Correlations of RMT Characteristic Polynomials and Integrability: Hermitean Matrices

Integrable theory is formulated for correlation functions of characteristic polynomials associated with invariant non-Gaussian ensembles of Hermitean random matrices. By embedding the correlation functions of interest into a more general theory of tau-functions, we (i) identify a zoo of hierarchical relations satisfied by tau-functions in an abstract infinite-dimensional space, and (ii) present a technology to translate these relations into hierarchically structured nonlinear differential equations describing the correlation functions of characteristic polynomials in the physical, spectral space. Implications of this formalism for fermionic, bosonic, and supersymmetric variations of zero-dimensional replica field theories are discussed at length. A particular emphasis is placed on the phenomenon of fermionic-bosonic factorisation of random-matrix-theory correlation functions.Comment: 62 pages, 1 table, published version (typos corrected

Abundance of unknots in various models of polymer loops

A veritable zoo of different knots is seen in the ensemble of looped polymer chains, whether created computationally or observed in vitro. At short loop lengths, the spectrum of knots is dominated by the trivial knot (unknot). The fractional abundance of this topological state in the ensemble of all conformations of the loop of $N$ segments follows a decaying exponential form, $\sim \exp (-N/N_0)$, where $N_0$ marks the crossover from a mostly unknotted (ie topologically simple) to a mostly knotted (ie topologically complex) ensemble. In the present work we use computational simulation to look closer into the variation of $N_0$ for a variety of polymer models. Among models examined, $N_0$ is smallest (about 240) for the model with all segments of the same length, it is somewhat larger (305) for Gaussian distributed segments, and can be very large (up to many thousands) when the segment length distribution has a fat power law tail.Comment: 13 pages, 6 color figure

Radio Galaxy Zoo: Cosmological Alignment of Radio Sources

We study the mutual alignment of radio sources within two surveys, FIRST and TGSS. This is done by producing two position angle catalogues containing the preferential directions of respectively $30\,059$ and $11\,674$ extended sources distributed over more than $7\,000$ and $17\,000$ square degrees. The identification of the sources in the FIRST sample was performed in advance by volunteers of the Radio Galaxy Zoo project, while for the TGSS sample it is the result of an automated process presented here. After taking into account systematic effects, marginal evidence of a local alignment on scales smaller than $2.5\deg$ is found in the FIRST sample. The probability of this happening by chance is found to be less than $2$ per cent. Further study suggests that on scales up to $1.5\deg$ the alignment is maximal. For one third of the sources, the Radio Galaxy Zoo volunteers identified an optical counterpart. Assuming a flat $\Lambda$CDM cosmology with $\Omega_m = 0.31, \Omega_\Lambda = 0.69$, we convert the maximum angular scale on which alignment is seen into a physical scale in the range $[19, 38]$ Mpc $h_{70}^{-1}$. This result supports recent evidence reported by Taylor and Jagannathan of radio jet alignment in the $1.4$ deg$^2$ ELAIS N1 field observed with the Giant Metrewave Radio Telescope. The TGSS sample is found to be too sparsely populated to manifest a similar signal