Upper Bounds for the Connective Constant of Self-Avoiding Walks

Alm, Sven Erick

doi:10.1017/s0963548300000547

Cited by 31 publications

(57 citation statements)

References 22 publications

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“…These bounds are sharper than the one obtained by Alm [2], µ tri < 4.277799, using n = 16 and m = 6 (Alm has since improved this to µ tri < 4.25152 [3]). For the hexagonal lattice we have c 100 = 2585241775338665938539885252 and c 99 = 1394474897269109512317080364, which gives us the upper bounds µ 1 = 1.871004, µ 2 = 1.869731 and µ a = 1.869836.…”

Section: Upper Boundsmentioning

confidence: 88%

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Improved lower bounds on the connective constants for two-dimensional self-avoiding walks

Jensen

2004

J. Phys. A: Math. Gen.

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Abstract. We calculate improved lower bounds for the connective constants for selfavoiding walks on the square, hexagonal, triangular, (4.82 ), and (3.12 2 ) lattices. The bound is found by Kesten's method of irreducible bridges. This involves using transfermatrix techniques to exactly enumerate the number of bridges of a given span to very many steps. Upper bounds are obtained from recent exact enumeration data for the number of self-avoiding walks and compared to current best available upper bounds from other methods.

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Section: Upper Boundsmentioning

confidence: 88%

“…The best current method for obtaining upper bounds is due to Alm [2] and it essentially requires one to enumerate the number of walks according to length n and a specified head and tail each of length m. More precisely Alm showed that…”

Section: Upper Boundsmentioning

confidence: 99%

Improved lower bounds on the connective constants for two-dimensional self-avoiding walks

Jensen

2004

J. Phys. A: Math. Gen.

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“…For the square lattice it has been shown rigorously that c ‫ޚ‬ 2 ≤ 2.7 (see [1]), which implies that we have no infinite self-avoiding path with bounded partial sums for p ∈ [0, 0.035) ∪ (0.965, 1].…”

Section: Theoremmentioning

confidence: 99%

Infinite paths with bounded or recurrent partial sums

Booth¹,

Meester²

2001

Probab Theory Relat Fields

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We consider problems of the following type. Assign independently to each vertex of the square lattice the value +1, with probability p, or −1, with probability 1 − p. We ask whether an infinite path π exists, with the property that the partial sums of the ±1s along π are uniformly bounded, and whether there exists an infinite path π' with the property that the partial sums along π' are equal to zero infinitely often. The answers to these question depend on the type of path one allows, the value of p and the uniform bound specified. We show that phase transitions occur for these phenomena. Moreover, we make a surprising connection between the problem of finding a path to infinity (not necessarily self-avoiding, but visiting each vertex at most finitely many times) with a given bound on the partial sums, and the classical Boolean model with squares around the points of a Poisson process in the plane. For the recurrence problem, we also show that the probability of finding such a path is monotone in p, for p ≥ 1 2 .

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“…But it is rigorously known that µ 2 ≤ 2.69576, µ 3 ≤ 4.756, µ 4 ≤ 6.832 [1], and it is expected that µ 2 = 2.638, µ 3 = 4.683, µ 4 = 6.775 [18]. The numerical values using Corollary 1 and the above upper bounds and expected values of µ ν are listed in the Table, and they are in good agreement with experimental results except for two dimensional cases.…”

Section: Lemma 2 the Representationsmentioning

confidence: 99%