Phase structure and asymptotic zero densities of orthogonal polynomials in the cubic model

Abstract

We apply the method we have described in a previous paper (2013) to determine the phase structure of asymptotic zero densities of the standard cubic model of non-hermitian orthogonal polynomials. We provide a complete description of the two phases: the one cut phase and the two cut phase, and analyze the phase transition processes of the types: splitting of a cut, birth and death of a cut.

Complex saddle points in the Gross-Witten-Wadia matrix model

“…[12] can be completely justified in the large-N limit by using techniques similar to those developed for the determination of the asymptotic support of the zeros of certain non-Hermitian families of orthogonal polynomials [17,18,21].…”

“…(6) determines a non-Hermitian holomorphic matrix model [19,20] which can be analyzed in the same way as the models recently considered in Refs. [17,18,21]. It should be noticed that Eq.…”

“…A detailed discussion of the techniques we have used to carry out this calculations can be found in Refs. [17,18,21], and we have summarized the results of this study schematically in Fig. 1.…”

“…[17,18] to study the arcs that support the asymptotic density of zeros in families of non-Hermitian orthogonal polynomials.…”

Complex saddles in the Gross-Witten-Wadia matrix model

“…Our strategy it to use directly the matrix eigenvalues as integration variables, thus keeping the simpler form of the potential and in effect allowing us to write and solve a system of equations similar to those used in Refs. [17,18] to study the arcs that support the asymptotic density of zeros in families of non-hermitian orthogonal polynomials.…”