Building on insights from the theory of integrable lattices, the integrability is claimed for nonlinear replica $\sigma$ models derived in the context of real symmetric random matrices. Specifically, the fermionic and the bosonic replica partition functions are proven to form a single (supersymmetric) Pfaff-KP hierarchy whose replica limit is shown to reproduce the celebrated nonperturbative formula for the density-density eigenvalue correlation function in the infinite-dimensional Gaussian orthogonal ensemble. Implications of the formalism outlined are briefly discussed.

• Received 15 April 2013

DOI:https://doi.org/10.1103/PhysRevE.88.030101