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, ρ_{T}=1/T∫_{0}^{T}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 ρ_{T} obeys the large deviation principle, P(ρ_{T}=ρ)∼e^{-TI(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=ρ_{c}^{(d)}, where ρ_{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.