Widths of crossings in Poisson Boolean percolation

TL;DR

The paper investigates the width of horizontal crossings in planar Poisson Boolean percolation, distinguishing occupied and vacant crossings across subcritical, critical, and supercritical regimes. By introducing two radius-enlargement couplings and leveraging Russo's formula, RSW theory, and near-critical scaling, it derives precise, regime-dependent bounds for the maximal crossing widths $\mathrm{w}_n$ (occupied) and $\mathrm{w}_n^*$ (vacant) conditioned on the existence of a crossing. Central to the results are the four-arm probabilities $\pi_4(n)$ and the critical window parameter $\alpha_n=1/(\pi_4(n)n^2)$, which govern how widths scale with $n$ and with the distance to criticality. The work provides both upper and lower bounds, yielding a detailed, quantitative picture of crossing geometry in continuum percolation and offering techniques potentially adaptable to more general random geometric models.

Abstract

We answer the following question: if the occupied (or vacant) set of a planar Poisson Boolean percolation model does contain a crossing of an $n\times n$ square, how wide is this crossing? The answer depends on the whether we consider the critical, sub- or super-critical regime, and is different for the occupied and vacant sets.

Figures (8)

• Figure 1: Continum percolation on $\mathbb{R}^2$.
• Figure 2: The width of an occupied (a) and vacant (b) crossing of an $n\times n$ square.
• Figure 3: Left: A configuration in $\tilde{\mathcal{C}}(R) \setminus {\mathcal{C}}(R+r)$ contains a horizontal occupied crossing of the blue square $[2r,2r + 2R]\times[-R,R]$, but no crossing of the slightly longer rectangle $[0,2r + 2R]\times[-R,R]$. Occupied crossing are depicted by bold lines, vacant ones by dashed lines. Right: The horizontal crossing of $[2r,2r + 2R]\times[-R,R]$ may be lengthened into one of $[2r,2r + 4R]\times[-R,R]$ at constant cost, due to the RSW theorem. This configuration belongs to $\tilde{\mathcal{C}}(R+kr) \setminus {\mathcal{C}}(R+(k+1)r)$ for any $0 \leq k < R/r$.
• Figure 4: Computing the width of a vacant crossing by enlarging the balls.
• Figure 5: When $\mathcal{O}^{(r)} \in \mathrm{cross}(n+ \sqrt{1 - r^2},n-1)$, one may identify a chain of points of $\eta$, each at a distance at most $2r$ from the previous, contained in $\mathbb{R} \times [-n,n]$, with the first and last within a distance $r$ of the left and right sides of the rectangle, respectively. The path $\gamma$ (bold black path) is obtained by interpolating linearly between these points, and potentially connecting the first and last points by horizontal lines to the sides of $[-n,n]^2$. The distance from $\gamma$ to $[\bigcup_{i=0}^k B_1(x_i)]^c$ is attained at the center of one of the segments $[x_{i-1},x_i]$.

Theorems & Definitions (23)

• Theorem 1.1: widths of vacant crossings
• Theorem 1.2: widths of occupied crossings
• Remark 1.3
• Proposition 2.1: Russo's formula
• Proposition 2.2: RSW
• Lemma 2.3
• proof
• Theorem 2.4: Crossings in near-critical percolation
• Lemma 3.1
• Remark 3.2