The Infinite Dyson Brownian Motion with $β=2$ Does Not Have a Spectral Gap

TL;DR

This work addresses whether the unlabelled infinite Dyson Brownian motion at inverse temperature $\beta=2$ admits a spectral gap for its Dirichlet form. The authors construct a one-parameter family of linear statistics $U_\sigma$ from the $\mathsf{sine}_2$ point process via $u_\sigma(x)=x e^{-x^2/(2\sigma^2)}$ and compute both the variance $\mathrm{Var}(U_\sigma)$ using the intensity and two-point correlation $\rho^{(2)}$, and an upper bound for the energy $\mathcal E(U_\sigma)$. They show $\mathrm{Var}(U_\sigma) \ge \frac{\sigma^2}{4\pi}\left(1-e^{-4\pi^2\sigma^2}\right)$ while $\mathcal E(U_\sigma) \le \sqrt{\pi}\,\sigma + \frac{3\sqrt{\pi}}{4}\,\sigma$, leading to $\inf_{F} \frac{\mathcal E(F)}{\mathrm{Var}(F)} \le \frac{\mathcal E(U_\sigma)}{\mathrm{Var}(U_\sigma)} \to 0$ as $\sigma\to\infty$. This yields a negative result: no spectral gap for any closed extension of the Dirichlet form when $\beta=2$. The concluding discussion connects this to variance growth of linear statistics and hyperuniformity, suggesting that a spectral gap could only arise under substantially slower variance growth than in the Poisson or sine$_2$ cases.

Abstract

We prove that the Dirichlet forms associated with the unlabelled infinite Dyson Brownian motion with the inverse temperature $β=2$ do not have a spectral gap.

Theorems & Definitions (3)

• Theorem
• proof : Proof of the Theorem \ref{['t:maind']}
• proof : Proof of \ref{['c:s']}