Non-Commutativity of the Zero Chemical Potential Limit and the Thermodynamic Limit in Finite Density Systems

Monte Carlo simulations of finite density systems are often plagued by the complex action problem. We point out that there exists certain non-commutativity in the zero chemical potential limit and the thermodynamic limit when one tries to study such systems by reweighting techniques. This is demonstrated by explicit calculations in a Random Matrix Theory, which is thought to be a simple qualitative model for finite density QCD. The factorization method allows us to understand how the non-commutativity, which appears at the intermediate steps, cancels in the end results for physical observables.Comment: 7 pages, 9 figure

Symmetry classes of disordered fermions

Building upon Dyson's fundamental 1962 article known in random-matrix theory as 'the threefold way', we classify disordered fermion systems with quadratic Hamiltonians by their unitary and antiunitary symmetries. Important examples are afforded by noninteracting quasiparticles in disordered metals and superconductors, and by relativistic fermions in random gauge field backgrounds. The primary data of the classification are a Nambu space of fermionic field operators which carry a representation of some symmetry group. Our approach is to eliminate all of the unitary symmetries from the picture by transferring to an irreducible block of equivariant homomorphisms. After reduction, the block data specifying a linear space of symmetry-compatible Hamiltonians consist of a basic vector space V, a space of endomorphisms in End(V+V*), a bilinear form on V+V* which is either symmetric or alternating, and one or two antiunitary symmetries that may mix V with V*. Every such set of block data is shown to determine an irreducible classical compact symmetric space. Conversely, every irreducible classical compact symmetric space occurs in this way. This proves the correspondence between symmetry classes and symmetric spaces conjectured some time ago.Comment: 52 pages, dedicated to Freeman J. Dyson on the occasion of his 80th birthda

Effect of Spatial Inhomogeneities on the Membrane Surface on Receptor Dimerization and Signal Initiation

Important signal transduction pathways originate on the plasma membrane, where microdomains may transiently entrap diffusing receptors. This results in a non-random distribution of receptors even in the resting state, which can be visualized as “clusters” by high resolution imaging methods. Here, we explore how spatial in-homogeneities in the plasma membrane might influence the dimerization and phosphorylation status of ErbB2 and ErbB3, two receptor tyrosine kinases that preferentially heterodimerize and are often co-expressed in cancer. This theoretical study is based upon spatial stochastic simulations of the two-dimensional membrane landscape, where variables include differential distributions and overlap of transient confinement zones (“domains”) for the two receptor species. The in silico model is parameterized and validated using data from single particle tracking experiments. We report key differences in signaling output based on the degree of overlap between domains and the relative retention of receptors in such domains, expressed as escape probability. Results predict that a high overlap of domains, which favors transient co-confinement of both receptor species, will enhance the rate of hetero-interactions. Where domains do not overlap, simulations confirm expectations that homo-interactions are favored. Since ErbB3 is uniquely dependent on ErbB2 interactions for activation of its catalytic activity, variations in domain overlap or escape probability markedly alter the predicted patterns and time course of ErbB3 and ErbB2 phosphorylation. Taken together, these results implicate membrane domain organization as an important modulator of signal initiation, motivating the design of novel experimental approaches to measure these important parameters across a wider range of receptor systems

Non-Hermitian Random Matrix Theory and Lattice QCD with Chemical Potential

In quantum chromodynamics (QCD) at nonzero chemical potential, the eigenvalues of the Dirac operator are scattered in the complex plane. Can the fluctuation properties of the Dirac spectrum be described by universal predictions of non-Hermitian random matrix theory? We introduce an unfolding procedure for complex eigenvalues and apply it to data from lattice QCD at finite chemical potential $\mu$ to construct the nearest-neighbor spacing distribution of adjacent eigenvalues in the complex plane. For intermediate values of $\mu$, we find agreement with predictions of the Ginibre ensemble of random matrix theory, both in the confinement and in the deconfinement phase.Comment: 4 pages, 3 figures, to appear in Phys. Rev. Let

Impossibility of spontaneously breaking local symmetries and the sign problem

Elitzur's theorem stating the impossibility of spontaneous breaking of local symmetries in a gauge theory is reexamined. The existing proofs of this theorem rely on gauge invariance as well as positivity of the weight in the Euclidean partition function. We examine the validity of Elitzur's theorem in gauge theories for which the Euclidean measure of the partition function is not positive definite. We find that Elitzur's theorem does not follow from gauge invariance alone. We formulate a general criterion under which spontaneous breaking of local symmetries in a gauge theory is excluded. Finally we illustrate the results in an exactly solvable two dimensional abelian gauge theory.Comment: Latex 6 page

Fermion determinants in matrix models of QCD at nonzero chemical potential

The presence of a chemical potential completely changes the analytical structure of the QCD partition function. In particular, the eigenvalues of the Dirac operator are distributed over a finite area in the complex plane, whereas the zeros of the partition function in the complex mass plane remain on a curve. In this paper we study the effects of the fermion determinant at nonzero chemical potential on the Dirac spectrum by means of the resolvent, G(z), of the QCD Dirac operator. The resolvent is studied both in a one-dimensional U(1) model (Gibbs model) and in a random matrix model with the global symmetries of the QCD partition function. In both cases we find that, if the argument z of the resolvent is not equal to the mass m in the fermion determinant, the resolvent diverges in the thermodynamic limit. However, for z =m the resolvent in both models is well defined. In particular, the nature of the limit $z \rightarrow m$ is illuminated in the Gibbs model. The phase structure of the random matrix model in the complex m and \mu-planes is investigated both by a saddle point approximation and via the distribution of Yang-Lee zeros. Both methods are in complete agreement and lead to a well-defined chiral condensate and quark number density.Comment: 27 pages, 6 figures, Late

Detecting transcriptionally active regions using genomic tiling arrays

We have developed a method for interpreting genomic tiling array data, implemented as the program TranscriptionDetector. Probed loci expressed above background are identified by combining replicates in a way that makes minimal assumptions about the data. We performed medium-resolution Anopheles gambiae tiling array experiments and found extensive transcription of both coding and non-coding regions. Our method also showed improved detection of transcriptional units when applied to high-density tiling array data for ten human chromosomes

The Fractal Geometry of Critical Systems

We investigate the geometry of a critical system undergoing a second order thermal phase transition. Using a local description for the dynamics characterizing the system at the critical point T=Tc, we reveal the formation of clusters with fractal geometry, where the term cluster is used to describe regions with a nonvanishing value of the order parameter. We show that, treating the cluster as an open subsystem of the entire system, new instanton-like configurations dominate the statistical mechanics of the cluster. We study the dependence of the resulting fractal dimension on the embedding dimension and the scaling properties (isothermal critical exponent) of the system. Taking into account the finite size effects we are able to calculate the size of the critical cluster in terms of the total size of the system, the critical temperature and the effective coupling of the long wavelength interaction at the critical point. We also show that the size of the cluster has to be identified with the correlation length at criticality. Finally, within the framework of the mean field approximation, we extend our local considerations to obtain a global description of the system.Comment: 1 LaTeX file, 4 figures in ps-files. Accepted for publication in Physical Review