[Paper Review] Expansion in $n^{-1}$ for percolation critical values on the $n$-cube and $Z^n$: the first three terms

Remco van der Hofstad, Gordon Slade|ArXiv.org|Jan 8, 2004

Stochastic processes and statistical mechanics13 references4 citations

TL;DR

This paper derives the first three terms of the $ n^{-1} $ expansion for the bond percolation critical probability on the $ n $-cube and $ \mathbb{Z}^n $ using the lace expansion. It proves $ p_c(\mathbb{G}) = \Omega^{-1} + \Omega^{-2} + \frac{7}{2}\Omega^{-3} + O(\Omega^{-4}) $ for both graphs, where $ \Omega = n $ for the $ n $-cube and $ \Omega = 2n $ for $ \mathbb{Z}^n $, extending prior results and simplifying earlier proofs for $ \mathbb{Z}^n $.

ABSTRACT

Let $p_c(\mathbb{Q}_n)$ and $p_c(\mathbb{Z}^n)$ denote the critical values for nearest-neighbour bond percolation on the $n$-cube $\mathbb{Q}_n = \{0,1\}^n$ and on $\Z^n$, respectively. Let $Ω= n$ for $\mathbb{G} = \mathbb{Q}_n$ and $Ω= 2n$ for $\mathbb{G} = \mathbb{Z}^n$ denote the degree of $\mathbb{G}$. We use the lace expansion to prove that for both $\mathbb{G} = \mathbb{Q}_n$ and $\mathbb{G} = \mathbb{Z}^n$, $p_c(\mathbb{G}) & = \cn^{-1} + \cn^{-2} + {7/2} \cn^{-3} + O(\cn^{-4}).$ This extends by two terms the result $p_c(\mathbb{Q}_n) = \cn^{-1} + O(\cn^{-2})$ of Borgs, Chayes, van der Hofstad, Slade and Spencer, and provides a simplified proof of a previous result of Hara and Slade for $\mathbb{Z}^n$.

Motivation & Objective

• Establish a higher-order asymptotic expansion for the critical percolation threshold on the $ n $-cube and $ \mathbb{Z}^n $, extending beyond the first-order result.
• Provide a simplified and unified proof for the critical value expansion on both $ \mathbb{Z}^n $ and the $ n $-cube using the lace expansion technique.
• Confirm that the first three coefficients in the $ \Omega^{-1} $ expansion are identical for both graphs, where $ \Omega $ is the graph degree.
• Address the open question of whether $ p_c({\mathbb{Q}}_n) = \frac{1}{n-1} $, showing it is asymptotically $ \frac{1}{n} + \frac{1}{n^2} + \frac{7}{2n^3} + O(n^{-4}) $, thus disproving equality.
• Explore the possibility of computing higher-order coefficients and investigate potential differences in the $ \Omega^{-4} $ term between the $ n $-cube and $ \mathbb{Z}^n $.

Proposed method

• The lace expansion is applied to both the $ n $-cube and $ \mathbb{Z}^n $, leveraging the general framework established by Hara and Slade for $ \mathbb{Z}^n $, but adapted to handle the finite structure of the $ n $-cube.
• Key estimates are derived using the generating function $ \chi(p) = \mathbb{E}_p |C(0)| $, which tracks the expected cluster size at the origin, and its behavior near the critical threshold.
• Critical values are defined via $ \chi(p_c) = \lambda_0 2^{n/3} $ for the $ n $-cube and $ \chi(p_c) < \infty $ for $ \mathbb{Z}^n $, ensuring uniqueness and enabling asymptotic analysis.
• Higher-order terms in the expansion are computed by analyzing the lace expansion's recursive structure, with error terms controlled via bounds on truncated correlation functions.
• Non-pivotal (NP) conditions in the expansion are shown to contribute only to error terms, allowing simplification of the main contributions to the critical probability.
• By comparing contributions from different bond configurations and cluster structures, the authors isolate the $ \Omega^{-1} $, $ \Omega^{-2} $, and $ \Omega^{-3} $ terms, yielding the full expansion.

Experimental results

Research questions

• RQ1 What is the asymptotic expansion of the bond percolation critical probability $ p_c $ on the $ n $-cube up to and including the $ \Omega^{-3} $ term?
• RQ2 How does the critical value on $ \mathbb{Z}^n $ compare to that on the $ n $-cube in the first three orders of $ \Omega^{-1} $, where $ \Omega $ is the graph degree?
• RQ3 Can the lace expansion be used to derive a simplified proof of the known critical value expansion for $ \mathbb{Z}^n $, and does it extend naturally to the $ n $-cube?
• RQ4 Is the critical value on the $ n $-cube equal to $ \frac{1}{n-1} $, as conjectured by Bollobás, Kohayakawa, and Łuczak?
• RQ5 Are the coefficients of the $ \Omega^{-4} $ term identical for $ \mathbb{Z}^n $ and the $ \mathbb{Q}_n $, or do they differ due to structural differences in the graphs?

Key findings

• The critical percolation threshold for $ \mathbb{Z}^n $ satisfies $ p_c(\mathbb{Z}^n) = \frac{1}{2n} + \frac{1}{(2n)^2} + \frac{7}{2(2n)^3} + O\left(\frac{1}{(2n)^4}\right) $ as $ n \to \infty $, confirming the first three terms of the $ \Omega^{-1} $ expansion.
• For the $ n $-cube, $ p_c({\mathbb{Q}}_n) = \frac{1}{n} + \frac{1}{n^2} + \frac{7}{2n^3} + O\left(\frac{1}{n^4}\right) $, with the error term depending on the choice of $ \lambda_0 $ in the definition $ \chi(p_c) = \lambda_0 2^{n/3} $.
• The first three coefficients in the $ \Omega^{-1} $ expansion are identical for both $ \mathbb{Z}^n $ and the $ n $-cube, despite their differing graph structures.
• The paper provides a simplified proof of the $ \mathbb{Z}^n $ result previously established by Hara and Slade, using a more mechanical application of the lace expansion.
• Numerical evidence suggests that the $ \Omega^{-4} $ coefficient likely differs between $ \mathbb{Z}^n $ and the $ n $-cube, though this term is not computed here.
• The method is generalizable to other graphs with convergent lace expansions, such as the Hamming cube $ \{0,1,\ldots,s\}^n $ for $ s \geq 2 $, though cycles in such graphs may affect higher-order coefficients.