Voter Model stability with respect to conservative noises
Gideon Amir, Omer Angel, Rangel Baldasso, Daniel de la Riva
TL;DR
This work analyzes the robustness of consensus in finite-graph voter models under two conservative dynamical noises: a swap-based interchange process on edge activations and Brownian perturbations of clock times. Using a graphical construction and a pivotal-flip framework tied to coalescing random walks, it derives explicit stability bounds that depend on graph parameters such as the maximum degree $\Delta_{n}$ and edge count $|E_{n}|$. For the interchange noise, the consensus remains stable with probability at least $1-\,O\big( t n \Delta_{n}/|E_{n}| \big)$; for Brownian perturbations, stability holds when $|E_{n}|\sqrt{s}$ is small, with probability bounded by $O\big( n \Delta_{n} \sqrt{s} \big)$. The results illuminate how local graph geometry and perturbation scales govern dynamical noise sensitivity versus stability in interacting particle systems, extending previous static notions to dynamical settings via pivotality and duality with coalescing random walks.
Abstract
The notions of noise sensitivity and stability were recently extended for the voter model. In this model, the vertices of a graph have opinions that are updated by uniformly selecting edges. We further extend stability results to different classes of perturbations. We consider two different types of noise: in the first one, an exclusion process is performed on the edge selections, while in the second, independent Brownian motions are applied to such a sequence. In both cases, we prove stability of the consensus opinion provided the noise is run for a short amount of time, depending on the underlying graph structure. This is done by analyzing the expected size of the pivotal set, whose definition differs from the usual one in order to reflect the change associated with these noises.