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