87 research outputs found
On 3-dimensional lattice walks confined to the positive octant
Many recent papers deal with the enumeration of 2-dimensional walks with
prescribed steps confined to the positive quadrant. The classification is now
complete for walks with steps in : the generating function is
D-finite if and only if a certain group associated with the step set is finite.
We explore in this paper the analogous problem for 3-dimensional walks
confined to the positive octant. The first difficulty is their number: there
are 11074225 non-trivial and non-equivalent step sets in
(instead of 79 in the quadrant case). We focus on the 35548 that have at most
six steps.
We apply to them a combined approach, first experimental and then rigorous.
On the experimental side, we try to guess differential equations. We also try
to determine if the associated group is finite. The largest finite groups that
we find have order 48 -- the larger ones have order at least 200 and we believe
them to be infinite. No differential equation has been detected in those cases.
On the rigorous side, we apply three main techniques to prove D-finiteness.
The algebraic kernel method, applied earlier to quadrant walks, works in many
cases. Certain, more challenging, cases turn out to have a special Hadamard
structure, which allows us to solve them via a reduction to problems of smaller
dimension. Finally, for two special cases, we had to resort to computer algebra
proofs. We prove with these techniques all the guessed differential equations.
This leaves us with exactly 19 very intriguing step sets for which the group
is finite, but the nature of the generating function still unclear.Comment: Final version, to appear in Annals of Combinatorics. 36 page
Walks in the Quarter Plane with Multiple Steps
We extend the classification of nearest neighbour walks in the quarter plane
to models in which multiplicities are attached to each direction in the step
set. Our study leads to a small number of infinite families that completely
characterize all the models whose associated group is D4, D6, or D8. These
families cover all the models with multiplicites 0, 1, 2, or 3, which were
experimentally found to be D-finite --- with three noteworthy exceptions.Comment: 12 pages, FPSAC 2015 submissio
Automatic Classification of Restricted Lattice Walks
We propose an experimental mathematics approach leading to the
computer-driven discovery of various structural properties of general counting
functions coming from enumeration of walks
A family of centered random walks on weight lattices conditioned to stay in Weyl chambers
Under a natural asumption on the drift, the law of the simple random walk on
the multidimensional first quadrant conditioned to always stay in the first
octant was obtained by O'Connell in [O]. It coincides with that of the image of
the simple random walk under the multidimensional Pitman transform and can be
expressed in terms of specializations of Schur functions. This result has been
generalized in [LLP1] and [LLP2] for a large class of random walks on weight
lattices defined from representations of Kac-Moody algebras and their
conditionings to always stay in Weyl chambers. In these various works, the
drift of the considered random walk is always assumed in the interior of the
cone. In this paper, we introduce for some zero drift random walks defined from
minuscule representations a relevant notion of conditioning to stay in Weyl
chambers and we compute their laws. Namely, we consider the conditioning for
these walks to stay in these cones until an instant we let tend to infinity. We
also prove that the laws so obtained can be recovered by letting the drift tend
to zero in the transitions matrices obtained in [LLP1]. We also conjecture our
results remain true in the more general case of a drift in the frontier of the
Weyl chamber
Conditioned one-way simple random walk and representation theory
We call one-way simple random walk a random walk in the quadrant Z_+^n whose
increments belong to the canonical base. In relation with representation theory
of Lie algebras and superalgebras, we describe the law of such a random walk
conditioned to stay in a closed octant, a semi-open octant or other types of
semi-groups. The combinatorial representation theory of these algebras allows
us to describe a generalized Pitman transformation which realizes the
conditioning on the set of paths of the walk. We pursue here in a direction
initiated by O'Connell and his coauthors [13,14,2], and also developed in [12].
Our work relies on crystal bases theory and insertion schemes on tableaux
described by Kashiwara and his coauthors in [1] and, very recently, in [5].Comment: 32 page
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