The Kardar–Parisi–Zhang equation and its matrix generalization

We study the problem of the condensate (stochastic average) origination for an auxiliary field in the Kardar–Parisi–Zhang equation and its matrix generalization. We cannot reliably conclude that there is a condensate for the Kardar–Parisi–Zhang equation in the framework of the one-loop approximation improved by the renormalization group method. The matrix generalization of the Kardar–Parisi–Zhang equation permits a positive answer to the question of whether there is a nonzero condensate, and the problem can be solved exactly in the large-$N$ limit.