International audienceWe prove that the sequence $(e^{\mathfrak{S}}_n)_{n\geq 0}$ of excursions in the quarter plane corresponding to a nonsingular step set~$\mathfrak{S} \subseteq \{0,\pm 1 \}^2$ with infinite group does not satisfy any nontrivial linear recurrence with polynomial coefficients. Accordingly, in those cases, the trivariate generating function of the numbers of walks with given length and prescribed ending point is not D-finite. Moreover, we display the asymptotics of $e^{\mathfrak{S}}_n$

We introduce a one-parameter family of massive Laplacian operators (∆ m(k) ) k∈[0,1) defined on isoradial graphs, involving elliptic functions. We prove an explicit formula for the inverse of ∆ m(k) , the massive Green function, which has the remarkable property of only depending on the local geometry of the graph, and compute its asymptotics. We study the corresponding statistical mechanics model of random rooted spanning forests. We prove an explicit local formula for an infinite volume Boltzmann measure, and for the free energy of the model. We show that the model undergoes a second order phase transition at k = 0, thus proving that spanning trees corresponding to the Laplacian introduced by Kenyon [Ken02] are critical. We prove that the massive Laplacian operators (∆ m(k) ) k∈(0,1) provide a one-parameter family of Z-invariant rooted spanning forest models. When the isoradial graph is moreover Z 2 -periodic, we consider the spectral curve of the characteristic polynomial of the massive Laplacian. We provide an explicit parametrization of the curve and prove that it is Harnack and has genus 1. We further show that every Harnack curve of genus 1 with (z, w) ↔ (z −1 , w −1 ) symmetry arises from such a massive Laplacian.

Abstract. Models of spatially homogeneous walks in the quarter plane Z 2 + with steps taken from a subset S of the set of jumps to the eight nearest neighbors are considered. The generating function (x, y, z) → Q(x, y; z) of the numbers q(i, j; n) of such walks starting at the origin and ending at (i, j) ∈ Z 2 + after n steps is studied. For all nonsingular models of walks, the functions x → Q(x, 0; z) and y → Q(0, y; z) are continued as multi-valued functions on C having infinitely many meromorphic branches, of which the set of poles is identified. The nature of these functions is derived from this result: namely, for all the 51 walks which admit a certain infinite group of birational transformations of C 2 , the interval ]0, 1/|S|[ of variation of z splits into two dense subsets such that the functions x → Q(x, 0; z) and y → Q(0, y; z) are shown to be holonomic for any z from the one of them and non-holonomic for any z from the other. This entails the non-holonomy of (x, y, z) → Q(x, y; z), and therefore proves a conjecture of Bousquet-Mélou and Mishna in [5].