The Hermitian two matrix model with an even quartic potential

Abstract

We consider the two matrix model with an even quartic potential W (y) = y 4 /4 + αy 2 /2 and an even polynomial potential V (x). The main result of the paper is the formulation of a vector equilibrium problem for the limiting mean density for the eigenvalues of one of the matrices M 1 . The vector equilibrium problem is defined for three measures, with external fields on the first and third measures and an upper constraint on the second measure. The proof is based on a steepest descent analysis of a 4 × 4 matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigenvalues of M 1 . Our results generalize earlier results for the case α = 0, where the external field on the third measure was not present.

Critical measures for vector energy: Global structure of trajectories of quadratic differentials

“…Returning to our motivation rooted in the interacting particle systems, recent works have showed that more general models, such as the hermitian matrix model with an external source [21][22][23], the two-matrix model [11,16,[32][33][34], some classes of non-intersecting paths [47,48] or the normal matrix model [26,52] require considering log gases with more than one group of charges. In this case, standard orthogonal polynomials are not enough, and we must turn to the so-called multiple orthogonal polynomials and to the associated vector equilibrium problems [39].…”

Non-intersecting squared Bessel paths with one positive starting and ending point

“…In recent papers [23,24] concerning eigenvalue distribution of a two-matrix model in random matrix theory, a vector equilibrium problem with three measures appears as well. In that situation, however, the first component of the minimizer is relevant to the limiting mean distribution.…”

Аппроксимации Эрмита - Паде И Ансамбли Совместно Ортогональных Многочленов

“…Основным инструментом в [90] является векторная задача равновесия для трех мер. В ней требуется минимизировать функционал энергии…”

“…Мы проведем его при α < 0, что соответствует двухъямному потенциалу W ; см. обсуждение всех случаев в [90].…”