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## Beringer Dorottya (Rényi Intézet): Local weak limits of random graphs and parameter continuity

 Hétfő, 8. Július 2019, 14:00 - 16:00
Abstract. The theory of local weak convergence provides a useful tool to study parameters of large finite graphs that are determined by the local structure of the graph. The observations of Liu, Slotine and Barabási (2011) suggested that the matching ratio of directed graphs, which is closely related to an important parameter in control theory, is in some cases essentially determined by the degrees of the graphs and converges along local weak convergent sequences. We give the precise formulation and rigorous proofs of these statements about the concentration and limiting properties of the matching ratio of random graphs. The results are joint work with Ádám Timár.

The question of parameter continuity with respect to local weak convergence arises also for infinite graphs. We examine Schramm's conjectures about the locality of percolation critical probability in the class of unimodular random graphs, which is a common generalization of invariant subgraphs of Cayley graphs and local weak limits of random graphs. We extend the definitions of critical probabilities to this class and investigate the relationship between the different notions of criticality. We give conditions that imply the convergence of the percolation critical probability along local weak convergent sequences, and we show by examples that the locality conjecture does not hold in the generality of unimodular random graphs. The results are joint work with Gábor Pete and Ádám Timár.

The talk is the internal defence of the PhD thesis.
Hely : Szeged, Aradi vértanúk tere 1., Szőkefalvi terem.