Conditioned stochastic particle systems and integrable quantum spin systems

TL;DR

This paper explores large deviation properties of stochastic particle systems via their connection to integrable quantum spin systems, providing explicit results and new insights into conditioned dynamics and universal features.

Contribution

It introduces new analytical results for conditioned stochastic particle systems, including explicit formulas and expansions, using their mapping to integrable quantum spin models.

Findings

Explicit expansion of no-absorption probability for symmetric exclusion process

Disorder independence in effective dynamics under large current conditioning

Universal features in annihilating and coalescing random walkers

Abstract

We consider from a microscopic perspective large deviation properties of several stochastic interacting particle systems, using their mapping to integrable quantum spin systems. A brief review of recent work is given and several new results are presented: (i) For the general disordered symmectric exclusion process (SEP) on some finite lattice conditioned on no jumps into some absorbing sublattice and with initial Bernoulli product measure with density $\rho$ we prove that the probability $S_\rho(t)$ of no absorption event up to microscopic time $t$ can be expressed in terms of the generating function for the particle number of a SEP with particle injection and empty initial lattice. Specifically, for the symmetric simple exclusion process on $\mathbb Z$ conditioned on no jumps into the origin we obtain the explicit first and second order expansion in $\rho$ of $S_\rho(t)$ and also to first order in $\rho$ the optimal microscopic density profile under this conditioning. For the disordered ASEP on the finite torus conditioned on a very large current we show that the effective dynamics that optimally realizes this rare event does not depend on the disorder, except for the time scale. For annihilating and coalescing random walkers we obtain the generating function of the number of annihilated particles up to time $t$, which turns out to exhibit some universal features.