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

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 its associated recursion $N \times N$ matrices either real symmetric $\beta=1$ or complex Hermitian $\beta=2$. In the continuum limit recursion, we show that matrix distribution converges inverse-Wishart ensemble matrices. full dynamics is solved using mapping $N$ fermions Morse potential, which are non-interacting for At finite eigenvalues exhibits tails, generalizing Kesten's results scalar case. density potential studied large $N$, power-law eigenvalue related properties so-called determinantal Bessel process describes hard edge universality For discrete free probability limit, obtain self-consistent equation stationary relation our recent works Rider Valk\'o, Grabsch Texier, as well Ossipov, discussed.