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An ergodic and isotropic zero-conductance model with arbitrarily strong local connectivity

Martin Heida, Benedikt Jahnel, Anh Duc Vu

Abstract

We exhibit a percolating ergodic and isotropic lattice model in all but at least two dimensions that has zero effective conductivity in all spatial directions and for all non-trivial choices of the connectivity parameter. The model is based on the so-called randomly stretched lattice where we additionally elongate layers containing few open edges.

An ergodic and isotropic zero-conductance model with arbitrarily strong local connectivity

Abstract

We exhibit a percolating ergodic and isotropic lattice model in all but at least two dimensions that has zero effective conductivity in all spatial directions and for all non-trivial choices of the connectivity parameter. The model is based on the so-called randomly stretched lattice where we additionally elongate layers containing few open edges.
Paper Structure (14 sections, 5 theorems, 31 equations, 3 figures)

This paper contains 14 sections, 5 theorems, 31 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setting and main result
  3. Construction: RSL and ERSL
  4. Checking non-conductance
  5. Parameters
  6. Finding bad layers
  7. Calculating the effective conductivity
  8. Background on discrete models for perforated domains
  9. Justification of discrete models replacing continuous problems
  10. The classical point of view.
  11. The exact solution point of view.
  12. Formulas for the effective conductivity
  13. Violation of the homogenisation conditions in heida3perforation
  14. Discussions and outlook

Key Result

Theorem 1

For any $d\geq2$ and $\bar{p}\in(0,1)$, there exists an ERSL -- a stationary ergodic nearest-neighbour bond percolation model on $\mathbb{Z}^{d}$ -- satisfying the following properties. In particular, the random walk on the ERSL is subdiffusive.

Figures (3)

  • Figure 1: Two realisations of the non-conductive medium, on slightly differing scales, given by the elongated randomly stretched lattice. Blue edges belong to the centrally placed green dot's cluster (restricted to the observation window).
  • Figure 2: Realisations of the RSL with parameters $p=0.65,q=0.3$ (left), the elongated version with $\sigma=0.25$ (middle), and the filled version $F_L(\text{RSL})$ with $L=2$ (right). Blue edges belong to the centrally placed green dot's connected component (before using the grey filling).
  • Figure 3: Realisation of a lattice model with zero conductivity in vertical direction. Here, $P_0$ is a uniform random variable.

Theorems & Definitions (13)

  • Theorem 1
  • Definition 2: Randomly stretched lattice (RSL)
  • Theorem 3: Existence of supercritical regime in the RSL, MR1761579MR2116736MR4634238rsl2023
  • proof
  • Definition 4: Elongated randomly stretched lattice (ERSL)
  • proof : Proof of Theorem \ref{['thm:main-thm']} Part (1) and (2)
  • Lemma 5: Probability of bad layers
  • proof
  • Lemma 6: Conductivity through bad layers
  • proof
  • ...and 3 more