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We simulate a supersymmetric matrix model obtained from dimensional reduction of 4d SU(N) super Yang-Mills theory (a 4d counter part of the IKKT model or IIB matrix model). The eigenvalue distribution determines the space structure. The measurement of Wilson loop correlators reveals a universal large N scaling. Eguchi-Kawai equivalence may hold in a finite range of scale, which is also true for the bosonic case. We finally report on simulations of a low energy approximation of the 10d IKKT model, where we omit the phase of the Pfaffian and look for evidence for a spontaneous Lorentz symmetry breaking., Comment: 4 pages, talk presented at LATTICE 2000 (Bangalore)

We propose a method for Monte Carlo simulations of systems with a complex action. The method has the advantages of being in principle applicable to any such system and provides a solution to the overlap problem. In some cases, like in the IKKT matrix model, a finite size scaling extrapolation can provide results for systems whose size would make it prohibitive to simulate directly., Comment: Lattice2003(nonzero), 3 pages, 4 figures, Proceedings for Lattice 2003, July 2003, Tsukuba, Japan

Using numerical simulations we establish a phase diagram for the model of the Ising matter interacting with three-dimensional simplicial quantum gravity.

A model of simplicial quantum gravity in 3 dimensions is described, and some preliminary numerical analysis of themodel is presented. The present knowledge of ergodic moves in 3 dimensions is reviewed, and the best approach for numerical analysis discussed.

We briefly discuss some recent analytical results in 4D simplicial quantum gravity: (i) The characterization of the singular structures sampling a neighborhood of the critical point; (ii) The transition between weak and strong coupling as driven by the dynamics of a polymeric interface; (iii) The possible connection between triangulations counting and the geometry of the moduli space of singular Euclidean structures on 4D manifolds.

I review the recent progress in 2d gravity and discuss the new numerical simulations for 2d gravity and for random surfaces in d > 2. The random surface theories of interest in d > 2 have extrinsic curvature terms, and for a finite value of the extrinsic curvature coupling there seems to be a second order phase transition where the string tension scales.

We study SU (2) gauge theory in the presence of a constant chromomagnetic background field. For large fields we observe qualitative agreement with the one-loop perturbative expression for the energy density.

We perform a numerical investigation of high temperature transitions involving the change of topological charge in the electroweak theory. 3-dimensional space is approximated by a lattice. The dynamics is followed in real time, not Euclidian time. Transitions were observed in the phase with spontaneous symmetry breaking as well as in the phase with restored symmetry. We determine the rate Λ of fermion number non-conservation in electroweak theory in the phase with restored symmetry. There is no suppression and Λ = κ(αWT)4 with κ ≈ 0.1 – 1.0.

We discuss a new attempt to determine the critical exponent γ for the bosonic string. The method uses a new way of calculating the partion function by Monte Carlo techniques. The algorithm also allows a partial vectorization. However, the simulation confirms older results, which are not in agreement with the analytically derived formula of KPZ. Possible reasons for this disagreement is discussed.

We study the zeros in the complex plane of the partition function for the Ising model coupled to $2d$ quantum gravity for complex magnetic field and for complex temperature. We compute the zeros by using the exact solution coming from a two matrix model and by Monte Carlo simulations of Ising spins on dynamical triangulations. We present evidence that the zeros form simple one-dimensional patterns in the complex plane, and that the critical behaviour of the system is governed by the scaling of the distribution of singularities near the critical point., Comment: 3 pages, 8 figures, latex2e, uses espcrc2.sty. Contribution to Lattice '97, Edinburgh, July 1997, to appear on Nucl. Phys. B (Proc. Suppl.)

We perform Monte Carlo simulations of 2-d dynamically triangulated surfaces coupled to Ising and three--states Potts model matter. By measuring spin-spin correlation functions as a function of the geodesic distance we provide substantial evidence for a diverging correlation length at $\beta_c$. The corresponding scaling exponents are directly related to the KPZ exponents of the matter fields as conjectured in [4] (NPB454(1995)313)., Comment: Talk presented at LATTICE96(gravity)

Employing baby universe surgery we study numerically the fractal structure of the intrinsic geometry of random surfaces coupled to matter field with c = 1. We simulate surfaces of the size up to 260.000 triangles. We observe the first indication that the branching and Haussdorff dimensions saturate. We estimate the branching dimension d(B) approximate to 2.6 and get a lower bound for the Hausdorff dimension d(H) > 3.0.

We address the question of spontaneous breaking of parity in three-dimensional euclidian SU(2) gauge-Higgs theory by Monte Carlo simulations. We observe no sign of spontaneous parity breaking. However, the presence of parity odd terms in the action can induce a phase transition to a parity odd ground state.

I review the basic properties of the theory of random surfaces. It is shown that the theory gives a correct description of strings in dimension d ≤ 1 and of matter fields with central charge c ≤ 1. In this region it coincides with the new results of Knizhnik, Polyakov and Zamolodchikov. Analytical and numerical results for the discretized model and for the standard continuum string model for d > 1 indicate a trivial theory in this region.

I discuss various attempts to connect the theory of discretized random surfaces to the continuum theory of bosonic strings and superstrings. A review of the numerical Monte Carlo results for the critical exponents of random surfaces is presented and it is argued that we do not have a simple connection between the bosonic string and the theory of random surface because of the tachyon. Random surfaces with extrinsic curvature might have correct scaling of the string tension and no tachyons and could therefore be candidates for a regularized version of the (bosonized) superstring.