P Bialas

Phase diagram of the mean field model of simplicial gravity

Abstract We discuss the phase diagram of the balls in boxes model, with a varying number of boxes. The model can be regarded as a mean-field model of simplicial gravity. We analyse in detail the case of weights of the form p(q) = q−β, which correspond to the measure term introduced in the simplicial quantum gravity simulations. The system has two phases: elongated (fluid) and crumpled. For β ϵ (2, ∞) the transition between these two phases is first-order, while for β ϵ (1, 2) it is continuous. The transition becomes softer when β approaches unity and eventually disappears at β = 1. We then generalise the discussion to an arbitrary set of weights. Finally, we show that if one introduces an additional kinematic bound on the average density of balls per box then a new condensed phase appears in the phase diagram. It bears some similarity to the crinkled phase of simplicial gravity discussed recently in models of gravity interacting with matter fields.

Large N limit of the IKKT matrix model

Using the dynamical triangulation approach we perform a numerical study of a supersymmetric random surface model that corresponds to the large N limit of the four-dimensional version of the IKKT matrix model. We show that the addition of fermionic degrees of freedom suppresses the spiky world-sheet configurations that are responsible for the pathological behaviour of the purely bosonic model. We observe that the distribution of the gyration radius has a power-like tail p(R) ~ R^{-2.4}. We check numerically that when the number of fermionic degrees of freedom is not susy-balanced, p(R) grows with

Effective sampling of random surfaces by baby universe surgery

R

Finite size scaling of the balls in boxes model

and the model is not well-defined. Numerical sampling of the configurations in the tail of the distribution shows that the bosonic degrees of freedom collapse to a one-dimensional tube with small transverse fluctuations. Assuming that the vertex positions can fluctuate independently within the tube, we give a theoretical argument which essentially explains the behaviour of p(R) in the different cases, in particular predicting p(R) ~ R^{-3} in the supersymmetric case. Extending the argument to six and ten dimensions, we predict p(R) ~ R^{-7} and p(R) ~ R^{-15}, respectively.Abstract Using the dynamical triangulation approach we perform a numerical study of a supersymmetric random surface model that corresponds to the large N limit of the four-dimensional version of the IKKT matrix model. We show that the addition of fermionic degrees of freedom suppresses the spiky world-sheet configurations that are responsible for the pathological behaviour of the purely bosonic model. We observe that the distribution of the gyration radius has a power-like tail p(R)∼R −2.4 . We check numerically that when the number of fermionic degrees of freedom is not SUSY-balanced, p(R) grows with R and the model is not well-defined. Numerical sampling of the configurations in the tail of the distribution shows that the bosonic degrees of freedom collapse to a one-dimensional tube with small transverse fluctuations. Assuming that the vertex positions can fluctuate independently within the tube, we give a theoretical argument which essentially explains the behaviour of p(R) in the different cases, in particular predicting p(R)∼R −3 in the supersymmetric case. Extending the argument to six and ten dimensions, we predict p(R)∼R −7 and p(R)∼R −15 , respectively.

High temperature 3D QCD: dimensional reduction at work

We propose a new, very efficient algorithm for sampling of random surfaces in the Monte Carlo simulations, based on so-called baby universe surgery, i.e. cutting and pasting of baby universe. It drastically reduces slowing down as compared to the standard local flip algorithm, thereby allowing simulations of large random surfaces coupled to matter fields. As an example we investigate the efficiency of the algorithm for 2d simplicial gravity interacting with a one-component free scalar field. The radius of gyration is the slowest mode in the standard local flip/shift algorithm. The use of baby universe surgery decrease the autocorrelation time by three order of magnitude for a random surface of 0.5 · 105 triangles, where it is found to be τint = 150 ± 31 sweeps.

Three dimensional finite temperature SU(3) gauge theory in the confined region and the string picture

Abstract We discuss the finite size behaviour in the canonical ensemble of the balls in boxes model. We compare theoretical predictions and numerical results for the finite size scaling of cumulants of the energy distribution in the canonical ensemble and perform a detailed analysis of the first and third order phase transitions which appear for different parameter values in the model.

Percolation and magnetization in the continuous spin Ising model

We investigate the three-dimensional SU(3) gauge theory at finite temperature in the framework of dimensional reduction. The large scale propel ties of this theory are expected to be conceptually more complicated than in four dimensions. The dimensionally reduced action is computed in closed analytical form. The resulting effective two-dimensional theory is studied numerically both in the electric and magnetic sector. We find that dimensional reduction works excellently down to temperatures of 1.5 times the deconfinement phase transition temperature and even on rather short length scales. We obtain strong evidence that for QCD(3), even at high temperature the colour averaged potential is represented by the exchange of a single state, at variance with the usual Debye screening picture involving a pair of electric gluons

A gauge theory of Wilson lines as a dimensionally reduced model of QCD3

Abstract We determine the correlation between Polyakov loops in three-dimensional SU ( 3 ) gauge theory in the confined region at finite temperature. For this purpose we perform lattice calculations for the number of steps in the temperature direction equal to six. This is expected to be in the scaling region of the lattice theory. We compare the results to the bosonic string model. The agreement is very good for temperatures T 0.7 T c , where T c is the critical temperature. In the region 0.7 T c T T c we enter the critical region, where the critical properties of the correlations are fixed by universality to be those of the two-dimensional three state Potts model. Nevertheless, by calculating the critical lattice coupling, we show that the ratio of the critical temperature to the square root of the zero temperature string tension, where the latter is taken from the literature, remains very near to the string model prediction.

THE WEAK-COUPLING LIMIT OF SIMPLICIAL QUANTUM GRAVITY

Abstract In the strong coupling limit the partition function of SU (2) gauge theory can be reduced to that of the continuous spin Ising model with nearest neighbour pair-interactions. The random cluster representation of the continuous spin Ising model in two dimensions is derived through a Fortuin–Kasteleyn transformation, and the properties of the corresponding cluster distribution are analyzed. It is shown that for this model, the magnetic transition is equivalent to the percolation transition of Fortuin–Kasteleyn clusters, using local bond weights. These results are also illustrated by means of numerical simulations.

Connected correlators in quantum gravity

We analyze a two-dimensional SU(3) gauge model of Wilson lines as a dimensionally reduced model of high temperature QCD(3). In contrast to perturbative dimensional reduction it has an explicit global Z(3) symmetry in the action. The phase diagram of the model is studied in the space of two free parameters used to describe the self interaction of the Wilson lines. In addition to the confinement-deconfinement transition, the model also exhibits a new Z(3)-breaking phase. These findings are obtained by numerical simulations, and supported by a perturbative calculation to one loop. A screening mass from Polyakov loop correlations is calculated numerically. It matches the known QCD(3) mass in a domain of parameters belonging to the normal deconfined phase