Dear All, 

we have two talks tomorrow.

Remember that we start 14:00 UTC which is 16:00 CET!

14:00-15:00 UTC Christina Goldschmidt

The stable graph: the scaling limit of critical random graphs with i.i.d. random degrees having power-law tails

Abstract: Consider a graph with label set \{1,2, \ldots,n\} chosen uniformly at random from those such that vertex i has degree D_i, where D_1, D_2, \ldots, D_n are i.i.d. strictly positive random variables. The condition for criticality (i.e. the threshold for the emergence of a giant component) in this setting is E[D^2] = 2 E[D], and we assume additionally that P(D = k) \sim c k^{-(\alpha + 2)} as k tends to infinity, for some \alpha \in (1,2).  In this situation, it turns out that the largest components have sizes on the order of n^{\alpha/(\alpha+1)}. Building on earlier work of Adrien Joseph, we show that the components have scaling limits which can be related to a forest of stable trees (à la Duquesne-Le Gall-Le Jan) via an absolute continuity relation.  This gives a natural generalisation of the scaling limit for the Erd\H{o}s-Renyi random graph (obtained in collaboration with Louigi Addario-Berry and Nicolas Broutin a few years ago, extending results of Aldous), which we call the stable graph.  This complements recent work on random graph scaling limits by various authors including Bhamidi, Broutin, Duquesne, van der Hofstad, van Leeuwaarden, Riordan, Sen, M. Wang and X. Wang.

15:00 - 16:00 UTC Bénédicte Haas  

Distributional properties of the stable graphs

Abstract: In this talk we will investigate some distributional properties of the connected components of the stable graphs introduced by Christina in the previous talk. We recall that for $\alpha \in (1,2]$, the $\alpha$-stable graph arises as the universal scaling limit of critical random graphs with i.i.d. degrees having an $\alpha$-dependent power-law tail behavior. Consider a connected component of such a graph. Our aim will be: (1) to describe the distribution of its kernel and more generally of its discrete finite-dimensional marginals, (2) to explicit its distribution as a collection of $\alpha$-stable trees glued on the kernel, and (3) present a line-breaking construction, in the same spirit as Aldous’ line-breaking construction of the Brownian CRT.

Based on a joint work with Christina Goldschmidt and Delphin Sénizergues.

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