Related papers: Reflected random walks and unstable Martin boundar…
The t-Martin boundary of a random walk on a half-space with reflected boundary conditions is identified. It is shown in particular that the t-Martin boundary of such a random walk is not stable in the following sense : for different values…
We determine the asymptotic behavior of the Green function for zero-drift random walks confined to multidimensional convex cones. As a consequence, we prove that there is a unique positive discrete harmonic function for these processes (up…
This paper investigates the asymptotic behavior of Green functions associated to partially homogeneous random walks in the quadrant $Z_+^2$. There are four possible distributions for the jumps of these processes, depending on the location…
We investigate reflected random walks in the quarter plane, with particular emphasis on the time spent along the reflection boundary axes. Assuming the drift of the random walk lies within the cone, the local time converges -- without the…
The complete representation of the Martin compactification for reflected random walks on a half-space $\Z^d\times\N$ is obtained. It is shown that the full Martin compactification is in general not homeomorphic to the ``radial''…
Given a probability measure on a finitely generated group, its Martin boundary is a natural way to compactify the group using the Green function of the corresponding random walk. For finitely supported measures in hyperbolic groups, it is…
We study asymptotic properties of the Green metric associated with transient random walks on countable groups. We prove that the rate of escape of the random walk computed in the Green metric equals its asymptotic entropy. The proof relies…
A complete representation of the Martin boundary of killed random walks on a half-space $\Z^{d-1}\times\N^*$ is obtained. In particular, it is proved that the corresponding Martin boundary is homemorphic to the half-sphere ${\cal S}^d_+ =…
The Martin compactification is investigated for a d-dimensional random walk which is killed when at least one of it's coordinates becomes zero or negative. The limits of the Martin kernel are represented in terms of the harmonic functions…
The Martin boundary associated with the simple random walk on an example of partially oriented lattice is shown to be trivial by computing fine estimates of the Green kernel.
We analyse how simple local constraints in two dimensions lead a defect to exhibit robust, non-transient, and tunable, subdiffusion. We uncover a rich dynamical phenomenology realised in ice- and dimer-type models. On the microscopic scale…
We study a random walk in random environment on the non-negative integers. The random environment is not homogeneous in law, but is a mixture of two kinds of site, one in asymptotically vanishing proportion. The two kinds of site are (i)…
Given a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green function of the corresponding random walk. It is known from the work of W. Woess that when a finitely supported…
We consider the random walks killed at the boundary of the quarter plane, with homogeneous non-zero jump probabilities to the eight nearest neighbors and drift zero in the interior, and which admit a positive harmonic polynomial of degree…
We consider a two dimensional reflecting random walk on the nonnegative integer quadrant. This random walk is assumed to be skip free in the direction to the boundary of the quadrant, but may have unbounded jumps in the opposite direction,…
We consider discrete (time and space) random walks confined to the quarter plane, with jumps only in directions $(i,j)$ with $i+j \geq 0$ and small negative jumps, i.e., $i,j \geq -1$. These walks are called singular, and were recently…
We compute the $t$-Martin boundary of two-dimensional small steps random walks killed at the boundary of the quarter plane. We further provide explicit expressions for the (generating functions of the) discrete $t$-harmonic functions. Our…
A complete representation of the Martin boundary of killed random walks on the quadrant ${\mathbb{N}}^*\times{\mathbb{N}}^*$ is obtained. It is proved that the corresponding full Martin compactification of the quadrant…
We study a random walk (Markov chain) in an unbounded planar domain whose boundary is described by two curves of the form $x_2 = a^+ x_1^{\beta^+}$ and $x_2 = -a^- x_1^{\beta^-}$, with $x_1 \geq 0$. In the interior of the domain, the random…
We consider a random walk on the first quadrant of the square lattice, whose increment law is, roughly speaking, homogeneous along a finite number of half-lines near each of the two boundaries, and hence essentially specified by…