Fine-Graining and Continuous Space Scaling Limit of the $H^{2|2}$ Model on the Hierarchical Lattice

TL;DR

This paper extends the coarse-graining of the $H^{2|2}$ model to a random Schr"odinger operator, introduces a fine-graining method, and studies its continuous space scaling limit, revealing a phase transition related to the recurrence of a Vertex Reinforced Jump Process.

Contribution

It introduces a fine-graining procedure for the $H^{2|2}$ model and establishes its continuous space scaling limit on the hierarchical lattice.

Findings

The scaling limit is a singular measure if the VRJP is recurrent.

The measure has an absolutely continuous component if the VRJP is transient.

The density of the absolutely continuous part is non-trivial and linked to an exponential martingale.

Abstract

We extend the exact coarse-graining result of Disertori, Merkl and Rolles~\cite{MR4517733} for the random field of $H^{2|2}$-model to the random Schr\"odinger operator representation of the $H^{2|2}$-model. We also introduce a fine-graining procedure as the reverse operation, and establish an associated exponential martingale property. Applying this fine-graining procedure to the $H^{2|2}$-model on the Dyson hierarchical lattice, we establish its continuous space scaling limit as a non-trivial random measure on $[0,1]$. This random measure is almost surely singular with respect to the Lebesgue measure if and only if the Vertex Reinforced Jump Process on the Dyson hierarchical lattice is recurrent. If the process is transient, the random measure almost surely has an absolutely continuous component. The density of this component is everywhere non-trivial and can be identified with the pointwise limit of an exponential martingale associated with the $H^{2|2}$-model on the Dyson hierarchical lattice.