The group and zero drift case

In several recent studies on random walks with small jumps in the quarter plane, it has been noticed that the so-called group of the walk governs the behavior of a number of quantities, in particular through its order. When the drift of the random walk is equal to 0, we have provided an effective criterion (see RA 2012) giving the order of this group. More generally, we showed that in all cases where the genus of the algebraic curve defined by the so-called kernel is 0, the group is infinite, except precisely for the zero drift case, where finiteness is quite possible.

Counting and asymptotics

The enumeration of planar lattice walks is a classical topic in combinatorics. For a given set $𝒮$ of allowed unit jumps (or steps), it is a matter of counting the number of paths starting from some point and ending at some arbitrary point in a given time, and possibly restricted to some regions of the plane.

Like in the probabilistic context, a common way of attacking these problems relies on the following analytic approach. Let $f(i,j,k)$ denote the number of paths in ${\mathbb{Z}}_{+}^2$ starting from $(0,0)$ and ending at $(i,j)$ at time $k$. In the case of small jumps (size at most one), the corresponding CGF

satisfies the functional equation

where $x,y,z$ are complex variables, $K(x,y,z)$ is a polynomial of degree 2 (both in $x$ and $y$), and linear in the time variable $z$ which plays somehow the role of a parameter. The question of the type of the associated counting generating functions, rational, algebraic, or holonomic (i.e. solution of a linear differential equation with polynomial coefficients), was solved whenever the group is finite (see RA 2010). When the group is infinite, the problem is still largely open.

Harmonic functions and more general jumps

The determination of Martin boundaries in the case of random walks is a longstanding problem, solved only in special situations. For homogeneous random walks in the quarter plane, stopped on the boundary (the axes), with upward jumps of size 1, and arbitrary downward jumps of size $d$, it turns out that the computation of harmonic functions is here plainly equivalent to find a positive function $H$ satisfying a functional equation of the form

Here the chief difficulty to make the reduction to a boundary value problem is to analyze the algebraic curve $L(x,y)=0$, which might be of arbitrary genus. Some examples lead us to conjecture the existence of a single real cut inside the unit disk, which should allow to get integral form solution.

Correction of papers