Matrix addition and the Dunkl transform at high temperature

Abstract

We develop a framework for establishing the Law of Large Numbers for the eigenvalues in the random matrix ensembles as the size of the matrix goes to infinity simultaneously with the beta (inverse temperature) parameter going to zero. Our approach is based on the analysis of the (symmetric) Dunkl transform in this regime. As an application we obtain the LLN for the sums of random matrices as the inverse temperature goes to 0. This results in a one-parameter family of binary operations which interpolates between classical and free convolutions of the probability measures. We also introduce and study a family of deformed cumulants, which linearize this operation.

v2: added sections 1.4 and 4.1; fixed minor error in claim 9.1; added references. v3: 59 pages; added Sec. 3.4; added Conjecture 1.3 and Lemma 4.5; fixed a typo in last eqns. of Sec. 9

References in corpus (25)

• Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
• Global spectrum fluctuations for the $\beta$-Hermite and $\beta$-Laguerre ensembles via matrix models
• Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability
• Invariant $\beta$-ensembles and the Gauss-Wigner crossover
• Generalized Gibbs Ensembles of the classical Toda chain
• Fluctuations of particle systems determined by Schur generating functions
• Lecture notes on Cherednik algebras
• Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory
• Invariant $\beta$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law
• Moments of the position of the maximum for GUE characteristic polynomials and for log-correlated Gaussian processes
• Some multivariate master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions
• From Sine kernel to Poisson statistics
• The combinatorics of associated Hermite polynomials
• Beta Jacobi ensembles and associated Jacobi polynomials
• Fourier transform on high-dimensional unitary groups with applications to random tilings
• The high temperature crossover for general 2D Coulomb gases
• The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles
• Polynomial convolutions and (finite) free probability
• Rank one HCIZ at high temperature: interpolating between classical and free convolutions
• Airy Point Process via Supersymmetric Lifts
• Gaussian fluctuations for products of random matrices
• Reflection length with two parameters in the asymptotic representation theory of type B/C and applications
• Limit theorems for moment processes of beta Dyson's Brownian motions and beta Laguerre processes
• Law of Large Numbers and Central Limit Theorems by Jack Generating Functions
• High-low temperature dualities for the classical $\beta$-ensembles

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• Classical beta ensembles and related eigenvalues processes at high temperature and the Markov--Krein transform
• Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures
• Limits of Probability Measures with General Coefficients