  This is a short summary of the phase structure of vector O(N) symmetric
quantum field theories in a singular limit, the double scaling limit.It is
motivated by the fact that summing up dynamically triangulated random surfaces
using Feynman graphs of the O(N) matrix model results in a genus expansion and
it provides,in some sense, a nonperturbative treatment of string theory when
the double scaling limit is enforced. The main point emphasized here is that
this formal singular limit, recently discussed mainly in d=0 O(N) matrix
models, has an intriguing physical meaning in d>2 O(N) vector theories. In this
limit all orders in {1\over N} are of equal importance since at each order
infrared divergences compensate for the decrease in powers of {1\over N}. The
infrared divergences are due to the tuning of the strength of the force {g \to
g_c} between the O(N) quanta so that a massless O(N) singlet appears in the
spectrum. At the critical dimension an interesting phase structure is revealed,
the massless excitation has the expected physical meaning: it is the Goldston
boson of spontaneous breaking of scale invariance - the dilaton.
