Large Deviations for Ablowitz-Ladik lattice, and the Schur flow

We consider the Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, and the Schur flow. We derive large deviations principles for the distribution of the empirical measures of the equilibrium measures for these ensembles. As a consequence, we deduce their almost sure convergence. Moreover, we are able to characterize their limit in terms of the equilibrium measure of the Circular, and the Jacobi beta ensemble respectively.


Introduction
The defocusing Ablowitz-Ladik (AL) lattice is the system of ODEs i 9 α j "´pα j`1`αj´1´2 α j q`|α j | 2 pα j´1`αj`1 q , (1.1) that describe the evolution of the complex functions α j ptq, j P Z and t P R, here 9 α j " dα j dt . We assume N -periodic boundary conditions α j`N " α j , for all j P Z. For simplicity, we consider the case N even, and, when not mentioned, the limits as N Ñ 8 is taken along N even. This system was introduced by Ablowitz and Ladik [2,3] as a spatial discretization of the defocusing Nonlinear Schrödinger Equation (NLS) iB t ψpx, tq "´1 2 B 2 x ψpx, tq`|ψpx, tq| 2 ψpx, tq.
The NLS is a well-known integrable model [36], and the Ablowitz-Ladik lattice is one of the several discretizations that preserve integrability [27]. It is straightforward to verify that the two quantities are constants of motion for the AL lattice. Since K p0q is conserved along the flow, it implies that if |α j p0q| ă 1 for all j " 1, . . . , N , then |α j ptq| ă 1 for all times. Thus, we can consider D N as our phase space, where D " tz P C | |z| ă 1u.
On this phase space we consider the symplectic form ω [15,18] ω " i N ÿ j"1 The corresponding Poisson bracket is defined for functions f, g P C 8 pD N q as Using this Poisson bracket, it is possible to rewrite the equations of motion (1) of the AL lattice in Hamiltonian form as 9 α j " tα j , H AL u, H AL pαq "´2 lnpK p0q q`K p1q`K p1q , here α " pα 1 , . . . , α N q.
Conserved quantities. As we already mentioned, the AL lattice is an integrable model: this was proved by Ablowitz and Ladik [1,2]. Specifically, they were able to obtain a Lax pair for the Ablowitz-Ladik lattice by discretizing the 2ˆ2 Zakharov-Shabat Lax pair of the cubic nonlinear Schrödinger equation. Nenciu and Simon in [26,28] constructed a new Lax pair for the Ablowitz-Ladik lattice, exploiting the connection of this system to the orthogonal polynomials on the unit circle. This link is the analogue of the well-known link between the Toda lattice and orthogonal polynomials on the real line (see e.g. [8]). This connection was also generalized to the non-commutative case [6].
Following [26,28], we construct the Lax matrix as follows. Consider the 2ˆ2 unitary matrices Ξpα j q " Ξ j "ˆα j ρ j ρ j´αj˙, ρ j " b 1´|α j | 2 , j " 1, . . . , N , and the NˆN matrices Now let us define the Lax matrix E " LM , (1.4) which has the following structure The matrix E is a periodic CMV matrix (after Cantero, Moral and Velazquez [7]). It is straightforward to verify that the equations of motions (1) are equivalent to the following Lax equation for the matrix E: 9 E " i " E, E``pE`q : ‰ , (1.5) where : stands for hermitian conjugate, and Remark 1.1. We notice that since all the Ξ j are unitary, then also E is unitary, this implies that all the eigenvalues λ j lie on the unit circle, and they can be written in terms of their argument, namely for all j " 1, . . . , N there exists a θ j P T :" r´π, πq such that In view of this identification, and in order to simplify the notations, for any function f pzq : BD Ñ R, we write f pθq in place of f pe iθ q when it is convenient. Further, we will write indifferently ş T f pθqdµpθq or ş BD f pzqdµpzq for any probability measure µ having support on the circle. Remark 1.2. We notice that pE`q :`p E : q`" E : and rE, E : s " 0 since E is unitary. Therefore, the Lax pair (1) can be rewritten in the equivalent form The formulation (1) implies that the quantities K pℓq " Tr´E ℓ¯, ℓ " 1, . . . , N´1, are constants of motion for the defocusing AL system (1). As in [25,29], we introduce the Generalized Gibbs ensemble for the Ablowitz-Ladik lattice, namely the following probability measure on the phase space D N dP V,β AL,N pα 1 , . . . , α N q " 1 Z AL N pV, βq N ź j"1 p1´|α j | 2 q β´1 1 tα j PDu expp´TrpV pEqqqd 2 α, (1.6) where V pe iθ q : T Ñ R is a continuous function, 1 A is the indicator function of the set A, and Z AL N pV, βq is the partition function of the system Furthermore, we consider the empirical measure µ N pEq of the eigenvalues e iθ 1 , . . . , e iθ N of the matrix E (1), namely here δ x is the delta function centred at x, furthermore, we notice that we can just consider the arguments θ 1 , . . . , θ N of the eigenvalues since the matrix E is unitary, see Remark 1.1.
Our main result is a large deviations principle (LDP) with good rate function for the sequence pµ N pEqq under the law P V,β AL,N (1). Namely, denoting by PpTq the set of probability measures on the Torus T endowed with the topology of weak convergence, there exists a function J V β : PpTq Ñ r0,`8s such that: We refer to [10] for a general introduction to large deviations. From this large deviations principle we are able to deduce that µ N pEq converges almost surely as N goes to infinity. a. the sequence µ N pEq under the law P V,β AL,N satisfies a large deviations principle at speed N with a good rate function J V β , b. J V β achieves its minimum at a unique probability measure ν V β , c. µ N pEq converges almost surely and in L 1 pTq towards ν V β .
Moreover, following [20,25,29], we are able to characterize the measure ν V β in terms of the equilibrium measure of the Circular beta ensemble at high temperature [21,22]. More precisely, consider the functional µ Þ Ñ f V β pµq given, for any µ P PpTq absolutely continuous with respect to Lebesgue measure and with density dµ dθ , by It is shown in [21], that the previous functional reaches its minimum for a unique absolutely continuous probability measure µ V β . Moreover in [25] it is proved that this measure is almost surely differentiable with respect to β. Exploiting this result, and Theorem 1.4 we are able to show that Thus, we obtain a unique characterization of the measure ν V β . In [25,29], the authors considered the GGE (1) with polynomial potential, and they were able to prove Theorem (1.5) for this particular class of potentials using a transfer operator technique. In this sense, we generalize their result, extending it to the class of continuous potentials.
In the last part of the manuscript, we consider another integrable model related to the Ablowitz-Ladik lattice, namely the so-called Schur flow [19]. Also for this system, the Lax matrix is E (1). Following the same construction as in the Ablowitz-Ladik lattice case, we define a GGE for this model. We are able to show analogous results to Theorem 1.4 and Theorem 1.5 for the Schur flow. The main difference is that in place of the Circular beta ensemble, we have the Jacobi one.
To give a wider overview of the relevant literature, we mention that, H. Spohn in [29], applying the theory of generalized hydrodynamics [12], argues that the correlation functions of the Ablowitz-Ladik lattice with respect to the GGE (1) show a ballistic behaviour. As we already mentioned, in [25] the authors rigorously proved Theorem 1.5 for polynomial potential V pzq. Moreover, they computed explicitly the density of states in the case V pzq " ηpz`zq, which corresponds to the classical Gibbs ensemble.
Lastly, it is worth to mention that this link between random matrix and integrable system was first noticed in [32]. In this paper the author considered the GGE for the Toda lattice, and he was able to study this ensemble, comparing it with the Gaussian beta ensemble [13]. We refer to [20,24,25,30,31,33] for subsequent developments.
In particular, our work was inspired by the recent paper [20]. In this paper, the authors obtained a large deviations principle for the Toda lattice, and obtain an analogous result to Theorem 1.5, where in place of the Circular beta ensemble, they had the Gaussian one.
The structure of the paper is the following. In Section 2, we prove the first point of Theorem 1.4. In Section 3, we collect some known results related to the Circular beta ensemble in the high-temperature regime. Moreover, we reformulate the already known large deviations principle for this ensemble in terms of the AL lattice. In Section 4, we conclude the proof of Theorem 1.4, and we prove Theorem 1.5. Section 5, is dedicated to the Schur flow, where we prove the analogue of Theorem 1.4 and Theorem 1.5 for this integrable model. Finally, we defer to the appendix the most technical results of our manuscript.
2 Existence of a Large deviations principle for the empirical measure of the Ablowitz-Ladik lattice The aim of this section is to prove the first point of Theorem 1.4, namely to show that, for N Q N ě 2 and even, the sequence of empirical measures µ N pEq " 1 N ř N j"1 δ e iθ j pEq satisfies a large deviations principle. The strategy of proof is the following. First, we show that if E is distributed according to P β AL,N :" P 0,β AL,N defined in (1), then the sequence of random probability measures pµ N pEqq N even satisfies a large deviations principle in PpTq, the space of probability measures on T, endowed with the topology of weak convergence. Since according to this toppology PpTq is compact, it suffices to show that the sequence pµ N pEqq N even satisfies a weak large deviations principle, see Remark 1.3. Then, applying Varadhan's Lemma [14, Theorem 1.2.1], we obtain the existence of a large deviations principle for arbitrary continuous V , i.e. the first point of Theorem 1.4.
We also notice that when V " 0 in (1) the α i 's are independent and identically distributed (i.i.d) with distribution Θ 2β`1 , where Θ ν is defined for ν ą 1 as the random variable such that for f : C Ñ R bounded and measurable Remark 2.1. We recall that for integer ν ě 2, such measure has the following geometrical interpretation: if u " pu 1 , . . . , u ν`1 q is chosen at random according to the surface measure on the unit sphere S ν in R ν`1 , then u 1`i u 2 is Θ ν distributed [23].
To show that the sequence pµ N pEqq N even satisfies a weak large deviations principle according to the law P β AL,N , we only need the α i 's to be i.i.d according to some law σ with supppσq Ď D. Thus, we just assume the latter hypothesis, and we prove the result in more generality.
Where the Lipschitz and the bounded variation norms are defined on the space of functions f : T Ñ R as The distance d is compatible with the weak convergence of probability measures [20]. We recall that for a NˆN matrix A, its empirical measure of eigenvalues is defined by where λ j pAq, j " 1, . . . , N , are the eigenvalues of A. The following Lemma, whose proof can be found in Appendix A, gives an upper bound on the distance of the empirical measures of two unitary matrices. • For f Lipschitz,ˇˇˇˇż As a consequence, dpµpAq, µpBqq ď min We are now in position to prove that the sequence µ N pEq with pα i q iě1 i.i.d with law σ, such that supppσq Ď D, satisfies a large deviations principle. The proof of the following Lemma follows the same line as the corresponding one in [20]. Lemma 2.3. Let pα i q iě1 be an i.i.d sequence of law σ with supppσq Ď D, and E be the associated matrix defined in (1). Then the sequence of empirical measures pµ N pEqq N even satisfies a large deviations principle in PpTq endowed with the topology of weak convergence.
Proof. We use a subadditivity argument to show that for any fixed µ P PpTq the following holds where B µ pδq :" tν P PpTq | dpµ, νq ă δu. Then, applying [9, Theorem 4.1.11], along with the fact that in our setting a weak LDP is equivalent to a full LDP, due to the compactness of PpTq, see remark 1.3, we conclude. The first step to prove the result is to approximate the matrix E (whose law we denote by E pN q ) by a diagonal block matrix of independent blocks. To this end, fix q P N even such that q ď N , write the euclidean division of N by q, N " kq`r with 0 ď r ă q. We consider M given by (1), and approximate it the following way.
1´|r α i | 2 , and the remaining block (of size rˆr) R is defined similarly: Following the same decomposition of N " kq`r we write L " diagpL 1 , . . . , L k , L k`1 q, with L i of size q for 1 ď i ď k and L k`1 of size r.
By the first point of Lemma 2.2 we deduce Moreover, we can rewrite µ N p r Eq as Using the independence of the blocks of r E, we deduce that Where we used the convexity of balls in the first inequality. This implies that, setting We now conclude as in [20,Lemma 2.3]. Let δ ą 0 and choose q in such a way that 8 q ď δ, so we deduce that The previous inequality holds true for all q big enough, so we conclude that From this last inequality we deduce that thus we obtain (2.1), and the conclusion follows applying [9, Theorem 4.1.11].
Since pPpTq, dq is compact, Lemma 2.3 automatically implies the existence of a strong large deviations principle. Furthermore, the corresponding rate function J ,which depends on the distribution σ of the entries of L and M, can be seen to be convex. We collect these results in the following proposition.
Proposition 2.4. In the same hypothesis and notations as in Lemma 2.3, the sequence of empirical measures pµ N pEqq N even satisfies a large deviations principle with good, convex rate function J : PpTq Ñ r0,`8s, i.e.
• The function J is convex and its level sets J´1pr0, asq, a ě 0, are compact, • For all F Ă PpTq closed, Proof. We already established all the claims except the fact that the function J is convex and that the level sets J´1pr0, asq are compact. The latter comes from the fact that these sets are closed, see [9,Theorem 4.1.11]. To prove the convexity of J, we follow the same argument as [20,Theorem 2.4]. Let µ 1 , µ 2 P PpTq. Since µ 2N pEq can be approximated by the sum of two independent µ N pEq's up to a mistake smaller than 4 N by the first point of Lemma 2.2, for δ ą 0 the following holds 2ˆδ`4 N˙˙, taking minus the logarithm of both sides, dividing by 2N , taking the limit for N going to infinity and then for δ to zero, we deduce that: which, together with the lower semi-continuity of J, implies the convexity of J, see [10, Lemma 4.1.21] .

Large deviations principle for the Ablowitz-Ladik lattice
Taking σ " Θ 2β`1 given by equation (2), Proposition 2.4 applies to pµ N pEqq N even , where E follows P β AL,N defined in (1). Thus, pµ N pEqq N even with law P β AL,N satisfies a large deviations principle, with a good convex rate function, that we denote by J β .
We can now state the existence of a large deviations principle for pµ N pEqq N even under P V,β AL,N for V continuous.
Corollary 2.5. Let β ą 0, and V : T Ñ R be continuous. Under P V,β AL,N the sequence pµ N pEqq N even fulfils a large deviations principle with good, convex rate function Proof. Let us write which ensures by [14,Theorem 1.2.3] that pµ N q satisfies a large deviations principle under P V,β AL,N with the announced rate function. Since the function J V β is an affine perturbation of J β , which is convex, J V β is also convex.
The first point of Theorem 1.4 is proven.
3 Circular β ensemble at high temperature In this section, we consider the Circular β ensemble, and we collect some known results that we exploit in our treatment. The aim of this section is to prove an alternative formulation of the large deviations principle for the Circular beta ensemble in the high-temperature regime, see Theorem 3.11 below. Our formulation allows us to relate the large deviations principle of the Coulomb gas with the one of Ablowitz-Ladik, proved in the previous section.

Large deviations principle for Circular β ensemble
Coulomb gas on the torus T " r´π, πq at temperature r β´1 are described by here V : T Ñ R is a continuous potential, and θ " pθ 1 , . . . , θ N q. When V " 0, Killip and Nenciu showed that dP 0, r β C,N is the law of the eigenvalues of a CMV matrix [22], see Theorem 3.4. In this manuscript, we are interested in the so-called high-temperature regime of this ensemble, namely the limit of number of particles N going to infinity with the constraint that r βN Ñ 2β ą 0. This regime was considered by Hardy and Lambert in [21], who proved the following large deviations principle for the measure µ N " 1 N ř N j"1 δ e iθ j , where the θ j are distributed according to (3.1).
Theorem 3.1. Let r β " 2β N , β ą 0 and assume V : T Ñ R to be continuous. Define for any µ P PpTq absolutely continuous with respect to the Lebesgue measure the functional i. the functional f V β pµq is strictly convex and achieves its minimal value at the unique probability measure µ V β absolutely continuous with respect to the Lebesgue measure; ii. the sequence pµ N q satisfies a large deviations principle in PpTq equipped with the weak topology at speed βN with rate function defined for absolutely continuous µ P PpTq with respect to Lebesgue measure by Exploiting this result, in [25] the authors deduced several useful properties of the minimizer µ V β , specifically they proved the following. ii. Let D be the distance on PpTq given by Then for any ε ą 0 there exists a finite constant C ε such that for all β, β 1 ą ε So, for any measure ν with zero mass we obtain the following Then, by Cauchy-Schwartz inequality, we deduce the following inequalityˇˇˇż Combining (3.2) and (3.3), we deduce that for any function f with finite ||f || 1 2 norm, the map

Relation with the large deviations principle of the Ablowitz-Ladik lattice
In the case V " 0, for any r β ą 0, Killip and Nenciu in [22] showed that the law P 0, r β C,N (3.1) coincides with the distribution of the eigenvalues of a certain CMV matrix. Specifically they proved the following: where the block Ξ j , j " 1, . . . , N´1, takes the form while Ξ 0 " p1q and Ξ N " pα N q are 1ˆ1 matrices. Define the NˆN sparse matrix

4)
and suppose that the entries α j are independent complex random variables with α j " Θ r βpN´jq`1 for 1 ď j ď N´1 and α N is uniformly distributed on the unit circle. Then the eigenvalues of E are distributed according to the Circular Ensemble (3.1) at temperatureβ´1.
To simplify the notation, we will denote by P To achive our goal, we need several technical results regarding the distribution Θ ν (2), and the CMV matrix E (3.4). First, in the next Lemma, we give a reprensentation of Θ ν in terms of Gaussian, and Chi distributions.
Proof. We identify C with R 2 and check that for any f : D Ñ R bounded and measurable, i.e. that for some constant c, For fixed y ą 0, we perform the diffeomorphic change of variables pu, vq " where X u,v denotes a Gaussian variable N p0, 1´pu 2`v2 qq. By [35], for some constant c ν independent of u, v. Substituting this last equality in the previous integral, we conclude.
To obtain our main result, we need some technical lemmas. Since they are based on standard techniques, we just state them here, and we defer their proofs to the appendix A.
The first one gives an estimate that we use combined with Lemma 2.2.
Lemma 3.7. Let N " 2k be even and A be a NˆN matrix. Then, where M, and L are defined in (1).
, and aphq "´1 2 lnphq`1, then there exists a constant K independent of h such that sup 0ăhă1 E rexppaphqZ h qs ď K . (3.7) Remark 3.9. Let h ă h 1 , and let Z h , Z h 1 be given by Z being a χ h 1´h variable independent of Y h . Then, because of the monotonicity of the function x Þ Ñ x ?
a`x 2 for a ą 0, we have almost surely Z h ď Z h 1 . We are now in position to give an alternate formulation of the large deviations principle for the sequence of measures pµ N pEqq under the law P 2β N Lemma 3.10. Let β ą 0. The law of the empirical measure pµ N pEqq N even under P 2β N C,N satisfies a large deviations principle at speed N and with a good rate function where J η is the rate function of Proposition 2.4 applied to σ " Θ 2η`1 .
From these two families of matrices, we can define a third one, namely E piq k " L piq k M piq k , i " 1, . . . , q. We notice that E piq k is distributed according to P β N´ik N AL,k , and that the E piq k , 1 ď i ď q, are independent.
Our aim is to prove that the empirical measure of the matrix C q where 0 rˆr is a null block of size rˆr, is an exponential approximation (see [9, Definition 4.2.14]) of the empirical measure of E " P 2β N C;N (3.4), that is, for any positive real number δ: where P denotes the coupling introduced in equation (3.2). In this way, we obtain the claim as an application of [9, Theorem 4.2.16]. The strategy of proof is the following. First we approximate C q N and E by two block diagonal matrices r C q N , r E q N respectively. Finally, we will prove that both r C q N , and r E q N approximate a third matrix B q N .
Consider another family of matrices p Ă M piq k q 1ďiďq of size kˆk, defined as where the matrices Ξ piq ℓ are defined in (3.2), while r Ξ piq 0 " p1q and r Ξ piq k " pα ik q are 1ˆ1 matrices, where the α ik are independent, uniformly distributed on the unit circle for all i " 1, . . . , q, and independent from pα ipk´1q`l q 1ďiďq,1ďℓďk´1 . Define the kˆk family of CMV matrices From the family of matrices p r E piq k q 1ďiďq , we define the block diagonal matrix: We claim that r C q N is such that rankpC q N´r C q N q ď 2q .
and for any δ ą 0 and sufficiently large N , we can take k such that 2 k ď δ 4 . Consider now another two families of matrices pL piq k q 1ďiďq , and pM piq k q 1ďiďq , constructed in the same way as pL piq k q 1ďiďq , and p Ă M piq k q 1ďiďq by means of independent variables r α pi´1qk`j , where each r α pi´1qk`j " Θ 2β N´pi´1qk´j N is coupled to α pi´1qk`j by equation (3.2), for all j " 0, . . . , k´1, and i " 1, . . . , q, and where r α ik " α ik for i " 1, . . . , q is uniformly distributed on the unit circle. Define the family of CMV matrices pE piq k q 1ďiďq as Define the block diagonal matrix E q N as: From the definition of P 2β N C,N and E q N , we conclude that for some E " P 2β N C,N , we have rankpE´E q N q ď 2q`r . As before, from the previous inequality we deduce that Finally, we define the matrix B q N as As a consequence, Moreover, combining Lemma 2.2 and Lemma 3.7 we deduce that Applying Lemma 3.8 point i., we deduce that where the last sum denotes the sum of independent random variables with law Z k´j N , defined in Lemma 3.8.
Thus, for N large enough such that 4 k ď δ 4 , we deduce that for any non-negative function apq´1q: Where in the last inequality we used Remark 3.9, namely, since k´j N ď 1 q , we have Setting aphq "´1 2 lnphq`1 and applying Lemma 3.8 point ii., we deduce that there exist constants r K and c ą 0, independent of q, such that And we obtain the claim.
We can apply the previous Lemma to study the case of continuous potential, indeed as a consequence of Varadhan's Lemma we obtain the main result of this section, namely

Proof of the main results
In this section, we conclude the proof of Theorem 1.4 and prove Theorem 1.5. The main tool to prove these theorems is the uniqueness of the minimizer of the rate function for the β ensemble. Define the free energies of the Ablowtiz-Ladik lattice and the Circular beta ensemble at high temperature as where g V β , and f V β are given by (2.5) and (3.11). We claim that Lemma 4.1. Let β ą 0, and V : T Ñ R continuous, then the following holds: a. the map β Ñ F C pV, βq is continuously differentiable on R˚. Moreover: b. for almost all β ą 0 there exists a unique minimizer ν V β of the functional J V β pµq, see Corollary 2.5, given by we recall that the measure µ V β is defined as the unique minimizer of the functional I V β in Theorem 3.11. Remark 4.2. Our definition of Free Energy is different from the one used in [25,29]. Indeed, in virtue of Varadhan's Lemma [14, Theorem 1.2.1], we have instead in [25,29], the authors defined the free energies as We notice that it is possible to recover one expression from the other since F AL pV, βq " r F AL pV, βq´r F AL p0, βq , To prove uniqueness of the minimizer ν V β , we need to consider a continuous family pµs q 0ăsăβ , where each µs minimizes J V s , see Corollary 2.5. We address the existence of such a family in the next Lemma, which we prove in the appendix A.
where for A Ă PpTq we denote A ε " tµ P PpTq | dpµ, Aq ď εu. Proof of Lemma 4.1. First, we notice that for any probability measure µ P PpTq, Theorem 3.11 implies Where we noticed that the Riemann sums indeed converge towards the integral since s Þ Ñ F AL pV, sβq is concave, this can be seen by applying Hölder inequality to equation (4.2).
To prove the first part of the claim, we show that the lower bound is achieved. For s P r0, 1s, let νs β be a minimizer of inf νPPpTq g V sβ pνq. From Lemma 4.3, we can choose νs β such that the map s Ñ νs β is continuous. This implies that µβ " ş 1 0 νs β ds is a well-defined probability measure on T. We claim that this measure minimizes f V β (3.11), and so I V β . Indeed, from Theorem 3.11, we deduce that  Combining (4)-(4), and performing the change of coordinates sβ " t we deduce that: Moreover, from Lemma 3.2 we deduce that the map β Ñ F C pV, βq is Lipschitz in β, and so almost surely differentiable. This implies that for almost all β ą 0 F AL pV, βq " B β pβF C pV, βqq .
Furthermore, we have just shown that I V β pµq " f V β pµq´inf νPPpTq f V β pνq reaches its minimum at ş 1 0 νs β ds. By uniqueness of the minimizer of I V β pµq, Theorem 3.1, we deduce that we have the equality between probability measures µ V β " Note that the function s Þ Ñ ş T f dνs is continuous, therefore by differentiating this equality, we get that νβ is the unique minimizer of J V β , which we denote by ν V β , and satisfies for f continuous ż proving point b.

The Schur Flow
In this section, we consider another integrable model, namely the Schur flow. Our goal is to show that is possible to obtain a similar result to the one that we presented for the Ablowitz-ladik lattice. Namely, we prove the existence of a large deviations principle for the Schur flow, and we relate its density of state to the one of the Jacobi beta ensemble in the high temperature regime.

Generalized Gibbs Ensemble
The Schur flow is the system of ODEs [19] 9 and, as before, we consider periodic boundary conditions, namely α j " α j`N for all j P Z.
In [3], it is argued that the continuum limit of (5.1) is the modified Korteweg-de Vries equation: x u´6u 2 B x u . We notice that, if one chooses an initial data such that α j p0q P R for all j " 1, . . . , N , then α j ptq P R for all times. Moreover, it is straightforeward to verify that K 0 " ś N j"1`1´| α j | 2˘i s conserved along the Schur flow. This implies that we can choose as phase space for the Schur flow the N -cube I N , where I :" p´1, 1q.
On this phase space, we consider the Poisson braket (1), so we can rewrite the Schur flow (5.1) in Hamiltonian form as It is well known that the Schur flow admits as Lax matrix the same one as the AL [19], namely E (1) is the Lax matrix of the Schur flow. This implies that the Ablowitz-Ladik's constants of motion are conserved also along the Schur flow (5.1).
Following the same construction made for the Ablowitz-Ladik lattice, on I N we define the finite volume limit GGE as dP V,β S,N pα 1 , . . . , α N q " where Z S N pV, βq is the partition function of the system Since according to the measure (5.1) the matrix E is real, its eigenvalues come in pairs [28], meaning that if e iθ j is an eigenvalue, then its conjugate e´i θ j is also an eigenvalue. This implies that for a system of size N even, there are just n " N {2 independent eigenvalues. Following the same idea as in [22], it is more convenient to restrict the argument of the eigenvalues in r0, πq and then consider x j " cospθ j q, j " 1, . . . , n. In these variables, the empirical spectral measure µ n pEq reads: As a corollary of Lemma 2.3 and Proposition 2.4, we obtain the existence of a large deviations principle for the sequence pµ n pEqq, namely: Corollary 5.1. Let V : I Ñ R be continuous. Under P V,β S,n the sequence pµ n pEqq fulfils a large deviations principle with good, convex rate function where K β pνq is the rate function of µ n under the law P 0,β S,n .

Jacobi beta ensemble in the high temperature regime
The Jacobi beta ensemble refers to the distribution of charges constrained to the segment I, and subjected to an external potential W pxq "´a lnp1´xq´b lnp1`xq`V pxq, here a, b ą´1 and W pxq P C 0 pIq. Specifically the joint distribution of these particles is dP pV, r βq In [22], Killip and Nenciu were able to show that the distribution (5.2) can be realized as the eigenvalues distribution of a particular CMV matrix, specifically they proved the following p1´α j q a`1´r β{4 p1`p´1q j`1 α j q b`1´r β{4 e TrV pEq dα j , and α 2n "´1, here Z N pV, βq is the normalization constant. Then all the eigenvalues of E come in pairs, meaning that if e iθ j is an eigenvalue, then also e´i θ j is one. Moreover, under the change of variables cospθ j q " x j , the x j s are distributed according to (5.2).
Remark 5.3. We notice that the previous proposition is not stated in this way in [22], but this equivalent formulation is more useful for our purpose.
Also in this case, we are interested in the high temperature regime for this ensemble. Specifically we consider the situation r β " 4β N " 2β n , and a " b "´1`r β 4 , in this regime dP pV, β n q J,n reads dP pV, 2β n q and dB pV, β n q n becomes dB pV, β n q n " 1 We mention that this particular regime was considered in [16,34]. In these papers the authors computed the density of states for this ensemble in the case V " 0.
We can apply [17, Corollary 1.3] to (5.2) to obtain a large deviations principle for the empirical measure µ n pEq " 1 n ř n j"1 δ x j . Specifically, we deduce that Proposition 5.4. For any continuous V : I Ñ R. The law of the empirical measures µ n pEq under dP pV, 2β n q J,n satisfies a large deviations principle at speed n in the space PpIq, with a good rate function µ Þ Ñ Q V β pµq given for µ absolutely continuous with respect to Lebesgue measure, and with density dµ dx , by 6) and Q V β pµq "`8 otherwise. We notice that the arguments in Section 3 and 4 can be applied also in this context with dP pV, 2β n q J,n in place of dP pV, 2β N q C,N , and dP V,β S,N in place of dP V,β AL,N . Hence, we deduce the following result Moreover, ν V β is absolutely continuous with respect to the Lebesgue measure, and it reads β is the unique minimizer of the functional q V β (5.4).
Finally, it is worth to mention that in the case V pxq " 0, it is possible to compute explicitly the densities of states for both the Jacobi beta ensemble at high temperature and for the Schur flow [16,24].

A Technical Results
In this appendix we collect the proof of all the technical results that we exploit along the proof of the main theorem. For reader convenience, we report here the statement of Lemmas.
As a consequence, dpµpAq, µpBqq ď min Proof. The first point is a consequence of the fact that the eigenvalues of A and B interlace on the unit circle. First, we order the eigenvalues λ 1 pAq, . . . , λ N pAq, λ 1 pBq, . . . , λ N pBq of A, B in such a way that´π ď argpλ 1 pAqq ď . . . ď argpλ N pAqq ă π , and analogously for B. Write B " pI N`p B´AqA´1qA and set U :" I N`p B´AqA´1. One checks that U is unitary, B " U A, and that rankpU´Iq " rankpB´Aq ": r. By [5, section 6, equation (85)], we deduce that for 1 ď j ď N argpλ j´r pAqq ď argpλ j pBqq ď argpλ j`r pAqq . (A.1) This means that λ j pBq lies on the anticlockwise arc pargpλ j´r pAqq, argpλ j`r pAqqq of the circle. If j´r ď 0 we identify λ j´r with λ j´r`N , and analogously for the case j`r ą N . It is a classical result (see [4]) to deduce from (A) thaťˇˇˇż for any f : T Ñ R such that ||f || BV ď 1. As a consequence, we obtain the first point.
The proof of the second point is the same as in the symmetric case, see [20, (16)]. Indeed, we only use the fact that a normal matrix is unitarily diagonalizable.

Proof of Lemma 3.7
Lemma A.2. Let N " 2k be even and A be a NˆN matrix. Then, where M, and L are defined in (1).
Proof. We will just prove the first point, since the proof of the second one follows the same lines.
, almost surely , , and aphq "´1 2 lnphq`1, then there exists a constant K independent of h such that Proof. First, we focus on claim i.. We recall that α ν , α ν`h are defined by From the previous equation, we deduce that where we used again equation (A) with a " 1´|α ν | 2 and b " 1´|α ν`h | 2 . Thus, point i. is proved.
To prove point ii., we find explicitly the law of Z h . Thus, we consider a continuous function f : p0, 1q Ñ R, and we compute: Performing the change of coordinates pu, vq " 1 px 2 1`x 2 2`y 2 q 1{2 px 1 , x 2 q, which is the same one that we performed in Lemma 3.5, we obtain that ż R 2ˆR`f¨ỳ We can now explicitly compute the integral in t. Moreover, we can express the remaining part of the integral in polar coordinates; namely, we apply the change of variables u " ρ cospθq, v " ρ sinpθq, obtaining that: ρf p a 1´ρ 2 q`1´ρ 2˘h 2´1 dρ ? Combining (A)-(A), with our choice of aphq "´1 2 lnphq`1, we deduce that there exists a constant K independent of h such that (ii.) holds.

Proof of Lemma 4.3
Lemma A.4. Let M V β " pJ V β q´1pt0uq be the set of minimizers of J V β . Then, β Þ Ñ M V β is continuous in the sense that for all ε ą 0, there exists δ ą 0 such that for all 0 ď h ď δ, M V β`h Ă pM V β q ε , where for A Ă PpTq we denote A ε " tµ P PpTq | dpµ, Aq ď εu. Proof. Let ε ą 0. We are going to show that for h ą 0 small enough, we havé which will ensure that J V β`h ą 0 on Since for any positive h and α P D N ś N j"1 p1´|α j | 2 q h ď 1 ,we deduce that for any A Ď D N Thus, to conclude we have just to prove that the function gpβq " lim N Ñ8 1 N ln`Z AL N pV, βq˘is continuous in β. Actually, we prove that this function is convex in β. Let 1{p`1{q " 1, and β 1 , β 2 P R`then Z AL NˆV , where in the last inequality we used Hölder inequality. This implies that so we obtain that inf rpM V β q ε s c J V β`h ą 0 .