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Coupling Invasion and First Passage Percolation

Aldo Morelli

TL;DR

This paper develops a coupling between invasion percolation and first passage percolation with $log$-$uniform$ passage times by setting $\tau_{K,e}=e^{K w(e)}$, enabling a translation of invasion events into short-path events via $T_K$. The main result shows that, for large $R$ and appropriately chosen $K$, the invaded set inside $B_R$ matches the sublevel set $\{v: T_K(0,v)<T_K(0,\partial B_R)\}$ with high probability, providing a new equivalent condition for a continuous phase transition in Bernoulli bond percolation. A corollary ties the continuity of $\theta_d(p_{c,d})$ to a FPP criterion on log-uniform times, and simulations in $\mathbb{Z}^2$ suggest a power-law-type decay for the probability of deeper invasion as a function of radius. Overall, the work offers a novel analytic bridge between invasion percolation and first passage percolation, with potential to leverage FPP techniques to advance understanding of bond-percolation transitions and their continuity properties.

Abstract

It is well known that a continuous phase transition in Bernoulli bond percolation on the integer lattice is equivalent to a vanishing probability a vertex is invaded in invasion percolation. We provide a coupling between invasion percolation and first passage percolation with log-uniform passage times. This yields a new equivalent condition for a continuous phase transition in bond percolation.

Coupling Invasion and First Passage Percolation

TL;DR

This paper develops a coupling between invasion percolation and first passage percolation with - passage times by setting , enabling a translation of invasion events into short-path events via . The main result shows that, for large and appropriately chosen , the invaded set inside matches the sublevel set with high probability, providing a new equivalent condition for a continuous phase transition in Bernoulli bond percolation. A corollary ties the continuity of to a FPP criterion on log-uniform times, and simulations in suggest a power-law-type decay for the probability of deeper invasion as a function of radius. Overall, the work offers a novel analytic bridge between invasion percolation and first passage percolation, with potential to leverage FPP techniques to advance understanding of bond-percolation transitions and their continuity properties.

Abstract

It is well known that a continuous phase transition in Bernoulli bond percolation on the integer lattice is equivalent to a vanishing probability a vertex is invaded in invasion percolation. We provide a coupling between invasion percolation and first passage percolation with log-uniform passage times. This yields a new equivalent condition for a continuous phase transition in bond percolation.

Paper Structure

This paper contains 10 sections, 7 theorems, 35 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bond Percolation
  4. Invasion Percolation
  5. First Passage Percolation
  6. Coupling and Main Theorem
  7. Proof Sketch
  8. Proof of Theorem \ref{['thm:coupling']}
  9. Proofs of Lemmas
  10. Simulation of First Passage Percolation

Key Result

Theorem 3.1

For any $\epsilon >0$, $r\ge 0$, there exists an $R_0=R_0(\epsilon, r)$ such that for all $R\ge R_0$ and where $K=K(R, \epsilon/2):=\tfrac{\log|E_R|}{\delta(R, \epsilon/2)}=O(\epsilon^{-1}R^{4d}\log R )$ as $R\to\infty, \epsilon\to 0^{+}$.

Figures (3)

  • Figure 1: An example configuration in the coupled process in $\mathbb{Z}^2$. Edges $f$ drawn in red are such that $e'_{\texttt{adj}}\le f<_{\texttt{IP}}e'$, edges $g$ drawn in blue are such that $g<_{\texttt{IP}}e'_{\texttt{adj}}$. In this case, as $e^{(1)}_{\texttt{adj}}$ is adjacent to the origin and $e^{(1)}_{\texttt{adj}}>_{\texttt{IP}}e'_{\texttt{adj}}$, we stop our process at $e^{(2)}_{\texttt{adj}}=e_0$.
  • Figure 2: Simulations recording the number $P$ of trials $(n=10000)$ where the vertex with coordinates $(x,y)$ had passage time less than the passage time to the given boundary. The level curve $P=0.1$ is given in red. Outside of the boundary, the proportions are $0$ as expected, and the level curves away from $0$ of both distributions appear to be circular.
  • Figure 3: On the left, figure \ref{['fig:slicecomparison']} records the proportion of trials for which the vertices $(Rx,0)$ for $x=k/R$ with $k\in \{-R, \dots, R\}$ had passage time at most $T_{K}(0, \partial B_R)$ for $R=100,200, 500,1000$. We see that the shapes resemble that of $1-|x|^{\alpha(R)}$ for some exponent $\alpha(R)$. On the right, figure \ref{['fig:regression']} shows in blue is the proportion of trials for which the vertices $(1000x,0)$ for $x=k/1000$ with $k\in \{-1000, \dots, 1000\}$ had passage time at most $T_{K}(0, \partial B_{1000})$. In black is a regression of the form $1-|x|^{\alpha}$ with $\alpha\approx 0.23$ and correlation coefficient $r=0.998$.

Theorems & Definitions (11)

  • Theorem 3.1
  • Corollary 3.2
  • Lemma 4.1
  • Lemma 4.2
  • proof : Proof of Theorem \ref{['thm:coupling']}
  • Lemma 5.1
  • proof
  • Lemma 5.1
  • proof
  • Lemma 5.1
  • ...and 1 more