Bergfinnur Durhuus

Diseases of triangulated random surface models, and possible cures

Abstract We discuss divergence problems in models of triangulated random surfaces whose action is given, for example, by the surface area. Our results uncover new difficulties in quantizing and regularizing the Nambu-Goto string. We also propose a new class of random surface models with more coercive actions which are expected to have improved behaviour and appear to be accessible to numerical simulations.

Three-dimensional simplicial quantum gravity and generalized matrix models

We consider a discrete model of Euclidean quantum gravity in three dimensions based on a summation over random simplicial manifolds. We derive some elementary properties of the model and discuss possible “matrix” models for 3-D gravity.

The appearance of critical dimensions in regulated string theories

We discuss a gaussian model of a discretized string. We analyze the d → ± ∞ limits and argue that there is a lower critical dimension, dc1, (∼ 2–4) and an upper critical dimension, dc2, (∼ 20–30) between which sensible continuum limits may exist.

Critical behaviour in a model of planar random surfaces

Abstract We solve a model of planar random surfaces exactly in the sense that, by assuming that the susceptibility diverges at a critical point, we determine the critical exponents and the Hausdorff dimension, and we show that the string tension does not tend to zero at the critical point. (The assumption that the susceptibility diverges has been verified numerically in 2 and 3 dimensions and proven for d = ∞.)

The Wess-Zumino action in two dimensions and non-abelian bosonization

Abstract Using recent results on fermionic determinants in two-dimensional non-abelian background fields we give a very simple path integral demonstration of the equivalence between the free Fermi theory in this background and a corresponding chiral Bose theory with Wess-Zumino action. The result is compared to previously proposed bosonization rules and certain limitations to the general validity of these are found.

REGULARIZED BOSONIC STRINGS NEED EXTRINSIC CURVATURE

Abstract We show that the string tension in a class of triangulated random surface models with gaussian action does not tend to zero at the critical point. This rules out the existence of a non-trivial continuum limit of these models. Furthermore, the proof seems to lead to the conclusion that the most natural (if not the only) way to avoid this feature of random surface models is by introducing asymptotically free interactions depending e.g. on the extrinsic curvature of the surface.

Summing over all genera for d > 1: a toy model☆

Abstract We consider a reduced model of discretized random surfaces which is believed to be a good approximation to the standard theory of discretized random surfaces for large dimensions d . In this model we observe multicritical behaviour. The relation between internal observables, in the sense of two-dimensional gravity, and external observables, in the sense of string theory, can be studied in detail. Furthermore, it is possible to perform explicitly the summation over all genera in this model.

Self-avoiding and planar random surfaces on the lattice

Abstract We study models of self-avoiding (SARS) and of planar (PRS) random surfaces on a (hyper-) cubic lattice. If Nγ(A) is the number of such surfaces with given boundary γ and area A, then Nγ(A) = exp(β0A + o(A)), where β0 is independent of γ. We prove that, for β > β0, the string tension is finite for the SARS model and strictly positive for the PRS model and that in both models the correlation length (inverse mass) is positive and finite. We discuss the possibility of the existence of a critical point and of a roughening transition. Estimates on intersection probabilities for random surfaces and connections with lattice gauge theories are sketched.

Local limit of labeled trees and expected volume growth in a random quadrangulation

Exploiting a bijective correspondence between planar quadrangulations and well-labeled trees, we define an ensemble of infinite surfaces as a limit of uniformly distributed ensembles of quadrangulations of fixed finite volume. The limit random surface can be described in terms of a birth and death process and a sequence of multitype Galton–Watson trees. As a consequence, we find that the expected volume of the ball of radius r around a marked point in the limit random surface is Θ(r4).

Matter fields with c > 1 coupled to 2d gravity☆

Abstract We solve a class of branched polymer models coupled to spin systems and show that they have no phase transition and are either always magnetized or never magnetized depending on the branching weights. By comparing these results with numerical simulations of two-dimensional quantum gravity coupled to matter fields with central charge c we provide evidence that for c sufficiently large ( c ⩾ 12) these models are effectively described by branched polymers. Moreover, the numerical results indicate a remarkable universality in the influence on the geometry of surfaces due to the interaction with matter. For spin systems this influence only depends on the total central charge.