Quantitative sub-ballisticity of self-avoiding walk on ℤd
Abstract
We prove quantitative sub-ballisticity for the self-avoiding walk on the lattice ℤd, d≥ 2. Namely, we obtain an explicit exponential rate of decay for the probability that a self-avoiding walk of length n reaches distance at least vn, in terms of v>0. As a consequence, we show that the displacement of an n-step self-avoiding walk is at most n/exp(clog* n) with very high probability. Previously, only a non-quantitative o(n) bound was known from the work of Duminil-Copin and Hammond. Along the way, we show that at criticality the partition function of bridges of length n is on average bounded by C/loglog n, which we believe to be of independent interest. Our approach is based on a quantitative analysis of the original nonconstructive proof of Duminil-Copin and Hammond using ideas from proof theory.
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BibTeX
@misc{arikok2026quantitative,
title = {{Quantitative sub-ballisticity of self-avoiding walk on $\mathbbZ^d$}},
author = {Ekin Arikök and Christoforos Panagiotis and Nicholas Pischke and Thomas Powell},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40520},
eprint = {2610.40520},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40520},
}
RIS
TY - UNPB
TI - Quantitative sub-ballisticity of self-avoiding walk on ℤ^d
AU - Ekin Arikök
AU - Christoforos Panagiotis
AU - Nicholas Pischke
AU - Thomas Powell
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40520
UR - https://arxiv.org/abs/2610.40520
AB - We prove quantitative sub-ballisticity for the self-avoiding walk on the lattice ℤ^d, d≥ 2. Namely, we obtain an explicit exponential rate of decay for the probability that a self-avoiding walk of length n reaches distance at least vn, in terms of v>0. As a consequence, we show that the displacement of an n-step self-avoiding walk is at most n/exp(clog^* n) with very high probability. Previously, only a non-quantitative o(n) bound was known from the work of Duminil-Copin and Hammond. Along the way, we show that at criticality the partition function of bridges of length n is on average bounded by C/loglog n, which we believe to be of independent interest. Our approach is based on a quantitative analysis of the original nonconstructive proof of Duminil-Copin and Hammond using ideas from proof theory.
ER -