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Alexander Its

Publications and source records attributed to Alexander Its.

At least 19 recordsLinked to original sources
arXiv2025

A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants II

In this article, we continue the development of the Riemann-Hilbert formalism for studying the asymptotics of Toeplitz+Hankel determinants with non-identical symbols, which we initiated in \cite{GI}. In \cite{GI}, we showed that the Riemann-Hilbert problem we formulated admits the Deift-Zhou nonlinear steepest descent analysis, but with a special restriction on the winding numbers of the associated symbols. In particular, the most natural case, namely zero winding numbers, is not allowed. A principal goal of this paper is to develop a framework that extends the asymptotic analysis of Toeplitz+Hankel determinants to a broader range of winding-number configurations. As an application, we consider the case in which the winding numbers of the Szeg\H{o}-type Toeplitz and Hankel symbols are zero and one, respectively, and compute the asymptotics of the norms of the corresponding system of orthogonal polynomials.

math-ph↗
arXiv2022

Harold Widom's work in random matrix theory

This is a survey of Harold Widom's work in random matrices. We start with his pioneering papers on the sine-kernel determinant, continue with his and Craig Tracy's groundbreaking results concerning the distribution functions of random matrix theory, touch on the remarkable universality of the Tracy-Widom distributions in mathematics and physics, and close with Tracy and Widom's remarkable work on the asymmetric simple exclusion process.

math-ph↗
arXiv2020

Asymptotics of bordered Toeplitz determinants and next-to-diagonal Ising correlations

We prove the analogue of the strong Szeg{\H o} limit theorem for a large class of bordered Toeplitz determinants. In particular, by applying our results to the formula of Au-Yang and Perk \cite{YP} for the next-to-diagonal correlations $\langle \sigma_{0,0}\sigma_{N-1,N} \rangle$ in the anisotropic square lattice Ising model, we rigorously justify that the next-to-diagonal long-range order is the same as the diagonal and horizontal ones in the low temperature regime. The anisotropy-dependence of the subleading term in the asymptotics of the next-to-diagonal correlations is also established. We use Riemann-Hilbert and operator theory techniques, independently and in parallel, to prove these results.

math-ph↗
arXiv2020

On $\beta=6$ Tracy-Widom distribution and the second Calogero-Painlev\'e system

The Calogero-Painlev\'e systems were introduced in 2001 by K. Takasaki as a natural generalization of the classical Painlev\'e equations to the case of the several Painlev\'e ``particles'' coupled via the Calogero type interactions. In 2014, I. Rumanov discovered the remarkable fact that a particular case of the second Calogero-Painlev\'e II equation describes the Tracy-Widom distribution function for the general beta-ensembles with even values of the parameter beta. Most recently, in 2017 work of M. Bertola, M. Cafasso, and V. Rubtsov, it was proven that all Calogero-Painlev\'e systems are Lax integrable, and hence their solutions admit a Riemann-Hilbert representation. This important observation has opened the door to rigorous, based on the Deift-Zhou nonlinear steepest descent method, asymptotic analysis of the Calogero-Painlev\'e equations. This in turn yields the possibility of rigorous evaluation of the asymptotic behavior of the Tracy-Widom distributions for the values of beta beyond the classical $\beta =1, 2, 4.$ In this work we shall start an asymptotic analysis of the Calogero-Painlev\'e system with a special focus on the Calogero-Painlev\'e system corresponding to $\beta = 6$ Tracy-Widom distribution function. The principle technical challenge is the implementation of the nonlinear steepest descent approach beyond the $2\times 2$ matrix dimension of the corresponding Riemann-Hilbert problem; in our case, it is $6\times 6$.

nlin.SI↗
arXiv2019

A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants

In this paper we will formulate $4\times4$ Riemann-Hilbert problems for Toeplitz+Hankel determinants and the associated system of orthogonal polynomials, when the Hankel symbol is supported on the unit circle and also when it is supported on an interval $[a,b]$, $0<a<b<1$. The distinguishing feature of this work is that in the formulation of the Riemann-Hilbert problem no specific relationship is assumed between the Toeplitz and Hankel symbols. We will develop nonlinear steepest descent methods for analysing these problems in the case where the symbols are smooth (i.e., in the absence of Fisher-Hartwig singularities) and admit an analytic continuation in a neighborhood of the unit circle (if the symbol's support is the unit circle). We will finally introduce a model problem and will present its solution requiring certain conditions on the ratio of Hankel and Toeplitz symbols. This in turn will allow us to find the asymptotics of the norms $h_n$ of the corresponding orthogonal polynomials and, in fact, the large $n$ asymptotics of the polynomials themselves. We will explain how this solvable case is related to the recent operator-theoretic approach in [Basor E., Ehrhardt T., in Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics, Oper. Theory Adv. Appl., Vol. 259, Birkh\"auser/Springer, Cham, 2017, 125-154, arXiv:1603.00506] to Toeplitz+Hankel determinants. At the end we will discuss the prospects of future work and outline several technical, as well as conceptual, issues which we are going to address next within the $4\times 4$ Riemann-Hilbert framework introduced in this paper.

math-ph↗
arXiv2018

A representation of joint moments of CUE characteristic polynomials in terms of Painleve functions

We establish a representation of the joint moments of the characteristic polynomial of a CUE random matrix and its derivative in terms of a solution of the sigma-Painleve V equation. The derivation involves the analysis of a formula for the joint moments in terms of a determinant of generalised Laguerre polynomials using the Riemann-Hilbert method. We use this connection with the sigma-Painleve V equation to derive explicit formulae for the joint moments and to show that in the large-matrix limit the joint moments are related to a solution of the sigma-Painleve III equation. Using the conformal block expansion of the tau-functions associated with the sigma-Painleve V and the sigma-Painleve III equations leads to general conjectures for the joint moments.

math-ph↗
arXiv2018

The sine process under the influence of a varying potential

We review the authors' recent work \cite{BDIK1,BDIK2,BDIK3} where we obtain the uniform large $s$ asymptotics for the Fredholm determinant $D(s,\gamma):=\det(I-\gamma K_s\upharpoonright_{L^2(-1,1)})$, $0\leq\gamma\leq 1$. The operator $K_s$ acts with kernel $K_s(x,y)=\sin(s(x-y))/(\pi(x-y))$ and $D(s,\gamma)$ appears for instance in Dyson's model \cite{Dyson2} of a Coulomb log-gas with varying external potential or in the bulk scaling analysis of the thinned GUE \cite{BP}.

math-ph↗
arXiv2017

On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo-Miwa-Ueno differential

Several distribution functions in the classical unitarily invariant matrix ensembles are prime examples of isomonodromic tau functions as introduced by Jimbo, Miwa and Ueno (JMU) in the early 1980s \cite{JMU}. Recent advances in the theory of tau functions \cite{ILP}, based on earlier works of B. Malgrange and M. Bertola, have allowed to extend the original Jimbo-Miwa-Ueno differential form to a 1-form closed on the full space of extended monodromy data of the underlying Lax pairs. This in turn has yielded a novel approach for the asymptotic evaluation of isomonodromic tau functions, including the exact computation of all relevant constant factors. We use this method to efficiently compute the tail asymptotics of soft-edge, hard-edge and bulk scaled distribution and gap functions in the complex Wishart ensemble, provided each eigenvalue particle has been removed independently with probability $1-\gamma\in(0,1]$.

math-ph↗
arXiv2016

On the Tracy-Widom$_\beta$ Distribution for $\beta=6$

We study the Tracy-Widom distribution function for Dyson's $\beta$-ensemble with $\beta = 6$. The starting point of our analysis is the recent work of I. Rumanov where he produces a Lax-pair representation for the Bloemendal-Vir\'ag equation. The latter is a linear PDE which describes the Tracy-Widom functions corresponding to general values of $\beta$. Using his Lax pair, Rumanov derives an explicit formula for the Tracy-Widom $\beta=6$ function in terms of the second Painlev\'e transcendent and the solution of an auxiliary ODE. Rumanov also shows that this formula allows him to derive formally the asymptotic expansion of the Tracy-Widom function. Our goal is to make Rumanov's approach and hence the asymptotic analysis it provides rigorous. In this paper, the first one in a sequel, we show that Rumanov's Lax-pair can be interpreted as a certain gauge transformation of the standard Lax pair for the second Painlev\'e equation. This gauge transformation though contains functional parameters which are defined via some auxiliary nonlinear ODE which is equivalent to the auxiliary ODE of Rumanov's formula. The gauge-interpretation of Rumanov's Lax-pair allows us to highlight the steps of the original Rumanov's method which needs rigorous justifications in order to make the method complete. We provide a rigorous justification of one of these steps. Namely, we prove that the Painlev\'e function involved in Rumanov's formula is indeed, as it has been suggested by Rumanov, the Hastings-McLeod solution of the second Painlev\'e equation. The key issue which we also discuss and which is still open is the question of integrability of the auxiliary ODE in Rumanov's formula. We note that this question is crucial for the rigorous asymptotic analysis of the Tracy-Widom function.

math-ph↗
arXiv2015

On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential II

In this paper we continue our analysis \cite{BDIK} of the determinant $\det(I-\gamma K_s),\gamma\in(0,1)$ where $K_s$ is the trace class operator acting in $L^2(-1,1)$ with kernel $K_s(\lambda,\mu)=\frac{\sin s(\lambda-\mu)}{\pi(\lambda-\mu)}$. In \cite{BDIK} various key asymptotic results were stated and utilized, but without proof: Here we provide the proofs (see Theorem 1.2 and Proposition 1.3 below).

math-ph↗
arXiv2015

Numerical Methods for the Discrete Map $Z^a$

As a basic example in nonlinear theories of discrete complex analysis, we explore various numerical methods for the accurate evaluation of the discrete map $Z^a$ introduced by Agafonov and Bobenko. The methods are based either on a discrete Painlev\'e equation or on the Riemann-Hilbert method. In the latter case, the underlying structure of a triangular Riemann-Hilbert problem with a non-triangular solution requires special care in the numerical approach. Complexity and numerical stability are discussed, the results are illustrated by numerical examples

math.NA↗
arXiv2015

Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge

We compute asymptotics for Hankel determinants and orthogonal polynomials with respect to a discontinuous Gaussian weight, in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support. Their behavior is described in terms of the Ablowitz-Segur family of solutions to the Painlev\'e II equation. Our results complement the ones in [Xu,Zhao,2011]. As consequences of our results, we conjecture asymptotics for an Airy kernel Fredholm determinant and total integral identities for Painlev\'e II transcendents, and we also prove a new result on the poles of the Ablowitz-Segur solutions to the Painlev\'e II equation. We also highlight applications of our results in random matrix theory.

math-ph↗
arXiv2015

Connection problem for the tau-function of the Sine-Gordon reduction of Painlev\'{e}-III equation via the Riemann-Hilbert approach

We evaluate explicitly, in terms of the Cauchy data, the constant pre-factor in the large $x$ asymptotics of the Painlev\'e III tau-function. Our result proves the conjectural formula for this pre-factor obtained recently by O. Lisovyy, Y. Tykhyy, and the first co-author with the help of the recently discovered connection of the Painlev\'e tau-functions with the Virasoro conformal blocks. Our approach does not use this connection, and it is based on the Riemann-Hilbert method.

math-ph↗
arXiv2014

On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential I

We study the determinant $\det(I-\gamma K_s), 0<\gamma <1$, of the integrable Fredholm operator $K_s$ acting on the interval $(-1,1)$ with kernel $K_s(\lambda, \mu)= \frac{\sin s(\lambda - \mu)}{\pi (\lambda-\mu)}$. This determinant arises in the analysis of a log-gas of interacting particles in the bulk-scaling limit, at inverse temperature $\beta=2$, in the presence of an external potential $v=-\frac{1}{2}\ln(1-\gamma)$ supported on an interval of length $\frac{2s}{\pi}$. We evaluate, in particular, the double scaling limit of $\det(I-\gamma K_s)$ as $s\rightarrow\infty$ and $\gamma\uparrow 1$, in the region $0\leq\kappa=\frac{v}{s}=-\frac{1}{2s}\ln(1-\gamma)\leq 1-\delta$, for any fixed $0<\delta<1$. This problem was first considered by Dyson in \cite{Dy1}.

math-ph↗
arXiv2014

Periodic problem for the nonlinear Schroedinger equation

The paper offers the method of discovering of some class of solutions for the nonlinear Schroedinger equation. An algorithm of constructive solving of the Cauchy periodic problem with a finite-gap initial condition was also obtained.

nlin.SI↗