On the Six-Vertex Model’s Free Energy

In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta <1$$\end{document}Δ<1. As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches 1/2. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=b=1$$\end{document}a=b=1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\ge 1$$\end{document}c≥1, and the rotational invariance of the six-vertex model and the Fortuin–Kasteleyn percolation.


Definition of the model.
The six-vertex model, first proposed by Pauling [36] in 1935 to study the residual entropy of ice, became the archetypical example of a planar integrable model with Lieb's solution of the model in 1967 in its anti-ferroelectric and ferroelectric phases [29][30][31][32] using the Bethe Ansatz. We refer to [3,33,39] for detailed expositions and reviews and to [2] for the most general solution. The six-vertex model on the torus is defined as follows. For N , M > 0 with N even, let T N ,M := (Z/N Z) × (Z/MZ) be the N by M torus. An arrow configuration ω is a choice of orientation for every edge of T N ,M . We say that ω satisfies the ice rule (or equivalently that it is a six-vertex configuration) if every vertex of T N ,M has two incoming and two outgoing edges in ω. These edges can be arranged in six different types around each vertex as depicted in Fig. 1, hence the name of the model. One may easily check that the icerule guarantees that each horizontal line of vertical edges contains the same number of up arrows. From now on, let (T N ,M ) (resp. (n) (T N ,M )) be the set of six-vertex configurations (resp. containing exactly n up arrows on each line). For parameters a 1 , a 2 , b 1 , b 2 , c 1 , c 2 ≥ 0, define the weight of a six-vertex configuration ω to be W 6V (ω) := a n 1 1 a n 2 2 b n 3 1 b n 4 2 c n 5 1 c n 6 2 , where n i is the number of vertices of T N ,M having type i in ω. In this paper, we choose to focus on a 1 = a 2 = a, b 1 = b 2 = b and c 1 = c 2 = c. Some of the results of this paper may extend to the asymmetric case and will be the object of a future paper.
Define In the analysis of the model, it is customary to introduce the parameter Below, we consider the region of parameters (a, b, c) such that < 1; see Fig. 2 for the phase diagram of the model.

1.2.
Main results for the symmetric six-vertex model. It appears convenient to adopt a parameterisation of the weights which makes transparent the connection with the algebraic Bethe Ansatz construction of the model's transfer matrix. We thus introduce auxiliary parameters θ ∈ (0, π), r ∈ R + and ζ such that 1 • for −1 < < 1, = − cos ζ with ζ ∈ (0, π), a sin ζ 2 := r sin (1 − θ π )ζ , b sin ζ 2 := r sin θζ π , c := 2r cos ζ 2 , • for = −1, a := 2r π −θ π , b := 2r θ π , c := 2r, • for < −1, = − cosh ζ with ζ ∈ R + , a sinh ζ 2 := r sinh (1 − θ π )ζ , b sinh ζ 2 := r sinh θζ π , c := 2r cosh ζ 2 . The first result goes back to Lieb [30][31][32] and Sutherland [38] and deals with the per-site free energy defined by N , a, b, c), (5) in which the limits may be taken in any order as established in [33]. The mentioned papers characterised the per-site free energy relying on the same strategy as the original paper [41] which deals with the XXZ quantum spin chain. At the time, the closed expressions for f (a, b, c) were derived under the hypothesis of the so-called condensation of Bethe roots. As will be discussed more precisely later on in the introduction, the condensation property has nowadays been rigorously established. Here, we develop an alternative technique for proving condensation which, on the one hand, turns out to be particularly effective for our goals and, on the other hand, allows one to go beyond what can be rigorously proven within the existing scope of techniques.
The value of f (1, 1, 1) was first obtained by Lieb and corresponds to the residual entropy of square ice.
Our second result deals with the following extension of the per-site free energy to generic values of n and N : (n) (T N ,M , a, b, c).
It provides a characterisation of the subleading corrections to f (n) N (a, b, c) as n, N → +∞ in such a way that n/N → 1/2. The condition on n and N appearing in the statement below is technical and takes its origin in the statements of the subsequent theorems in this paper. Theorem 2. For N ≥ 2 even and a, b, c ≥ 0 leading to < 1 (c.f. (1)), there exist constants C, C(ζ ), C (ζ, θ ) ∈ (0, ∞) such that for every using the parametrisation (2)- (4), we have where o(1) means a quantity tending to 0 as n/N tends to 1/2.
Notice that for ∈ [−1, 1), (8) only gives meaningful information when N 2 − n exceeds √ N . This extension has important applications for the six-vertex model and other related models. The six-vertex model lies at the crossroads of a vast family of two-dimensional lattice models; for instance, it has been related to the dimer model, the Ising and Potts models, Fortuin-Kasteleyn (FK) percolation, the loop O(n) models, the Ashkin-Teller models, random permutations, stochastic growth models, quantum spin chains, to cite but a few examples. Among such links, one can use the Baxter-Kelland-Wu mapping between the six-vertex model and FK percolation [4] to deduce from Theorem 2 and the dichotomy result of [13] that the phase transition of FK percolation on the square lattice is continuous if the cluster weight q satisfies 1 ≤ q ≤ 4, and is discontinuous for q > 4. We refer to the papers where the results were proved (using alternative methods) for additional details [13,14]. It should be mentioned that the continuity result of [13] may be deduced directly from Theorem 2 using the same procedure as in [18]. In the same spirit, the results can be used to derive dimerisation properties of the anti-ferromagnetic Heisenberg chain [1].
A second application of our results is related to the height function h of the six-vertex model, which can be proved to be localised (meaning that the variance of h(x) − h(y) is bounded uniformly in |x − y|) whenever a = b = 1 and c > 2, and delocalised (meaning that the variance of h(x) − h(y) tends to infinity logarithmically fast in |x − y|) when a = b = 1 and 1 ≤ c ≤ 2. We refer to [16,18,21] for more details. It is conjectured more generally that the height-function is localised when < −1, and delocalised when −1 ≤ < 1. This property is closely related to the existence of a massive ( < −1) and massless (−1 ≤ < 1) regime in the XXZ spin-1/2 Heisenberg chain; there, the ground state correlation functions of local operators at distance m decay exponentially fast in m → +∞ in the massive regime and algebraically in m in the massless regime. Indeed, one can show that the XXZ spin-1/2 Heisenberg Hamiltonian ground state generating function of the longitudinal spin-spin correlations does coincide with the generating function of variances of the height function of the six vertex model. Thus, the power-law decay of the correlators in the XXZ chain translates to the logarithmic growth of the variance of the height function for the six-vertex model.
Finally, another use of our results is in [17], where a refined version of Theorem 2 (see Sect. 8) is employed to show that the correlations of the height function of the six-vertex model are invariant under rotations in the scaling limit, when a = b = 1 and √ 3 ≤ c ≤ 2. This rotation invariance should in fact hold for every c ∈ [1,2] (but this has not been proven yet) and be wrong for c > 2 (when the height function localises, as discussed above). The argument of [17] involves the FK percolation representation of the six-vertex model, and the rotational invariance result also applies to critical FK percolation on Z 2 with cluster weight q ∈ [1, 4].

Transfer matrix of the six-vertex model and the Bethe Ansatz.
In order to understand the large scale asymptotics of Z (n) (T N ,M , a, b, c) with 0 ≤ n ≤ N /2, one introduces the transfer matrix V N = V N (a, b, c) (that we do not write explicitly here; see e.g. [7]) defined as an endomorphism of the 2 N -dimensional real vector space spanned by the basis In particular, one finds that where V  a, b, c). Standard arguments of rigorous statistical mechanics, see e.g. [33], allow one to conclude that where N (a, b, c) and     N (a, b, c) is not Perron-Frobenius, but it may be shown that its single largest eigenvalue is The coordinate Bethe Ansatz, introduced by Bethe [6] in 1931, provides mathematicians and physicists with a powerful way of obtaining eigenvalues of one-dimensional quantum models and of the transfer matrices of certain two-dimensional lattice models. In particular, Orbach [35] put it in a form allowing one to study the eigenvalues of the XXZ spin-1/2 Heisenberg chain, a model sharing the same eigenvectors as the six-vertex transfer matrix, see [34] for the explanation of this last fact. Further, since the visionary work of the Leningrad School [19], the coordinate Bethe Ansatz has been put into a fully algebraic framework, called nowadays the algebraic Bethe Ansatz, which is deeply connected with the representation theory of quantum groups. This picture strongly simplified the analysis of integrable models.
We now summarise the program corresponding to the implementation of the Bethe Ansatz to understand the asymptotics of the largest eigenvalue of V a, b, c). The survey [15] contains an elementary derivation of Bethe's Ansatz intended for probabilists, and is a useful reference for most of what is discussed above.

The Bethe Ansatz approach to the dominant eigenvalue
Step 1. Fix distinct integers or half-integers, depending on the parity of n, (n 1 , . . . , n n ) and consider a solution λ λ λ = (λ 1 , . . . , λ n ) ∈ R n to the following set of equations called the logarithmic Bethe equations where p and ϑ are defined in Appendix A. While these functions do depend on and have quite different expressions in the regimes < −1, = −1 and | | < 1, we shall keep this dependence implicit. The coordinates of solutions to (10) are called Bethe roots.
Step 2. Consider the vector for which ψ(x|λ λ λ) is defined for every x with |x| = n by where S n is the symmetric group on n elements, ε(σ ) is the signature of the permutation σ , and The Bethe Ansatz guarantees that for a solution to (10) which has pairwise distinct coordinates that lie away from the singularities of p and ϑ, where (n) N (λ λ λ) is given by the formula in which and M(λ) := Step 3. Show that for the specific choice of (half-)integers n i ≡ I i := i − n+1 Step 4. Perform a large n, N asymptotic expansion of the formula in (14) to conclude.
Note that the Bethe equations ensure that the Bethe roots are not poles of L or M, so that   At this stage, implementing the above program rigorously requires particular attention at certain points, namely: In Step 1, for a given choice of n n n = (n 1 , . . . , n n ), one must prove the existence of solutions to (10). In the regime < 1, Yang and Yang [41] proved the existence of Bethe roots when n i = I i as above. Then Griffiths [22] established the existence of solutions to a certain class of (half-)integers n n n. More recently, Kozlowski [28] established the existence of solutions, as well as their uniqueness when N is large enough, for a wide class of (half-)integers n n n describing the so-called particle-hole excitations.
In Step 2, in order to conclude from (13) that is non-zero. This was shown to hold for the solution λ λ λ associated with n i = I i by Yang and Yang [41]. For solutions having pairwise distinct coordinates that are associated with other choices of (half-)integers n n n and which satisfy some form of condensation, c.f. later on, the non-vanishing of (n) N (λ λ λ) for N large enough may be proved using the determinant representation for the norm of (n) N (λ λ λ), which was conjectured in [20,27] and rigorously proven in [26,37].
In Step 3, one should argue that the vector (n) N (λ λ λ) obtained using the specific choice of (half-)integers I i is indeed the Perron-Frobenius eigenvector of V (n) N (a, b, c). This was first conjectured by Hulthén [25] and was established by Yang and Yang [41]. Checking that (n) N (λ λ λ) is the Perron-Frobenius eigenvector is reasonably simple for equal to 0 or −∞ (for = −∞ and general n this actually does require some effort). In order to extend the result to an interval of values of , one may prove the continuity or analyticity of (n) N (λ λ λ) as a function of . If continuity is used, then one should additionally prove that To be more precise, we should first introduce the continuum Bethe equation whose solution allows one to characterise the limiting measure. For q ∈ [0, ∞] when | | ≤ 1 and q ∈ [0, π/2] when < −1, define ρ(·|q) as the solution (the unique solvability was thoroughly discussed, by different methods, in [12,28,42]) to the linear integral equation with K := 1 2π ϑ and ξ := 1 2π p . When n/N → m ∈ [0, 1/2] as n, N → +∞, the point measure L (λ λ λ) N associated with the solution λ λ λ to (10) corresponding to the choice of (half-)integers n i = I i converges weakly towards ρ λ|Q(m) The existence and uniqueness of Q(m) has been first proven in [12]. We also refer to "Appendixes B and D" for a proof of Q(m)'s existence. The uniqueness of Q(m) may be obtained as a consequence of Theorem 3 below, and will be discussed thereafter. For future reference, it may be useful to note that Q is increasing and Q(1/2) = π/2 when < −1 and Q(1/2) = ∞ when | | ≤ 1. Condensation of Bethe roots was first proven when 0 < < 1 by Gusev [24] for any m using convex analysis tools. Much later, Dorlas and Samsonov [10] used different convex analysis techniques to prove the same result and were also able to prove condensation for any m and < − 0 with 0 large enough, viz. perturbatively around = −∞. More recently, Kozlowski [28] proved condensation for any value of m ∈ [0, 1/2] and ∈ (−∞, 1), in particular away from the region where convexity or perturbative arguments are applicable. That proof relied on developing a rigorous approach to dealing with the non-linear integral equations governing the so-called counting function of the Bethe roots that were introduced and handled, on a loose level of rigour in [5,8,40]. The non-linear integral equation method allowed to rigorously establish the condensation of Bethe roots associated with a large class of (half-)integers in (10), not only n i = I i , as well as to go beyond the limiting value, and to compute an all order asymptotic expansion in N for f (μ)dL (λ λ λ) N (μ) for any < −1 and m ∈ [0, 1/2], as well as for any −1 ≤ < 1 and m ∈ [0, 1/2). However, owing to the lack of certain compactness properties, the non-linear integral equation method does not allow one to reach rigorously 2 an estimate beyond o(1) for when −1 ≤ < 1 and m = 1/2. In this work, we develop a method which allows one to estimate (17) up to a O(1/N ) for −1 < < 1 and m = 1/2 and up to a O(ln N /N ) for = −1 and m = 1/2. Reaching these values of the parameters in the model plays a very important role for the results obtained in [16][17][18] and this stresses the significance of our result.

Results for Bethe's equations.
For n ≤ N /2, we will henceforth always consider the sequence of (half)-integers appearing in (10). For < 1, recall that we are interested in the solutions λ λ λ = (λ 1 , . . . , λ n ) ∈ R n to where p and ϑ are defined in "Appendix A". We will also require that solutions are • strictly ordered, meaning λ i < λ i+1 for every 1 ≤ i < n.
The first main result of this section is the existence of solutions to (19) without any assumption on n ≤ N /2 or = −1, with a quantitative control on how condensed these solutions are.
Theorem 3 (Existence of condensed solutions to discrete Bethe equations when = −1). There exists a constant C > 0 such that for every n ≤ N /2 and every ∈ (−∞, −1)∪(−1, 1), there exists a symmetric strictly ordered solution λ λ λ = (λ 1 , . . . , λ n ) to (19), which satisfies for every f : R → R with integrable derivative. Above, ζ is related to as in (2)-(4), and we introduced the shorthand notation A solution satisfying (20) will be referred to as condensed. Note that the condensation is fairly quantitative but that the control degenerates when is approaching −1. We refer to Theorem 6 below for the treatment of the case = −1.
The second theorem will be devoted to the existence of an analytic family of such solutions. The existence of a continuous family of solutions has been previously proven in [41]. Yet, we could not identify any use of the continuity property which warrants mentioning this stronger statement. On the contrary, a property that seems crucial for applications to Bethe's Ansatz is the property of analyticity in of the Bethe roots. Analyticity may also be directly inferred from the results of [28] for < −1 and all m as well as for −1 ≤ < 1 and m ∈ [0, 1/2). In this paper, we extend these analyticity results (by another range of arguments) up to m = 1/2 in the sense described by Theorem 4 below.
We are currently unable to prove, with our method, the existence of an analytic solution for arbitrary n ≤ N /2 over the whole intervals (−∞, −1) and (−1, 1). We refer to the remarks in Sect. 3.2 for more details. However, this fact appears to be closely related to the expected property that the model undergoes a phase transition of infinite order at = −1.
Our next result states that the eigenvalue (14) obtained from the Bethe roots provided by Theorem 4 is indeed the Perron-Frobenius eigenvalue of V The last two theorems have the following direct consequence. For = −1 and n ≤ N /2, consider the solution λ λ λ( ) provided by Theorem 4. Since the functions log |L(x)| and log |M(x)| are differentiable, the condensation and symmetry imply that and a similar expression for M. When a > b, one may check that the contribution to (n) N λ λ λ( ) issuing from the L term is larger than the one issuing from the M term. This allows one to deduce from the transfer matrix formalism and Theorem 5 that 3 as long as n, N , are in one the cases where Theorem 4 holds and a ≥ b. Theorems 1 and 2 follow from (22) once one can estimate efficiently the right-hand side. At the core of this estimate is the following observation going back to [42]. Let K be the operator acting on L 2 (I ), where I = R for | | ≤ 1 and We refer to R as the resolvent, and to its integral kernel R as the resolvent kernel. Then, (15) is equivalent to the linear integral equation where ρ = (id − R)ξ . The resolvent kernel R and ρ are best expressed through their Fourier transforms/coefficients 4 We refer to "Appendix A" for the explicit formulae. Due to the definition of Q, we have that I = [−Q(1/2), Q(1/2)], and thus ρ(λ) = ρ λ|Q( 1 2 ) . The rewriting of (15) as (23) has the advantage of putting emphasis on the perturbative structure of the equation for q located in the vicinity of Q( 1 2 ). Up to now, our results were always stated for belonging to strict subintervals of (−∞, −1) or (−1, 1). We conclude this section with a result dealing with the case = −1.
In Remark 19, we will also see from the proof that one can obtain a solution λ λ λ(−1) of (19) with = −1 by taking the limit of 1 ζ λ λ λ( ) when ζ tends to 0. This solution also satisfies (24).
Organization The paper is split into seven further sections and several appendixes. In Sects. 2 and 3, we present the proofs of Theorems 3 and 4, respectively. The sections themselves start with general considerations and are then divided into the different cases 0 ≤ < 1, −1 < ≤ 0 and < −1, as these exhibit different features. Sections 4 and 5 contain the proofs of Theorems 5 and 6, respectively.
Building upon these results, Theorems 1 and 2 are proved in Sects. 6 and 7, respectively. These sections are divided between the cases | | < 1 and < −1 as these correspond to different behaviours. 4 When | | ≤ 1, we consider square-integrable functions on R. For F in L 2 (R), the Fourier transform of F is given by Then, for f ∈ L 2 π (R), define the Fourier coefficients f : Z → C by Finally, Sect. 8 presents a refined version of Theorem 2. While being interesting in its own right, this result is mostly useful in our subsequent paper [17].
The first Appendix lists the different definitions of functions in order to have a place conveniently gathering all the formulae. The three other Appendixes gather properties of ρ(·|q) and (15) so as to not overburden the rest of the text.

Proof of Theorem 3
In this whole section, fix ∈ (−∞, −1) ∪ (−1, 1) and n ≤ N /2. Recall that N is even and that I i = i − n+1 2 for 1 ≤ i ≤ n. Below, we introduce the notion of an interlaced solution which will be useful in the proof. For q > 0 (with q ≤ π/2 when < −1) and introduce the quantile (x|q) given by the formula Note that (x|q) is unambiguously defined since ρ(λ|q) > 0. Due to the definition of ρ(·|q) (see "Appendix A" and (23)), x 0 is equal to infinity for < −1 and is finite, but larger than or equal to π/2 for ≥ −1. In the latter case, in order to avoid unnecessarily heavy notation, we set (x|q) = +∞ for x ≥ x 0 and −∞ for x ≤ −x 0 . Note also that by definition of q, c.f. (16) and (21), we have that q = (n + 1 2 |q) = − ( 1 2 |q).
We say that λ λ λ is (k, q)-strictly interlaced if the strict inequalities hold.

Remark 8.
When k = 1, this corresponds to a perfect interlacement between the λ i and the quantiles of the measure ρ(λ|q)dλ.
The interest of this notion of interlacement becomes apparent in the following lemma which states that (k, q)-interlaced solutions satisfy (20), provided k ≤ C/ζ . Lemma 9 (From interlacement to quantitative condensation). Fix k ≥ 1 and q ∈ R + . For every (k, q)-interlaced λ λ λ and every f : R → R with integrable derivative, with Proof. By (25), the integral of ρ(·|q) between ( j − 1 2 |q) and ( j + 1 2 |q) is 1 N , as long as both arguments are inbetween −x 0 and x 0 . Thus, we find where we invoked the following inequality, valid for In the case when f is non-decreasing (non-increasing works in the same way), (k, q)interlacement gives The lower-bound holds for n ≥ j ≥ (k + 3)/2 while the upper one for 1 ≤ j ≤ n − k+1 2 . Summing the left-hand side over j ≥ (k + 3)/2, bounding from below the remaining sums of f (λ j ) by k+1 2 f (λ 1 ), and the missing piece of integral by − k+1 2 f (n + 1 2 |q) gives the lower bound on the difference. The upper bound follows from analogous considerations.
The core of the proof of Theorem 3 will be to construct solutions of (19) that lie in the subset of R n given by with q given by (21), as fixed points of a well-chosen function. This will be done by picking k = k( ) and R = R( , N ) carefully, and then proving that the closure k,R of k,R is mapped to k,R by this function. Then, the Brouwer fixed point theorem implies the existence of a fixed point for this function, which is a solution to (19) due to the choice of the function. While the proof is fairly similar in the different regimes, some tiny differences still exist and we therefore divide it between the cases ∈ [0, 1), ∈ (−1, 0], and < −1. Let us mention that a fixed point method was already used, although in a slightly different manner, for proving the existence of Bethe roots in [22]. Remark 10. The reason for distinguishing between ≥ 0 and < 0 issues from the fact that λ → ϑ(λ) given in "Appendix A" is respectively decreasing and increasing. Furthermore, when < 0, dividing between < −1 and > −1 comes from the small caveat that the image of ϑ is an interval of length strictly smaller than 2π when > −1 and equal to 2π when < −1.
Remark 11. We expect the existence of (1, q)-interlaced solutions to (19) for every < 1, even though we are currently unable to prove this fact for n close to N /2 (when n/N ≤ 1/2 − , this follows readily from the results established in [28]). Below are plots of as a function of the system size N , for n = N /2 and different ≥ −1. One sees that the quantity remains bounded by 1/2, meaning that the solution is (1, +∞)-interlaced.
Consider the map : R n → R n , λ λ λ → μ μ μ for which μ i is defined for every 1 ≤ i ≤ n by where p and ϑ are given in "Appendix A". This function is well-defined as the map is continuous strictly increasing (here the fact that ≥ 0 ensures that ϑ is decreasing) and tends to ±ϒ with ϒ : (recall that n ≤ N /2 and that ζ ≤ π/2 in this case), there exists a constant R = R( , N ) such that |μ i | < R for every i. From now on, we fix this constant and show that maps k,R onto k,R . The Brouwer fixed point theorem then implies that has a fixed point (since k,R is a compact convex set), which is a strictly (k, q)-interlaced symmetric strictly ordered solution of (19). Note that the choice (30) of k ensures that k ≤ C/ζ for some universal constant C, which implies (20) through Lemma 9.
Let μ μ μ = (λ λ λ) for some λ λ λ ∈ k,R . That μ μ μ is strictly ordered and symmetric is obvious from (31), whose left-hand side is strictly increasing in μ i . We therefore only need to check the strict (k, q)-interlacement, which is a direct consequence of the following sequence of inequalities: The equality in (32) is due to the following identity, valid for every x and q, which is the integrated version of (15) (recall that ϑ and p are odd). The first inequality in (32) is an application of the definition (31) of μ i together with Lemma 9 applied to the monotone function ϑ(μ i − ·); it is also useful to recall that q = (n + 1 2 |q) = − (1 − 1 2 |q), due to the definitions of q and (·|q).
We are unable to use the map from the previous subsection as ϑ is now increasing. We therefore change the map slightly and consider the map : The map is again well-defined as p is continuous, strictly increasing, and p(R) is equal to [ζ − π, π − ζ ], while for any λ λ λ ∈ R n , (we use that |ϑ| ≤ π −2ζ , |I i | ≤ (n −1)/2) which ensures that the left-hand side of (34) lies in the range of p since n ≤ N /2. Thus, as before, we may find R = R( , N ) large enough such that |μ i | < R for every i. Again, we wish to prove that is mapping k,R to k,R , which will imply the existence of a fixed point, and therefore a strictly (k, q)-interlaced symmetric, strictly ordered solution to (19). Note that definition (30) for k entails that k ≤ C/ζ for some constant C > 0, which implies (20) by applying Lemma 9.
Fix λ λ λ ∈ k,R and set μ μ μ := (λ λ λ). The strict monotonicity and the fact that μ i ∈ (−R, R) are immediate consequences of the definition of and the choice of R, and we do not give further details. Lemma 9 applied to the decreasing function ϑ(λ i − ·) implies that Observe now that, due to (30), the maximum above is smaller than |2π − 4ζ | < 2π k k+1 . Since in addition λ i was assumed smaller than (i + k 2 |q), and since ϑ is increasing, we conclude that where the last equality follows from (33) and the definition of (i + k 2 |q). Since p is increasing, we get that μ i < (i + k 2 |q). Similarly, one proves that μ i > (i − k 2 |q).
For small values of N , the existence of a fixed point of (or equivalently of a solution to (19)) that is not necessarily (k, R)-interlaced is easily obtained. Its condensation may be derived by adjusting the constant C in (20). Henceforth we focus on values of N above a threshold independent of chosen below.
Let μ μ μ = (λ λ λ) for some λ λ λ ∈ k,R . As in the previous part, it is immediate that μ μ μ is symmetric and strictly ordered. One should still establish the boundedness and the strict (k, q)-interlacement of μ μ μ. We will argue that the former is a direct consequence of the later. We thus first establish interlacement.
To do so, one should start by establishing a generalisation of Lemma 9 to the case of a function g : [0, +∞) → R which is monotonous on [0, π/2]. Here, we only treat the case of n even and leave the details of n odd to the reader, since it only leads to minor modifications. We claim that, for any such function g, where The inequality above is obtained in the same way as Lemma 9, so we provide no further details.
. Then a direct computation shows that the functions ϑ sym (λ i , ·) for i = 1, . . . , n are monotonous on [0, π]. Applying (36) to ϑ sym (λ i , ·) we find It follows from the lower bound ρ(x|q) ≥ ρ(x) ≥ 1 2ζ established in Lemma 29 of "Appendix D", that for each j Therefore, the (k, q)-interlacement of λ λ λ allows one to infer that λ j = q + O( k N ), with the O(.) here and below being uniform in j = n − k−1 2 , . . . , n and < −1. Hence, since q ≤ π/2, any λ j appearing in the definition of m + [g] exceeds π/2 by at most O( k N ). A direct computation shows that π/2 is a local extremum of μ → ϑ sym (λ, μ) on [π/2 − η, π/2 + η] for some η > 0 independent of and λ. Thus, we conclude that for all N large enough (which we will assume henceforth for reasons described at the start of the proof), every 2n ≤ N and i ∈ {1 + n 2 , . . . , n}, Plugging the above into (37), we find Invoking the choice of k and the fact that λ → ϑ(λ) is increasing on R gives that This yields the lower bound for the (k, q)-interlacement of μ μ μ. The upper bound is obtained in an analogous way. Finally, the (k, q)-interlacement of μ μ μ and the upper bound (n + k 2 |q) − (n + 1 2 |q) ≤ N and thus, by symmetry, that μ i ∈ (−R, R) with R := π 2 + ζ(k−1) N . The latter establishes that ( k,R ) ⊂ k,R . As before, we deduce through Brouwer's fixed point theorem that admits a fixed point, which provides a solution to (19) satisfying the conditions of Theorem 3. The validity of (20) is due to Lemma 9 and the fact that k ≤ C/ζ for some universal constant C; the latter follows directly from (35) and an upper bound on ϕ( π 4 ) easily obtained from the definition of ϑ and "Appendix A.3".

Proof of Theorem 4
Fix n ≤ N /2. Since the dependence on plays a role in this argument, we recall it in the subscript of the map T : We recall that p and ϑ appearing above do depend on , c.f. "Appendix A". The zeroes of T correspond to the solutions to (19) for . The following proposition will play a key role in the proof of Theorem 4. (30)

Proposition 12. Let k be as defined by
we have that dT is invertible at λ λ λ for any (k, q)-interlaced, ordered, symmetric λ λ λ.
With this proposition at hand, we are in position to prove the theorem.
Proof of Theorem 4. Taking into account the definition of T and introducing ( ) := k( ),R( ,N ) , with k,R given by (29) and R( , N ) as constructed in Sects. 2.1, 2.2 and 2.3, depending on the value of , we can restate the theorem as the existence of an analytic family → λ λ λ( ) such that λ λ λ( ) ∈ ( ) satisfies T (λ λ λ( )) = 0 for every . Consider some 0 for which we are in the possession of λ λ λ( 0 ) ∈ ( 0 ) satisfying T 0 (λ λ λ( 0 )) = 0. Using Proposition 12, the implicit function theorem for analytic functions gives the existence of an analytic family → λ λ λ( ) ∈ R n such that T (λ λ λ( )) = 0 in a small neighbourhood of 0 . Continuity implies that by reducing the neighbourhood if need be, we can further assume that λ λ λ( ) ∈ ( ). Also note that a continuous limit, as tends to some 1 , of λ λ λ( ) ∈ ( ) with T (λ λ λ( )) = 0 converges to λ λ λ( 1 ) ∈ ( 1 ) with T 1 (λ λ λ( 1 )) = 0. But, we saw in the previous section that solutions to (19) in ( 1 ) are necessarily in ( 1 ). Together with the previous paragraph, this implies the existence of an analytic family of solutions on any open interval on which the conditions of Proposition 12 hold, and which contains at least one value for which there exists a solution λ λ λ ∈ ( ) to (19). The intervals of Theorem 4 are indeed such that the conditions of Proposition 12 hold; the existence of solutions for some in these intervals is ensured by Theorem 3 (or alternatively by Lemmata 16 and 17, see below).
To prove the uniqueness of the solutions for all , it suffices to prove it for a single value 1 in each of the two intervals of Theorem 4. Indeed, assuming the existence of multiple solutions at some value of , the argument above implies the existence of multiple analytic families of solutions in the whole interval. These families may not cross inside the interval, due to the implicit function theorem, and would therefore contradict the uniqueness at 1 . We choose to check the uniqueness of solutions for 1 = 0 and 1 a very large negative number. This is done by solving (19) explicitly for = 0 and = −∞, then extending the property to large negative numbers by continuity; see Lemmata 16 and 17 below for more details.
We now focus on the proof of Proposition 12, and divide it into three subsections depending on the range of as before. Note that since K = 1 2π ϑ and ξ = 1 2π p , the matrix dT (λ λ λ) can be evaluated as 3.1. Proof of Proposition 12 when 0 ≤ < 1. For any λ λ λ ∈ R n , the matrix dT (λ λ λ) is symmetric (since K is even) and positive definite: since K ≤ 0 and ξ > 0, as these are derivatives of decreasing and strictly increasing functions, respectively (see also "Appendix A"). As a consequence, dT (λ λ λ) is invertible.
is strictly convex, and has therefore at most one extremum which, if it exists, is its minimum. The existence thereof can be obtained in at least three ways. Either one uses the fixed point theorem in the previous section, or one checks that S tends to infinity as soon as one of the λ i tends to infinity (this is slightly technical), or finally one observes that at = 0 there is an explicit solution and that the implicit function theorem guarantees that this solution extends into an analytic function on 0 ≤ < 1.

3.2.
Proof of Proposition 12 when −1 < < 0. We remind to the reader that in this regime we restrict our attention to n satisfying where k is given by (30) while C (which is independent of N and k) is yet to be determined. Also, we recall that q is given by (21). Fix λ λ λ as in the proposition. The matrix dT (λ λ λ) is no longer obviously positive definite and therefore not obviously invertible. In order to prove invertibility, we rather show that the matrix A defined by is diagonal dominant 5 hence invertible (note that Proposition 24(i) of "Appendix B" gives that ρ(λ|q) > 0 on R and therefore the matrix A is well-defined). The invertibility of dT (λ λ λ) follows trivially from that of A. Checking diagonal dominance relies on two computations. On the one hand, Lemma 9 applied to λ → K (λ j − λ) (since λ λ λ is (k, q)-interlaced) together with (15) gives On the other hand, Lemma 9 applied to the function where I k := [ (1 − k 2 |q), (n + k 2 |q)]. Now, we estimate the error terms (meaning the terms with factor k N ) in (41) and (42) separately. Below, the constants C i are independent of everything else. We use analytic properties of the solution to the continuum Bethe Equation (15) that are proved in Proposition 24 of "Appendix B". The assumption (40) plays an essential role in what follows.
We start by estimating the error in (41). Using the fact that |ρ (λ)| ≤ πρ(λ)/ζ (see "Appendix A") and further invoking Proposition 24(i), we find: Furthermore, by Proposition 24(iii) Combining the two last displayed equations, we infer a lower bound Then Proposition 24(i), the monotonicity of ρ(·) on (−∞, 0] and the symmetry and (k, q)-interlacement of λ λ λ give that Now, since K is unimodal, even, and has limits 0 at ±∞, K L 1 (R) = 2K (0). Using this and the previous paragraph, we find where the last inequality is obtained by observing that K (0) ≤ C 0 /ζ for some ζindependent constant C 0 . We now turn to the error term in (42). First, we have that for 1 ≤ j ≤ n, t ∈ I k , and N large enough, where in the second inequality we used Proposition 24(ii).

Now, note that by Proposition 24(i) and (45), for every t
Also, Proposition 24(iii) gives that Plugging (49) and (50) in (48) and then integrating over I k , we find that By choosing C appropriately in (40), we may assume that the parenthesis in the righthand side above is bounded by a uniform constant. Using the previously mentioned facts that K L 1 (R) and ζ K L 1 (R) are bounded by a constant independent of ζ , we conclude that the bracket above, when multiplied by ζ , is bounded uniformly in ζ . Thus, we may bound the error term of (42) as: Plug (47) and (51) in (41) and (42), respectively, to find that Taking C large enough in the statement of the proposition ensures that A is indeed diagonal dominant.

Remark 14.
The difficulty in proving that A i j is diagonally dominant comes from the estimates involving j close to 1 or n as approximating sums by integrals is not efficient for these values of j. Another way of seeing this is that when j is far from 1 and n then ρ(λ j |q) is larger and therefore the error term is smaller. Nonetheless, numerics suggest that the matrix is diagonal dominant for every −1 < < 1 and n ≤ N /2, as shown on the plots of m := min{A ii − j =i |A i j | : 1 ≤ i ≤ n} as a function of the system-size N , for n = N /2 and at four different values of .
Remark 15. Since the differential is non-zero at = 0 (it is diagonal since K ≡ 0), we obtain in particular the existence of an analytic family of solutions for every n ≤ N /2 for | | ≤ 0 with 0 small enough.

Proof of Proposition 12 when < −1. Fix λ λ λ as in the proposition. Again, dT (λ λ λ)
is not obviously positive definite. At this point, we may use a symmetrization trick like in the proof of Theorem 3 for < −1. This argument was presented in [14] and we refer to this paper for a full proof. Here, we present an alternative proof that we find to be of some interest. We show that dT (λ λ λ) is invertible by estimating the large N behaviour of det[dT (λ λ λ)] with the help of Lemma 9. To start with, observe that where I k := [ 1 − k 2 |q , n + k 2 |q ] is a subinterval of [−π, π] uniformly bounded in n ≤ N /2.
Since ρ(λ|q) ≥ 2 ζ > 0 on R (by Lemma 29), the above ensures that the matrix is well-defined for any n, N with N large enough. Then, introduce an integral operator M acting on L 2 ( (−q, q]) with the integral kernel where J i := i − 1 2 |q , i + 1 2 |q is a partition of the integration domain ∪ n i=1 J i = (−q, q] as can be inferred from the identities Introduce the integral operator K on L 2 ((−q, q]) characterised by the integral kernel K (λ − μ). Both M and K are trace class. Indeed, M is of finite rank while K has smooth kernel and acts on functions supported on a compact interval [11]. Moreover, we have Here we used the estimate (52). We now estimate the Hilbert-Schmidt norm of K − M.
One starts with the representation for the kernel By using that x + k|q − x|q = O k/N uniformly in x, the mean-value theorem and (k, q)-interlacement of λ λ λ, one gets that Then, the estimate (52) allows one to conclude that  ([−q, q]) for any q ∈ [0, π/2] (this is for instance a consequence of the proofs of Propositions 23, 26, and 28), this entails that det Id +M = 0 for N large enough.

Proof of Theorem 5
Let us start by stating two lemmata.
for any r and θ .
Proof. For = 0, the unique solution to (19) is given by It is then a matter of elementary computation to show that the entries of (n) N are strictly positive, which concludes the proof of the lemma. N (a, b, c).

Remark 18. The analyticity of
→ λ λ λ( ) was only used once in the proof above, namely to show that the vector (n) N (λ λ λ( )) is non-zero for (almost) all . It is non-trivial that this property holds for all , N and n. The norm of (n) N (λ λ λ( )) has been argued to be given in terms of the determinant of dT (λ λ λ) in [20,27] and this was proven in [26,37]. The results reads for some explicit non-zero function f . Therefore, proving that the vector is non-zero amounts to proving that the differential of T is invertible which, as shown above, automatically implies analyticity.
In conclusion, proving analyticity of the solutions and using the strategy above, rather than proving their continuity and separately that the resulting vector is non-zero, bypasses the use of (54) and contains no additional complications.

Proof of Theorem 6
Let k = C 0 log N and n ≤ N /2 − C 1 k 2 . The constants C 0 and C 1 will be chosen large enough in the course of the proof.
To start, we follow the argument of Theorem 3 in Sect. 2.2 to guarantee that for each −1 < < 0, every (k, q)-interlaced solution is strictly interlaced. For the proof to work, we need to check that In order to do that, remark that the value of the extremum of ϑ and the (k, q)-interlacement imply that owing that the maximum of the difference is attained at the midpoint. Then, by using interlacement Further, upon using the explicit expression for ϑ given in "Appendix A", one gets .
The last inequality above follows via simple trigonometric manipulations from tanh(y) ≤ y, cot(ζ ) ≤ 1/ζ and the fact that q → (x|q) is decreasing for x < (n + 1)/2 in this regime of . The latter is easily inferred by taking the q derivative of the equation defining (x|q). Using (46), the above then leads to .
Overall, we deduce that there exits a constant c 3 > 0 independent of everything such that for every −1 < < 0, We deduce (55) by fixing C 0 large enough. In particular, we obtain the equivalent of Theorem 3, namely that for every −1 < < 0, there exists a (k, q)-interlaced strictly ordered symmetric solution λ λ λ( ) to (19).
We now need to prove that the family λ λ λ( ) can be assumed to be analytic. We follow the argument of Sect. 3.2, with minor changes which we describe next. As in Sect. 3.2 we may use (55) to deduce (41) and (42). It remains to bound the error terms. We start with (42) This can be made smaller than 1/(4k) by choosing C large enough. The same argument applies to the error term in (41) and the proof follows.
To prove an analogue of Theorem 3 for = −1, one may employ the bounds of Sect. 2.2 and rescale all variables by 1/ζ . This allows one to conclude that maps 1 ζ¯ k,R onto 1 ζ k,R and so, by taking the ζ → 0 limit, yields a Brouwer fixed point for the map | =−1 . The unique fixed point of this map coincides, by construction, with any sub-sequential limit of 1 ζ λ λ λ( ) as ζ tends to 0. Thus, such sub-sequential limits are unique, which shows that 1 ζ λ λ λ( ) does converge to a (C 0 log N , q)-interlaced, strictly ordered, symmetric solution λ λ λ(−1) to (19) for = −1.

Proof of Theorem 1
We prove the statement for a > b and = −1. The results extends to all a ≥ b ≥ 0 and c ≥ 0 with < 1, since the left and right sides of (6) are Lipschitz in each coordinate of (a, b, c). The particular expression for = −1 is obtained by taking the limit either from above or below in (6).
We start with the expression T N ,M , a, b, c), where in the first equality we restrict our attention to n ≤ N /2 thanks to the symmetry n ←→ N − n corresponding to the symmetry under the reversal of all arrows. The last equality uses the classical fact that f (n) N (a, b, c) is maximal for n = N /2, see [14,Lemma 3.6] or [31] for a proof.
For 0 ≤ < 1 or < −1, we apply (22) and evaluate (14) at the n = N /2 groundstate's Bethe roots to get We claim that the same holds for −1 < < 0. Notice however that we do not have access to f a, b, c) in this case. However, the inequality 7 Z (n) (T N ,M , a, b, c M , a, b, c) is sufficient to obtain (57) as we can apply (22)  We shall only focus on the evaluation of the first term and split the proof in two depending on whether | | < 1 or < −1. We leave to the reader the verification that the first term does indeed dominate the second when a > b. 6.1. Case | | < 1. When | | < 1, we use the Fourier transform on R. Define 7 The displayed inequality can be obtained from the easy observation (already made in a number of papers, see e.g. [18]) that there exists a map from configurations with n + 1 up arrows per row to configurations with n up arrows per row constructed by choosing a path of oriented edges cycling around the torus in the vertical direction with length smaller than M N/n (such a path always exists and there are at most N 4 M N/n choices for it) and reversing all arrows on it. This changes the weights, hence the factor min{a, b, c}/ max{a, b, c}. and observe the exact expressions given in "Appendix A" for the different functions and their Fourier transforms. Recalling that ρ is even and Q(1/2) = +∞, we find that dt.
Using that dt some algebra and the change of variables t → 2t give the result.
Remark 20. For the special case a = b = c = 1, we may compute the integral directly. After a fairly elementary computation, we recover the classical result of Lieb [31] f (1, The particular expression for a = b = 1 and c = 2 is obtained from (6) by direct computation.

Proof of Theorem 2
7.1. Focusing on the asymptotic in the q variable. We claim that it suffices to estimate the asymptotic behaviour of as q Q(1/2), with L (x) defined in (58).
The asymptotics of Q(n/N ) as n/N approaches 1/2 are given by Propositions 25, 27 and Lemma 30 of "Appendixes B, C and D", respectively, and read lim m→1/2 The estimation of (60) is obtained differently for −1 ≤ < 1 and < −1 and does not refer anymore to the discrete Bethe equation. Deriving Theorem 2 from the asymptotics of (60) obtained below and those for Q mentioned above is a matter of simple algebra, which we do not detail further.
Remark 21. The constant C was explicitly computed in [12]. We do not need the precise value here and therefore work with this weaker and simpler result. We provide however in Proposition 25 of "Appendix B" an integral representation for C in terms of the solution to a Wiener-Hopf equation on R + .

Case
Using (23), then reorganizing the integrals (in particular using that L and R are even), and then passing to Fourier gives that In the last identity we use that is integrable on a neighbourhood of R and has exponential decay. We now perform two elementary residue computations. Fix π/ζ < β < 3π/ζ and s ∈ iδ + R for δ > 0 very small. Since Res t=±iπ/ζ [ G] = ∓2i sin θ and there is no other pole of G in the strip {z ∈ C : Im(z) < β} (and G tends to 0 at infinity), we get where and similarly Putting these two displayed equations in the first one gives that We first justify that the second term is a O(e −βq ). Clearly, ψ (±) q ∈ L 2 (R) and ψ (±) q L 2 (R) ≤ C uniformly in q. Furthermore, it is established in Proposition 25 of "Appendix B" that given we find that and T is the unique solution of the integral equation By interpolation theorems for L p spaces, we get that T and δT belong to L 2 (R) with norms uniformly bounded in q. Therefore, ρ(·|q) ∈ L 2 (R) with a norm controlled uniformly in q. All of this ensures that the last term in (62) is indeed O(e −βq ).
Since ρ(·|q) is even and 1 π/ζ +is is the Fourier transform of e, we find Plugging x + q in (63), and performing an asymptotic expansion of (63) (at first order) enables us to recast (62) as where we used the fact that β can be taken strictly larger than 2π/ζ . Then, by using • the fact that as λ tends to infinity, which follows from the integral equation satisfied by T , the fact that T ∈ L 1 (R) and similar estimates for the behaviour of R(λ) when λ → ±∞, one readily infers that Thus, all-in-all, we get that with q = Q(n/N ), Note that the constant is strictly positive since T > 0 on R.
7.3. Case = −1. We omit the proof as it is the same as in the previous section, using "Appendix C" instead of "Appendix B".
7.4. Case < −1. Following the same reasoning as in the previous section gives Above, the o(1) term is as q → π/2. It remains to prove that the following quantity is strictly positive: The above is the convolution of the inverse Fourier transforms of tanh[nζ ]/(2n) and sinh [2nζ θ π ]/ sinh [nζ ] evaluated at π/2. Both of these inverse Fourier transforms may be shown to be positive using Poisson summation, which in turn implies the positivity of the above expression.

A Refined Version of Theorem 2 (Under Additional Conditions)
In this section, we prove a sharper version of Theorem 2 under mild conditions on a, b, c and n, N .

Theorem 22.
For N ≥ 2 and a = b and c ≥ 0 leading to | | < 1, there exists a constant C = C(ζ ) < ∞ such that for every n ≤ 1 where o(1) is a quantity tending to zero as n/N tends to 1/2.
The improvement with respect to Theorem 2 is that the O(1/N ) is replaced with the more precise O( 1 ζ(N −2n)N ). This will be particularly useful in [17], where it is used to prove that N | f (n) This convergence is expected to hold for all n = N /2 + o(N ), when ∈ [−1, 1).
Proof of Theorem 22. Due to the form of (66), it suffices to prove the statement for N large enough. Fix for now n and N as in the theorem; we will see later which bounds are needed on N .
Proof. The constant C 2 will be chosen at the end of the proof; it will be apparent that it is independent of n or N . For = 0, the result is obvious as the explicit (and unique) solution of the discrete Bethe Equation satisfies Diff = 0. Assume that there exists ∈ (−1, 1) such that Diff = C 2 ζ(N −2n) ≤ 1 2 . Using (33) in the first equality and then (19) in the second, we find Now, we use that for K = 1 2π ϑ , |K | ≤ C ζ |K | and K L 1 [R] = 1 − 2ζ π (see the "Appendix" again). Apply Claim 1 to 1 2π ϑ(λ i − x) (and bound the L 1 norm on [−q, q] by the L 1 norm on R) to get where C 0 is the constant given by Claim 1, and C 0 depends on C 0 , but not on C 2 . Since this applies to all i, we conclude that Choose now C 2 so that C 2 > π 2 C 0 . Then (67) contradicts our assumption on Diff, and we conclude that there exists no ∈ (−1, 1) with Diff = C 2 ζ(N −2n−C 0 ) . By the continuity of Diff as a function of and considering the fact that Diff = 0 for = 0, we conclude that Diff < C 2 ζ(N −2n−C 0 ) for all ∈ (−1, 1). We are now in a position to conclude the proof of Theorem 22. Let C = C 2 be given by Claim 2 and fix a, b, c as in the theorem. By taking N large enough, we may assume that the value corresponding to (a, b, c) is contained in the domain in which λ λ λ is defined for any n ≤ N /2 − C/ζ (see Theorem 4). Then the dominant Eigenvalue may be expressed as where L (·) is the function defined in (58). Claims 1 and 2 give 1 N log Furthermore, Sects. 7.1 and 7.2 give that The above implies (66) by choosing C large enough.
Acknowledgements. The first author was funded by the ERC CriBLaM. The second author was funded by ERC Project LDRAM : ERC-2019-ADG Project 884584 The third author was funded by the Swiss FNS. The first, third, fourth and fifth authors were partially funded by the NCCR SwissMap and the Swiss FNS. The first and fourth authors thank Matan Harel for inspiring discussions at the beginning of the project.
Funding Open access funding provided by University of Fribourg.
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We will also be interested in the following Fourier coefficients (with relevant extensions at t = 0) The functions p and ϑ are then defined as the odd smooth functions on R that have 1 2π ξ and 1 2π K as derivatives. In particular, on (− π 2 , π 2 ), they are equal to Moreover, the following direct consequences of the formulas above are used in the text: p is increasing and maps R to R; ϑ is increasing; ϑ([−π/2, π/2]) = [−π, π] and ϑ extends to R as a quasi-periodic continuous function. The function K is even, unimodal, and has zero limits at ±∞. We stress that these formulae do not extend, per se, beyond (− π 2 , π 2 ). We will also be interested in the following Fourier coefficients, when 2θ/π < 1, , (e −|n|· 1− 2θ π ζ − e −|n|(1+ 2θ π )ζ ) = π n e −|n|ζ sinh 2ζ n θ π if n = 0, and L (0) = 2θζ .

Proposition 23 (Existence of solutions to
Proof. Direct computation shows that the operator K on It follows that Id +K is invertible and the solution ρ(λ|q) is unique and lies in L 1 (R)∩ L ∞ (R) with a uniform bound on the norm. Since K (x) is smooth in x and q, the Fredholm series representation for the resolvent of Id +K [23] allows one to infer that (x, q, ) → ρ(x|q) is smooth.
. The lower bound of (i) was first established in [12].
Proof. Recall that R = K /(1+ K ). Following [42], one gets that R ≥ 0 since R = ρ K / ξ and therefore in which ρ is obviously positive while where the second equality follows from a straightforward residue computation.
so that the version (23) of (15) immediately gives that This expression and the fact that R ≥ 0 gives the lower bound of (i). For the upper bound, we isolate the first term in the sum and then use operator bounds to get that To prove (ii), differentiate (23) with respect to λ and then integrate by parts to obtain that ρ (·|q) satisfies the functional equation: In particular since R ∞ < C/ζ and ρ(q|q) < 2ρ(q). Finally, using |ρ (x)| ≤ π ζ ρ(x) we obtain the desired bound for a well-chosen value of C.
We now focus on (iii). The definition of m, (23), and We finish with the properties necessary to obtain Theorem 2 for | | < 1. Note that, in fact, one may solve (73) in terms of a scalar Riemann-Hilbert problem by implementing the Wiener-Hopf method. However, we will not need such a precise information on T and will thus establish Proposition 25 by more elementary means.
For Item (iii), recall the exact representation for 1 2 − m given in (72). Then, by virtue of (74) and upon using the symmetry of the operator T , one gets Here, we used that e has support on R + . The part involving δT can be directly estimated when q → +∞ to give O(b(2q) + e −2 π ζ q ) Using the explicit expression for ρ(λ), it is easy to see that Further, by using the integral equation satisfied by T, one gets that Thus, by substituting Q(m) in place of q in the above estimates and using that Q(m) → +∞ as m → 1 2 by virtue of Item (iii) of Proposition 24, one obtains that The constant on the right-hand side is strictly positive since T is given by a sum of strictly positive terms.
Proof. The proof valid for | | < 1 does not generalise directly since estimating that the operator K has · L 1 →L 1 -norm strictly less than 1 demands more effort, see [28]. However, by working with (23) instead of (15), one readily checks that the fact that the operator U has norm smaller than 1 2 , which gives the existence of solutions. The rest of the proof is the same.
The following proposition gives the necessary properties for the proof of Theorem 2 when = −1. The proof is the same as for | | < 1. Note that, above, R −1 stands for the resolvent kernel to the operator Id +K at = −1 and acting on L 2 (R).
Then, it remains to use that R L 1 ([−π/2,π/2] = 1/2 so as to conclude as in the = −1. Finally, the existence of Q(m) follows from the same argument as in the | | < 1 case.