Frequently dense harmonic functions and universal martingales on trees

On a large class of infinite trees T T , we prove the existence harmonic functions alttext="h"> h encoding="application/x-tex">h with respect to suitable transient transition operators P"> P encoding="application/x-tex">P that satisfy following universal property: is Poisson transform martingale on end-point boundary alttext="normal upper Omega"> Ω encoding="application/x-tex">\Omega (equipped measure induced by ) such that, for every measurable function alttext="f"> f encoding="application/x-tex">f it contains subsequence converging in measure. Moreover, visits open set positive lower density.