[Paper Review] Brownian Web and Oriented Percolation: Density Bounds

Anish Sarkar, Rongfeng Sun|arXiv (Cornell University)|Nov 10, 2011

Stochastic processes and statistical mechanics7 references3 citations

TL;DR

This paper establishes a new convergence criterion for the Brownian web using a dual density bound that avoids reliance on the FKG inequality, offering a more robust alternative to prior methods. It proves that the spatial density of rightmost infinite open paths in supercritical oriented percolation decays as $\frac{2+o(1)}{\sigma\sqrt{\pi t}}$ under diffusive scaling, confirming a precise asymptotic decay rate.

ABSTRACT

In a recent work, we proved that under diffusive scaling, the collection of rightmost infinite open paths in a supercritical oriented percolation configuration on the space-time lattice Z^2 converges in distribution to the Brownian web. In that proof, the FKG inequality played an important role in establishing a density bound, which is a part of the convergence criterion for the Brownian web formulated by Fontes et al (2004). In this note, we illustrate how an alternative convergence criterion formulated by Newman et al (2005) can be verified in this case, which involves a dual density bound that can be established without using the FKG inequality. This alternative approach is in some sense more robust. We will also show that the spatial density of the collection of rightmost infinite open paths starting at time 0 decays asymptotically in time as c/\sqrt{t} for some c>0.

Motivation & Objective

• To provide an alternative convergence criterion for the Brownian web that avoids the FKG inequality, enhancing robustness.
• To establish a dual density bound for the collection of rightmost infinite open paths in supercritical oriented percolation.
• To derive the precise asymptotic decay rate of the spatial density of these paths over time.
• To verify weak convergence of rescaled path collections to the Brownian web using a novel approach based on negative correlation and vague convergence.

Proposed method

• Utilizes an alternative convergence criterion from Newman et al. [NRS05], which relies on a dual density bound instead of FKG inequality.
• Applies diffusive scaling $S_\epsilon$ to the collection of rightmost infinite open paths $\Gamma$ in oriented percolation on $\mathbb{Z}^2_{\text{even}}$.
• Introduces the random set $\mathcal{R}_0(n)$, representing the rightmost paths at time $n$ starting from time 0, to avoid future dependence issues in path dependence.
• Employs Reimer’s inequality to establish negative correlation between events $\{i \in \mathcal{R}_0(n)\}$ and $\{j \in \mathcal{R}_0(n)\}$ for $i < j$, enabling tightness and weak convergence.
• Uses vague convergence of random counting measures to strengthen weak convergence of rescaled path sets to the Brownian web.
• Applies Fatou’s lemma and translation invariance to derive lower and upper bounds on path existence probabilities, leading to the asymptotic density result.

Experimental results

Research questions

• RQ1 Can the convergence of rightmost infinite paths in oriented percolation to the Brownian web be established without relying on the FKG inequality?
• RQ2 What is the precise asymptotic decay rate of the spatial density of rightmost infinite open paths in supercritical oriented percolation?
• RQ3 How can a dual density bound be constructed and verified in the absence of FKG-based estimates?
• RQ4 Can weak convergence of rescaled path collections to the Brownian web be strengthened using negative correlation and vague convergence?

Key findings

• The dual density bound can be established without the FKG inequality, providing a more robust alternative to the convergence criterion in Fontes et al. [FINR04].
• The spatial density of rightmost infinite open paths starting at time 0 decays asymptotically as $\frac{2+o(1)}{\sigma\sqrt{\pi t}}$ for large $t$.
• Weak convergence of $S_{1/\sqrt{n}}\Gamma_0$ to the standard Brownian web $\mathcal{W}_0$ holds, with convergence of the rescaled path sets at time 1 in the vague topology.
• The limit of $\frac{\sigma\sqrt{n}}{2}\mathbb{P}(\Gamma_0(n) \cap \{0,1\} \neq \emptyset)$ as $n \to \infty$ is bounded below by $\frac{1}{\sqrt{\pi}}$, matching the intensity of the Brownian web.
• The use of $\mathcal{R}_0(n)$ instead of $\Gamma_0(n)$ enables the application of Reimer’s inequality due to disjoint edge dependence, which is not possible for $\Gamma_0(n)$ due to future dependence.
• The convergence of $S_{1/\sqrt{n}}\mathcal{R}_0$ to $\mathcal{W}_0$ as random counting measures in the vague topology is established, leading to the precise asymptotic density result.