Seed thresholds and degree variance in heterogeneous bootstrap percolation with growing degrees

Abstract

We study bootstrap percolation with independent vertex thresholds taking values one and two, with threshold-one probability $(1-c/d)/d$ for fixed $c>0$. The seed set is chosen uniformly among sets of a prescribed deterministic size, independently of the graph and thresholds. We prove threshold statements at fixed relative margins. For uniform simple graphs with prescribed nonnegative integer degrees of even sum and exact mean $d$, assume $\max_i|d_i-d|\le C\sqrt d$ and $c+1-v_n\ge\kappa>0$, where $v_n=\operatorname{Var}(D)/d$ and $C,\kappa$ are fixed. When $d\to\infty$ and $d=o(n^{1/7})$, the leading seed scale is $n(c+1-v_n)^2/(2d^4)$, without requiring a limit of $v_n$. At fixed relative margins below and above this scale, the final active set has size $O(n/d^3)$ and $n-o(n)$, respectively, with high probability. A separate result for $G(n,d/n)$ holds when $d\to\infty$ and $d^5/n\to0$, and gives scale $nc^2/(2d^4)$. Thus regular and independent-edge graphs have different coefficients at the same asymptotic mean degree. Local exploration estimates yield explicit inactive remainders and survive conditioning on simplicity in the prescribed model. We also quantify the precision obstruction to static inclusion transfer and compute deterministic response-barrier corrections, without identifying a shrinking random critical window.