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, ρT = 1/T∫₀T δ( 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 → ∞, the local time density ρT obeys the large deviation principle, P(ρT= ρ) ~ e-T I(ρ), where the rate function $I(ρ)$ is analytic everywhere for d≤ 4. In contrast, for $d>4$, $I(ρ)$ becomes nonanalytic at a specific point ρ=ρc⁽ᵈ⁾, where ρc⁽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.