45 research outputs found
Asymptotics of lattice walks via analytic combinatorics in several variables
We consider the enumeration of walks on the two dimensional non-negative
integer lattice with short steps. Up to isomorphism there are 79 unique two
dimensional models to consider, and previous work in this area has used the
kernel method, along with a rigorous computer algebra approach, to show that 23
of the 79 models admit D-finite generating functions. In 2009, Bostan and
Kauers used Pad\'e-Hermite approximants to guess differential equations which
these 23 generating functions satisfy, in the process guessing asymptotics of
their coefficient sequences. In this article we provide, for the first time, a
complete rigorous verification of these guesses. Our technique is to use the
kernel method to express 19 of the 23 generating functions as diagonals of
tri-variate rational functions and apply the methods of analytic combinatorics
in several variables (the remaining 4 models have algebraic generating
functions and can thus be handled by univariate techniques). This approach also
shows the link between combinatorial properties of the models and features of
its asymptotics such as asymptotic and polynomial growth factors. In addition,
we give expressions for the number of walks returning to the x-axis, the
y-axis, and the origin, proving recently conjectured asymptotics of Bostan,
Chyzak, van Hoeij, Kauers, and Pech.Comment: 10 pages, 3 tables, as accepted to proceedings of FPSAC 2016 (without
conference formatting
On the denominators of the Taylor coefficients of G-functions
Let be a
-function, and, for any , let denote the least
integer such that are all
algebraic integers. By definition of a -function, there exists some constant
such that for all . In practice, it is
observed that always divides where
, are positive integers and is an
integer. We prove that this observation holds for any -function provided the
following conjecture is assumed: {\em Let be a number field, and
be a -operator; then the generic radius
of solvability is equal to 1, for all finite places of
except a finite number.} The proof makes use of very precise
estimates in the theory of -adic differential equations, in particular the
Christol-Dwork Theorem. Our result becomes unconditional when is a
geometric differential operator, a special type of -operators for which the
conjecture is known to be true. The famous Bombieri-Dwork Conjecture asserts
that any -operator is of geometric type, hence it implies the above
conjecture
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
Analytic approach for reflected Brownian motion in the quadrant
Random walks in the quarter plane are an important object both of
combinatorics and probability theory. Of particular interest for their study,
there is an analytic approach initiated by Fayolle, Iasnogorodski and Malyshev,
and further developed by the last two authors of this note. The outcomes of
this method are explicit expressions for the generating functions of interest,
asymptotic analysis of their coefficients, etc. Although there is an important
literature on reflected Brownian motion in the quarter plane (the continuous
counterpart of quadrant random walks), an analogue of the analytic approach has
not been fully developed to that context. The aim of this note is twofold: it
is first an extended abstract of two recent articles of the authors of this
paper, which propose such an approach; we further compare various aspects of
the discrete and continuous analytic approaches.Comment: 19 pages, 5 figures. Extended abstract of the papers arXiv:1602.03054
and arXiv:1604.02918, to appear in Proceedings of the 27th International
Conference on Probabilistic, Combinatorial and Asymptotic Methods for the
Analysis of Algorithms, Krakow, Poland, 4-8 July 2016 arXiv admin note: text
overlap with arXiv:1602.0305