Geometry of the vacant set left by random walk on random graphs, Wright's constants, and critical random graphs with prescribed degrees. Issue 3 (25th July 2019)

• Record Type: Journal Article
• Authors: Bhamidi, Shankar; Sen, Sanchayan
• Is Part Of: Random structures & algorithms. Volume 56:Issue 3(2020)
• Page Start: 676
• Page End: 721
• Publication Date: 2019-07-25
• Subjects: Brownian excursions -- configuration model -- continuum random trees -- critical random graphs -- graph enumeration -- graphs with prescribed degree sequence -- Gromov‐Hausdorff distance -- Gromov‐weak topology -- multiplicative coalescent -- vacant sets -- Wright's constants; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20880
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract : We provide an explicit algorithm for sampling a uniform simple connected random graph with a given degree sequence. By products of this central result include: (1) continuum scaling limits of uniform simple connected graphs with given degree sequence and asymptotics for the number of simple connected graphs with given degree sequence under some regularity conditions, and (2) scaling limits for the metric space structure of the maximal components in the critical regime of both the configuration model and the uniform simple random graph model with prescribed degree sequence under finite third moment assumption on the degree sequence. As a substantive application we answer a question raised by Černý and Teixeira study by obtaining the metric space scaling limit of maximal components in the vacant set left by random walks on random regular graphs.