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SUMMARY:Random Walks on Groups: Shuffling Cards and the Cutoff Phenomenon 
 - Oliver Matheau-Raven\, University of York
DTSTART:20191018T140000Z
DTEND:20191018T150000Z
UID:TALK131092@talks.cam.ac.uk
CONTACT:Liam Jolliffe
DESCRIPTION:Random walks on Groups is an area of mathematics that has flou
 rished since the 80s with beautiful connections between algebra and probab
 ility being discovered. In this talk we present an overview of the techniq
 ues that enable\nthe use of irreducible representations of a group G to st
 udy the convergence time of random walks on G to the uniform distribution.
  This convergence often becomes sharp if we consider a family of random wa
 lks on G_n where |G_n| tends to infinity as n tends to infinity\, we call 
 this the cutoff phenomenon. We specialise to two examples of random walks 
 on S_n\; the random transpositions shuffle\, and the one-sided transpositi
 on shuffle. The random transposition shuffle is defined by our hands indep
 endently choosing cards each to transpose every step. The one-sided transp
 osition shuffle restricts our hands to cross each\nother when choosing car
 ds. The former is a classical problem studied by Diaconis and Shahshahani\
 , who proved it exhibits a cutoff at time (n/2)log(n). The latter is a rec
 ent problem\, and in joint work with Michael Bate and\nStephen Connor we u
 ncovered a remarkable branching\nstructure for the eigenspaces involving Y
 oung diagrams and paths between them. After analysis of the eigenvalues we
  prove a cutoff at time (n)log(n).
LOCATION:CMS\, MR9
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