# Direct evaluation of dynamical large-deviation rate functions using a variational ansatz

## TL;DR

This paper introduces a variational importance sampling method to efficiently bound and compute large-deviation rate functions for dynamical observables in Markov chains, without prior knowledge of rare behaviors.

## Contribution

It presents a simple, physically transparent variational ansatz-based importance sampling technique for directly estimating large-deviation rate functions in continuous-time Markov models.

## Key findings

- Bounds are tighter than Level 2.5 large deviation approximations.
- The method can recover exact rate functions with proper ansatz corrections.
- Applied successfully to network and lattice models from literature.

## Abstract

We describe a simple form of importance sampling designed to bound and compute large-deviation rate functions for time-extensive dynamical observables in continuous-time Markov chains. We start with a model, defined by a set of rates, and a time-extensive dynamical observable. We construct a reference model, a variational ansatz for the behavior of the original model conditioned on atypical values of the observable. Direct simulation of the reference model provides an upper bound on the large-deviation rate function associated with the original model, an estimate of the tightness of the bound, and, if the ansatz is chosen well, the exact rate function. The exact rare behavior of the original model does not need to be known in advance. We use this method to calculate rate functions for currents and counting observables in a set of network- and lattice models taken from the literature. Straightforward ansatze yield bounds that are tighter than bounds obtained from Level 2.5 of large deviations via approximations that involve uniform scalings of rates. We show how to correct these bounds in order to recover the rate functions exactly. Our approach is complementary to more specialized methods, and offers a physically transparent framework for approximating and calculating the likelihood of dynamical large deviations.