  The dissipative Hofstadter model describes the quantum mechanics of a charged
particle in two dimensions subject to a periodic potential, uniform magnetic
field, and dissipative force. Its phase diagram exhibits an SL(2,Z) duality
symmetry and has an infinite number of critical circles in the
dissipation/magnetic field plane. In addition, multi-critical points on a
particular critical circle correspond to non-trivial solutions of open string
theory. The duality symmetry is expected to provide relations between
correlation functions at different multi-critical points. Many of these
correlators are contact terms. However we expect them to have physical
significance because under duality they transform into functions that are
non-zero for large separations of the operators. Motivated by the search for
exact, regulator independent solutions for these contact terms, in this paper
we derive many properties and symmetries of the coordinate correlation
functions at the special multi-critical points. In particular, we prove that
the correlation functions are homogeneous, piecewise-linear functions of the
momenta, and we prove a weaker version of the anticipated duality
transformation. Consequently, the possible forms of the correlation functions
are limited to lie in a finite dimensional linear space. We treat the potential
perturbatively and these results are valid to all orders in perturbation
theory.
