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February 2023 Diffusivity of a walk on fractures of a hypertorus
Piet Lammers
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 59(1): 208-229 (February 2023). DOI: 10.1214/22-AIHP1257


This article studies discrete height functions on the discrete hypertorus. These are functions on the vertices of this hypertorus graph for which the derivative satisfies a specific condition on each edge. We then perform a random walk on the set of such height functions, in the spirit of Diffusivity of a random walk on random walks, a work of Boissard, Cohen, Espinasse, and Norris. The goal is to estimate the diffusivity of this random walk in the mesh limit. It turns out that each height functions is characterised by a number of so-called fractures of the hypertorus. These fractures are then studied in isolation; we are able to understand their asymptotic behaviour in the mesh limit due to the recent understanding of the associated random surfaces. This allows for an asymptotic reduction to a one-dimensional continuous system consisting of gcdn parts where nNd is the fundamental parameter of the original model. We then prove that the diffusivity of the random walk tends to 1/(1+2gcdn) in this mesh limit.

Cet article étudie des fonctions de hauteur discrètes sur l’hypertore discret. Il s’agit de fonctions définies sur les sommets de l’hypertore dont la dérivée satisfait une certaine condition en chaque arête. Nous considérons une marche aléatoire sur cet ensemble de fonctions, à l’instar des travaux de Boissard, Cohen, Espinasse et Norris dans leur article Diffusivity of a random walk on random walks. L’objectif est d’estimer la diffusivité de cette marche aléatoire dans la limite d’échelle. Nous montrons que toute fonction de hauteur est caractérisée par le nombre de fractures qu’elle induit sur l’hypertore. Nous étudions ensuite ces fractures ; il est possible de comprendre leur comportement asymptotique dans la limite d’échelle grâce à de récents travaux sur les surfaces aléatoires qui leur sont associées. Cela permet de réduire notre étude asymptotique à un système continu à une dimension, constitué de pgcdn parties, où nNd est un paramètre fondamental du modèle initial. Nous montrons alors que la diffusivité de la marche aléatoire converge vers 1/(1+2pgcdn) dans cette limite d’échelle.

Funding Statement

The author was supported by the Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, the UK Engineering and Physical Sciences Research Council grant EP/L016516/1.


The author would like to thank James Norris for the inspiration to study walks on height functions, and for the supervision of this project. The author would also like to thank the anonymous referee for their constructive feedback on the manuscript. Finally, many thanks to François Jacopin for translating the abstract into French.


Download Citation

Piet Lammers. "Diffusivity of a walk on fractures of a hypertorus." Ann. Inst. H. Poincaré Probab. Statist. 59 (1) 208 - 229, February 2023.


Received: 11 September 2020; Revised: 29 October 2021; Accepted: 9 February 2022; Published: February 2023
First available in Project Euclid: 16 January 2023

Digital Object Identifier: 10.1214/22-AIHP1257

Primary: 60J10
Secondary: 60F05

Keywords: central limit theorem , Markov chain , Martingale approximation , Random walk

Rights: Copyright © 2023 Association des Publications de l’Institut Henri Poincaré


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Vol.59 • No. 1 • February 2023
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