Crystallization of random trigonometric polynomials

We give a precise measure of the rate at which repeated differentiation of a random trigonometric polynomial causes the roots of the function to approach equal spacing. This can be viewed as a toy model of crystallization in one dimension. In particular we determine the asymptotics of the distribution of the roots around the crystalline configuration and find that the distribution is not Gaussian.Comment: 10 pages, 3 figure

Randomly incomplete spectra and intermediate statistics

By randomly removing a fraction of levels from a given spectrum a model is constructed that describes a crossover from this spectrum to a Poisson spectrum. The formalism is applied to the transitions towards Poisson from random matrix theory (RMT) spectra and picket fence spectra. It is shown that the Fredholm determinant formalism of RMT extends naturally to describe incomplete RMT spectra.Comment: 9 pages, 2 figures. To appear in Physical Review

Quantum Chaotic Dynamics and Random Polynomials

We investigate the distribution of roots of polynomials of high degree with random coefficients which, among others, appear naturally in the context of "quantum chaotic dynamics". It is shown that under quite general conditions their roots tend to concentrate near the unit circle in the complex plane. In order to further increase this tendency, we study in detail the particular case of self-inversive random polynomials and show that for them a finite portion of all roots lies exactly on the unit circle. Correlation functions of these roots are also computed analytically, and compared to the correlations of eigenvalues of random matrices. The problem of ergodicity of chaotic wave-functions is also considered. For that purpose we introduce a family of random polynomials whose roots spread uniformly over phase space. While these results are consistent with random matrix theory predictions, they provide a new and different insight into the problem of quantum ergodicity. Special attention is devoted all over the paper to the role of symmetries in the distribution of roots of random polynomials.Comment: 33 pages, Latex, 6 Figures not included (a copy of them can be requested at [email protected]); to appear in Journal of Statistical Physic

Random polynomials, random matrices, and LL-functions

We show that the Circular Orthogonal Ensemble of random matrices arises naturally from a family of random polynomials. This sheds light on the appearance of random matrix statistics in the zeros of the Riemann zeta-function.Comment: Added background material. Final version. To appear in Nonlinearit

An improvement of the Berry--Esseen inequality with applications to Poisson and mixed Poisson random sums

By a modification of the method that was applied in (Korolev and Shevtsova, 2009), here the inequalities $$\rho(F_n,\Phi)\le\frac{0.335789(\beta^3+0.425)}{\sqrt{n}}$$ and $$\rho(F_n,\Phi)\le \frac{0.3051(\beta^3+1)}{\sqrt{n}}$$ are proved for the uniform distance $\rho(F_n,\Phi)$ between the standard normal distribution function $\Phi$ and the distribution function $F_n$ of the normalized sum of an arbitrary number $n\ge1$ of independent identically distributed random variables with zero mean, unit variance and finite third absolute moment $\beta^3$. The first of these inequalities sharpens the best known version of the classical Berry--Esseen inequality since $0.335789(\beta^3+0.425)\le0.335789(1+0.425)\beta^3<0.4785\beta^3$ by virtue of the condition $\beta^3\ge1$, and 0.4785 is the best known upper estimate of the absolute constant in the classical Berry--Esseen inequality. The second inequality is applied to lowering the upper estimate of the absolute constant in the analog of the Berry--Esseen inequality for Poisson random sums to 0.3051 which is strictly less than the least possible value of the absolute constant in the classical Berry--Esseen inequality. As a corollary, the estimates of the rate of convergence in limit theorems for compound mixed Poisson distributions are refined.Comment: 33 page

Valuing Health Gain from Composite Response Endpoints for Multisystem Diseases

Objectives: This study aimed to demonstrate how to estimate the value of health gain after patients with a multisystem disease achieve a condition-specific composite response endpoint. Methods: Data from patients treated in routine practice with an exemplar multisystem disease (systemic lupus erythematosus) were extracted from a national register (British Isles Lupus Assessment Group Biologics Register). Two bespoke composite response endpoints (Major Clinical Response and Improvement) were developed in advance of this study. Difference-in-differences regression compared health utility values (3-level version of EQ-5D; UK tariff) over 6 months for responders and nonresponders. Bootstrapped regression estimated the incremental quality-adjusted life-years (QALYs), probability of QALY gain after achieving the response criteria, and population monetary benefit of response. Results: Within the sample (n = 171), 18.2% achieved Major Clinical Response and 49.1% achieved Improvement at 6 months. Incremental health utility values were 0.0923 for Major Clinical Response and 0.0454 for Improvement. Expected incremental QALY gain at 6 months was 0.020 for Major Clinical Response and 0.012 for Improvement. Probability of QALY gain after achieving the response criteria was 77.6% for Major Clinical Response and 72.7% for Improvement. Population monetary benefit of response was £1 106 458 for Major Clinical Response and £649 134 for Improvement. Conclusions: Bespoke composite response endpoints are becoming more common to measure treatment response for multisystem diseases in trials and observational studies. Health technology assessment agencies face a growing challenge to establish whether these endpoints correspond with improved health gain. Health utility values can generate this evidence to enhance the usefulness of composite response endpoints for health technology assessment, decision making, and economic evaluation