Rate of convergence of the critical point of the memory-τ self-avoiding walk in dimensions d>4

2023/06/24 by Noe Kawamoto, Kawamoto, Noe
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2306.13936

openalex publication_date 2023/06/24 · openalex created_date 2023/06/28 · openalex updated_date 2026/08/01

Abstract

We consider spread-out models of the self-avoiding walk and its finite-memory version, known as the memory-τ walk, which prohibits loops whose length is at most τ, in dimensions d>4. The critical point is defined as the radius of convergence of the generating function for each model. It is known that the critical point of the memory-τ walk is non-decreasing in τ and converges to that of the self-avoiding walk as τ tends to infinity. In this paper, we study the rate at which the critical point of the memory-τ walk converges to that of the self-avoiding walk and show that the order is τ^-(d-2)/2. The proof relies on the lace expansion, introduced by Brydges and Spencer.

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