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In the context of lattice walk enumeration in cones, we consider the numberof walks in the quarter plane with fixed starting and ending points, prescribedstep-set and given length. After renormalization, this number may beinterpreted as a discrete heat kernel in the quadrant. We propose a new methodto compute complete asymptotic expansions of these numbers of walks as theirlength tends to infinity, based on two main ingredients: explicit expressionsfor the underlying generating functions in terms of elliptic Jacobi thetafunctions along with a duality known as Jacobi transformation. This dualityallows us to pass from a classical Taylor expansion of the series to anexpansion at the critical point of the model. We work through two examples.First, we present our approach on the well-known Kreweras model, which isalgebraic, 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 theassociated complete asymptotic expansion. In this second case, we prove theexistence of logarithmic terms in the asymptotic expansion, and we relate thecoefficients appearing in the expansion to polyharmonic functions. To ourknowledge, this is the first time that logarithmic terms have been observed inthe asymptotics of a class of lattice walks confined to a quadrant.

Logarithmic terms in discrete heat kernel expansions in the quadrant