Forum de Physique Statistique à l'École Normale Supérieure

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Mathis Gueneau 

7 October, 11h, Conf IV 

Title: Large Deviations in Switching Diffusion: Trajectory Condensation and Diffusivity BBP Transition

Abstract: We consider a Brownian particle whose diffusivity changes at Poissonian switching times, with each new value drawn independently from a bounded distribution W(D). Assuming that W(D) \sim C(D_{\max}-D)^\nu near its upper edge, we study rare ballistic displacements x(t)= yt, far beyond the typical diffusive scale. Our previous work [1] obtained the corresponding large-deviation rate function and revealed a rich structure of dynamical transitions depending on the value of \nu.

Here, we ask a complementary question: what do the trajectories realizing these rare displacements actually look like? We characterize the distribution of the displacement accumulated during a single interval between two switches, conditioned on the endpoint x(t)=yt. We show that the transitions in the rate function correspond to a dynamical condensation phenomenon. For moderate displacements, the total displacement is shared among O(t) microscopic increments. Beyond a critical regime, however, a single interval produces a macroscopic O(t) increment carrying a finite fraction of the total displacement.

A complementary picture emerges in the moment-generating function. There, the transition corresponds to a change from collective contributions of many diffusivity states to the selection of diffusivities close to D_{\max}. By reformulating the tilted dynamics as a rank-one random matrix problem, we identify this transition with a BBP-type spectral transition. Thus, the diffusivity BBP-type selection and trajectory condensation at fixed displacement provide two complementary descriptions of the same underlying mechanism.

Finally, we test the predicted condensate structure using rare-event sampling and find excellent agreement with the analytical results.

[1] M. Guéneau, S. N. Majumdar, and G. Schehr, Phys. Rev. Lett. 135, 067102 (2025)

[2] M. Guéneau, T. Shorlepp, S. N. Majumdar, and G. Schehr, in preparation (2026)

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