Simulations of Discrete Random Geometries: Simplicial Quantum Gravity and Quantum String Theory

I investigate two discrete models of random geometries, namely simplicial quantum gravity and quantum string theory. In four-dimensional simplicial quantum gravity, I show that the addition of matter gauge fields to the model is capable of changing its phase structure by replacing the branched polymers of the pure gravity model with a new phase that has a negative string susceptibility exponent and a fractal dimension of four. Some of the results are derived from a strong coupling expansion of the model, a technique which is used here for the first time in this context. In quantum string theory, I study a discrete version of the IIB superstring. I show that the divergences encountered in the discretization of the bosonic string are eliminated in the supersymmetric case. I give theoretical arguments for the appearance of one-dimensional structures in the region of large system extents that manifest as a power-law tail in the link length distribution; this is confirmed by numerical simulations of the model. I also examine a lower-dimensional version of the IKKT matrix model, in which a similar effect can be observed.