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On cable-graph percolation between dimensions 2 and 3

Pierre-François Rodriguez, Wen Zhang

TL;DR

The paper analyzes cable-graph percolation for the Gaussian free field on 2D slabs of height h_N, bridging two- and three-dimensional behaviors by compactifying one dimension and studying the resulting slab model. A decomposition of the field into two- and three-dimensional components, coupled with precise Green’s function and capacity estimates on slabs, reveals a spectrum of fixed points and a plateau in the one-arm decay in the delocalized phase. It provides sharp lower and upper bounds for crossing probabilities, leveraging obstacle-based methods, killed Green’s functions, and loop-soup isomorphisms to handle the mixed-dimensional regime. The results illuminate a logarithmic-to-polynomial transition between dimensions 2 and 3 and introduce a detailed framework that can accommodate a wide range of slab heights, including thin limits, with implications for universality classes beyond polynomial scaling. Overall, the work advances a rigorous, parameter-robust ε-expansion perspective for cable-system percolation in mixed-dimensional settings.

Abstract

We consider the Gaussian free field on two-dimensional slabs with a thickness described by a height $h$ at spatial scale $N$. We investigate the radius of critical clusters for the associated cable-graph percolation problem, which depends sensitively on the parameter $h$. Our results unveil a whole family of new "fixed points", which interpolate between recent results from arXiv:2303.03782 in two dimensions and from arXiv:2405.17417 and arXiv:2406.02397 in three dimensions, and describe critical behaviour beyond those regimes. In the delocalised phase, the one-arm decay exhibits a "plateau", i.e. it doesn't depend on the speed at which the variance of the field diverges in the large-$N$ limit. Our methods rely on a careful analysis of the interplay between two- and three-dimensional effects for the underlying random walk, which manifest themselves in a corresponding decomposition of the field.

On cable-graph percolation between dimensions 2 and 3

TL;DR

The paper analyzes cable-graph percolation for the Gaussian free field on 2D slabs of height h_N, bridging two- and three-dimensional behaviors by compactifying one dimension and studying the resulting slab model. A decomposition of the field into two- and three-dimensional components, coupled with precise Green’s function and capacity estimates on slabs, reveals a spectrum of fixed points and a plateau in the one-arm decay in the delocalized phase. It provides sharp lower and upper bounds for crossing probabilities, leveraging obstacle-based methods, killed Green’s functions, and loop-soup isomorphisms to handle the mixed-dimensional regime. The results illuminate a logarithmic-to-polynomial transition between dimensions 2 and 3 and introduce a detailed framework that can accommodate a wide range of slab heights, including thin limits, with implications for universality classes beyond polynomial scaling. Overall, the work advances a rigorous, parameter-robust ε-expansion perspective for cable-system percolation in mixed-dimensional settings.

Abstract

We consider the Gaussian free field on two-dimensional slabs with a thickness described by a height at spatial scale . We investigate the radius of critical clusters for the associated cable-graph percolation problem, which depends sensitively on the parameter . Our results unveil a whole family of new "fixed points", which interpolate between recent results from arXiv:2303.03782 in two dimensions and from arXiv:2405.17417 and arXiv:2406.02397 in three dimensions, and describe critical behaviour beyond those regimes. In the delocalised phase, the one-arm decay exhibits a "plateau", i.e. it doesn't depend on the speed at which the variance of the field diverges in the large- limit. Our methods rely on a careful analysis of the interplay between two- and three-dimensional effects for the underlying random walk, which manifest themselves in a corresponding decomposition of the field.

Paper Structure

This paper contains 10 sections, 32 theorems, 184 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Green's function $g_N$ on the slab
  3. Capacity estimates on slabs
  4. Line
  5. Ball
  6. Lower bounds and precise asymptotics in the "thin" case
  7. Killed estimates and Harnack inequality
  8. Upper bounds
  9. Heat kernel approximation
  10. Properties of $K_0$

Key Result

Theorem 1.1

Let For all $N, R\geq 1$ such that ${R} \leq \Cr{C:range}N$, we have that In particular, for all $N \geq 2$,

Figures (1)

  • Figure 1: The map $h \mapsto \tfrac{1}{\theta^h}$ with $\theta^h\equiv \theta^h_N(N)$ in Theorem \ref{['thm:critical_connect']}.

Theorems & Definitions (71)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Corollary 1.6: Intrinsic scale in the recurrent regime
  • proof
  • Theorem 2.1
  • Remark 2.2
  • Proposition 2.3
  • ...and 61 more