A tridiagonal matrix-valued process with stochastic resetting for arbitrary Dyson index $\beta>0$

Abstract

We introduce a symmetric tridiagonal matrix-valued process ($\beta$-TMP) $H(t)$ whose diagonal entries $H_{k,k}(t)$ evolve independently via an Ornstein-Uhlenbeck process starting at the origin and the off-diagonal entries $H_{k,k+1}(t)$ evolve independently via the Cox-Ingersoll-Ross process, starting at the origin and with parameters that depend on the row index $k$. We show that the joint distribution of the entries of the matrix can be computed exactly at all times and moreover, the joint distribution of its $N$ real eigenvalues can be computed exactly at all times too. We then subject this time-evolving matrix-valued process to stochastic resetting with rate $r$ in two different settings: (i) simultaneous resetting of the matrix entries to the origin with rate $r$ ($\beta$-SRTMP process) and (ii) independent resetting of the matrix entries to the origin with rate $r$ ($\beta$-IRTMP process). We show that the joint distribution of the eigenvalues of the $\beta$-SRTMP process at long times can be computed analytically and it coincides with the joint distribution of the positions of the resetting Dyson Brownian motion in its stationary state for arbitrary $\beta>0$. For the $\beta$-IRTMP stationary ensemble, computing analytically the joint distribution of eigenvalues or even the average density of eigenvalues is difficult. However, generating the stationary $\beta$-IRTMP ensemble numerically is relatively straightforward and we compare its numerical average eigenvalue density to the corresponding analytical results for the $\beta$-SRTMP stationary ensemble with same parameter values, showing that they are quite different from each other. Finally, we provide a simple and concrete application of this tridiagonal matrix-valued process in computing the annealed partition function of a disordered quantum tight-binding Hamiltonian on a one-dimensional lattice.