Mean First Passage Time of the Symmetric Noisy Voter Model with Arbitrary Initial and Boundary Conditions

TL;DR

This paper derives exact formulas for the mean first passage time in a symmetric noisy voter model with arbitrary initial and boundary conditions, revealing how boundary placement and independent transition rates influence asymmetry and confirming Kramers' law at high transition rates.

Contribution

It provides novel analytical expressions for mean first passage time under asymmetric boundary conditions and demonstrates the conditions for symmetry and the applicability of Kramers' law.

Findings

Mean first passage time depends on boundary placement and initial conditions.

Symmetry occurs only when boundaries are equidistant from the midpoint.

Kramers' law holds at high independent transition rates.

Abstract

Models of imitation and herding behavior often underestimate the role of individualistic actions and assume symmetric boundary conditions. However, real-world systems (e.g., electoral processes) frequently involve asymmetric boundaries. In this study, we explore how arbitrarily placed boundary conditions influence the mean first passage time in the symmetric noisy voter model, and how individualistic behavior amplifies this asymmetry. We derive exact analytical expressions for mean first passage time that accommodate any initial condition and two types of boundary configurations: (i) both boundaries absorbing, and (ii) one absorbing and one reflective. In both scenarios, mean first passage time exhibits a clear asymmetry with respect to the initial condition, shaped by the boundary placement and the rate of independent transitions. Symmetry in mean first passage time emerges only when absorbing boundaries are equidistant from the midpoint. Additionally, we show that Kramers' law holds in both configurations when the rate of independent transitions is large. Our analytical results are in excellent agreement with numerical simulations, reinforcing the robustness of our findings.