  We continue our study of string theory in a background that interpolates
between $AdS_3$ in the infrared and a linear dilaton spacetime $R^{1,1}\times
R_\phi$ in the UV. This background corresponds via holography to a $CFT_2$
deformed by a certain irrelevant operator of dimension $(2,2)$. We show that
for two point functions of local operators in the infrared CFT, conformal
perturbation theory in this irrelevant operator has a finite radius of
convergence in momentum space, and one can use it to flow up the
renormalization group. The spectral density develops an imaginary part above a
certain critical value of the spectral parameter; this appears to be related to
the non-locality of the theory. In position space, conformal perturbation
theory has a vanishing radius of convergence; the leading non-perturbative
effect is an imaginary part of the two point function.
