Fluctuating Dimension in a Discrete Model for Quantum Gravity Based on the Spectral Principle

The spectral principle of Connes and Chamseddine is used as a starting point to define a discrete model for Euclidean quantum gravity. Instead of summing over ordinary geometries, we consider the sum over generalized geometries where topology, metric and dimension can fluctuate. The model describes the geometry of spaces with a countable number $n$ of points, and is related to the Gaussian unitary ensemble of Hermitian matrices. We show that this simple model has two phases. The expectation value $$, the average number of points in the universe, is finite in one phase and diverges in the other. We compute the critical point as well as the critical exponent of$$. Moreover, the space-time dimension $\delta$ is a dynamical observable in our model, and plays the role of an order parameter. The computation of $$ is discussed and an upper bound is found, $< 2$.Comment: 10 pages, no figures. Third version: This new version emphasizes the spectral principle rather than the spectral action. Title has been changed accordingly. We also reformulated the computation of the dimension, and added a new reference. To appear in Physical Review Letter

Correlation functions of eigenvalues of multi-matrix models, and the limit of a time dependent matrix

We consider the correlation functions of eigenvalues of a unidimensional chain of large random hermitian matrices. An asymptotic expression of the orthogonal polynomials allows to find new results for the correlations of eigenvalues of different matrices of the chain. Eventually, we consider the limit of the infinite chain of matrices, which can be interpreted as a time dependent one-matrix model, and give the correlation functions of eigenvalues at different times.Comment: Tex-Harvmac, 27 pages, submitted to Journ. Phys.

Universal parametric correlations in the transmission eigenvalue spectra of disordered conductors

We study the response of the transmission eigenvalue spectrum of disordered metallic conductors to an arbitrary external perturbation. For systems without time-reversal symmetry we find an exact non-perturbative solution for the two-point correlation function, which exhibits a new kind of universal behavior characteristic of disordered conductors. Systems with orthogonal and symplectic symmetries are studied in the hydrodynamic regime.Comment: 10 pages, written in plain TeX, Preprint OUTP-93-36S (University of Oxford), to appear in Phys. Rev. B (Rapid Communication

Theory of random matrices with strong level confinement: orthogonal polynomial approach

Strongly non-Gaussian ensembles of large random matrices possessing unitary symmetry and logarithmic level repulsion are studied both in presence and absence of hard edge in their energy spectra. Employing a theory of polynomials orthogonal with respect to exponential weights we calculate with asymptotic accuracy the two-point kernel over all distance scale, and show that in the limit of large dimensions of random matrices the properly rescaled local eigenvalue correlations are independent of level confinement while global smoothed connected correlations depend on confinement potential only through the endpoints of spectrum. We also obtain exact expressions for density of levels, one- and two-point Green's functions, and prove that new universal local relationship exists for suitably normalized and rescaled connected two-point Green's function. Connection between structure of Szeg\"o function entering strong polynomial asymptotics and mean-field equation is traced.Comment: 12 pages (latex), to appear in Physical Review

Asymptotic Level Spacing of the Laguerre Ensemble: A Coulomb Fluid Approach

We determine the asymptotic level spacing distribution for the Laguerre Ensemble in a single scaled interval, $(0,s)$, containing no levels, E_{\bt}(0,s), via Dyson's Coulomb Fluid approach. For the $\alpha=0$ Unitary-Laguerre Ensemble, we recover the exact spacing distribution found by both Edelman and Forrester, while for $\alpha\neq 0$, the leading terms of $E_{2}(0,s)$, found by Tracy and Widom, are reproduced without the use of the Bessel kernel and the associated Painlev\'e transcendent. In the same approximation, the next leading term, due to a ``finite temperature'' perturbation (\bt\neq 2), is found.Comment: 10pp, LaTe

Random matrix ensembles with an effective extensive external charge

Recent theoretical studies of chaotic scattering have encounted ensembles of random matrices in which the eigenvalue probability density function contains a one-body factor with an exponent proportional to the number of eigenvalues. Two such ensembles have been encounted: an ensemble of unitary matrices specified by the so-called Poisson kernel, and the Laguerre ensemble of positive definite matrices. Here we consider various properties of these ensembles. Jack polynomial theory is used to prove a reproducing property of the Poisson kernel, and a certain unimodular mapping is used to demonstrate that the variance of a linear statistic is the same as in the Dyson circular ensemble. For the Laguerre ensemble, the scaled global density is calculated exactly for all even values of the parameter $\beta$, while for $\beta = 2$ (random matrices with unitary symmetry), the neighbourhood of the smallest eigenvalue is shown to be in the soft edge universality class.Comment: LaTeX209, 17 page

Lives before and after Stonehenge: An osteobiographical study of four prehistoric burials recently excavated from the Stonehenge World Heritage Site

Osteobiographies of four individuals whose skeletal remains were recovered in 2015–16 from the Stonehenge World Heritage Site are constructed, drawing upon evidence from funerary taphonomy, radiocarbon dating, osteological study, stable isotope analyses, and microscopic and biomolecular analyses of dental calculus. The burials comprise an adult from the Middle Neolithic period, immediately prior to the building of Stonehenge, and two adults and a perinatal infant dating from the Middle Bronze Age, shortly after the monument ceased to be structurally modified. The two Middle Bronze Age adults were closely contemporary, but differed from one another in ancestry, appearance and geographic origin (key components of ethnicity). They were nevertheless buried in very similar ways. This suggests that aspects they held in common (osteological analysis suggests perhaps a highly mobile lifestyle) were more important in determining the manner of deposition of their bodies than any differences between them in ethnicity. One of these individuals probably came from outside Britain, as perhaps did the Middle Neolithic adult. This would be consistent with the idea that the Stonehenge landscape had begun to draw people to it from beyond Britain before Stonehenge was constructed and that it continued to do so after structural modification to the monument had ceased

Breakdown of universality in multi-cut matrix models

We solve the puzzle of the disagreement between orthogonal polynomials methods and mean field calculations for random NxN matrices with a disconnected eigenvalue support. We show that the difference does not stem from a Z2 symmetry breaking, but from the discreteness of the number of eigenvalues. This leads to additional terms (quasiperiodic in N) which must be added to the naive mean field expressions. Our result invalidates the existence of a smooth topological large N expansion and some postulated universality properties of correlators. We derive the large N expansion of the free energy for the general 2-cut case. From it we rederive by a direct and easy mean-field-like method the 2-point correlators and the asymptotic orthogonal polynomials. We extend our results to any number of cuts and to non-real potentials.Comment: 35 pages, Latex (1 file) + 3 figures (3 .eps files), revised to take into account a few reference

A Brownian Motion Model of Parametric Correlations in Ballistic Cavities

A Brownian motion model is proposed to study parametric correlations in the transmission eigenvalues of open ballistic cavities. We find interesting universal properties when the eigenvalues are rescaled at the hard edge of the spectrum. We derive a formula for the power spectrum of the fluctuations of transport observables as a response to an external adiabatic perturbation. Our formula correctly recovers the Lorentzian-squared behaviour obtained by semiclassical approaches for the correlation function of conductance fluctuations.Comment: 19 pages, written in RevTe

Effective Lagrangians and Chiral Random Matrix Theory

Recently, sum rules were derived for the inverse eigenvalues of the Dirac operator. They were obtained in two different ways: i) starting from the low-energy effective Lagrangian and ii) starting from a random matrix theory with the symmetries of the Dirac operator. This suggests that the effective theory can be obtained directly from the random matrix theory. Previously, this was shown for three or more colors with fundamental fermions. In this paper we construct the effective theory from a random matrix theory for two colors in the fundamental representation and for an arbitrary number of colors in the adjoint representation. We construct a fermionic partition function for Majorana fermions in Euclidean space time. Their reality condition is formulated in terms of complex conjugation of the second kind.Comment: 27 page