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

Yichao Huang, Jinglin Wang, Xiaolin Zeng

TL;DR

This work builds a rigorous bridge between the H^{2|2}-model and VRJP on the Dyson hierarchical lattice by developing a reverse fine-graining renormalization and an exponential martingale framework. It establishes a continuum-space scaling limit, yielding a random measure on [0,1] whose singularity is dictated by VRJP recurrence, and provides a detailed ergodic and mixing analysis that underpins a zero-one law for the limiting field. The combination of coarse- and fine-graining, hierarchical lattice analysis, and martingale methods yields a precise characterization of when the limiting measure is singular or carries an absolutely continuous component, tying spectral-type behavior to probabilistic long-time dynamics. The results illuminate non-perturbative aspects of the H^{2|2}-theory on hierarchical structures and advance understanding of localization-delocalization phenomena via a probabilistic continuum limit.

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 Schrödinger 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.

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

TL;DR

This work builds a rigorous bridge between the H^{2|2}-model and VRJP on the Dyson hierarchical lattice by developing a reverse fine-graining renormalization and an exponential martingale framework. It establishes a continuum-space scaling limit, yielding a random measure on [0,1] whose singularity is dictated by VRJP recurrence, and provides a detailed ergodic and mixing analysis that underpins a zero-one law for the limiting field. The combination of coarse- and fine-graining, hierarchical lattice analysis, and martingale methods yields a precise characterization of when the limiting measure is singular or carries an absolutely continuous component, tying spectral-type behavior to probabilistic long-time dynamics. The results illuminate non-perturbative aspects of the H^{2|2}-theory on hierarchical structures and advance understanding of localization-delocalization phenomena via a probabilistic continuum limit.

Abstract

We extend the exact coarse-graining result of Disertori, Merkl and Rolles~\cite{MR4517733} for the random field of -model to the random Schrödinger operator representation of the -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 -model on the Dyson hierarchical lattice, we establish its continuous space scaling limit as a non-trivial random measure on . 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 -model on the Dyson hierarchical lattice.

Paper Structure

This paper contains 19 sections, 12 theorems, 94 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Coarse and fine-graining properties of the $H^{2|2}$-model
  3. Definition of the $H^{2|2}$-model via its random Schrödinger operator representation
  4. Green function and random walk expansion of the $H^{2|2}$-model
  5. Coupling with different pinning positions
  6. Coarse-graining procedure
  7. Fine-graining procedure
  8. Wired boundary coupling and infinite graph
  9. A one-step martingale property
  10. The $H^{2|2}$-model on the Dyson hierarchical lattice
  11. Definition of the hierarchical $H^{2|2}$-model
  12. Coarse-graining procedure on the hierarchical $H^{2|2}$-model
  13. Fine-graining coupling on the hierarchical $H^{2|2}$-model
  14. Some known estimates of the hierarchical $H^{2|2}$-model
  15. Continuous space scaling limit of the hierarchical $H^{2|2}$-model
  16. ...and 4 more sections

Key Result

Theorem 3

Consider the $H^{2|2}$-model (with boundary condition $\eta$) on a finite graph $\mathcal{G}$ by the data of its random Green function operator $G=H^{-1}$. If $U\subset V$ is a subset of vertices indistinguishable from outside, then one can effectively replace $U$ by a single vertex $u$ and coarse-g The resulting matrix $G'$ has the same law of the Green operator defined by the $H^{2|2}$-model (wi

Figures (2)

  • Figure 1: Hierarchical distance and couping.
  • Figure 2: Sequence of $\varphi^{(n)}_{x}=e^{\widetilde{u}^{(n)}_{i}}$ converges to $\varphi_{x}$

Theorems & Definitions (39)

  • Remark 1
  • Definition 2: Indistinguishability
  • Theorem 3
  • Remark 4
  • proof
  • Remark 5
  • proof : Alternative proof of the first part of Theorem \ref{['th:CoarseGraining']}
  • Theorem 6
  • proof
  • Theorem 7
  • ...and 29 more