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].

Abstract. -Spatially homogeneous random walks in (Z + ) 2 with non-zero jump probabilities at distance at most 1, with non-zero drift in the interior of the quadrant and absorbed when reaching the axes are studied. Absorption probabilities generating functions are obtained and the asymptotic of absorption probabilities along the axes is made explicit. The asymptotic of the Green functions is computed along all different infinite paths of states, in particular along those approaching the axes.Résumé (Marches aléatoires dans Z 2 + avec un drift non nul, absorbées au bord) Dans cet article, nous étudions les marches aléatoires du quart de plan ayant des sauts à distance au plus un, avec un drift non nul à l'intérieur et absorbées au bord. Nous obtenons de façon explicite les séries génératrices des probabilités d'absorption au bord, puis leur asymptotique lorsque le site d'absorption tend vers l'infini. Nous calculons également l'asymptotique des fonctions de Green le long de toutes les trajectoires, en particulier selon celles tangentes aux axes.Texte reçu le 9 avril 2009, accepté le 6 novembre 2009.

For Brownian motion in a (two-dimensional) wedge with negative drift and oblique reflection on the axes, we derive an explicit formula for the Laplace transform of its stationary distribution (when it exists), in terms of Cauchy integrals and generalized Chebyshev polynomials. To that purpose we solve a Carleman-type boundary value problem on a hyperbola, satisfied by the Laplace transforms of the boundary stationary distribution.

The Z-invariant Ising model [3] is defined on an isoradial graph and has coupling constants depending on an elliptic parameter k. When k = 0 the model is critical, and as k varies the whole range of temperatures is covered. In this paper we study the corresponding dimer model on the Fisher graph, thus extending our papers [7,8] to the full Z-invariant case. One of our main results is an explicit, local formula for the inverse of the Kasteleyn operator. Its most remarkable feature is that it is an elliptic generalization of [8]: it involves a local function and the massive discrete exponential function introduced in [10]. This shows in particular that Z-invariance, and not criticality, is at the heart of obtaining local expressions. We then compute asymptotics and deduce an explicit, local expression for a natural Gibbs measure. We prove a local formula for the Ising model free energy. We also prove that this free energy is equal, up to constants, to that of the Z-invariant spanning forests of [10], and deduce that the two models have the same second order phase transition in k. Next, we prove a self-duality relation for this model, extending a result of Baxter to all isoradial graphs. In the last part we prove explicit, local expressions for the dimer model on a bipartite graph corresponding to the XOR version of this Z-invariant Ising model. √ 1−k 2 on the dual isoradial graph G * yield the same probability measure on polygon configurations of the graph G. The elliptic parameters k and k * can be interpreted as parametrizing dual temperatures, see Section 4.2 and also [11,47].where Z Ising (G, J) is the normalizing constant known as the Ising partition function.A polygon configuration of G is a subset of edges such that every vertex has even degree; let P(G) denote the set of polygon configurations of G. Then, the high temperature expansion [36,37] of the Ising model partition function gives the following identity: Z Ising (G, J) = 2 |V| e∈E cosh J e P∈P(G) e∈P tanh J e .

In the 1970s, William Tutte developed a clever algebraic approach, based on certain "invariants", to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past 20 years to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps, taken in {−1, 0, 1} 2 .We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic. This includes Gessel's famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity follows almost automatically.Then, we move to a complex analytic viewpoint that has already proved very powerful, leading in particular to integral expressions for the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions for the generating function, and a proof that this series is D-algebraic (that is, satisfies polynomial differential equations).

With an appendix by Sandro Franceschi. 32 pages, 9 figures.International audienceWe propose a new approach for finding discrete harmonic functions in the quarter plane with Dirichlet conditions. It is based on solving functional equations that are satisfied by the generating functions of the values taken by the harmonic functions. As a first application of our results, we obtain a simple expression for the harmonic function that governs the asymptotic tail distribution of the first exit time for random walks from the quarter plane. As another corollary, we prove, in the zero drift case, the uniqueness of the discrete harmonic function