Transition matrix from a random walk

TL;DR

This paper introduces a method to derive transition probability matrices from a given random walk, testing its accuracy across various scenarios including non-ergodic and detailed balance-violating systems.

Contribution

It presents a novel reverse approach to estimate transition matrices from observed paths, applicable even in non-ergodic and complex systems.

Findings

The method accurately reconstructs transition matrices from paths.

It performs well with random and metastable configurations.

It handles systems violating detailed balance.

Abstract

Given a random walk a method is presented to produce a matrix of transition probabilities that is consistent with that random walk. The method is a kind of reverse application of the usual ergodicity and is tested by using a transition matrix to produce a path and then using that path to create the estimate. The two matrices and their predictions are then compared. A variety of situations test the method, random matrices, metastable configurations (for which ergodicity often does not apply) and explicit violation of detailed balance.