Excursion sets of stable random fields

Studying the geometry generated by Gaussian and Gaussian- related random fields via their excursion sets is now a well developed and well understood subject. The purely non-Gaussian scenario has, however, not been studied at all. In this paper we look at three classes of stable random fields, and obtain asymptotic formulae for the mean values of various geometric characteristics of their excursion sets over high levels. While the formulae are asymptotic, they contain enough information to show that not only do stable random fields exhibit geometric behaviour very different from that of Gaussian fields, but they also differ significantly among themselves.Comment: 35 pages, 1 figur

High level excursion set geometry for non-Gaussian infinitely divisible random fields

We consider smooth, infinitely divisible random fields $(X(t),t\in M)$, $M\subset {\mathbb{R}}^d$, with regularly varying Levy measure, and are interested in the geometric characteristics of the excursion sets $$A_u=\{t\in M:X(t)>u\}$$ over high levels u. For a large class of such random fields, we compute the $u\to\infty$ asymptotic joint distribution of the numbers of critical points, of various types, of X in $A_u$, conditional on $A_u$ being nonempty. This allows us, for example, to obtain the asymptotic conditional distribution of the Euler characteristic of the excursion set. In a significant departure from the Gaussian situation, the high level excursion sets for these random fields can have quite a complicated geometry. Whereas in the Gaussian case nonempty excursion sets are, with high probability, roughly ellipsoidal, in the more general infinitely divisible setting almost any shape is possible.Comment: Published in at http://dx.doi.org/10.1214/11-AOP738 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonlinear evolution of the plasma beatwave: Compressing the laser beatnotes via electromagnetic cascading

The near-resonant beatwave excitation of an electron plasma wave (EPW) can be employed for generating the trains of few-femtosecond electromagnetic (EM) pulses in rarefied plasmas. The EPW produces a co-moving index grating that induces a laser phase modulation at the difference frequency. The bandwidth of the phase-modulated laser is proportional to the product of the plasma length, laser wavelength, and amplitude of the electron density perturbation. The laser spectrum is composed of a cascade of red and blue sidebands shifted by integer multiples of the beat frequency. When the beat frequency is lower than the electron plasma frequency, the red-shifted spectral components are advanced in time with respect to the blue-shifted ones near the center of each laser beatnote. The group velocity dispersion of plasma compresses so chirped beatnotes to a few-laser-cycle duration thus creating a train of sharp EM spikes with the beat periodicity. Depending on the plasma and laser parameters, chirping and compression can be implemented either concurrently in the same, or sequentially in different plasmas. Evolution of the laser beatwave end electron density perturbations is described in time and one spatial dimension in a weakly relativistic approximation. Using the compression effect, we demonstrate that the relativistic bi-stability regime of the EPW excitation [G. Shvets, Phys. Rev. Lett. 93, 195004 (2004)] can be achieved with the initially sub-threshold beatwave pulse.Comment: 13 pages, 11 figures, submitted to Physical Review

Quantum Criticality in Dimerized Spin Ladders

We analyze a possibility of quantum criticality (gaplessness) in dimerized antiferromagnetic two- and three-leg spin-1/2 ladders. Contrary to earlier studies of these models, we examine different dimerization patterns in the ladder. We find that ladders with the columnar dimerization order have lower zero-temperature energies and they are always gapped. For the staggered dimerization order, we find the quantum critical lines, in agreement with earlier analyses. The bond mean-field theory we apply, demonstrates its quantitative accuracy and agrees with available numerical results. We conclude that unless some mechanism for locking dimerization into the energetically less favorable staggered configuration is provided, the dimerized ladders do not order into the phase where the quantum criticality occurs.Comment: 7 pages, 9 figure

Peculiarities of evolutions of elastic-plastic shock compression waves in different materials

In the paper, we discuss such unexpected features in the wave evolution in solids as strongly nonlinear uniaxial elastic compression in a picosecond time range, a departure from self-similar development of the wave process which is accompanied with apparent sub-sonic wave propagation, changes of shape of elastic precursor wave as a result of variations in the material structure and the temperature, unexpected peculiarities of reflection of elastic-plastic waves from free surface

Nonlinear Dynamics of Dipoles in Microtubules: Pseudo-Spin Model

We perform a theoretical study of the dynamics of the electric field excitations in a microtubule by taking into consideration the realistic cylindrical geometry, dipole-dipole interactions of the tubulin-based protein heterodimers, the radial electric field produced by the solvent, and a possible degeneracy of energy states of individual heterodimers. The consideration is done in the frames of the classical pseudo-spin model. We derive the system of nonlinear dynamical ordinary differential equations of motion for interacting dipoles, and the continuum version of these equations. We obtain the solutions of these equations in the form of snoidal waves, solitons, kinks, and localized spikes. Our results will help to a better understanding of the functional properties of microtubules including the motor protein dynamics and the information transfer processes. Our considerations are based on classical dynamics. Some speculations on the role of possible quantum effects are also made.Comment: 14 pages, 15 figures. The high resolution figure files are available by reques