arXiv subjects

Alexander Its

Publications and source records attributed to Alexander Its.

27 records · Page 2Linked to original sources

arXiv2013

Asymptotics of a cubic sine kernel determinant

We study the one parameter family of Fredholm determinants $\det(I-γK_{\textnormal{csin}}),γ\in\mathbb{R}$ of an integrable Fredholm operator $K_{\textnormal{csin}}$ acting on the interval $(-s,s)$ whose kernel is a cubic generalization of the sine kernel which appears in random matrix theory. This Fredholm determinant appears in the description of the Fermi distribution of semiclassical non-equilibrium Fermi states in condensed matter physics as well as in random matrix theory. Using the Riemann-Hilbert method, we calculate the large $s$-asymptotics of $\det(I-γK_{\textnormal{csin}})$ for all values of the real parameter $γ$.

nlin.SI↗

arXiv2012

Asymptotics of a Fredholm determinant involving the second Painlevé transcendent

We study the determinant $\det(I-K_{\textnormal{PII}})$ of an integrable Fredholm operator $K_{\textnormal{PII}}$ acting on the interval $(-s,s)$ whose kernel is constructed out of the $Ψ$-function associated with the Hastings-McLeod solution of the second Painlevé equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the Unitary Ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann-Hilbert method, we evaluate the large $s$-asymptotics of $\det(I-K_{\textnormal{PII}})$.

math-ph↗

arXiv2012

Initial boundary value problem for the focusing NLS equation with Robin boundary condition: half-line approach

We consider the initial boundary value problem for the focusing nonlinear Schrödinger equation in the quarter plane $x>0,t>0$ in the case of decaying initial data (for $t=0$, as $x\to +\infty$) and the Robin boundary condition at $x=0$. We revisit the approach based on the simultaneous spectral analysis of the Lax pair equations [Fokas, 2002; Fokas, Its, Sung, 2005] and show that the approach can be implemented without any a priori assumptions on the long-time behavior of the boundary values.

nlin.SI↗

arXiv2011

Asymptotics of the instantons of Painleve I

The 0-instanton solution of Painlevé I is a sequence $(u_{n,0})$ of complex numbers which appears universally in many enumerative problems in algebraic geometry, graph theory, matrix models and 2-dimensional quantum gravity. The asymptotics of the 0-instanton $(u_{n,0})$ for large $n$ were obtained by the third author using the Riemann-Hilbert approach. For $k=0,1,2,...$, the $k$-instanton solution of Painlevé I is a doubly-indexed sequence $(u_{n,k})$ of complex numbers that satisfies an explicit quadratic non-linear recursion relation. The goal of the paper is three-fold: (a) to compute the asymptotics of the 1-instanton sequence $(u_{n,1})$ to all orders in $1/n$ by using the Riemann-Hilbert method, (b) to present formulas for the asymptotics of $(u_{n,k})$ for fixed $k$ and to all orders in $1/n$ using resurgent analysis, and (c) to confirm numerically the predictions of resurgent analysis. We point out that the instanton solutions display a new type of Stokes behavior, induced from the tritronquée Painlevé transcendents, and which we call the induced Stokes phenomenon. The asymptotics of the 2-instanton and beyond exhibits new phenomena not seen in 0 and 1-instantons, and their enumerative context is at present unknown.

math.CA↗

arXiv2009

Painléve III and a singular linear statistics in Hermitian random matrix ensembles I

In this paper, we study a certain linear statistics of the unitary Laguerre ensembles, motivated in part by an integrable quantum field theory at finite temperature. It transpires that this is equivalent to the characterization of a sequence of polynomials orthogonal with respect to the weight $$ w(x)=w(x,s):=x^{\al}\rme^{-x}\rme^{-s/x}, \quad 0\leq x<\infty, \al>0, s>0, $$ namely, the determination of the associated Hankel determinant and recurrence coefficients. Here $w(x,s)$ is the Laguerre weight $x^{\al}\:\rme^{-x}$ 'perturbed' by a multiplicative factor $\rme^{-s/x},$ which induces an infinitely strong zero at the origin. For polynomials orthogonal on the unit circle, a particular example where there are explicit formulas, the weight of which has infinitely strong zeros, was investigated by Pollazcek and Szegö many years ago. Such weights are said to be 'singular' or irregular due to the violation of the Szegö condition. In our problem, the linear statistics is a sum of the reciprocal of positive random variables $\{x_j:j=1,..,,n\};$ $\sum_{j=1}^{n}1/x_j.$ We show that the moment generating function, or the Laplace transform of the probability density function of this linear statistics is expressed as the ratio of Hankel determinants and as an integral of the combination of a particular third Painlevé function.

math.CA↗

arXiv2008

Total integrals of global solutions to Painleve II

We evaluate the total integral from negative infinity to positive infinity of all global solutions to the Painleve II equation on the real line. The method is based on the interplay between one of the equations of the associated Lax pair and the corresponding Riemann-Hilbert problem. In addition, we evaluate the total integral of a function related to a special solution to the Painleve V equation. As a corollary, we obtain short proofs of the computation of the constant terms of the limiting gap probabilities in the edge and the bulk of the Gaussian Orthogonal and Gaussian Symplectic Ensembles that were obtained recently in [4] and [18]. We also evaluate the total integrals of certain polynomials of the Painleve functions and their derivatives. These polynomials are the densities of the first integrals of the modified Korteweg-de Vries equation. We discuss the relations of the formulae we have obtained to the classical trace formulae for the Dirac operator on the line.

math.CA↗

arXiv2004

Asymptotics of the partition function of a random matrix model

We prove a number of results concerning the large $N$ asymptotics of the free energy of a random matrix model with a polynomial potential $V(z)$. Our approach is based on a deformation $τ_tV(z)$ of $V(z)$ to $z^2$, $0\le t<\infty$ and on the use of the underlying integrable structures of the matrix model. The main results include (1) the existence of a full asymptotic expansion in powers of $N^{-2}$ of the recurrence coefficients of the related orthogonal polynomials, for a one-cut regular $V$; (2) the existence of a full asymptotic expansion in powers of $N^{-2}$ of the free energy, for a $V$, which admits a one-cut regular deformation $τ_tV$; (3) the analyticity of the coefficients of the asymptotic expansions of the recurrence coefficients and the free energy, with respect to the coefficients of $V$; (4) the one-sided analyticity of the recurrent coefficients and the free energy for a one-cut singular $V$; (5) the double scaling asymptotics of the free energy for a singular quartic polynomial $V$.

math-ph↗

arXiv2002

Double scaling limit in the random matrix model: the Riemann-Hilbert approach

We prove the existence of the double scaling limit in the unitary matrix model with quartic interaction, and we show that the correlation functions in the double scaling limit are expressed in terms of the integrable kernel determined by the psi-function for the Hastings-McLeod solution to the Painlevé II equation. The proof is based on the Riemann-Hilbert approach.

math-ph↗

arXiv1999

Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model

We derive semiclassical asymptotics for the orthogonal polynomials P_n(z) on the line with respect to the exponential weight \exp(-NV(z)), where V(z) is a double-well quartic polynomial, in the limit when n, N \to \infty. We assume that ε\le (n/N) \le λ_{cr} - εfor some ε> 0, where λ_{cr} is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut. Simultaneously we derive semiclassical asymptotics for the recursive coefficients of the orthogonal polynomials, and we show that these coefficients form a cycle of period two which drifts slowly with the change of the ratio n/N. The proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix Riemann-Hilbert problem. As an application of the semiclassical asymptotics of the orthogonal polynomials, we prove the universality of the local distribution of eigenvalues in the matrix model with the double-well quartic interaction in the presence of two cuts.

math-ph↗