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SUMMARY:Counting lattice walks confined to cones
DTSTART;VALUE=DATE-TIME:20190503T080000Z
DTEND;VALUE=DATE-TIME:20190503T103000Z
DTSTAMP;VALUE=DATE-TIME:20190419T005903Z
UID:indico-event-3372@indico.math.cnrs.fr
DESCRIPTION:The study of lattice walks confined to cones is a very lively
topic in combinatorics and in probability theory\, which has witnessed ric
h developments in the past 20 years. 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\, the
re are |S|^n walks of length n that start from the origin and take their s
teps in S. But how many of them remain the the cone C?\nOne of the motivat
ions for studying such questions is that lattice walks are ubiquitous in v
arious mathematical fields\, where they encode important classes of object
s: in discrete mathematics (permutations\, trees\, words...)\, in statisti
cal physics (polymers...)\, in probability theory (urns\, branching proces
ses\, systems of queues)\, among other fields.\nThe systematic study of th
ese counting problems started about 20 years ago. Beforehand\, only sporad
ic cases had been solved\, with the exception of walks with small steps co
nfined to a Weyl chamber\, for which a general reflection principle had be
en developed. Since then\, several approaches have been combined to unders
tand how the choice of the steps and of the cone influence the nature of t
he 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 c
onsists of "small" steps\, it is now understood when A(t) is rational\, al
gebraic\, or when it satisfies a linear\, or a non-linear\, differential e
quation. Even in this simple case\, the classification involves tools comi
ng from an attractive variety of fields: algebra on formal power series\,
complex analysis\, computer algebra\, differential Galois theory\, to cite
just a few. And much remains to be done\, for other cones and sets of ste
ps.\n\n \n\nAfter the break\, the second talk will be on:\n\nWalks in the
quadrant: Tutte's invariant method\n \n\nThe common starting point to al
l approaches in the (exact) enumeration of quadrant walks is a functional
equation that defines a three variable power series\, counting these walks
according to their length and the coordinates of their endpoint. Similar
equations appeared in the seventies in the work of William Tutte on the en
umeration of properly coloured planar maps. In particular\, Tutte devoted
ten years (and as many papers) to the enumeration of properly coloured tri
angulations\, for which he established\, at the end\, a non-linear differe
ntial equation for the main generating function.\nIn this talk\, we will s
how how to apply (and then to extend to a more analytic framework) Tutte's
ideas to treat in the uniform manner two kinds of quadrant problems:\n- t
he 4 that have an algebraic generating function\n- the 9 that have a diffe
rentially algebraic (but not differentially finite) generating function.\n
Tutte's notion of "invariants" is central to this solution.\nThese results
have been obtained in collaboration with Olivier Bernardi and Kilian Rasc
hel.\n\n \n\nhttps://indico.math.cnrs.fr/event/3372/
LOCATION:ENS de Lyon\, site Monod Amphi B
URL:https://indico.math.cnrs.fr/event/3372/
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