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Percolative MDLA: phase transition and critical behaviour
Abstract
In this paper we introduce the model of Percolative MDLA (PMDLA) on ℤ2 which engulfs the entire occupied percolation component of each particle that reaches the aggregate. We prove a phase transition of finite time explosion, as a function of the particle density λ, where the critical value equals λc=−log(1−pcsite(ℤ2)). Below λc, the process is nonexplosive and has a linear upper bound on its radius. For 2/3<λ<λc, it also has a linear lower bound on the growth rate. Above λc, an infinite component is swallowed in finite time a.s. At λc, every update is finite, but the updates accumulate in finite time, causing a blowup. Excluding criticality, the results are generalized to all higher dimensions.
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BibTeX
@misc{procaccia2026percolative,
title = {{Percolative MDLA: phase transition and critical behaviour}},
author = {Eviatar B. Procaccia and Haoyu Wang and Yuan Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19622},
eprint = {2610.19622},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19622},
}
RIS
TY - UNPB
TI - Percolative MDLA: phase transition and critical behaviour
AU - Eviatar B. Procaccia
AU - Haoyu Wang
AU - Yuan Zhang
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19622
UR - https://arxiv.org/abs/2610.19622
AB - In this paper we introduce the model of Percolative MDLA (PMDLA) on ℤ^2 which engulfs the entire occupied percolation component of each particle that reaches the aggregate. We prove a phase transition of finite time explosion, as a function of the particle density λ, where the critical value equals λ_c=−log(1−p_c^site(ℤ^2)). Below λ_c, the process is nonexplosive and has a linear upper bound on its radius. For 2/3<λ<λ_c, it also has a linear lower bound on the growth rate. Above λ_c, an infinite component is swallowed in finite time a.s. At λ_c, every update is finite, but the updates accumulate in finite time, causing a blowup. Excluding criticality, the results are generalized to all higher dimensions.
ER -