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Department of Mathematical Sciences

# Seminar Archives

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 (\`a 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 which I obtained in collaboration with Louigi Addario-Berry and Nicolas Broutin a few years ago (extending results of Aldous), and complements recent work on random graph scaling limits of various authors including Bhamidi, Broutin, Duquesne, van der Hofstad, van Leeuwaarden, Riordan, Sen, M. Wang and X. Wang.