Thordur Jonsson

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.

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 = ∞.)

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.

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.

The Spectral Dimension of Generic Trees

Abstract We define generic ensembles of infinite trees. These are limits as N→∞ of ensembles of finite trees of fixed size N, defined in terms of a set of branching weights. Among these ensembles are those supported on trees with vertices of a uniformly bounded order. The associated probability measures are supported on trees with a single spine and Hausdorff dimension dh=2. Our main result is that the spectral dimension of the ensemble average is ds=4/3, and that the critical exponent of the mass, defined as the exponential decay rate of the two-point function along the spine, is 1/3.

THE SPECTRAL DIMENSION OF THE BRANCHED POLYMER PHASE OF TWO-DIMENSIONAL QUANTUM GRAVITY

Abstract The metric of two-dimensional quantum gravity interacting with conformal matter is believed to collapse to a branched polymer metric when the central charge c > 1. We show analytically that the spectral dimension, ds, of such a branched polymer phase is 4 3 . This is in good agreement with numerical simulations for large c.

Regularized strings with extrinsic curvature

Abstract We analyse models of discretized string theories, where the path integral over world sheet variables is regularized by summing over triangulated surfaces. The inclusion of curvature in the action is a necessity for the scaling of the string tension. We discuss the physical properties of models with extrinsic curvature terms in the action and show that the string tension vanishes at the critical point where the bare extrinsic curvature coupling tends to infinity. Similar results are derived for models with intrinsic curvature.

Kinematical numerical study of the crumpling transition in crystalline surfaces

Abstract By analyzing the tangent-tangent correlation in a model of crystalline surfaces with in action depending on the extrinsic curvature, we derive constraints on the possible behaviour of the Hausdorff dimension as a function of the extrinsic-curvature coupling constant λ. For λ sufficiently small, the Hausdorff dimension is infinite, but jumps to a value than smaller than or equal to 4 a critical value λ c of λ. For λ above the critical value there is a number of possilbilities, but our numerical data favour a continuously varying Hausdorff dimension, changing from 4 at λ c to 2 for λ going to infinity.

Random walks on combs

We develop techniques to obtain rigorous bounds on the behaviour of random walks on combs. Using these bounds, we calculate exactly the spectral dimension of random combs with infinite teeth at random positions or teeth with random but finite length. We also calculate exactly the spectral dimension of some fixed non-translationally invariant combs. We relate the spectral dimension to the critical exponent of the mass of the two-point function for random walks on random combs, and compute mean displacements as a function of walk duration. We prove that the mean first passage time is generally infinite for combs with anomalous spectral dimension.