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A reduction of the $θ(p_c) = 0$ problem to a conjectured inequality

Gady Kozma, Shahaf Nitzan

Abstract

We conjecture a new correlation-like inequality for percolation probabilities and support our conjecture with numerical evidence and a few special cases which we prove. This inequality, if true, implies that there is no percolation at criticality on the Euclidean lattice, for any dimension bigger than one.

A reduction of the $θ(p_c) = 0$ problem to a conjectured inequality

Abstract

We conjecture a new correlation-like inequality for percolation probabilities and support our conjecture with numerical evidence and a few special cases which we prove. This inequality, if true, implies that there is no percolation at criticality on the Euclidean lattice, for any dimension bigger than one.
Paper Structure (16 sections, 27 theorems, 168 equations, 4 figures)

This paper contains 16 sections, 27 theorems, 168 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. The van den Berg-Häggström-Kahn inequality
  5. Some cases where the inequality holds
  6. Small $A$
  7. $A$ close to 0
  8. $\theta(p_c)$
  9. Miscellenia
  10. Montone cluster properties
  11. Coefficients
  12. An inductive approach
  13. An application of the theory of boolean functions
  14. Candidates
  15. Simple reductions
  16. ...and 1 more sections

Key Result

Theorem 1

Let $G$ be a finite graph and $A, B\subset G$. Let $f$ and $g$ be two increasing functions defined on the cluster of $A$. Then The inequality is reversed if either $f$ or $g$ are decreasing (but not both). Contrariwise, if $f$ and $g$ are two increasing functions defined on the clusters of $A$ and $B$ respectively then: Again, of course, if either $f$ or $g$ are decreasing then the inequality is

Figures (4)

  • Figure 1: The geometry of lemma \ref{['lem:shrink']}.
  • Figure 2: The set $Q_v$, $M_v$, $E$ and $H$.
  • Figure 3: The set $\mathcal{E}_i$ is in a dotted line while $\mathcal{E}_{i+1}$ is dot-dashed. There are two sets denoted $H_x^{j_x}$ in the picture --- they simply relate to different $x$, the right and top ones. There is nothing 'wrong' with the fact that no $H$ appear in the picture below --- this would happen if both boxes below were already in $X_i$ when $w$ and $v$ were first explored, or if some $j$ were 0.
  • Figure 4: A directed percolation counterexample.

Theorems & Definitions (63)

  • Conjecture 1
  • Conjecture 2
  • Theorem : van den Berg, Häggström and Kahn (BHK)
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • ...and 53 more