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→∞ and deviations at most C√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 Ta 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∼ d2 be the quadratic response coefficient. If b>0 and nb3/K→∞, 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=√(nb/K), where αdet is the exact empirical response-barrier center and Φ is the standard normal distribution function. Put a0=(n/K)1/3. If d4/n→0, a/a0→ x>0 and λ=ba0→ℓ∈ℝ, then ℙ(Ta≥ n/2)→ p(x,ℓ), 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.
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BibTeX
@misc{rodionov2026finitesize,
title = {{Finite-size ignition laws near a vanishing branching deficit on near-regular simple graphs}},
author = {Aleksandr Rodionov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38912},
eprint = {2610.38912},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38912},
}
RIS
TY - UNPB
TI - Finite-size ignition laws near a vanishing branching deficit on near-regular simple graphs
AU - Aleksandr Rodionov
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38912
UR - https://arxiv.org/abs/2610.38912
AB - We study heterogeneous bootstrap percolation on uniform simple graphs with prescribed degrees of mean d→∞ and deviations at most C√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∼ d^2 be the quadratic response coefficient. If b>0 and nb^3/K→∞, 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=√(nb/K), where α_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→0, a/a_0→ x>0 and λ=ba_0→ℓ∈ℝ, then ℙ(T_a≥ n/2)→ p(x,ℓ), 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.
ER -