TY - JOUR
T1 - Critical behaviour of self-avoiding walk in five or more dimensions
AU - Hara, Takashi
AU - Slade, Gordon
N1 - Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.
PY - 1991/10
Y1 - 1991/10
N2 - We use the lace expansion to prove that in five or more dimensions the standard self-avoiding walk on the hypercubic (integer) lattice behaves in many respects like the simple random walk. In particular, it is shown that the leading asymptotic behaviour of the number of n-step self-avoiding walks is purely exponential, that the mean square displacement is asymptotically linear in the number of steps, and that the scaling limit is Gaussian, in the sense of convergence in distribution to Brownian motion. Some related facts are also proved. These results are optimal, according to the widely believed conjecture that the self-avoiding walk behaves unlike the simple random walk in dimensions two, three and four.
AB - We use the lace expansion to prove that in five or more dimensions the standard self-avoiding walk on the hypercubic (integer) lattice behaves in many respects like the simple random walk. In particular, it is shown that the leading asymptotic behaviour of the number of n-step self-avoiding walks is purely exponential, that the mean square displacement is asymptotically linear in the number of steps, and that the scaling limit is Gaussian, in the sense of convergence in distribution to Brownian motion. Some related facts are also proved. These results are optimal, according to the widely believed conjecture that the self-avoiding walk behaves unlike the simple random walk in dimensions two, three and four.
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U2 - 10.1090/S0273-0979-1991-16085-4
DO - 10.1090/S0273-0979-1991-16085-4
M3 - Comment/debate
AN - SCOPUS:84967728367
VL - 25
SP - 417
EP - 423
JO - Bulletin of the American Mathematical Society
JF - Bulletin of the American Mathematical Society
SN - 0273-0979
IS - 2
ER -