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An expansion for self-interacting random walks
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In this article, the authors derived a perturbation expansion for general self-interacting random walks, where steps are made on the basis of the history of the path, and showed that the expansion gives rise to useful formulae for the speed and variance of the random walk, when these quantities are known to exist.Abstract:
We derive a perturbation expansion for general self-interacting random walks, where steps are made on the basis of the history of the path. Examples of models where this expansion applies are reinforced random walk, excited random walk, the true (weakly) self-avoiding walk, loop-erased random walk, and annealed random walk in random environment.
In this paper we show that the expansion gives rise to useful formulae for the speed and variance of the random walk, when these quantities are known to exist. The results and formulae of this paper have been used elsewhere by the authors to prove monotonicity properties for the speed (in high dimensions) of excited random walk and related models, and certain models of random walk in random environment.
We also derive a law of large numbers and central limit theorem (with explicit error terms) directly from this expansion, under strong assumptions on the expansion coefficients. The assumptions are shown to be satisfied by excited random walk in high dimensions with small excitation parameter, a model of reinforced random walk with underlying drift and small reinforcement parameter, and certain models of random walk in random environment under strong ellipticity conditions. This is the extended version of the paper, where we provide all proofs.read more
Citations
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Positively and negatively excited random walks on integers, with branching processes
TL;DR: In this paper, the authors consider excited random walks on the integers with a bounded number of i.i.d. cookies per site which may induce drifts both to the left and to the right.
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Ballistic phase of self-interacting random walks
Dmitry Ioffe,Yvan Velenik +1 more
TL;DR: In this paper, a unified approach to a study of ballistic phase for a large family of selfinteracting random walks with a drift and self-interacting polymers with an external stretching force is presented.
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Central Limit Theorem For The Excited Random Walk In Dimension $d\geq 2$
Jean Bérard,Alejandro F. Ramírez +1 more
TL;DR: In this article, it was shown that a law of large numbers and a central limit theorem hold for the excited random walk model in every dimension of the model, and that the central limit is tight in the case of excited random walks.
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Monotonicity for excited random walk in high dimensions
TL;DR: In this paper, it was shown that the drift for excited random walk in dimension $d$ is monotone in the excitement parameter when $d\ge 9$ when d = 0, 1.
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Central Limit Theorem for the Excited Random Walk in dimension $d \geq 2$
Jean Bérard,Alejandro F. Ramírez +1 more
TL;DR: In this paper, it was shown that a law of large numbers and a central limit theorem hold for the excited random walk model in every dimension of the model, and that the central limit is tight in the case of excited random walks.
References
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Book
Large Deviations Techniques and Applications
Amir Dembo,Ofer Zeitouni +1 more
TL;DR: The LDP for Abstract Empirical Measures and applications-The Finite Dimensional Case and Applications of Empirically Measures LDP are presented.
Journal ArticleDOI
A survey of random processes with reinforcement
TL;DR: The models surveyed in this paper include generalized Polya urns, reinforced random walks, interacting urn models, and continuous reinforced processes, with a focus on methods and results, with sketches provided of some proofs.
Journal ArticleDOI
Self-Avoiding Walk in 5 or More Dimensions
David C. Brydges,Thomas Spencer +1 more
TL;DR: In this paper, it was shown that for a T step weakly self-avoiding random walk in five or more dimensions the variance of the endpoint is of orderT and the scaling limit is gaussian, asT→∞.
Journal ArticleDOI
Phase transition in reinforced random walk and RWRE on trees
TL;DR: In this paper, the authors show that the reinforced random walk can vary from transient to recurrent, depending on the value of an adjustable parameter measuring the strength of the feedback, which is calculated at the phase transition.
Journal ArticleDOI
Excited Random Walk
Itai Benjamini,David B. Wilson +1 more
TL;DR: In this paper, it was shown that an excited random walk on a polygonal grid is transient iff $d > 1, where d is the number of vertices in the grid.