Loop-weighted walk with parameter \lambda \geq 0 is a non-Markovian model of random walks that is related to the loop O(N) model of statistical mechanics. A walk receives weight \lambda^{k} if it contains k loops; whether this is a reward or punishment for containing loops depends on the value of \lambda . 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 esse…