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Fluctuations of the local times of the self-repelling random walk with directed edges

Part of: Stochastic processes Special processes Limit theorems Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  15 September 2023

Laure Marêché
Laure Marêché*
Affiliation:
Institut de Recherche Mathématique Avancée, UMR 7501 Université de Strasbourg et CNRS
*
*Postal address: 7 rue René Descartes, 67000 Strasbourg, France. Email address: laure.mareche@math.unistra.fr
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Abstract

In 2008, Tóth and Vető defined the self-repelling random walk with directed edges as a non-Markovian random walk on $\unicode{x2124}$: in this model, the probability that the walk moves from a point of $\unicode{x2124}$ to a given neighbor depends on the number of previous crossings of the directed edge from the initial point to the target, called the local time of the edge. Tóth and Vető found that this model exhibited very peculiar behavior, as the process formed by the local times of all the edges, evaluated at a stopping time of a certain type and suitably renormalized, converges to a deterministic process, instead of a random one as in similar models. In this work, we study the fluctuations of the local times process around its deterministic limit, about which nothing was previously known. We prove that these fluctuations converge in the Skorokhod $M_1$ topology, as well as in the uniform topology away from the discontinuities of the limit, but not in the most classical Skorokhod topology. We also prove the convergence of the fluctuations of the aforementioned stopping times.

Keywords

Self-interacting random walksself-repelling random walk with directed edgeslocal timesfunctional limit theoremsfluctuations

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G50: Sums of independent random variables; random walks 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82C41: Dynamics of random walks, random surfaces, lattice animals, etc.
Type
Original Article
Information
Advances in Applied Probability , Volume 56 , Issue 2 , June 2024 , pp. 545 - 586
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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