14-1 Charles Radin, Kui Ren, Lorenzo Sadun

The asymptotics of large constrained graphs (3371K, pdf) Jan 9, 14

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Abstract. We show, through local estimates and simulation, that if one constrains simple graphs by their densities e of edges and t of triangles, then asymptotically (in the number of vertices) for over 95% of the possible range of those densities there is a well- defi ned typical graph, and it has a very simple structure: the vertices are decomposed into two subsets V1 and V2 of fixed relative sizes c and 1 - c, and there are well- defi ned probabilities of edges, gjk, between vj in Vj , and vk in Vk. Furthermore the four parameters c, g11, g22 and g12 are smooth functions of ( e,t) except at two smooth phase transition curves.

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