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, $T \to \infty$, the local time density $\rho_T$ obeys the large deviation principle, $P(\rho _T= \rho) \sim e^{-T I(\rho)}$, where the rate function $I(\rho)$ is analytic everywhere for $d\leq 4$. In contrast, for $d>4$, $I(\rho)$ becomes nonanalytic at a specific point $\rho=\rho_c^{(d)}$, where $\rho_c^{(d)}=d(d-4)/(2d-4)$ 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.