[Paper Review] Percolative properties of Brownian interlacements and its vacant set

Xinyi Li|arXiv (Cornell University)|Oct 26, 2016

Stochastic processes and statistical mechanics31 references3 citations

TL;DR

This paper establishes that Brownian interlacements in $\mathbb{R}^d$, $d \geq 3$, are well-connected: any two $r$-sausages in the interlacement set can be linked via at most $\lceil(d-2)/2\rceil$ intermediate sausages almost surely. It further proves a non-trivial percolation phase transition for the vacant set, with a sharp threshold $\alpha_r^* = \alpha_1^* r^{2-d}$ separating percolation (for $\alpha < \alpha_r^*$) from non-percolation (for $\alpha > \alpha_r^*$).

ABSTRACT

In this article we investigate the percolative properties of Brownian interlacements, a model introduced by Alain-Sol Sznitman in arXiv:1209.4531, and show that: the interlacement set is "well-connected", i.e., any two "sausages" in $d$-dimensional Brownian interlacements, $d\geq 3$, can be connected via no more than $\lceil (d-4)/2 ceil$ intermediate sausages almost surely; while the vacant set undergoes a non-trivial percolation phase transition when the level parameter varies.

Motivation & Objective

• To investigate the percolative connectivity of Brownian interlacements in $\mathbb{R}^d$, $d \geq 3$, focusing on the structure of the interlacement set and its graph of intersecting $r$-sausages.
• To establish the existence of a non-trivial percolation phase transition in the vacant set of Brownian interlacements as the level parameter $\alpha$ varies.
• To determine the critical threshold $\alpha_r^*$ for percolation in the vacant set and analyze its scaling behavior with respect to the radius $r$.
• To explore the uniqueness of the unbounded cluster in the vacant set and the sharpness of the phase transition, drawing analogies to discrete random interlacements.

Proposed method

• Construct a random geometric graph $G_{\alpha,r}$ where vertices represent trajectories in Brownian interlacements and edges connect trajectories whose $r$-neighborhoods intersect.
• Prove that the diameter of $G_{\alpha,r}$ is almost surely $\lceil(d-2)/2\rceil$, implying that any two sausages are connected via at most $\lceil(d-2)/2\rceil - 1$ intermediate sausages.
• Use scaling properties of Brownian interlacements to relate the critical threshold $\alpha_r^*$ for radius $r$ to the threshold $\alpha_1^*$ at radius 1, yielding $\alpha_r^* = \alpha_1^* r^{2-d}$.
• Apply a multi-scale renormalization scheme involving nested boxes $B_k$, with events $A_n^\alpha$ and $\widehat{A}_n^\alpha$ to control the probability of non-percolation in large regions.
• Use exponential decay estimates on the probability of bad events $B_{k,x}^\alpha$ and $\overline{B}_{k,x}^\alpha$ to show that $\mathbb{P}[(\widehat{A}_n^\alpha)^c] \to 0$ as $n \to \infty$, implying percolation for $\alpha < \widehat{\alpha}$.
• Leverage the scaling invariance of Brownian motion and the Poissonian nature of the interlacement process to derive the critical threshold and its dependence on $r$.

Experimental results

Research questions

• RQ1 What is the maximal graph distance between any two $r$-sausages in the Brownian interlacement set, and does this distance grow with dimension?
• RQ2 Does the vacant set of Brownian interlacements undergo a non-trivial percolation phase transition as the level parameter $\alpha$ increases?
• RQ3 Is the critical threshold $\alpha_r^*$ for vacant set percolation sharp, and how does it scale with the radius $r$?
• RQ4 Does the unbounded cluster in the vacant set remain unique in the supercritical regime, and can techniques from discrete random interlacements be adapted to prove this?
• RQ5 How do the percolation thresholds for the vacant set in slabs $F_R = \mathbb{R}^2 \times [0,R]^{d-2}$ relate to the full-space threshold $\alpha_r^*$?

Key findings

• The diameter of the graph $G_{\alpha,r}$ formed by intersecting $r$-sausages in Brownian interlacements is almost surely $\lceil(d-2)/2\rceil$, proving that the interlacement set is well-connected with bounded path lengths.
• For all $\alpha, r > 0$, the interlacement set $\mathcal{I}_r^\alpha$ is $\mathbb{P}$-almost surely connected, as a direct consequence of the bounded graph diameter.
• The vacant set $\mathcal{V}_r^\alpha$ undergoes a non-trivial percolation phase transition at a finite critical threshold $\alpha_r^* = \alpha_1^* r^{2-d}$, with percolation occurring if $\alpha < \alpha_r^*$ and non-percolation if $\alpha > \alpha_r^*$.
• The critical threshold $\alpha_r^*$ is finite and strictly positive for all $r > 0$, and its scaling with $r$ is explicitly determined by the dimension-dependent factor $r^{2-d}$.
• For $\alpha < \widehat{\alpha}$, the vacant set percolates not only in $\mathbb{R}^d$ but also in any slab $F_R = \mathbb{R}^2 \times [0,R]^{d-2}$, suggesting a robustness of percolation in lower-dimensional slices.
• The results suggest that $\alpha_r^*$ may be sharp, with $\widetilde{\alpha}_r \leq \alpha_r^*$, where $\widetilde{\alpha}_r$ is the threshold for percolation in a plane, and $\lim_{R \to \infty} \widetilde{\alpha}_{R,r} = \alpha_r^*$, indicating a possible phase transition in the geometry of the unbounded cluster.