  In the context of lattice walk enumeration in cones, we consider the number
of walks in the quarter plane with fixed starting and ending points, prescribed
step-set and given length. After renormalization, this number may be
interpreted as a discrete heat kernel in the quadrant. We propose a new method
to compute complete asymptotic expansions of these numbers of walks as their
length tends to infinity, based on two main ingredients: explicit expressions
for the underlying generating functions in terms of elliptic Jacobi theta
functions along with a duality known as Jacobi transformation. This duality
allows us to pass from a classical Taylor expansion of the series to an
expansion at the critical point of the model. We work through two examples.
First, we present our approach on the well-known Kreweras model, which is
algebraic, and show how to obtain a complete asymptotic expansion in this case.
We then consider a more generic (so-called infinite group) model, and find the
associated complete asymptotic expansion. In this second case, we prove the
existence of logarithmic terms in the asymptotic expansion, and we relate the
coefficients appearing in the expansion to polyharmonic functions. To our
knowledge, this is the first time that logarithmic terms have been observed in
the asymptotics of a class of lattice walks confined to a quadrant.
