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 on finite boxes in $\ZZ^d$ with $d\ge 3$, 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.

Major revision of earlier draft. Theorems concerning random-length random walk two-point functions are both significantly sharper and more general (two additional choices of boundary now considered). In addition to the plateau theorem, there is now a sharp result for short walks. New numerical results added for Ising and SAW model to demonstrate universality of random-length random walk results

References in corpus (3)

• Finite-size scaling above the upper critical dimension
• Non-vanishing boundary effects and quasi-first order phase transitions in high dimensional Ising models
• Finite size scaling of the 5D Ising model with free boundary conditions