Phase Transition in the Aldous-Shields Model of Growing Trees

We study analytically the late time statistics of the number of particles in a growing tree model introduced by Aldous and Shields. In this model, a cluster grows in continuous time on a binary Cayley tree, starting from the root, by absorbing new particles at the empty perimeter sites at a rate proportional to c^{-l} where c is a positive parameter and l is the distance of the perimeter site from the root. For c=1, this model corresponds to random binary search trees and for c=2 it corresponds to digital search trees in computer science. By introducing a backward Fokker-Planck approach, we calculate the mean and the variance of the number of particles at large times and show that the variance undergoes a `phase transition' at a critical value c=sqrt{2}. While for c>sqrt{2} the variance is proportional to the mean and the distribution is normal, for c<sqrt{2} the variance is anomalously large and the distribution is non-Gaussian due to the appearance of extreme fluctuations. The model is generalized to one where growth occurs on a tree with $m$ branches and, in this more general case, we show that the critical point occurs at c=sqrt{m}.Comment: Latex 17 pages, 6 figure

First Passage Properties of the Erdos-Renyi Random Graph

We study the mean time for a random walk to traverse between two arbitrary sites of the Erdos-Renyi random graph. We develop an effective medium approximation that predicts that the mean first-passage time between pairs of nodes, as well as all moments of this first-passage time, are insensitive to the fraction p of occupied links. This prediction qualitatively agrees with numerical simulations away from the percolation threshold. Near the percolation threshold, the statistically meaningful quantity is the mean transit rate, namely, the inverse of the first-passage time. This rate varies non-monotonically with p near the percolation transition. Much of this behavior can be understood by simple heuristic arguments.Comment: 10 pages, 9 figures, 2-column revtex4 forma

Multiphoton radiative recombination of electron assisted by laser field

In the presence of an intensive laser field the radiative recombination of the continuum electron into an atomic bound state generally is accompanied by absorption or emission of several laser quanta. The spectrum of emitted photons represents an equidistant pattern with the spacing equal to the laser frequency. The distribution of intensities in this spectrum is studied employing the Keldysh-type approximation, i.e. neglecting interaction of the impact electron with the atomic core in the initial continuum state. Within the adiabatic approximation the scale of emitted photon frequencies is subdivided into classically allowed and classically forbidden domains. The highest intensities correspond to emission frequencies close to the edges of classically allowed domain. The total cross section of electron recombination summed over all emitted photon channels exhibits negligible dependence on the laser field intensity.Comment: 14 pages, 5 figures (Figs.2-5 have "a" and "b" parts), Phys.Rev.A accepted for publication. Fig.2b is presented correctl

Combinatorial Markov chains on linear extensions

We consider generalizations of Schuetzenberger's promotion operator on the set L of linear extensions of a finite poset of size n. This gives rise to a strongly connected graph on L. By assigning weights to the edges of the graph in two different ways, we study two Markov chains, both of which are irreducible. The stationary state of one gives rise to the uniform distribution, whereas the weights of the stationary state of the other has a nice product formula. This generalizes results by Hendricks on the Tsetlin library, which corresponds to the case when the poset is the anti-chain and hence L=S_n is the full symmetric group. We also provide explicit eigenvalues of the transition matrix in general when the poset is a rooted forest. This is shown by proving that the associated monoid is R-trivial and then using Steinberg's extension of Brown's theory for Markov chains on left regular bands to R-trivial monoids.Comment: 35 pages, more examples of promotion, rephrased the main theorems in terms of discrete time Markov chain

Almost uniform sampling via quantum walks

Many classical randomized algorithms (e.g., approximation algorithms for #P-complete problems) utilize the following random walk algorithm for {\em almost uniform sampling} from a state space $S$ of cardinality $N$: run a symmetric ergodic Markov chain $P$ on $S$ for long enough to obtain a random state from within $\epsilon$ total variation distance of the uniform distribution over $S$. The running time of this algorithm, the so-called {\em mixing time} of $P$, is $O(\delta^{-1} (\log N + \log \epsilon^{-1}))$, where $\delta$ is the spectral gap of $P$. We present a natural quantum version of this algorithm based on repeated measurements of the {\em quantum walk} $U_t = e^{-iPt}$. We show that it samples almost uniformly from $S$ with logarithmic dependence on $\epsilon^{-1}$ just as the classical walk $P$ does; previously, no such quantum walk algorithm was known. We then outline a framework for analyzing its running time and formulate two plausible conjectures which together would imply that it runs in time $O(\delta^{-1/2} \log N \log \epsilon^{-1})$ when $P$ is the standard transition matrix of a constant-degree graph. We prove each conjecture for a subclass of Cayley graphs.Comment: 13 pages; v2 added NSF grant info; v3 incorporated feedbac

Role of electromagnetically induced transparency in resonant four-wave-mixing schemes.

Published versio

Exact eigenvalue spectrum of a class of fractal scale-free networks

The eigenvalue spectrum of the transition matrix of a network encodes important information about its structural and dynamical properties. We study the transition matrix of a family of fractal scale-free networks and analytically determine all the eigenvalues and their degeneracies. We then use these eigenvalues to evaluate the closed-form solution to the eigentime for random walks on the networks under consideration. Through the connection between the spectrum of transition matrix and the number of spanning trees, we corroborate the obtained eigenvalues and their multiplicities.Comment: Definitive version accepted for publication in EPL (Europhysics Letters

Exact field ionization rates in the barrier suppression-regime from numerical TDSE calculations

Numerically determined ionization rates for the field ionization of atomic hydrogen in strong and short laser pulses are presented. The laser pulse intensity reaches the so-called "barrier suppression ionization" regime where field ionization occurs within a few half laser cycles. Comparison of our numerical results with analytical theories frequently used shows poor agreement. An empirical formula for the "barrier suppression ionization"-rate is presented. This rate reproduces very well the course of the numerically determined ground state populations for laser pulses with different length, shape, amplitude, and frequency. Number(s): 32.80.RmComment: Enlarged and newly revised version, 22 pages (REVTeX) + 8 figures in ps-format, submitted for publication to Physical Review A, WWW: http://www.physik.tu-darmstadt.de/tqe

Political branding: sense of identity or identity crisis? An investigation of the transfer potential of the brand identity prism to the UK Conservative Party

Brands are strategic assets and key to achieving a competitive advantage. Brands can be seen as a heuristic device, encapsulating a series of values that enable the consumer to make quick and efficient choices. More recently, the notion of a political brand and the rhetoric of branding have been widely adopted by many political parties as they seek to differentiate themselves, and this has led to an emerging interest in the idea of the political brand. Therefore, this paper examines the UK Conservative Party brand under David Cameron’s leadership and examines the applicability of Kapferer’s brand identity prism to political branding. This paper extends and operationalises the brand identity prism into a ‘political brand identity network’ which identifies the inter-relatedness of the components of the corporate political brand and the candidate political brand. Crucial for practitioners, this model can demonstrate how the brand is presented and communicated to the electorate and serves as a useful mechanism to identify consistency within the corporate and candidate political brands