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SUMMARY:The phase transition in Achlioptas processes - Oliver Riordan (Uni
 versity of Oxford)
DTSTART:20160711T104500Z
DTEND:20160711T113000Z
UID:TALK66699@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:The classical random graph process starts with a fixed set of&
 nbsp\;<i>n</i> vertices and no edges. Edges are then added one-by-one\, un
 iformly at random. One of the most interesting features of this process\, 
 established by Erd&#x151\;s and R&eacute\;nyi more than 50 years ago\, is 
 the&nbsp\;<i>phase transition</i> near&nbsp\;<i>n</i>/2 edges\, where a si
 ngle `giant&#39\; component emerges from a sea of small components. This e
 xample serves as a starting point for understanding phase transitions in a
  wide variety of other contexts. Around 2000\, Dimitris Achlioptas suggest
 ed an innocent-sounding variation of the model: at each stage two edges ar
 e selected at random\, but only one is added\, the choice depending on (ty
 pically) the sizes of the components it would connect. This may seem like 
 a small change\, but these processes do not have the key independence prop
 erty that underlies our understanding of the classical process. One can as
 k many questions about Achlioptas processes\; the most interesting concern
  the phase transition: does the critical value change from&nbsp\;<i>n</i>/
 2? Is the nature of the transition the same or not? I will describe a numb
 er of results on these questions from joint work with Lutz Warnke. <br>
LOCATION:Seminar Room 1\, Newton Institute
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