  The transresistance and the Hall transresistance of dirty two-dimensional
bi-layer systems in the presence of tunneling bridges (pinholes) are studied
theoretically. We find, at weak magnetic field, a non-zero Hall
transresistance. In a geometry where the pinholes are concentrated in the
middle of the sample, a quantum process gives the dominant contribution to both
the ordinary transresistance and the Hall transresistance. Arising from the
interplay between Coulomb repulsion, disorder and tunneling, the quantum
contribution increases in a singular way as the temperature decreases.
