Dimensionality-induced dynamical phase transition in the large deviation of local time density for Brownian motion

We study the fluctuation properties of the local time density, ${\rho }_T=\frac{1}{T}{\int }_0^T\delta (r(t)-1)dt$, spent by a $d$-dimensional Brownian particle at a spherical shell of unit radius, where $r(t)$ denotes the radial distance from the particle to the origin. In the large observation time limit, $T\to \mathrm{\infty }$, the local time density ${\rho }_T$ obeys the large deviation principle, $P({\rho }_T=\rho )\sim e^{-TI(\rho )}$, where the rate function $I(\rho )$ is analytic everywhere for $d\le 4$. In contrast, for $d>4$, $I(\rho )$ becomes nonanalytic at a specific point $\rho ={\rho }_c^{(d)}$, where ${\rho }_c^{(d)}=d(d-4)\mathrm{/}(2d-4)$ depends solely on dimensionality. The singularity signals the occurrence of a first-order dynamical phase transition in dimensions higher than four. Such a transition is accompanied by temporal phase separations in the large deviations of Brownian trajectories. Finally, we validate our theoretical results using a rare-event simulation approach.