The singular Gaussian scaling limit for a tagged particle in the infinite-dimensional Dyson model at $β=2$

Abstract

We prove a logarithmic central limit theorem for a tagged particle in infinite-dimensional Dyson Brownian motion at inverse temperature $β=2$, starting from the sine$_2$ point process. The fluctuation is of order $\sqrt{\log t}$, which is subdiffusive and substantially slower than the polynomial fluctuations of classical short-range interacting particle systems, reflecting the rigidity induced by long-range logarithmic repulsion. We further identify the scaling limit as the singular Gaussian field corresponding to the zero-Hurst parameter, which provides a realisation of the singular Gaussian field as a scaling limit of a tagged particle raised by Dürr, Goldstein and Lebowitz (Comm. Pure Appl. Math. 38 (1985)).