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

2025/05/02 by Jinglin Wang, Wang, Jinglin, Xiaolin Zeng +1
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Quantum Chromodynamics and Particle Interactions #Stochastic processes and financial applications #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2505.01337

openalex publication_date 2025/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \(W\), and at the critical spectral dimension \(d=2\) for sufficiently strong reinforcement, i.e., sufficiently small \(W\). 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 \(H^2|2\) 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.

Citations

• Fine-Graining and Continuous Space Scaling Limit of the H^2|2 Model on the Hierarchical Lattice
• Fluctuations in the non-linear supersymmetric hyperbolic sigma model with long-range interactions
• Transience of vertex-reinforced jump processes with long-range jumps
• The non-linear supersymmetric hyperbolic sigma model on a complete graph with hierarchical interactions
• Power-law decay of weights and recurrence of the two-dimensional VRJP
• Polynomial localization of the 2D-Vertex Reinforced Jump Process
• A note on recurrence of the Vertex reinforced jump process and fractional moments localization
• A supersymmetric approach to martingales related to the\n vertex-reinforced jump process
• A random Schrödinger operator associated with the Vertex Reinforced Jump Process on infinite graphs
• The Vertex Reinforced Jump Process and a Random Schrödinger operator on finite graphs
• Edge-reinforced random walk, Vertex-Reinforced Jump Process and the supersymmetric hyperbolic sigma model

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• Fine-Graining and Continuous Space Scaling Limit of the H^2|2 Model on the Hierarchical Lattice
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• Power-law decay of weights and recurrence of the two-dimensional VRJP
• ℍ^2|2-model and Vertex-Reinforced Jump Process on Regular Trees: Infinite-Order Transition and an Intermediate Phase
• Potts models on hierarchical lattices and Renormalization Group dynamics II: examples and numerical results