A new method for computing the expected hitting time between arbitrary different configurations of the multiple-urn Ehrenfest model

TL;DR

This paper introduces a novel method to compute the expected hitting time between different configurations in a multiple-urn Ehrenfest model, providing exact calculations and confirming a recent conjecture.

Contribution

It develops a new approach using stopping times to calculate expected hitting times in a generalized Ehrenfest model, solving an open conjecture.

Findings

Derived explicit formulas for expected hitting times

Confirmed the conjecture on moving all balls between two specific urns

Extended the analysis to arbitrary urn configurations

Abstract

We study a multiple-urn version of the Ehrenfest model. In this setting, we denote the n urns by Urn 1 to Urn n, where n>=2. Initially, M balls are randomly placed in the n urns. At each subsequent step, a ball is selected and put into the other n-1 urns with equal probability. The expected hitting time leading to a change of the M balls' status is computed using the method of stopping times. As a corollary, we obtain the expected hitting time of moving all the M balls from Urn 1 to Urn 2. This proves a conjecture which was recently made in Chen et al.(2017).