# 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