Large spin large deviations for interacting particle systems

Abstract

We propose a novel framework for analyzing the sample-path large deviations of interacting particle systems directly on a lattice by considering a large-spin limit ($s \to \infty$). Through an appropriate scaling of variables, the process concentrates around its deterministic trajectory, governed by the lattice diffusion equation. We quantify the exponential cost of observing atypical trajectories by explicitly computing the path-space large deviation rate function from the scaled microscopic generator, based on a theory due to Feng and Kurtz. The rate function is described in terms of a Hamiltonian on a lattice in terms of classical spin variables. Notably, for our main model, the symmetric harmonic process, the corresponding Hamiltonian is identified with the classical integrable non-compact Ishimori spin chain. Furthermore, we show that our large spin large deviation theory is fully consistent with traditional Macroscopic Fluctuation Theory (MFT), smoothly recovering the standard MFT Hamiltonian in the spatial continuous limit. We also perform a Poisson-noise Martin–Siggia–Rose (MSR) computation to derive the non-linear lattice Hamiltonian directly from microscopic dynamics, use a relative entropy argument to obtain the density-current action, and establish the lattice fluctuation theorem alongside its symmetry origin. Overall, our framework provides a naturally regularized, lattice analogue of MFT that circumvents spatial ultraviolet singularities and possible continuous density blow-ups. We also apply our new scheme to partial symmetric exclusion process, symmetric inclusion processes, Brownian energy process and integrable heat conduction model, showing how their large-deviation Hamiltonians relate to classical compact and non-compact spin chains.