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arXiv · 2008.00913

Two-point functions of random-length random walk on high-dimensional boxes

Youjin Deng·Timothy M. Garoni·Jens Grimm·Zongzheng Zhou
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Abstract

We study the two-point functions of a general class of random-length random walks on finite boxes in $\ZZ^d$ with $d\ge3$, and provide precise asymptotics for their behaviour. We show that the finite-box two-point function is asymptotic to the infinite-lattice two-point function when the typical walk length is $o(L^2)$, but develops a plateau when the typical walk length is $\Omega(L^2)$. We also numerically study walk length moments and limiting distributions of the self-avoiding walk and Ising model on five-dimensional tori, and find that they agree asymptotically with the known results for self-avoiding walk on the complete graph, both at the critical point and also for a broad class of scaling windows/pseudocritical points. Furthermore, we show that the two-point function of the finite-box random-length random walk, with walk length chosen via the complete graph self-avoiding walk, agrees numerically with the two-point functions of the self-avoiding walk and Ising model on five-dimensional tori. We conjecture that these observations in five dimensions should also hold in all higher dimensions.

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Youjin Deng, Timothy M. Garoni, Jens Grimm, Zongzheng Zhou. 2020-08-03. Two-point functions of random-length random walk on high-dimensional boxes. https://arxiv.org/abs/2008.00913

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