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 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.

scholarly journals NoBLE for Lattice Trees and Lattice Animals

2021 ◽  
Vol 185 (2) ◽  
Author(s):  
Robert Fitzner ◽  
Remco van der Hofstad
Keyword(s):  
Random Walk ◽  
Nearest Neighbor ◽  
Mean Field ◽  
Critical Dimension ◽  
Simple Random Walk ◽  
Computer Assisted ◽  
High Dimensions ◽  
Lace Expansion ◽  
Upper Critical Dimension ◽  
Lattice Trees

AbstractWe study lattice trees (LTs) and animals (LAs) on the nearest-neighbor lattice $${\mathbb {Z}}^d$$ Z d in high dimensions. We prove that LTs and LAs display mean-field behavior above dimension $$16$$ 16 and $$17$$ 17 , respectively. Such results have previously been obtained by Hara and Slade in sufficiently high dimensions. The dimension above which their results apply was not yet specified. We rely on the non-backtracking lace expansion (NoBLE) method that we have recently developed. The NoBLE makes use of an alternative lace expansion for LAs and LTs that perturbs around non-backtracking random walk rather than around simple random walk, leading to smaller corrections. The NoBLE method then provides a careful computational analysis that improves the dimension above which the result applies. Universality arguments predict that the upper critical dimension, above which our results apply, is equal to $$d_c=8$$ d c = 8 for both models, as is known for sufficiently spread-out models by the results of Hara and Slade mentioned earlier. The main ingredients in this paper are (a) a derivation of a non-backtracking lace expansion for the LT and LA two-point functions; (b) bounds on the non-backtracking lace-expansion coefficients, thus showing that our general NoBLE methodology can be applied; and (c) sharp numerical bounds on the coefficients. Our proof is complemented by a computer-assisted numerical analysis that verifies that the necessary bounds used in the NoBLE are satisfied.

A connective constant for loop-erased self-avoiding random walk

1983 ◽  
Vol 20 (02) ◽  
pp. 264-276
Author(s):  
Gregory F. Lawler
Keyword(s):  
Random Walk ◽  
Uniform Distribution ◽  
Weak Sense ◽  
Simple Random Walk ◽  
Connective Constant

A ‘connective constant' is defined for self-avoiding random walk derived by erasing loops from simple random walk. For d ≧ 5, it is shown that this distribution on n-step self-avoiding paths approaches a uniform distribution in a weak sense.

A connective constant for loop-erased self-avoiding random walk

1983 ◽  
Vol 20 (2) ◽  
pp. 264-276 ◽  
Author(s):  
Gregory F. Lawler
Keyword(s):  
Random Walk ◽  
Uniform Distribution ◽  
Weak Sense ◽  
Simple Random Walk ◽  
Connective Constant

A ‘connective constant' is defined for self-avoiding random walk derived by erasing loops from simple random walk. For d ≧ 5, it is shown that this distribution on n-step self-avoiding paths approaches a uniform distribution in a weak sense.

The size of the region occupied by one type in an invasion process

1976 ◽  
Vol 13 (02) ◽  
pp. 355-356 ◽  
Author(s):  
Aidan Sudbury
Keyword(s):  
Random Walk ◽  
Simple Random Walk ◽  
Rectangular Lattice ◽  
Invasion Process

Particles are situated on a rectangular lattice and proceed to invade each other's territory. When they are equally competitive this creates larger and larger blocks of one type as time goes by. It is shown that the expected size of such blocks is equal to the expected range of a simple random walk.

Self-avoiding walk enumeration via the lace expansion

2007 ◽  
Vol 40 (36) ◽  
pp. 10973-11017 ◽  
Author(s):  
Nathan Clisby ◽  
Richard Liang ◽  
Gordon Slade
Keyword(s):  
Lace Expansion ◽  
Self Avoiding Walk

On coupling of random walks and renewal processes

1996 ◽  
Vol 33 (1) ◽  
pp. 122-126
Author(s):  
Torgny Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
One Dimensional ◽  
Continuous State Space ◽  
Continuous State ◽  
Efficient Coupling ◽  
Blackwell's Renewal Theorem

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

A Simple Random Walk Model for Dairy Cow Calving Time Prediction

2021 ◽  
Author(s):  
Thi Thi Zin ◽  
Pyke Tin ◽  
Pann Thinzar Seint ◽  
Kosuke Sumi ◽  
Ikuo Kobayashi ◽  
...  
Keyword(s):  
Random Walk ◽  
Dairy Cow ◽  
Simple Random Walk ◽  
Random Walk Model ◽  
Time Prediction

scholarly journals Diamond aggregation

2010 ◽  
Vol 149 (2) ◽  
pp. 351-372
Author(s):  
WOUTER KAGER ◽  
LIONEL LEVINE
Keyword(s):  
Random Walk ◽  
Growth Model ◽  
Power Law ◽  
Simple Random Walk ◽  
Internal Diffusion ◽  
Diffusion Limited Aggregation ◽  
Directional Bias ◽  
Model Based ◽  
Diffusion Limited ◽  
Logarithmic Fluctuations

AbstractInternal diffusion-limited aggregation is a growth model based on random walk in ℤd. We study how the shape of the aggregate depends on the law of the underlying walk, focusing on a family of walks in ℤ2 for which the limiting shape is a diamond. Certain of these walks—those with a directional bias toward the origin—have at most logarithmic fluctuations around the limiting shape. This contrasts with the simple random walk, where the limiting shape is a disk and the best known bound on the fluctuations, due to Lawler, is a power law. Our walks enjoy a uniform layering property which simplifies many of the proofs.

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