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

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 $⟨x^{2n}(t)⟩$ at any finite time $t$, and for any $W(D)$ with finite moments $⟨D^n⟩$. For $t\gg 1$, 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 $I(y=x\mathrm{/}t)$. Our analytical predictions are validated numerically with high precision, achieving accuracy up to ${10}^{-2000}$.