VRJP recurrence and fractional-moment decay for the \(H^{2|2}\) model's effective field on the hierarchical lattice

Abstract

We prove recurrence of the vertex-reinforced jump process on the hierarchical lattice for spectral dimension \(d<2\) for every value of the conductance parameter \(\overline{W}\), and at the critical spectral dimension \(d=2\) for sufficiently strong reinforcement, i.e., sufficiently small \(\overline{W}\). The key estimate is a fractional-moment bound for the Green's function of the associated random Schr\"odinger operator, expressed as geometric decay across hierarchical scales for the effective \(H^{2|2}\) field. The proof combines the fractional-moment method with an exact hierarchical coarse-graining identity, which turns the path expansion into a recursion over block scales and controls the combinatorial growth created by long-range edges. Together with existing long-range-order results in the regime \(d>2\), these estimates identify the recurrent side of the hierarchical VRJP phase diagram, leaving only the weak-reinforcement critical regime open.