Condensation of the roots of real random polynomials on the real axis

We introduce a family of real random polynomials of degree n whose coefficients a_k are symmetric independent Gaussian variables with variance = e^{-k^\alpha}, indexed by a real \alpha \geq 0. We compute exactly the mean number of real roots for large n. As \alpha is varied, one finds three different phases. First, for 0 \leq \alpha \sim (\frac{2}{\pi}) \log{n}. For 1 < \alpha < 2, there is an intermediate phase where grows algebraically with a continuously varying exponent, \sim \frac{2}{\pi} \sqrt{\frac{\alpha-1}{\alpha}} n^{\alpha/2}. And finally for \alpha > 2, one finds a third phase where \sim n. This family of real random polynomials thus exhibits a condensation of their roots on the real line in the sense that, for large n, a finite fraction of their roots /n are real. This condensation occurs via a localization of the real roots around the values \pm \exp{[\frac{\alpha}{2}(k+{1/2})^{\alpha-1} ]}, 1 \ll k \leq n.Comment: 13 pages, 2 figure

Critical properties of superconducting Ba1-xKxFe2As2

Magnetisation and magnetoresistance measurements have been carried out on superconducting Ba1-xKxFe2As2 samples with x=0.40 and 0.50. From low field magnetization data carried out at different temperatures below TC, HC1 has been extracted. The plot of HC1 versus temperature shows an anomalous increase at low temperatures. From high field magnetization hysterisis measurements carried out in fields up to 16 T at 4.2 K and 20 K, the critical current density has been evaluated using the Bean critical state model. The JC determined from the high field data is >104A/cm2 at 4.2 K and 5 T. The superconducting transitions were also measured resistively in increasing applied magnetic fields up to 12 Tesla. From the variation of the TC onset with applied field, dHC2/dT at TC was obtained to be -7.708 T/K and -5.57 T/K in the samples with x=0.40 and 0.50.Comment: 14 pages; 7 figure

Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State

A recent conjecture regarding the average of the minimum eigenvalue of the reduced density matrix of a random complex state is proved. In fact, the full distribution of the minimum eigenvalue is derived exactly for both the cases of a random real and a random complex state. Our results are relevant to the entanglement properties of eigenvectors of the orthogonal and unitary ensembles of random matrix theory and quantum chaotic systems. They also provide a rare exactly solvable case for the distribution of the minimum of a set of N {\em strongly correlated} random variables for all values of N (and not just for large N).Comment: 13 pages, 2 figures included; typos corrected; to appear in J. Stat. Phy

Statistical distribution of quantum entanglement for a random bipartite state

We compute analytically the statistics of the Renyi and von Neumann entropies (standard measures of entanglement), for a random pure state in a large bipartite quantum system. The full probability distribution is computed by first mapping the problem to a random matrix model and then using a Coulomb gas method. We identify three different regimes in the entropy distribution, which correspond to two phase transitions in the associated Coulomb gas. The two critical points correspond to sudden changes in the shape of the Coulomb charge density: the appearance of an integrable singularity at the origin for the first critical point, and the detachement of the rightmost charge (largest eigenvalue) from the sea of the other charges at the second critical point. Analytical results are verified by Monte Carlo numerical simulations. A short account of some of these results appeared recently in Phys. Rev. Lett. {\bf 104}, 110501 (2010).Comment: 7 figure

Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation

We study various statistical properties of real roots of three different classes of random polynomials which recently attracted a vivid interest in the context of probability theory and quantum chaos. We first focus on gap probabilities on the real axis, i.e. the probability that these polynomials have no real root in a given interval. For generalized Kac polynomials, indexed by an integer d, of large degree n, one finds that the probability of no real root in the interval [0,1] decays as a power law n^{-\theta(d)} where \theta(d) > 0 is the persistence exponent of the diffusion equation with random initial conditions in spatial dimension d. For n \gg 1 even, the probability that they have no real root on the full real axis decays like n^{-2(\theta(2)+\theta(d))}. For Weyl polynomials and Binomial polynomials, this probability decays respectively like \exp{(-2\theta_{\infty}} \sqrt{n}) and \exp{(-\pi \theta_{\infty} \sqrt{n})} where \theta_{\infty} is such that \theta(d) = 2^{-3/2} \theta_{\infty} \sqrt{d} in large dimension d. We also show that the probability that such polynomials have exactly k roots on a given interval [a,b] has a scaling form given by \exp{(-N_{ab} \tilde \phi(k/N_{ab}))} where N_{ab} is the mean number of real roots in [a,b] and \tilde \phi(x) a universal scaling function. We develop a simple Mean Field (MF) theory reproducing qualitatively these scaling behaviors, and improve systematically this MF approach using the method of persistence with partial survival, which in some cases yields exact results. Finally, we show that the probability density function of the largest absolute value of the real roots has a universal algebraic tail with exponent {-2}. These analytical results are confirmed by detailed numerical computations.Comment: 32 pages, 16 figure

Intricacies of Strain and Magnetic Field Induced Charge Order Melting in Pr0.5Ca0.5MnO3 Thin Films

Thin films of the half doped manganite Pr0.5Ca0.5MnO3 were grown on (100) oriented MgO substrates by pulsed laser deposition technique. In order to study the effect of strain on the magnetic field induced charge order melting, films of different thicknesses were prepared and their properties were studied by x-ray diffraction, electrical resistivity and magnetoresistance measurements. A field induced charge order melting is observed for films with very small thicknesses. The charge order transition temperature and the magnetic filed induced charge order melting are observed to depend on the nature of strain that is experienced by the film.Comment: 18 pages, (including 6 figures