Loop-weighted walk with parameter is a non-Markovian model of random walks that is related to the loop model of statistical mechanics. A walk receives weight if it contains loops; whether this is a reward or punishment for containing loops depends on the value of . A challenging feature of loop-weighted walk is that it is not purely repulsive, meaning the weight of the future of a walk may either increase or decrease if the past is forgotten. Repulsion is typically an essential property for lace expansion arguments. This article circumvents the lack of repulsion and proves, for any , that loop-weighted walk is diffusive in high dimensions by lace expansion methods.
Cite this article
Tyler Helmuth, Loop-weighted walk. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 3 (2016), no. 1, pp. 55–119DOI 10.4171/AIHPD/25