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The critical percolation window in uniformly grown random graphs
We study graphs in which vertices 1,…,n arrive sequentially, and in which vertex j connects independently to earlier vertices with probability β/j. Bollobás, Janson, and Riordan (‘05) proved that the phase transition in β beyond which a giant component appears is of infinite order, in sharp contrast to the classical mean-field transition in Erdős–Rényi random graphs.
In this talk we determine the asymptotic order of the largest component at the critical value β_c. Moreover, for sequences β_n → β_c, we describe the critical window and uncover several surprising features, including a secondary phase transition and bounded susceptibility (average component size) throughout the window.
The proofs rely on a coupling between a component exploration process and a branching random walk with two killing barriers.
Joint work with Pascal Maillard and Peter Mörters.
Date:
16 February 2026, 14:00
Venue:
Mathematical Institute, Woodstock Road OX2 6GG
Venue Details:
L5
Speaker:
Joost Jorritsma (University of Oxford)
Organising department:
Department of Statistics
Organisers:
Matthias Winkel (Department of Statistics, University of Oxford),
Julien Berestycki (University of Oxford),
Christina Goldschmidt (Department of Statistics, University of Oxford),
James Martin (Department of Statistics, University of Oxford)
Part of:
Probability seminar
Booking required?:
Not required
Audience:
Members of the University only
Editor:
Christina Goldschmidt
