Stochastic quantization of matrix and lattice gauge models

We introduce a stochastic diffusion equation and the Fokker-Planck equation for various matrix models including the $\mathrm{U}\mathrm{}\left(N\right)\mathrm{}\times \mathrm{U}\mathrm{}\left(N\right)\mathrm{}$ chiral model and lattice gauge theories. It is shown how to calculate various $\mathrm{U}\mathrm{}\left(N\right)\mathrm{}$ integrals using the stochastic equation. In particular, in the external-field problem, the exact large-$N$ result (in the weak-coupling region) is reproduced and a $\frac{1}{N^2}$ correction is computed. Also, the order parameter $〈\det U〉$ is calculated up to order $\frac{1}{{\beta }^2}$. In the $\mathrm{U}\mathrm{}\left(N\right)\mathrm{}\times \mathrm{U}\mathrm{}\left(N\right)\mathrm{}$ chiral model, the large-$N$ reduction and quenching is done in the context of stochastic quantization, and the semiclassical results of Bars, Gunaydin, and Yankielowicz for the free energy and the two-point correlation function are derived. In lattice gauge theory, a very simple way of deriving the complete Schwinger-Dyson equations from the Fokker-Planck equation is demonstrated and, as in the chiral model, a reduced, quenched stochastic equation is derived.