本文研究 Harish-Chandra--Itzykson--Zuber 张量积分在特征大小 N 趋大时的渐近行为，通过分析双参数缩放假设揭示非平凡渐近状态，为多体量子系统纠缠性质及随机局部测量提供应用。
摘要：
我们考虑了最近引入的 Harish-Chandra--Itzykson--Zuber 积分向张量的推广，并讨论了当张量的特征大小 N 趋于很大时的渐近行为。这项研究要求我们对外部张量随 N 的缩放做出假设。我们分析了一类双参数的渐近缩放假设，揭示了几个非平凡的渐近状态。该研究对于分析多体量子系统的纠缠性质具有相关性。我们讨论了我们的结果在该领域的潜在应用，特别是在随机局部测量的背景下。
英文摘要：
We consider the recently introduced generalization of the Harish-Chandra--Itzykson--Zuber integral to tensors and discuss its asymptotic behavior when the characteristic size N of the tensors is taken to be large. This study requires us to make assumptions on the scaling with N of the external tensors. We analyze a two-parameter class of asymptotic scaling ansätze, uncovering several non-trivial asymptotic regimes. This study is relevant for analyzing the entanglement properties of multipartite quantum systems. We discuss potential applications of our results to this domain, in particular in the context of randomized local measurements.
We apply the renormalized coupling constants and Virasoro constraints to derive the Itzykson-Zuber Ansatz on the form of the free energy in topological 2D gravity. We also treat the topological 1D gravity and the Hermitian one-matrix models in the same fashion. Some uniform behaviors are discovered in this approach.
We reconsider the large $N$ asymptotics of Harish-Chandra-Itzykson-Zuber integrals. We provide, using Dyson's Brownian motion and the method of instantons, an alternative, transparent derivation of the Matytsin formalism for the unitary case. Our method is easily generalized to the orthogonal and symplectic ensembles. We obtain an explicit solution of Matytsin's equations in the case of Wigner matrices, as well as a general expansion method in the dilute limit, when the spectrum of eigenvalues spreads over very wide regions.
This article is devoted to the asymptotic expansion of the generalized Harish Chandra-Itzykson-Zuber matrix integral for non-unitary symmetries characterized by a parameter beta(as usual beta =1,2 and 4 correspond to the orthogonal, unitary and symplectic group integrals). A WKB-expansion for f is derived from the heat kernel differential equation, for general values of k and beta. From an expansion in terms of zonal polynomials, one obtain an expansion in powers of the tau's for beta=1, and generalizations are considered for general beta. A duality relation, and a transformation of products o…
We investigate spectral properties of a Hermitised random matrix product which, contrary to previous product ensembles, allows for eigenvalues on the full real line. We prove that the eigenvalues form a bi-orthogonal ensemble, which reduces asymptotically to the Hermite Muttalib-Borodin ensemble. Explicit expressions for the bi-orthogonal functions as well as the correlation kernel are provided. Scaling the latter near the origin gives a limiting kernel involving Meijer G-functions, and the functional form of the global density is calculated. As a part of this study, we introduce a new matrix …
Concise review of the basic properties of unitary matrix integrals. They are studied with the help of the three matrix models: the ordinary unitary model, Brezin-Gross-Witten model and the Harish-Charndra-Itzykson-Zuber model. Especial attention is paid to the tricky sides of the story, from De Wit-t'Hooft anomaly in unitary integrals to the problem of correlators with Itzykson-Zuber measure. Of technical tools emphasized is the method of character expansions. The subject of unitary integrals remains highly under-investigated and a lot of new results are expected in this field when it attracts…
In this paper we return to the classical work by Brézin, Itzykson, Parisi and Zuber, in which, among other things, the authors explicitly calculated the coefficients of the topological expansion in the cubic random matrix model in genus 0. Our main goal will be to rigorously prove the results of Brézin, Itzykson, Parisi and Zuber and to obtain an explicit formula for the coefficients of the topological expansion in genus 1. We will also prove some formulae and asymptotic results for the coefficients of the topological expansion in higher genera.
Recently the correlation functions of the so-called Itzykson-Zuber/Harish-Chandra integrals were computed (by one of the authors and collaborators) for all classical groups using an integration formula that relates integrals over compact groups with respect to the Haar measure and Gaussian integrals over a maximal nilpotent Lie subalgebra of their complexification. Since the integration formula a posteriori had the same form for the classical series, a conjecture was formulated that such a formula should hold for arbitrary semisimple Lie groups. We prove this conjecture using an abstract Lie-t…
In this note, we study the asymptotic of spherical integrals, which are analytical extension in index of the normalized Schur polynomials for $β=2$ , and of Jack symmetric polynomials otherwise. Such integrals are the multiplicative counterparts of the Harish-Chandra-Itzykson-Zuber (HCIZ) integrals, whose asymptotic are given by the so-called $R$-transform when one of the matrix is of rank one. We argue by a saddle-point analysis that a similar result holds for all $β>0$ in the multiplicative case, where the asymptotic is governed by the logarithm of the $S$-transform. As a consequence of this…
We consider solvable matrix models. We generalize Harish-Chandra-Itzykson-Zuber and certain other integrals (Gross-Witten integral and integrals over complex matrices) using the notion of tau function of matrix argument. In this case one can reduce the matrix integral to the integral over eigenvalues, which in turn is certain tau function. The resulting tau function may be analyzed either by the method of orthogonal polynomials or by the Schur functions expansion method.
We find new representations for Itzykson-Zuber like angular integrals for arbitrary beta, 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).
In this paper, we prove a longstanding conjecture on the asymptotic behavior of a pair of oscillatory matrix integrals: the Harish-Chandra/Itzykson-Zuber (HCIZ) integral, and the Brezin-Gross-Witten (BGW) integral. The main result gives a complete asymptotic expansion of these integrals for small complex parameters. The coefficients of these asymptotic expansions are generating functions for monotone Hurwitz numbers sorted by genus.
We present a new proof of Harish-Chandra's formula $$Π(h_1) Π(h_2) \int_G e^{\langle \mathrm{Ad}_g h_1, h_2 \rangle} dg = \frac{ [ \! [ Π, Π] \!] }{|W|} \sum_{w \in W} ε(w) e^{\langle w(h_1),h_2 \rangle},$$ where $G$ is a compact, connected, semisimple Lie group, $dg$ is normalized Haar measure, $h_1$ and $h_2$ lie in a Cartan subalgebra of the complexified Lie algebra, $Π$ is the discriminant, $\langle \cdot, \cdot \rangle$ is the Killing form, $[ \! [ \cdot, \cdot ] \!]$ is an inner product that extends the Killing form to polynomials, $W$ is a Weyl group, and $ε(w)$ is the sign of $w \in W$…
