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Shadow and percolation II: discrete and continuous landscapes with correlations

David Vernotte

TL;DR

This work studies shadow percolation for planar Gaussian landscapes by introducing the slope field $\alpha^f$ to identify lit and shadow regions. The authors develop finite-range approximations and a two-scale renormalization framework to transfer local crossing events into global percolation structure, handling both continuous landscapes with correlations and discrete, correlated fields. They establish a nontrivial phase transition: for continuous fields with sufficiently fast decay ($\beta>\frac{5}{2}$) there exists $\ell_1>0$ such that $\{\alpha^f\le\ell\}$ contains a unique unbounded component while $\{\alpha^f\ge\ell\}$ has no unbounded component for every $\ell>\ell_1$, and a parallel discrete result shows corresponding transitions for $\alpha^X$. The methods combine Gaussian-field approximation, renormalization, and gluing arguments to overcome lack of standard percolation properties, offering insights into phase transitions in correlated planar random fields with geometric illumination interpretations.

Abstract

In this paper we consider a discrete or continuous landscape with correlations and we consider a source of light (a sun) at infinity emitting parallel rays of light making a slope l with the horizontal plane. Depending on the value of l some portions of the landscape may be lit by the sun or be in the shadow. Under some assumptions, we show that if l is big enough then there exists a giant component of light. However, if l>0 is small enough and if we are in discrete case, then there exists a giant component of shadow. We relate this problem to the study of the percolation properties of a new random planar field.

Shadow and percolation II: discrete and continuous landscapes with correlations

TL;DR

This work studies shadow percolation for planar Gaussian landscapes by introducing the slope field to identify lit and shadow regions. The authors develop finite-range approximations and a two-scale renormalization framework to transfer local crossing events into global percolation structure, handling both continuous landscapes with correlations and discrete, correlated fields. They establish a nontrivial phase transition: for continuous fields with sufficiently fast decay () there exists such that contains a unique unbounded component while has no unbounded component for every , and a parallel discrete result shows corresponding transitions for . The methods combine Gaussian-field approximation, renormalization, and gluing arguments to overcome lack of standard percolation properties, offering insights into phase transitions in correlated planar random fields with geometric illumination interpretations.

Abstract

In this paper we consider a discrete or continuous landscape with correlations and we consider a source of light (a sun) at infinity emitting parallel rays of light making a slope l with the horizontal plane. Depending on the value of l some portions of the landscape may be lit by the sun or be in the shadow. Under some assumptions, we show that if l is big enough then there exists a giant component of light. However, if l>0 is small enough and if we are in discrete case, then there exists a giant component of shadow. We relate this problem to the study of the percolation properties of a new random planar field.

Paper Structure

This paper contains 9 sections, 23 theorems, 199 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Comments on Theorem \ref{['thm:continuous_l_grand']} and Theorem \ref{['thm:discrete_correlated']}.
  3. Strategy of proof
  4. Acknowledgements:
  5. Preliminaries
  6. A renormalization argument
  7. Proof of Theorem \ref{['thm:continuous_l_grand']}
  8. Proof of Theorem \ref{['thm:discrete_correlated']}
  9. Gluing constructions for planar percolation

Key Result

Theorem 1.4

Let $f=q\ast W$ where $q$ satisfies Assumption a:q for some $\beta>\frac{5}{2}$. There exists $\ell_1\in ]0,\infty[$ (depending on $q$) such that, for any $\ell>\ell_1$, almost surely, the set $\{\alpha^f \leq \ell\}$ contains a unique unbounded connected component and the set $\{\alpha^f \geq \ell\

Figures (4)

  • Figure 1: Illustration of $\{\alpha^f>\ell\}$ when $f$ is the Bargmann-Fock field and $\ell=0.3$. In black, the points $x$ such that $\alpha^f(x)>\ell$, in white the points $x$ such that $\alpha^f(x)\leq \ell$.
  • Figure 2: Illustration of the gluing construction in Lemma \ref{['lemma:grow_rectangle']}, in blue the horizontal crossings of $\mathcal{R}_1$ and $\mathcal{R}_2$ and in red the vertical crossing of $\mathcal{R}_3$.
  • Figure 3: Illustration of the gluing construction in Lemma \ref{['lemma:cross_imply_giant']}.
  • Figure 4: Illustration of the gluing construction in the proof of Proposition \ref{['prop:crossings']}.

Theorems & Definitions (55)

  • Definition 1.1
  • Example 1.3
  • Theorem 1.4
  • Theorem 1.6
  • Definition 2.1
  • Proposition 2.2: see MV20, Sev21
  • Proposition 2.3: see AT07 for instance
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 45 more