A phase transition in the Bakry-Émery gradient estimate for Dyson Brownian motion

Abstract

In this paper, we find a gap between the lower bound of the Bakry-Émery $N$-Ricci tensor ${\rm Ric}_N$ and the Bakry-Émery gradient estimate ${\sf BE}$ in the space associated with the finite-particle Dyson Brownian motion (DBM) with inverse temperature $0<β<1$. Namely, we prove that, for the weighted space $(\mathbb R^n, w_β)$ with $w_β=\prod_{i<j}^n |x_i-x_j|^β$ and any $N\in[n+\fracβ{2}n(n-1),+\infty]$, $β\ge 1 \implies {\rm Ric}_N \ge 0 \ \& \ {\sf BE}(0,N)$ hold; $0 < β< 1 \implies {\rm Ric}_N \ge 0$ holds while ${\sf BE}(0,N)$ does not, which shows a phase transition of the Dyson Brownian motion regarding the Bakry-Émery curvature bound in the small inverse temperature regime.