[Paper Review] Critical percolation on random regular graphs

Asaf Nachmias, Yuval Peres|ArXiv.org|Jul 19, 2007

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper establishes mean-field behavior in critical bond percolation on random $d$-regular graphs for fixed $d \geq 3$ as $n \to \infty$. It identifies a scaling window of width $n^{-1/3}$ around the critical threshold $p_c = 1/(d-1)$, where component sizes scale as $n^{2/3}$, and derives the limiting joint distribution of component sizes. Outside the window, the largest component is concentrated at $\sim \varepsilon(n)n$ in the supercritical regime and at $\sim \varepsilon(n)^{-2} \log(n \varepsilon(n)^3)$ in the subcritical regime, with a duality between regimes.

ABSTRACT

We describe the component sizes in critical independent p-bond percolation on a random d-regular graph on n vertices, where d \geq 3 is fixed and n grows. We prove mean-field behavior around the critical probability p_c=1/(d-1). In particular, we show that there is a scaling window of width n^{-1/3} around p_c in which the sizes of the largest components are roughly n^{2/3} and we describe their limiting joint distribution. We also show that for the subcritical regime, i.e. p = (1-eps(n))p_c where eps(n)=o(1) but \eps(n)n^{1/3} tends to infinity, the sizes of the largest components are concentrated around an explicit function of n and eps(n) which is of order o(n^{2/3}). In the supercritical regime, i.e. p = (1+\eps(n))p_c where eps(n)=o(1) but eps(n)n^{1/3} tends to infinity, the size of the largest component is concentrated around the value (2d/(d-2))\eps(n)n and a duality principle holds: other component sizes are distributed as in the subcritical regime.

Motivation & Objective

• To determine whether critical bond percolation on random $d$-regular graphs exhibits mean-field behavior, as posed by Itai Benjamini.
• To characterize the scaling window of width $n^{-1/3}$ around the critical probability $p_c = 1/(d-1)$, where component sizes are of order $n^{2/3}$.
• To describe the limiting joint distribution of component sizes within the critical window.
• To analyze the behavior of component sizes in the subcritical and supercritical regimes, including concentration and duality.
• To establish sharp upper and lower bounds on component sizes in all regimes, using martingale and branching process approximations.

Proposed method

• Analyzes $G(n,d,p)$, a random $d$-regular graph with independent $p$-bond percolation, for fixed $d \geq 3$ and large $n$.
• Uses a scaling window $p = \frac{1 + \lambda n^{-1/3}}{d-1}$ to study critical behavior, with $\lambda \in \mathbb{R}$.
• Applies martingale central limit theorems to show convergence of rescaled component size processes to a Brownian motion with drift.
• Derives bounds on component sizes via branching process approximations and conditional expectation techniques.
• Employs a duality principle: component size distributions in the supercritical regime mirror those in the subcritical regime under parameter transformation.
• Uses continuous linear interpolation of discrete processes and weak convergence to establish functional limit theorems for component size processes.

Experimental results

Research questions

• RQ1 Does critical percolation on random $d$-regular graphs exhibit mean-field behavior, as seen in the Erdős–Rényi model?
• RQ2 What is the width of the critical scaling window for component size fluctuations in random $d$-regular graphs?
• RQ3 What is the limiting joint distribution of component sizes within the critical window?
• RQ4 How do component sizes behave in the subcritical and supercritical regimes, particularly outside the scaling window?
• RQ5 Is there a duality between the subcritical and supercritical regimes in terms of component size distributions?

Key findings

• The critical window for $p$ is of width $n^{-1/3}$, centered at $p_c = 1/(d-1)$, where component sizes scale as $n^{2/3}$.
• Within the critical window, the largest component size $|\mathcal{C}_1|$ satisfies $\mathbf{P}(|\mathcal{C}_1| \geq A n^{2/3}) \leq C(\lambda,d) e^{-c(\lambda,d) A^3} / A$, indicating exponential tail bounds.
• For $p = (1 - \varepsilon(n))/ (d-1)$ with $\varepsilon(n) \to 0$ and $\varepsilon(n) n^{1/3} \to \infty$, the largest component is concentrated at $\sim \frac{d-2}{d-1} \psi_n(\varepsilon(n))$, where $\psi_n(\varepsilon) = 2 \varepsilon^{-2} \log(n \varepsilon^{-3})$.
• For $p = (1 + \varepsilon(n))/ (d-1)$ with $\varepsilon(n) \to 0$ and $\varepsilon(n) n^{1/3} \to \infty$, the largest component is concentrated at $\sim \frac{2d}{d-2} \varepsilon(n) n$.
• A duality principle holds: the distribution of component sizes in the supercritical regime matches that of the subcritical regime under $\varepsilon \to -\varepsilon$.
• The rescaled component size process $n^{-1/3} Y_{(n^{2/3} t)}$ converges weakly to a Brownian motion with drift $B^{\lambda}(t)$, where $B^{\lambda}$ is a standard Brownian motion with drift $\lambda t - \frac{(d-2)t^2}{2d(d-1)}$.