VRJP recurrence and fractional-moment decay for the $H^{2\mathrm{\mid }2}$ model's effective field on the hierarchical lattice

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 $\bar{W}$, and at the critical spectral dimension $d=2$ for sufficiently strong reinforcement, i.e., sufficiently small $\bar{W}$. The key estimate is a fractional-moment bound for the Green’s function of the associated random Schrödinger operator, expressed as geometric decay across hierarchical scales for the effective $H^{2\mathrm{\mid }2}$ 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.