New scaling of Itzykson-Zuber integrals

Matrix product ensembles of Hermite-type and the hyperbolic Harish-Chandra-Itzykson-Zuber integral

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 …

Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability

We consider integrals on unitary groups $U_d$ of the form $$\int_{U_d}U_{i_1j_1}... U_{i_qj_q}U^*_{j'_{1}i'_{1}} ... U^*_{j'_{q'}i'_{q'}}dU$$ We give an explicit formula in terms of characters of symmetric groups and Schur functions, which allows us to rederive an asymptotic expansion as $d\to\infty$. Using this we rederive and strenghthen a result of asymptotic freeness due to Voiculescu. We then study large $d$ asymptotics of matrix model integrals and of the logarithm of Itzykson-Zuber integrals and show that they converge towards a limit when considered as power series. In particular we gi…

Sampling Matrices from Harish-Chandra-Itzykson-Zuber Densities with Applications to Quantum Inference and Differential Privacy

Given two $n \times n$ Hermitian matrices $Y$ and $Λ$, the Harish-Chandra-Itzykson-Zuber (HCIZ) distribution on the unitary group $\text{U}(n)$ is $e^{\text{tr}(UΛU^*Y)}dμ(U)$, where $μ$ is the Haar measure on $\text{U}(n)$. The density $e^{\text{tr}(UΛU^*Y)}$ is known as the HCIZ density. Random unitary matrices distributed according to the HCIZ density are important in various settings in physics and random matrix theory. However, the basic question of efficient sampling from the HCIZ distribution has remained open. We present two efficient algorithms to sample matrices from distributions th…

Rank one HCIZ at high temperature: interpolating between classical and free convolutions

We study the rank one Harish-Chandra-Itzykson-Zuber integral in the limit where $\frac{N β}{2} \to c $, called the high temperature regime and show that it can be used to construct a promising one-parameter interpolation, with parameter $c$ between the classical and the free convolution. This $c$-convolution has a simple interpretation in terms of another associated family of distribution indexed by $c$, called the Markov-Krein transform: the $c$-convolution of two distributions corresponds to the classical convolution of their Markov-Krein transforms. We derive first cumulants-moments relatio…

耦合球形积分的渐近性：自由傅里叶变换的算子值推广

On the Asymptotic of Coupled Spherical Integrals: An Operator-Valued Extension of the Free Fourier Transform

本文研究耦合HCIZ积分的渐近指数，用算子值R变换刻画其归一化渐近集中指数，将耦合HCIZ积分作为经典拉普拉斯变换的算子值自由类比，还可分解高秩耦合HCIZ积分并为耦合自旋玻璃研究提供支撑。

The rank 1 real Wishart spiked model

In this paper, we consider N-dimensional real Wishart matrices Y in the class $W_{\mathbb{R}}(Σ,M)$ in which all but one eigenvalues of $Σ$ is 1. Let the non-trivial eigenvalue of $Σ$ be $1+τ$, then as N, $M\rightarrow\infty$, with $M/N=γ^2$ finite and non-zero, the eigenvalue distribution of $Y$ will converge into the Marchenko-Pastur distribution inside a bulk region. When $τ$ increases from zero, one starts to see a stray eigenvalue of Y outside of the support of the Marchenko-Pastur density. As the this stray eigenvalue leaves the bulk region, a phase transition will occur in the largest e…