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Massive Majorana fermion coupled to two-dimensional gravity and the random-lattice Ising model. (English) Zbl 1177.81093

Theor. Math. Phys. 147, No. 3, 755-776 (2006); translation from Teor. Mat. Fiz. 147, No. 3, 372-398 (2006).
Summary: We consider the partition function of the two-dimensional free massive Majorana fermion coupled to the quantized metric of the spherical topology. By adding an arbitrary conformal “spectator” matter, we gain control over the total matter central charge. This provides an interesting continuously parameterized family of critical points and also allows making a connection with the semiclassical limit. We use the Liouville field theory as the effective description of the quantized gravity. The spherical scaling function is calculated approximately, but (we believe) to a good numerical precision, in almost the whole domain of the spectator parameter. An impressive comparison with the predictions of the exactly solvable matrix model yields a more general model of random-lattice statistics, which is most probably not solvable by the matrix-model technique but reveals a more general pattern of critical behavior. We hope that numerical simulations or series extrapolation will be able to reveal our family of scaling functions.

MSC:

81T10 Model quantum field theories
81T27 Continuum limits in quantum field theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
83C80 Analogues of general relativity in lower dimensions
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