×

The drift of a one-dimensional self-repellent random walk with bounded increments. (English) Zbl 0810.60095

Summary: Consider a one-dimensional walk \((S_ k)_ k\) having steps of bounded size, and weight the probability of the path with some factor \(1 - \alpha \in (0,1)\) for every single self-intersection up to time \(n\). We prove that \(S_ n/n\) converges towards some deterministic number called the effective drift of the self-repellent walk. Furthermore, this drift is shown to tend to the basic drift as \(\alpha\) tends to 0 and, as \(\alpha\) tends to 1, to the self-avoiding walk’s drift which is introduced in [author, ibid. 96, No. 4, 521-543 (1993; Zbl 0792.60097)]. The main tool of the present paper is a representation of the sequence of the local times as a functional of a certain Markov process.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
58E30 Variational principles in infinite-dimensional spaces
60F10 Large deviations
60G50 Sums of independent random variables; random walks

Citations:

Zbl 0792.60097
Full Text: DOI

References:

[1] de Acosta, A.: Upper bounds for large deviations of dependent random vectors. Z. Wahrscheinlichkeitstheor. Verw. Geb.69, 551-565 (1985) · Zbl 0547.60033 · doi:10.1007/BF00532666
[2] Aldous, D. J.: Self-intersections of 1-dimensional random walks. Probab. Theory Relat. Fields72, 559-587 (1986) · Zbl 0602.60055 · doi:10.1007/BF00344721
[3] Bolthausen, E.: On self-repellent one-dimensional random walks. Probab. Theory Relat. Fields86, 423-441 (1990) · Zbl 0691.60060 · doi:10.1007/BF01198167
[4] Deuschel, J.-D., Stroock, D.W.: Large deviations. Academic Press, Boston (1989) · Zbl 0705.60029
[5] Ellis, R.: Entropy, large deviations, and statistical mechanics (Grundlehren der mathematischen Wissenschaften 271) Springer-Verlag, New York (1984)
[6] Freed, K. F.: Polymers as self-avoiding walks. Ann. Probab.9, 537-556 (1981) · Zbl 0468.60097 · doi:10.1214/aop/1176994359
[7] de Gennes, P. G.: Scaling concepts in polymer physics. Cornell University Press, Ithaca (1979)
[8] Greven, A., den Hollander, F.: A variational characterization of the speed of a one-dimensional self-repellent random walk. Ann. Appl. Probab.3, 1067-1099 (1993) · Zbl 0784.60094 · doi:10.1214/aoap/1177005273
[9] v. d. Hofstad, R.: Scaling for a random polymer. Master thesis, University of Utrecht (1993)
[10] König, W.: The drift of a one-dimensional self-avoiding random walk. Probab. Theory Relat. Fields.96, 521-543 (1993) · Zbl 0792.60097 · doi:10.1007/BF01200208
[11] Kusuoka, S.: Asymptotics of the nolymer measure in one dimension. In: Infinite dimensional analysis and stochastic processes. Sem. Meet. Bielefeld 1983. Albeverio, S. (ed.) (Res. Notes Math. 124) Pitman Advanced Publishing Program IX. Boston, Pitman, 66-82 (1985)
[12] Madras, N., Slade, G.: The self-avoiding walk. Birkhäuser, Boston (1993) · Zbl 0780.60103
[13] Seneta, E.: Non-negative matrices and markov chains. Springer, New York (1981) · Zbl 0471.60001
[14] Slade, G.: The self-avoiding walk. The Mathematical Intelligencer16, 29-35 (1994) · Zbl 0795.60065 · doi:10.1007/BF03026612
[15] Varadhan, S.R.S.: Asymptotic probabilities and differential equations. Commun. Pure Appl. Math.XIX, 261-286 (1966) · Zbl 0147.15503 · doi:10.1002/cpa.3160190303
[16] J. Westwater: On Edward’s model for polymer chains. In: Trends and developments in the eighties, Albeverio, S., Blanchard, P. (eds.) Bielefeld encounters in Math. Phys.IV/V (1982/83). World Science, Singapore (1984). · Zbl 0498.60095
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.