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

We study the diffusion of a particle with a time-dependent diffusion coefficient 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 ⟨x^{2n}(t)⟩ at any finite time t, and for any W(D) with finite moments ⟨D^{n}⟩. For t≫1, we demonstrate that the cumulants ⟨x^{2n}(t)⟩_{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}.