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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.
