A supersymmetric quantum perspective on the explicit large deviations for reversible Markov jump processes, with applications to pure and random spin chains (2403.11525v2)

Abstract: The large deviations at various levels that are explicit for Markov jump processes satisfying detailed-balance are revisited in terms of the supersymmetric quantum Hamiltonian $H$ that can be obtained from the Markov generator via a similarity transformation. We first focus on the large deviations at level 2 for the empirical density ${\hat p}(C) $ of the configurations $C$ seen during a trajectory over the large time-window $[0,T]$ and rewrite the explicit Donsker-Varadhan rate function as the matrix element $I^{[2]}[{\hat p}(.) ] = \langle \sqrt{ {\hat p} } \vert H \vert \sqrt{{\hat p}} \rangle $ involving the square-root ket $\vert \sqrt{{\hat p}} \rangle $. [The analog formula is also discussed for reversible diffusion processes as a comparison.] We then consider the explicit rate functions at higher levels, in particular for the joint probability of the empirical density ${\hat p}(C) $ and the empirical local activities ${\hat a}(C,C') $ characterizing the density of jumps between two configurations $(C,C')$. Finally, the explicit rate function for the joint probability of the empirical density ${\hat p}(C) $ and of the empirical total activity ${\hat A} $ that represents the total density of jumps of a long trajectory is written in terms of the two matrix elements $ \langle \sqrt{ {\hat p} } \vert H \vert \sqrt{{\hat p}} \rangle$ and $\langle \sqrt{ {\hat p} } \vert H^{off} \vert \sqrt{{\hat p}} \rangle $, where $H^{off} $ represents the off-diagonal part of the supersymmetric Hamiltonian $H$. This general formalism is then applied to pure or random spin chains with single-spin-flip or two-spin-flip transition rates, where the supersymmetric Hamiltonian $H$ correspond to quantum spin chains with local interactions involving Pauli matrices of two or three neighboring sites.