Supercritical Frozen Erdős-Rényi and Uniform Random Forests

by Vincent Viau (LAGA, Univ. Sorbonne Paris-Nord)

Wednesday Nov 12, 2025, 2:00 PM → 3:00 PM Europe/Paris

Amphithéâtre Léon Motchane (IHES)

Seed Seminar of Mathematics and Physics

Fall' 25 : Random Forests and Fermionic Field Theories  

The frozen Erdős-Rényi random graph is a variant of the standard dynamical Erdős-Rényi random graph that prevents the creation of the giant component by freezing the evolution of connected components with a unique cycle. The formation of multicyclic components is forbidden, and the growth of components with a unique cycle is slowed down, depending on a parameter p∈[0,1] that quantifies the slowdown. In this talk, we will study the fluid limit of the main statistics of this process, that is their functional convergence as the number of vertices of the graph becomes large and after a proper rescaling, to the solution of a system of differential equations. The proof is based on the free forest property of the frozen model: the forest part of the graph is a uniform random forest. In order to prove the fluid limit results, we will explain how to study and count forests using conditioned random walks.

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