相关论文: Exact solution of some quarter plane walks with in…

We use Galois theory of difference equations to study the nature of the generating series of (weighted) walks in the quarter plane with genus zero kernel curve. Using this approach, we prove that the generating series do not satisfy any…

The study of lattice walks restricted to the first quadrant has shed a lot of interest in the past twenty years. In particular, there has been an important effort to classify models of weighted walks with small steps with respect to the…

In the field of enumeration of weighted walks confined to the quarter plane, it is known that the generating functions behave very differently depending on the chosen step set; in practice, the techniques used in the literature depend on…

In the past fifteen years, the enumeration of lattice walks with steps takenin a prescribed set S and confined to a given cone, especially the firstquadrant of the plane, has been intensely studied. As a result, the generating functions…

In the first part of this paper, we enumerate exactly walks on the square lattice that start from the origin, but otherwise avoid the non positive horizontal half-axis. We call them "walks on the slit plane". We count them by their length,…

We study planar walks that start from a given point (i\_0, j\_0), take their steps in a finite set S, and are confined in the first quadrant of the plane. Their enumeration can be attacked in a systematic way: the generating function Q(x,…

Models of spatially homogeneous walks in the quarter plane ${\bf Z}_+^{2}$ with steps taken from a subset $\mathcal{S}$ of the set of jumps to the eight nearest neighbors are considered. The generating function $(x,y,z)\mapsto Q(x,y;z)$ of…

In the 1970s, 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…

In the quarter plane, five lattice path models with unit steps have resisted the otherwise general approach of Fayolle, Rachel, and Kurkova. Here we consider these five models, called the singular models, and prove that the generating…

We investigate a functional equation which resembles the functional equation for the generating function of a lattice walk model for the quarter plane. The interesting feature of this equation is that its orbit sum is zero while its…

We continue the enumeration of plane lattice walks with small steps avoiding the negative quadrant, initiated by the first author in 2016. We solve in detail a new case, namely the king model where all eight nearest neighbour steps are…

We consider weighted small step walks in the positive quadrant, and provide algebraicity and differential transcendence results for the underlying generating functions: we prove that depending on the probabilities of allowed steps, certain…

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,…

The kernel method is an essential tool for the study of generating series of walks in the quarter plane. This method involves equating to zero a certain polynomial, the kernel polynomial, and using properties of the curve, the kernel curve,…

We present two classes of random walks restricted to the quarter plane whose generating function is not holonomic. The non-holonomy is established using the iterated kernel method, a recent variant of the kernel method. This adds evidence…

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…

We continue the enumeration of plane lattice paths avoiding the negative quadrant initiated by the first author in [Bousquet-M{\'e}lou, 2016]. We solve in detail a new case, the king walks, where all 8 nearest neighbour steps are allowed.…

In the book [FIM], original methods were proposed to determine the invariant measure of random walks in the quarter plane with small jumps, the general solution being obtained via reduction to boundary value problems. Among other things, an…

We refine necessary and sufficient conditions for the generating series of a weighted model of a quarter plane walk to be differentially algebraic. In addition, we give algorithms based on the theory of Mordell-Weil lattices, that, for each…

Many recent papers deal with the enumeration of 2-dimensional walks with prescribed steps confined to the positive quadrant. The classification is now complete for walks with steps in $\{0, \pm 1\}^2$: the generating function is D-finite if…