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

In this paper, we find a gap between the lower bound of the Bakry-Émery $N$-Ricci tensor ${\mathrm{R}\mathrm{i}\mathrm{c}}_N$ and the Bakry-Émery gradient estimate $\mathsf{B}\mathsf{E}$ 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 $w_{\beta }={\prod }_{i<j}^n\mathrm{\mid }{x}_i-x_j{\mathrm{\mid }}^{\beta }$ and any $N\in [n+\frac{\beta }{2}n(n-1),+\mathrm{\infty }]$, $\beta \ge 1⟹{\mathrm{R}\mathrm{i}\mathrm{c}}_N\ge 0\mathrm{\&}\mathsf{B}\mathsf{E}(0,N)$ hold; $0<\beta <1⟹{\mathrm{R}\mathrm{i}\mathrm{c}}_N\ge 0$ holds while $\mathsf{B}\mathsf{E}(0,N)$ does not, which shows a phase transition of the Dyson Brownian motion regarding the Bakry-Émery curvature bound in the small inverse temperature regime.