Tridiagonal Models for Dyson Brownian Motion

Abstract

In this paper, we consider tridiagonal matrices the eigenvalues of which evolve according to $\beta$-Dyson Brownian motion. This is the stochastic gradient flow on $\mathbb{R}^n$ given by, for all $1 \leq i \leq n,$ \[ d\lambda_{i,t} = \sqrt{\frac{2}{\beta}}dZ_{i,t} - \biggl( \frac{V'(\lambda_i)}{2} - \sum_{j: j \neq i} \frac{1}{\lambda_i - \lambda_j} \biggr)\,dt \] where $V$ is a constraining potential and $\left\{ Z_{i,t} \right\}_1^n$ are independent standard Brownian motions. This flow is stationary with respect to the distribution \[ \rho^{\beta}_N(\lambda) = \frac{1}{Z^{\beta}_N} e^{-\frac{\beta}{2} \left( -\sum_{1 \leq i \neq j \leq N} \log|\lambda_i - \lambda_j| + \sum_{i=1}^N V(\lambda_i) \right) }. \] The particular choice of $V(t)=2t^2$ leads to an eigenvalue distribution constrained to lie roughly in $(-\sqrt{n},\sqrt{n}).$ We study evolution of the entries of one choice of tridiagonal flow for this $V$ in the $n\to \infty$ limit. On the way to describing the evolution of the tridiagonal matrices we give the derivative of the Lanczos tridiagonalization algorithm under perturbation.