Universality in the Two Matrix Model with a Monomial Quartic and a General Even Polynomial Potential

Abstract

In this paper we studied the asymptotic eigenvalue statistics of the 2 matrix model with the probability measurein the case where W = y 4 4 and V is a general even polynomial. We studied the correlation kernel for the eigenvalues of the matrix M 1 in the limit as n → ∞. We extended the results of Duits and Kuijlaars in [14] to the case when the limiting eigenvalue density for M 1 is supported on multiple intervals. The results are achieved by constructing the parametrix to a Riemann-Hilbert problem obtained in [14] with theta functions and then showing that this parametrix is well-defined for all n by studying the theta divisor.

Universality and critical behaviour in the chiral two-matrix model

“…This representation allows us to derive the large n limit of the correlation kernel in the case of a quadratic potential W (y) = y 2 /2 + αy by performing a Deift/Zhou steepest descent analysis on RH problem 2.2. As will be clear from the analysis, this corresponds to a quartic potential in the non-chiral two-matrix model studied by Duits-Kuijlaars-Mo in [22,23,39]. We will largely follow the line of thought in these works, however, at several places complications will arise.…”

Аппроксимации Эрмита - Паде И Ансамбли Совместно Ортогональных Многочленов

“…Случай потенциала W степени 4 был недавно рассмотрен в [89]- [91]. В рабо-те [89] метод наискорейшего спуска Дейфта-Чжоу применялся к задаче Рима-на-Гильберта в случае, когда…”

“…Это позволило провести точный асимпто-тический анализ ядра K (n) 11 при n → ∞, что привело, в частности, к результатам о локальной универсальности, обычным для теории случайных матриц и фор-мулируемым в терминах синус-ядра и ядра Эйри. В работе [89] рассматривался только случай рода нуль, но это ограничение было отброшено в [91], где рас-сматривался случай более высокого рода. Здесь имеется в виду род римановой поверхности (спектральной кривой), связанной с задачей и определенной в [89] при помощи векторной задачи равновесия аналогично обсуждениям в п.…”

Painlevé Kernels in Hermitian Matrix Models

“…By using the Riemann-Hilbert approach for the associated biorthogonal polynomials, we recently analyzed the asymptotic behavior of the two matrix model with one quartic potential [22,23,25,26,49]. Here we will report on that progress.…”

“…In Section 3 we discuss the definition of the Hermitian two matrix model in its general form and the relation with certain biorthogonal polynomials. In Section 4 we first discuss the vector equilibrium problem that was a key ingredient for the asymptotic analysis [25,26,49] of the Riemann-Hilbert problem for the biorthogonal polynomials. Second, we present a phase diagram and a new critical phenomenon for the quartic/quadratic case that we analyzed in [22].…”