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SUMMARY:Edge- and vertex-reinforced random walks with super-linear reinfor
 cement on infinite graphs - Codina Cotar (University College London)
DTSTART:20161216T114500Z
DTEND:20161216T123000Z
UID:TALK69527@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:<span>Co-author: Debleena Thacker (Lund University)<br></span>
 <br>In this talk we introduce a new simple but powerful general technique 
 for the study of edge- and vertex-reinforced processes with super-linear r
 einforcement\, based on the use of order statistics for the number of edge
 \, respectively of vertex\, traversals. The technique relies on upper boun
 d estimates for the number of edge traversals\, proved in a different cont
 ext by Cotar and Limic [Ann. Appl. Probab. (2009)] for finite graphs with 
 edge reinforcement. We apply our new method both to edge- and to vertex-re
 inforced random walks with super-linear reinforcement on arbitrary infinit
 e connected graphs of bounded degree. We stress that\, unlike all previous
  results for processes with super-linear reinforcement\, we make no other 
 assumption on the graphs.<br><br>For edge-reinforced random walks\, we com
 plete the results of Limic and Tarres [Ann. Probab. (2007)] and we settle 
 a conjecture of Sellke [Technical Report 94-26\, Purdue University (1994)]
  by showing that for any reciprocally summable reinforcement weight functi
 on w\, the walk traverses a random attracting edge at all large times.<br>
 <br>For vertex-reinforced random walks\, we extend results previously obta
 ined on Z by Volkov [Ann. Probab. (2001)] and by Basdevant\, Schapira and 
 Singh [Ann. Probab. (2014)]\, and on complete graphs by Benaim\, Raimond a
 nd Schapira [ALEA (2013)]. We show that on any infinite connected graph of
  bounded degree\, with reinforcement weight function w taken from a genera
 l class of reciprocally summable reinforcement weight functions\, the walk
  traverses two random neighbouring attracting vertices at all large times.
 <br><br>Related Links<ul><li><a target="_blank" rel="nofollow">https://arx
 iv.org/abs/1509.00807</a>&nbsp\;- Webpage where the paper can be found</li
 ></ul>
LOCATION:Seminar Room 1\, Newton Institute
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