Random Matrix Theory and Classical Statistical Mechanics: Spin Models

We present a statistical analysis of spectra of transfer matrices of classical lattice spin models; this continues the work on the eight-vertex model of the preceding paper. We show that the statistical properties of these spectra can serve as a criterion of integrability. It provides also an operational numerical method to locate integrable varieties. In particular, we distinguish the notions of integrability and criticality considering the two examples of the three-dimensional Ising critical point and the two-dimensional three-state Potts critical point. For complex spectra which appear frequently in the context of transfer matrices, we show that the notion of independence of eigenvalues for integrable models still holds.