# Self-Avoiding Walk is Sub-Ballistic

@article{DuminilCopin2013SelfAvoidingWI,
title={Self-Avoiding Walk is Sub-Ballistic},
author={Hugo Duminil-Copin and Alan Hammond},
journal={Communications in Mathematical Physics},
year={2013},
volume={324},
pages={401-423}
}
• Published 2 May 2012
• Mathematics
• Communications in Mathematical Physics
We prove that self-avoiding walk on $${\mathbb{Z}^d}$$Zd is sub-ballistic in any dimension d ≥ 2. That is, writing $${\| u \|}$$‖u‖ for the Euclidean norm of $${u \in \mathbb{Z}^d}$$u∈Zd, and $${\mathsf{P_{SAW}}_n}$$PSAWn for the uniform measure on self-avoiding walks $${\gamma : \{0, \ldots, n\} \to \mathbb{Z}^d}$$γ:{0,…,n}→Zd for which γ0 = 0, we show that, for each v > 0, there exists $${\varepsilon > 0}$$ε>0 such that, for each $${n \in \mathbb{N}, \mathsf{P_{SAW}}_n \big( {\rm max}\big… ### Bounding the number of self-avoiding walks: Hammersley–Welsh with polygon insertion • Mathematics The Annals of Probability • 2020 Let c_n = c_n(d) denote the number of self-avoiding walks of length n starting at the origin in the Euclidean nearest-neighbour lattice \mathbb{Z}^d. Let \mu = \lim_n c_n^{1/n} denote the ### Self-avoiding walks and polygons on hyperbolic graphs We prove that for the d-regular tessellations of the hyperbolic plane by k-gons, there are exponentially more self-avoiding walks of length n than there are self-avoiding polygons of length ### Self-avoiding walk on nonunimodular transitive graphs We study self-avoiding walk on graphs whose automorphism group has a transitive nonunimodular subgroup. We prove that self-avoiding walk is ballistic, that the bubble diagram converges at ### Non-coincidence of quenched and annealed connective constants on the supercritical planar percolation cluster In this paper, we study the abundance of self-avoiding paths of a given length on a supercritical percolation cluster on$$\mathbb{Z }^d$$Zd. More precisely, we count$$Z_N$$ZN, the number of ### A Lower Bound for the End-to-End Distance of the Self-Avoiding Walk • N. Madras • Mathematics Canadian Mathematical Bulletin • 2014 Abstract For an N -step self-avoiding walk on the hypercubic lattice {{Z}^{d}} , we prove that the meansquare end-to-end distance is at least {{N}^{4/(3d)}} times a constant. This implies that ### The Critical Fugacity for Surface Adsorption of Self-Avoiding Walks on the Honeycomb Lattice is$${1+\sqrt{2}}$$1+2 • Mathematics • 2011 In 2010, Duminil-Copin and Smirnov proved a long-standing conjecture of Nienhuis, made in 1982, that the growth constant of self-avoiding walks on the hexagonal (a.k.a. honeycomb) lattice is ### Convergence of the self-avoiding walk on random quadrangulations to SLE_{8/3} on \sqrt{8/3}-Liouville quantum gravity • Mathematics Annales Scientifiques de l'École Normale Supérieure • 2021 We prove that a uniform infinite quadrangulation of the half-plane decorated by a self-avoiding walk (SAW) converges in the scaling limit to the metric gluing of two independent Brownian half-planes ### ON THE PROBABILITY THAT SELF-AVOIDING WALK ENDS AT A GIVEN POINT • Mathematics • 2013 We prove two results on the delocalization of the endpoint of a uniform self-avoiding walk on Z^d for d>1. We show that the probability that a walk of length n ends at a point x tends to 0 as n tends ### The scaling limit of the weakly self-avoiding walk on a high-dimensional torus • Mathematics • 2021 How long does a self-avoiding walk on a discrete d -dimensional torus have to be before it begins to behave diﬀerently from a self-avoiding walk on Z d ? We consider a version of this question for ## References SHOWING 1-10 OF 31 REFERENCES ### Self-Avoiding Walks on Hyperbolic Graphs • Mathematics Combinatorics, Probability and Computing • 2005 It is proved that on all but at most eight graphs of the hyperbolic plane, there are exponentially fewer self-avoiding polygons than there are SAWs, and it is proven that \gamma is finite on allhyperbolic graphs. ### The Critical Fugacity for Surface Adsorption of Self-Avoiding Walks on the Honeycomb Lattice is$${1+\sqrt{2}}1+2

• Mathematics
• 2011
In 2010, Duminil-Copin and Smirnov proved a long-standing conjecture of Nienhuis, made in 1982, that the growth constant of self-avoiding walks on the hexagonal (a.k.a. honeycomb) lattice is

### Self-avoiding walk in five or more dimensions I. The critical behaviour

• Mathematics
• 1992
We use the lace expansion to study the standard self-avoiding walk in thed-dimensional hypercubic lattice, ford≧5. We prove that the numbercn ofn-step self-avoiding walks satisfiescn~Aμn, where μ is

### The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$

• Mathematics
• 2010
We provide the first mathematical proof that the connective constant of the hexagonal lattice is equal to $\sqrt{2+\sqrt 2}$. This value has been derived non rigorously by B. Nienhuis in 1982, using

### On the Number of Self‐Avoiding Walks

Let χn be the number of self‐avoiding walks on the integral points in Euclidean d space and γn the number of n‐stepped self‐avoiding polygons. It is shown that χn+2/χn−β2 and γ2n+3/γ2n+1−β2 tend to

### Ornstein-Zernike Behaviour And Analyticity Of Shapes For Self-Avoiding Walks On Z^d

1 Abstract. We derive precise Ornstein-Zernike asymptotics for the decay of the two-point function in any direction of the simple self-avoiding walk on the integer lattice Z d in any dimension d 2

### Critical behaviour of self-avoiding walk in five or more dimensions.

• Mathematics
• 1991
We use the lace expansion to prove that in five or more dimensions the standard self-avoiding walk on the hypercubic (integer) lattice behaves in many respects like the simple random walk. In

### On the scaling limit of planar self-avoiding walk

• Mathematics
• 2002
A planar self-avoiding walk (SAW) is a nearest neighbor random walk path in the square lattice with no self-intersection. A planar self-avoiding polygon (SAP) is a loop with no self-intersection. In

### Anisotropic self-avoiding walks

• Mathematics
• 2000
We consider a model of self-avoiding walks on the lattice Zd with different weights for steps in each of the 2d lattice directions. We find that the direction-dependent mass for the two-point

### Supercritical self-avoiding walks are space-filling

• Physics
• 2011
We consider random self-avoiding walks between two points on the boundary of a finite subdomain of Z^d (the probability of a self-avoiding trajectory gamma is proportional to mu^{-length(gamma)}). We