相关论文: On Differentially Algebraic Generating Series for …

In the present paper we study the nature of the trivariate generating series of weighted walks in the quarter plane. Combining the results of this paper to previous ones, we complete the proof of the following theorem. The series satisfies…

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…

In the present paper, we introduce a new approach, relying on the Galois theory of difference equations, to study the nature of the generating series of walks in the quarter plane. Using this approach, we are not only able to recover many…

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…

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…

We consider inhomogeneous lattice walk models in a half-space and in the quarter plane. For the models in a half-space, we show by a generalization of the kernel method to linear systems of functional equations that their generating…

We study the nature of the generating series of some models of walks with small steps in the three quarter plane. More precisely, we restrict ourselves to the situation where the group is infinite, the kernel has genus one, and the step set…

In the present paper, we use difference Galois theory to study the nature of the generating function counting walks with small steps in the quarter plane. These series are trivariate formal power series $Q(x,y,t)$ that count the number of…

Lattice paths in the quarter plane have led to a large and varied set of results in recent years. One major project has been the classification of step sets according to the properties of the corresponding generating functions, and this has…

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…

This work considers lattice walks restricted to the quarter plane, with steps taken from a set of cardinality three. We present a complete classification of the generating functions of these walks with respect to the classes algebraic,…

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…

Let S be a subset of {-1,0,1}^2 not containing (0,0). We address the enumeration of plane lattice walks with steps in S, that start from (0,0) and always remain in the first quadrant. A priori, there are 2^8 problems of this type, but some…

The article provides an explicit algebraic expression for the generating function of walks on graphs. Its proof is based on the scattering theory for the differential Laplace operator on non-compact graphs.

Let S be a finite subset of Z^2. A walk on the slit plane with steps in S is a sequence (0,0)=w_0, w_1, ..., w_n of points of Z^2 such that w_{i+1}-w_i belongs to S for all i, and none of the points w_i, i>0, lie on the half-line H= {(k,0):…

We consider walks on the edges of the square lattice $\mathbb Z^2$ which obey \emph{two-step rules,} which allow (or forbid) steps in a given direction to be followed by steps in another direction. We classify these rules according to a…

We address the enumeration of walks with small steps confined to a two-dimensional cone, for example the quarter plane, three-quarter plane or the slit plane. In the quarter plane case, the solutions for unweighted step-sets are already…

The set of random walks with different step sets (of short steps) in the quarter plane has provided a rich set of models that have profoundly different integrability properties. In particular, 23 of the 79 effectively different models can…

In this article we obtain new expressions for the generating functions counting (non-singular) walks with small steps in the quarter plane. Those are given in terms of infinite series, while in the literature, the standard expressions use…

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…