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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.13936 |
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Table of Contents:
- 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.