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We provide an explicit algorithm for sampling a uniform simple connectedrandom graph with a given degree sequence. By products of this central resultinclude:  (i) continuum scaling limits of uniform simple connected graphs with givendegree sequence and asymptotics for the number of simple connected graphs withgiven degree sequence under some regularity conditions, and  (ii) scaling limits for the metric space structure of the maximal componentsin the critical regime of both the configuration model and the uniform simplerandom graph model with prescribed degree sequence under finite third momentassumption on the degree sequence.  As a substantive application we answer a question raised by Cerny andTeixeira by obtaining the metric space scaling limit of maximal components inthe vacant set left by random walks on random regular graphs.

Geometry of the vacant set left by random walk on random graphs, Wright's constants, and critical random graphs with prescribed degrees