Streched exponential in non-linear stochastic filed theories

Abstract

We consider the time dependent two point function, <\phi_q (t) \phi_-q (0)> in non-linear stochastic field theories, for which the KPZ equation serves as a prototype. In particular we consider the small q's and long times such that \omega_q t>>1 (\omega_q being the corresponding decay rate). We find that, since the generic case has \omega_q \propto q^\mu for small q where \mu>1, the decay of the two point function is given by a streched exponential in \omega_qt multiplied by a factor of t, <\phi_q (t) \phi_-q (0)> \propto t^{\beta_d}exp[-\gamma(\omega_qt)^{1/\mu}], where \beta_d=(d-1)/2\mu, d is the dimensionality of space and \gamma a dimensionless constant.