Zdzislaw Burda

Focusing on the fixed point of 4D simplicial gravity

Our earlier renormalization group analysis of simplicial gravity is extended. A high-statistics study of the volume and coupling constant dependence of the cumulants of the node distribution is carried out. It appears that the phase transition of the theory is of first order, contrary to what is generally believed.

Universal microscopic correlation functions for products of independent Ginibre matrices

We consider the product of n complex non-Hermitian, independent random matrices, each of size NxN with independent identically distributed Gaussian entries (Ginibre matrices). The joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single Ginibre matrix, but with a more complicated weight given by a Meijer G-function depending on n. Using the method of orthogonal polynomials we compute all eigenvalue density correlation functions exactly for finite N and fixed n. They are given by the determinant of the corresponding kernel which we construct explicitly. In the large-N limit at fixed n we first determine the microscopic correlation functions in the bulk and at the edge of the spectrum. After unfolding they are identical to that of the Ginibre ensemble with n=1 and thus universal. In contrast the microscopic correlations we find at the origin differ for each n>1 and generalise the known Bessel-law in the complex plane for n=2 to a new hypergeometric kernel 0_F_n-1.

Condensation in the backgammon model

Abstract We analyse the properties of a very simple “balls-in-boxes” model which can exhibit a phase transition between a fluid and a condensed phase, similar to the behaviour encountered in models of random geometries in one, two and four dimensions. This model can be viewed as a generalization of the backgammon model introduced by Ritort as an example of glassy behaviour without disorder.

Uncorrelated random networks.

We define a statistical ensemble of nondegenerate graphs, i.e., graphs without multiple-connections and self-connections between nodes. The node degree distribution is arbitrary, but the nodes are assumed to be uncorrelated. This completes our earlier publication [Phys. Rev. 64, 046118 (2001)] where trees and degenerate graphs were considered. An efficient algorithm generating nondegenerate graphs is constructed. The corresponding computer code is available on request. Finite-size effects in scale-free graphs, i.e., those where the tail of the degree distribution falls like n(-beta), are carefully studied. We find that in the absence of dynamical internode correlations the degree distribution is cut at a degree value scaling like N(gamma), with gamma=min[1/2,1/(beta-1)], where N is the total number of nodes. The consequence is that, independently of any specific model, the internode correlations seem to be a necessary ingredient of the physics of scale-free networks observed in nature.

Appearance of mother universe and singular vertices in random geometries

We discuss a general mechanism that drives the phase transition in the canonical ensemble in models of random geometries. As an example we consider a solvable model of branched polymers where the transition leading from tree– to bush–like polymers relies on the occurrence of vertices with a large number of branches. The source of this transition is a combination of the constraint on the total number of branches in the canonical ensemble and a nonlinear one–vertex action. We argue that exactly the same mechanism, which we call constrained mean–field, plays the crucial role in the phase transition in 4d simplicial gravity and, when applied to the effective one–vertex action, explains the occurrence of both the mother universe and singular vertices at the transition point when the system enters the crumpled phase. ITFA-96-29, BI-TP 96/32, hep-lat/9608030

Phase transition in fluctuating branched geometry

Abstract We study grand-canonical and canonical properties of the model of branched polymers proposed in [1]. We show that the model has a fourth order phase transition and calculate critical exponents. At the transition the exponent γ of the grand-canonical ensemble, analogous to the string susceptibility exponent of surface models, γ ∼ 0.3237525… is the first known example of positive γ which is not of the form 1 n , n = 2, 3, …. We show that a slight modification of the model produces a continuos spectrum of γs in the range (0, 1 2 ] and changes the order of the transition.

Geometry of reduced supersymmetric 4D Yang–Mills integrals

We study numerically the geometric properties of reduced supersymmetric non-compact SU(N) Yang-Mills integrals in D=4 dimensions, for N = 2,3, ..., 8. We show that in the range of large eigenvalues of the matrices A^mu, the original D-dimensional rotational symmetry is spontaneously broken and the dominating field configurations become one-dimensional, as anticipated by studies of the underlying surface theory. We also discuss possible implications of our results for the IKKT model.We study numerically the geometric properties of reduced supersymmetric non-compact SU(N) Yang-Mills integrals in D = 4 dimensions, for N = 2, 3,...,8. We show that in the range of large eigenvalues of the matrices A(mu). the original D-dimensional rotational symmetry is spontaneously broken and the dominating field configurations become one-dimensional, as anticipated by studies of the underlying surface theory. We also discuss possible implications of our results for the IKKT model

4d simplicial quantum gravity interacting with gauge matter fields

The effect of coupling non-compact

Universal microscopic correlation functions for products of truncated unitary matrices

U(1)

Spectral moments of correlated Wishart matrices.

gauge fields to four dimensional simplicial quantum gravity is studied using strong coupling expansions and Monte Carlo simulations. For one gauge field the back-reaction of the matter on the geometry is weak. This changes, however, as more matter fields are introduced. For more than two gauge fields the degeneracy of random manifolds into branched polymers does not occur, and the branched polymer phase seems to be replaced by a new phase with a negative string susceptibility exponent