Matrix Kesten recursion, inverse-Wishart ensemble and fermions in a Morse potential

The random variable 1 + z (1) + z (1) z (2) + horizontal ellipsis 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 x 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 Valko, Grabsch and Texier, as well as Ossipov, is discussed.

Automated summary

This passage discusses the extension of random variables and their associated recursion to matrices, showing the matrix distribution properties in the continuum limit, and investigates the relationship between the distribution of eigenvalues and the density of fermions in the finite case.