Fermionic quantum gravity

We study the statistical mechanics of random surfaces generated by N x N on e-matrix integrals over anti-commuting variables, These Grassmann-Yalued ma trix models are shown to be equivalent to N x N unitary versions of general ized Penner matrix models. We explicitly solve for the combinatorics of 't Hooft diagrams of the matrix integral and develop an orthogonal polynomial formulation of the statistical theory. An examination of the large N and do uble scaling limits of the theory shows that the genus expansion is a Borel summable alternating series which otherwise coincides with two-dimensional quantum gravity in the continuum limit. We demonstrate that the partition functions of these matrix models belong to the relativistic Toda chain inte grable hierarchy, The corresponding string equations and Virasoro constrain ts are derived and used to analyse the generalized KdV flow structure of th e continuum limit. (C) 2001 Elsevier Science B.V. All rights reserved.