Abstract
We present a simple model of a random walk with partial memory, which we call the random memory walk. We introduce this model motivated by the belief that it mimics the behavior of the once-reinforced random walk in high dimensions and with small reinforcement. We establish the transience of the random memory walk in dimensions three and higher, and show that its scaling limit is a Brownian motion.
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To the memory of Vladas Sidoravicius
The author “Alexandre Stauffer” was supported by EPSRC Fellowship EP/N004566/1.
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Angel, O., Crawford, N., Kozma, G.: Localization for linearly edge reinforced random walks. Duke Math. J. 163(5), 889–921 (2014)
Dembo, A., Huang, R., Sidoravicius, V.: Walking within growing domains: recurrence versus transience. Electron. J. Probab. 19, 20 pp. (2014)
Dembo, A., Huang, R., Sidoravicius, V.: Monotone interaction of walk and graph: recurrence versus transience. Electron. Commun. Probab. 19, 12 pp. (2014)
Collevecchio, A.: One the transience of processes defined on Galton-Watson trees. Ann. Probab. 34(3), 870–878 (2006)
Collevecchio, A., Kious, D., Sidoravicius, V.: The Branching–Ruin number and the critical parameter of once? Reinforced random walk on trees. Commun. Pure Appl. Math. 73(1), 210–236 (2020)
Davis, B.: Reinforced random walk. Probab. Theory Relat. Fields 84(2), 203–229 (1990)
Disertori, M., Sabot, C., Tarrès, P.: Transience of edge-reinforced random walk. Commun. Math. Phys. 339(1), 121–148 (2015)
Durrett, R., Kesten, H., Limic, V.: Once edge-reinforced random walk on a tree. Probab. Theory Relat. Fields 122(4), 567–592 (2002)
Holmes, M.P.: The scaling limit of senile reinforced random walk. Electron. Commun. Probab. 14, 104–115 (2009)
Holmes, M.P., Sakai, A.: Senile reinforced random walks. Stoch. Process. Appl. 117, 1519–1539 (2007)
Huang, R.: On random walk on growing graphs. Ann. Inst. H. Poincaré Probab. Stat. 55, 1149–1162 (2019)
Kious, D., Sidoravicius, V.: Phase transition for the once-reinforced random walk on Zd-like trees. Ann. Probab. 46(4), 2121–2133 (2018)
Kious, D., Schapira, B., Singh, A.: Once reinforced random walk on Z x Gamma (2018). Preprint, arXiv:1807.07167
Kozma, G.: Reinforced random walks (2012). Preprint, arXiv:1208.0364
Sabot, C., Tarrès, P.: Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model. J. Eur. Math. Soc. 17(9), 2353–2378 (2015)
Sabot, C., Zeng, X.: A random Schrödinger operator associated with the Vertex Reinforced Jump Process on infinite graphs. J. Am. Math. Soc. 32, 311–349 (2019)
Sellke, T.: Recurrence of reinforced random walk on a ladder. Electron. J. Probab. 11, 301–310 (2006)
Spitzer, F.: Principles of Random Walk, 2nd edn. Springer, Berlin (1976)
Sznitman, A.-S.: Slowdown estimates and central limit theorem for random walks in random environment. J. Eur. Math. Soc. 2, 93–143 (2000)
Sznitman, A.-S., Zerner, M.: A law of large numbers for random walks in random environment. Ann. Probab. 27(4), 1851–1869 (1999)
Vervoort, M.: Reinforced random walks (2002)
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Fribergh, A., Kious, D., Sidoravicius, V., Stauffer, A. (2021). Random Memory Walk. In: Vares, M.E., Fernández, R., Fontes, L.R., Newman, C.M. (eds) In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius. Progress in Probability, vol 77. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-60754-8_20
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DOI: https://doi.org/10.1007/978-3-030-60754-8_20
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