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

Abstract

We study a new hermitian one-matrix model containing a logarithmic Penner's type term and another term, which can be obtained as a limit from logarithmic terms. For small coupling, the potential has an absolute minimum at the origin, but beyond a certain value of the coupling the potential develops a double well. For a higher critical value of the coupling, the system undergoes a large N third-order phase transition.

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

“…With this deformation, the GWW transition extends to a critical line |λ cr (τ )|, given in (3.16), and in the new phase eigenvalues get distributed into two separated, symmetric cuts. This is the counterpart of the third order phase transition found in the dual Hermitian model [11]. An interesting question is if all phases appearing in the whole (λ, τ ) parameter space have a counterpart in the Hermitian model of [11].…”

“…If we restrict ourselves to the "physical" τ > 0 region, the phase structure seems to be simpler, with a single phase transition from region VI to region VII, which can be thought of as the extension of the GWW phase transition in the presence of the τ coupling (analogous to the liquid-vapor phase transition, in a phase diagram that includes temperature and pressure). The transition should be of the third order, being the counterpart of the third-order phase transition found in [11] in the dual Hermitian model. We leave these problems for future work.…”

“…This phase transition is nothing but the unitary model version of the phase transition described in [11] for the corresponding Hermitian matrix model (2.4). 4 The eigenvalue density may be found from the two-cut solution in [11] by using the duality map between Hermitian and unitary models [21] (see also [19]). However, in this paper we will not explore this region.…”

“…In this work, we shall study a generalized version of the model, emphasizing the analysis of its phase structure in a number of scaling limits. To obtain our model, we start with the Hermitian matrix model introduced in [11], with potential…”

“…However, in this paper we will not explore this region. Like in [11], the critical line occurs when the eigenvalue density vanishes at α = 0 (see fig. 8b).…”