Abstract
We consider a nearest-neighbor random walk on ℤ, for which the probability of jumping along a bond of the lattice is proportional to exp[−g. (number of previous jumps along that bond)k], withg>0,k∈(0,1]. After a review of earlier results obtained for the casek=1 we outline the generalizations fork∈(0,1), obtaining a whole range of anomalous diffusion limits.
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References
D. Amit, G. Parisi, and L. Peliti, Asymptotic behaviour of the ‘true’ self-avoiding walk,Phys. Rev. B 27: 1635/1645 (1983).
P. Biane and M. Yor, Valeurs principals associées aux temps locaux browniens,Bull. Sci. Math. 111: 23–101 (1987).
P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968).
K. Itô and H. P. McKean,Diffusion Processes and Their Sample Paths (Springer-Verlag, 1965).
H. Kesten, M. V. Kozlov, and F. Spitzer, A limit law for random walk in random environment,Compositio Mathematica 30: 145–168 (1975).
F. B. Knight, Random walks and a sojourn density process of Brownian motion,Trans. AMS 109: 56–86 (1963).
G. Lawler,Intersections of Random Walks (Birkhäuser, Basel, 1991).
T. Lindvall, Weak convergence of probability measures and random functions in the function spaceD[0, ∞),J. Appl. Prob. 10: 109–121 (1973).
N. Madras and G. Slade,The Self-Avoiding Walk (Birkhäuser, Basel, 1993).
L. Peliti and L. Pietronero, Random walks with memory,Nuovo Cimento 10: 1–33 (1987).
D. Revuz and M. Yor,Continuous Martingales and Brownian Motion (Springer-Verlag, 1991).
B. Tóth, ‘True’ self-avoiding walk with bond repulsion on ℤ: Limit theorems,Ann. Prob., to appear.
B. Tóth,Generalized Ray-Knight Theory and Limit Theorems for Self-Interacting Random Walks, in preparation.
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Dedicated to Oliver Penrose on the occasion of his 65th birthday
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Tóth, B. ‘True’ self-avoiding walks with generalized bond repulsion on ℤ. J Stat Phys 77, 17–33 (1994). https://doi.org/10.1007/BF02186830
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DOI: https://doi.org/10.1007/BF02186830