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

Source: [Ordering Kinetics of the two-dimensional voter model with long-range interactions](<https://www.sota2.com/research/paper/68fa46c11e209131ce922be0>)

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

Data updated: 2025-10-25T05:58:06.488Z

Authors: Federico Corberi, Luca Smaldone

Published: 2023-12-01T17:25:40.000Z

## Abstract

We study analytically the ordering kinetics of the two-dimensional long-range voter model on a two-dimensional lattice, where agents on each vertex take the opinion of others at distance $r$ with probability $P(r) \\propto r^{-\\al}$. The model is characterized by different regimes, as $\\al$ is varied. For $\\al \> 4$ the behaviour is similar to that of the nearest-neighbor model, with the formation of ordered domains of a typical size growing as $L(t) \\propto \\sqrt{t}$, until consensus is reached in a time or order $N\\ln N$, $N$ being the number of agents. Dynamical scaling is violated due to an excess of interfacial sites whose density decays as slow as $\\rho(t) \\propto 1/\\ln t$. Sizable finite-time corrections are also present, which are absent in the case of nearest-neighbors interactions. For $0\<\\al \\leq 4$ standard scaling is reinstated, and the correlation length increases algebraically as $L(t)\\propto t^{1/z}$, with $1/z=2/\\al$ for $3\<\\al\<4$ and $1/z=2/3$ for $0\<\\al\<3$. In addition, for $\\al \\le 3$, $L(t)$ depends on $N$ at any time $t\>0$. Such coarsening, however, only leads the system to a partially ordered metastable state where correlations decay algebraically with distance, and whose lifetime diverges in the $N\\to \\infty$ limit. In finite systems consensus is reached in a time of order $N$ for any $\\al \<4$.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2312.00743>)

[PDF](<https://arxiv.org/pdf/2312.00743.pdf>)