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Weakly self-avoiding walk on a high-dimensional torus

Emmanuel Michta and Gordon Slade

Vol. 4 (2023), No. 2, 331–375
DOI: 10.2140/pmp.2023.4.331
Abstract

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 d? We consider a version of this question for weakly self-avoiding walk on a torus in dimensions d > 4. On d for d > 4, the partition function for n-step weakly self-avoiding walk is known to be asymptotically purely exponential, of the form Aμn, where μ is the growth constant for weakly self-avoiding walk on d. We prove the identical asymptotic behaviour Aμn on the torus (with the same A and μ as on d) until n reaches order V 12, where V is the number of vertices in the torus. This shows that the walk must have length of order at least V 12 before it “feels” the torus in its leading asymptotics. Our results support the conjecture that the behaviour of the partition function does change once n reaches V 12, and we relate this to a conjectural critical scaling window which separates the dilute phase n V 12 from the dense phase n V 12. To prove the conjecture and to establish the existence of the scaling window remains a challenging open problem. The proof uses a novel lace expansion analysis based on the “plateau” for the torus two-point function obtained in previous work.

Keywords
self-avoiding walk, lace expansion, torus plateau, phase transition
Mathematical Subject Classification
Primary: 60K35, 82B27, 82B41
Milestones
Received: 2 August 2021
Revised: 1 September 2022
Accepted: 24 January 2023
Published: 31 May 2023
Authors
Emmanuel Michta
Department of Mathematics
University of British Columbia
Vancouver
Canada
Gordon Slade
Department of Mathematics
University of British Columbia
Vancouver
Canada