Bootstrap Percolation on Homogeneous Trees Has 2 Phase Transitions

We study the threshold θ bootstrap percolation model on the homogeneous tree with degree b+1, 2≤θ≤b, and initial density p. It is known that there exists a nontrivial critical value for p, which we call p _f , such that a) for p>p _f , the final bootstrapped configuration is fully occupied for almost every initial configuration, and b) if p<p _f , then for almost every initial configuration, the final bootstrapped configuration has density of occupied vertices less than 1. In this paper, we establish the existence of a distinct critical value for p, p _c , such that 0<p _c <p _f , with the following properties: 1) if p≤p _c , then for almost every initial configuration there is no infinite cluster of occupied vertices in the final bootstrapped configuration; 2) if p>p _c , then for almost every initial configuration there are infinite clusters of occupied vertices in the final bootstrapped configuration. Moreover, we show that 3) for p<p _c , the distribution of the occupied cluster size in the final bootstrapped configuration has an exponential tail; 4) at p=p _c , the expected occupied cluster size in the final bootstrapped configuration is infinite; 5) the probability of percolation of occupied vertices in the final bootstrapped configuration is continuous on [0,p _f ] and analytic on (p _c ,p _f ), admitting an analytic continuation from the right at p _c and, only in the case θ=b, also from the left at p _f . L.R.G. Fontes partially supported by the Brazilians CNPq through grants 475833/2003-1, 307978/2004-4 and 484351/2006-0, and FAPESP through grant 04/07276-2. R.H. Schonmann partially supported by the American N.S.F. through grant DMS-0300672.     

Power-Law Behavior in Geometric Characteristics of Full Binary Trees

Natural river networks exhibit regular scaling laws in their topological organization. Here, we investigate whether these scaling laws are unique characteristics of river networks or can be applicable to general binary tree networks. We generate numerous binary trees, ranging from purely ordered trees to completely random trees. For each generated binary tree, we analyze whether the tree exhibits any scaling property found in river networks, i.e., the power-laws in the size distribution, the length distribution, the distance-load relationship, and the power spectrum of width function. We found that partially random trees generated on the basis of two distinct types of deterministic trees, i.e., deterministic critical and supercritical trees, show contrasting characteristics. Partially random trees generated on the basis of deterministic critical trees exhibit all power-law characteristics investigated in this study with their fitted exponents close to the values observed in natural river networks over a wide range of random-degree. On the other hand, partially random trees generated on the basis of deterministic supercritical trees rarely follow scaling laws of river networks.     

Large Deviations for Subcritical Bootstrap Percolation on the Erdős–Rényi Graph

Journal of Statistical Physics - We show that certain billiard flows on planar billiard tables with horns can be modeled as suspension flows over Young towers (Ann. Math. 147:585–650, 1998)...     

Continuous-Time Quantum Walk on Integer Lattices and Homogeneous Trees

This paper is concerned with the continuous-time quantum walk on ℤ, ℤ ^d , and infinite homogeneous trees. By using the generating function method, we compute the limit of the average probability distribution for the general isotropic walk on ℤ, and for nearest-neighbor walks on ℤ ^d and infinite homogeneous trees. In addition, we compute the asymptotic approximation for the probability of the return to zero at time t in all these cases.     

Connectivity Properties of Continuum Percolation Processes on ℝ2

A new approach is introduced in order to estimate the critical parameters of continuous percolation of overlapping disks in ² when the centres of the disks are Poisson distributed. Better insights on relevant parameters near criticality are found. Moreover, introducing a suitable connectivity length, the model is able to describe continuous percolation of any type of geometrical objects in ².     

Femtosecond OPO based on LBO pumped by a frequency-doubled Yb-fiber laser-amplifier system for CARS spectroscopy

We present detailed investigations of a femtosecond green-pumped optical parametric oscillator (OPO) based on lithium triborate. As pump source, a frequency-doubled Yb-fiber laser-amplifier system is used. The OPO generates signal pulses tunable over a spectral range from 780 to 940 nm and idler pulses tunable from 1630 to 1190 nm. More than 250 mW are generated in the signal beam and more than 300 mW in the idler beam. Without dispersion compensation chirped signal pulses with a pulse duration between 100 and 250 fs are measured. Using this system for coherent anti-Stokes Raman scattering spectroscopy, vibrational resonances between 1110 and 6760 cm⁻¹ can be excited. Due to the chirped pulses, a spectral resolution of 100 cm⁻¹ is achieved, which is 2.5 times higher compared to an excitation with time-bandwidth limited pulses.     

Metastable states for the Becker-Döring cluster equations

The Becker-Döring equations, in whichc _l (t) can represent the concentration ofl-particle clusters or droplets in (say) a condensing vapour at timet, are $$\begin{array}{*{20}c} {{{dc_l (t)} \mathord{\left/ {\vphantom {{dc_l (t)} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = J_{l - 1} (t) - J_l (t)} & {(l = 2,3,...)} \\ \end{array} $$ with $$J_l (t): = a_l c_1 (t)c_l (t) - b_{l + 1} c_{l + 1} (t)$$ and eitherc ₁=const. (‘case A’) or \(\rho : = \sum\limits_1^\infty {lc_l } \) =const. (‘case B’). The equilibrium solutions arec _l =Q _l z ^l , where \(Q_l : = \prod\limits_2^l {({{a_{r - 1} } \mathord{\left/ {\vphantom {{a_{r - 1} } {b_r }}} \right. \kern-0em} {b_r }})} \) . The density of the saturated vapour, defined as \(\rho _s : = \sum\limits_1^\infty {lQ_l z_s ^l } \) , wherez _s is the radius of convergence of the series, is assumed finite. It is proved here that, subject to some further plausible conditions on the kinetic coefficientsa _l andb _l , there is a class of “metastable” solutions of the equations, withc ₁?z _s small and positive, which take an exponentially long time to decay to their asymptotic steady states. (An “exponentially long time” means one that increases more rapidly than any negative power of the given value ofc ₁?z _s (or, in caseB,ρ?ρ _s ) as the latter tends to zero). The main ingredients in the proof are (i) a time-independent upper bound on the solution of the kinetic equations (this upper bound is a steady-state solution of case A of the equations, of the type used in the Becker-Döring theory of nucleation), and (ii) an upper bound on the total concentration of particles in clusters greater than a certain critical size, which (with suitable initial conditions) remains exponentially small until the time becomes exponentially large.     

A Lagrangian Formalism Based on the Velocity-Determined Virtual Displacements for Systems with Nonholonomic Constraints

We develop a Lagrangian formulation for classical systems with a general nonholonomic constraints by utilizing the so-called velocity-determined virtual-displacement conditions, i.e. by assuming the virtual displacements to be along the direction of the velocities in a special reference frame. It is shown that our general scheme encompasses as special cases the Chetaev and Voronets approaches when the constraints are homogeneous or linear in relative velocities.     

Comments on “Barut-Girardello Coherent States for the Parabolic Cylinder Functions”

In Chenaghlou and Faizy (Int. J. Theor. Phys. 2008), the authors claim that they have constructed the Barut-Girardello coherent states for the parabolic cylinder functions. However, we point out here that by introducing these coherent states, Schrödinger was able to put forth the idea of “coherent states of the quantum harmonic oscillator” over eighty years ago. These coherent states are derived not only from the Barut-Girardello eigenvalue equation, but also from the Schrödinger and the Klauder-Perelomov approaches. Thus, contrary to their claim, the authors have not introduced new coherent states. In particular, a wide range of the parabolic cylinder functions do not form an orthonormal basis.     

Lower Bounds on the Squashed Entanglement for Multi-Party System

Squashed entanglement is a promising entanglement measure that can be generalized to multipartite case, and it has all of the desirable properties for a good entanglement measure. In this paper we present computable lower bounds to evaluate the multipartite squashed entanglement. We also derive some inequalities relating the squashed entanglement to other entanglement measures.     

Hyperscaling for Oriented Percolation in $$$$ Space–Time Dimensions

Consider nearest-neighbor oriented percolation in \(d+1\) space–time dimensions. Let \(\rho ,\eta ,\nu \) be the critical exponents for the survival probability up to time t, the expected number of vertices at time t connected from the space–time origin, and the gyration radius of those vertices, respectively. We prove that the hyperscaling inequality \(d\nu \ge \eta +2\rho \), which holds for all \(d\ge 1\) and is a strict inequality above the upper-critical dimension 4, becomes an equality for \(d=1\), i.e., \(\nu =\eta +2\rho \), provided existence of at least two among \(\rho ,\eta ,\nu \). The key to the proof is the recent result on the critical box-crossing property by Duminil-Copin et al. [6].     

Resistance Scaling and the Number of Spanning Trees in Self-Similar Lattices

We consider the problem of enumerating spanning trees in self-similar lattices, motivated by recent work of Chang, Chen and Yang, who determined explicit formulae in the case of Sierpiński graphs and some of their generalizations. The aim of this note is to show that their results hold in more generality and that there is a strong relation between this enumeration problem and resistance scaling on self-similar lattices.     

Fluctuations of Matrix Elements of Regular Functions of Gaussian Random Matrices

We find the limit of the variance and prove the Central Limit Theorem (CLT) for the matrix elements φ _jk (M), j,k=1,…,n of a regular function φ of the Gaussian matrix M (GOE and GUE) as its size n tends to infinity. We show that unlike the linear eigenvalue statistics Tr φ(M), a traditional object of random matrix theory, whose variance is bounded as n→∞ and the CLT is valid for Tr φ(M)−E{Tr φ(M)}, the variance of φ _jk (M) is O(1/n), and the CLT is valid for . This shows the role of eigenvectors in the forming of the asymptotic regime of various functions (statistics) of random matrices. Our proof is based on the use of the Fourier transform as a basic characteristic function, unlike the Stieltjes transform and moments, used in majority of works of the field. We also comment on the validity of analogous results for other random matrices.     

Fokker–Planck Equation for a Metastable Time Dependent Potential

We use the Fokker–Planck equation to study the diffusion process driven for a metastable potential within a temporal dependence. This potential is characterized by the existence of a barrier that increases with time and reduces the particle diffusion. Escape rate across the barrier for different values of diffusion coefficient is analyzed. The results are also associated with the diffusion process through ion channels in biological system.     

Solution epitaxy of gallium-doped ZnO on p-GaN for heterojunction light-emitting diodes

We report white light emission from a Ga-doped ZnO/p-GaN heterojunction light-emitting diode which was fabricated by growing gallium-doped ZnO film on the p-GaN in water at 90°C. As determined from Ga-doped ZnO films grown on (111) oriented MgAl₂O₄ spinel single crystal substrates, thermal treatment at 600°C in nitrogen ambient leads to a carrier concentration of 3.1×10²⁰ cm⁻³ (and carrier mobility of 28 cm²/Vs) which is two orders of magnitude higher than that of the undoped films. Electroluminescence emissions at wavelengths of 393 nm (3.155 eV) and 529.5 nm (2.4 eV) were observed under forward bias in the heterojunction diode and white light could be visibly observed. The high concentration of electrons supplied from the Ga-doped ZnO films helped to enhance the carrier recombination and increase the light-emitting efficiency of the heterojunction diode.     

On the Essential Spectrum of Schrödinger Operators on Trees

It is known that the essential spectrum of a Schrödinger operator H on \(\ell ^{2}\left (\mathbb {N}\right )\) is equal to the union of the spectra of right limits of H. The natural generalization of this relation to \(\mathbb {Z}^{n}\) is known to hold as well. In this paper we generalize the notion of right limits to general infinite connected graphs and construct examples of graphs for which the essential spectrum of the Laplacian is strictly bigger than the union of the spectra of its right limits. As these right limits are trees, this result is complemented by the fact that the equality still holds for general bounded operators on regular trees. We prove this and characterize the essential spectrum in the spherically symmetric case.     

Law of Large Numbers for Non-Local Functions on Probabilistic Cellular Automata

We prove a multi-dimensional version of the law of large numbers for invariant measures of a large class of probabilistic cellular automata, whose transition probabilities satisfy some inequalities, which are known to assure their ergodicity. In some non-ergodic cases analogous results have been obtained for local functions. We deal with a larger class of functions, which includes some non-local ones.