  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 essential property for lace
expansion arguments. This article circumvents the lack of repulsion and proves,
via the lace expansion, that for any $\lambda\geq 0$ loop-weighted walk is
diffusive in high dimensions.
