Abstract
We investigate the asymptotic behaviour of a class of self-interacting nearest neighbour random walks on the one-dimensional integer lattice which are pushed by a particular linear combination of their own local time on edges in the neighbourhood of their current position. We prove that in a range of the relevant parameter of the model such random walkers can be eventually confined to a finite interval of length depending on the parameter value. The phenomenon arises as a result of competing self-attracting and self-repelling effects where in the named parameter range the former wins.
References
Amit D., Parisi G., Peliti L.: Asymptotic behavior of the ‘true’ self-avoiding walk. Phys. Rev. B 27, 1635–1645 (1983)
Erschler, A., Tóth, B., Werner, W.: Some locally self-interacting walks on the integers. In: Festschrift in Honour of Jürgen Gärtner and Erwin Bolthausen. Springer, Berlin (to appear, 2011). http://arxiv.org/abs/1011.1102
Obukhov S.P., Peliti L.: Renormalisation of the “true” self-avoiding walk. J. Phys. A 16, L147–L151 (1983)
Peliti L., Pietronero L.: Random walks with memory. Riv. Nuovo Cimento 10, 1–33 (1987)
Pemantle R.: Vertex-reinforced random walk. Probab. Theory Relat. Fields 92, 117–136 (1992)
Pemantle R., Volkov S.: Vertex-reinforced random walk on \({{\mathbb Z}}\) has finite range. Ann. Probab. 27, 1368–1388 (1999)
Tarrès P.: Vertex-reinforced random walk on \({{\mathbb Z}}\) eventually gets stuck on five points. Ann. Probab. 32, 2650–2701 (2004)
Tarrès, P., Tóth, B., Valkó, B.: Diffusivity bounds for 1d Brownian polymers. Ann. Probab. (to appear, 2011). http://arxiv.org/abs/0911.2356
Tóth B.: ‘True’ self-avoiding walk with bond repulsion on \({{\mathbb Z}}\): limit theorems. Ann. Probab. 23, 1523–1556 (1995)
Tóth, B.: Self-interacting random motions. In: Proceedings of the 3rd European Congress of Mathematics, vol. 1, pp. 555–565. Birkhauser, Barcelona 2000 (2001)
Tóth B., Werner W.: The true self-repelling motion. Probab. Theory Relat. Fields 111, 375–452 (1998)
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Erschler, A., Tóth, B. & Werner, W. Stuck walks. Probab. Theory Relat. Fields 154, 149–163 (2012). https://doi.org/10.1007/s00440-011-0365-4
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DOI: https://doi.org/10.1007/s00440-011-0365-4
Keywords
- Self-interacting random walk
- Local time
- Trapping
Mathematics Subject Classification (2010)
- 60K37
- 60K99
- 60J55