Autocorrelation of Random Matrix Polynomials

We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than large-matrix asymptotic approximations. They also mirror exactly autocorrelation formulae conjectured to hold for L-functions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of L-functions

Scale invariant correlations and the distribution of prime numbers

Negative correlations in the distribution of prime numbers are found to display a scale invariance. This occurs in conjunction with a nonstationary behavior. We compare the prime number series to a type of fractional Brownian motion which incorporates both the scale invariance and the nonstationary behavior. Interesting discrepancies remain. The scale invariance also appears to imply the Riemann hypothesis and we study the use of the former as a test of the latter.Comment: 13 pages, 8 figures, version to appear in J. Phys.

The RTC: A Practical Guide to the Receivership/Conservatorship Process and the Resolution of Failed Thrifts

In response to a growing crisis in the thrift industry, Congress enacted the Financial Institutions Reform, Recovery, and Enforcement Act of 1989 ( FIRREA or Act ). The crisis was evidenced by the failure of over 500 thrifts between 1980 and 1988-more than three and one-half times as many in the previous forty-five years combined. In 1988 alone, the Federal Savings and Loan Insurance Corporation ( FSLIC, which prior to FIRREA insured most of the thrift industry\u27s deposits) merged or liquidated over 200 insolvent thrifts, and the U.S. Government\u27s General Accounting Office ( GAO ) estimated in 1989 that at least 338 additional thrifts were insolvent as of December 31, 1988. Despite the attempted recapitalization of the FSLIC through enactment of the Competitive Equality Banking Act of 1987, the insurance fund held by FSLIC was inadequate to allow the FSLIC and the Federal Home Loan Bank Board ( FHLBB or Bank Board ) to close these insolvent thrifts. As a result, these thrifts continued to operate and to incur massive losses

Packing Returning Secretaries

We study online secretary problems with returns in combinatorial packing domains with $n$ candidates that arrive sequentially over time in random order. The goal is to accept a feasible packing of candidates of maximum total value. In the first variant, each candidate arrives exactly twice. All $2n$ arrivals occur in random order. We propose a simple 0.5-competitive algorithm that can be combined with arbitrary approximation algorithms for the packing domain, even when the total value of candidates is a subadditive function. For bipartite matching, we obtain an algorithm with competitive ratio at least $0.5721 - o(1)$ for growing $n$, and an algorithm with ratio at least $0.5459$ for all $n \ge 1$. We extend all algorithms and ratios to $k \ge 2$ arrivals per candidate. In the second variant, there is a pool of undecided candidates. In each round, a random candidate from the pool arrives. Upon arrival a candidate can be either decided (accept/reject) or postponed (returned into the pool). We mainly focus on minimizing the expected number of postponements when computing an optimal solution. An expected number of $\Theta(n \log n)$ is always sufficient. For matroids, we show that the expected number can be reduced to $O(r \log (n/r))$, where $r \le n/2$ is the minimum of the ranks of matroid and dual matroid. For bipartite matching, we show a bound of $O(r \log n)$, where $r$ is the size of the optimum matching. For general packing, we show a lower bound of $\Omega(n \log \log n)$, even when the size of the optimum is $r = \Theta(\log n)$.Comment: 23 pages, 5 figure

Boundary conditions associated with the Painlev\'e III' and V evaluations of some random matrix averages

In a previous work a random matrix average for the Laguerre unitary ensemble, generalising the generating function for the probability that an interval $(0,s)$ at the hard edge contains $k$ eigenvalues, was evaluated in terms of a Painlev\'e V transcendent in $\sigma$-form. However the boundary conditions for the corresponding differential equation were not specified for the full parameter space. Here this task is accomplished in general, and the obtained functional form is compared against the most general small $s$ behaviour of the Painlev\'e V equation in $\sigma$-form known from the work of Jimbo. An analogous study is carried out for the the hard edge scaling limit of the random matrix average, which we have previously evaluated in terms of a Painlev\'e \IIId transcendent in $\sigma$-form. An application of the latter result is given to the rapid evaluation of a Hankel determinant appearing in a recent work of Conrey, Rubinstein and Snaith relating to the derivative of the Riemann zeta function

Nonaffine rubber elasticity for stiff polymer networks

We present a theory for the elasticity of cross-linked stiff polymer networks. Stiff polymers, unlike their flexible counterparts, are highly anisotropic elastic objects. Similar to mechanical beams stiff polymers easily deform in bending, while they are much stiffer with respect to tensile forces (``stretching''). Unlike in previous approaches, where network elasticity is derived from the stretching mode, our theory properly accounts for the soft bending response. A self-consistent effective medium approach is used to calculate the macroscopic elastic moduli starting from a microscopic characterization of the deformation field in terms of ``floppy modes'' -- low-energy bending excitations that retain a high degree of non-affinity. The length-scale characterizing the emergent non-affinity is given by the ``fiber length'' $l_f$, defined as the scale over which the polymers remain straight. The calculated scaling properties for the shear modulus are in excellent agreement with the results of recent simulations obtained in two-dimensional model networks. Furthermore, our theory can be applied to rationalize bulk rheological data in reconstituted actin networks.Comment: 12 pages, 10 figures, revised Section II

On the Propagation of Slip Fronts at Frictional Interfaces

The dynamic initiation of sliding at planar interfaces between deformable and rigid solids is studied with particular focus on the speed of the slip front. Recent experimental results showed a close relation between this speed and the local ratio of shear to normal stress measured before slip occurs (static stress ratio). Using a two-dimensional finite element model, we demonstrate, however, that fronts propagating in different directions do not have the same dynamics under similar stress conditions. A lack of correlation is also observed between accelerating and decelerating slip fronts. These effects cannot be entirely associated with static local stresses but call for a dynamic description. Considering a dynamic stress ratio (measured in front of the slip tip) instead of a static one reduces the above-mentioned inconsistencies. However, the effects of the direction and acceleration are still present. To overcome this we propose an energetic criterion that uniquely associates, independently on the direction of propagation and its acceleration, the slip front velocity with the relative rise of the energy density at the slip tip.Comment: 15 pages, 6 figure

Polymer chain stiffness versus excluded volume: A Monte Carlo study of the crossover towards the wormlike chain model

When the local intrinsic stiffness of a polymer chain varies over a wide range, one can observe both a crossover from rigid-rod-like behavior to (almost) Gaussian random coils and a further crossover towards self-avoiding walks in good solvents. Using the pruned-enriched Rosenbluth method (PERM) to study self-avoiding walks of up to $N_b=50000$ steps and variable flexibility, the applicability of the Kratky-Porod model is tested. Evidence for non-exponential decay of the bond-orientational correlations $<\cos \theta (s) >$ for large distances $s$ along the chain contour is presented, irrespective of chain stiffness. For bottle-brush polymers on the other hand, where experimentally stiffness is varied via the length of side-chains, it is shown that these cylindrical brushes (with flexible backbones) are not described by the Kratky-Porod wormlike chain model, since their persistence length is (roughly) proportional to their cross-sectional radius, for all conditions of practical interest.Comment: 6 pages, 5 figures, to be published in Europhys. Lett. (2010

Adaptive Importance Sampling in General Mixture Classes

In this paper, we propose an adaptive algorithm that iteratively updates both the weights and component parameters of a mixture importance sampling density so as to optimise the importance sampling performances, as measured by an entropy criterion. The method is shown to be applicable to a wide class of importance sampling densities, which includes in particular mixtures of multivariate Student t distributions. The performances of the proposed scheme are studied on both artificial and real examples, highlighting in particular the benefit of a novel Rao-Blackwellisation device which can be easily incorporated in the updating scheme.Comment: Removed misleading comment in Section