Checkered surfaces and complex matrices

We investigate a large- N matrix model involving general complex matrices. It can be reinterpreted as a model of two hermitian matrices with specific couplings, and as a model of positive definite hermitian matrices. Large- N perturbation theory generates dynamical triangulations in which the triangles can be chequered (i.e. coloured so that neighbours are opposite colours). On a sphere there is a simple relation between such triangulations and those generated by the single hermitian matrix model. For the torus (and a quartic potential) we solve the counting problem for the number of triangulations that cannot be chequered. The critical physics of chequered triangulations is the same as that of the hemitian matrix model. We show this explicitly by solving non-perturbatively pure two-dimensional “chequered” gravity. The interpretative framework given here applies to a number of other generalisations of the hermitian matrix model.

statistical mechanics
matrix model
expansion 1/N
perturbation theory
space-time: sphere
space-time: torus
space-time: triangulation
continuum limit
critical phenomena

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