Loop-Erased Random Walk on a Torus in Dimensions 4 and Above

Abstract

We show that the statistics of loop erased random walks above the upper critical dimension, 4, are different between the torus and the full space. The typical length of the path connecting a pair of sites at distance L, which scales as L2 in the full space, changes under the periodic boundary conditions to L d /2. The results are precise for dimensions ≥5; for the dimension d=4 we prove an upper bound, conjecturally sharp up to subpolyonmial factors.