Exact equivalences and phase discrepancies between random matrix ensembles

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 . Contents 1. Introduction 1 2. Random matrix ensembles on the unit circle 5 3. Random matrix ensembles on the real line 14 4. Outlook 23 Appendix A. Solution to saddle point equations on the unit circle 24 Appendix B. Free energies 31 Appendix C. Solution to the saddle point equations on the real line 33 References 37

Deformed Cauchy random matrix ensembles and large $N$ phase transitions

“…The particular case with A = N was recently studied in section 3 of [30], where it originated from a family of unitary matrix models through the map to the unit circle. This case is special, as we shall discuss, and in the large N theory corresponds to the marginal case for stability of the model.…”

“…This case is special, as we shall discuss, and in the large N theory corresponds to the marginal case for stability of the model. When = 0, the potential (3) describes a Cauchy ensemble, studied in [31,30].…”

“…The integral is a complicated generalization of the Selberg's integral, with singularities at different places. The particular case A = N was studied recently in [30].…”

Multiple phases in a generalized Gross-Witten-Wadia matrix model

“…An interesting question is if all phases appearing in the whole (λ, τ ) parameter space have a counterpart in the Hermitian model of [11]. The correspondence in the phase structure of unitary and Hermitian matrix models in the large N limit involves subtle issues that have been recently discussed in [23]. In the present model, the main difference will occur in the τ < 0 region.…”

Phases of unitary matrix models and lattice QCD2

“…The term with coefficient A corresponds to the logarithmic term in (1) (taking into account a shift due to the Jacobian of the sterographic map), while the term with coefficient B gives rise to the GWW term. The model generalizes the Hermitian model appeared in [8], derived from the ν = 0 case (see [9,10] for earlier studies on related models). The logarithmic term appears in the mathematical literature in the study of Cauchy random matrix ensembles [11].…”