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

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.