Summary
We prove that a self-avoiding random walk on the integers with bounded increments grows linearly. We characterize its drift in terms of the Frobenius eigenvalue of a certain one parameter family of primitive matrices. As an important tool, we express the local times as a two-block functional of a certain Markov chain, which is of independent interest.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aldous, D.J.: Self-intersections of 1-dimensional random walks. Probab. Theory Relat. Fields72, 559–587 (1986)
Bolthausen, E.: On self-repellent one-dimensional random walks. Probab. Theory Relat. Fields86, 423–441 (1990)
Deuschel, J.-D., Stroock, D.W.: Large deviations. Boston: Academic Press 1989
Ellis, R.: Entropy, large deviations, and statistical mechanics. Grundlehren der mathematischen Wissenschaften vol. 271. New York: Springer 1984
Freed, K.F.: Polymers as self-avoiding walks. Ann. Appl. Probab.9, 537–556 (1981)
Greven, A., den Hollander, F.: A variational characterization of the speed of a one-dimensional self-repellent random walk. Ann. Appl. Probab. (to appear)
König, W.: The drift of a one-dimensional self-repellent random walk with bounded increments (in preparation)
Kusuoka, S.: Asymptotics of the polymer measure in one dimension. Infinite dimensional analysis and stochastic processes, Sem. Meet. Bielefeld 1983. Albeverio, S. (ed.). Res. Notes Math. vol. 124, pp. 66–82. Pitman Advanced Publishing Program. Boston: Pitman 1985
Seneta, E.: Non-negative matrices and Markov chains. New York: Springer 1981
Varadhan, S.R.S.: Asymptotic probabilities and differential equations. Commun. Pure Appl. Math.XIX, 261–286 (1966)
Westwater, J.: On Edward's model for polymer chains. Trends and developments in the eighties. Albeverio, S., Blanchard, P. (eds.). Bielefeld encounters in Math./Phys. Singapore: World Scientific 1984