Matrix Kesten Recursion, Inverse-Wishart Ensemble and Fermions in a Morse Potential

TL;DR

This paper extends Kesten's scalar recursion to matrix form, showing convergence to the inverse-Wishart ensemble, and maps the problem to fermions in a Morse potential, revealing heavy-tailed eigenvalue distributions.

Contribution

It introduces a matrix recursion generalizing Kesten's variable, solves the associated dynamics via a fermion mapping, and analyzes eigenvalue distributions and heavy tails in the large N limit.

Findings

Matrix recursion converges to inverse-Wishart ensemble.

Eigenvalue distributions exhibit heavy tails at finite N.

Large N fermion density relates to determinantal Bessel process.

Abstract

The random variable $1+z_1+z_1z_2+\dots$ appears in many contexts and was shown by Kesten to exhibit a heavy tail distribution. We consider natural extensions of this variable and its associated recursion to $N \times N$ matrices either real symmetric $\beta=1$ or complex Hermitian $\beta=2$. In the continuum limit of this recursion, we show that the matrix distribution converges to the inverse-Wishart ensemble of random matrices. The full dynamics is solved using a mapping to $N$ fermions in a Morse potential, which are non-interacting for $\beta=2$. At finite $N$ the distribution of eigenvalues exhibits heavy tails, generalizing Kesten's results in the scalar case. The density of fermions in this potential is studied for large $N$, and the power-law tail of the eigenvalue distribution is related to the properties of the so-called determinantal Bessel process which describes the hard edge universality of random matrices. For the discrete matrix recursion, using free probability in the large $N$ limit, we obtain a self-consistent equation for the stationary distribution. The relation of our results to recent works of Rider and Valk\'o, Grabsch and Texier, as well as Ossipov, is discussed.