  We study the effect of localized modes in lattices of size N with parity-time
(PT) symmetry. Such modes are arranged in pairs of quasi-degenerate levels with
splitting delta exp{-N/xi}, where \xi is their localization length. The level
"evolution" with respect to the PT breaking parameter gamma shows a cascade of
bifurcations during which a pair of real levels becomes complex. The
spontaneous PT symmetry breaking occurs at gamma min(delta), thus resulting in
an exponentially narrow exact PT phase. As N/xi decreases, it becomes more
robust with gamma (1/N)^2 and the distribution P(gamma) changes from log-normal
to semi-Gaussian. Our theory can be tested in the frame of optical lattices.
