF][START_REF] Pond | Interfacial structure of {10 11} twins and twinning dislocations in titanium[END_REF][START_REF] Wang | Nucleation of elementary { 1 0 1 1} and { 1 0 1 3} twinning dislocations at a twin boundary in hexagonal close-packed crystals[END_REF], but the stability of the corresponding two-layer twin was so far unknown.

The stacking fault energy deduced from ab initio calculations is ∆E = 163 mJ m -2 . It is lower than the energy of the prismatic stacking fault (∆E = 211 mJ m -2 ) [START_REF] Clouet | Screw dislocation in zirconium: An ab initio study[END_REF]. Compared to the ab initio value, the EAM potential overestimates the pyramidal stacking fault energy (∆E = 243 mJ m -2 ). This leads to a higher energy than for the prismatic fault (∆E = 135 mJ m -2 )

Peierls barrier in the pyramidal plane

Considering the results for the generalized stacking faults, we conclude that it is important to include both pyramidal π 1L and π 1D planes in our study. We thus investigate in this part the glide of an a screw dislocation in first order pyramidal planes, for both π 1L and π 1D planes.

EAM

Starting from a dislocation in its equilibrium configuration, i.e. initially spread in a prismatic plane (Fig. 7(a)), we calculated the energy encountered by the dislocation to overcome a Peierls valley in the pyramidal plane, moving to a final equilibrium state where the dislocation spreads in the next prismatic plane. We used the NEB method to calculate the Peierls barrier in both pyramidal π 1L and π 1D planes with the EAM potential. The initial path of the dislocation is obtained by a linear interpolation between the initial and the final states with the cut created by the dislocation glide localized in the chosen first order pyramidal plane.

The results are shown in Fig. 5. The energy barrier in the π 1L plane is twice higher than in the π 1D plane. Thus, according to the EAM potential, it is easier for the dislocation to glide in the π 1D plane than in the π 1L plane.

The energy path obtained in the π 1D plane shows a local minimum at halfway across the migration. This minimum corresponds to an intermediate metastable configuration of the screw dislocation (Fig. 7(b)), to be described in more details below.

Ab initio

We only consider dislocation glide in the π 1D plane for the ab initio calculations. This is motivated by the ab initio results for the generalized stacking faults, showing that the π 1D plane is easier to shear in the [1 210] direction than the π 1L plane (Fig. 3), as well as by the Peierls barrier obtained with the EAM potential, showing that dislocation glide in the π 1D plane costs less energy than in the π 1L plane (Fig. 5).

Ab initio calculations of the Peierls barrier in the π 1D plane were performed for different simulation cell sizes. The minimum energy paths obtained are illustrated in Fig. 6. They all show a local minimum halfway across the migration, in agreement with the EAM results. This minimum corresponds to the same intermediate metastable configuration of the screw dislocation as found with the EAM potential. The ab initio energy barrier is twice lower than with the EAM potential. The difference in energy between ab initio and EAM is related to the fact that the Mendelev potential overestimates the energy of the pyramidal metastable stacking fault. As a consequence this empirical potential leads to a higher Peierls barrier in the pyramidal plane.

Metastable configuration of the screw dislocation in zirconium

Both ab initio calculations and the EAM potential showed that pyramidal glide involves an intermediate metastable configuration of the screw dislocation appearing halfway across the migration. In the following, a detailed description of this metastable configuration is proposed. Core structure Fig. 7 shows the core structure of the two possible configurations obtained for the a screw dislocation with ab initio calculations and with the EAM potential. The differential displacement maps have been superimposed to the Nye tensor distribution calculated following the method of Hartley and Mishin [START_REF] Hartley | Characterization and visualization of the lattice misfit associated with dislocation cores[END_REF]. Only the screw component of the Nye tensors is plotted in each structure. This Burgers vector density is deduced from the position variation of the nearest neighbors for each corresponding atom. In a perfect hcp structure each atom has twelve nearest neighbors forming a defined pattern. However in a faulted structure, this number of nearest neighbors can be different with different corresponding patterns. In the figure, atoms belonging to a pattern that corresponds to a prismatic stacking fault have been plotted as squares while those belonging to a pattern that corresponds to the pyramidal twin described before are plotted as diamonds. We can see through Fig. 7 that ab initio calculations and EAM potential results are in good agreement with the same metastable core obtained in both cases.

The equilibrium configuration of the dislocation (Fig. 7(a) and (c)) shows a spread in the prismatic plane in agreement with the literature [START_REF] Clouet | Screw dislocation in zirconium: An ab initio study[END_REF][START_REF] Bacon | Atomic-scale modeling of dislocations and related properties in the hexagonal-close-packed metals[END_REF]. The dissociation of the dislocation into two partials is illustrated by two local extrema in the Nye tensor distribution and the prismatic stacking fault in the core is highlighted by the squares showing the atoms involved in the fault.

Analyzing the displacement maps of the metastable core structure (Fig. 7(b) and (d)), we show that the dislocation is spread in three different crystallographic planes at the same time: in the core center, the dislocation lies in a pyramidal plane while it lies in two adjacent prismatic planes at the extremities. The two central atoms sketched by black and white diamonds (Fig. 7(b) and (d)) witness of the presence of the pyramidal twin pattern, while the squares result from a local shearing in the prismatic plane.

The pyramidal spreading in the core center is explained by the metastable stacking fault evidenced above in the pyramidal π 1D plane with a fault vector f = 1/2 a + b e . The central part of the core thus corresponds to an elementary two layer twin of a : Differential displacement map of the screw dislocation in its equilibrium and metastable configuration deduced from ab initio calculations and EAM potential. The black and white symbols corresponds to atoms depending on their position along the a = [1 210] direction in the perfect crystal, while the blue arrows corresponds to the differential displacement between two columns of atoms in the a direction after relaxation with dislocations. The pink cross indicates the dislocation center obtained by symmetry. The screw component of the Nye tensor distribution is superimposed. Squares correspond to atoms in a prismatic fault neighborhood while diamonds correspond to atoms in the pyramidal twin neighborhood. (color online) finite extension. Since the screw component of the fault vector, 1/2 a, is identical in both the prismatic and pyramidal faults, there is no discontinuity in the screw direction at the intersection between the faults. The pyramidal fault is thus bordered at its intersections with the prismatic fault by two edge disconnections of slip vectors ± b e , while the prismatic faults end with screw partial dislocations with 1/2 a Burgers vectors. This can be seen through the Nye tensor distribution plot in Fig. 7(b) and (d) where two partial dislocations are distinguished in two neighboring prismatic planes.

The metastable core may thus be described as two partial dislocations spread in two adjacent prismatic planes, separated by two prismatic stacking faults and a pyramidal nanotwin in-between [START_REF] Chaari | First principles study of secondary slip in zirconium[END_REF]. The prismatic stacking faults are linked to the nanotwin by stair-rods forming a dipole of disconnections ( b e ). The corresponding decomposition of the total Burgers vector is Fig. 8 summarizes, for different cell sizes, the excess energy of the metastable configuration, with respect to the energy of the equilibrium configuration fully dissociated in the prismatic plane. With the EAM potential, this excess energy is always positive, confirming that the metastable core is less favorable than the prismatic configuration. Convergence of the results is obtained for simulation cells containing more than 2000 atoms (Fig. 8(a)), with an excess energy of ∆E = 24 meV/ Å. Different simulation cell shapes lead to different convergence behaviors. At small sizes, an upper bound of the converged excess energy is obtained with shapes defined by m = n, whereas a lower bound is obtained with m = 2n.

Ab initio calculations lead to a lower excess energy (Fig. 8(b)). As a consequence, stability inversion was observed for very small simulation cells (4 × 8 cell containing only 128 atoms). However, larger simulation cells confirm that the configuration partially spread in the pyramidal plane is metastable. Since ab initio calculations are more expensive and limited to few hundred atoms, it was not possible to reach a converged value of the energy. However, the same dependence of the convergence rate with the cell shape was observed with the ab initio calculations and with the EAM potential. An upper limit of the energy is thus given by the m = n cells and a lower limit by the m = 2n cells. Our ab initio calculations therefore lead to an excess energy ∆E = 3.2 ± 1.6 meV/ Å.

Conclusion

In the present work, based on generalized stacking fault calculations in the first order pyramidal plane, we demonstrated that shearing along the [1 210] direction inside a dense pyramidal plane costs less energy than between two corrugated planes. In addition, calculations showed a metastable stacking fault in the pyramidal plane, which corresponds to an elementary pyramidal two-layer twin. This metastable stacking fault is at the origin of the new metastable core configuration of the a screw dislocation in zirconium, which appears halfway across the migration path when the dislocation glides in a pyramidal plane. This metastable configuration presents an unusual core structure with an incipient two-layer twin in its center.

We conclude that there are two possible configurations of the screw dislocation in zirconium. The one with the lower core energy, is dissociated in the prismatic plane and responsible for the easy glide in this plane [START_REF] Clouet | Screw dislocation in zirconium: An ab initio study[END_REF]. The second configuration is metastable and appears during pyramidal and basal slips [START_REF] Chaari | First principles study of secondary slip in zirconium[END_REF]. The results show a good agreement between ab initio calculations and the Mendelev empirical potential since, qualitatively, both lead to the same glide mechanism and metastable core. This work is a first step toward understanding cross-slip in zirconium, showing a new and unexpected relation between dislocation glide and twinning, two essential motors for hcp plasticity.

Figure 1 :

 1 Figure 1: Hexagonal close-packed structure showing the different potential glide planes for a screw dislocation of Burgers vector a = 1/3 [1 210]. A projection perpendicular to a is shown on the right, where atoms are sketched by circles with a color depending on their (1 210) prismatic plane. (color online)

Figure 2 :

 2 Figure 2: Generalized staking fault energy in the first order pyramidal plane (10 11) calculated with ab initio. The crystal is sheared either (a) in the loose plane π 1L , or (b) in the dense plane π 1D . The red arrows show the fault vector a/3 [1 210] in (a) and its decomposition in (b) corresponding to the metastable stacking fault obtained after a full relaxation of atoms in the π 1D plane. (color online)

Figure 3 :

 3 Figure 3: Generalized staking fault energy in the first order pyramidal plane (10 11) along the [1 210] direction calculated with ab initio and with the EAM potential. The crystal is sheared either (a) in the loose plane π 1L , or (b) in the dense plane π 1D . (color online)

Figure 4 :

 4 Figure 4: Atomic structure of the metastable pyramidal stacking fault: The displacement in the shearing a = [1 210] direction is shown by the projection of atoms in this direction, where atoms are sketched by black and white symbols depending on their position at half or full a vector respectively. The blue arrows show atom displacements perpendicular to the shearing direction in the (1 210) plane with a magnification factor of 3. The metastable fault corresponds to a pyramidal {10 11} twin bordered by two mirror planes M and M' sketched by pink dashed lines. The blue line S corresponds to the shearing plane. The mirror planes symmetry is highlighted by a broken line corresponding to the corrugated prismatic plane, in red in the parent hcp crystal and in black in the twinned crystal. Atoms with a neighborhood corresponding to the twinned crystal are sketched by diamonds, while the circles corresponds to atoms in the parent crystal. (color online)

Figure 5 :

 5 Figure 5: Energy barrier encountered by a screw dislocation dissociated in a prismatic plane and gliding in pyramidal π 1L and π 1D planes calculated with the EAM potential. (color online)

Figure 6 :

 6 Figure 6: Energy barrier encountered by a screw dislocation dissociated in a prismatic plane when gliding in a pyramidal π 1D plane. Ab initio calculations are performed for different simulation cell sizes n × m. (color online)

Figure 7

 7 Figure7: Differential displacement map of the screw dislocation in its equilibrium and metastable configuration deduced from ab initio calculations and EAM potential. The black and white symbols corresponds to atoms depending on their position along the a = [1 210] direction in the perfect crystal, while the blue arrows corresponds to the differential displacement between two columns of atoms in the a direction after relaxation with dislocations. The pink cross indicates the dislocation center obtained by symmetry. The screw component of the Nye tensor distribution is superimposed. Squares correspond to atoms in a prismatic fault neighborhood while diamonds correspond to atoms in the pyramidal twin neighborhood. (color online)

Figure 8 :

 8 Figure 8: Difference between the energy of the metastable configuration and the equilibrium configuration of the screw dislocation for different simulation cell sizes n × m obtained with the EAM potential on the left and with ab initio calculations on the right. (color online)

Acknowledgments

This work was performed using HPC resources from GENCI-CINES, -CCRT and -IDRIS (Grants 2013-096847). The authors also acknowledge PRACE for awarding access to the Marenostrum resources based in Barcelona Supercomputing Center (project DIMAIM).