Self-Avoiding Walk is Sub-Ballistic

2012/05/02 by Hugo Duminil‐Copin, Hugo Duminil-Copin, Alan Hammond · 60 citations
Biochemistry, Genetics and Molecular Biology · Engineering · Mathematics · Medicine · Neuroscience · Physics and Astronomy · #Advanced Neuroimaging Techniques and Applications #Aeronautics #Engineering #Functional Brain Connectivity Studies #Protein Structure and Dynamics #math-ph #math.CO #math.MP #math.PR

paper · pdf · doi:10.1007/s00220-013-1811-1

published in Communications in Mathematical Physics 324(2), 401-423 (Springer Science+Business Media) · 27 pages and four figures

arxiv created 2012/05/02 · openalex publication_date 2013/10/12 · arxiv updated 2015/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We prove that self-avoiding walk on Z^d is sub-ballistic in any dimension d at least two. That is, writing ||u|| for the Euclidean norm of u ∈ Z^d, and SAW_n for the uniform measure on self-avoiding walks gamma:0,...,n → Z^d for which gamma₀ = 0, we show that, for each v > 0, there exists c > 0 such that, for each positive integer n, SAW_n (max || gamma_k || : k ∈ 0,...,n > v n) < e^{- c n}.

Citations

• Bridge Decomposition of Restriction Measures
• Lectures on Self-Avoiding Walks

Cited by

• Non-coincidence of Quenched and Annealed Connective Constants on the supercritical planar percolation cluster
• Self-avoiding walk, spin systems and renormalization
• The Critical Fugacity for Surface Adsorption of Self-Avoiding Walks on the Honeycomb Lattice is 1+√(2) 1 + 2
• A lattice point counting approach for the study of the number of self-avoiding walks on ℤ^d