Some properties of angular integrals

Abstract

Abstract:We find new representations for Itzykson-Zuber like angular integrals for arbitrary β, in particular for the orthogonal group O(n), the unitary group U(n) and the symplectic group Sp(2n). We rewrite the Haar measure integral, as a flat Lebesge measure integral, and we deduce some recursion formula on n. The same methods gives also the Shatashvili's type moments. Finally we prove that, in agreement with Brezin and Hikami's observation, the angular integrals are linear combinations of exponentials whose coefficients are polynomials in the reduced variables (x i − x j )(y i − y j ).

Tensor models from the viewpoint of matrix models: the cases of loop models on random surfaces and of the Gaussian distribution

“…Even the simplest integral over U (N ) in the Gaussian case was shown to be of degree 4, implying that the exponential is not of the Itzykson-Zuber type. It may be that techniques not considered in the present paper work, like the one developed in [26], and this will be investigated elsewhere. We hope that the relationship we have established between tensor models and loop models can lead to fruitful cross-fertilization.…”

LargeNexpansions and Painlevé hierarchies in the Hermitian matrix model

“…These critical models are related to conformal (2m, 1) minimal models [16,17]. Note that there is a difference between the definitions of order of a singular solution in the one-cut (definition 1) and two-cut (definition 2) cases, which in turn entails a difference in the statements of our main results (theorems 2 and 3).…”

“…These critical matrix models correspond to conformal (2m−1, 2) minimal models [7,16]. Let us thus consider a model that (locally) for T < T c is in the regular one-cut case and that for the critical temperature T = T c the corresponding hodograph equation (61) has a singular solution r c of order m ≥ 2.…”

“…However, most of the existing work, specially for the two-cut case, deals only with particular examples [5,7,12,19,20,21,22] and does not offer a general characterization of the specific member of the Painlevé hierarchy associated to a critical model in terms of the coupling constants g and of the critical value of the recurrence coefficient. More recent works [16,17] do prove the occurrence of the full Painlevé hierarchies in the context of critical matrix models, but again they do not give an explicit correspondence between Painlevé equations and critical models.…”

“…Critical matrix models correspond to points T c where the asymptotic free energy ceases to be analytic as a function of T . Some of these critical models are related to conformal (p, q) minimal models [7,15,16,17] and have asymptotic eigenvalue densities with rational singularities ρ(x) ∼ (x − α) p/q (8) near one of the endpoints α of J. It has been known for a long time in the physics literature that the double scaling limit of critical matrix models in the one-cut and twocut cases reveals the presence of equations belonging to the Painlevé I and Painlevé II hierarchies respectively (see [18] for a description of these hierarchies).…”