Positive eigenfunctions of the Laplacian andconformal densities on homogeneous trees

In this paper, we study some asymptotic aspects of the positive eigenfunctions of the combinatorial Laplacian associated to a homogeneous tree. The results are inspired by results of Dennis Sullivan concerning ?harmonic functions on the hyperbolic spaces H I n and contained in the paper Sul]. Let k be an integer 3 and X the homogeneous tree of degree k, that is, the unique simply connected simplicial complex of dimension 1 in which every vertex belongs to exactly k edges. X is equipped with the length metric in which every edge is isometric to the unit interval 0; 1]. The distance in X between two points x and y is denoted by jx ? yj. We denote by @X the boundary (at innnity) of X, that is, the set of ends of X. Recall that the set X @X has a natural topology which makes it a compact space in which X sits as a dense open subspace.