Spectral statistics for unitary transfer matrices of binary graphs

Quantum graphs have recently been introduced as model systems to study the spectral statistics of linear wave problems with chaotic classical limits. It is proposed here to generalise this approach by considering arbitrary, directed graphs with unitary transfer matrices. An exponentially increasing contribution to the form factor is identified when performing a diagonal summation over periodic orbit degeneracy classes. A special class of graphs, so-called binary graphs, is studied in more detail. For these, the conditions for periodic orbit pairs to be correlated (including correlations due to the unitarity of the transfer matrix) can be given explicitly. Using combinatorial techniques it is possible to perform the summation over correlated periodic orbit pair contributions to the form factor for some low--dimensional cases. Gradual convergence towards random matrix results is observed when increasing the number of vertices of the binary graphs.Comment: 18 pages, 8 figure

Binary tree summation Monte Carlo simulation for Potts models

In this talk, we briefly comment on Sweeny and Gliozzi methods, cluster Monte Carlo method, and recent transition matrix Monte Carlo for Potts models. We mostly concentrate on a new algorithm known as "binary tree summation". Some of the most interesting features of this method will be highlighted - such as simulating fractional number of Potts states, as well as offering the partition function and thermodynamic quantities as functions of temperature in a single run.Comment: 9 pages, 2 figures, for StatPhys-Taiwan 2002 conferenc

Compactness properties of weighted summation operators on trees - the critical case

The aim of this paper is to provide upper bounds for the entropy numbers of summation operators on trees in a critical case. In a recent paper [10] we elaborated a framework of weighted summation operators on general trees where we related the entropy of the operator with those of the underlying tree equipped with an appropriate metric. However, the results were left incomplete in a critical case of the entropy behavior, because this case requires much more involved techniques. In the present article we fill the gap left open in [10]. To this end we develop a method, working in the context of general trees and general weighted summation operators, which was recently proposed in [9] for a particular critical operator on the binary tree. Those problems appeared in natural way during the study of compactness properties of certain Volterra integral operators in a critical case

Long-Term Evolution of Massive Black Hole Binaries. II. Binary Evolution in Low-Density Galaxies

We use direct-summation N-body integrations to follow the evolution of binary black holes at the centers of galaxy models with large, constant-density cores. Particle numbers as large as 400K are considered. The results are compared with the predictions of loss-cone theory, under the assumption that the supply of stars to the binary is limited by the rate at which they can be scattered into the binary's influence sphere by gravitational encounters. The agreement between theory and simulation is quite good; in particular, we are able to quantitatively explain the observed dependence of binary hardening rate on N. We do not verify the recent claim of Chatterjee, Hernquist & Loeb (2003) that the hardening rate of the binary stabilizes when N exceeds a particular value, or that Brownian wandering of the binary has a significant effect on its evolution. When scaled to real galaxies, our results suggest that massive black hole binaries in gas-poor nuclei would be unlikely to reach gravitational-wave coalescence in a Hubble time.Comment: 13 pages, 8 figure

Psychometric precision in phenotype definition is a useful step in molecular genetic investigation of psychiatric disorders

Affective disorders are highly heritable, but few genetic risk variants have been consistently replicated in molecular genetic association studies. The common method of defining psychiatric phenotypes in molecular genetic research is either a summation of symptom scores or binary threshold score representing the risk of diagnosis. Psychometric latent variable methods can improve the precision of psychiatric phenotypes, especially when the data structure is not straightforward. Using data from the British 1946 birth cohort, we compared summary scores with psychometric modeling based on the General Health Questionnaire (GHQ-28) scale for affective symptoms in an association analysis of 27 candidate genes (249 single-nucleotide polymorphisms (SNPs)). The psychometric method utilized a bi-factor model that partitioned the phenotype variances into five orthogonal latent variable factors, in accordance with the multidimensional data structure of the GHQ-28 involving somatic, social, anxiety and depression domains. Results showed that, compared with the summation approach, the affective symptoms defined by the bi-factor psychometric model had a higher number of associated SNPs of larger effect sizes. These results suggest that psychometrically defined mental health phenotypes can reflect the dimensions of complex phenotypes better than summation scores, and therefore offer a useful approach in genetic association investigations