This Chapter outlines the replica approach in Random Matrix Theory. Both fermionic and bosonic versions of the replica limit are introduced and its trickery is discussed. A brief overview of early heuristic treatments of zero-dimensional replica field theories is given to advocate an exact approach to replicas. The latter is presented in two elaborations: by viewing the $\beta=2$ replica partition function as the Toda Lattice and by embedding the replica partition function into a more general theory of $\tau$ functions.
Categories: cond-mat.dis-nn hep-th math-ph math.MP nlin.SI