# Constant Martin kernel and amenability

Consider a finitely supported random walk on a discrete group G such that the support generates $$G$$ as a semigroup. The Martin kernels are then non-negative functions on the product $$G \times M$$ where $$M$$ denotes the Martin boundary of the random walk. Does anyone know an example with $$G$$ amenable such that there does not exist a point m in the Martin boundary for which the function $$K( . , m)$$ is constant $$1$$? And does anyone know an example of a non-amenable group for which there is an element $$m$$ in the Martin boundary such that the corresponding Martin kernel $$K( . ,m)$$ is constant $$1$$?

This is a partial answer to your second question. If the function 1 is minimal, then every bounded harmonic function is constant. To reformulate, if there exists a point $$m$$ in the minimal Martin boundary such that $$K(\cdot, m)$$ is constant 1,then the Poisson boundary is trivial, which cannot happen if the group is non-amenable. So you have to look at non-minimal points.