Diffusivity of a walk on fractures of a hypertorus. (English. French summary) Zbl 1508.60076

Summary: 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 [E. Boissard et al., Random Struct. Algorithms 47, No. 2, 267–283 (2015; Zbl 1325.60066)]. 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 \(\gcd\, \mathbf{n}\) parts where \(\mathbf{n}\in\mathbb{N}^d\) is the fundamental parameter of the original model. We then prove that the diffusivity of the random walk tends to \(1/(1+2\gcd\, \mathbf{n})\) in this mesh limit.


60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks


Zbl 1325.60066
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