New spectral relations between products and powers of isotropic random matrices

We show that the limiting eigenvalue density of the product of n identically distributed random matrices from an isotropic unitary ensemble (IUE) is equal to the eigenvalue density of n-th power of a single matrix from this ensemble, in the limit when the size of the matrix tends to infinity. Using this observation one can derive the limiting density of the product of n independent identically distributed non-hermitian matrices with unitary invariant measures. In this paper we discuss two examples: the product of n Girko-Ginibre matrices and the product of n truncated unitary matrices. We also provide an evidence that the result holds also for isotropic orthogonal ensembles (IOE).Comment: 8 pages, 3 figures (in version 2 we added a figure and discussion on finite size effects for isotropic orthogonal ensemble

Heavy Mesons in a Random Instanton Gas

We analyze the correlation function of a meson with one heavy and one light quark in inverse powers of the heavy quark mass $m_Q$ using a succession of Foldy-Wouthuysen-type transformations prior to radiative corrections. We evaluate the correlator to order $m_Q^0$ in a random and dilute gas of instantons, using the planar approximation. We show, in leading order in the density, that the heavy quark mass is shifted to the order $m_Q^0$ and that the induced interaction between the heavy and light quarks is attractive. We also find it to be an order of magnitude smaller than the 't Hooft interaction between two light quarks. The shift in the heavy quark mass is related to the perimeter law of large Wilson loops. The relevance of these results for general hadronic correlators with heavy quarks is discussed.Comment: 13 pages, SUNY-NTG-94-3

Heavy Hadrons and QCD Instantons

Heavy hadrons are analyzed in a random and dilute gas of instantons. We derive the instanton-induced interactions between heavy and light quarks at next to leading order in the heavy quark mass and in the planar approximation, and discuss their effects on the hadronic spectrum. The role of these interactions in the formation of exotic hadrons is also discussed.Comment: 26 pages, REVTeX, 2 tables, 5 figures, uses FEYNMAN.st

The Penn State - Torun Centre for Astronomy Planet Search stars. II. Lithium abundance analysis of the Red Giant Clump sample

Using the sample of 348 stars from the PennState-Torun Centre for Astronomy Planet Search, for which uniformly determined atmospheric parameters are available, with chemical abundances and rotational velocities presented here, we investigate various channels of Li enrichment in giants. Our work is based on the HET/HRS spectra. The A(Li) was determined from the 670.8nm line, while we use a more extended set of lines for alpha-elements abundances. In a series of K-S tests, we compare Li-rich giants with other stars in the sample. We also use available IR photometric and kinematical data in search for evidence of mass-loss. We investigate properties of the most Li-abundant giants in more detail by using multi-epoch precise radial velocities. We present Li and alpha-elements abundances, as well as vsini for 348 stars. We detected Li in 92 stars, of which 82 are giants. 11 of them show significant Li abundance A(Li)>1.4 and 7 of them are Li-overabundant objects, according to criterion of A(Li)>1.5 and their location on HR diagram, including two giants with Li abundances close to meteoritic level. For another 271 stars, upper limits of A(Li) are presented. We show that Li-rich giants are among the most massive stars from our sample and show larger than average effective temperatures. They are indistinguishable from the complete sample in terms of their distribution of luminosity, [Fe/H], vsini, and alpha-elements abundances. Our results do not point out to one specific Li enrichment mechanism operating in our sample of giants. On the contrary, in some cases, we cannot identify fingerprints of any of known scenarios. We show, however, that the 4 most Li-rich giant in our sample either have low-mass companions or have RV variations at the level of ~100 m/s, which strongly suggests that the presence of companions is an important factor in the Li-enrichment processes in giants.Comment: Accepted for publication in A&A, 13 figures, 11 tables, 26 page

A Proof of Tarski’s Fixed Point Theorem by Application of Galois Connections

Two examples of Galois connections and their dual forms are considered. One of them is applied to formulate a criterion when a given subset of a complete lattice forms a complete lattice. The second, closely related to the first, is used to prove in a short way the Knaster-Tarski’s fixed point theore

A simple toy model for effective restoration of chiral symmetry in excited hadrons

A simple solvable toy model exhibiting effective restoration of chiral symmetry in excited hadrons is constructed. A salient feature is that while physics of the low-lying states is crucially determined by the spontaneous breaking of chiral symmetry, in the high-lying states the effects of chiral symmetry breaking represent only a small correction. Asymptotically the states approach the regime where their properties are determined by the underlying unbroken chiral symmetry.Comment: This is the published version of this paper. Note that the title has changed from earlier versions as has the abstract. The emphasis is slightly different from previous versions but the essential physical content is the sam

Mobility and asymmetry effects in one-dimensional rock-paper-scissors games

As the behavior of a system composed of cyclically competing species is strongly influenced by the presence of fluctuations, it is of interest to study cyclic dominance in low dimensions where these effects are the most prominent. We here discuss rock-paper-scissors games on a one-dimensional lattice where the interaction rates and the mobility can be species dependent. Allowing only single site occupation, we realize mobility by exchanging individuals of different species. When the interaction and swapping rates are symmetric, a strongly enhanced swapping rate yields an increased mixing of the species, leading to a mean-field like coexistence even in one-dimensional systems. This coexistence is transient when the rates are asymmetric, and eventually only one species will survive. Interestingly, in our spatial games the dominating species can differ from the species that would dominate in the corresponding nonspatial model. We identify different regimes in the parameter space and construct the corresponding dynamical phase diagram.Comment: 6 pages, 5 figures, to appear in Physical Review

Anomalous Chiral Fermi Surface

We provide a geometrical argument for the emergence of a Wess-Zumino-Witten (WZW) term for a Fermi surface threaded by a Berry curvature. In the presence of external fields, the gauged WZW term yields a chiral (triangle) anomaly for the fermionic current at the edge of the Fermi surface. Fermion number is conserved though since the Berry curvatures occur always in pairs with opposite (monopole) charge. The anomalous vector and axial currents for a a fermionic fluid at low temperature threaded by pairs of Berry curvatures are discussed. The leading temperature correction to the chiral vortical effect in a slowly rotating Fermi surface threaded by a Berry curvature maybe tied to the gravitational anomaly.Comment: 4 pages; version to appear in PR

Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices (The Extended Version)

We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit, eventually verified numerically. These densities are encoded in the form of the so called M-transforms, for which polynomial equations are found. We exploit the methods of planar diagrammatics, enhanced to the non-Hermitian case, and free random variables, respectively; both are described in the appendices. As particular results of these two main equations, we find the singular behavior of the spectral densities near zero. Moreover, we propose a finite-size form of the spectral density of the product close to the border of its eigenvalues' domain. Also, led by the striking similarity between the two main equations, we put forward a conjecture about a simple relationship between the eigenvalues and singular values of any non-Hermitian random matrix whose spectrum exhibits rotational symmetry around zero.Comment: 50 pages, 8 figures, to appear in the Proceedings of the 23rd Marian Smoluchowski Symposium on Statistical Physics: "Random Matrices, Statistical Physics and Information Theory," September 26-30, 2010, Krakow, Polan