Exact equivalences and phase discrepancies between random matrix ensembles. (English) Zbl 1459.60014

Santilli, Leonardo; Tierz, Miguel

J. Stat. Mech. Theory Exp. 2020, No. 8, Article ID 083107, 49 p. (2020).

Summary: 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 \(N\).

MSC:

60B20	Random matrices (probabilistic aspects)
15B52	Random matrices (algebraic aspects)
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
82C23	Exactly solvable dynamic models in time-dependent statistical mechanics

Keywords:

matrix models; random matrix theory and extensions; dimers; quantum phase transitions

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