Hundred Thousand Degree Gas in the Virgo Cluster of Galaxies

The physical relationship between low-excitation gas filaments at ~10^4 K, seen in optical line emission, and diffuse X-ray emitting coronal gas at ~10^7 K in the centers of many galaxy clusters is not understood. It is unclear whether the ~10^4 K filaments have cooled and condensed from the ambient hot (~10^7 K) medium or have some other origin such as the infall of cold gas in a merger, or the disturbance of an internal cool reservoir of gas by nuclear activity. Observations of gas at intermediate temperatures (~10^5-10^6 K) can potentially reveal whether the central massive galaxies are gaining cool gas through condensation or losing it through conductive evaporation and hence identify plausible scenarios for transport processes in galaxy cluster gas. Here we present spectroscopic detection of ~10^5 K gas spatially associated with the H-alpha filaments in a central cluster galaxy, M87 in the Virgo Cluster. The measured emission-line fluxes from triply ionized carbon (CIV 1549 A) and singly ionized helium (HeII 1640 A) are consistent with a model in which thermal conduction determines the interaction between hot and cold phases.Comment: 10 pages, 2 figures; to appear in ApJ

Continuous Association Schemes and Hypergroups

Classical finite association schemes lead to a finite-dimensional algebras which are generated by finitely many stochastic matrices. Moreover, there exist associated finite hypergroups. The notion of classical discrete association schemes can be easily extended to the possibly infinite case. Moreover, the notion of association schemes can be relaxed slightly by using suitably deformed families of stochastic matrices by skipping the integrality conditions. This leads to larger class of examples which are again associated to discrete hypergroups. In this paper we propose a topological generalization of the notion of association schemes by using a locally compact basis space $X$ and a family of Markov-kernels on $X$ indexed by a further locally compact space $D$ where the supports of the associated probability measures satisfy some partition property. These objects, called continuous association schemes, will be related to hypergroup structures on $D$. We study some basic results for this new notion and present several classes of examples. It turns out that for a given commutative hypergroup the existence of an associated continuous association scheme implies that the hypergroup has many features of a double coset hypergroup. We in particular show that commutative hypergroups, which are associated with commutative continuous association schemes, carry dual positive product formulas for the characters. On the other hand, we prove some rigidity results in particular in the compact case which say that for given spaces $X,D$ there are only a few continuous association schemes

Dynamical correlation functions of one-dimensional superconductors and Peierls and Mott insulators

I construct the spectral function of the Luther-Emery model which describes one-dimensional fermions with one gapless and one gapped degree of freedom, i.e. superconductors and Peierls and Mott insulators, by using symmetries, relations to other models, and known limits. Depending on the relative magnitudes of the charge and spin velocities, and on whether a charge or a spin gap is present, I find spectral functions differing in the number of singularities and presence or absence of anomalous dimensions of fermion operators. I find, for a Peierls system, one singularity with anomalous dimension and one finite maximum; for a superconductor two singularities with anomalous dimensions; and for a Mott insulator one or two singularities without anomalous dimension. In addition, there are strong shadow bands. I generalize the construction to arbitrary dynamical multi-particle correlation functions. The main aspects of this work are in agreement with numerical and Bethe Ansatz calculations by others. I also discuss the application to photoemission experiments on 1D Mott insulators and on the normal state of 1D Peierls systems, and propose the Luther-Emery model as the generic description of 1D charge density wave systems with important electronic correlations.Comment: Revtex, 27 pages, 5 figures, to be published in European Physical Journal

Central limit theorems for multivariate Bessel processes in the freezing regime

Multivariate Bessel processes are classified via associated root systems and positive multiplicity constants. They describe the dynamics of interacting particle systems of Calogero-Moser-Sutherland type. Recently, Andraus, Katori, and Miyashita derived some weak laws of large numbers for these processes for fixed positive times and multiplicities tending to infinity. In this paper we derive associated central limit theorems for the root systems of types A, B and D in an elementary way. In most cases, the limits will be normal distributions, but in the B-case there are freezing limits where distributions associated with the root system A or one-sided normal distributions on half-spaces appear. Our results are connected to central limit theorems of Dumitriu and Edelman for beta-Hermite and beta-Laguerre ensembles

A brief introduction to Luttinger liquids

I give a brief introduction to Luttinger liquids. Luttinger liquids are paramagnetic one-dimensional metals without Landau quasi-particle excitations. The elementary excitations are collective charge and spin modes, leading to charge-spin separation. Correlation functions exhibit power-law behavior. All physical properties can be calculated, e.g. by bosonization, and depend on three parameters only: the renormalized coupling constant $K_{\rho}$, and the charge and spin velocities. I also discuss the stability of Luttinger liquids with respect to temperature, interchain coupling, lattice effects and phonons, and list important open problems.Comment: 10 pages, 2 figures, to be published in the Proceedings of the International Winterschool on Electronic Properties of Novel Materials 2000, Kirchberg, March 4-11, 200

Product formulas for a two-parameter family of Heckman-Opdam hypergeometric functions of type BC

In this paper we present explicit product formulas for a continuous two-parameter family of Heckman-Opdam hypergeometric functions of type BC on Weyl chambers $C_q\subset \mathbb R^q$ of type $B$. These formulas are related to continuous one-parameter families of probability-preserving convolution structures on $C_q\times\mathbb R$. These convolutions on $C_q\times\mathbb R$ are constructed via product formulas for the spherical functions of the symmetric spaces $U(p,q)/ (U(p)\times SU(q))$ and associated double coset convolutions on $C_q\times\mathbb T$ with the torus $\mathbb T$. We shall obtain positive product formulas for a restricted parameter set only, while the associated convolutions are always norm-decreasing. Our paper is related to recent positive product formulas of R\"osler for three series of Heckman-Opdam hypergeometric functions of type BC as well as to classical product formulas for Jacobi functions of Koornwinder and Trimeche for rank $q=1$

Dispersion and limit theorems for random walks associated with hypergeometric functions of type BC

The spherical functions of the noncompact Grassmann manifolds $G_{p,q}(\mathbb F)=G/K$ over the (skew-)fields $\mathbb F=\mathbb R, \mathbb C, \mathbb H$ with rank $q\ge1$ and dimension parameter $p>q$ can be described as Heckman-Opdam hypergeometric functions of type BC, where the double coset space $G//K$ is identified with the Weyl chamber $C_q^B\subset \mathbb R^q$ of type B. The corresponding product formulas and Harish-Chandra integral representations were recently written down by M. R\"osler and the author in an explicit way such that both formulas can be extended analytically to all real parameters $p\in[2q-1,\infty[$, and that associated commutative convolution structures $*_p$ on $C_q^B$ exist. In this paper we introduce moment functions and the dispersion of probability measures on $C_q^B$ depending on $*_p$ and study these functions with the aid of this generalized integral representation. Moreover, we derive strong laws of large numbers and central limit theorems for associated time-homogeneous random walks on $(C_q^B, *_p)$ where the moment functions and the dispersion appear in order to determine drift vectors and covariance matrices of these limit laws explicitely. For integers $p$, all results have interpretations for $G$-invariant random walks on the Grassmannians $G/K$. Besides the BC-cases we also study the spaces $GL(q,\mathbb F)/U(q,\mathbb F)$, which are related to Weyl chambers of type A, and for which corresponding results hold. For the rank-one-case $q=1$, the results of this paper are well-known in the context of Jacobi-type hypergroups on $[0,\infty[$.Comment: Extended version of arXiv:1205.4866; some corrections to prior version. Accepted for publication in J. Theor. Proba