Equilibrium measures on trees

On a metric space (X,d) one can define a set function called capacity which has motivations coming from Physics and which plays a deep role in potential theory and geometric measure theory. It is well known that to any compact subset E of X can be associated a probability measure m on X, called equilibrium measure for E, such that m(E)=cap(E). These measures at present are not well understood. We will present a characterization of equilibrium measures when X is a locally finite tree of infinite depth.