Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlev\'e II equation

We study multiplicative statistics for the eigenvalues of unitarily-invariant Hermitian random matrix models. We consider one-cut regular polynomial potentials and a large class of multiplicative statistics. We show that in the large matrix limit several associated quantities converge to limits which are universal in both the potential and the family of multiplicative statistics considered. In turn, such universal limits are described by the integro-differential Painlev\'e II equation, and in particular they connect the random matrix models considered with the narrow wedge solution to the KPZ equation at any finite time.


Introduction
Random matrix theory has proven over time to be a powerful modern tool in mathematics and physics. With widespread applications in different areas such as engineering, statistical mechanics, probability, number theory, to mention only a few, its theory is rich and has been under intense development in the past thirty or so years. In a sense, much of the success of random matrix theory has been due to its exact solvability, or integrability, turning them into touchstones for predicting and confirming complex phenomenon in nature.
One of the most celebrated results in random matrix theory is the convergence of fluctuations of the largest eigenvalue towards the Tracy-Widom law F GUE . This result was first obtained by Tracy and Widom [58] for matrices from the Gaussian Unitary Ensemble (GUE), who also showed that F GUE is expressible in terms of a particular solution to the Painlevé II equation (shortly PII). Their findings sparked numerous advances in mathematics and physics, which began from the extension to several other matrix models but shortly afterwards widespread beyond the realm of random matrices.
Starting with the celebrated Baik-Deift-Johansson Theorem [5], the distribution F GUE has been identified as the limiting one-point distribution for the fluctuations of a wide range of different probabilistic models. One of the most ubiquitous of such models is the KPZ equation, introduced in the 1980s by Kardar, Parisi and Zhang. Despite numerous developments surrounding it, exactly solving it remained an outstanding open problem until the early 2010s, where four different groups of researchers [2,26,39,57] independently found an exact solution for its so-called narrow wedge solution. Amongst these works, Amir, Corwin and Quastel [2] found the one-point distribution for the height function of the KPZ solution, showing that it relates to a distribution found a little earlier by Johansson in a grand canonical Gaussian-type matrix model [45], and further characterizing it in terms of the integro-differential Painlevé II equation. The latter is an extension of the PII differential equation, and almost as an immediate consequence the authors of [2] also obtained that this one-point distribution, in the large time limit, converges to F GUE itself.
In much inspired by [5,44] and later by [2,26,39,57], it has been realized that several stochastic growth models share an inherent connection with statistics of integrable point processes, in what is formally established as an identity between a transformation of the growth model and statistics for the point process. To our knowledge, the very first instance of such relation appears in the work of Borodin [15] which connects higher spin vertex model with Macdonald measures. By taking appropriate limit of such connection, it was later found that the KPZ equation is connected to the Airy 2 point process [18], ASEP is related to the discrete Laguerre Ensemble, stochastic six vertex model is connected in the same way to the Meixner ensemble, or yet the Krawtchouk ensemble [19].
As a common feature to these connections, the underlying correspondences establish that the so-called q-Laplace transform of the associated height function coincides with some multiplicative statistics of the point process. The latter, in turn, admits exact solvability, and it is widely believed that a long list of new insights on the growth models can be obtained by studying the corresponding multiplicative statistics of the point processes.
This program has already been taken to a great start for the KPZ equation, and exploring its connection with the Airy 2 point process Corwin and Ghosal [30] were able to obtain bounds for the lower tail of the KPZ equation. Shortly afterwards, such bounds were improved by Cafasso and Claeys [24] with Riemann-Hilbert methods common in random matrix theory.
Our major goal is to take on the program of understanding multiplicative statistics for random particle systems, and carry out its detailed asymptotic analysis for one of the most inspiring models, namely eigenvalues of random matrices. While fluctuations of linear statistics of the eigenvalues of random matrices has been extensively studied in the past, the study of multiplicative statistics has only been carried over in particular instances, remarkably when dealing with products and quotients of characteristic polynomials, see [1,6,38] and the references therein for a complete account.
In this work, we consider the Hermitian matrix model with an arbitrary one-cut regular potential V , and associate to it a general family of multiplicative statistics on its eigenvalues, indexed by a function Q satisfying certain regularity conditions. Our findings show that when the number of eigenvalues is large such multiplicative statistics become universal: in the large matrix limit they converge to a multiplicative statistics of the Airy 2 point process which is independent of V and Q. This limiting statistics admits a characterization in terms of a particular solution to the integro-differential Painlevé II equation, and it is the same quantity that connects the KPZ equation and the Airy 2 point process. So, in turn, we find that random matrix theory can recast the narrow wedge solution to KPZ equation for finite time in an universal way.
The random matrix statistics that we study are associated to a deformed orthogonal polynomial ensemble, also indexed by Q, which we analyze. As we learn from earlier work of Borodin and Rains [20] which was recently rediscovered and greatly extended by Claeys and Glesner [29] (and which we also briefly explain later on), this deformed ensemble is a conditional ensemble of a marked process associated to the original random matrix model. We show that the correlation kernel for this point process converges to a kernel constructed out of the same solution to the integro-differential PII equation that appeared in [2]. This kernel is again universal in both V and Q, and turns out to be the kernel of the induced conditional process on the marked Airy 2 point process. Naturally, there are orthogonal polynomials and their norming constants and recurrence coefficients associated to this deformed ensemble. With our approach we also obtain similar universality results for such quantities, showing that they are indeed universal in V and Q and also connect to the integro-differential PII in a neat way.
Beyond the concrete results, with this work we also hope to shed light into the rich structure underlying multiplicative statistics for eigenvalues of random matrices. Much of the recent relevance of Painlevé equations is due to its appearance in random matrix theory, see [40] for an overview of several of these connections. There has been a growing recent interest in integro-differential Painlevé-type equations [22,23,25,27,46,51], and our results place the integro-differential PII as a central universal object in random matrix theory as well.
We scale the multiplicative statistics to produce a critical behavior at a soft edge of the matrix model, and consequently the core of our asymptotic analysis lies within the construction of a local approximation to all the quantities near this critical point. Our main technical tool is the application of the Deift-Zhou nonlinear steepest descent method to the associated Riemann-Hilbert problem (shortly RHP), and the mentioned local approximation is the so-called construction of a local parametrix. In our case, a novel feat is that this local parametrix construction is performed in a two-step way, first with the construction of a model problem varying with large parameter, and second with the asymptotic analysis of this model problem. In the latter, a RHP recently studied by Cafasso and Claeys [24] (see also the subsequent works [25,27]) which is related to the lower tail of the KPZ equation shows up, and it is this RHP that ultimately connects all of our considered quantities to the integro-differential PII.
The choice of scaling of our multiplicative statistics is natural, illustrative but not exhaustive. As we point out later, with our approach it becomes clear that other scalings could also be analyzed, say for instance scaling around a bulk point, or yet soft/hard edge points with critical potentials, and indicate that other integrable systems extending the integro-differential PII may emerge.
When Q is linear, with a straightforward change of parameters L Q n reduces to E x∈X 1 1 + ζq x , (2.5) where the expectation is over the set X of configurations of points (that is, for us X = Λ (n) ) and q ∈ (0, 1) should be viewed as a parameter of the model and, in general, ζ ∈ C is a free parameter. The expression (2.5) may be viewed as a transformation of the point process, where ζ ∈ C becomes the spectral variable of this transformation, and the matching ζ = e −s motivates the distinguished role of s in (2.3). In the context of random particle systems, this particular multiplicative statistics is associated to the notion of a q-Laplace transform [16][17][18][19] that we already mentioned in the Introduction, and it has been one of the key quantities in several outstanding recent progresses in asymptotics for random particle systems [24,30,43]. We work under the following assumptions.
(i) The potential V is a nonconstant real polynomial of even degree and positive leading coefficient, and its equilibrium measure µ V is one-cut regular, we refer to Section 8.1 below for the precise definitions. Performing a shift on the variable, without loss of generality we assume that the right-most endpoint of supp µ V is at the origin, so that for some a > 0. (ii) The function Q is real-valued over the real line, and analytic on a neighborhood of the real axis.
We also assume that it changes sign at the right-most endpoint of supp µ V , with Although t in Assumption 2.1-(ii) will have the interpretation of time, we stress that in this paper it will be kept fixed within a compact of (0, +∞) rather than being made large or small.
For our results and throughout the whole work, we also talk about uniformity of several error terms with respect to t in the sense that we now explain. Because Q is analytic on a neighborhood of the real axis, analytic continuation shows that it is completely determined by its derivatives Q (k) (0), k ≥ 0. When we say that some error is uniform in t within a certain range, we mean uniform when we vary Q as a function of −Q (0) = t while keeping all other derivatives Q (k) (0), k ≥ 2, fixed.
The condition in Assumption 2.1-(i) is standard in random matrix theory and they are known to hold when, say, V is a convex function [56]. The one-cut assumption is made just for ease of presentation, as it simplifies the Riemann-Hilbert analysis at the technical level considerably. On the other hand, the regularity condition is used substantially in our arguments, but is standard in Random Matrix Theory literature and holds true generically [49]. Most of our results are of local nature near the right-most endpoint of supp µ V and could be shown to hold true for multi-cut potentials near regular endpoints as well, with appropriate but non-essential modifications.
Assumption 2.1-(ii) should be seen as specifying enough regularity on the multiplicative statistics, here indexed by this factor Q. Because of condition (ii), we have the pointwise convergence 0, x > 0, 1, x < 0, (2.8) which means that the introduction of the factor σ n in the original weight e −nV has the effect of producing an interpolation between this original weight and its cut-off version χ (−∞,0) e −nV , where from here onward χ J is the characteristic version of a set J. Comparing the Euler-Lagrange conditions on the equilibrium problem induced by the weights e −nV and χ (−∞,0) e −nV , the observation we just made heuristically indicates that the factor σ n does not change the global behavior of eigenvalues. This may also be rigorously confirmed as an immediate consequence of our analysis, but we do not elaborate on this end. On the other hand, introducing a local coordinate u = −z/n 2/3 near the origin, the approximation e −s−n 2/3 Q(z) ≈ e −(tu+s) goes through, and we see that there is a competition between the term s and Q(z) that affects the local behavior of the weight at the scale O(n −2/3 ) near the origin, which is the same scale for nontrivial fluctuations of eigenvalues around the same point. The main results that we are about to state concern obtaining the asymptotic behavior as n → ∞ of several quantities of the model, and in particular they showcase how this term Q affects the local scaling regime of the eigenvalues near the origin and leads to connections with the integro-differential Painlevé II equation as already mentioned. A central object in this paper is the multiplicative statistics where the expectation is over the Airy 2 point process with random configuration of points {a j } [55]. The quantity L Ai admits two remarkable characterizations, which are also of particular interest to us. The first is the formulation via a Fredholm determinant, namely where K Ai T is the integral operator on L 2 (−s, ∞) acting with the finite temperature (or fermi-type) deformation of the Airy kernel K Ai T , defined by The term 'temperature' stems from the connection between the KPZ equation and the random polymer models. Despite the name finite temperature, the parameter T here corresponds to the time in the KPZ equation, see (2.12) below. The Fredholm determinant det I − K Ai T appeared for the first time in the work of Johansson [45] as the limiting process of a grand canonical (that is, when the number of particles/size of matrix is also random) version of a Gaussian random matrix model, and interpolates between the classical Airy kernel when T → +∞ and the Gumbel distribution when T → 0 + with s scaled appropriately. In [45,Remark 1.13] Johansson already raises the question on whether a related classical (that is, not grand canonical) matrix model has limiting local statistics that interpolate between Gumbel and Tracy-Widom, as a similar feature to det I − K Ai T . Since then, other works have found det I − K Ai T to be the limiting distribution for fluctuations around the largest particle of a point process [10,31,32,51]. In common, these works consider specific models rather than obtaining det I − K Ai T as the universal limit for a whole family of particle systems. Another characterization of L Ai is via a Tracy-Widom type formula that relates it to the integrodifferential PII. It reads where Φ solves the integro-differential Painlevé II equation for any δ > 0. This characterization has been obtained in the already mentioned work by Amir, Corwin and Quastel [2], in connection with the narrow wedge solution to the KPZ equation, and following the work [18] by Borodin and Gorin has the interpretation that we now describe. For H(X, T ) being the Hopf-Cole solution to the KPZ equation with narrow wedge initial data at the space-time point (X, T ), introduce the rescaled random variable Based on the previous works [2,26,39,57], in [18] the identity between the height function of the KPZ equation and the multiplicative statistics L Ai (s, T ) is identified. This is an instance of matching formulas relating growth processes with determinantal point processes that we already mentioned at the Introduction. One of the key aspects of this representation is that the Airy 2 point process is determinantal, and consequently its statistics can be studied using techniques from exactly solvable/integrable models. Indeed Equation 2.12 is the starting point taken by Cafasso and Claeys [24], who then connected L Ai (s, T ) to a RHP that will also play a major role for us. Recently, Cafasso, Claeys and Ruzza [25] also obtained an independent proof of the representation (2.10), extending it to more general multiplicative statistics of the Airy 2 point process. Other proofs and extensions of this integro-differential equation have also been recently found in related contexts [22,23,46]. Also, by exploring (2.12) the tail behavior of the KPZ equation has become rigorously accessible in various asymptotic regimes [24,25,27,30]. As a first result, we prove that the multiplicative statistics L Ai is the universal limit of L Q n (s).
Findings on random matrix theory surrounding the Tracy-Widom distribution have inspired an enormous development in the KPZ universality theory. For the KPZ equation one of the major developments can be phrased by saying that the fluctuations of the height function for the narrow wedge solution coincide, in the large time limit, with the β = 2 Tracy-Widom law from random matrix theory. Theorem 2.2 is saying that the connection between random matrix theory and the KPZ equation can be recast already at any finite time, and not only for Gaussian models but also universally in V and Q. Similar connection exists [8] between the solution of the KPZ equation in half-space under the Robin boundary condition and Airy 1 point process which, in turn, in the large time limit relate this KPZ solution to GOE matrices.
The constant c V is determined from (8.2) and (8.5) below. It is the first derivative of a conformal map near the origin, which is constructed out of the equilibrium measure for V . Ultimately, we make a conformal change of variables of the form ζ ≈ c V z/n 2/3 , which in turn identifies In light of (2.9), this explains the evaluation s = −sc V /t and T = t 3 /c 3 V on the right-hand side of (2.13).
We emphasize that the error term in (2.13) is not in sharp form. In Section 3.1 we explain how this term arises from our techniques. We do not have indications regarding whether the true optimal error would be O(n −2/3 ) (or of any polynomial order) or if it should involve, say, logarithmic corrections.
Our next results concern limiting asymptotic formulas for the matrix model underlying L Q n , starting with the partition function Z Q n from (2.2). For a polynomial V , its deformation has the property that V 1 = V and V ∞ = z 2 is Gaussian. Under the assumption that V t is one-cut regular for every t ≥ 1, Bleher and Its [14] proved that an expansion of the form holds, where Z GUE n is the GUE partition function for V ∞ = z 2 and e V 0 and e V 1 are functions analytic on the coefficients of V . In fact, their result ensures a full asymptotic expansion in inverse powers of n 2 , see also [9,13,41] for important earlier work obtaining similar results under different conditions, and also the more recent contributions [12,21]. As an immediate corollary to Theorem 2.2 we obtain some terms in the asymptotic expansion of the deformed partition function (2.4). Corollary 2.3. Suppose V and Q satisfy Assumptions 2.1 and fix s 0 > 0 and t 0 > 0. In addition, assume that the potential V t in (2.14) is one-cut regular for any t ≥ 1. For any ν ∈ (0, 2/3), the partition function admits an expansion of the form which is valid uniformly for s ≥ −s 0 and t 0 ≤ t ≤ 1/t 0 , and the coefficients e V 0 and e V 1 are as in (2.15). Proof. Follows from the expansion (2.15), Theorem 2.2 and (2.3).
The order of error (2.16) is not n −2 as in (2.15) but weaker and not sharp. This phenomenon can be traced back to the fact that σ n has infinitely many poles accumulating on the real axis as n → ∞, see the discussion in Section 3.1 below. A similar error order was obtained in [14, Theorem 9.1 and Equation (9.68)], in a transitional regime from a one-cut to two-cut potential, and where the role played here by L Ai is replaced by the GUE Tracy-Widom distribution itself.
From the general theory of unitarily invariant random matrix models, it is known that the density appearing in (2.3) admits a determinantal form. Setting 17) this means that the identity holds true for a function of two variables K Q n (x, y) satisfying certain properties, known as the correlation kernel of the eigenvalue density on the left-hand side. The correlation kernel is not unique, but in the present setup it may be taken to be the Christoffel-Darboux kernel for the orthogonal polynomials for the weight ω n , as we introduce in detail in (9.3), and whenever we talk about K Q n we mean this Christoffel-Darboux kernel. In particular, K Q n = K Q n (· | s) does depend on both Q and s. Our second result proves universality of the kernel K Q n , showing that its limit depends solely on s and t = −Q (0), but not on other aspects on Q, and relates to the integro-differential PII. For its statement, it is convenient to introduce the new set of variables being the solution to the integro-differential Painlevé II equation in (2.11) and the variables s, t and S, T related by (2.18), we set and introduce the kernel Theorem 2.4. Assume that V and Q satisfy Assumptions 2.1 and fix s 0 > 0 and t 0 ∈ (0, 1). With the estimate holds true uniformly for u, v in compacts of R, and uniformly for s ≥ −s 0 and t 0 ≤ t ≤ 1/t 0 .
In the recent work [29], Claeys and Glesner developed a general framework for certain conditional point processes, which in particular yields a probabilistic interpretation of the kernel K Q n as we now explain. For a point process Λ, we add a mark 0 to a point λ ∈ Λ with probability σ n (λ) and a mark 1 with complementary probability 1 − σ n (λ). This induces a decomposition of the point process Λ = Λ 0 ∪ Λ 1 , where Λ j is the set of eigenvalues with mark j. We then consider the induced point process Λ obtained from Λ upon conditioning that Λ 1 = ∅, that is, that all points have mark 0.
When applied to the eigenvalue point process Λ = Λ (n) induced by the distribution (2.1), the theory developed in [29] shows that Λ (n) is a determinantal point process with correlation kernel proportional to ω n (x) 1/2 ω n (y) 1/2 K Q n (x, y) which, in turn, generates the same point process as the left-hand side of (2.20), see [29,Sections 4 and 5]. A comparison of the RHP that characterizes the kernel K ∞ (see Section 5.1 below, in particular (5.16)) with the discussion in [29,Section 5.2] shows that K ∞ is a (renormalized) correlation kernel for the marked point process {a k } of the Airy 2 point process {a k } with the marking function (1 + e −s+tλ ) −1 . So Theorem 2.4 assures that the conditional process on the marked eigenvalues converges, at the level of rescaled correlation kernels, to the conditional process on the marked Airy 2 point process.
We also obtain asymptotics for the norming constant γ (n,Q) n−1 (s) for the (n − 1)-th monic orthogonal polynomial for the weight ω n (x | s) (see (9.2) for the definition), showing that its first correction term depends again solely on s, t, and also relates to the integro-differential Painlevé II equation.
Theorem 2.5. Suppose that V and Q satisfy Assumptions 2.1 and fix s 0 > 0 and t 0 ∈ (0, 1). The norming constant has asymptotic behavior γ (n,Q) uniformly for s ≥ −s 0 and t 0 ≤ t ≤ 1/t 0 , where p(s, t) = P(S, T), and the function P relates to the solution Φ from (2.11) via Our approach also yields asymptotic formulas for the orthogonal polynomials and their recurrence coefficients, and relate them to the integro-differential Painlevé II equation as well, but for the sake of brevity we do not state them.
3. About our approach: issues and extensions 3.1. Issues to be overcome.
Our main tool for obtaining all of our results is the Fokas-Its-Kitaev [42] Riemann-Hilbert Problem (RHP) for orthogonal polynomials (shortly OPs) that encodes the correlation kernel K Q n , the norming constants γ (n,Q) n−1 (s) 2 and ultimately also the multiplicative statistics L Q n , and its asymptotic analysis via the Deift-Zhou nonlinear steepest descent method [35,37]. The overall arch of this asymptotic analysis is the usual one, summarized in the diagram in Figure 1, and we now comment on its major steps.  Figure 1. Schematic diagram for the steps in the asymptotic analysis of the RHP for orthogonal polynomials. There is also an Airy local parametrix used near z = −a which we omit in this diagram.
Starting with the RHP for OPs that we name Y, in the first step we transform Y → T with the introduction of the g-function (or, equivalently, the φ-function), and this is done so with the help of the equilibrium measure for V that accounts only for the part e −nV of the weight ω n . In the second step, we open lenses with a transformation T → S as usual.
The third step is the construction of the global parametrix G. In our case, in this construction we also have to account for the perturbation σ n of the weight e −nV , so a Szegö function-type construction is used.
The fourth step is the construction of local parametrices at the endpoints z = −a, 0 of supp µ V , with the goal of approximating all the quantities locally near these endpoints. This is accomplished by, first, considering a change of variables z → ζ = n 2/3 ψ(z) after the conformal map ψ chosen appropriately for each endpoint and, then, constructing the solution to a model RHP Φ(ζ) in the ζ-plane. Following these steps, the local parametrix at the left edge z = −a of supp µ V is standard and utilizes Airy functions.
The construction of the local parametrix at the right edge z = 0 is, however, a lot more involved. As we mentioned earlier, the factor σ n affects asymptotics of local statistics near the origin. In fact, the weight σ n has singularities precisely at the points of the form This means that for s n 2/3 there are infinitely many poles of σ n accumulating near the real axis. As such, in this case for large n the perturbed weight ω n fails to be analytic in any fixed neighborhood of the origin. If we were to consider only s → +∞ fast enough, one could still push the standard RHP analysis further with the aid of Airy local parametrices, at the cost of a worse error estimate. However, when s = O(1) we have poles accumulating too fast to the real axis, and a different asymptotic analysis has to be accomplished, in particular a new local parametrix is needed.
When changing coordinates z → ζ near z = 0, the model problem Φ = Φ n obtained is then ndependent. This is so because the jump of the local parametrix involves σ n , and consequently in the process of changing variables the resulting model problem has a jump that involves a transformation of σ n itself. This is in contrast with usual constructions with, say, Airy, Bessel or Painlevé-type parametrices, where the jumps can be turned into piecewise constant in the z-plane and, hence, also remain piecewise constant in the ζ-plane. Another feature of the RHP for the model problem Φ n is that its jump is not analytic on the whole plane, and instead it is analytic only in a growing (with n) disk, and for a fixed n we can only ensure that its jump matrix is C ∞ in a neighborhood of the jump contour.
All in all, this means that carrying out the asymptotic analysis of Φ n (ζ) as n → ∞ is also needed. As we said, the jump for Φ n involves a transformation of σ n , so ultimately also depends on the function Q from (2.17). But it turns out that as n → ∞, we have the convergence Φ n → Φ 0 in an appropriate sense, where Φ 0 is independent of Q. This limiting Φ 0 is the solution to a RHP that appeared recently in connection with the KPZ equation [24] and which was later shown to connect with the integrodifferential PII in the recent work of Claeys, Cafasso and Ruzza [25,27]. For this reason we term it the id-PII RHP.
With the construction of the global and local parametrices, the asymptotic analysis is concluded in the usual way, by patching them together and obtaining a new RHP for a matrix function R. This matrix R, in turn, solves a RHP whose jump is asymptotically close to the identity, and consequently R can be found perturbatively.
After concluding this asymptotic analysis, we undress the transformations R → · · · → Y and obtain asymptotic expressions for the wanted quantities. For the kernel K Q n and the norming constant γ (n,Q) n−1 (s) 2 , after this undressing Theorems 2.4 and 2.5 follow in a standard manner. However, to obtain (2.13) quite some extra work is needed. When dealing with statistics of matrix models via OPs, one of the usual approaches is to extract the needed information via the partition function and its relation with the norming constants via a product formula, see for instance (9.6) below. Usually this is accomplished via some differential identity or with careful estimate of each term in the product formula, see for instance the works [4,12,14,47] and their references for explorations along these lines. In virtue of the relation (2.3) this was in fact our original attempt, but several technical issues arise. Instead, at the end we express L Q n directly as a weighted double integral of K Q n (x, x | s) in the variables in x and s, this is done in Proposition 9.1 below. The x-integral takes place over the whole real line, which means that when we undress R → Y we obtain a formula for L Q n involving global and all local parametrices. The integral in s extends to +∞, which is one of the main reasons why in our main statements we also keep track of uniformity of errors when s → +∞. We then have to estimate the double integral, accounting for exponential decays of most of the terms but also exact cancellations of some other terms. Ultimately, the whole analysis leads to a leading contribution coming solely from a portion of the integral that arises from the model problem Φ n . With a further asymptotic analysis of the later integral we obtain an integral solely of Φ 0 which then yields Theorem 2.2.
The convergence Φ n → Φ 0 is treated as a separate issue, and to achieve it we need several information about this id-PII parametrix Φ 0 . As a final outcome, we obtain that Φ n is close to Φ 0 with an error term of the form O(n −ν ), for any ν ∈ (0, 2/3). But, in much due to the non-analyticity of the jump matrix for Φ n , we are not able to achieve a sharp order O(n −2/3 ) unless further conditions were placed on Q. This non-optimal error explains the appearance of the same error order in (2.13). In the course of this asymptotic analysis we rely substantially in [25]. Among other needed info, we also borrow from the same work the connection of Φ 0 with the integro-differential PII. In the same work, the authors actually show that Φ 0 relates to particular solutions to the KdV equation that reduce to the integro-differential PII. As such, Theorems 2.4 and 2.5 could be phrased in terms of a solution to the KdV rather than to the integro-differential PII. We opt to phrase them with the latter because this formulation encodes that all self-similarities have already been accounted for.
If we were to assume that the jump matrix for Φ n were piecewise analytic on the whole plane and not merely C ∞ , we could deform Φ n to a family of RHPs considered in [25]. With this in mind, the analysis of the convergence Φ n → Φ 0 is inspired by several aspects in this just mentioned work but, as we already said, here we are forced to work under different conditions on the jump matrix. In particular, one could adapt the methods in [25] to actually prove that Φ n does too relate to an ndependent solution to the integro-differential PII. Consequently, with a careful inspection of our work one could show that Theorems 2.2, 2.4 and 2.5 admit versions with n-dependent leading terms. For instance, relating the norming constant γ (n,Q) n−1 (s) 2 with the model problem Φ n one could obtain an asymptotic formula of the form where the n-dependent function p n is obtained from Φ n and relates to a n-dependent solution Φ n to the integro-differential PII. In fact, with standard arguments one could improve the formula above to a full asymptotic expansion in powers of n −1/3 , with bounded but n-dependent coefficients. Underlying our arguments there is the statement that p n = p + O(n −ν ) for any ν ∈ (0, 2/3), which then yields Theorem 2.5. But as a drawback, although one could potentially improve (2.21) and also obtain the term of order n −2/3 explicitly, it is not possible to obtain the O(n −1 ) term in (2.21) unless one improves the error O(n −ν ) in the convergence Φ n → Φ 0 to a sharp error O(n −2/3 ).

Possible extensions.
Most of our approach may be extended to potentials V for which the equilibrium measure µ V is critical, and also under different conditions on Q as we now explain.
Apart from technical adaptations in several steps of the RHP for OPs which are nowadays well understood, our analysis carries over to potentials V for which the equilibrium measure µ V is regular but multicut, with the same conditions on Q when µ V has the origin as its right-most endpoint.
When, say, the density µ V vanishes to a higher power at a soft edge and/or Q changes sign with an arbitrary odd vanishing order at the same soft edge, we need to replace the power n 2/3 in σ n by another appropriate power to modify the local statistics near this point in a non-trivial critical manner. Once this is done, the asymptotic analysis of the RHP for OPs that we perform carries over mostly with minor modifications, and the only major issue to overcome is in the construction of a new local parametrix Φ n near this soft edge point and its corresponding asymptotic analysis. In this case, we expect that Φ n → Φ 0 for a new function Φ 0 . It is relatively simple to write a RHP that should be satisfied by this Φ 0 , and we expect it to be related to the KdV hierarchy [28] but with nonstandard initial data. It would be interesting to see if the particular solutions obtained this way reduce to integro-differential hierarchies of Painlevé equations, in the same spirit of the recent works [23,46].
One could also consider similar statistics to (2.3) with a Q that vanishes at a bulk point of supp µ V . We do expect that most of our work carries through to this situation, at least when we impose V to be again one-cut regular and Q to vanish quadratically at a point inside supp µ V . The main issue that should arise is again on the construction of the local parametrix near this point, and its corresponding asymptotic analysis. This model should lead to multiplicative statistics of the Sine kernel (and the higher order generalizations of it). Similar considerations go through to hard-edge models, leading to multiplicative statistics of the Bessel process. To our knowledge, such multiplicative statistics of Bessel and Sine have not been considered in the literature so far. However, finite temperature versions of the Sine and Bessel kernels do have appeared, see for instance [10,11,31,45].

Organization of the paper.
The paper is organized in two parts. In the first part, we deal with a family of RHPs Φ τ that contains the model RHP Φ n needed in the asymptotic analysis of OPs. In Section 4 we introduce Φ τ formally. In Sections 5 and 6 we discuss the RHP Φ 0 , which is a particular case of Φ τ , and review several of its properties, translating results from [24,25] to our notation and needs. In Section 7 we prove the convergence Φ τ → Φ 0 and of related quantities in the appropriate sense. The latter section contains all the needed results for the asymptotic analysis of the RHP for OPs, and concludes the first part of this paper.
The second part of the paper is focused on the asymptotic analysis of the RHP for OPs. In Section 8 we discuss several aspects that relate to the equilibrium measure. In Section 9 we introduce the Christoffel-Darboux kernel K Q n and related quantities, and display how they relate to the RHP for OPs. In particular, in Proposition 9.1 we write L Q n directly to as an integral of the kernel K Q n , a result which may be of independent interest. In Section 10 we perform the asymptotic analysis of the RHP for the OPs. In Section 11 use the conclusions from Sections 10 and 7 and prove Theorems 2.4 and 2.5. Also from the results from Sections 10 and 7 and assuming additional technical estimates, the proof of Theorem 2.2 is given in Section 11. Such remaining technical estimates are also ultimately a consequence of the RHP analysis, but their proofs are rather cumbersome and postponed to Section 12.
For the remainder of the paper it is convenient to denote so E jk is a 2 × 2 matrix with the (j, k)-entry equals 1 and all other entries zero. With this notation, the Pauli matrices, for instance, take the form In particular, for any reasonably regular scalar function f , the spectral calculus yields These notations will be used extensively in the coming sections.
14 P. GHOSAL AND G. SILVA orienting Σ 0 from the origin to ∞, and the remaining arcs from ∞ to the origin, see Figure 2.
The model RHP we are about to introduce depends on a function h : Σ → C used to describe its jump. For the moment we assume These conditions are present only to ensure the RHP below is well posed and are far from optimal, but enough for our purposes. Later on we will impose more conditions on this function h, these conditions will be tailored to our later needs regarding the asymptotic analysis of OPs. The associated RHP asks for finding a 2 × 2 matrix-valued function Φ with the following properties.
Along the interior of the arcs of Σ the function Φ admits continuous boundary values where and Given h, it is not at all obvious that the RHP above has a solution and how to describe it. We study this model problem when h = h τ depends on an additional large parameter τ , in a way that appears naturally in the asymptotic analysis of the orthogonal polynomials mentioned earlier. For large values of τ , we then prove that the solution Φ exists and is asymptotically close to a model RHP that appeared recently [24] and that we discuss in a moment.

The model RHP with admissible data.
For us, we need to consider the model problem Φ = Φ(· | h) with functions h = h τ satisfying certain properties which are formally introduced in the next definition.
where H is defined on a neighborhood S of Σ and satisfies the following properties.
(i) The function H is independent of τ and s, of class C ∞ on S and real-valued along R.
(ii) H is analytic on a disk D δ (0) ⊂ S centered at the origin, and its unique zero on (iii) There exist constants η, η > 0 for which Conditions (i)-(ii), and also the bounds in (iii) involving η, are natural in our setup. The bound H(w) > − ηw 3/2− is present for technical reasons, and it plays a role only for the proof of Lemma 7.4, allowing us to write certain estimates in a cleaner matter. It could be removed, at the cost of slightly more complicated error terms in the mentioned Lemma. For our purposes, namely to use Φ = Φ(· | h τ ) as a local parametrix with an appropriate h τ , this condition is satisfied anyway (this will be accomplished in Proposition 8.3), so we include it in our definition here as well, as it simplifies our analysis.
In the course of the analysis for the RHP for the orthogonal polynomials discussed in Section 9, the function H will be a transformation of the function Q appearing in the deformed weight (2.17), and the parameter t that we defined here will play the same role as the one in the definition (2.7).
Given an admissible h τ , we denote We are interested in the asymptotic analysis for Φ τ as τ → +∞ and s ≥ −s 0 , for any s 0 > 0, and t > 0 kept fixed within a compact of the positive axis.
We now explain in an ad hoc manner the appearance of a RHP for the integro-differential equation, which also relates to the KPZ equation. Definition 4.1-(ii) gives that H has an expansion of the form This means that any admissible function h τ satisfies In particular, the convergence holds true uniformly in compacts as τ → ∞. This indicates that the solution Φ τ should converge to the solution of the model problem obtained from h 0 . The RHP-Φ 0 relates to the integro-differential PII and is a rescaled version of an RHP that appears in the description of the narrow wedge solution to the KPZ equation, as we discuss in the next section in detail.

The RHP for the integro-differential RHP
appeared for the first time in the work of Cafasso and Clayes [24] (this is the RHP-Ψ in Section 2 therein) in connection with the narrow wedge solution to the KPZ equation as we explain in a moment, in Section 5.1. To avoid confusion with the related quantities that we are about to introduce, we term it the KPZ RHP. In virtue of the identity which follows from (4.6) and (5.1), we also have the correspondence and we refer to Φ 0 as the id-PII RHP. For the record, we state the existence of Φ 0 formally as a result.
Proposition 5.1. For any s ∈ R and any t > 0, the solution Φ 0 exists and is unique. Furthermore, for any fixed s 0 > 0 and t 0 ∈ (0, 1), the solution Φ 0,+ (ζ) remains bounded for ζ in compacts of R and Proof. It is a consequence of [24, Section 2] that the solution Φ (KPZ) (· | s, T ) exists and is unique, for any s ∈ R and T > 0, and from the correspondence (5.2) the existence and uniqueness of Φ 0 is thus granted.
For the boundedness, we start from the representation which follows from the L p theory of RHPs (see [34]). The jump matrix admits an analytic continuation to any neighborhood of the real axis, and this analytic continuation remains bounded in compacts, also uniformly for s ≥ −s 0 and t 0 ≤ t ≤ 1/t 0 (see for instance (5.9) for the exact expression). With these observations in mind, the claimed boundedness follows from standard arguments. We skip additional details, but refer to the proof of Theorem 7.1, in particular (7.16) et seq., for similar arguments in a more involved context.
In this section we collect several results on Φ 0 that were obtained in [24,25] and which will be needed later.
But before proceeding, a word of caution. As we said, the RHP-Φ (KPZ) appeared first in [24], but was also studied in the subsequent work [25]. The meanings for the variables s, x and t in these two works are not consistent, but we need results from both of them. Comparing to the work [24] by Cafasso and Claeys, the correspondence is This correspondence is consistent with (5.2). On the other hand, when comparing to the subsequent work [25] by Cafasso, Claeys and Ruzza, the correspondence between notations is where T, S are as in (2.18).
In our asymptotic analysis, the most convenient choice of variables to work with is the choice (s, t) and the correspondence (S, T) from (2.18) that we have already been using, and which leads to the RHP Φ 0 as we introduced. Nevertheless, we will need to collect results from both mentioned works, and when the need arises we refer to the correspondences of variables (5.3)-(5.4).
On the other hand, when making correspondence with integrable systems, in particular the integrodifferential Painlevé II equation, it is more convenient to work with the variables S and T as in (2.18).

Properties of the id-PII parametrix.
In this section we describe many of the findings from [24,25], in a way suitably adapted to our notation and needs. In particular the connection of Φ 0 introduced in (4.7) with the integro-differential Painlevé II equation is described in this section. For was shown in [24, Theorem 2.1] and will also be useful for us. With (5.2) we now rewrite this identity in terms of Φ 0 . With the principal branch of the argument, set This function relates to ∆ (KPZ) in (5.5) via and (5.6) rewrites as For further reference, it is now convenient to state the RHP for Φ 0 explicitly.
To compare with [25] we perform a transformation of this RHP. All the calculations that follow already take into account the correspondence (5.4) between the notation in the mentioned work and our notation.
As a consequence, and with the change of variables (s, t) → (S, T) from (2.18), we obtain that for some functions Q = Q(S, T), R = R(S, T), P = P(S, T) and q = q(s, t) = Q(S, T), r = r(s, t) = R(S, T), p = p(s, t) = P(S, T), (5.13) the asymptotic behavior (5.12) improves to Stressing that the correspondence (5.4) is in place, the functions P and Q satisfy the relation [25,Equation (3.14)] In addition, P and Φ are related through the identity (2.22) which, in turn, implies that Φ is the solution to the integro-differential Painlevé II equation in (2.11).
It is convenient to write some quantities of Φ 0 directly in terms of the just introduced functions. The first identity we need for later is which follows from (5.11) and (5.15) after a straightforward calculation, accounting also that det Φ 0 = det Ψ 0 ≡ 1.
The second relation we need is an improvement of the asymptotics of Φ 0 in (5.10). With the coefficients and the functions q, r, p in (5.13), introduce After some cumbersome but straightforward calculations, the asymptotics (5.14) improves (5.10) to

Bounds on the id-PII RHP
We need to obtain certain asymptotic bounds on Φ 0 in different regimes. These bounds will be used later to show that the model problem Φ τ converges, as τ → +∞, to Φ 0 as already indicated in (4.5) et seq. We split these necessary estimates in the next subsections, depending on the regime we are.
In what follows, for a matrix-valued function M = (M jk ) and a contour Σ ⊂ C, we also use the pointwise matrix norm and the matrix L p norm (possibly also with p = ∞) where the measure is always understood to be the arc-length measure. In particular, for any two given matrices M 1 and M 2 the inequality is satisfied. Similar straightforward inequalities involving L 1 , L 2 and L ∞ and the pointwise norm (6.1) also hold, and will be used without further mention. Sometimes we also write to identify that possible convergences are taking place in various norms simultaneously. In a similar manner we define norms · L p 1 ∩L p 2 ∩L p 3 involving three (or possibly more) function spaces.
The first asymptotic regime we consider is where t 0 ∈ (0, 1) is any given value, and s 0 = s 0 (t 0 ) > 0 will be made sufficiently large depending on t 0 > 0, but independent of t within the range above. With (5.4) in mind, this is a particular case of the singular regime in [25]. For this asymptotic regime, we need the following result.
hold true for any s ≥ s 0 and any t ∈ The proof of Proposition 6.2 is a recollection of the analysis in [25], so before going into the details we need to review some further notions from their work. Introduce The exact form of the matrix Y is not important for us, but we can interpret this last equality as a defining identity for Y. What is important for us is that Y is analytic off the real axis, with a jump matrix J Y on R which admits an analytic continuation to a neighborhood of the axis. The small norm theory for Y in our regime of interest was carried out in [25, Lemma 5.1 and Section 5.2]. As a consequence, we obtain that for any hold for any s ≥ s 0 and any t ∈ [t 0 , 1/t 0 ]. Also as a consequence of the small norm theory, we obtain the expression We combine this last identity with the fact that J Y admits an analytic continuation in a neighborhood of R, and learn that there exists M = M (t 0 ) > 0 for which Proof of Proposition 6.1. For ζ ∈ Σ 0 = (0, ∞), we use (6.5) and the definition of Φ Ai in Using the bound (6.6), the continuity and the known asymptotics as ζ → ∞ of the Airy function and its derivative, the claim along Σ 0 follows. The claim for ζ ∈ Σ j with j = 1, 2, 3 follows in exactly the same explicit manner, we skip the details.
In the non-asymptotic regime, we fix any t 0 ∈ (0, 1) and s 0 > 0 and seek for bounds of certain entries of Φ 0 which are valid uniformly within the range For the next result, we recall the matrix norm introduced in (6.1).

P. GHOSAL AND G. SILVA
Proof. The asymptotic behavior as ζ → ∞ in the RHP-Φ 0 is valid uniformly up to the boundary values Φ 0,± as well, and also uniformly when the parameters s and t vary within compact sets, implying that Combined with the continuity of the boundary value Φ 0,+ with respect to both ζ and also s, t, the first estimate follows. The remaining estimates are completely analogous.

Asymptotic analysis for the model problem with admissible data
We now carry out the asymptotic analysis as τ → +∞ of Φ τ introduced in (4.5). For that, we fix s 0 > 0 and t 0 ∈ (0, 1) and work under the assumption that During this section, h τ always denotes an admissible function in the sense of Definition 4.1, and Φ τ is the solution to the associated RHP.
We also talk about uniformity of error terms in the parameter t ranging on a compact interval K ⊂ (0, ∞), and by this we mean the following. The solution Φ τ depends on the parameter t via the derivative H (0) = −t, see Definition 4.1. We view H = H t as varying with t while keeping all the remaining derivatives H (k) (0), k = 1 fixed. By analyticity this determines H uniquely at D δ (0), but not outside this disk. We then consider H t outside D δ (0) to be any extension from D δ (0) that satisfies Definition 4.1 with the additional requirement that the constants η, η and in (iii) may depend on K but are independent of t ∈ K. Of course, for each H t extended this way there corresponds a solution Φ τ of the associated RHP. By uniformity in t ∈ K we mean that the error may depend on K and the corresponding values η, η and , but is valid for any Φ τ obtained with an extension H t constructed with the explained requirement.
The asymptotic analysis itself makes use of somewhat standard arguments and objects in the RHP literature. Some consequences of this asymptotic analysis will be needed later, and we now state them.
The first such consequence is the existence of a solution with asymptotic formulas relating quantities of interest with the corresponding quantities in the id-PII RHP.
For the proof of these results, we compare Φ τ with the solution Φ 0 of the id-PII RHP via the Deift-Zhou nonlinear steepest descent method. The required asymptotic analysis itself is carried out Section 7.1, and the proofs of Theorems 7.1 and 7.2 are completed in Section 7.2.

Asymptotic analysis.
For Φ 0 as introduced in (5.2) and whose properties were discussed in Section 5.1, we perform the transformation Then Ψ τ satisfies the following RHP.
Along the interior of the arcs of Σ the function Ψ τ admits continuous boundary values Ψ τ,± related by Ψ τ, where h 0 is as in (4.6), the jump matrix J Ψτ is 0 the residues at ∞ of Φ τ and Φ 0 , respectively, the matrix Ψ τ has the asymptotic behavior The next step is to verify that the jump matrix decays to the identity in the appropriate norms. The terms in the jump that come from Φ 0 are precisely the ones we already estimated in Sections 6.1 and 6.2, so it remains to estimate the terms involving the λ-functions. The basic needed estimate is the following lemma. Lemma 7.3. Fix ν ∈ (0, 1/2) and t 0 ∈ (0, 1). The estimate holds true uniformly for |ζ| ≤ τ ν and uniformly for t 0 ≤ t ≤ 1/t 0 , where the error term is independent of s ∈ R.
Proof. The Definition (4.1) of admissibility of h τ ensures that for τ sufficiently large, we can expand the term H(ζ/τ ) in power series near the origin and obtain the expansion valid uniformly for |ζ| ≤ τ ν , t 0 ≤ t ≤ 1/t 0 , and with error independent of s ∈ R. Recalling that h 0 (ζ) = s − tζ, the proof is complete.
We are now able to prove the appropriate convergence of J Ψτ to the identity matrix. We split the analysis into three lemmas, corresponding to different pieces of the contour Σ. In the results that follow we use the matrix norm notations introduced in (6.1)-(6.3).
Next, we prove the equivalent result along the pieces of Σ which are not on the real line.
where the implicit error term is independent of s and uniform for t ∈ [t 0 , 1/t 0 ]. On the other hand, from the explicit form of h 0 and Definition 4.1-(iii), We combine this inequality with Propositions 6.1 and 6.2 and use them on (7.5). The conclusion is that there exist M > 0, η 1 , η 2 > 0 and τ 0 > 0, depending on ν, t 0 , s 0 , for which the inequality .10) is valid for every τ ≥ τ 0 , s ≥ −s 0 and t ∈ [t 0 , 1/t 0 ]. The definition (4.1) of the contours Σ 1 and Σ 3 assure us that Re ζ 3/2 > 0 and Re ζ < 0 on these contours. After possibly changing the values of the constants η 1 , η 2 and M , the result follows.
Finally, we now handle the jump on the negative axis. Proof. The initial step is to rewrite the last line of (7.5) as The identities are trivial, and because h 0 and h τ are real-valued along Σ 2 = (−∞, 0), these equalities give For |ζ| ≤ τ ν we use Lemma 7.3 and estimate whereas for ζ ≤ −τ ν we use instead the definition of h 0 in (4.6) and Definition 4.1-(iii) and write We combine these two inequalities with Propositions 6.1 and 6.2, and apply them to (7.11). As a result, we learn that there exist M > 0, η > 0, τ 0 > 0 for which the estimate is valid for any s ≥ −s 0 , t ∈ [t 0 , 1/t 0 ], τ ≥ τ 0 . After possibly changing the values of η, M , the result follows from standard arguments.
Now that we controlled the asymptotic behavior for the jump matrix J Ψτ , we are ready to obtain small norm estimates for Ψ τ itself. We summarize these estimates in the next result. For that, we recall the matrix norm notations introduced in (6.1),(6.2),(6.3).
Its boundary value Ψ τ,− exists along Σ, and satisfies the estimate for any κ ∈ (0, 1), where the error term, for a given κ, is uniform for s ≥ −s 0 and t ∈ [t 0 , 1/t 0 ]. For τ sufficiently large, the solution Ψ τ admits the representation Still for τ sufficiently large, Ψ τ satisfies Proof. The small norm estimates provided by Lemmas 7.4, 7.5 and 7.6 allow us to apply the small norm theory for Riemann-Hilbert problems (see for instance [34,37]), and the claims follow with standard methods. We stress that for this statement we only need the L 2 and L ∞ estimates from the aforementioned lemmas, but the L 1 estimates provided by them will be useful later.

Proof of main results of the section.
We are ready to prove the main results of this section.
To prove the asymptotic formula (7.3) we follow arguments presented in [48,Theorem 3.1] and [3,Lemma 2], with minor modifications to handle the uniformity on the unbounded set |x| ≤ τ −ν as claimed.
First off, the jump matrix J Ψτ is C ∞ on Σ, in particular Hölder continuous, implying that Ψ τ extends continuously to its boundary values Ψ τ,± . Accounting also for the behavior of Ψ τ at ∞ and combining with the maximum principle, where the constant M τ is finite. For a point x ∈ Σ \ {0} and ε > 0, we consider the arcs C ± ε (x) of the disk centered at x and radius ε which are on the ± side of Σ. We then set , with the orientation induced from Σ. We deform contour in the integral representation (9.4) and then send z → x, obtaining that From standard estimates and using (7.16), the just written equation yields and therefore Lemmas 7.4, 7.5 and 7.6 provide L p estimates for J Ψτ − I along Σ. Exploring that Σ ± is obtained from Σ after a small deformation around the point x = O(τ −ν ), it is straightforward to see that the same estimates hold in Σ ± , which can be summarized as for some constants η, M > 0 which may depend on s 0 , t 0 , τ 0 but are independent of s ≥ s 0 , t ∈ [t 0 , 1/t 0 ], τ ≥ τ 0 , see (7.9), (7.10) and (7.12). Combining with (7.18), we conclude in particular that M τ ≤ 2, for s, t, τ in the same range of values. Having in mind (7.4), this bound on M τ applied to (7.17) is enough to ensure that Ψ τ,± − I = O(τ −ν ), but to obtain the decay in x for the error claimed in (7.3) a little more care is needed as follows. First off, we split the integral in (7.17) into two, namely along For the integral over J x , we estimate as where we used (7.19), the fact that x = O(τ −κ ) and again the bound M τ ≤ 2. Observing that |ζ −x| ≥ ε along the remaining piece, a similar argument yields and again from (7.19) we see that the right-hand side above decays exponentially in x when x → ∞ and is O(τ −κ ) when τ → ∞. From (7.17) we thus obtain uniformly for x ∈ Σ, |x| ≤ τ −ν , and uniformly for s ≥ −s 0 , t 0 ≤ t ≤ 1/t 0 . In virtue of (7.4), this proves (7.3).
Remark 7.8. For admissible functions h τ , the asymptotics (4.3) as ζ → ∞ of Φ τ is valid uniformly in τ, t and s, in the sense that for any t 0 ∈ (0, 1) and any s 0 > 0, there exist K > 0 and R > 0 such that and we emphasize that K and R are independent of s, t. To see that this is true, in virtue of (7.4) it is enough to show that the asymptotics (5.10) and (7.6) are uniform in the same sense, we indicate the proof for the latter and the former is analogous.
To finish this section, it remains to prove Theorem 7.2, and for that end we first establish a lemma.
Proof. We prove (7.21) which is slightly more technical because the integrand depends on τ , the estimate (7.20) follows in a similar manner.
Having in mind Remark 7.8, we use the expansion (4.3) to estimate Observe that the factor χ (−∞,0) (x) e ) is bounded, thanks to the facts that x 3/2 + ∈ iR and Re H > 0 for x < 0 (see Definition 4.1-(iii)). The identity above can be differentiated, and using it we obtain the crude bound which is non-optimal in x, but will be enough for the coming estimates, and which is valid uniformly for u ≥ −s 0 , t ∈ [t 0 , 1/t 0 ] as τ → ∞. Thus, to conclude the result it is enough to estimate each of the integrals for some η > 0. In a similar manner, we also obtain that where now for the last equality we used Definition 4.1-(iii) along Σ 0 = (0, ∞).

The underlying equilibrium measure and related quantities
As we mentioned earlier, one of the main objects we will analyze is the RHP for the orthogonal polynomials associated to (2.3)-(2.4). In this analysis, the model problem we discussed in the last few sections plays a central role, and another important quantity is the associated equilibrium measure and related objects that we discuss in this section.

The equilibrium measure.
A major role in our calculations is played by the equilibrium measure for the polynomial potential V , which is the unique probability measure µ V on R for which the quantity attains its minimum over all Borel probability measures µ supported on R. Its existence and uniqueness is assured by standard results, see for instance [56], and its regularity that we now discuss is of particular relevance. The measure µ V is supported on a finite union of bounded intervals and is absolutely continuous with respect to the Lebesgue measure [36]. Following Assumption 2.1-(i), we assume that µ V is one-cut, that is, it is supported on a single interval that we take to be of the form supp µ V = [−a, 0], a > 0, and regular, meaning that the density of µ V vanishes as a square-root at a neighborhood of the endpoints, does not vanish on (−a, 0), and the Euler-Lagrange equations are valid with strict inequality outside the support, for some constant V ∈ R. The notions just introduced are consistent with the notions and notations that we already introduced and used in Section 2. A transformation of the equilibrium measure of particular interest is its Cauchy transform, Using the Euler-Lagrange identity, it can be shown that C µ V satisfies an algebraic equation of the form for some polynomial h V which does not vanish on [−a, 0] [50,52].
We also associate to the equilibrium measure its φ function The next result summarizes some properties of φ that will be needed later.
where V is as in (8.1). (vi) The function φ satisfies the estimate Proof. The proof is standard using the properties of the equilibrium measure, see for instance [33].

The conformal map.
Finally, using φ we construct a conformal map ψ, introduced formally with the next result.
Proposition 8.2. The function , is a conformal map from a neighborhood of the origin to a disk D 2δ (0), with U 0 . . = ψ −1 (D δ (0)), and admits an expansion of the form Proof. The proof is also standard, and follows essentially from Proposition 8.1-(iv). We omit the details.
In the previous proposition, the factor 2δ instead of δ is chosen just for later convenience, for the statement of Proposition 8.3. Later on, we use ψ only over the smaller neighborhood D δ (0).
As it is customary in RHP analysis, at a later stage we will need to glue the model problem as a local parametrix for the original RHP for orthogonal polynomials. This gluing procedure is done, in our case, using the conformal map ψ. In usual situations, the jump matrices of the model local problem are piecewise constant or yet homogeneous, and as such this procedure of using the conformal map does not significantly alter them. However, in our situation the jump involves the function Q, and consequently the jump will be altered by the conformal map in a nontrivial way.
With the next result we introduce the necessary quantities needed to keep track of this transformation. Recall the half rays Σ j , j = 0, 1, 2, 3, which were introduced in (4.1). For the next statement, we talk about neighborhoods of Σ j , by which we mean open connected sets that contain Σ j \ {0} in their interior. (ii) The function H Q is C ∞ on S, and is an extension of Q • ψ −1 from D δ (0), that is, The function H Q is analytic on D δ (0), extends continuously up to the boundary of D δ (0) and satisfies uniformly for |w| ≤ δ, where we recall that t and c V are as in (2.7) and (8.5), respectively. (iv) For some constants η > η > 0, the function H Q satisfies the estimates − η|w| ≤ Re H Q (w) ≤ −η|w| for every w ∈ S 0 \ D δ (0),
Proof. We construct the set S j as tubular neighborhoods of Σ j away from the origin, and as disks near the origin, namely By choosing δ > 0 sufficiently small, in particular smaller than δ > 0, property (i) is immediate. The function is obviously analytic on D δ (0), satisfies (iii) and admits an extension to the larger open set D 2δ (0). A standard argument using partitions of unity allows us to extend it to the sets S j 's as claimed by (ii), also making sure that it satisfies (iv).

Associated orthogonal polynomials
The first and arguably major step towards understanding L Q n (s) is to study several quantities related to the orthogonal polynomials for the varying weight (2.17), as we introduce next.

Orthogonal polynomials and related quantities.
Denote by P k = P These polynomials depend on ω n , so ultimately also on Q, but we refrain from stressing this dependence in the notation. We also denote by γ We associate to the orthogonal polynomials their Christoffel-Darboux kernel, stressing that we are not including the weight ω n in this definition. In particular, the identity holds true for any s ∈ R and follows immediately from (9.2). In a similar manner, we introduce the related quantities for the undeformed weight e −nV . The partition function Z n already appeared in (2.2), the orthogonal polynomials P k = P The orthogonal polynomials P (n,s) k vary continuously with s, which is a consequence of Heine's formula [33,Equation (3.10)]. In particular, when taking the limit s → +∞ we have that x k ω n (x) → x k e −nV both uniformly in compacts and also in L 1 , and |x| k ω n (x) ≤ |x| k e −nV (x) . Thus, dominated convergence then gives that all the just introduced undeformed quantities are recovered from their deformed versions in the limit s → +∞. This means, for instance, that the Christoffel-Darboux kernel K n and the partition function Z n are recovered via K n (x, y) = K n (x, y | s = +∞) and Z n = Z n (s = +∞). (9.5) The next result will be key into transforming asymptotics for the orthogonal polynomials to asymptotics for L Q n (s) itself.
Proposition 9.1. The identity holds true for every s ∈ R.
Remark 9.2. While we were finishing this manuscript, the work [29] was posted to the ArXiv. Therein, they also derive the formula (9.6) in more general terms, using the underlying RHP for IIKS-type integrable operators, see the first displayed formula in page 28 therein. Similar formulas play a fundamental role in the recent works [24,25], see for instance (5.6) above. Our proof of (9.6) relies solely on orthogonality properties, so we decided to present it nevertheless.
Proof. The equality is standard in random matrix theory. From this identity, (9.2) and the orthogonality relations we derive the deformation formula which in fact is valid for general weights depending on an additional parameter s. To our knowledge, this last identity was first observed by Krasovsky [47,Equation (14)]. We fix constants L > s > 0 and integrate the identity above, We want to interchange the order of integration in the above. The derivative ∂ s K Q n is a polynomial of degree at most 2n − 2 in x, and from Heine's formula for orthogonal polynomials we see that the polynomial coefficients of K n are continuous functions of s. Therefore, for given s, L there exists a constant M = M (s, L) > 0 for which the pointwise bound is valid for every u ∈ [s, L]. Together with the inequalities 0 ≤ ω n (x) ≤ e −nV (x) , this bound ensures that we can interchange order of integration in (9.7). After integration by parts, we then obtain From the identity (9.4) the two single integrals cancel one another. The integrand of the double integral is positive, so by Tonelli's Theorem we can interchange order of integration. After this interchange, we take the limit L → +∞ and use (9.5) and (2.3) to conclude the proof.

The Riemann-Hilbert Problem for orthogonal polynomials.
We are ready to introduce the RHP for orthogonal polynomials for the weight ω n in (2.17). During this section, we keep using the matrix notation that was already used in previous sections, recall for instance (3.1),(3.2) and (3.3).
The RHP for orthogonal polynomials for the weight (2.17) asks for finding a 2 × 2 matrix-valued function Y with the following properties.

Y-2. The function Y has continuous boundary values
Observe that Y = Y (n) (· | s, Q) depends on the index n and also on s and Q, although we do not make this dependence explicit in our notation. As shown by Fokas, Its and Kitaev [42], for each n the RHP above has a unique solution, which is explicitly given by where γ (n,Q) k (s) and P (n,s) k are as in (9.1) and (9.2). In particular, from this identity we obtain the relation where Y (n,1) = Y (n,1) (s, Q) is the matrix determined from the more detailed expansion Also, the Christoffel-Darboux kernel (9.3) can be recast directly from Y from the identity x, y ∈ R, x = y. (9.10) In the confluent limit x = y, this formula yields The remainder of this paper is dedicated to applying the Deift-Zhou method for this RHP and collecting its consequences, analysis which will ultimately lead to the proofs of our main results.

The RHP analysis for the orthogonal polynomials
With all the preliminary work completed, we are finally at the stage of performing the asymptotic analysis for the RHP-Y for orthogonal polynomials that was introduced in Section 9. Most of the transformations are standard, so we go over them quickly and without much detail. Care will be taken in the construction of the parametrices, which are the steps where the introduction of the factor σ n plays a major role. We also remind the reader that the function V and Q are always assumed to satisfy Assumptions 2.1.
The function σ n = σ n (z | s, Q) depends on s ∈ R and Q, and as such in all the steps below several quantities will also depend on these parameters. Nevertheless, in most of the work that follows the parameter s and the function Q do not play a major role so we omit them in our notations unless when needed to avoid confusion.

First transformation: normalization at infinity.
Recall the function φ introduced in (8.3). The first transformation, which has the effect of normalizing the RHP as z → ∞, takes the form From the RHP for Y and the properties from Proposition 8.1, we obtain that T satisfies the following RHP.
T-3. As z → ∞, where the coefficient T 1 is and we recall that Y (n,1) and φ ∞ were introduced in (9.9) and in Proposition 8.1-(v), respectively.

Second transformation: opening of lenses.
From the identities just written for J T and Proposition 8.1-(ii), it follows that the diagonal entries of J T are highly oscillatory on (−a, 0) as n → ∞. In the second transformation of the RHP we perform the so-called opening of lenses, which has the effect of moving this oscillatory behavior to a region where it becomes exponentially decaying.
Define regions G ± on the ±-side of (−a, 0) (the lenses, see Figure 3), assuming in addition that for U 0 as in Proposition 8.2 these regions satisfy which can always be achieved because ψ : is conformal from a neighborhood of U 0 to a neighborhood of D δ (0). The function σ n has no zeros and may have singularities, but these are all poles due to the analyticity of Q in a neighborhood of the real axis. Therefore, the fraction 1/σ n is analytic on a neighborhood of the real axis. We use this fraction to transform With Γ S . . = R ∪ ∂G + ∪ ∂G − , and using the jump properties of φ listed in Proposition 8.1, the matrix S satisfies the following RHP.

S-3.
As z → ∞, S-4. The matrix S remains bounded near the points z = −a, 0. Before moving to the construction of the mentioned parametrices, we conclude this section with the needed estimate for the jump matrix J S away from [−a, 0]. For that, recall the matrix norm notation introduced in (6.1),(6.2),(6.3).
For some ε > 0, and possibly reducing U 0 if necessary, there is an η > 0 such that as n → ∞.
Proof. From Proposition 8.1-(iv) we obtain that for any ε > 0 there is a constant η > 0 for which On the other hand, the jump conditions on Proposition 8.1-(ii) combined with Cauchy-Riemann equations imply in a standard way that Re φ ≤ −η along the lipses of the lenses and away from the endpoints −a and 0, as long as the lens stay within a positive but small distance from the interval [−a, 0]. From these pointwise estimates, the growth of Re φ as z → ±∞ and the fact that σ n remains bounded on R and 1/σ n grows at most with O(e cn 2/3 ) on compacts of C, the claimed L p estimates follow in a standard manner.

Global parametrix.
The global parametrix problem, obtained after neglecting the jumps of S that are exponentially close to the identity, is the following RHP. For z ∈ (−a, 0), it satisfies the jump G-4. G has square-integrable singularities at z = −a, 0.
The construction of the global parametrix follows standard techniques. First, one introduces a function that we denote q(z), with the aim at transforming the RHP for G to a RHP with constant jumps. Then, by diagonalizing the resulting jump matrix, we further reduce the problem to two scalarvalued RHPs. With the help of Plemelj's formula, we then solve these scalar RHPs, and by tracing back all the transformations we recover the matrix G itself.
The procedure just described is standard in RHP literature, see for instance [7, Appendix A.1], so we refrain from completing it in detail and instead only describe the final form of the solution.
The function σ n does not vanish and is real and positive over the real axis, so its real logarithm over the real axis is well defined. With this in mind, introduce where (·) 1/2 stands for the principal branch of the square root and √ · is reserved for the standard positive real root of positive real numbers. This function q depends on n but we do not make this dependence explicit for ease of notation. It is analytic on C \ [−a, 0], and it is chosen to satisfy the jump condition Furthermore, standard calculations show that with coefficients given by Next, set 4) which is consistent with (4.4), and introduce where σ 2 is the second Pauli matrix (recall (3.2)).
Then the solution to the global parametrix RHP-G is This solution G satisfies Recall that U 0 denotes the neighborhood of the origin given in Proposition 8.2. We will also need some control on q inside U 0 .
For the next result, set (10.8) which are n-independent quantities. The index β does not have any specific meaning for what comes later, but it arises naturally from the asymptotic analysis resulting in the following result.  Proof. The estimate for q 0 (n) follows immediately from an application of Proposition A.2. The estimates for q(z) and q (z) also follow from Proposition A.2, once we observe that the integrals defining them can be slightly deformed to the upper/lower half plane in a neighborhood of the unique point in the intersection ∂U 0 ∩ (−a, 0).
Finally, using the first part of the statement and the fact that M is bounded for z ∈ ∂U 0 and independent of n, we expand the exponentials in series and write and the last claim follows after rearranging the terms in this expansion.

Local Parametrix near −a.
The local parametrix P = P (a) near z = −a is constructed in a neighborhood of z = −a which without loss of generality can be taken to be the disk D δ (−a) of radius δ around a, and it is the solution to the following RHP. P (a) -1. The matrix P (a) : D δ (−a) \ Γ S → C 2×2 is analytic. P (a) -2. For z ∈ Γ S ∩ ∂D δ (−a), it satisfies the jump P The asymptotic condition P (a) -3. above will be improved to (10.13) below. From the conditions on Q, we know that there exists a value η > 0 for which This value is uniform for t ∈ [t 0 , 1/t 0 ], for any t 0 ∈ (0, 1) fixed, and it is independent of s ∈ R. In particular, once we fix s 0 > 0 and assume that s ≥ −s 0 , from this inequality we obtain | e −s−n 2/3 Q(z) | ≤ e −n 2/3 η , |z + a| < δ, for large enough n.
This way, for n > 0 sufficiently large the function σ n admits an analytic continuation to the whole disk D δ (−a), and this continuation does not have zeros on the same disk. Thus, a branch of log σ n is well defined in a neighborhood of z = −a, and the just mentioned estimate also shows that log σ n (z) = O(e −ηn 2/3 ), n → ∞, (10.9) uniformly for z in a neighborhood of z = −a and s ≥ −s 0 . With this in mind, the parametrix P (a) can be constructed explicitly out of Airy functions in a standard way, see for instance [33,Section 7.6]. Since it involves a somewhat nonstandard matching analytic prefactor that accounts for σ n , we briefly go over this construction.
Recall the contour Σ introduced in (4.1). With appropriate Airy functions, we construct a 2 × 2 matrix Ψ Ai , which is analytic on C \ Σ and satisfies In fact, Ψ Ai can be obtained with a modification of the matrix Φ Ai which we previously used in (6.4). We will not need its explicit form, so we do not write it down explicitly.
Using the properties of φ we construct a conformal map ϕ from a neighborhood of −a to a neighborhood of the origin, with ϕ(−a) = 0, ϕ (−a) < 0 and 2 3 With standard arguments (see for instance the proof of Proposition 10.5 below for similar arguments), one shows that the matrix is analytic on a neighborhood of z = −a. The local parametrix then takes the form As a result, the error term in fact takes on the stronger form which is valid uniformly for z ∈ ∂D δ (−a) and uniformly for s ≥ −s 0 and t ∈ [t 0 , 1/t 0 ], for any s 0 > 0 and t 0 ∈ (0, 1) fixed.

Local Parametrix near the origin.
The local parametrix near the origin requires the model problem from Section 4. Recall the neighborhood U 0 of the origin introduced in Proposition 8.2. The initial local parametrix we seek for should be the solution to the following RHP. P (0) -1. The matrix P (0) : U 0 \ Γ S → C 2×2 is analytic. P (0) -2. For z ∈ Γ S ∩ U 0 , it satisfies the jump P To construct the solution P (0) required above, some work is needed. Aiming at removing φ from the jump of P (0) , we change this RHP-P with the transformation (10.14) Then the matrix L, should it exist, must satisfy the following RHP. , 0), Based on the usual way of matching the local parametrix with a model problem, one is tempted to moving the non-constant part of the jump -namely σ n -to the behavior at ∂U 0 as well. This would be done so including a term of the form σ σ 3 /2 n = e σ 3 log σn/2 into the transformation P → L, in much the same way we did in (10.12). However, as we discussed in Section 3.1, for any s ∈ R fixed there are poles of σ n accumulating too fast near the origin, so σ n fails to be analytic in any small neighborhood of the origin and we have to stick to the non-constant jumps as above.
The RHP-L has a solution if, and only if, RHP-P (0) has a solution. Such solutions need not be unique, as one could possibly improve on the asymptotic matching conditions on ∂U 0 . The goal of the rest of this section is to describe a solution L, and consequently a solution P (0) related by (10.14), with a more explicit control of the error term in RHP-L-3. For that, we use the model problem thoroughly studied in Sections 4 and 7.
The construction that follows needs several quantities that appeared before. These are the conformal map ψ appearing in Proposition 8.2, the function H Q introduced with the help of Proposition 8.3, the model RHP solution Φ = Φ(· | h) introduced in Section 4 and further discussed in Section 7, and the constant q 0 and matrices U 0 and M(z) from (10.3)-(10.5). With all these quantities at hand, we set With the identification τ = n 2/3 and thanks to Proposition 8.3, the function h n becomes admissible in the sense of Definition 4.1, so the notation for the corresponding solution Φ n = Φ(· | h n ) chosen above is consistent with the solution Φ τ (ζ) τ =n 2/3 in (4.5). For later reference, we keep track of the expansion Proposition 10.3. Fix s 0 > 0 and t 0 ∈ (0, 1). There exists n 0 = n 0 (s 0 , t 0 ) for which for any s ≥ −s 0 and any t ∈ [t 0 , 1/t 0 ] the matrix Φ n exists for every n ≥ n 0 . This matrix satisfies the jump Furthermore, for the matrix the asymptotic expansion Φ n (z) = I + 1 holds true uniformly for z ∈ ∂U 0 and uniformly for s ≥ −s 0 and t ∈ [t 0 , 1/t 0 ].
Proof. The existence of Φ n is granted by the first claim of Theorem 7.1. The jumps for Φ n,+ follow from the jumps in (4.2), the definition of h n taken in (10.16), the correspondence (8.6) and the conformality of the change of variables ζ = n 2/3 ψ(z). The asymptotic expansion for Φ n is immediate from (4. 3) The introduction of the additional notation Φ n , which plays the role of the local parametrix in the variable ζ, is convenient for later calculations. At that moment, some of its properties will be needed, and we keep track of these properties with the next result. For the formal statement, we recall that Φ(ξ | S, T) is the solution to the integro-differential PII that already appeared in (2.11) and (5.15), and Φ (1) 0 is the residue matrix from (5.17) that collects the functions P(S, T) and Q(S, T) which, in turn, are related to Φ as explained in (5.13) et seq.. holds true uniformly for ζ ∈ Σ with |ζ| ≤ n Proof. It is immediate from the identification τ = n 2/3 , Equations (10.16) and Theorem 7.1.
Next, we now verify that L given as in (10.16) indeed solves the RHP-L.
Proof. First we prove that F is analytic. For that, notice that a jump for it may come only from the factors M and ψ 1/4 , and therefore only possibly in the interval (−a, 0)∩U 0 . However, along this interval it is simple to compute that (ψ 1/4 ) ± = e ±πi/4 |ψ| 1/4 , and also that so F is indeed analytic across (−a, 0) ∩ U 0 . In principle, F may have an isolated singularity at 0, but because M(z), ϕ(z) 1/4 = O(z 1/4 ) as z → 0, we see that z = 0 is a removable singularity. In virtue of the definition of L in (10.16) and the jump for Φ n from Proposition 10.3, the analyticity of F is enough to conclude that L satisfies RHP.L-1.
Knowing that F is analytic, the jump for L is precisely the same as the jump for Φ n , and by Proposition 10.3 we thus have that L satisfies RHP.L-2. Finally, to the asymptotic condition (10.20). For that, we use the asymptotic condition for Φ n given by Proposition 10.3 and the definition of F and write , n → ∞, and where the error is uniform for z ∈ ∂U 0 and s, t as claimed. Since ∂U 0 remains within a positive distance from the unique zero z = 0 of ψ, the function |ψ 1/4 | remains bounded from below away from zero, so the corresponding conjugation of the error by the term ψ σ 3 /4 does not change the order of the error. Next, the conjugation by n σ 3 /6 contributes at most to an error of order n 1/3 , and only in the (2, 1)-entry. The remaining term M is bounded along ∂U 0 , so it can be commuted with the error term above without changing its order, leading to (10.21). And (10.21) is indeed an improvement of the asymptotic condition RHP.L-3, because from Lemma 10.2 we know that M = (I + O(n −1/3 ))G uniformly as claimed.
We now trace back the transformation L → P (0) and construct the latter as keeping track that Φ n and F n are introduced in (10.16). With this construction and thanks to Lemma 10.2 and (10.21), the matching condition RHP.P (0) -3 is The final transformation combines the local and global parametrices to remove the non-decaying jumps from S.
orienting ∂U 0 and ∂D δ (−a) in the clockwise direction. With P (a) being the local parametrix near −a, P (0) the local parametrix near the origin and G the global parametrix, we introduce a last parametrix P with unified notation as The final transformation S → R is then With this transformation, the jumps that S has in common with the parametrices G and P get canceled, and we remain with jumps only away from [−a, 0] and U . With the matrix R satisfies the following RHP. R-1. The matrix R : C \ Γ R → C 2×2 is analytic.
To conclude that R is asymptotically close to the identity, we control its jumps. We use the matrix norm notation introduced in (6.1)-(6.3).
Proposition 10.6. Fix s 0 > 0 and t 0 ∈ (0, 1). There is η > 0 for which the jump matrix for R satisfies the estimates Proof. The function G remains uniformly bounded away from the interval [−a, 0], and the first claimed decay then follows from (10.24) and Proposition 10.1.
Using (10.13) and (10.24), we obtain the uniform estimate of the jump along ∂D δ (−a) directly. Since ∂D δ (−a) is bounded, the L 1 and L 2 estimates also follow. Finally, estimate for the jump along ∂U 0 follows similarly, once we recall (10.23).
With Proposition 10.6 at hand, the small norm theory of Riemann-Hilbert problems yields  (10.26) where the error terms are all uniform for z, w on the same connected components of C \ Γ R , and they are also uniform for s ≥ −s 0 and t 0 ≤ t ≤ 1/t 0 .
Proof. The arguments are standard, so we only sketch them. The small norm theory for RHPs ensures the representation This ensures the estimates for R and R away from Γ R and, because the jump matrix is analytic on a neighborhood of Γ R , we are able to extend these estimates also to Γ R . To obtain the estimate involving z and w, we then write, with the help of Cauchy's formula, ds, (10.28) where the integral is over a contour on C \ Γ R encircling both w and z. The right-hand side is now I + O((w − z)n −1/3 ) by the estimate on R already proven.

P. GHOSAL AND G. SILVA
Finally, the estimate for R 1 follows expanding (10.27) as z → ∞, and then using that the jump matrix decays to the identity at least as O(n −1 ) away from ∂U 0 .
Later on, we also need the first term in the asymptotic expansion for R, we state it as a separate result.

29)
the matrix R has the expansion where the error term is uniform for z ∈ U 0 , and it is also uniform for s ≥ −s 0 and t 0 ≤ t ≤ 1/t 0 .
Proof. Starting from the representation (10.27), we write and the result follows combining (10.24) and (10.23) and performing straightforward calculations.
All the transformations Y → T → S → R involve only analytic factors in their construction. As such, we can actually slightly deform the contour Γ R in all these steps, so that in fact the estimates in the previous result are valid everywhere on C, interpreting them as with boundary values when z, w ∈ Γ R . We will use this fact without further warning.
With this theorem at hand, in the next sections we recover the needed asymptotic formulas for the proof of our main results.

Proof of main results
The next step in the direction of concluding our asymptotic analysis is to unravel the transformations which are related by Then the result of the unfolding of the transformations is We split the proofs of our results in the next few sections.

Proof of Theorem 2.4.
Thanks to (11.1) and Theorem 10.7, the expression (9.10) reduces to the asymptotic formula 2) valid as n → ∞, x, y ∈ R and uniformly for s, t as in Theorem 10.7.
For the value c V > 0 introduced in (8.5), we scale x = u n and y = v n as in (2.19), where u, v are on any given compact of the real axis. For such values of u, for n large enough the points u n are always on the neighborhood U 0 of the origin where P = P (0) , and from (10.22) we simplify and P + Λ n e 1 = F n Φ n,+ ∆ n e 1 e nφ + .
where all the quantities above are evaluated at u n . The next step is to plug these identities into (11.2), when doing so we also use the estimates which follows from (10.16), (10.4) and (8.5), and its immediate consequence which is obtained with arguments similar to (10.28). These estimates are uniform for u, v in compacts of R, and are also uniform for s ≥ −s 0 and t ∈ [t 0 , 1/t 0 ]. Equation (11.2) then simplifies to e −n(V (un)+V (vn))/2 K Q n (u n , v n ) = for any ν ∈ (0, 2/3), with the error being valid uniformly for u in compacts of R, and also uniformly for s ≥ −s 0 , t 0 ≤ t ≤ 1/t 0 . Also, thanks to Theorem 5.1 we know that Φ 0,+ (u | s, c H ) ±1 remain bounded for u in compacts and s ≥ −s 0 , t 0 ≤ t ≤ 1/t 0 . Therefore, as n → ∞. In addition, the estimate 1 is valid uniformly for u in compacts, uniformly for s ∈ R and t 0 ≤ t ≤ 1/t 0 , and is immediate from (2.17). Combining everything, and denoting χ + = χ [0,∞) , we obtained the asymptotic formula e −n(V (un)+V (vn))/2 which is valid uniformly for u, v in compacts of R, and also uniformly for s ≥ −s 0 and t 0 ≤ t ≤ 1/t 0 .
With ∆ 0 as in (5.7), this identity rewrites as and the proof of Theorem 2.4 is now completed using Equation (5.16).
11.2. Proof of Theorem 2.5. Now that the RHP asymptotic analysis is completed, the proof of Theorem 2.5 follows standard steps. In our case, there is an additional cancellation that has to be accounted for at a later step, in virtue of the presence of the factor q in both the global and local parametrices, see (10.6) and (10.23). So we opt for presenting the detailed calculation.
Starting from the representation (9.8), we unravel the transformations and obtain 4) with R 1 as in (10.25), and where for the second identity we used the definition of G 1 in (10.7) and the estimate for q 0 from Lemma 10.2.
It remains to estimate R 1 , which we do so starting from (10.26). Using Cauchy-Schwartz, Propositions 10.6 and Theorem 10.7, we write The matrix J R is in (10.24), and combining its explicit expression along ∂U 0 with (10.23) and (10.5), after a cumbersome but straightforward calculation we arrive at where we recall that m, q (1) and L 1 are given in (10.4), (10.8) and (10.20), respectively. Using this explicit expression for L 1 , we see that Both functions m and ψ are analytic in a neighborhood of the origin, and vanish linearly therein. Combining in addition with (8.5), we see that the product m(z) 1/2 ψ(z) 1/2 admits an analytic continuation near the origin, with the expansion Thus, computing residues .

UNIVERSALITY OF HERMITIAN RANDOM MATRICES AND THE INT-DIFF PII 49
Similarly, using now (10.8) we obtain Hence, and (11.4) updates to γ (n,Q) uniformly for s ≥ −s 0 and t 0 ≤ t ≤ 1/t 0 . We now need only to apply (10.18) with any ν ≥ 1/3 to complete the proof.

Proof of Theorem 2.2.
Unlike the already proven major results, the proof of Theorem 2.2 is not a straightforward consequence of the steepest descent analysis concluded with Theorem 10.7. It does rely substantially on Theorem 10.7, but several other inputs are also needed along the way. Equipped with (9.10), the idea is to account for the different approximations for K Q n on the different components 0)), R ∪ U 0 , (−a, −δ, −a + δ) and (−a + δ, 0) \ U 0 that arise from the RHP, and integrate each such approximation. With Proposition 9.1 in mind, we are thus able to recover asymptotics for L Q n itself. As one would expect, it turns out that in this process the terms that arise away from U 0 become all exponentially negligible, and only the contribution from U 0 survives in the leading contribution. The contribution that arrives this way involves Φ n , and we further need to split it into different parts and still account for some exact cancellations to arrive at the leading asymptotic contribution. We postpone this analysis to the next section, where it is split into several different lemmas, and summarize the outcome with the next result. For its statement, we recall that h n and Φ n were introduced in (10.16), and we also set ∆ n (x) . . = I + (1 + e −hn(u) )χ (0,+∞)(x) E 21 , x ∈ R.
In words, Proposition 11.1 is saying that the major contribution to L Q n comes from the neighborhood U 0 , so from the local parametrix Φ n . But according to the developments from the previous sections, this local parametrix is close to the id-PII parametrix Φ 0 , and we are ready to conclude our last major result.
Proof of Theorem 2.2. We combine Propositions 9.1 and (11.1) to obtain valid as n → ∞ and uniformly for s ≥ −s 0 and t 0 ≤ t ≤ 1/t 0 . Next, with the identifications (10.16)- (10.17) in mind, we estimate the integral on the right-hand side with the help of Theorem 7.2 and conclude the proof.

Technical Lemmas
It remains to prove Proposition 11.1, analysis which we split into several technical lemmas in this section.
Our starting point is the integral representation for L Q n from Proposition 9.1 with the asymptotic information for K Q n provided by the RHP analysis. For x ∈ R, set Recalling (9.11), the unwrap of the transformations of the RHP yields the identity We now need to integrate each of the terms on the right-hand side above, first in x ∈ R and then in u ∈ (s, +∞). Each term will be analyzed individually, also depending on whether we integrate x in the bulk, each of the edges or away from the support [−a, 0] of the equilibrium measure. To simplify notation, it is convenient to introduce the additional notation for each relevant integral, and for an arbitrary set J ⊂ R denote and which are functions of u ∈ R as well. With ε 0 , ε 1 > 0 being determined by (−ε 0 , ε 1 ) = U 0 ∩ R, the split is immediate, and with the next series of lemmas we estimate each of the terms on the right-hand side.
On the interval (a + δ, −ε 0 ) the nontrivial entry of Λ n is e 2nφ + /σ n and, because Q < 0 and φ + is purely imaginary in this interval, this quotient is bounded. Also, away from the endpoints −a and 0 we have P ≡ G. Both P and G, and their x-derivatives, decay as x → ±∞ and remain bounded as n → ∞, all uniformly in u and t as claimed. All of these facts combined together, we obtain that for some constants K > 0 and n 0 ≥ 1 Next, we now use that Q ≥ 0 on the interval (−∞, 0) to bound which is valid for any n ≥ 1, u ∈ R, t > 0. Thus, combining everything we obtain and using Proposition 8.1-(iv),(v), this proves the bound for I 1 (−∞, −a − δ).
For the second integral, we term φ + is oscillatory, so to obtain the exponential decay in the x-integral we have to argue differently and as follows. The function is strictly increasing on (−∞, 0) and strictly decreasing on (0, +∞). Because Q > 0 on (−∞, 0) and Q(0) = 0, by reducing U 0 if necessary we can assume without loss of generality that Q(x) ≥ Q(−ε 0 ) for every x ∈ [−a, −ε 0 ]. This way, v = u + n 2/3 Q(x) ≥ v 0 . . = u + n 2/3 Q(−ε 0 ) and, because u is assumed to be bounded from below, we can make sure that v 0 > 0 for every u. Therefore and upon integration and using Proposition 8.1-(ii), we obtain for every n ≥ n 0 and t, u as claimed.
For the estimate for I 2 (−a, −ε 0 ) we use again the last inequality in (12.6) and also that both φ + and Q are continuous, and hence bounded, on (−a, −ε 0 ). This concludes the proof.
For the integral over (ε 1 , +∞), it is easier to actually estimate its u-integral directly.  Proof. With the bound (12.5) we see that it is enough to estimate the integral where for the equality we used Tonelli's Theorem to interchange the order of integration, and then integrated exactly. The term (1 + e s+n 2/3 Q(x) ) −1 is bounded by 1, and using Proposition 8.1-(iv), (v) we see that the integral of e −2nφ is O(e −ηn ) for some η > 0 independent of s and t, which concludes the proof.
Next, we analyze the contribution coming from a neighborhood of the endpoint z = a.
Proof. On the neighborhood D δ (−a) the function P = P (a) is the Airy local parametrix (10.12), which involves the function Φ Ai evaluated at the argument ζ = n 2/3 ϕ(z). In the ζ-plane, we fix R > 0 for which the asymptotic expansion (10.10) is valid for |ζ| > R, and split the analysis into two cases, namely for |n 2/3 ϕ(z)| ≤ R and |n 2/3 ϕ(z)| ≥ R, and for and now analyze each integral on the right-hand side separately. For |n 2/3 ϕ(z)| ≤ R, the terms Φ Ai,+ (ζ = n 2/3 ϕ(z)) and Φ Ai,+ (ζ = n 2/3 ϕ(z)) consist of continuous functions evaluated inside the compact interval [−R, R], and therefore they are bounded. By the same reason, the expression (10.11) shows that F is bounded for |n 2/3 ϕ(z)| ≤ R. On the other hand, without further analysis we obtain that F may grow at most as O(n 1/6 ). Finally, combining with (10.9) and the fact that the determinant of P is identically 1, we conclude that for |n 2/3 ϕ(x)| ≤ R, which is valid as n → ∞ and uniformly for x ∈ (−a − δ, −a + δ), also uniformly for u ≥ s and t 0 ≤ t ≤ 1/t 0 . We use (12.7) back into (12.1), and combined with the fact that |σ −1 n | ≤ 2 for x ≤ (−∞, 0) the result is that where the last error term is a scalar error, valid uniformly in x, u, t as before. Integrating, we obtain that for some absolute constant K > 0, where for the second inequality we used that 0 < σ n ≤ 1 and for the last estimate we used that Q is strictly positive on (−a − δ, −a + δ) ⊃ A n . Next, we consider the case |n 2/3 ϕ(z)| ≥ R. In such situation, the asymptotics (10.10) take place, and when combined with (10.11) they yield P + (x) = G + (x) e −σ 3 log σn(x)/2 I + O(n −1 ) e σ 3 log σn(x)/2 = G + (x) I + O(n −1 ) . (12.9) where for the last equality we used (10.9). The matrix P + is bounded as x → 0, whereas G is not, but the cancellation that leads to this boundedness of P is not captured by this asymptotics. Nevertheless, as we need some uniform control over x, we now account for the behavior of x → 0 in a rough manner as follows. We are assuming that |n 2/3 ϕ(x)| ≥ R, and because ϕ is conformal with ϕ(0) = 0 this means that |x + a| ≥ c/n 2/3 , for some fixed c > 0. The function G has a fourth-root singularity at x = 0, and therefore we arrive at the rough estimate P + (x) = O(n 1/6 ), n → ∞.
(12.10) The estimate (12.9) can be differentiated term by term. With arguments similar to the ones we just applied, we arrive at the rough estimate P + (x) = O(n 7/6 ), n → ∞. (12.11) These latter two estimates are valid uniformly for u ≥ s and t 0 ≤ t ≤ 1/t 0 as n → ∞. Finally, on the interval (−a, 0) the factor φ + is purely imaginary, implying that Λ ±1 n remains bounded therein. All combined, we obtained that uniformly for x ∈ B n and u, t as claimed. Proceeding as in (12.8) we obtain a bound for I 1 (B n ) and complete the proof.
It remains to analyze the two integrals I 1 (−ε 0 , ε 1 ) and I 2 (−ε 0 , 0). The hardest analysis is the integral I 1 which, as will turn out, contains both the leading contribution, a term to cancel I 2 and asymptotically negligible terms. For ease of presentation, we explore the expression for A in (12.1) as a sum and split (12.12) where, for any measurable set J ⊂ R, 1 + e u+n 2/3 Q(x) dx, (12.13) and analyze each of these terms separately. For the estimation of J 1 , it is also easier to perform the u-integral, just as we did in Lemma 12.2.
Proof. The idea is similar to the proof of Lemma 12.3. We fix a number R > 0 for which the asymptotic expansion (4.3) for Φ = Φ n is valid for |ζ| ≥ R, uniformly in t, s as claimed (see Remark 7.8). Then, introduce (12.15) We now find bounds for the integrands, with separate arguments for each of C n and D n .
Recalling (10.22), on the interval (−ε 0 , ε 1 ) the parametrix is P = P (0) = F n Φ n e nφσ 3 , with F n and Φ n as in (10.16). Using the definition of F n in (10.16) and Theorem 10.7, we express  A careful inspection on (10.29) shows that ( R 1 (z)) 12 is independent of z, so R has zero (1, 2)-entry. Therefore we conclude that F −1 n R −1 R F n = O(n −1/3 ), stressing that this is valid for x ∈ C n . Also, from (10.19) we learn that Φ n (x) = O(1) uniformly on C n . Everything we have so far combined yields that Along (−a, 0) we have Q > 0 so that 1/σ n remains bounded uniformly, and consequently ∆ n = O(1) in the same interval. Using this information in the last displayed equation, we thus obtain that uniformly for u ≥ s and t 0 ≤ t ≤ 1/t 0 , and where the error term is uniform for x ∈ C n . Next, along D n we now use the asymptotic expansion (4.3) for Φ = Φ n and the definition of F n and obtain that and similarly all valid as n → ∞, uniformly for x ∈ D n and uniformly in u, t as claimed With the same arguments that we applied in (12.16), we obtain along D n as well The factor φ + is purely imaginary on (−a, 0) and positive on [0, +∞), so the term e −2nφ + in the integrand above is bounded by a uniform constant. Combining with (12.17), we conclude Finally, we now integrate in u and use Tonelli's Theorem to interchange the order of integration, obtaining just like in the proof of Lemma 12.2 that which concludes the proof.
To conclude, it remains to show the bound for R. For that, we use again the sets C n and D n from (12.15) and analyze the integral over each of these sets separately.
Along C n , the convergence (10.19) ensures Φ n,+ remains bounded uniformly, and combined with the boundedness of all the other terms on the whole interval (−ε 0 , ε 1 ) we obtain dx.
The function ψ is conformal, and consequently |z| ≤ cn −2/3 for z ∈ C n and some constant c independent of t, u, n. In particular, this ensures that n 2/3 Q(x) ≤ −c for every x ∈ C n and a constantc > 0, and therefore σ n (x) 1 + e u+n 2/3 Q(x) ≤ 1 1 + e u−c ≤ e −u+c .
Finally, for the integral over (0, ε 1 ) we now have that φ ≥ 0 in this interval, and it is independent of u, so once again interchanging order of integration we obtain 0 ≤ and now changing variables v = nφ (which is well defined in this interval because of the local behavior (8.4)) we see that the integral on the right-most side is O(n −1 ). This completes the proof.
To conclude, it remains to prove Proposition 11.1.

Appendix A. Laplace-type integrals
For smooth enough functions f, g, consider the Laplace-type integral F (t) . . = a 0 g(x) log(1 + e −y−tf (x) )dx, a ∈ (0, +∞]. We are interested in the asymptotic behavior of F (t) as t → +∞ while y also scales with t. The first result we need is a version of the usual Watson's Lemma.
Lemma A.1. Fix f (x) ≡ x. Suppose also that g ∈ L 1 (0, a), and for some δ > 0 and κ ∈ [0, 1) it is of the form g(x) = g(x) x κ , 0 < x < δ, where g is C ∞ in a neighborhood of the origin. Assume that For y ≥ 0 and k ∈ Z, k ≥ 1, F k admits the alternative expression in terms of the polylog function Li s , F k (y) = −kΓ(k) Li 2+k (− e −y ).
Proof. The proof follows along the lines of the proof of the classical Watson's Lemma for Laplace transforms, see for instance [54], so we go over it without much detail. First off, we fix δ > 0 for which g admits a Taylor expansion of order N in a neighborhood of the interval (−δ, δ), and write for some η > 0, where the error term is obtained using that g ∈ L 1 (0, ∞) and that the function x → log(1 + e −y−tx ) is increasing. The value η > 0 is independent of y, and the error is uniform for y satisfying (A.1). For the integral on the right-hand side, we expand g in Taylor series and obtain Using (A.1) we see that the last integral is again O(e −ηt ) uniformly for y. Also, it is immediate that under the restriction (A.1) each coefficient F β (y) is bounded in y, and this concludes the proof of the asymptotic formula. The alternative representation of F k (y) in terms of polylogs follows by integration by parts.
Next, we move to more general exponents.
Proposition A.2. Suppose that f is C ∞ in a neighborhood of the origin, with a unique global minimum on [0, a] at x = 0 with f (0) = 0, f (0) > 0, and that g is as in Lemma A.1. In addition, suppose that y satisfies (A.1) for some M > 0 fixed. Then F (t) admits an expansion of the form F (t) = N k=0 1 t k+1−κ g (k) (0)F k−κ (y) + O(t −N −2+κ ), t → ∞, for any N ≥ 1, where the error term is uniform for y satisfying (A.1), F k 's are as in Lemma A.1 and the function g is C ∞ and satisfies the identity g(f (x)) = f (x) κ g(x), |x| sufficiently small.
Proof. Because f is assumed to be C ∞ and to have a unique global minimum at x = 0, by the Inverse Function Theorem it admits an inverse f −1 in a neighborhood of the origin. With the change of variables u = f (x), and setting u 0 . . = f (δ), g(x) . . = x κ g(f −1 (x)) (f −1 (x)) κ = x κ g(f −1 (x)), for δ > 0 sufficiently small the integral is turned into Proof. We prove the result for f (x) = x, the general case follows along the same lines of the proof of Proposition A.2.
Fix δ > 0 so that g is C ∞ on [0, δ]. By replacing g with gχ [0,a] we may assume that a = +∞. We can then bound ∞ δ g(u) log(1 + e −y−tu )du ≤ g L 1 (0,∞) e −ηt α −tδ = O(e −δt/2 ), as we are assuming that α < 1. Proceeding as in the proof of Proposition A.1, we obtain The integrals appearing in the sum and in the bound for the error above are of the same form, and now we study them. From (A.3) we see that by taking t large enough we can always assume that the point u t . . = −y/t where the exponent y + ut changes sign belongs to the interval (0, δ). We split the integrals on (0, δ) into the integrals over (0, u t ) and (u t , δ). For the first one, we then express it as   Using that y + tδ → +∞ and the bounds (A.3), it is straightforward to see that the first integral on the right-hand side is exponentially close to ϕ (2) k while the second integral is exponentially close to ϕ