On rolling, tunneling and decaying in some large N vector models

Various aspects of time-dependent processes are studied within the large N approximation of O(N) vector models in three dimensions. These include the rolling of fields, the tunneling and decay of vacua. We present an exact solution for the quantum conformal case and find a solution for more general potentials when the total change of the value of the field is small. Characteristic times are found to be shorter when the time dependence of the field is taken into account in constructing the exact large N effective potentials. We show that the different approximations yield the same answers in the regions of the overlap of the validity. A numerical solution of this potential reveals a tunneling in which the bubble that separates the true vacuum from the false one is thick.