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

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.