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Shadow and percolation I: discrete landscapes with independence

David Vernotte

TL;DR

This work introduces a novel percolation model derived from a planar random height field $X$ on $\mathbb{Z}^2$ and a sun with slope $\ell$, reformulating visibility as the excursion set of $\alpha(u)=\sup_{r\ge1}\frac{X(u+re_1)-X(u)}{r}$. Under i.i.d. $X$ with density tails, the authors prove a two-phase transition: for small $\ell$ there is almost surely an unbounded shadow cluster (where $\alpha\ge\ell$), while for large $\ell$ there is an unbounded sun-lit cluster (where $\alpha\le\ell$). The analysis navigates the lack of standard percolation properties (no FKG, long-range dependence) by two complementary strategies: a stochastic domination approach in the large-$\ell$ regime and a geometric, ordering-based Peierls argument in the small-$\ell$ regime, supplemented by a truncation technique and decomposition to control dependencies. These results extend percolation theory to dependent, non-FKG settings and illuminate how horizon-level geometry in random landscapes yields robust phase behavior with potential extensions to correlated landscapes.

Abstract

Let X be a planar random field on Z^2 which we interpret as a random height function describing some landscape of montains. We consider a source of light (a sun) located at infinity in a direction parallel with an axis od Z^2 and emitting rays which are all parallel and make a slope l with the horizontal plane. Given the value of l some montains of the landscape will be lit by the sun and other will be in the shadow of some higher mountain. Under some assumptions on X, including and independence assumption, we prove that this model may present two different phases depending on l. When l>0 is small enough then, almost surely, there exists an unbounded cluster of points in the shadow. However, if l is big enough then, almost surely, there exists an unbounded cluster of points lit by the sun. We reformulate this problem in terms of percolation of a field alpha which has a simple definition (in terms of X) but that does not present many of the nice properties usually found in percolation models such as FKG inequality, invariance by rotation or finite range correlations.

Shadow and percolation I: discrete landscapes with independence

TL;DR

This work introduces a novel percolation model derived from a planar random height field on and a sun with slope , reformulating visibility as the excursion set of . Under i.i.d. with density tails, the authors prove a two-phase transition: for small there is almost surely an unbounded shadow cluster (where ), while for large there is an unbounded sun-lit cluster (where ). The analysis navigates the lack of standard percolation properties (no FKG, long-range dependence) by two complementary strategies: a stochastic domination approach in the large- regime and a geometric, ordering-based Peierls argument in the small- regime, supplemented by a truncation technique and decomposition to control dependencies. These results extend percolation theory to dependent, non-FKG settings and illuminate how horizon-level geometry in random landscapes yields robust phase behavior with potential extensions to correlated landscapes.

Abstract

Let X be a planar random field on Z^2 which we interpret as a random height function describing some landscape of montains. We consider a source of light (a sun) located at infinity in a direction parallel with an axis od Z^2 and emitting rays which are all parallel and make a slope l with the horizontal plane. Given the value of l some montains of the landscape will be lit by the sun and other will be in the shadow of some higher mountain. Under some assumptions on X, including and independence assumption, we prove that this model may present two different phases depending on l. When l>0 is small enough then, almost surely, there exists an unbounded cluster of points in the shadow. However, if l is big enough then, almost surely, there exists an unbounded cluster of points lit by the sun. We reformulate this problem in terms of percolation of a field alpha which has a simple definition (in terms of X) but that does not present many of the nice properties usually found in percolation models such as FKG inequality, invariance by rotation or finite range correlations.

Paper Structure

This paper contains 14 sections, 11 theorems, 98 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Symmetries of the domain
  3. Symmetries of the random field
  4. Stationarity
  5. Independence
  6. Conditioning
  7. FKG property
  8. Comments on Theorem \ref{['a4:thm:principal']} and open questions
  9. Strategy of proof
  10. Notations
  11. Acknowledgements:
  12. The regime of large $\ell$
  13. The regime of small $\ell$
  14. Recovering $X$ from $\alpha$

Key Result

Theorem 1.4

Let $X : \mathbb{Z}^2 \to \mathbb{R}$ be such that Assumption a4:a:a1 holds and let $\alpha$ be defined as in a4:eq:def_alpha_general. We denote by $\mu$ the law of $X(0)$. In the context of site percolation on $\mathbb{Z}^2$ the following holds.

Figures (1)

  • Figure 1: Illustration of a configuration $\alpha$ on the hexagonal lattice when the entries of $X$ are independent standard Gaussian random variables. In black, $\alpha>\ell$. In white, $\alpha<\ell$. In red, the biggest component of $\alpha>\ell$. In blue the biggest component of $\alpha<\ell$.

Theorems & Definitions (38)

  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Remark 2.2
  • Corollary 2.3
  • proof : Proof of Corollary \ref{['a4:cor:discrete_indep']}
  • Definition 2.4
  • Remark 2.5
  • proof
  • Definition 2.7
  • ...and 28 more