Bootstrap percolation on a generalized Hamming cube

We consider the $r$-neighbor bootstrap percolation process on the graph with vertex set $V={0,1}^n$ and edges connecting the pairs at Hamming distance $1,2,\dots,k$, where $k\ge 2$. We find asymptotics of the critical probability of percolation for $r=2,3$. In the deterministic setting, we obtain several results for the size of the smallest percolating set for $k\ge 2$.

💡 Research Summary

The paper studies r‑neighbor bootstrap percolation on a family of graphs that generalize the n‑dimensional hypercube. The vertex set is V={0,1}ⁿ and two vertices are joined by an edge whenever their Hamming distance is at most k, where k≥2 is a fixed integer. This graph, denoted Qₙ,ₖ, contains all edges of the usual hypercube (k=1) and adds longer‑range connections, making the graph much denser as k grows.

The authors focus on two infection thresholds, r=2 and r=3, and determine the critical infection probability p_c(Qₙ,ₖ,r) that separates a regime where percolation (full infection) occurs with high probability from one where it fails with high probability. Their main results (Theorems 1 and 2) show that the thresholds are not sharp (the transition window has the same order as the threshold itself) but coarse, in contrast with the sharp threshold known for the ordinary hypercube when r=2.

For r=2 they prove that the relevant scale is p≈2^{-n^{2‑n‑k}}. More precisely, if p≫2^{-n^{2‑n‑k}} then the probability of percolation tends to 1, while if p≪2^{-n^{2‑n‑k}} it tends to 0. When p=c·2^{-n^{2‑n‑k}} with a fixed constant c>0, the probability converges to 1−exp(−c²/(2(2k)!)). Moreover, the number of unordered vertex pairs (x,y) with Hamming distance ≤2k that are both initially infected converges in distribution to a Poisson random variable with mean λ=c²/(2(2k)!).

For r=3 the analogous scale is p≈2^{-n^{3‑n‑k}}. At the critical window p=c·2^{-n^{3‑n‑k}} the percolation probability converges to 1−exp(−c³·(2k choose k)/(6(2k)!k!)), and the count of unordered triples (x,y,z) with all pairwise distances ≤2k converges to Poisson(λ) with λ equal to the same constant. Thus, the emergence of percolation is governed by the appearance of a small “balanced” configuration (a pair for r=2, a triple for r=3) in the random initial set, exactly as in the classic theory of strictly balanced subgraphs in Erdős–Rényi random graphs.

The proof strategy hinges on a deterministic characterization: Lemma 5 shows that if the initial set contains an (r−1)-dimensional sub‑cube, then the infection spreads to the whole graph in a few steps because every vertex has many neighbors within distance k. Consequently, percolation is equivalent to the existence of a distance‑≤2k pair (or triple) of initially infected vertices. The authors then compute the expectation and variance of the random variable X counting such configurations. By carefully bounding covariances between overlapping configurations they establish Var(X)=o(E