[Paper Review] percolation on finite graphs

Itaï Benjamini|ArXiv.org|Jun 4, 2001

Stochastic processes and statistical mechanics8 references3 citations

TL;DR

This paper investigates the emergence of a giant connected component in percolation on finite transitive graphs, establishing conditions under which a unique giant component forms with high probability as the graph size grows. It proves that under weak concentration of measure and logarithmic diameter, supercritical percolation (p = p_G + ε) yields a unique linear-sized component, extending insights from random graphs and infinite percolation theory to finite transitive structures.

ABSTRACT

The asymptotic study of percolation on finite transitive graphs is considered. Several questions and very few answers regarding percolation on finite graphs are presented.

Motivation & Objective

• To understand the phase transition in percolation on finite transitive graphs, particularly the emergence of a giant connected component.
• To bridge the gap between percolation on infinite graphs and random graphs by studying intermediate finite transitive graphs.
• To establish conditions under which the threshold for a giant component is sharp and bounded away from 1.
• To explore the asymptotic behavior of component sizes and thresholds in sequences of finite transitive graphs converging to infinite graphs.
• To investigate the relationship between graph geometry (diameter, expansion, isoperimetry) and percolation thresholds in finite transitive settings.

Proposed method

• Uses a weak concentration of measure condition: for large sets A_n, B_n in G_n, their distance is o(diameter(G_n)).
• Applies a path-counting argument to show exponentially many edge-disjoint short paths exist between large sets A_n and B_n.
• Estimates the probability that none of these paths are open using the independence of path openness under p = p_G + ε.
• Employs the FKG inequality and group-theoretic arguments (e.g., word length bounds in Cayley graphs) to derive lower bounds on connectivity probabilities.
• Adapts techniques from infinite graph percolation and random graph theory, including isoperimetric constants and exponential intersection tail (EIT) properties.
• Uses the concept of p^G_α as the threshold where a component of size α|G| exists with probability 1/2, and studies its asymptotic behavior.

Experimental results

Research questions

• RQ1 Under what conditions on finite transitive graphs does supercritical percolation (p = p_G + ε) yield a unique giant component with high probability as |G| → ∞?
• RQ2 How do the percolation thresholds p^G_α for different α ∈ (0,1) behave asymptotically, and do they remain bounded away from each other?
• RQ3 Can the threshold p^G for a component of size |G|/2 be bounded away from 1 under geometric constraints like logarithmic diameter and bounded degree?
• RQ4 Does the percolation probability on finite transitive graphs converge to the infinite cluster probability on the limiting infinite graph?
• RQ5 What is the asymptotic size of the largest component at criticality, and does it follow the n^{2/3} scaling seen in critical Erdős–Rényi graphs?

Key findings

• Under weak concentration of measure, logarithmic diameter, and bounded degree, for any ε > 0 and C > 0, P_{p^G + ε}(unique component of size ≥ C|G|) → 1 as n → ∞.
• The threshold p^G for a component of size |G|/2 is bounded away from 1 if the diameter satisfies diam(G) < |G| / log|G|, suggesting a sharp geometric condition.
• For Cayley graphs, the probability that two vertices are in the same large component is bounded below by (Cq/2)^{6/qC}, where q is the probability a vertex is in a component of size ≥ C|G|.
• The conjecture that p^G_n_α → p_c(G) as n → ∞ holds for sequences converging to the 3-regular tree T_3, where p_c(T_3) = 1/2.
• In the hypercube {0,1}^d, the threshold for a giant component is 1/d, supporting the conjecture that thresholds converge to the critical probability of the limiting infinite graph.
• The largest component at criticality is conjectured to be of size O(n^{2/3}) for k-regular graphs at p = (k-1)^{-1}, analogous to critical random graphs.