The Self-Avoiding-Walk and Percolation Critical Points in High Dimensions

1995 ◽  
Vol 4 (3) ◽  
pp. 197-215 ◽  
Author(s):  
Takashi Hara ◽  
Gordon Slade
Keyword(s):  
Critical Point ◽  
Error Bound ◽  
The Self ◽  
Nearest Neighbour ◽  
High Dimensions ◽  
Lace Expansion ◽  
Connective Constant ◽  
Rigorous Error ◽  
Self Avoiding Walk ◽  
Self Avoiding Walks

We prove the existence of an asymptotic expansion in the inverse dimension, to all orders, for the connective constant for self-avoiding walks on ℤd. For the critical point, defined as the reciprocal of the connective constant, the coefficients of the expansion are computed through orderd−6, with a rigorous error bound of orderd−7Our method for computing terms in the expansion also applies to percolation, and for nearest-neighbour independent Bernoulli bond percolation on ℤdgives the 1/d-expansion for the critical point through orderd−3, with a rigorous error bound of orderd−4The method uses the lace expansion.

THE LACE EXPANSION FOR SELF-AVOIDING WALK IN FIVE OR MORE DIMENSIONS

1992 ◽  
Vol 04 (02) ◽  
pp. 235-327 ◽  
Author(s):  
TAKASHI HARA ◽  
GORDON SLADE
Keyword(s):  
Random Walk ◽  
Critical Point ◽  
Simple Random Walk ◽  
Companion Paper ◽  
Computer Assisted ◽  
Lace Expansion ◽  
Connective Constant ◽  
Elementary Methods ◽  
Self Avoiding Walk ◽  
Proof Of Convergence

This paper is a continuation of the companion paper [14], in which it was proved that the standard model of self-avoiding walk in five or more dimensions has the same critical behaviour as the simple random walk, assuming convergence of the lace expansion. In this paper we prove the convergence of the lace expansion, an upper and lower infrared bound, and a number of other estimates that were used in the companion paper. The proof requires a good upper bound on the critical point (or equivalently a lower bound on the connective constant). In an appendix, new upper bounds on the critical point in dimensions higher than two are obtained, using elementary methods which are independent of the lace expansion. The proof of convergence of the lace expansion is computer assisted. Numerical aspects of the proof, including methods for the numerical evaluation of simple random walk quantities such as the two-point function (or lattice Green function), are treated in an appendix.

The number of polygons on a lattice

1961 ◽  
Vol 57 (3) ◽  
pp. 516-523 ◽  
Author(s):  
J. M. Hammersley
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
End Points ◽  
Unit Distance ◽  
Connective Constant ◽  
Image Position ◽  
Self Avoiding Walk ◽  
Self Avoiding Walks

In this paper an n-stepped self-avoiding walk is defined to be an ordered sequence of n + 1 mutually distinct points, each with (positive, negative, or zero) integer coordinates in d-dimensional Euclidean space (where d is fixed and d ≥ 2), such that any two successive points in the sequence are neighbours, i.e. are unit distance apart. If further the first and last points of such a sequence are neighbours, the sequence is called an (n + 1)-sided self-avoiding polygon. Clearly, under this definition a polygon must have an even number of sides. Let f(n) and g(n) denote the numbers of n-stepped self-avoiding walks and of n-sided self-avoiding polygons having a prescribed first point. In a previous paper (3), I proved that there exists a connective constant K such thatHere I shall prove the truth of the long-standing conjecture thatI shall also show that (2) is a particular case of an expression for the number of n-stepped self-avoiding walks with prescribed end-points, a distance o(n) apart, this being another old and popular conjecture.

scholarly journals New lower bounds on the self-avoiding-walk connective constant

1993 ◽  
Vol 72 (3-4) ◽  
pp. 479-517 ◽  
Author(s):  
Takashi Hara ◽  
Gordon Slade ◽  
Alan D. Sokal
Keyword(s):  
Lower Bounds ◽  
The Self ◽  
Connective Constant ◽  
Self Avoiding Walk

scholarly journals Erratum: New lower bounds on the self-avoiding-walk connective constant

1995 ◽  
Vol 78 (3-4) ◽  
pp. 1187-1188 ◽  
Author(s):  
Takashi Hara ◽  
Gordon Slade ◽  
Alan D. Sokal
Keyword(s):  
Lower Bounds ◽  
The Self ◽  
Connective Constant ◽  
Self Avoiding Walk

scholarly journals Expansion in High Dimension for the Growth Constants of Lattice Trees and Lattice Animals

2013 ◽  
Vol 22 (4) ◽  
pp. 527-565 ◽  
Author(s):  
YURI MEJÍA MIRANDA ◽  
GORDON SLADE
Keyword(s):  
Error Estimates ◽  
High Dimension ◽  
Integer Lattice ◽  
Nearest Neighbour ◽  
Lattice Animals ◽  
Lace Expansion ◽  
Lattice Trees ◽  
Growth Constants ◽  
The One ◽  
Rigorous Error

We compute the first three terms of the 1/d expansions for the growth constants and one-point functions of nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice $\mathbb{Z}^d$, with rigorous error estimates. The proof uses the lace expansion, together with a new expansion for the one-point functions based on inclusion–exclusion.

Sign in / Sign up

Export Citation Format

Share Document