[Paper Review] On the critical value function in the divide and color model

András Bálint, Vincent Beffara+1 more|arXiv (Cornell University)|Sep 15, 2011

Stochastic processes and statistical mechanics11 references9 citations

TL;DR

This paper investigates the continuity and monotonicity of the critical coloring value $ r_c^G(p) $ in the divide and color (DaC) model on graphs, showing that for $ \mathbb{Z}^2 $, $ r_c^G(p) $ is continuous on $[0, \frac{1}{2})$, and constructing bounded-degree graphs where $ r_c^G(p) $ is discontinuous at $ p = \frac{1}{2} $, thereby resolving questions on locality in percolation.

ABSTRACT

The divide and color model on a graph $G$ arises by first deleting each edge of $G$ with probability $1-p$ independently of each other, then coloring the resulting connected components (\emph{i.e.}, every vertex in the component) black or white with respective probabilities $r$ and $1-r$, independently for different components. Viewing it as a (dependent) site percolation model, one can define the critical point $r_c^G(p)$. In this paper, we mainly study the continuity properties of the function $r_c^G$, which is an instance of the question of locality for percolation. Our main result is the fact that in the case $G=\mathbb Z^2$, $r_c^G$ is continuous on the interval $[0,1/2)$; we also prove continuity at $p=0$ for the more general class of graphs with bounded degree. We then investigate the sharpness of the bounded degree condition and the monotonicity of $r_c^G(p)$ as a function of $p$.

Motivation & Objective

• To understand the dependence of the critical coloring parameter $ r_c^G(p) $ on the edge parameter $ p $ in the divide and color model.
• To investigate the continuity properties of $ r_c^G(p) $, particularly at $ p = 0 $ and at $ p = \frac{1}{2} $, in relation to graph structure.
• To examine the sharpness of the bounded-degree condition for continuity, and to determine whether $ r_c^G(p) $ is monotonic in $ p $.
• To construct explicit examples of graphs where $ r_c^G(p) $ exhibits discontinuities below the bond percolation threshold.
• To establish connections between the DaC model and the locality problem in percolation theory.

Proposed method

• Uses stochastic domination arguments to derive bounds on $ r_c^G(p) $ in terms of $ r_c^G(0) $, the critical value for site percolation.
• Applies Kolmogorov's 0-1 law and coupling techniques to analyze the probability of infinite black clusters in the DaC model.
• Employs a recursive construction of graphs $ D_n $ and defines a limit graph $ G = \Gamma_D $ using a sequence of such graphs to study critical behavior.
• Leverages Lemma 15 to relate the limit of cluster probabilities in finite graphs to the critical value in the infinite limit graph.
• Uses Lemma 16 to construct a sequence of bounded-degree graphs with sharp threshold behavior at $ p = \frac{1}{2} $, enabling discontinuity in $ r_c^G(p) $.
• Analyzes the asymptotic behavior of $ f^{D_n}(p,r) $ and $ h^{D_n}(p) $ to derive limits that determine $ r_c^G(p) $ via the critical threshold condition.

Experimental results

Research questions

• RQ1 Is the critical value function $ r_c^G(p) $ continuous at $ p = 0 $ for graphs of bounded degree?
• RQ2 Can discontinuities in $ r_c^G(p) $ occur at $ p > 0 $, even when the graph has bounded degree?
• RQ3 How does the bounded-degree assumption affect the continuity of $ r_c^G(p) $, and is it sharp?
• RQ4 Does the DaC model exhibit locality in the sense that small perturbations in edge retention probability $ p $ lead to small changes in $ r_c^G(p) $?
• RQ5 What is the relationship between the critical value $ r_c^G(p) $ and the underlying bond percolation structure on $ \mathbb{Z}^2 $?

Key findings

• For $ G = \mathbb{Z}^2 $, the critical value function $ r_c^G(p) $ is continuous on the interval $[0, \frac{1}{2})$.
• For any graph $ G $ with bounded degree, $ r_c^G(p) $ is continuous at $ p = 0 $, as shown via stochastic domination bounds.
• There exists a graph $ G $ with $ p_c^G > 0 $ such that $ r_c^G(p) $ is discontinuous at $ p = 0 $, demonstrating that bounded degree is a necessary condition for continuity at 0.
• A bounded-degree graph $ G $ exists with $ p_c^G > \frac{1}{2} $ such that $ r_c^G(p) $ is discontinuous at $ p = \frac{1}{2} $, constructed using a sequence of graphs with sharp threshold behavior.
• The limit of $ f^{D_n}(p,r) $ as $ n \to \infty $ satisfies $ \lim_{n \to \infty} f^{D_n}(p,r) = p + (1-p)r $ for $ p > 0 $, which implies $ r_c^G(p) \to \frac{1}{2} $ as $ p \to 0^+ $, while $ r_c^G(0) = \frac{1}{\sqrt{2}} $, confirming discontinuity at 0.
• For $ p = \frac{1}{2} $, the critical value satisfies $ r_c^G(\frac{1}{2}) \leq r_1 < r_0 \leq r_c^G(p) $ for all $ p < \frac{1}{2} $, where $ r_0 $ solves $ r(1 + r) = 1 $ and $ r_1 = \frac{1}{2} $, proving discontinuity at $ \frac{1}{2} $.