Large Deviations in Switching Diffusion: from Free Cumulants to Dynamical Transitions

TL;DR

This paper analyzes the large deviations of a particle's position in a switching diffusion process, deriving exact moments, cumulants, and rate functions, revealing dynamical transitions and validating results numerically with extraordinary precision.

Contribution

It provides an exact analytical framework for large deviations in switching diffusions, linking cumulants to free cumulants and uncovering dynamical transitions in the rate function.

Findings

Cumulants grow linearly with time and relate to free cumulants of $W(D)$

Explicit large deviation rate functions show rich behaviors and dynamical transitions

Numerical validation achieves precision up to $10^{-2000}$

Abstract

We study the diffusion of a particle with a time-dependent diffusion constant $D(t)$ that switches between random values drawn from a distribution $W(D)$ at a fixed rate $r$. Using a renewal approach, we compute exactly the moments of the position of the particle $\langle x^{2n}(t) \rangle$ at any finite time $t$, and for any $W(D)$ with finite moments $\langle D^n \rangle$. For $t \gg 1$, we demonstrate that the cumulants $\langle x^{2n}(t) \rangle_c$ grow linearly with $t$ and are proportional to the free cumulants of a random variable distributed according to $W(D)$. For specific forms of $W(D)$, we compute the large deviations of the position of the particle, uncovering rich behaviors and dynamical transitions of the rate function $I(y=x/t)$. Our analytical predictions are validated numerically with high precision, achieving accuracy up to $10^{-2000}$.