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

Abstract

We study the two-point functions of a general class of random-length random walks (RLRWs) on finite boxes in Zd with d⩾3 , and provide precise asymptotics for their behaviour. We show that in a finite box of side length L, the two-point function is asymptotic to the infinite-lattice two-point function when the typical walk length is o(L2) , but develops a plateau when the typical walk length is Ω(L2) . 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 the 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 RLRW, 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.