Finite-size ignition laws near a vanishing branching deficit on near-regular simple graphs

Abstract

We study heterogeneous bootstrap percolation on uniform simple graphs with prescribed degrees of mean $d\to\infty$ and deviations at most $C\sqrt d$. Thresholds are independently one with probability $(1-c/d)/d$ and two otherwise; seeds form an independent uniform set of prescribed size $a$. Let $T_a$ be the terminal number of active vertices and $b$ be one minus the mean number of threshold-one descendants of a reached vertex. Let $K\sim d^2$ be the quadratic response coefficient. If $b>0$ and $nb^3/K\to\infty$, we prove, uniformly over seed counts $a$, $ \mathbb{P}(T_a\ge n/2)=Φ(\frac{a-nα_{\mathrm{det}}}{w})+o(1)$ with $w=\sqrt{nb/K}$, where $α_{\mathrm{det}}$ is the exact empirical response-barrier center and $Φ$ is the standard normal distribution function. Put $a_0=(n/K)^{1/3}$. If $d^4/n\to0$, $a/a_0\to x>0$ and $λ=ba_0\to\ell\in\mathbb{R}$, then $\mathbb{P}(T_a\ge n/2)\to p(x,\ell)$, the one-sided Brownian parabolic survival probability. A sequential absolute-error law joins these regimes. A graphical counterexample shows that replacing the exact center by its leading term can cause limiting probability error $1/2$. The proof balances adaptive early-exploration transfer against late concentration at a tunable physical horizon, despite exponentially rare simplicity. We also prove quantitative and mesoscopic-seed refinements in explicitly narrower regimes. The only external graph-enumeration input is a stated reduction of McKay--Wormald's estimate; the probabilistic estimates for the cascade argument are proved in the text and appendices.