# Kinetics of the one-dimensional voter model with long-range interactions

Source: [Kinetics of the one-dimensional voter model with long-range interactions](<https://www.sota2.com/research/paper/68fa47371e209131ce92648d>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T06:12:15.305Z

Authors: Federico Corberi, Claudio Castellano

Published: 2023-09-28T15:24:50.000Z

## Abstract

The one-dimensional long-range voter model, where an agent takes the opinion of another at distance $r$ with probability $\\propto r^{-\\alpha}$, is studied analytically. The model displays rich and diverse features as $\\alpha$ is changed. For $\\alpha \>3$ the behavior is similar to the one of the nearest-neighbor version, with the formation of ordered domains whose typical size grows as $R(t)\\propto t^{1/2}$ until consensus (a fully ordered configuration) is reached. The correlation function $C(r,t)$ between two agents at distance $r$ obeys dynamical scaling with sizeable corrections at large distances $r\>r^\*(t)$, slowly fading away in time. For $2\< \\alpha \\le 3$ violations of scaling appear, due to the simultaneous presence of two lengh-scales, the size of domains growing as $t^{(\\alpha-2)/(\\alpha-1)}$, and the distance $L(t)\\propto t^{1/(\\alpha-1)}$ over which correlations extend. For $\\alpha \\le 2$ the system reaches a partially ordered stationary state, characterised by an algebraic correlator, % $C(r)\\propto r^{-(2-\\alpha)}$, whose lifetime diverges in the thermodynamic limit of infinitely many agents, so that consensus is not reached. For a finite system escape towards the fully ordered configuration is finally promoted by development of large distance correlations. In a system of $N$ sites, global consensus is achieved after a time $T \\propto N^2$ for $\\alpha\>3$, $T \\propto N^{\\alpha-1}$ for $2\<\\alpha \\le 3$, and $T \\propto N$ for $\\alpha \\le 2$.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2309.16517>)

[PDF](<https://arxiv.org/pdf/2309.16517.pdf>)