# 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} }

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…

## 35 Citations

### Bounding the number of self-avoiding walks: Hammersley–Welsh with polygon insertion

- MathematicsThe 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

- Mathematics
- 2019

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

- MathematicsThe Annals of Probability
- 2019

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

- Mathematics
- 2012

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

- MathematicsCanadian 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…

### Positive speed self-avoiding walks on graphs with more than one end

- MathematicsJ. Comb. Theory, Ser. A
- 2020

### Convergence of the self-avoiding walk on random quadrangulations to SLE$_{8/3}$ on $\sqrt{8/3}$-Liouville quantum gravity

- MathematicsAnnales 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…

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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…

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