# Matrix Addition and the Dunkl Transform at High Temperature

@article{BenaychGeorges2022MatrixAA,
title={Matrix Addition and the Dunkl Transform at High Temperature},
author={Florent Benaych-Georges and Cesar Cuenca and Vadim Gorin},
journal={Communications in Mathematical Physics},
year={2022}
}
• Published 8 May 2021
• Mathematics
• Communications in Mathematical Physics
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…
3 Citations
Rank one HCIZ at high temperature: interpolating between classical and free convolutions
• Mathematics
SciPost Physics
• 2022
We study the rank one Harish-Chandra-Itzykson-Zuber integral in the limit where \frac{N\beta}{2} \to cNβ2→c, called the high-temperature regime and show that it can be used to construct a promising
Limits of Probability Measures With General Coefficients
With operators on formal series in $x_i$, $1\leq i\leq N$, which are symmetric in $N-1$ of the $x_i$, probability measures can be studied through Bessel generating functions. These operators are used
Operators on Bessel Generating Functions With General Coefficients UROP+ Final Paper, Summer 2021
• Mathematics
• 2021
With operators on formal series in xi, 1 ≤ i ≤ N , which are symmetric in N−1 of the xi, Bessel generating funtions can be studied. These operators are used with the Dunkl transform on Bessel

## References

SHOWING 1-10 OF 78 REFERENCES
Polynomial convolutions and (nite) free probability
We introduce a nite version of free probability and show the link between recent results using polynomial convolutions and the traditional theory of free probability. One tool for accomplishing this
Time Global Mild Solutions of Navier-Stokes-Oseen Equations
• V. Trinh
• Mathematics
Acta Mathematica Scientia
• 2021
In this paper we prove the existence and uniqueness of time global mild solutions to the Navier-Stokes-Oseen equations, which describes dynamics of incompressible viscous fluid flows passing a
Reflection length with two parameters in the asymptotic representation theory of type B/C and applications
• Mathematics
• 2021
We introduce a two-parameter function φq+,q− on the infinite hyperoctahedral group, which is a bivariate refinement of the reflection length. We show that this signed reflection function φq+,q− is
Limit theorems for moment processes of beta Dyson's Brownian motions and beta Laguerre processes
• Mathematics
• 2021
In the regime where the parameter beta is proportional to the reciprocal of the system size, it is known that the empirical distribution of Gaussian beta ensembles (resp. beta Laguerre ensembles)
The classical β-ensembles with β proportional to 1/N: From loop equations to Dyson’s disordered chain
• Mathematics
Journal of Mathematical Physics
• 2021
In the classical β-ensembles of random matrix theory, setting β = 2α/N and taking the N → ∞ limit gives a statistical state depending on α. Using the loop equations for the classical β-ensembles, we
Rank one HCIZ at high temperature: interpolating between classical and free convolutions
• Mathematics
SciPost Physics
• 2022
We study the rank one Harish-Chandra-Itzykson-Zuber integral in the limit where \frac{N\beta}{2} \to cNβ2→c, called the high-temperature regime and show that it can be used to construct a promising
Airy Point Process via Supersymmetric Lifts
A supersymmetric lift of a symmetric function is a special sequence of doubly symmetric lifts which satisfies a cancellation property relating neighboring lifts in the sequence. We obtain contour
Limit theorems and soft edge of freezing random matrix models via dual orthogonal polynomials
• Mathematics
• 2020
$N$-dimensional Bessel and Jacobi processes describe interacting particle systems with $N$ particles and are related to $\beta$-Hermite, $\beta$-Laguerre, and $\beta$-Jacobi ensembles. For fixed $N$
Beta Jacobi Ensembles and Associated Jacobi Polynomials
• Mathematics, Computer Science
Journal of Statistical Physics
• 2021
In the regime where beta N is the system size, the empirical distribution of the eigenvalues converges weakly to a limiting measure which belongs to a new class of probability measures of associated Jacobi polynomials, making use of the random matrix model.