The mean field stubborn voter model

TL;DR

This paper studies how heavy-tailed waiting times affect the voter model on a complete graph, deriving a new scaling limit and explicit consensus probabilities based on the tail index.

Contribution

It introduces a novel scaling limit for the voter model with heavy-tailed waiting times and characterizes the limiting behavior and consensus probabilities explicitly.

Findings

Derived a new scaling limit for the voter model with heavy-tailed waiting times.

Explicitly calculated consensus probabilities depending on the tail index.

Showed the coalescing system of random walks comes down from infinity.

Abstract

We analyse the effect of agent-dependent heavy-tailed waiting times in the voter model on the complete graph with $N$ vertices. We derive a novel scaling limit and show the existence of a limiting infinite voter model on the slowest updating sites. We further derive the consensus probabilities in the limit model explicitly. In the mean-field setting, the limit is determined by the extreme-value landscape of the waiting times and depends only on the tail index. To obtain these results, we study the coalescing system of random walks that is dual to the limit voter model and prove, among other auxiliary results, that it comes down from infinity.