A phase transition in the Bakry-\'Emery gradient estimate for Dyson Brownian motion

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