Self-avoiding walk is ballistic on graphs with more than one end

2024/03/19 by Florian Lehner, Lehner, Florian, Christian Lindorfer +3
Computer Science · #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Parallel Computing and Optimization Techniques #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2403.13121

openalex publication_date 2024/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We prove that on any transitive graph G with infinitely many ends, a self-avoiding walk of length n is ballistic with extremely high probability, in the sense that there exist constants c,t>0 such that ℙ_n(d_G(w₀,w_n)≥ cn)≥ 1-e^-tn for every n≥ 1. Furthermore, we show that the number of self-avoiding walks of length n grows asymptotically like μ_wⁿ, in the sense that there exists C>0 such that μ_wⁿ≤ c_n≤ Cμ_wⁿ for every n≥ 1. Our results extend more generally to quasi-transitive graphs with infinitely many ends, satisfying the additional technical property that there is a quasi-transitive group of automorphisms of G which does not fix an end of G.

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