相关论文: Free probability and combinatorics
We study the coefficients in the expansion of Jack polynomials in terms of power sums. We express them as polynomials in the free cumulants of the transition measure of an anisotropic Young diagram. We conjecture that such polynomials have…
We compute the generating series for the simplest class of bi-free cumulants, beyond free cumulants, the two-bands bi-free cumulants of a pair of a left and a right variable. We also consider two-faced systems with a commutation condition…
In this paper additive bi-free convolution is defined for general Borel probability measures, and the limiting distributions for sums of bi-free pairs of selfadjoint commuting random variables in an infinitesimal triangular array are…
A result of Hoskins and Steinerberger [Int. Math. Res. Not., (13):9784-9809, 2022] states that repeatedly differentiating a random polynomials with independent and identically distributed mean zero and variance one roots will result, after…
We consider (self-adjoint) families of infinite matrices of noncommutative random variables such that the joint distribution of their entries is invariant under conjugation by a free quantum group. For the free orthogonal and…
This paper centers around two basic problems of topological coincidence theory. First, try to measure (with help of Nielsen and minimum numbers) how far a given pair of maps is from being loose, i.e. from being homotopic to a pair of…
The system of one-dimensional symmetric simple random walks, in which none of walkers have met others in a given time period, is called the vicious walker model. It was introduced by Michael Fisher and applications of the model to various…
Conditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric $\{\pm 1\}$-matrix is irreducible. This addresses a question raised by Eberhard in recent work. The main…
We consider a possible generalization of the random matrix theory, which involves the maximization of Tsallis' $q$-parametrized entropy. We discuss the dependence of the spacing distribution on $q$ using a non-extensive generalization of…
Let A be a unital $C^*$-algebra, given together with a specified state $\phi:A \to C$. Consider two selfadjoint elements a,b of A, which are free with respect to $\phi$ (in the sense of the free probability theory of Voiculescu). Let us…
Noncommutative invariant theory is a generalization of the classical invariant theory of the action of $SL(2,\IC)$ on binary forms. The dimensions of the spaces of invariant noncommutative polynomials coincide with the numbers of certain…
In [5], O. Bauer interpreted the chordal Loewner equation in terms of non-commutative probability theory. We follow this perspective and identify the chordal Loewner equations as the non-autonomous versions of evolution equations for…
We characterize the limiting second order distributions of certain independent complex Wigner and deterministic matrices using Voiculescu's notions of freeness over the diagonal. If the Wigner matrices are Gaussian, Mingo and Speicher's…
This is the first installment of a series of papers whose aim is to lay a foundation for homotopy probability theory by establishing its basic principles and practices. The notion of a homotopy probability space is an enrichment of the…
A statistic can be a function of multiple samples. There is little existing work on asymptotic theory for such statistics when group membership is random. We propose a flexible framework that can handle both deterministic and random…
Situations in many fields of research, such as digital communications, nuclear physics and mathematical finance, can be modelled with random matrices. When the matrices get large, free probability theory is an invaluable tool for describing…
Quantum field theories with quenched disorder are so hard to study that even exactly solvable free theories present puzzling aspects. We consider a free scalar field $\phi$ in $d$ dimensions coupled to a random source $h$ with quenched…
Some tools and ideas are interchanged between random matrix theory and multivariate statistics. In the context of the random matrix theory, classes of spherical and generalised Wishart random matrix ensemble, containing as particular cases…
The aim of this paper is to show how free probability theory sheds light on spectral properties of deformed matricial models and provides a unified understanding of various asymptotic phenomena such as spectral measure description,…
We study matricial approximations of master fields we constructed in a previous work. These approximations (in non-commutative distribution) are obtained by extracting blocks of a Brownian unitary diffusion (with entries in $\mathbb{R},…