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, ${\ensuremath{\rho}}_{T}=\frac{1}{T}{\ensuremath{\int}}_{0}^{T}\ensuremath{\delta}(r(t)\ensuremath{-}1)dt$, 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, $T\ensuremath{\rightarrow}\ensuremath{\infty}$, the local time density ${\ensuremath{\rho}}_{T}$ obeys the large deviation principle, $P({\ensuremath{\rho}}_{T}=\ensuremath{\rho})\ensuremath{\sim}{e}^{\ensuremath{-}TI(\ensuremath{\rho})}$, where the rate function $I(\ensuremath{\rho})$ is analytic everywhere for $d\ensuremath{\le}4$. In contrast, for $d>4, I(\ensuremath{\rho})$ becomes nonanalytic at a specific point $\ensuremath{\rho}={\ensuremath{\rho}}_{c}^{(d)}$, where ${\ensuremath{\rho}}_{c}^{(d)}=d(d\ensuremath{-}4)/(2d\ensuremath{-}4)$ 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.