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Two different matrix models for QCD with a non-vanishing quark chemical potential are shown to be equivalent by mapping the corresponding partition functions. The equivalence holds in the phase with broken chiral symmetry. It is exact in the limit of weak non-Hermiticity, where the chemical potential squared is rescaled with the volume. At strong non-Hermiticity it holds only for small chemical potential. The first model proposed by Stephanov is directly related to QCD and allows to analyze the QCD phase diagram. In the second model suggested by the author all microscopic spectral correlation functions of complex Dirac operators can be calculated in the broken phase. We briefly compare those predictions to complex Dirac eigenvalues from quenched QCD lattice simulations.

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The asymmetric ABAB-matrix model describes the transfer matrix of three-dimensional Lorentzian quantum gravity. We study perturbatively the scaling of the ABAB-matrix model in the neighborhood of it’s symmetric solution and deduce the associated renormalization of three-dimensional Lorentzian quantum gravity.

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The density of states contains all informations on energetic quantities of a statistical system, such as the mean energy, free energy, entropy, and specific heat. As a specific application, we consider in this work a simple lattice model for heteropolymers that is widely used for studying statistical properties of proteins. For short chains, we have derived exact results from conformational enumeration, while for longer ones we developed a multicanonical Monte Carlo variant of the nPERM-based chain growth method in order to directly simulate the density of states. For simplification, only two types of monomers with respective hydrophobic (H) and polar (P) residues are regarded and only the next-neighbour interaction between hydrophobic monomers, being nonadjacent along the chain, is taken into account. This is known as the HP model for the folding of lattice proteins.

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We discuss the geometry of trees endowed with a causal structure using the conventional framework of equilibrium statistical mechanics. We show how this ensemble is related to popular growing network models. In particular we demonstrate that on a class of afine attachment kernels the two models are identical but they can differ substantially for other choice of weights. We show that causal trees exhibit condensation even for asymptotically linear kernels. We derive general formulae describing the degree distribution, the ancestor–descendant correlation and the probability that a randomly chosen node lives at a given geodesic distance from the root. It is shown that the Hausdorff dimension \(d_{\rm H}\) of the causal networks is generically infinite.

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In the recent years, field theory on non-commutative (NC) spaces has attracted a lot of attention. Most literature on this subject deals with perturbation theory, although the latter runs into grave problems beyond one loop. Here we present results from a fully non-perturbative approach. In particular, we performed numerical simulations of the \(\lambda \phi ^{4}\) model with two NC spatial coordinates, and a commutative Euclidean time. This theory is lattice discretized and then mapped onto a matrix model. The simulation results reveal a phase diagram with various types of ordered phases. We discuss the suitable order parameters, as well as the spatial and temporal correlators. The dispersion relation clearly shows a trend towards the expected IR singularity. Its parameterization provides the tool to extract the continuum limit.

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We study the complex \(|\psi |^4\) theory in three dimensions and compare our numerical results with a recently proposed mean-field like approximation. The mean-field result, which predicts a first-order phase transition in parts of the phase diagram, cannot be confirmed. To get a closer look inside this discrepancy, we introduce a generalized Hamiltonian with an additional fugacity term. With this modification we can show that the complex \(|\psi |^4\) theory can indeed be tuned to undergo a first-order phase transition by varying the strength of the new term in the generalized Hamiltonian.

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We implement fermions on 2D dynamical random triangulation and determine the spectrum of the Dirac operator. We study the dependence of the spectrum on the hopping parameter and use finite size analysis to determine critical exponents. The results for regular, for Euclidean and for Lorentzian lattices are briefly presented.

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In a given market, financial covariances capture the intra-stock correlations and can be used to address statistically the bulk nature of the market as a complex system. We provide a statistical analysis of three SP500 covariances with evidence for raw tail distributions. We study the stability of these tails against reshuffling for the SP500 data and show that the covariance with the strongest tails is robust, with a spectral density in remarkable agreement with random Lévy matrix theory. We study the inverse participation ratio for the three covariances. The strong localization observed at both ends of the spectral density is analogous to the localization exhibited in the random Lévy matrix ensemble. We discuss two competitive mechanisms responsible for the occurrence of an extensive and delocalized eigenvalue at the edge of the spectrum: (a) the Lévy character of the entries of the correlation matrix and (b) a sort of off-diagonal order induced by underlying inter-stock correlations. (b) can be destroyed by reshuffling, while (a) cannot. We show that the stocks with the largest scattering are the least susceptible to correlations, and likely candidates for the localized states. We introduce a simple model for price fluctuations which captures behavior of the SP500 covariances. It may be of importance for assets diversification.

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We derive analytically one-loop corrections to the effective Polyakov-line operator in the branched-polymer approximation of the reduced four-dimensional supersymmetric Yang–Mills integrals.

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We review recent results concerning finite size corrections to the Ising model free energy in lattices with non-trivial topology and curvature. From conformal field theory considerations two distinct universal terms are expected, a logarithmic term determined by the system curvature and a scale invariant term determined by the system shape and topology. Both terms have been observed numerically, using the Kasteleyn Pfaffian method, for lattices with topologies ranging from the sphere to that of a genus two surface. The constant term is shown to be expressible in terms of Riemann theta functions while the logarithmic correction reproduces the theoretical prediction by Cardy and Peschel for singular metrics.

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We present a simple construction of a probability measure on rooted infinite planar trees as a limit of a sequence of uniform measures on finite trees. We compute the conditional probability measure on the set of trees containing a given finite tree and use this to determine the distribution of the number of vertices at a given distance from the root, and thereby the Hausdorff dimension associated with this measure. The construction can be generalised to other ensembles of infinite discrete structures. We indicate, in particular, how it can be adapted in a straight forward manner to obtain a probability measure on infinite planar surfaces by using a certain correspondence between quadrangulated surfaces and so-called well labelled trees. The Hausdorff dimension of this measure turns out to be \(4\). Details of these latter results will appear elsewhere.

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QCD can be described in a certain kinematical regime by an effective string theory. This string must couple to background chiral fields in a chirally invariant manner, thus taking into account the true chirally non-invariant QCD vacuum. By requiring conformal symmetry of the string and the unitarity constraint on chiral fields we reconstruct the equations of motion for the latter ones. These provide a consistent background for the propagation of the string. By further requiring locality of the effective action we recover the Lagrangian of non-linear sigma model of pion interactions. The prediction is unambiguous and parameter-free. The estimated chiral structural constants of Gasser and Leutwyler fit very well the phenomenological values.

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We present the one and two-loop quantum corrections to the earlier proposed string theory whose world-sheet action measures the linear sizes of the surfaces by the square root of the extrinsic curvature. We find in this model the usual conformal anomaly. Moreover, the one-loop perturbative analysis shows that the dynamics of this model is determined by a reduced number of degrees of freedom compared to the usual string. We point out that this model does not receive any quantum corrections around its classical trajectory. Finally we show that the constraint, relating the induced metric with the string fields, is enforced by radiative corrections and it does not allow the generation of the Polyakov–Kleinert smooth string.

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We describe a matrix analogy of the log-normal random walk, in the large \(N\) (size of the matrix) limit. In particular, we present an exact result for the infinite product of random matrices, corresponding to the multiplicative diffusion triggered by Ginibre–Girko ensemble. We observe the emergence of a “topological phase transition”, when a hole develops in the complex eigenvalue spectrum, after some critical diffusion time \(\tau _{\rm crit}\) is reached.

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Noncommutative conformal transformations are constructed on noncommutative \(\mathbb {R}^4\) and used to derive the Seiberg–Witten differential equation.

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Many small proteins fold in a two-state manner, the rate-limiting step being the passage of the free-energy barrier separating the unfolded state from the native one. The free-energy barrier is, however, weak or absent for the fastest-folding proteins. Here a simple diffusion picture for such proteins is discussed. It is tested on a model protein that makes a three-helix bundle. Assuming the motion along individual reaction coordinates to be diffusive on timescales beyond the reconfiguration time for a single helix, it is found that the relaxation time can be predicted within a factor of two. It is also shown that melting curves for this protein to a good approximation can be described in terms of a simple two-state system, despite the absence of a clear free-energy barrier.

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In this talk I describe some applications of random matrix models to the study of \({\cal N}\!=\!1\) supersymmetric Yang–Mills theories with matter fields in the fundamental representation. I review the derivation of the Veneziano–Yankielowicz–Taylor/Affleck–Dine–Seiberg superpotentials from constrained random matrix models (hep-th/0211082), a field theoretical justification of the logarithmic matter contribution to the Veneziano–Yankielowicz–Taylor superpotential (hep-th/0306242) and the random matrix based solution of the complete factorization problem of Seiberg–Witten curves for \({\cal N}=2\) theories with fundamental matter (hep-th/0212212).

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The relevance of quenched, uncorrelated disorder coupling to the local energy density, its paradigm being the random-bond model, is judged by the Harris criterion. A generalization of the underlying argument to the case of spatially correlated disorder, exemplified by quasi-crystals, has been given by Luck. We address the question, whether a relevance criterion of this type is applicable to the case of spin models coupled to different kinds of random graphs. The geometrical fluctuation exponent appearing in Luck’s criterion is precisely determined for the cases of two-dimensional Poissonian Voronoï–Delaunay random lattices and planar, “fat” \(\phi ^3\) Feynman diagrams. While previous work for the latter graphs is in accord with the determined relevance threshold, a preliminary analysis of the results of a Monte Carlo simulation of the three-states Potts model on Poissonian Voronoï lattices presented here does not meet the expectations from the relevance criterion.

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We present results of high-statistics Monte Carlo simulations of the three-dimensional Edwards–Anderson Ising spin-glass model. The study is performed with the multi-overlap algorithm, a non-Boltzmann sampling technique which is specifically tailored for sampling rare-event states. This enabled us to study the free-energy barriers \(F^q_{\rm B}\) in the probability densities \(P_J(q)\) of the Parisi overlap parameter \(q\) and the far tail region of the disorder averaged density \(P(q) = [P_J(q)]_{\rm av}\). In the latter case we find support for extreme order statistics over many orders of magnitude. A comparative study of the three-dimensional pure Ising model shows that this property is special to spin glasses.

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Although the notion of entropy lies at the core of statistical mechanics, it is not often used in statistical mechanical models to characterize phase transitions, a role more usually played by quantities such as various order parameters, specific heats or susceptibilities. The relative entropy induces a metric, the so-called information or Fisher–Rao metric, on the space of parameters and the geometrical invariants of this metric carry information about the phase structure of the model. In various models the scalar curvature, \({\cal R}\), of the information metric has been found to diverge at the phase transition point and a plausible scaling relation postulated. For spin models the necessity of calculating in non-zero field has limited analytic consideration to one-dimensional, mean-field and Bethe lattice Ising models. We extend the list somewhat in the current note by considering the one-dimensional Potts model, the two-dimensional Ising model coupled to two-dimensional quantum gravity and the three-dimensional spherical model. We note that similar ideas have been applied to elucidate possible critical behaviour in families of black hole solutions in four space-time dimensions.

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We discuss geometric paths and review the theory of continuous trees which has been developed in the last 12 years. We explain the relation of continuum trees to the extensively studied discrete trees.

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We review how the identification of gauge theory operators representing string states in the \(pp\)-wave/BMN correspondence and their associated anomalous dimension reduces to the determination of the eigenvectors and the eigenvalues of a simple quantum mechanical Hamiltonian and analyze the properties of this Hamiltonian. Furthermore, we discuss the role of random matrices as a tool for performing explicit evaluation of correlation functions.

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The Krakow-Orsay collaboration has applied methods borrowed from equilibrium statistical mechanics and analytic combinatorics to study the geometry of random networks. Results contained in a series of recent publications and concerning networks that are either uncorrelated or causal are briefly overviewed.

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A unified treatment of the distribution of eigenvalues for different matrix ensembles is given with the help of parametric dynamics of levels and symplectic reduction of it.

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The article presents a selected set of recent results from numerical investigations of QCD at finite temperature. It is focused (i) on the present understanding of thermodynamic properties of QCD in the presence of dynamical quarks at various masses and at small yet phenomenologically relevant values for the baryon density and (ii) on a fairly new approach to studying thermal hadronic excitations.

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It is well-known that the sum over topologies in quantum gravity is ill-defined, due to a super-exponential growth of the number of geometries as a function of the space-time volume, leading to a badly divergent gravitational path integral. Not even in dimension 2, where a non-perturbative quantum gravity theory can be constructed explicitly from a (regularized) path integral, has this problem found a satisfactory solution. In the present work, we extend a previous 2d Lorentzian path integral, regulated in terms of Lorentzian random triangulations, to include space-times with an arbitrary number of handles. We show that after the imposition of physically motivated causality constraints, the combined sum over geometries and topologies is well-defined and possesses a continuum limit which yields a concrete model of space-time foam in two dimensions.

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We discuss some of the similarities between strings and particles with extrinsic curvature. We shall highlight the appearance of extra classical symmetries that appear in particular actions.

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We study the performance of scale free Internet-like networks and compare them to a classical random graph based network. The scaling of the traffic load with the nodal degree is established, and confirmed in a numerical simulation of the TCP traffic. The scaling allows us to estimate the link capacity upgrade required making and extra connection to an existing node.

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The gonihedric spin model was first introduced as the action for a discretized tensionless string in a discretized embedding space. Afterwards was found that there are interesting features on the dynamical behavior of this model in 3 dimensions (as it was first formulated) that make us think on glassy spin model without inherent disorder. Extensive simulations have been carried out in the 3-dimensional model. In the following I will report on a work composed of two different but related parts. The first part is a numerical study through Monte Carlo simulations of the dynamical properties of the 2 dimensional version of the model (i.e. the loop model), which is much simpler due to the fact that it has trivial thermodynamical properties. The second part consists on an analytical approach of this 2-dimensional loop model coupled to gravity. We solve partially the associated two-matrix model via a reduction to an equivalent one matrix model and saddle point methods with the last one-matrix model.

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The Kauffman model describes a particularly simple class of random Boolean networks. Despite the simplicity of the model, it exhibits complex behavior and has been suggested as a model for real world network problems. This work is based on an earlier paper where we introduced a novel approach to analyzing attractors in random Boolean networks. Applying this approach to Kauffman networks, we prove that the average number of attractors grows faster than any power law with system size.

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The duality map between gauge theories and strings suggests that when the gauge theory is in the weak coupling regime the dual string tension effectively tends to zero, \(\alpha ' \rightarrow \infty \). This observation of Sundborg and Witten initiates a fresh interest to the old problem of tensionless limit of standard string theory and to the description of its genuine symmetries. We approach this problem formulating tensionless string theory by means of geometrical concept of surface perimeter. The perimeter action uniquely leads to a tensionless theory.

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We demonstrate how to generalize two of the most well-known random graph models, the classic random graph, and random graphs with a given degree distribution, by the introduction of hidden variables in the form of extra degrees of freedom, color, applied to vertices or stubs (half-edges). The color is assumed unobservable, but is allowed to affect edge probabilities. This serves as a convenient method to define very general classes of models within a common unifying formalism, and allows for a non-trivial edge correlation structure.

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We review the last year progress in understanding supersymmetric SU(2) Yang–Mills quantum mechanics in the \(D=4\) and \(10\) space-time dimensions. The four dimensional system is now well under control and the precise spectrum is obtained in all channels. In \(D=10\) some new results are also available.