4 problems

Recurrence conjecture. The model is recurrent in $d=2$ even with weak reinforcement.

Let the Dyson hierarchical lattice have critical dimension $2$, and let $\overline{W}$ be any inverse temperature. Consider the vertex-reinforced jump process (VRJP) on this lattic…

Fix $d\ge 3$ and consider the $d$-dimensional lattice ${\mathbb Z}^d$ with nonnegative symmetric edge weights $W_{ij}$ satisfying … and … The associated VRJP is known in the source…

Let $({\mathbb Z}^d,W)$ be the lattice with long-range interactions described above, where the edge weights are not necessarily locally finite but satisfy the stated long-range con…