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Author van der Hofstad, Remco
Source SpringerLink
Content type Text
Publisher Springer-Verlag
File Format PDF
Copyright Year ©2001
Language English
Subject Domain (in DDC) Social sciences ♦ Sociology & anthropology
Abstract We prove ballistic behaviour in dimension one fora model of weakly self-avoiding walks where loops of length m are penalized by a factor e −β/mp with p∈ [0, 1] and β sufficiently large. Furthermore, we prove that the fluctuations around the linear drift satisfy a centrallimit theorem. The proof uses a variant of the lace expansion, together with an inductive analysis of the arising recursion relation.In particular, we derive the law of large numbers, first obtainedby Greven and den Hollander, and the central limit theorem, firstobtained by König, for the weakly self-avoiding walk (p = 0 and β > 0).Their proofs use large deviation theory for the Markov description of the local times of one-dimensional simple random walk.It is the first time that the lace expansion is used to provebehaviour that is not diffusive. It has previously been used by van der Hofstad, den Hollander and Sladeto prove diffusive behaviour in dimension d for p≥ 0 such that p > and β > 0 sufficiently small.The lace expansion presented here compares the above weaklyself-avoiding walk to strictly self-avoiding walk in dimension one, obtained when β = ∞, and shows that the difference in behaviour is small when β is large.
ISSN 01788051
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 2001-03-01
Publisher Place Berlin/Heidelberg
e-ISSN 14322064
Journal Probability Theory and Related Fields
Volume Number 119
Issue Number 3
Page Count 39
Starting Page 311
Ending Page 349

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Source: SpringerLink