  The vertex-reinforced jump process (VRJP) is a form of self-interacting
random walk in which the walker is biased towards returning to previously
visited vertices with the bias depending linearly on the local time at these
vertices. We prove that, for any initial bias, the weights sampled from the
magic formula on a two-dimensional graph decay at least at a power-law rate.
Via arguments of Sabot and Zeng, the result implies that the VRJP is recurrent
in two dimensions for any initial bias.
