  We construct a one-to-one continuous map from the Morse boundary of a
hierarchically hyperbolic group to its Martin boundary. This construction is
based on deviation inequalities generalizing Ancona's work on hyperbolic
groups. This provides a possibly new metrizable topology on the Morse boundary
of such groups. We also prove that the Morse boundary has measure 0 with
respect to the harmonic measure unless the group is hyperbolic.
