Analytic calculation of anomalous scaling in random shell models for a passive scalar

An exact nonperturbative calculation of the fourth-order anomalous correction to the scaling behavior of a random shell model for passive scalars is presented. Importance of ultraviolet (UV) and infrared (IR) boundary conditions on the inertial scaling properties are determined. We find that anomalous behavior is given by the null space of the inertial operator and we prove strong UV and IR independence of the anomalous exponent. A limiting case where diffusive behavior can influence inertial properties is also presented.