Jean-Yves Fortin
email: jean-yves.fortin@univ-lorraine.fr

Singular self-energy for itinerant electrons in a dilute Ising spin bath

Keywords: numbers: 05.50.+q, 71.10.-w, 71.27.+a, 75.10.Hk

We consider itinerant spinless electrons moving as defects in a dilute two-dimensional Ising spin system, leading to an effective interaction mediated by spin fluctuations. Coupled selfconsistent equations are analyzed after expressing the spin and fermion operators in terms of Grassmann variables. The effective mass, density of states, and specific heat are evaluated. We found that at low temperature and low electron density the effective mass is reduced, whereas in the critical region it sharply diverges. At higher temperature, the fermions behave like a Fermi liquid with a small enhanced mass.

Introduction

The interplay between spin fluctuations and itinerant holes or electrons is important in understanding the nature of quasiparticles in complex systems where cooperative phenomena are dominant. For example, in hole doped copper oxide material La 1.6-x Nd 0.4 Sr x CuO 4 , hole segregation was observed and interpreted as stripe modulation [1] and this spatial structure is assumed to be responsible for the suppression of superconductivity. In general mobile defects in spin environment have been studied theoretically [2,3,4], and itinerant electrons coupled to a spin bath are known to generate domain structures [5,6,7]. Quantum Monte Carlo methods were investigated in order to determine the magnetic properties of coupled lattices between spins and fermions in models for metal-insulator interfaces [8]. It is expected that in the case of coupled layers itinerant electrons will suppress the magnetic order, and spin fluctuations will induce correlation between electrons. In this paper we would like to analyze the coupling between spinless electrons and ferromagnetic Ising spins in two-dimensional lattices where itinerant electrons occupy empty sites and interact via indirect spin fluctuations. We would like to study in particular how spin fluctuations modify the effective mass.

The paper is organized as follow. In section 2, we define the Hamiltonian of the model, then we use a Grassmannian representation of the spin sector in presence of holes or defects to obtain an effective fermionic action. Then the electronic part of the action is also transformed into a Grassmann integral representation at finite temperature. In section 3 we present the selfconsistent equations for the model using an approximation of the Green function and self-energy.

Finally in section 4 we study the physical solutions and evaluate the effective mass, density of states, and specific heat.

Model

We consider a two-dimensional lattice of size L 2 = N where each site r is occupied either by a Ising spin σ r = ±1 or an electron or hole with Fermi statistics. Spins are interacting with a ferromagnetic coupling J > 0. The electrons are represented by a tight-binding model and we do not consider their Coulomb interaction. A spin-hole configuration in a N = 5 × 5 system is represented in Figure 1 with a single hole defect and a domain wall (red dashed line) separating two regions of opposite spin polarity due for example to opposite boundary conditions. The indirect interaction between holes and spins can be understood by the fact that there exist hole configurations which lower the domain wall excess energy. These configurations have the hole attached to the domain wall, which gives in Figure 1(a) an energy gain of 4J compared to configuration (b) where the hole sits strictly inside the region of spins up. We therefore expect an effective interaction between holes and spins where spin fluctuations play an essential role in the electron or hole dynamics. The Hamiltonian describing the interaction between spins and electrons can be written in terms of fermionic operators (c r , c † r ) and fermion numbers n r = c † r c r as

H = - J 2 r,δ=±x,±y σ r σ r+δ (1 -n r )(1 -n r+δ ) -t r,δ=±x,±y c † r+δ c r , ( 1
)
where t is the kinetic energy for the itinerant electrons. We first consider the fermionic action of the Ising spin system in absence of holes, H 0 = -1 2 J r,δ=±x,±y σ r σ r+δ , and the representation of its partition function Z 0 in terms of Grassmann variables a r and ār [START_REF] Plechko | Anticommuting integrals and fermionic field theories for two-dimensional ising models[END_REF][START_REF] Plecko | [END_REF]

Z 0 = Tr e -βH 0 = r dā r da r e S 0 , (2)
where the Grassmannian action S 0 is defined by

S 0 = r S r , S r = a r ār + u(a r + ār )(a r-x -ār-y ) -u 2 a r-x ār-y + 2 ln cosh(βJ), (3)
In this expression u = tanh(βJ) = tanh(J/k B T ) and r runs over the lattice. The action S 0 is quadratic in Grassmann variables and was derived in detail by Plechko [11,12,[START_REF] Plecko | [END_REF]13] who studied various geometries of the lattice. We now consider a dilute system where empty sites are replaced by itinerant spinless electrons. In presence of vacancies (or empty sites) at different locations r i , the partition function is modified accordingly [14,[START_REF] Plechko | Fermionic path integrals and two-dimensional ising model with quenched site disorder Path Integrals from PeV to TeV: 50 Years after Feynman[END_REF][START_REF] Plechko | [END_REF] Z h [{r i }] = r dā r da r δ nr,0 e Sr + δ nr,1 a r ār ,

where n r = 1 when a hole is present at site r = r i , and n r = 0 otherwise, when the site is occupied by a spin. This partition function can be rewritten as

Z h [{r i }] = r dā r da r 1 + a r ār e -Sr -1 n r e S 0 . (5)
We first diagonalize S 0 , using the Fourier space transformation a r = L -1 k a k e ik.r , ār = L -1

k āk e -ik.r , and express the partition function Z 0 as a product over decoupled momentum dependent factors, after relabeling k x → -k x , see Appendix A:

S 0 = 2N ln cosh(βJ) + k ′ [g k a k āk + ḡk a -k ā-k -2iu sin k x a k a -k + 2iu sin k y āk ā-k ] , g k = 1 -u(e ikx + e iky ) -u 2 e ikx+iky , (6)
Z 0 = cosh(βJ) 2N k ′ g k ḡk -4u 2 sin k x sin k y = cosh(βJ) 2N k ′ (1 + u 2 ) 2 -2u(1 -u 2 )(cos k x + cos k y ) .
The prime symbol corresponds to the set of momenta k located in half of the Brillouin zone, the opposite momenta -k completing the zone. One can write an effective action for (5) by introducing nilpotent commuting variables b r such that b 2 r = 0 and

Z h [{r i }] = r dā r da r db r exp S 0 + r b r (1 -n r ) + r b r n r a r ār e -Sr . (7)
One should notice that b r can always be written as a product of two Grassmann variables. The term a r ār e -Sr = a r ār +u 2 a r a r-x ār ār-y will be approximated in the following by the quadratic part since the quartic part contains second order derivatives which will be assumed to be negligible for small momenta. From Z h we can now construct the total partition function Z = Tr e -βH = e S , where the action S includes the Ising part, the electron kinetic energy, and the hole constraint in equation (1). It can be expressed as a Grassmannian functional integral by substituting the Fermi operators with dimensionless Grassmann variables c r → c r (τ ) and c † r → cr (τ ) [START_REF] Altland | Condensed Matter Field Theory 2[END_REF][START_REF] Lichtenstein | Path integrals and dual fermions The Physics of Correlated Insulators[END_REF], where τ is an imaginary time variable taking values between 0 and

β Z = r dā r da r db r r,τ dc r (τ )dc r (τ ) exp S 0 + r b r 1 2 -β -1 β 0 dτ cr (τ )c r (τ ) + r b r 1 2 + β -1 β 0 dτ cr (τ )c r (τ ) a r ār + r β 0 dτ -c r (τ )∂ τ c r (τ ) + t δ cr+δ (τ )c r (τ ) . ( 8
)
We have used the following prescription in the case where products of two Fermi operators on the same site are encountered:

c † r c r = (c † r c r -c r c † r + 1)/2 → cr (τ )c r (τ ) + 1/2.
In the following section we study the extrema of this action using Lagrange multipliers.

Self-consistent equations

We consider the total action S in equation ( 8) with antisymmetric boundary conditions c r (0) = -c r (β) and cr (0) = -c r (β). These operators can be expanded using the Matsubara frequencies

ω n = (2n + 1)π/β, n = 0, ±1, ±2, • • •, for which we have c r (τ ) = β -1/2 ωn e -iωnτ c r (ω n ), cr (τ ) = β -1/2 ωn e iωnτ cr (ω n ). (9)
This allows us to write the total action as

S = S 0 + r b r 1 2 -β -1 ωn cr (ω n )c r (ω n ) + r b r 1 2 + β -1 ωn cr (ω n )c r (ω n ) a r ār + r ωn (iω n + µ)c r (ω n )c r (ω n ) + t δ cr+δ (ω n )c r (ω n ) , (10)
n r = 1 2 + β -1 ωn cr (ω n )c r (ω n ),
where we have introduced a chemical potential µ. To simplify the problem, we would like to express the action in terms of order parameters satisfying self-consistent mean-field equations.

The idea is to replace cr

(ω n )c r (ω n ) by a Green function G r (ω n ) with conjugate variable Σ r (ω n ),
or self-energy. After integrating over b r , we can write the following effective action

S eff = S 0 + r ln 1 2 -β -1 ωn G r (ω n ) + r 1 + 2β -1 ωn G r (ω n ) 1 -2β -1 ωn G r (ω n ) a r ār + r,ωn Σ r (ω n ) G r (ω n ) -cr (ω n )c r (ω n ) + r ωn (iω n + µ)c r (ω n )c r (ω n ) + t δ cr+δ (ω n )c r (ω n ) . (11)
From this expression, we then compute the effective thermodynamical potential -βΩ = ln( e S eff ) which is given by

-βΩ = 2N ln cosh(βJ) + r ln 1 2 -β -1 ωn G r (ω n ) + r,ωn Σ r (ω n )G r (ω n ) + 1 2 ln det A 0 + V r,r ′ + ωn ln det [iω n + µ -Σ r (ω n )]δ r,r ′ + t δ δ r ′ ,r+δ r,r ′
, where A 0 is the matrix corresponding to the Ising part in the real space:

S 0 = 1 2 r,r ′ Ψ ⊤ r (A 0 ) r,r ′ Ψ r ′
, and V the local potential due to the hole contribution

(V ) r,r ′ = δ r,r ′ 1 + 2β -1 ωn G r (ω n ) 1 -2β -1 ωn G r (ω n ) iσ 2 , (12)
in the basis of vectors Ψ r = (a r , ār ) ⊤ (see Appendix B). If we extremize Ω with respect to the parameters Σ r (ω n ) and G r (ω n ), we obtain a set of self-consistent equations for the different order parameters, in particular

G r (ω n ) = - δ δΣ r (ω n ) ln det [iω n + µ -Σ r (ω n )]δ r,r ′ + t δ δ r ′ ,r+δ r,r ′ , 1 1 -n r -βΣ r (ω n ) = β 2 δ δG r (ω n ) ln det A 0 + V r,r ′ . ( 13
)
Since the system is invariant by translation, we may seek solutions of the form G r (ω n ) = G(ω n ), Σ r (ω n ) = Σ, and n r = n e the particle density. We can therefore simplify ( 13) and obtain under this approximation the following equations

G(ω n ) = 1 N k 1 iω n + µ -Σ -ǫ k , ǫ k = 2t (2 -cos k x -cos k y ) , βΣ = 1 1 -n e - F (v) (1 -n e ) 2 , F (v) ≡ 1 N k ′ 2v + g k + ḡk (g k + v)(ḡ k + v) -4u 2 sin k x sin k y , n e = 1 2 + β -1 ωn G(ω n ),
v = 1 + 2β -1 ωn G(ω n ) 1 -2β -1 ωn G(ω n ) = n e 1 -n e , (14)
where we have shifted µ → µ -4t for convenience, so that for small momenta ǫ k ≃ tk 2 . Σ depends only on µ and the temperature. The exact expression of F (v) in the thermodynamical limit is given in Appendix C, equation (C.10), in terms of elliptic functions. In Figure 2 we have plotted F (v) for several temperature values or u, as well as the asymptotic value at high temperature (dashed line) whose expression is given by equation (C.11). The sum over the Matsubara frequencies of the Green functions

G(ω n ) can be performed since Σ is independent of ω n ωn G(ω n ) = β 2N k tanh β 2 (µ -Σ -ǫ k ) . ( 15
)
The potential is finally given by

- β N Ω = 2 ln cosh(βJ) + ln(1 -n e ) + n e - 1 2 βΣ + 1 N k ln cosh β 2 (µ -Σ -ǫ k ) (16) + 1 2N k ln (1 + v) 2 + u 2 (2 + u 2 ) -2u(1 + v -u 2 )(cos k x + cos k y ) -2u 2 v cos(k x + k y ) .

Effective mass and singular behavior of the self-energy

When the density is small n e ≃ 0, we have the approximations v ≃ n e and Σ ≃ k B T [1 -F (0)]. F (0) ≃ 1/4 when T → 0 and F (0) ≃ 1 -3u 4 when T → ∞. In the latter case, Σ ≃ 3J 4 /k 3 B T 3 and is positive. We can notice that due to the dilution of the spins with electrons, the singular part of Ω gives a critical temperature (β c J) -1 satisfying tanh(

β c J) = √ 2 + v -1 = u c (v)
and which is lower than the Ising value u c (v) ≥ u c (0) deduced from the singular behavior of Z 0 when k ≃ 0. We can notice that this critical temperature vanishes when u c (v) = 1 at the density n e = 2/3 ≃ 0.67 given by the saddle point approximation. Efficient Monte Carlo algorithms give a lower value close to n e = 0.59 for the percolation threshold [START_REF] Newman | [END_REF]20] above which the critical transition disappears. When fermions are coupled separately with Ising lattices, the ferromagnetic coupling is known to decrease the critical temperature whereas in the antiferromagnetic case it increases it [8]. Near the Fermi surface, the normalized effective mass m * with respect to the bare mass of the particles, which is defined as m e = 2 /2t, is equal to m * = 1 -∂Σ/∂µ [21,[START_REF] Mahan | Many-particle physics 3rd ed Physics of solids and liquids[END_REF][START_REF] Asgari | [END_REF]24]. Using this definition, we first derive the self-energy in (14) with respect to µ

β ∂Σ ∂µ = n ′ e (1 -n e ) 4 (1 -n e ) 2 -2(1 -n e )F (v) -F ′ (v) = n ′ e Λ, (17)
with F ′ (v) = dF (v)/dv and the density of states n ′ e = ∂n e /∂µ n ′ e ≡ 1 -

∂Σ ∂µ β 4N k 1 -tanh 2 β 2 (µ -Σ -ǫ k ) ≡ 1 - ∂Σ ∂µ β ñ′ e 4 . (18)
We can then express the effective mass as m * = (1 + ñ′ e Λ/4) -1 , with 0 < ñ′ e < 1. The value of the effective mass relatively to unity depends therefore on the sign of the function Λ. At high temperature where u ≪ 1 we have Λ ≃ -6u 4 (1 -n e ) < 0. This follows from the series expansion

F (v) ≃ (1 + v) -1 -3u 4 (1 + v) -4
, see equation (C.11). We then obtain the following approximation for the effective mass at high temperature, where u ≪ 1 and ñ′ e ≃ 1

m * ≃ 1 + 3 2 u 4 (1 -n e ) > 1, (19)
which is always greater than unity. In the following we will take t = 1 for the numerical evaluations. At low temperature, or u 1, we find that Λ > 0 for parameter u above a threshold value u = u * > u c which depends on n e , see top inset of Figure 3. In this region m * < 1 and u * goes to unity when the density approaches a critical value n * e ≃ 0.545, using N = 500 2 lattice points. The low temperature correlated spin fluctuations reduce the effective mass, whereas they tend to increase it at higher temperatures. At criticality the mass diverges, as evidenced in Figure 3. This can be interpreted as the fact that large spin correlations lead to the localization of electrons. In the low temperature regime, the mass is less that unity and tends to decrease as n e increases from zero with a minimum around n e ≃ 0.425, see bottom inset of Figure 3 the existence of the crossover value u * can be given by comparing the two characteristic lengths of the system, which are the average distance between the particles n -1/2 e and the spin correlation length ξ defined by

ξ -2 = (1 + v -2u -u 2 ) 2 u (1 + v -u 2 )(1 + v + 2uv -u 2 ) ( 20
)
This quantity is extracted from the constant term of the free energy singular part at low momenta k, see equations ( 16) and (C.2), after expansion and diagonalization of the quadratic form, and which can be identified as the inverse square of the correlation length. We propose to characterize the transition at u * by the fact that both lengths are of the same order ξ ∼ n -1/2 e which gives an equation for u * as function of n e . We then obtain a similar behavior to that of the curve in the first inset of Figure 3, with a critical density value n * e ≃ 0.51 close to the numerical result 0.545. Spin correlations in the low temperature regime are therefore efficient in reducing the effective particle mass as long as their typical length does not exceed the average distance between particles, or when the particle density is low enough. This can be compare with two-dimensional electron systems involving Coulomb interaction in silicon where an electron density increase leads instead to a reduction of the effective mass, whereas at lower electron density there is a sharp increase of the mass below a threshold value [21,25,[START_REF] Asgari | [END_REF]26].

The density of states n ′ e is displayed in Figure 4 as function of µ and for several values of u. At low temperature (u ≃ 1) a peak in density develops which is due to the singular part of the self-energy. The peak actually coincides with the divergence of the effective mass of Figure 3, due to the fact that n ′ e is proportional to m * . The location of this peak moves to higher energy as the temperature decreases. In Figure 5 we have plotted the specific heat

C v /k B = -T ∂ 2 Ω/∂ 2 T
for different values of µ. For low chemical potential or low electron density the specific heat is dominated by the spin fluctuations which is characterized by an exponential decay near zero temperature and logarithmic divergence near the singularity at u c . As µ (or density) increases the contribution from the electrons which is linear in T at low temperature becomes dominant whereas the singular peak vanishes at a value µ < 6.

Conclusion

We have studied a two-dimensional system of itinerant Fermi particles embedded inside a dilute Ising system or spin bath using Grassmannian path integral technique. Particles and spins are interacting through the principle of volume exclusion which yields an effective interaction between the electrons. This interaction originates from the spin fluctuations. At low temperature, the spin fluctuations tend to lower the mass of the particles whereas in the high temperature regime the independent spin fluctuations lead to a small mass increase proportional to 1/T 4 , which makes the electronic sector a Fermi liquid. At very low temperature, we expect that there are still spin fluctuations as some of the electron kinetic energy t can be exchanged for spin fluctuations mediated by J. At the critical point of the dilute Ising system, the long range fluctuations lead to a sharp increase of the mass which can be considered as a localized phase. As a result the density of states and specific heat present a similar singular behavior as function of the energy and temperature respectively. the Jacobian is unity

a k = 2iuα sin k y 1 Reg k + √ ∆ k b -k + 1 Reg k - √ ∆ k b k āk = 2iuα sin k x 1 Reg k + √ ∆ k b-k + 1 Reg k - √ ∆ k bk a -k = α b-k + bk , ā-k = α (b -k + b k ) 0 0.5 1
∆ k = (Reg k ) 2 -4u 2 sin k x sin k y , α 2 = u sin k x sin k y ∆ k 1/2 . (A.1)
This transformation leads to an action that is simply the sum of two independent fermionic oscillators with frequencies Ω k and Ω -k in the reduced Brillouin zone

S 0 = 2N ln cosh(βJ) + k ′ Ω k b k bk + Ω -k b -k b-k , Ω k = 2u sin k x sin k y √ ∆ k + iImg k √ ∆ k -Reg k , Ω -k = -2u sin k x sin k y √ ∆ k -iImg k √ ∆ k + Reg k . (A.2)
The partition function Z 0 for the Ising model is therefore equivalent to

Z 0 = k ′ d bk db k db -k d b-k e S 0 = cosh(βJ) 2N k ′ Ω k Ω -k = cosh(βJ) 2N k ′ |g k | 2 -4u 2 sin(k x ) sin(k y ) = cosh(βJ) 2N k ′ (1 + u 2 ) 2 -2u(1 -u 2 )(cos k x + cos k y ) . (A.3)
Appendix B. Perturbation scheme for the Green function

The Ising unperturbed action S 0 , equation ( 3), can be written, up to the constant term 2N ln cosh(βJ), as

S 0 = 1 2 r,r ′ Ψ ⊤ r (A 0 ) r,r ′ Ψ r ′ , (B.1)
with the 2 × 2 matrix elements

(A 0 ) r,r ′ = u(δ r ′ ,r-x -δ r ′ ,r+x ) δ r ′ ,r -uδ r ′ ,r+x -uδ r ′ ,r-y -u 2 δ r ′ ,r+x-y -δ r ′ ,r + uδ r ′ ,r-x + uδ r ′ ,r+y + u 2 δ r ′ ,r-x+y u(δ r ′ ,r+y -δ r ′ ,r-y ) ,
and Ψ r = (a r , ār ) ⊤ . The interaction part with the holes can be expressed as a local potential in this new basis

r 1 + 2β -1 ωn G r (ω n ) 1 -2β -1 ωn G r (ω n ) a r ār = 1 2 r Ψ ⊤ r V r Ψ r , (B.2)
with the 2 × 2 local matrix

V r = 0 v r -v r 0 = v r iσ 2 , v r = 1 + 2β -1 ωn G r (ω n ) 1 -2β -1 ωn G r (ω n ) . (B.3)
The partial derivatives of ln det A that appears in the saddle point equations (13), with the global block matrix A = A 0 + V and (V ) r,r ′ = V r δ r,r ′ , can be expressed as a series expansion in powers of V . These derivatives with respect to G r (ω n ) are both proportional to the derivative with respect to v r and we define the quantity

G r ≡ δ δv r ln det A = δ δv r tr ln A (B.4) = δ δv r tr ln A 0 + ln(1 + A -1 0 V ) = δ δv r tr(A -1 0 V ) - 1 2 tr(A -1 0 V A -1 0 V ) + • • • .
Let T r be the block matrix for which the diagonal element at position r is equal to iσ 2 , or (T r ) s,s ′ = δ s,r δ s ′ ,r iσ 2 . Then V = r v r T r , and one has the following expansion

G r = tr(A -1 0 T r ) - s v s tr(A -1 0 T r A -1 0 T s ) - s,s ′ v s v s ′ tr(A -1 0 T r A -1 0 T s A -1 0 T s ′ ) + • • •
Since the elements (A 0 ) r,r ′ depend only on the difference rr ′ , we define the following Fourier transform (A 0 ) r,r ′ = k ( Ã0 ) k e ik.(r ′ -r) , with k = (-k x , k y ) and

( Ã0 ) k = -2iu sin k x g k -ḡ k -2iu sin k y . (B.5)
The Fourier elements of the inverse matrix A -1 0 are then given by

( Ã-1 0 ) k = 1 D 0 (k x , k y ) -2iu sin k y -g k ḡk -2iu sin k x , D 0 (k x , k y ) = (1 + u 2 ) 2 -2u(1 -u 2 )(cos k x + cos k y ). (B.6)
One obtains, after some algebra, the expression for the Fourier transform of the quantity G r

Gk = k ′ tr{( Ã-1 0 ) k ′ iσ 2 } -ṽk k ′ tr{( Ã-1 0 ) k ′ iσ 2 tr( Ã-1 0 ) k ′ -k iσ 2 } + • • • (B.7) Appendix C. Expression of F (v) in term of complete elliptic integrals
The expression of F (v) in equation ( 14) defined in terms of a double discrete sum converges, in the thermodynamical limit N → ∞, to the following double integral

F (v) = 1 4π 2 π -π π -π dk x dk y 1 + v -u(cos k x + cos k y ) -u 2 cos(k x + k y ) D(k x , k y ) , (C.1)
where the denominator D(k x , k y ) is equal to

D(k x , k y ) = (1 + v) 2 + u 2 (2 + u 2 ) -2u(1 + v -u 2 )(cos k x + cos k y ) -2u 2 v cos(k x + k y ). (C.2)
We can rearrange the expression for F (v) using symmetries such as D(k x , k y ) = D(k y , k x ) so that As before, after integrating first over k x and performing the previous change of variable on k y , we obtain [27]

F (v) = 1 2v - 1 -v 2 + u 2 (2 + u 2 ) 8π 2 v
K 2 = 8π |α| 1 [(r 2 1 -1)(r 2 2 -1)] 1/4 1 + A 0 1 -A 0 K(q) -Π - (1 -A 0 ) 2 4A 0 , q , A 2 0 = (r 2 1 -1)(r 2 2 -1) [(r 1 + 1)(r 2 + 1)] 2 , (C.9)
where Π is the complete elliptic integral of the third kind. Finally, we can express F (v) as function of K 1 and K 2

F (v) = 1 2v - 1 v 1 -v 2 + u 2 (2 + u 2 ) (1 + v) 2 + u 2 (2 + u 2 ) K 1 8π 2 + 1 v u(1 -u 2 ) (1 + v) 2 + u 2 (2 + u 2 ) K 2 2π 2 .
(C.10)

In particular, in the high temperature regime T → ∞ or u ≃ 0, we find the following asymptotic limit

F (v) ≃ 1 1 + v - 3u 4 (1 + v) 4 .
(C.11)

Figure 1 :

 1 Figure 1: Spin configurations separating two regions of opposite polarity, when one hole is located along the domain wall (a) and outside (b). The energy E a for the spin configuration (a) has a lower energy than the energy E b in (b), since E b -E a = 4J.

Figure 2 :

 2 Figure 2: Plot of function F (v) in the thermodynamical limit, equation (C.10), for several temperature values or u = tanh(βJ). At v = 0 the function becomes singular with respect to u for the critical value u = √ 2 -1 ≃ 0.414. The dashed line is the asymptotic value F (v) = (1 + v) -1 in the high temperature regime, see equation (C.11). For comparison, dark green circles are values from the discrete formula defined in equation (14) with u = 0.3 and N = 2500 lattice points.

Figure 3 :

 3 Figure 3: Effective mass m * as function of u = tanh(βJ) for several values of n e and t = 1. First inset: Threshold temperature or u * defined as the temperature below which the mass becomes less that unity as function of n e . Red line is an interpolation which shows that m * > 1 for all temperature above the critical value n * e ≃ 0.55. Second inset: Effective mass at low temperature (u = 0.999) as function of n e . We have chosen N = 500 2 lattice sites for evaluating the chemical potential.

Figure 4 :

 4 Figure 4: Density of states as function of µ for several temperature values. From bottom to top, u increases (T decreases) from 0.2 to 0.95 (see arrow) by increments of 0.05. Each curve is shifted upward for clarity.

Figure 5 :

 5 Figure 5: Specific heat for several value of the chemical potential.

1 ( 1 + v) 2 + u 2 ( 2 + u 2 )dk x dk y 1 -dk x dk y cos k y 1 -

 11222211 x dk y cos k y D(k x , k y ) . (C.3)The computation of F (v) is reduced to evaluating two double integrals. The first integral givesπ x dk y D(k x , k y ) ≡ K α(cos k x + cos k y ) -β cos(k x + k y ) , v -u 2 ) (1 + v) 2 + u 2 (2 + u 2 ) , β = 2u 2 v (1 + v) 2 + u 2 (2 + u 2 ) . (C.5)After integrating over k x , we find thatK 1 = 2π α π -π dk y (r 1 -cos k y )(r 2 -cos k y ) r 1 = 1 + β + α 2 + 2β(1 + β) α , r 2 = 1 + β -α 2 + 2β(1 + β) α , (C.6)with r 1 > r 2 > 1. A standard change of variable k y → tan(k y /2) leads to the evaluation of the following complete elliptic integral of the first kind[START_REF] Byrd | Handbook of elliptic integrals for engineers and scientists 2[END_REF] x dk y cos k yD(k x , k y ) ≡ K 2 (1 + v) 2 + u 2 (2 + u 2 ) α(cos k x + cos k y ) -β cos(k x + k y ) . (C.8)

Appendix A. Representation of the fermionic action in terms of fermionic oscillator

The Ising action S 0 , equation (6), can be diagonalized in order to express the quadratic action as a fermionic oscillator. Let us consider the following change of Grassmannian variables for which


I Bailleul

A Brault

L Coutin

Young and rough differential inclusions

We define in this work a notion of Young differential inclusion dz t P F pz t qdx t , for an α-Hölder control x, with α ą 1{2, and give an existence result for such a differential system. As a by-product of our proof, we show that a bounded, compact-valued, γ-Hölder continuous set-valued map on the interval r0, 1s has a selection with finite p-variation, for p ą 1{γ. We also give a notion of solution to the rough differential inclusion

for an α-Hölder rough path X with α P `1 3 , 1 2 ‰ , a set-valued map F and a single-valued one form G. Then, we prove the existence of a solution to the inclusion when F is bounded and lower semi-continuous with compact values, or upper semi-continuous with compact and convex values.

-Introduction 1.-Setting

One of the motivations for considering differential inclusions comes from the study of differential equations with discontinuous coefficients. In the setting of a possibly timedependent widely discontinuous vector field on R d , it makes sense to replace the dynamical prescription 9 z t " f pt, z t q, by 9 z t P F pt, z t q, where F pt, zq is here the closed set of cluster points ot F ps, wq, as ps, wq converges to pt, zq. This somehow accounts for the impossiblity to make measurements with absolute precision. It also makes sense to take for F pt, zq the convex hull of the former set. A set-valued application F is a map from r0, T s ˆRd into the power set of R d . Different natural setvalued extensions of f may have different regularity properties; in any case, switching from the differential equation prescription to the differential inclusion formulation somehow allows to account for the uncertainty in the modeling. A Caratheodory solution of the differential inclusion 9 z t P F pt, z t q, t P r0, T s, z 0 " ξ P R d

(1.1) is an absolutely continuous path z started from ξ, whose derivative 9 z satisfies 9 z t P F pt, z t q at almost all times t P r0, T s. Existence of solutions of differential inclusions and their properties were widely studied; see for instance J.P. Aubin and A. Cellina's book [START_REF] Aubin | Differential inclusions. Set-valued maps and viability theory[END_REF] for an authoritative pedagogical treatment. Equation (1.1) has at least one solution under two kinds of assumptions on the set valued map F : either F is upper semicontinuous with compact convex values, or F is bounded and lower semicontinuous with compact values.

Recall that a set-valued function F on r0, T s ˆRd is said to be upper semicontinuous if one can associate to every spacetime point ps, wq an open neighbourhood U of ps, wq and an open neighbourhood V of F ps, wq such that F pU q Ă V , and F is said to be lower semicontinuous if for any ps, wq, any w 1 P F ps, wq and any neighbourhood V pw 1 q of w 1 , there exists a neighbourhood U ps, wq of ps, wq such that F pt, xq X V pw 1 q ‰ H, for all pt, xq P U ps, wq. The existence proofs for solutions of the differential inclusion (1.1) generally rely on either approximation schemes or a fixed point reformulation. Differential equations, and their extensions into differential inclusions, are not the only kind of natural dynamics. Some extensions to stochastic cases where also studied, after the pioneering works of Aubin-da Prato [START_REF] Aubin | Stochastic Viability and Invariance[END_REF][START_REF] Aubin | Stochastic Nagumo's viability theorem[END_REF][START_REF] Aubin | The Viability Theorem for Stochastic Differential Inclusions[END_REF][START_REF] Aubin | Stochastic invariance for differential inclusions[END_REF] and Kisielewicz [START_REF] Kisielewicz | Differential inclusions and optimal control[END_REF][START_REF] Kisielewicz | Strong and weak solutions to stochastic inclusions[END_REF][START_REF] Kisielewicz | M Stochastic differential inclusions and applications[END_REF]. The works [START_REF] Aubin | Stochastic Viability and Invariance[END_REF][START_REF] Aubin | Stochastic Nagumo's viability theorem[END_REF][START_REF] Aubin | The Viability Theorem for Stochastic Differential Inclusions[END_REF][START_REF] Aubin | Stochastic invariance for differential inclusions[END_REF] were essentially motivated by studying viability questions in a stochastic setting. M. Kisielewicz in [START_REF] Kisielewicz | Differential inclusions and optimal control[END_REF] defined the notion of solution and obtained the existence of stochastic differential inclusions of the form

X t ´Xs P ż t s F pr, X r qdr `ż t s Gpr, X r qdW r (1.2)
where W is an R -valued Brownian motion, F is a lower semicontinuous set-valued random map with values in R d and G is a lower semicontinuous set-valued map with values in LpR d , R q. In [START_REF] Kisielewicz | Strong and weak solutions to stochastic inclusions[END_REF], M. Kisielewicz also studied the case where equation (1.2) also contains an additional compound Poisson random measure term; the case of semimartingale drivers was also investigated. All proofs fundamentally rely on the isometry property of stochastic integration with respect to Brownian motion or compound Poisson measures. There are however a number of interesting non-semimartingale random processes of practical relevance, such as Mandelbrot's fractional Brownian motion [START_REF] Mandelbrot | Fractional Brownian motions, fractional noises and applications[END_REF] or random Fourier series [START_REF] Friz | The Jain-Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory[END_REF].

Their sample paths are α-Hölder continuous, for a regularity exponent α P p0, 1q. One can use T. Lyons' work [START_REF] Lyons | Differential equations driven by rough signals. I. An extension of an inequality of L. C. Young[END_REF] to solve differential equations driven by a fractional Brownian motion, for α ą 1{2; it relies on the notion of Young integral [START_REF] Young | An inequality of Hölder type, connected with Stieltjes integration[END_REF]. The case 1{4 ď α ď 1{2 is much more involved and can be handled using Lyons' theory of rough paths [START_REF] Lyons | Differential equations driven by rough signals[END_REF]. In [START_REF] Levakov | Existence of solutions of stochastic differential inclusions with standard and fractional Brownian motions[END_REF], A. Levakov and M. Vas'kovskii obtain the existence of solutions to stochastic differential inclusions of the form

X t ´Xs P ż t s F pr, X r qdr `ż t s G 1 pr, X r qdW r `ż t s G 2 pr, X r qdW FBM r
where W FBM is a fractional Brownian motion with α-Hölder continuous sample paths, with α ą 1{2, and the set valued map G 2 is a globally Lipschitz function of x that takes values in the set of nonempty compact convex subsets of R d and satisfies a local δ-Hölder condition as a function of time, with δ ą 1 ´α.

-Young differential inclusions

Motivated by control problems, the first aim of this article is to define and prove the existence of solutions of Young differential inclusions

z t " ξ `ż t 0 v r dx r , v r P F pz r q, (1.3)
where x is an R -valued α-Hölder continuous control, with α ą 1{2, and F is a γ -Hölder continuous set valued map with compact values, for a regularity exponent γ P `1 α ´1, 1 ˘. We do not require that F takes values in convex sets. The notion of a solution to a Young differential inclusion involves a number of elementary results on Young integrals that are recalled in Appendix A.

Notations. We gather here a number of notations that are used throughout the work.

' For any integer m, a point a P R m and a positive real number R, we denote by Bpa, Rq the closed ball of center a and radius R in R m .

' For α P p0, 1s and U, V Banach spaces, we denote by C α pU, V q the space of α-Hölder continuous functions from U to V . We write }x} α for the α-Hölder semi-norm of a path x P C α pU, V q. We set x s,t :" x t ´xs , for s, t P U .

' Given a set S, we denote by 2 S the power set of S, 9 2 S for 2 S ztHu and K `S˘t he compact sets of 9 2 S . We denote by LpR , R d q the space of linear maps from R to R d . Recall that endowing the space 2 LpR ,R d q with the Hausdorff pseudo-metric turns the space K `LpR , R d q ˘into a complete metric space. For F : R d Ñ KpLpR , R d qq a γ-Hölder set-valued map, we denote again by }F } γ the γ-Hölder semi-norm of F . (See Chapter 1, Section 5 in [START_REF] Aubin | Differential inclusions. Set-valued maps and viability theory[END_REF]) ' If f is a single-valued map of p-variation, we denote by }f } p´var the p-variation semi-norm of f -more on this notation in Appendix A.

Definition -Let x be an element of C α pr0, T s, R q, with α P `1 2 , 1 ‰ , and F : R d Þ Ñ 9 2 LpR ,R d q be a set-valued map. A solution to the Young differential inclusion

dz t P F pz t qdx t , z 0 " ξ P R d (1.4)
is a pair of paths pz, vq, defined on the time interval r0, T s, ' with v a LpR , R d q-valued path of finite p-variation such that α `1 p ą 1,

' and for all 0 ď t ď T ,

v t P F pz t q and z t " ξ `ż t 0 v s dx s .
The letter v is chosen for "velocity". The integral ş t 0 v s dx s makes sense as a Young integral under the assumption α `1 q ą 1 -see Appendix A.

1.1. Theorem -Let a positive time horizon T be given and x P C α pr0, T s, R q with α P `1 2 , 1 ‰ . Let F : R d Þ Ñ K `LpR , R d q ˘be a bounded set-valued map with nonempty compact values and γ-Hölder, for a regularity exponent γ P `1 α ´1, 1 ˘. Then, for any initial condition ξ P R d , the Young differential inclusion

dz t P F pz t qdx t ,
has a solution path started from ξ, defined over the time interval r0, T s.

For a single-valued map, this is a consequence of Young's original result [START_REF] Young | An inequality of Hölder type, connected with Stieltjes integration[END_REF][START_REF] Lyons | Differential equations driven by rough signals. I. An extension of an inequality of L. C. Young[END_REF]. Unlike the setting of ordinary differential equations, no uniqueness is to be expected in the present setting. Our regularity condition on the set-valued map F is the same as in the ordinary differential setting. However, there may be no Hölder, or even continuous, selection of Fsee Proposition 8.2 in [START_REF] Chistyakov | On maps of bounded p-variation with p ą 1[END_REF], so existence results for Young differential inclusions do not follow from existence results for Young differential equations. We refer the reader to Chapters 1 and 2 of [START_REF] Aubin | Differential inclusions. Set-valued maps and viability theory[END_REF] for the basics on differential inclusions. Note that since continuous paths with finite p-variation can be reparamatrized into 1{p-Hölder paths, the result of Theorem 1.1 holds for continuous paths x with finite p-variation 1 ď p ă 2.

-A selection result

As a by-product of our proof of Theorem 1.1, we show in Section 3 that a bounded, nonempty compact-valued, γ-Hölder continuous set-valued map on the interval r0, 1s has a selection with finite p-variation, for p ą 1{γ. 1.2. Theorem -Pick γ P p0, 1s, and let F : r0, 1s Þ Ñ KpR d q, be a bounded, γ-Hölder nonempty compact set-valued map. Then, for any p ą 1{γ and any ξ P F p0q, there exists a map f : r0, 1s Þ Ñ R d , of finite p-variation, such that f ptq P F ptq, for all 0 ď t ď 1, with f p0q " ξ, and which furthermore satisfies the estimate for a constant C γ,p depending only on γ, p

}f } p´var ď C γ,p }F } γ .
This statement partly answers an open question in Chistyakov and Galkin's work [START_REF] Chistyakov | On maps of bounded p-variation with p ą 1[END_REF] see Remark 8.1 therein.

-Rough differential inclusions

Stochastic analysis has undergone a real change under the impulse of T. Lyons' theory of rough paths [START_REF] Lyons | Differential equations driven by rough signals[END_REF]. It provides a deep understanding of stochastic differential equations and disentangles completely in that setting probabilistic and dynamical matters. As a first step to establishing a full theory of rough differential inclusions, the second aim of the present work is to deal with rough differential equations perturbed by set-valued drifts. Fix a finite positive time horizon T . We refer here the reader to Appendix A for basics on rough paths and rough integrals, in the setting of controlled paths; it suffices here to say that controlled paths are needed to make sense of the rough integral that appears in the next definition.

For an integer k ě 1 and γ P r0, 1s, let us denote by C k,γ b the space of bounded maps G from R d to LpR , R d q which are k-times differentiable, with bounded derivatives, and have a γ-Hölder k-th derivative. We endow C k,γ b with the norm

}G} C k,γ b :" k ÿ i"0 }D i G} 8 `}G} γ 1 γą0 ,
where D i G is the i-th derivative of G. By convention D 0 G :" G. Definition -Let T be a fixed positive time horizon. Pick ξ P R d , let Ω be a subset of R ˆRd containing r0, T s ˆtξu. Let X be a weak geometric α-Hölder rough path, with 1{3 ă α ď 1{2. Let F : Ω Ñ 9

2 R d , be a set-valued map, and G P C 1,γ b with p2 `γqα ą 1. An R d -valued path z started from ξ is said to be a solution to the rough differential inclusion

dz t P F pt, z t qdt `Gpz t qdX t , (1.5)
if z is part of a path pz, z 1 q controlled by X, and there exists an absolutely continuous path x from r0, T s to R d starting from zero, such that 9

x t P F pt, z t q at Lebesgue almost all times 0 ď t ď T , and one has

z t " ξ `xt `ż t 0 Gpz s qdX s at all times.
The regularity assumption on G is the optimal classical regularity assumption required to have a well-posed rough differential equation when F is null. The technics used to prove the existence of a solution to equation (1.5) are different depending on the regularity of the set-valued drift F . We first deal with the case where the drift is upper semicontinuous. We need in that case to assume that F takes values in the set of convex compact subsets of R d . 1.3. Theorem -Let X be a weak geometric α-Hölder rough path, with 1{3 ă α ď 1{2, and let G be a C 2,γ b one form, with αp2 `γq ą 1. Let F be an upper semicontinuous set-valued drift. Assume further that F is locally bounded, and takes it values in the set of nonempty compact convex subsets of R d . Then there exists a time horizon T ˚P p0, T q and a solution z to the rough differential inclusion (1.5), defined on the time interval r0, T ˚s.

In contrast to the preceding statement, we do not need to assume that F has convex images in the case where it is lower semicontinuous; we assume instead a mild boundedness assumption on F . 1.4. Theorem -Let X be a weak geometric α-Hölder rough path, with 1{3 ă α ď 1{2, and let G be a C 1,γ b one form, with αp2 `γq ą 1. Assume that the set-valued drift F is defined on a closed subset Ω of R ˆRd , where it is lower semicontinuous. Assume further that F takes values in the set KpR d q of nonempty compact subsets of R d , and that there is a positive constant L such that r0, T s ˆB`ξ , LT α ˘Ă Ω, and

› › F pt, aq › › ď L, @pt, aq P r0, T s ˆB`ξ , LT α ˘.
Then there exists a time horizon T ˚P p0, T q and a solution z to the rough differential inclusion (1.5) stated at ξ, defined on the time interval r0, T ˚s.

Note that the regularity of G is lower in Theorem 1.4 than in Theorem 1.3. When G " 0 in Theorem 1.3 and Theorem 1.4, we recover classical conditions for existence of solutions to differential inclusions -see e.g. the classical reference [START_REF] Aubin | Differential inclusions. Set-valued maps and viability theory[END_REF]. In Theorem 1.4, when F " t0u, we recover the optimal condition of regularity on G for existence of solutions to a rough differential equation [START_REF] Davie | Differential equations driven by rough paths: an approach via discrete approximation[END_REF].

The work has been organized as follows. Section 2 is dedicated to proving Theorem 1.1 on Young differential inclusions, following the above strategy. Theorem 1.2 is proved in Section 3, and Section 4 is dedicated to the proof of Theorem 1.4.

-Young differential inclusions

This section is dedicated to proving Theorem 1.1. Given q ě 1 and a finite dimensional vector space E, we denote by V q `r0, T s, E ˘the space of E-valued paths with finite qvariation on the time interval r0, T s. Refer to Appendix A.

We fix α P p 1 2 , 1s once and for all in this section, and work in the setting of Theorem 1.1. The pattern of proof of Theorem 1.1 goes as follows. Recall from Appendix A that the space of paths with finite p-variation is endowed with the p-variation norm } ¨}p-var,8 , defined in (A.1). Given a positive finite time horizon T , we define the dyadic partitions π pmq " tt m i u of the interval r0, T s, with t m i :" i2 ´mT , 0 ď i ď 2 m and m ě 1. Let γ be given in Theorem 1.1 and p ě 1 be such that γ p `α ą 1. We construct in Section 2.1 an approximate solution to the problem on a sufficiently small time interval r0, T s, under the form of a pair pz m , v m q such that ' v m has } ¨}p-var,8 -norm uniformly bounded in m, and is equioscillating, ' v m t P F pz m t q for all dyadic times, and

z m t " ξ `ż t 0 v m u dx u , 0 ď t ď T,
with ξ P R d . It follows then from the first item that the sequence v m has a converging subsequence v m k with limit some v, for the } ¨}q-var,8 -norm, for any q ą p with γ q `α ą 1. The continuity statement from Corollary A.2 implies then that z m k converges in } ¨}α,8 -norm to the path z :" ξ `ş¨0 v u dx u . One gets the fact that v t P F pz t q, for all 0 ď t ď T , from the fact that F is bounded and takes values in closed sets. Thus pz, vq is a solution in C α `r0, T s, R d ˘ˆV q `r0, T s, LpR , R d q ˘. The existence of a solution to the inclusion defined up to the initial time horizon is a consequence of the fact that the previous existence time does not depend on ξ. We now turn to the details.

-Construction of the approximate solution

For t P Ť mě0 π pmq , set
M ptq :" min j; t P π pjq ( and define the ancestor sptq of t as sptq :" max

! s P π pM ptq´1q ; s ă t ) . (2.1)
For each m ě 0, we construct the path z m on r0, t m i`1 s, and v m on r0, t m i`1 q, recursively on 0 ď i ď 2 m ´1 where t m i`1 " 2 ´mT pi `1q. The construction is not inductive on m. ' For m " 0, choose v 0 0 P F pξq, and set v 0 t " v 0 0 , z 0 t " ξ `v0 0 x 0,t , @t P r0, T s.

' Pick m ě 1.We set v m t :" v 0 0 , @t P r0, T 2 ´mq, z m t :" ξ `v0 0 x 0,t , @t P r0, T 2 ´ms.

This starts the induction over

0 ď i ď 2 m ´1. If z m : r0, t m i s Ñ R d and v m : r0, t m i q Ñ R d , have been constructed, use the Hölder continuity of F to choose v m t m i P F pz m t m i q such that ˇˇv m t m i ´vm spt m i q ˇˇď }F } γ ˇˇz m t m i ´zm spt m i q ˇˇγ, and set v m t :" v m t m i , @t P rt m i , t m i `T 2 ´mq, z m t :" z m t m i `vm t m i x t m i t , @t P rt m i , t m i `T 2 ´ms. If t m i `2´m T " T , set v m T " v m t m i .
We have z m " ξ `şt 0 v m u dx u , as a consequence of the fact that v m is constant along the intervals of the partition π pmq of r0, T s. The next section is dedicated to proving a uniform p-variation bound on a small time interval satisfied uniformly by the paths v m .

-Study of the p-variation norm of v m in a small time interval

This section is dedicated to proving the following intermediate result.

2.1. Proposition -Pick 1 1`γ ă β ă α, and set T 0 as equal to

min # 1, `2}F } 8 }x} α ˘´2 α´β , ˆ2}F } γ }x} α 1 ´2´pα`βγ´1q ˙´2 α´β , ˆ}F } γ 1 ´2´γβ ˙´4 γpα´βq + . (2.2)
Pick p ą 1{pγβq. Then we have, for any S ď T 0 ,

}v m } p´var,r0,Ss ď ˆ1 1 ´2´γβp ˙1{p S pα´βqγ 4 `γβ . (2.
3)

The proof of Proposition 2.1 proceeds in several steps. We first give a discrete integral representation of z m that proves useful. 2.2. Lemma -Pick 0 ď n ď m, and two consecutive points s, t in π pnq . Then, setting ' One then sees by induction on k that setting as in the statement of the Lemma

s k i " s `i2 ´n´k T , we have z m s,t " v m s x s,t `m´n´1 ÿ k"0 2 k ´1 ÿ i"0 v m s k i ,s k`1 2i`1 x s k`1 2i`1 ,s k i`1 . Proof. For 0 ď s ď t ď T , set µ m s,t :" µ m,
s k i " s `i2 ´kpt ´sq " s `i2 ´n´k T, one has µ m,pkq s,t " 2 k ´1 ÿ i"0 µ m s k i ,s k i`1 and µ m,pk`1q s,t ´µm,pkq s,t " 2 k ´1 ÿ i"0 v m s k i ,s k`1 2i`1 x s k`1 2i`1 ,s k i`1
.

(2.5)

Summing equation (2.5) for k from 0 to m ´n, and using identity (2.4) gives the identity of the Lemma.

2.3. Corollary -Pick 1 1`γ ă β ă α, and set

T 1 :" min # 1, p2}F } 8 }x} α q ´2 α´β , ˆ2}F } γ }x} α 1 ´2´pα`βγ´1q ˙´2 α´β + .
Then we have for any 0 ď S ď T 1 and any 0 ď n ď m, the m-uniform bound

sup rs,tsPπ pnq |z m s,t | ď S α´β 2 pS2 ´nq β ; (2.6)
the supremum is over consecutive points s, t of π pnq .

Proof. The proof is again by descending induction on n P t0, . . . , mu. We first have for two consecutive points s, t of π pmq the estimate |z m s,t | ď }F } 8 }x} α pS2 ´mq α , so (2.6) holds true for m " n since 0 ď S ď T 1 . Assume now that (2.6) has been proved for n ď m, an let s, t be two consecutive points of π pn´1q . We use the representation formula

z m s,t " v m s x s,t `m´n`1 ÿ k"0 2 k ´1 ÿ i"0 v m s k i ,s k`1 2i`1 x s k`1 2i`1 ,s k i`1
from Lemma 2.2, with s k i " s `i2 ´n´k`1 S. Note that M ps k`1 2i`1 q " n `k, here, and the ancestor sps k`1 2i`1 q of s k`1 2i`1 is s k i . Then, using (2.6) for n `k, we have ˇˇv m

s k i ,s k`1 2i`1 ˇˇď }F } γ ˇˇz m s k i ,s k`1 2i`1 ˇˇγ ď }F } γ S pα´βq 2
γ `S2 ´n´k ˘βγ , so we obtain for |z m st | the upper bounds

}F } 8 }x} α `2´n`1 S ˘α `m´n`1 ÿ k"0 2 k S pα´βq 2 γ `S2 ´n´k ˘βγ }x} α `S2 ´n´k ˘α ď }F } 8 }x} α `2´n`1 S ˘α `}x} α }F } γ S α`γβ`p α´βq 2 γ 2 ´npα`βγq 1 ´2´pα`βγ´1q . The choice of S ď T 1 ensures that }F } 8 }x} α p2 ´n`1 Sq α ď 1 2 S α´β 2 p2 ´pn´1q Sq β
and since α `γβ ą 1 ą α and S ă 1 (2.7)

}x} α }F } γ S α`γβ`p α´βq 2 γ 2 ´npα`βγq 1 ´2´pα`βγ´1q ď
Proof. Note that if n ě m and s and t are two consecutive points of π pnq , the function v m has null oscillation on the interval rs, tq, since it is constant on the intervals of the partition π pmq . Let then take n ď pm ´1q. Set s 0 " s and define a finite sequence ps i q i"0,...,m´n setting s i`1 " s i `2´pn`i`1q S, if s i `2´pn`i`1q S ď t, and s i`1 " s i , otherwise. Then s i P π pn`iq , for any i, and either s i`1 " s i or its ancestor sps i`1 q is s i . We then have from the uniform estimate (2.6) on z m s,t the bound

ˇˇv m s i ,s i`1 ˇˇď }F } γ ˇˇz m s i ,s i`1 ˇˇγ ď }F } γ ´S α´β 2 pS2 ´n´i q β ¯γ .
We obtain (2.7) summing these inequalities for i from 0 to pm´nq, and from the definition of T 0 .

Proof of Proposition 2.1. Take 0 ď S ď T 0 , and let π " ps i q N i"0 be a partition of the interval r0, Ss.

The following partition is a kind of greedy partition of the set t0, ..., N ´1u, in terms of the size of corresponding increments in the above formula. Let then set I 0 :" ! P t0, .., N ´1u; Dj P t0, ..., 2 m ´1u, ps , s `1s Ă rt m j , t m j`1 q

) , I 1 :" ! P t0, .., N ´1u; 2 ´1 P ps , s `1s

) , and, for 2 ď j ď m, set I j :" ! P t0, .., N ´1u; Dt P π pjq zπ pj´1q X ps , s `1s

) .

We define a partition of t0, . . . , N ´1u setting K 0 :" I 0 and

K j :" I j z j´1 ď k"0 I k ,
for 2 ď j ď m. Note that K j has at most 2 j elements.

' For P K 1 , we know from Corollary 2.4 on oscillations of v m , with n " 0, that

ˇˇv m s l`1 ´vm s l ˇˇď S pα´βqγ 4
`γβ .

' For P K 0 , one has v m s j ,s j`1 " 0, since v m is constant over the intervall of π pmq .

' For P K j , there exists t P π pjq zπ pj´1q such that s ă t ď s `1, and for all u P π pj´1q , one has either u ď s , or u ą s `1. Then rs , s `1q Ă rt ´S2 ´j , t `S2 ´j q, and using Corollary 2.4 with j ´1, one gets

ˇˇv m s `1 ´vm s ˇˇď S pα´βqγ 4 `S2 ´pj´1q ˘γβ .
Taking p ą 1{pγβq, one then has

N ´1 ÿ i"0 ˇˇv m s i`1 ´vm s i ˇˇp ď 2S pα´βqγp 4 m ÿ j"0 2 j `S2 ´pj´1q ˘γβp ď 2S pα´βqγp 4 S γβp 1 1 ´2´γβp , from which (2.3) follows }v m } p´var,r0,Ss ď ˆ1 1 ´2´γβp ˙1{p S pα´βqγ 4
`γβ .

-Local and global existence for solutions

We finally give the proof of Theorem 1.1 in this section. We first prove the existence of a solution to the differential inclusion (1.4) on the time interval r0, T 0 s, for the time T 0 defined in (2.2) in Proposition 2.1. Since the definition of T 0 does not involve the initial condition ξ of the dynamics, we obtain by concatenation a solution to the inclusion defined over the whole interval r0, T s. ' Recall that a sequence of bounded functions py m q mě0 from compact segment ra, bs into a compact set K is said to be equioscillating if, and only if, one can associate to any positive ε a finite partition pJ k q 0ďkďr of ra, bs into subintervals such that Oscpy m , J k q ď ε, uniformly in k, m. The Ascoli-Arzela-type convergence theorem from Theorem 5 of section 4 of chap 0 of Aubin and Cellina's book [START_REF] Aubin | Differential inclusions. Set-valued maps and viability theory[END_REF], ensure the existence of a subsequence py m k q kě0 that converges uniformly to some limit bounded function y. Since the family pv m q mě0 is bounded by }F } 8 , and equioscillating, from Corolary 2.4, it has a uniformly converging subsequence, with limit v.

Pick then q ą p such that 1 q `α ą 1. Since all the v m have the same starting point, we have the elementary interpolation bound › › v m ´vn › › q´var,r0,T 0 s ď `2}v m ´vn } 8,r0,T 0 s ˘q´p q `}v m } p´var,r0,T 0 s `}v n } p´var,r0,T 0 s ˘p{q , on which we read off the convergence of a subsequence z m k of the v m in q-variation norm to its limit, as a consequence of the uniform bound from Proposition 2.1. The convergence to v of this subsequence is thus in the sense of the } ¨}q´var,r0,T 0 s,8 -norm.

' The continuity result on Young integrals recalled in Corollary A.2 from Appendix A implies then the convregence in the norm } ¨}α,8 of z m k on r0, T 0 s to the path z defined by the equation

z t " ξ `ż t 0 v u dx u .
' It remains to prove that v t P F pz t q, for all times 0 ď t ď T 0 . For a dyadic time t P Ť mě0 π pmq , then for k big enough, one has v m k t P F pz m k t q. Then, since F is γ-Hölder, one has

d `vt , F pz t q ˘ď ˇˇv t ´vm k t ˇˇ`d `F pz m k t q, F pz t q ˘ď ˇˇv t ´vm k t ˇˇ`}F } γ ˇˇz m k t ´zt ˇˇγ ;
so one gets dpv t , F pz t qq " 0, and v t P F pz t q, since F pz t q is closed. For an non-dyadic time t P r0, T 0 s, there exists two consecutive points in π pmq such that t P ru, vr, for every m ě 0. Then,

d `vt , F pz t q ˘ď ˇˇv t ´vm t ˇˇ`ˇˇv m t ´vm u ˇˇ`d `vm u , F pz m u q ˘`d `F pz m u q, F pz u q d`F pz u q, F pz t q ˘,
› v ´vm k › › 8,r0,T s `pT 0 2 ´mk q γβ `}F } γ ´› › z ´vm k › › 8,r0,T s `}z} α,r0,T s pT 0 2 ´mk q γα ¯.
So one gets dpv t , F pz t qq " 0, and v t P F pz t q, since F pz t q is closed.

-A selection result

As a by product of the proof of Theorem 1.1, we obtain the selection result of Theorem 1.2 wich partially answers Remark 8.1 of Chistyakov and Galkin's work [START_REF] Chistyakov | On maps of bounded p-variation with p ą 1[END_REF]. This section is dedicated to proving Theorem 1.2.

We use the same notations an in Section 2. Define for each non-negative integer m the partition π m :" tt m i u i"0..2 m of the interval r0, 1s, with t m i :" i2 ´m. We define as follows a path x m : r0, 1s Þ Ñ R d , on each sub-interval r0, t m i q, recursively over i. ' For m " 0, set x 0 ptq " x 0 , for all 0 ď t ď 1. ' For m ě 1, set first x m ptq " x 0 , on r0, t m 1 q, and assuming x m has been constructed on the time interval r0, τ q, set x m T " x m τ ´2´m , if τ " T , otherwise choose x m τ P F pτ q such that d ´xm τ ; x m spτ q ¯ď }F } γ 2 p´M pτ q`1qγ and set x m t " x m τ , for τ ď t ă τ `2´m . We first prove that we have for all r, m the estimate

max tPπ r sup sPrt,t`2 ´r q ˇˇx m s,t ˇˇď }F } γ 1 ´2´γ 2 ´rγ . (3.1)
By construction, x m is constant on the each interval of π m , so if r ě m, t P π r and s P rt, t `2´r q, then x m s,t " 0 and (3.1) holds true. Let then consider the case where r ă m. We define a finite sequence of times ps i q i"0,...,m´r , with s 0 " t, such that for 0 ď i ď pm ´r ´1q, we have s i`1 " s i `2´r´i´1 , if s i `2´r´i´1 ď s, and s i`1 " s i otherwise. Then, for s i P π r`i , for i ě 1, and either sps i`1 q " s i or s i`1 " s i . The path x m is constructed in such a way as to have

ˇˇx m s i ,s i`1 ˇˇď }F } γ 2 ´pr`i´1qγ .
Summing these estimates for 0 ď i ď m ´r, gives (3.1), and

max tPπ r Osc `xm , rt, t `2´r q ˘ď 2}F } γ 1 ´2´γ 2 ´rγ . (3.2)
Since x m is constant along the sub-intervals of the partition π m it is enough, to compute the p-variation of x m along any partition π " ts i , i " 0, ..., N u of r0, 1s, to assume that all the partition times s i P π pmq . Let then define as follows be a finite partition pK j q j"0,...,m of t0, . . . , N ´1u, with possibly empty sets : ) , 2 ď j ď m, with K 0 :" I 0 and

I 0 :" ! P t0,
K j`1 :" I j`1 z ˜j ď k"0 I k ¸.
For P K 1 , using (3.2) for r " 0 we bound

ˇˇx m s `1 ´xm s ˇˇď 2}F } γ 1 ´2´γ . (3.3)
For P K j , then there exists t P π j zπ j´1 such that

s ă t ď s `1,
and any u P π j´1 satisfies either u ď s or u ą s `1. Then rs , s `1q Ă rt ´2´j , t `2´j q, and using inequality (3.2) for j ´1, we see that

ˇˇx m s `1 ´xm s ˇˇď 2}F } γ 1 ´2´γ 2 ´pj´1qγ . (3.4)
The number of indices P K j is at most 2 j . Using the fact that the family pK j q 1ďjďm defines a partition of t0, . . . , N ´1u, and adding inequalities (3.3) and (3.4) for q ą 1{γ, we then have

N ´1 ÿ i"0 ˇˇx m s i`1 ´xm s i ˇˇq ď ˆ2}F } γ 2 γ 1 ´2´γ ˙q m ÿ j"0 2 j 2 ´jγq ď ˆ2}F } γ 2 γ 1 ´2´γ ˙q 1 1 ´2´γq`1
and we derive

}x m } q´var ď 2}F } γ 2 γ 1 ´2´γ
1 p1 ´2´γq`1 q 1{q . From Helly' selection principle, see e.g. Theorem 6.1 in Chistyakov and Galkin's work [START_REF] Chistyakov | On maps of bounded p-variation with p ą 1[END_REF], the sequence x m has a convergent subsequence in p-variation, for any p ą q. We identify the path f from the statement as such a limit. One proves that f is a selection of F in the same way as we proved that v t P F pz t q in Section 2.3, using the fact that F ptq is closed for all times, and the regularity properties of F .

-Rough differential inclusions

This section is dedicated to the proof of Theorem 1.3 and Theorem 1.4. We refer the reader to Appendix A for basics on rough paths theory. All we need to know is recalled there. The reader will notice that the proof of Theorem 1.3 is much shorter than the proof of Theorem 1.4. This is due to the fact that we somehow assume much more in Theorem 1.3, asking in particular that the set-valued drift has compact convex images. Together with the assumed upper semicontinuity, this allows for the use of powerful approximate selection theorems that greatly simplify the matter. See e.g. Section 1 of Chapter 2 in [START_REF] Aubin | Differential inclusions. Set-valued maps and viability theory[END_REF] for the differential inclusion case.

-Upper semicontinuous drift

This section is dedicated to proving Theorem 1.3; we work in this section in the setting of that statement. Here, X is an α-Hölder weak geometric rough path with α P p1{3, 1{2s and G P C 2,γ b with αp2 `γq ą 1. We recall that according to the notation introduced in the Introduction, C 1 pr0, T s, R d q is the space of R d -valued Lipschitz paths defined on r0, T s. Given x P C 1 pr0, T s, R d q, we denote by ψpt, xq the solution path to the rough differential equation

y t " ξ `xt `ż t 0 Gpy s qdX s , (4.1)
with fixed initial condition ξ. The controlled solution path comes under the form of a pair `yt , Gpy t q ˘. Classical results from rough paths theory ensure that ψ is a continuous function from C 1 pr0, T s, R d q with values in the space C α pr0, T s, R d q -see e.g. Theorem 3 of [START_REF] Coutin | Sensitivity of rough differential equations: an approach through the Omega lemma[END_REF]. This is where we need the assumption that G is p2 `γq-Hölder, rather than just p1 `γq-Hölder, as in the proof of Theorem 1.4 given in the next section. For each 0 ď t ď T , one has ψpt, xq " ψpt, xq, for any other x P C 1 pr0, T s, R d q that coincides with x on the time interval r0, ts. We can then work in the setting of differential inclusions with memory -see e.g. Section 7 in Chapter 4 of Aubin & Cellina's book [START_REF] Aubin | Differential inclusions. Set-valued maps and viability theory[END_REF]. Let us define Fpt, xq :" F `t, ψpt, xq ˘.

The function F is upper semicontinuous on R `ˆC 1 pr0, T s, R d q, with values in the set of convex compact subsets of R d . We can then use the obvious variant of Theorem 1 in Section 7 in Chapter 4 in [START_REF] Aubin | Differential inclusions. Set-valued maps and viability theory[END_REF], with no constraint and Kptq " R d for all t with the notations therein, to get the existence of a time T ˚P p0, T s, and a Lipschitz path x, defined on the time interval r0, T ˚s, such that one has for almost all t P r0, T ˚s

9 x t P F `t, x ˘.
This condition is equivalent to saying that the path y from (4.1) solves the rough differential inclusion dy t P F pt, y t q `Gpy t qdX t .

(Note that the convexity assumption on the pointwise images of F is essential for the use Theorem 1 in Section 7 in Chapter 4 of Aubin-Cellina's book [START_REF] Aubin | Differential inclusions. Set-valued maps and viability theory[END_REF].)

-Lower semicontinuous drift

This section is dedicated to proving Theorem 1.4. We take advantage in that task of the approach to Filipov's theorem given by Bressan in [START_REF] Bressan | Directionally continuous selections and differential inclusions[END_REF], based on a selection theorem of independent interest. We recall the setting before embarking on the proof of Theorem 1.4.

Recall also that a cone of R m is a subset Γ of R m such that Γ X p´Γq " t0u, and λa P Γ, if λ ě 0 and a P Γ. As an example, given a positive constant M , the set

Γ M :" ! pt, xq P R ˆRd ; t ě 0, }x} ď tM ) Ă R d`1 is a cone of R d`1 . Definition -Let Γ be a cone of R m . A map h : R m Ñ R d is said to be Γ-continuous at point a P R m , if for any ε ą 0, there is δ ą 0 such that › › hpbq ´hpaq › › ă ε, for any b P Bpa, δq X pa `Γq.
We say that h is Γ-continuous on a subset S Ă R m , if h is Γ-continuous at any point of S. The relevance of the notion of Γ-continuity in the setting of differential inclusions is a consequence of the following selection result, due to Bressan, Theorem 1 in [START_REF] Bressan | Directionally continuous selections and differential inclusions[END_REF].

4.1. Theorem -Let H : R m Ñ 9
2 R d be lower semicontinuous set-valued map with non-empty closed values. Then, for any cone Γ Ă R m , there is a Γ-continuous selection of H.

For a positive finite constant L, and pβ, ξq P p0, 1s ˆRd , set

E β L :" ! u P C 0 pr0, T s, R d q ; }u} β ď L, u 0 " ξ ) ,
where }u} β is the β-Hölder norm of u, for 0 ă β ă 1, and }u} 1 is the Lipschitz norm of u. The set E β L is a compact subset of `C0 pr0, T s, R d q, } ¨}8 ˘. We do not emphasize the dependence on ξ and T in the notation. 4.2. Lemma -Assume the set-valued map F is defined on a closed subset Ω of R ˆRd where it is lower semicontinuous and takes values in the set KpR d q of nonempty compact subsets of R d . Assume further that there is a positive constant L such that r0, T s ˆB`ξ , LT β ˘Ă Ω, and

› › F pt, xq › › ď L, for all pt, xq P r0, T s ˆB`ξ , LT β ˘.
Then there exists a continuous map

φ : E β L Ñ E 1 L
such that one has dφpuqptq dt P F pt, uptqq, for almost all t P r0, T s, (

for any u P E β L . Proof. We first extend F into a lower semicontinuous map F ˚defined on R ˆRd , setting We have }f pt, u t q} ď L, since u P E β L , and given the assumptions on F . So φpuq P E 1 L , and one has indeed the inclusion (4.2) for almost all times.

F ˚pt, xq " Bp0, Lq, if pt, xq R Ω. Pick M ą L,
One can proceed as follows to show the continuity of the map φ. Since f is Γ Mcontinuous, one can associate to any time t P r0, T s a positive η t such that for all s P rt, t `ηt sXr0, T s and }a ´ut } ď M η t , one has

› › f ps, aq ´f pt, u t q › › ď ε 4T . (4.3)
Write Leb for Lebesgue measure on the real line. From Lemma 1 in Bressan's work [START_REF] Bressan | Directionally continuous selections and differential inclusions[END_REF],

there is a finite family ˆ"t i , t i `η

J i :" « t i , t i `ˆδ M ´L ˙1 α ¸.

Then

Leb

˜N ď

i"1

J i ¸ď N ˆδ M ´L ˙1 α ď ε 8L . ( 4
› › vptq ´upt i q › › ď › › vptq ´uptq › › `› › uptq ´upt i q › › ď δ `Lpt ´ti q β ď M pt ´ti q β while › › u t ´ut i › › ď Lpt ´ti q β ă M pt ´ti q β .
Thus, with (4.3) we obtain that, for t P IzJ, we have

› › f pt, v t q ´f pt, u t q › › ď › › f pt, v t q ´f pt i , u t i q › › `› › f pt i , u t i q ´f pt, u t q › › ď ε 2T .
Since f is bounded by L on r0, T s ˆB`z 0 , LT β ˘, it follows that we have for any t P r0, T s the estimate › › φpvqptq ´φpuqptq

› › ď ż T 0 › › f pt, v s q ´f pt, u s q › › ds ď ż IzJ › › f ps, v s q ´f ps, u s q › › ds `żJ Ť r0,T szI › › f ps, v s q ´f pt, u s q › › ds ď T ε 2T `´LebpJq `Lebpr0, T szIq ¯2L ď ε 2 `´ε 8L `ε 8L ¯2L ď ε.
We use from now on the notations on controlled paths recalled in Appendix A. Recall the one form G in Theorem 1.4 is assumed to be C 1,γ b , with 1 ă αp2 `γq ă 3α. Choose a constant β P p1{3, αq such that βp2 `γq ą 1.

We define a ball in the space of paths controlled by X P C α `r0, T s, R ˘, using the β-norm

B β L :" ! py, y 1 q P D β,2β pr0, T s, R d q ; › › py, y 1 q › › β,2β ď L, y 0 " ξ, y 1 0 " Gpξq ) .
It follows from Ascoli-Arzela theorem that this ball is a compact convex subset of

C 0 `r0, T s, R d ˘ˆC 0 `r0, T s, LpR , R d q ˘,
with both factor endowed with the uniform norm. For py, y 1 q P D β,2β pr0, T s, R d q, and t P r0, T s, we set Φpy, y 1 qptq :" ˆφpyqptq `ż t 0 Gpy s qdX s , Gpy t q ˙.

We prove below that Φ is a continuous map from B β L into itself, provided T is small enough. See point (a) for the stability and point (b) for the continuity. It follows then from Schauder fixed point theorem that Φ has a fixed point pz, z1 q in B β L . One gets the fact that pz, z1 q P D α,2α pr0, T s, R d q from the equation that it satisfies, as (A.3) yields the estimate

}z} α ďLT 1´α `}Gpzq} 8 }X} α `}Gpzq 1 } 8 }X} 2α `C}X} β › › pGpzq, Gpzq 1 q › › β,p1`γqβ T p2`γqβ´α , from which we get }z 1 } α ď }G} C 1 b }z} α , and finally for R z st :" zst ´z 1 s X st › › R z › › 2α ďLT 1´2α `}Gpzq 1 } 8 }X} 2α `C}X} β › › pGpzq, Gpzq 1 q › › β,p1`γqβ T p2`γqβ´2α
. The triple `pz, z1 q, φpzq ˘is then a solution of the rough differential inclusion (1.5).

(a) Stability -Write R y st for y st ´y1 s X st , for any 0 ď s ď t ď T , and similary for z where pz, z 1 q " Φpy, y 1 q. On the one hand, for py, y 1 q P B β L , one has

}y} β ď }y 1 } 8 }X} β `}R y } 2β T β ď p › › y 1 0 › › `› › y 1 › › β T β q }X} β `}R y } 2β T β ď `}Gpξq} `LT β ˘T α´β }X} α `LT β .
We then have y P E β L . On another hand, according to Proposition A.

› › › › ď L|t ´s| `}Gpyq 1 } 8 }X} 2β |t ´s| 2β `C}X} β › › pGpyq, Gpyq 1 q › › β,βpγ`1q |t ´s| βp2`γq ď ´LT 1´2β `}Gpyq 1 } 8 }X} 2α T α´β `C}X} β › › pGpyq, Gpyq 1 q › › β,βpγ`1q T βγ ¯|t ´s| 2β .
Using the estimate of the size of the nonlinear image of a controlled path given in (A.2), one has eventually for }R z } 2β the upper bound

´LT 1´2β `}G} C 1,0 b }y} 8 }X} β T α´β `C}X} α }G} C 1,γ b p1 `}Gpξq} `Lq 2 T βγ ď T α´β ´L `}G} C 1,0 b }y} 8 }X} β `C}X} α }G} C 1,γ b p1 `}Gpξq} `Lq 2 ¯(4.6)
where we use the inequalities T α´β ě T 1´2β , T βγ in the last line. Since

}R y } 2β ď L,
we also have

}y} 8 ď }ξ} `}Gpξq} }X} β T β `LT 2β . (4.7)
Hence, we obtain from (4.6) and (4.7) the upper bound

T α´β ´L `}σ} C 1,0 b }X} β p}ξ} `}Gpξq} }X} β T β `LT 2β q `C}X} α }G} C γ´1 b p1 `}Gpξq} `Lq 2 ¯, (4.8)
for }R z } 2β . We control the β-Hölder norm of z 1 as follows

}z 1 } β ď }G} C 1,0 b }y} β ď }G} C 1,0 b ´}y 1 } 8 }X} β `}R y } 2β T β ď }G} C 1,0 b ´`› › y 1 0 › › `}y 1 } β T β ˘}X} α T α´β `}R y } 2β T β ď }G} C 1,0 b ´`}Gpξq} `LT β ˘}X} α T α´β `LT β ď }G} C 1,0 b ´`}Gpξq} `LT β ˘}X} α `L¯T α´β (4.9)
With T small enough, we obtain from (4.8) and (4.9) that }pz, z 1 q} α,2α ď L. This shows indeed that Φ sends B β L into itself. (b) Continuity -The continuity of Φ on B β L , for the uniform topology on B β L , is a direct consequence of the continuity of φ and the continuity estimate (A.3) on rough integrals.

A -Basics on Young and rough integrals

Let p ě 1 be given. Recall that an R d -valued path x defined on the time interval ra, bs. We say that x has finite p-variation over that interval rs, ts Ă ra, bs if }y} p p-var,rs,ts :" sup

ÿ i |y s i`1 ´ys i | p ă 8,
where the supremum runs over the set of finite partitions ts i u of the interval rs, ts. We denote by V p pra, bs, R d q the set of R d -valued path with finite p-variation over ra, bs and by C 1{p pra, bs, R d q the set of R d -valued path 1 p -Hölder continuous on ra, bs. Note that an element of V p pra, bs, R d q need not be continuous, while C 1{p pra, bsR d q Ă V p pra, bs, R d q. Also, a continuous path with finite p-variation has a reparametrized version that is 1 p -Hölder over its interval of definition. Refer to [START_REF] Chistyakov | On maps of bounded p-variation with p ą 1[END_REF] for a reference. We endow the space of paths with finite p-variation with the norm }y} p-var,8,ra,bs :" }y} p-var,ra,bs `}y} 8,ra,bs , (A.1)

where }¨} 8,ra,bs denotes the uniform norm over ra, bs. Similarly, for α P r0, 1s, we define a norm on the space of R d -valued α-Hölder functions defined on the interval ra, bs, setting }x} α,8,ra,bs :" }x} α,ra,bs `}x} 8 ,

where }¨} α,ra,bs is the classical semi-norm of α-Hölder paths. For these norms and seminorms, we omit to precise the interval ra, bs when the norms and semi-norms are taken over the whole interval of the path is defined. Given a ď b, set ∆ a,b :" tps, tq; a ď s ď t ď bu. A control over the interval ra, bs is a map ω : ∆ a,b Þ Ñ R `that is null on the diagonal, is non-increasing, resp. non-decreasing, as a function of each of its first, resp. second, argument, and is sub-additive ωps, uq `ωpu, tq ď ωps, tq, @a ď s ď u ď t ď b.

Denote by ωps, t ´q the left limit in t of the non-decreasing function ωps, ¨q. A control is said to be regular if it is continuous in a neighbourhood of the diagonal. As an example, for any path y P V p pra, bs, R d q, the function ω y ps, tq :" }y} 1{p p-var,rs,ts is a control. For an R d -valued α-Hölder path x, with 0 ă α ă 1, the function ω x ps, tq :" }x} 1{α α,rs,ts is a regular control.

From the present day perspective, Gubinelli' sewing lemma [START_REF] Gubinelli | Controlling rough paths[END_REF] offers an easy road to constructing the Young integral -see also [START_REF] Feyel | Curvilinear integrals along enriched paths[END_REF]. We give here a variation due to Friz and Zhang [START_REF] Friz | Differential equations driven by rough paths with jumps[END_REF], tailor-made to our needs. Given a map µ : ∆ a,b Þ Ñ R d and a finite partition π " ts i u of the interval ra, bs, set

µ π :" ÿ i µ s i`1 s i .
A.1. Proposition -Let a map µ : ∆ a,b Þ Ñ R d be given. If there exists positive exponents α 1 , α 2 , with α 1 `α2 ą 1 and controls ω 1 , ω 2 , with ˇˇµ ts ´pµ tu `µus q ˇˇď ω 1 ps, uq α 1 ω 2 pu, tq α 2 , @a ď s ď u ď t ď b, with ω 2 regular, then, for any interval rs, ts Ă ra, bs, the µ πst converge to some limit I t s pµq as the mesh of the partition π ts of rs, ts tends to 0, and one has ˇˇI t s pµq ´µπst ˇˇď Cω 1 ps, t ´qα 1 ω 2 ps, tq α 2 , for some positive constant C depending only on α 1 , α 2 .

Given x P C α pra, bs, R q and y P V p `ra, bs, LpR , R d q ˘, set µ st :" y s px t ´xs q, and note that µ ts ´pµ tu `µus q " py s ´yu qpx t ´xu q, s ď u ď t, so one has for any interval rs, ts the estimate ˇˇµ ts ´pµ tu `µus q ˇˇď }y} 1{p p-var,rs,ts }x} α α,rs,ts .

A.2. Corollary -If 1 q `α ą 1, the Riemann sums µ πst associated with the preceding two-index map µ converge to some limit which we denote by ş t s y u dx u . One has › › › › ż 0 y u dx u › › › › α,rs,ts À }y} q´var,rs,ts }x} α,rs,ts , and for any ε ą 0 with 1´ε q `α ą 1, and y, y 1 P V q `ra, bs, LpR , R d q ˘, one has › › › › ż 0 y u dx u ´ż 0 y 1 u dx u › › › › α,rs,ts À }y ´y1 } ε 8,rs,ts ˆ´}y} 1´ε q´var,8,rs,ts `}y 1 } 1´ε q´var,8,rs,ts ¯}x} α,rs,ts . We refer the reader to the lecture notes [START_REF] Caruana | Differential Equations Driven by Rough Paths[END_REF][START_REF] Friz | A course on rough paths[END_REF][START_REF] Bailleul | A flow-based approach to rough differential equations[END_REF] for pedagogical accounts of rough paths theory. The following definitions and propositions will be sufficient for our needs here. Definition -Let α P p1{3, 1{2q. We say that X :" pX, Xq is an α-Hölder rough path if ' X is an R -valued α-Hölder path on a time interval r0, T s, ' X is a path from r0, T s 2 to R b R such that }X} 2α :" sup s,tPr0,T s,s‰t }X st } |t ´s| 2α ă `8, ' we have X rt ´Xrs ´Xst " X rs b X st , for all 0 ď r ď s ď t ď T . The formula }X} α :" }X} α `}X} 2α defines a seminorm on the space of α-Hölder rough paths. Definition -Let α P p0, 1s, θ ą α and X P C α pr0, T s, R q. A path y P C α pr0, T s, R d q is said controlled by X, with a remainder of order θ, if there is a path

y 1 P C θ´α `r0, T s, LpR , R d q ˘,
and a map R : r0, T s 2 Ñ R d , such that }R} θ :" sup s,tPr0,T s,t‰s › › R y s,t › › |t ´s| θ ă 8. and y st " y 1 s X st `Rst . We denote by D α,θ pr0, T s, R d q the space of such pairs py, y 1 q, and talk of py, y 1 q as a controlled path. We make no reference to the reference path X in the notation as there is no risk of confusion. We define a semi-norm on D α,θ pr0, T s, R d q setting › › py, y 1 q › › α,θ :" › › y 1 › › θ´α `}R y } θ ; a norm is defined by the formula › › py, y 1 q › › α,θ :" }y 0 } `› › y 1 0 › › `› › py, y 1 q › › α,θ . The next proposition says that the class of controlled paths is stable by nonlinear maps. See Corollary 3 and Proposition 4 of Gubinelli's original statement [START_REF] Gubinelli | Controlling rough paths[END_REF].

A.3. Proposition -For py, y 1 q P D α,θ pr0, T s, R d q, and f P C 1,ε b with ε P p0, 1s, one has `f pyq, f pyq 1 ˘:" `f pyq, Df pyqy 1 ˘P D α,θ 1 , with θ 1 :" minpθ, αp1 `εqq, and › › pf pyq, f pyq 1 q

› › α,θ 1 À T }f } C 1,ε b ˆ› › y 1 0 › › `› › py, y 1 q › › α,θ `"› › y 1 0 › › `› › py, y 1 q › › α,θ ı 1`ε `"› › y 1 0 › › `› › py, y 1 q › › α,θ ı θ 1 {α ˙, (A.2)
for an implicit multiplicative constant depending only on T , that decreases to 0 when T goes to 0. For a controlled path py, y 1 q P D α,θ pr0, T s, R d q, with α `θ ą 1, it is elementary to see that the two-index map µ st :" y s X st `y1 s X st , satisfies the estimate ˇˇµ st ´µsu ´µut ˇˇÀ |t ´s| α`θ , for all 0 ď s ď u ď t ď T , so one can use the sewing lemma to make sense of the integral ż 0 y s dX s as the additive functional associated to µ.

2 ` 4 `

 24 2´pn´1q S ˘β; this closes the descending induction step. Recall the oscillation Oscpv, Iq of a function v : I Þ Ñ R d , is defined by the formula Oscpv, Iq :" sup a,bPI `vpbq ´vpaq ˘.The uniform control of the oscillation of the v m provided by the next statement is necessary to use the compactness result on the set of bounded functions equipped with uniform norm stated in Theorem 5, Section 4, Chapter 0 of Aubin and Cellina's book[START_REF] Aubin | Differential inclusions. Set-valued maps and viability theory[END_REF]. Recall the definition of T 0 ď T 1 from the statement of Proposition 2.1. The notation rs, ts P π pnq used below stands for two consecutive points s, t in π pnq . 2.4. Corollary -For any 0 ď S ď T 0 , we have, for any 0 ď n ď m, sup rs,tsPπ pnq Osc `vm , rs, tq ˘ď S pα´βqγ S2´n˘γ β .

  and use Theorem 4.1 to pick a Γ M -continuous selection f of F ˚. For u P E β L , and 0 ď t ď T , set φpuqptq :" ξ `ż t 0 f ps, u s qds.

'

  We first prove by descending induction on n that for 0 ď n ď m, and two consecutive points s, t in π pnq , we have

							p0q s,t	" v m s x s,t
	and, for k ě 1,
			µ	m,pkq s,t	:" µ	m,pk´1q s, t`s 2	`µm,pk´1q t`s 2 ,t	.
		z m s,t " µ m,pm´nq s,t	" µ	m,pkq s,t	,	@k ě m ´n.	(2.4)
		z m s,t " z m s, s`t 2	`zm s`t 2 ,t " µ m,pm´n`1q s,t	.
	For k ě pm ´nq, we have
	µ m,pkq s,t	" µ m,pk´1q s, s`t 2	`µm,pk´1q s`t 2 ,t	" µ m,pm´nq s, s`t 2	`µm,pm´nq s`t 2 ,t	" µ	m,pm´n`1q s,t	;
	this closes the inductive proof of identity 2.4.

This identity holds true when n " m as a consequence of the definition of the objects. Assume that (2.4) holds true for n ď m, and let s, t be two consecutive points in π pn´1q . Since ps, s`t 2 q and p s`t 2 , tq are two consecutive pairs of points in π pnq , we have, from the definition of µ m,pkq and induction,

  3, one has `Gpyq, Gpyq 1 ˘P D β,βpγ`1q , because βγ ă 2β. Using (A.3) and the fact that }X} 2β ď T α´β }X} 2α , one has for any s, t P r0, T s, Gpy r qdX r ´Gpy s qX st

	}R z st } "	› › z st	´z1 s X st	› ›
				ż t
	ď }φpyq st }	`› › ›
				›	s

  It satisfies, for any s, t P r0, T s, the estimate

	› ż t			›
	› › ›	s	y r dX r ´ys X st	´y1 s X st	› › › À }X} α	› › py, y 1 q › › α,θ |t ´s| α`θ .	(A.3)

Acknowledgement

The second author of this article thanks the Center for Mathematical Modeling, Conicyt fund AFB 170001.


Christophe Morlaas
email: christophe.morlaas@enac.fr

Johan Duplouy

Hervé Aubert

4-Port Vector Antenna for MIMO Applications

come

I. INTRODUCTION

Wireless telecommunication systems cover more and more all modern human activity fields. Moreover, the congestion of the radio frequency spectrum implies to optimize the spectral efficiency of these systems. Using Multiple Input Multiple Output (MIMO) antenna systems presents a great interest to increase data rate while guarantying spectral efficiency [START_REF] Vaughan | Antenna diversity in mobile communications[END_REF]. MIMO antennas exploit polarization diversity and pattern diversity to decrease the signal correlation due to propagation channel and antennas coupling [START_REF] Dong | Simulation of mimo channel capacity with antenna polarization diversity[END_REF]. Some works have demonstrated the capability of increasing the channel capacity using polarization diversity due to six uncorrelated signals based on three electric dipoles and three magnetic dipoles. This radiating element is known as vector antenna (VA) [START_REF] Andersen | The mimo cube-a compact mimo antenna[END_REF], [START_REF] Svantesson | On capacity and correlation of multi-antenna systems employing multiple polarizations[END_REF]. Moreover, VAs are very attractive to estimate the direction of arrival (DoA) of incident electromagnetic (EM) fields.

The intrinsic characteristics of the VA previously reported for radio-goniometry in [START_REF] Duplouy | Wideband and reconfigurable vector antenna using radiation pattern diversity for 3-d direction-of-arrival estimation[END_REF] seem relevant for MIMO applications.

In this paper, we show that the MIMO performances of this VA are very good for wideband frequency operation. These performances are obtained for 3D-half-space coverage.

II. ANTENNA PRESENTATION

The antenna reported in [START_REF] Duplouy | Wideband and reconfigurable vector antenna using radiation pattern diversity for 3-d direction-of-arrival estimation[END_REF], [START_REF] Duplouy | Reconfigurable grounded vector antenna for 3-d electromagnetic directionfinding applications[END_REF] is shown in Fig. 1. It is composed of two orthogonal and colocated semi-circular arrays of Vivaldi antennas above a metallic ground plane. Each sub-array contains 4 Vivaldi antennas connected by mean of slotlines to a dual-port feeding through a splitter. One 4-Vivaldi sub-array is then connected to ports labeled 1 and 2, while the orthogonal sub-array is connected to ports labeled 3 and 4. These slotlines associated with each Vivaldi antenna are electromagnetically coupled through a microstrip-to-slot transition ended by a circular cavity of 5.5 mm radius and an 80 • radial stub of 5.5 mm radius.

According to simulation results, the bandwidth at VSWR 2.6 is of 8.72:1 (1.28 GHz to 11.34 GHz). The isolation S ij parameters are plotted in Fig. 2. The values are lower than -20 dB for frequencies above 1.89 GHz.

III. MIMO PERFORMANCES

The main metrics for MIMO antenna systems are reported in [START_REF] Sharawi | Printed multi-band mimo antenna systems and their performance metrics [wireless corner[END_REF]. Here we report the antenna performances through the diversity characteristics, the total active reflection coefficient and the channel capacity losses.

A. The envelop correlation coefficient

The envelop correlation coefficient (ECC ij ) [START_REF] Sharawi | Printed multi-band mimo antenna systems and their performance metrics [wireless corner[END_REF] relates the ability of the MIMO antenna to discriminate the multipaths through its diversity in terms of radiation pattern, source location and polarization. ECC ij between port i and j = i is plotted in Fig. 3 from far field radiation pattern relation from 1.3 GHz to 9.8 GHz. This relation takes into account the radiated field coupling. Values lower than 0.06 are obtained from 1.6 GHz and up to 10 GHz (according to the 3GPP technical specification, an ECC of 0.3 is usually considered as acceptable for 4G wireless systems).

B. Total active reflection coefficient

Because in MIMO systems all antennas operate simultaneously, it is important to consider the impedance matching by taking into account the EM coupling between antennas. the so-called total active reflection coefficient (TARC) is used to characterize bandwidth of the MIMO antenna by including the efficiency. The corresponding voltage standing wave ratio (V SW R) is calculated according to [START_REF] Valderas | Uwb portable printed monopole array design for mimo communications[END_REF] from the set of 50 excitations with random phases from 0 to 2π (see Fig. 4). The total bandwidth is found to be 6.17:1 (1.85 GHz to 11.43 GHz) with a V SW R which does not exceed 2.3.

C. The channel capacity losses

The channel capacity losses (CCL) is computed by taking into account the total efficiency through the TARC as follows [START_REF] Valderas | Uwb portable printed monopole array design for mimo communications[END_REF]:

CCL = -2log 2 (1 -T ARC 2 ) -log 2 (Det(Ψ R )) (1)
Where Ψ R denotes the receiving antenna correlation matrix obtained from the correlation coefficients computed in section III-A and Det(Ψ R ) designates the determinant of this matrix. The Fig. 5 displays the variation of CCL with frequency. It can be observed that CCL is lower than 0.4 bits/s/Hz in the entire bandwidth, i.e., from 1.85 GHz to 11.43 GHz.

IV. CONCLUSION

A Vivaldi array antenna originally designed for radio-goniometry applications has been investigated in MIMO context. The main metrics used for characterizing the MIMO antenna performances have been evaluated. The isolation between the antenna ports are lower than -20 dB, the envelop correlation coefficient is lower than 0.06, the TARC bandwidth for VSWR lower than 2.3 is of 6.17:1 (1.85 GHz to 11.43 GHz) and the channel capacity losses are lower than 0.4 bits/s/Hz. As expected, this antenna is a good candidate for wideband MIMO applications with a very good angular coverage (semi-spherical coverage).

Fig. 1 .

 1 Fig. 1. Proposed MIMO antenna.

Fig. 2 .

 2 Fig. 2. Simulated isolation parameters of the 4-port MIMO antenna.

Fig. 3 .

 3 Fig. 3. Envelop correlation coefficient between port i and j = i of the 4-port MIMO antenna vs. frequency.

Fig. 4 .

 4 Fig. 4. V SW R of the total active reflection coefficient for 50 random phase excitation states (from 0 to 2π).

Fig. 5 .

 5 Fig. 5. Channel capacity losses in Bits /s/Hz vs. frequency.