Abstract
IfŜ(n) is the position of the self-avoiding random walk in ℤd obtained by erasing loops from simple random walk, then it is proved that the mean square displacementE(¦Ŝ(n)¦2) grows at least as fast as the Flory predictions for the usual SAW, i.e., at least as fast asn 3/2 ford=2 andn 6/5 ford=3. In particular, if the mean square displacement of the usual SAW grows liken 1.18... ind=3, as expected, then the loop-erased process is in a different universality class.
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References
A. Brandt,Math. Comp. 20:473–499 (1966).
D. Brydges and T. Spencer,Commun. Math. Phys. 97:125–148 (1985).
C. Domb, inStochastic Processes in Chemical Physics, K. E. Shuler, ed. (Wiley, New York, 1969), pp. 229–260.
H. Kesten and F. Spitzer,J. Analyse Math. 11:285–322 (1963).
G. Lawler,Duke Math. J. 47:655–694 (1980).
G. Lawler,Duke Math. J. 53:249–269 (1986).
G. Lawler,Contemp. Math. 41:281–289 (1985).
G. Lawler, Low density expansion for a two-state random walk in a random environment, to appear.
G. Lawler,J. Appl. Prob. 20:264–276 (1983).
G. Lawler,J. Phys. A 20:4565–4568 (1987).
J. W. Lyklema and C. Evertsz, inFractals in Physics, L. Pietronero and E. Tosatti, eds. (Elsevier, 1986), pp. 87–92.
A. Renyi,Probability Theory (North-Holland, Amsterdam, 1970).
G. Slade,Commun. Math. Phys., to appear.
F. Spitzer,Principles of Random Walk, 2nd ed. (Springer-Verlag, New York, 1976).
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Lawler, G.F. Loop-erased self-avoiding random walk in two and three dimensions. J Stat Phys 50, 91–108 (1988). https://doi.org/10.1007/BF01022989
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DOI: https://doi.org/10.1007/BF01022989