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| Format: | Preprint |
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2023
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| Online-Zugang: | https://arxiv.org/abs/2306.13936 |
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| _version_ | 1866917206399385600 |
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| author | Kawamoto, Noe |
| author_facet | Kawamoto, Noe |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_13936 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rate of convergence of the critical point of the memory-$τ$ self-avoiding walk in dimensions $d>4$ Kawamoto, Noe Probability Mathematical Physics 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. |
| title | Rate of convergence of the critical point of the memory-$τ$ self-avoiding walk in dimensions $d>4$ |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2306.13936 |