Abstract
We study certain self-interacting walks on the set of integers that choose to jump to the right or to the left randomly but influenced by the number of times they have previously jumped along the edges in the finite neighbourhood of their current position (in this paper, typically, we will discuss the case where one considers the neighbouring edges and the next-to-neighbouring edges). We survey a variety of possible behaviours, including some where the walk is eventually confined to an interval of large length. We also focus on certain “asymmetric” drifts, where we prove that with positive probability, the walks behave deterministically on large scale and move like \(n\mapsto c\sqrt{n}\) or like n↦clogn.
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In this terminology originating in the physics literature, “true” refers to the fact that this is a true walk – the distributions of the first n steps is consistent with the distribution of the first n + 1 steps – and this is not always the case for measures on self-avoiding or self-repelling walks; the term “myopic” random walk has also been used.
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Acknowledgements
BT thanks the kind hospitality of Ecole Normale Supérieure, Paris, where part of this work was done. The research of BT is partially supported by the Hungarian National Research Fund, grant no. K60708. WW’s research was supported in part by ANR-06-BLAN-00058. The cooperation of the authors is facilitated by the French–Hungarian bilateral mobility grant Balaton/02/2008.
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Erschler, A., Tóth, B., Werner, W. (2012). Some Locally Self-Interacting Walks on the Integers. In: Deuschel, JD., Gentz, B., König, W., von Renesse, M., Scheutzow, M., Schmock, U. (eds) Probability in Complex Physical Systems. Springer Proceedings in Mathematics, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23811-6_13
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DOI: https://doi.org/10.1007/978-3-642-23811-6_13
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