  Local and global scaling solutions for $O(N)$ symmetric scalar field theories
are studied in the complexified field plane with the help of the
renormalisation group. Using expansions of the effective action about small,
large, and purely imaginary fields, we obtain and solve exact recursion
relations for all couplings and determine the $3d$ Wilson-Fisher fixed point
analytically. For all $O(N)$ universality classes, we further establish that
Wilson-Fisher fixed point solutions display singularities in the complex field
plane, which dictate the radius of convergence for real-field expansions of the
effective action. At infinite $N$, we find closed expressions for the
convergence-limiting singularities and prove that local expansions of the
effective action are powerful enough to uniquely determine the global
Wilson-Fisher fixed point for any value of the fields. Implications of our
findings for interacting fixed points in more complicated theories are
indicated.
