Abstract
We study two types of random matrix ensembles that emerge when considering the same probability measure on partitions. One is the Meixner ensemble with a hard wall and the other are two families of unitary matrix models, with weight functions that can be interpreted as characteristic polynomial insertions. We show that the models, while having the same exact evaluation for fixed values of the parameter, may present a different phase structure. We find phase transitions of the second and third order, depending on the model. Other relationships, via direct mapping, between the unitary matrix models and continuous random matrix ensembles on the real line, of Cauchy–Romanovski type, are presented and studied both exactly and asymptotically. The case of orthogonal and symplectic groups is studied as well and related to Wronskians of Chebyshev polynomials, that we evaluate at large N.
Highlights
We further elaborate on this whole notion by studying other equivalences, with continuous random matrix ensembles on the real line, via direct mapping this time
We study two types of random matrix ensembles that emerge when considering the same probability measure on partitions
One is the Meixner ensemble with a hard wall and the other are two families of unitary matrix models, with weight functions that can be interpreted as characteristic polynomial insertions
Summary
In the present Section, we consider the asymptotic behaviour of the unitary matrix models defined in eq (1.6) and (1.7), when the rank K is large and N1, N2 scale with K. We comment on the already known aspects of the exact solvability of some of the models above
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