Large Deviations in Switching Diffusion: Dynamical Condensation and Diffusivity BBP Transition

Abstract

We consider a Brownian particle with position $x(t)$ whose diffusivity changes at Poissonian switching times, with each new value drawn independently from a bounded distribution $W(D)$. Assuming $W(D)\sim C(D_{\max}-D)^\nu$ near its upper edge, we study the large-deviation regime $x(t)\simeq yt$ with $y = O(1)$ at large times, far beyond the typical diffusive scale. In a recent work Gu\'eneau et al., Phys. Rev. Lett. 135, 067102 (2025), the rate function of the displacement was obtained through a renewal approach and it was shown to display several singularities. In this paper, we carry out a detailed study of the corresponding dynamical phase transitions and trajectory condensation. For moderate $y$, the imposed displacement is shared among $O(t)$ displacement increments accumulated during successive intervals between switches. Beyond a first threshold $y_-$, one interval produces an increment of order $t$ carrying a finite fraction of the total displacement, signaling dynamical condensation. For $\nu>1$, a second threshold $y_+>y_-$ is reached when this macroscopic increment carries asymptotically the entire imposed displacement, whereas for $0<\nu\leq1$ the two thresholds merge, $y_-=y_+$. In turn, the moment-generating function of $x(t)$ displays a complementary transition from collective contributions of many diffusivities to selection of states near $D_{\max}$. We show that this transition is governed by the same edge mechanism as the Baik-Ben Arous-P\'ech\'e (BBP) transition, indicating that diffusivity selection and trajectory condensation are two manifestations of the same phenomenon. We validate our predictions for the dynamical condensation against numerical results from rare event sampling.