  We consider potential theory on Bratteli diagrams arising from Macdonald
polynomials. The case of Hall-Littlewood polynomials is particularly
interesting; the elements of the diagram are partitions, the branching
multiplicities are integers, the combinatorial dimensions are Green's
polynomials, and the Jordan form of a randomly chosen unipotent upper
triangular matrix over a finite field gives rise to a harmonic function. The
case of Schur functions yields natural deformations of the Young lattice and
Plancharel measure. Many harmonic functions are constructed and algorithms for
sampling from the underlying probability measures are given.
