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Fires on large recursive trees

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  • Marzouk, Cyril

Abstract

We consider random dynamics on a uniform random recursive tree with n vertices. Successively, in a uniform random order, each edge is either set on fire with some probability pn or fireproof with probability 1−pn. Fires propagate in the tree and are only stopped by fireproof edges. We first consider the proportion of burnt and fireproof vertices as n→∞, and prove a phase transition when pn is of order lnn/n. We then study the connectivity of the fireproof forest, more precisely the existence of a giant component. We finally investigate the sizes of the burnt subtrees.

Suggested Citation

  • Marzouk, Cyril, 2016. "Fires on large recursive trees," Stochastic Processes and their Applications, Elsevier, vol. 126(1), pages 265-289.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:1:p:265-289
    DOI: 10.1016/j.spa.2015.08.006
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    References listed on IDEAS

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    1. Drossel, B. & Schwabl, F., 1992. "Self-organized criticality in a forest-fire model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 191(1), pages 47-50.
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