Wilson Fermions on a Randomly Triangulated Manifold

A general method of constructing the Dirac operator for a randomly triangulated manifold is proposed. The fermion field and the spin connection live, respectively, on the nodes and on the links of the corresponding dual graph. The construction is carried out explicitly in 2-d, on an arbitrary orientable manifold without boundary. It can be easily converted into a computer code. The equivalence, on a sphere, of Majorana fermions and Ising spins in 2-d is rederived. The method can, in principle, be extended to higher dimensions.Comment: 18 pages, latex, 6 eps figures, fig2 corrected, Comment added in the conclusion sectio

The Cost of Address Translation

Modern computers are not random access machines (RAMs). They have a memory hierarchy, multiple cores, and virtual memory. In this paper, we address the computational cost of address translation in virtual memory. Starting point for our work is the observation that the analysis of some simple algorithms (random scan of an array, binary search, heapsort) in either the RAM model or the EM model (external memory model) does not correctly predict growth rates of actual running times. We propose the VAT model (virtual address translation) to account for the cost of address translations and analyze the algorithms mentioned above and others in the model. The predictions agree with the measurements. We also analyze the VAT-cost of cache-oblivious algorithms.Comment: A extended abstract of this paper was published in the proceedings of ALENEX13, New Orleans, US

Evidence for Asymptotic Safety from Dimensional Reduction in Causal Dynamical Triangulations

We calculate the spectral dimension for a nonperturbative lattice approach to quantum gravity, known as causal dynamical triangulations (CDT), showing that the dimension of spacetime smoothly decreases from approximately 4 on large distance scales to approximately 3/2 on small distance scales. This novel result may provide a possible resolution to a long-standing argument against the asymptotic safety scenario. A method for determining the relative lattice spacing within the physical phase of the CDT parameter space is also outlined, which might prove useful when studying renormalization group flow in models of lattice quantum gravity.Comment: 21 pages, 8 figures, 4 tables. Typos corrected, 3 tables added. Conclusions unchanged. Conforms with version published in JHE

Eigenvalue density of empirical covariance matrix for correlated samples

We describe a method to determine the eigenvalue density of empirical covariance matrix in the presence of correlations between samples. This is a straightforward generalization of the method developed earlier by the authors for uncorrelated samples. The method allows for exact determination of the experimental spectrum for a given covariance matrix and given correlations between samples in the limit of large N and N/T=r=const with N being the number of degrees of freedom and T being the number of samples. We discuss the effect of correlations on several examples.Comment: 12 pages, 5 figures, to appear in Acta Phys. Pol. B (Proceedings of the conference on `Applications of Random Matrix Theory to Economy and Other Complex Systems', May 25-28, 2005, Cracow, Polan

The Universe from Scratch

A fascinating and deep question about nature is what one would see if one could probe space and time at smaller and smaller distances. Already the 19th-century founders of modern geometry contemplated the possibility that a piece of empty space that looks completely smooth and structureless to the naked eye might have an intricate microstructure at a much smaller scale. Our vastly increased understanding of the physical world acquired during the 20th century has made this a certainty. The laws of quantum theory tell us that looking at spacetime at ever smaller scales requires ever larger energies, and, according to Einstein's theory of general relativity, this will alter spacetime itself: it will acquire structure in the form of "curvature". What we still lack is a definitive Theory of Quantum Gravity to give us a detailed and quantitative description of the highly curved and quantum-fluctuating geometry of spacetime at this so-called Planck scale. - This article outlines a particular approach to constructing such a theory, that of Causal Dynamical Triangulations, and its achievements so far in deriving from first principles why spacetime is what it is, from the tiniest realms of the quantum to the large-scale structure of the universe.Comment: 31 pages, 5 figures; review paper commissioned by Contemporary Physics and aimed at a wider physics audience; minor beautifications, coincides with journal versio

Correlation functions and critical behaviour on fluctuating geometries

We study the two-point correlation function in the model of branched polymers and its relation to the critical behaviour of the model. We show that the correlation function has a universal scaling form in the generic phase with the only scale given by the size of the polymer. We show that the origin of the singularity of the free energy at the critical point is different from that in the standard statistical models. The transition is related to the change of the dimensionality of the system.Comment: 10 Pages, Latex2e, uses elsart.cls, 1 figure include