arXiv subjects
Publications and source records attributed to J. Ambjorn.
We consider a random surface representation of the three-dimensional Ising model.The model exhibit scaling behaviour and a new critical index $\k$ which relates $\g_{string}$ for the bosonic string to the exponent $\a$ of the specific heat of the 3d Ising model is introduced. We try to determine $\k$ by numerical simulations.
We consider a discrete model of euclidean quantum gravity in four dimensions based on a summation over random simplicial manifolds. The action used is the Einstein-Hilbert action plus an $R^2$-term. The phase diagram as a function of the bare coupling constants is studied in the search for a sensible continuum limit. For small values of the coupling constant of the $R^2$ term the model seems to belong to the same universality class as the model with pure Einstein-Hilbert action and exhibits the same phase transition. The order of the transition may be second or higher. The average curvature is positive at the phase transition, which makes it difficult to understand the possible scaling relations of the model.
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 ($c\geq 12$) 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.
By numerical simulations in {\it real time} we provide evidence in favour of sphaleron like transitions in the hot, symmetric phase of the electroweak theory. Earlier performed observations of a change in the Chern-Simons number are supplemented with a measurement of the lowest eigenvalues of the three-dimensional staggered fermion Dirac operator and observations of the spatial extension of energy lumps associated with the transition. The observations corroborate on the interpretation of the change in Chern-Simons numbers as representing continuum physics, not lattice artifacts. By combining the various observations it is possible to follow in considerable detail the time-history of thermal fluctuations of the classical gauge-field configurations responsible for the change in the Chern-Simons number.
We analyze numerically the critical properties of a two-dimensional discretized random surface with extrinsic curvature embedded in a three-dimensional space. The use of the toroidal topology enables us to enforce the non-zero external extension without the necessity of defining a boundary and allows us to measure directly the string tension. We show that a phase transition from the crumpled phase to the smooth phase observed earlier for a spherical topology appears also for a toroidal surface for the same finite value of the coupling constant of the extrinsic curvature term. The phase transition is characterized by the vanishing of the string tension. We discuss the possible non-trivial continuum limit of the theory, when approaching the critical point. Numerically we find a value of the critical exponent $\n$ to be between .38 and .42. The specific heat, related to the extrinsic curvature term seems not to diverge (or diverge slower than logarithmically) at the critical point.
We describe an iterative scheme which allows us to calculate any multi-loop correlator for the complex matrix model to any genus using only the first in the chain of loop equations. The method works for a completely general potential and the results contain no explicit reference to the couplings. The genus $g$ contribution to the $m$--loop correlator depends on a finite number of parameters, namely at most $4g-2+m$. We find the generating functional explicitly up to genus three. We show as well that the model is equivalent to an external field problem for the complex matrix model with a logarithmic potential.
We investigate the phase structure of three-dimensional quantum gravity coupled to an Ising spin system by means of numerical simulations. The quantum gravity part is modelled by the summation over random simplicial manifolds, and the Ising spins are located in the center of the tetrahedra, which constitute the building blocks of the piecewise linear manifold. We find that the coupling between spin and geometry is weak away from the critical point of the Ising model. At the critical point there is clear coupling, which however does not seem to change the first order transition between the ``hot'' and ``cold'' phase of three dimensional simplicial quantum gravity observed earlier.
We show how the stochastic stabilization provides both the weak coupling genus expansion and a strong coupling expansion of 2d quantum gravity. The double scaling limit is described by a hamiltonian which is unbounded from below, but which has a discrete spectrum.
We develop an iterative algorithm for the genus expansion of the hermitian $N\times N$ one-matrix model ( = the Penner model in an external field). By introducing moments of the external field, we prove that the genus $g$ contribution to the $m$-loop correlator depends only on $3g-2+m$ lower moments ($3g-2$ for the partition function). We present the explicit results for the partition function and the one-loop correlator in genus one. We compare the correlators for the hermitian one-matrix model with those at zero momenta for $c=1$ CFT and show an agreement of the one-loop correlators for genus zero.
We consider the stochastic quantization scheme for a non-perturbative stabilization of 2D quantum gravity and prove that it does not satisfy the KdV flow equations. It therefore differs from a recently suggested matrix model which allows real solutions to the KdV equations. The behaviour of the Fermi energy, the free energy and macroscopic loops in the stochastic quantization scheme are elucidated.