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DTSTAMP:20250708T005314Z
DESCRIPTION:Counting lattice walks confined to cones\n\nThe study of lattic
 e walks confined to cones is a very lively topic in combinatorics and in p
 robability theory\, which has witnessed rich developments in the past 20 y
 ears. In a typical problem\, one is given a finite set of allowed steps S 
 in Z^d\, and a cone C in R^d. Clearly\, there are |S|^n walks of length n 
 that start from the origin and take their steps in S. But how many of them
  remain the the cone C?\n	\n	One of the motivations for studying such questi
 ons is that lattice walks are ubiquitous in various mathematical fields\, 
 where they encode important classes of objects: in discrete mathematics (p
 ermutations\, trees\, words...)\, in statistical physics (polymers...)\, i
 n probability theory (urns\, branching processes\, systems of queues)\, am
 ong other fields.The systematic study of these counting problems started a
 bout 20 years ago. Beforehand\, only sporadic cases had been solved\, with
  the exception of walks with small steps confined to a Weyl chamber\, for 
 which a general reflection principle had been developed. Since then\, seve
 ral approaches have been combined to understand how the choice of the step
 s and of the cone influence the nature of the counting sequence a(n)\, or 
 of the the associated series A(t)=sum a(n) t^n. For instance\, if C is the
  first quadrant of the plane and S only consists of 'small' steps\, it is 
 now understood when A(t) is rational\, algebraic\, or when it satisfies a 
 linear\, or a non-linear\, differential equation. Even in this simple case
 \, the classification involves tools coming from an attractive variety of 
 fields: algebra on formal power series\, complex analysis\, computer algeb
 ra\, differential Galois theory\, to cite just a few. And much remains to 
 be done\, for other cones and sets of steps.This talk will survey these re
 cent developments\, and conclude a series of talks by the author\, in the 
 framework of the Aisenstadt chair.\n
DTSTART:20181005T200000Z
DTEND:20181005T210000Z
LOCATION:room 6254\, CA\, Pav. André-Aisenstadt
SUMMARY:Mireille Bousquet-Mélou\, CNRS - Université de Bordeaux
URL:/mathstat/channels/event/mireille-bousquet-melou-c
 nrs-universite-de-bordeaux-290207
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