Irreducible compositions and the first return to the origin of a random walk

Let $n = b_1 + ... + b_k = b_1' + \cdot + b_k'$ be a pair of compositions of $n$ into $k$ positive parts. We say this pair is {\em irreducible} if there is no positive $j < k$ for which $b_1 + ... b_j = b_1' + ... b_j'$. The probability that a random pair of compositions of $n$ is irreducible is shown to be asymptotic to $8/n$. This problem leads to a problem in probability theory. Two players move along a game board by rolling a die, and we ask when the two players will first coincide. A natural extension is to show that the probability of a first return to the origin at time $n$ for any mean-zero variance $V$ random walk is asymptotic to $\sqrt{V/(2 \pi)} n^{-3/2}$. We prove this via two methods, one analytic and one probabilistic

Non-perturbative calculations for the effective potential of the PTPT symmetric and non-Hermitian (−gϕ4)(-g\phi^{4}) field theoretic model

We investigate the effective potential of the $PT$ symmetric $(-g\phi^{4})$ field theory, perturbatively as well as non-perturbatively. For the perturbative calculations, we first use normal ordering to obtain the first order effective potential from which the predicted vacuum condensate vanishes exponentially as $G\to G^+$ in agreement with previous calculations. For the higher orders, we employed the invariance of the bare parameters under the change of the mass scale $t$ to fix the transformed form totally equivalent to the original theory. The form so obtained up to $G^3$ is new and shows that all the 1PI amplitudes are perurbative for both $G\ll 1$ and $G\gg 1$ regions. For the intermediate region, we modified the fractal self-similar resummation method to have a unique resummation formula for all $G$ values. This unique formula is necessary because the effective potential is the generating functional for all the 1PI amplitudes which can be obtained via $\partial^n E/\partial b^n$ and thus we can obtain an analytic calculation for the 1PI amplitudes. Again, the resummed from of the effective potential is new and interpolates the effective potential between the perturbative regions. Moreover, the resummed effective potential agrees in spirit of previous calculation concerning bound states.Comment: 20 page

Exact PT-Symmetry Is Equivalent to Hermiticity

We show that a quantum system possessing an exact antilinear symmetry, in particular PT-symmetry, is equivalent to a quantum system having a Hermitian Hamiltonian. We construct the unitary operator relating an arbitrary non-Hermitian Hamiltonian with exact PT-symmetry to a Hermitian Hamiltonian. We apply our general results to PT-symmetry in finite-dimensions and give the explicit form of the above-mentioned unitary operator and Hermitian Hamiltonian in two dimensions. Our findings lead to the conjecture that non-Hermitian CPT-symmetric field theories are equivalent to certain nonlocal Hermitian field theories.Comment: Few typos have been corrected and a reference update

Use of Equivalent Hermitian Hamiltonian for PTPT-Symmetric Sinusoidal Optical Lattices

We show how the band structure and beam dynamics of non-Hermitian $PT$-symmetric sinusoidal optical lattices can be approached from the point of view of the equivalent Hermitian problem, obtained by an analytic continuation in the transverse spatial variable $x$. In this latter problem the eigenvalue equation reduces to the Mathieu equation, whose eigenfunctions and properties have been well studied. That being the case, the beam propagation, which parallels the time-development of the wave-function in quantum mechanics, can be calculated using the equivalent of the method of stationary states. We also discuss a model potential that interpolates between a sinusoidal and periodic square well potential, showing that some of the striking properties of the sinusoidal potential, in particular birefringence, become much less prominent as one goes away from the sinusoidal case.Comment: 11 pages, 8 figure

Chaotic systems in complex phase space

This paper examines numerically the complex classical trajectories of the kicked rotor and the double pendulum. Both of these systems exhibit a transition to chaos, and this feature is studied in complex phase space. Additionally, it is shown that the short-time and long-time behaviors of these two PT-symmetric dynamical models in complex phase space exhibit strong qualitative similarities.Comment: 22 page, 16 figure

Vacuum Stability of the wrong sign (−ϕ6)(-\phi^{6}) Scalar Field Theory

We apply the effective potential method to study the vacuum stability of the bounded from above $(-\phi^{6})$ (unstable) quantum field potential. The stability ($\partial E/\partial b=0)$ and the mass renormalization ($\partial^{2} E/\partial b^{2}=M^{2})$ conditions force the effective potential of this theory to be bounded from below (stable). Since bounded from below potentials are always associated with localized wave functions, the algorithm we use replaces the boundary condition applied to the wave functions in the complex contour method by two stability conditions on the effective potential obtained. To test the validity of our calculations, we show that our variational predictions can reproduce exactly the results in the literature for the $\mathcal{PT}$-symmetric $\phi^{4}$ theory. We then extend the applications of the algorithm to the unstudied stability problem of the bounded from above $(-\phi^{6})$ scalar field theory where classical analysis prohibits the existence of a stable spectrum. Concerning this, we calculated the effective potential up to first order in the couplings in $d$ space-time dimensions. We find that a Hermitian effective theory is instable while a non-Hermitian but $\mathcal{PT}$-symmetric effective theory characterized by a pure imaginary vacuum condensate is stable (bounded from below) which is against the classical predictions of the instability of the theory. We assert that the work presented here represents the first calculations that advocates the stability of the $(-\phi^{6})$ scalar potential.Comment: 21pages, 12 figures. In this version, we updated the text and added some figure

The Isophotal Structure of Early-Type Galaxies in the SDSS: Dependence on AGN Activity and Environment

We study the dependence of the isophotal shape of early-type galaxies on their absolute B-band magnitude, their dynamical mass, and their nuclear activity and environment, using an unprecedented large sample of 847 early-type galaxies identified in the SDSS by Hao et al (2006). We find that the fraction of disky galaxies smoothly decreases with increasing luminosity. The large sample allows us to describe these trends accurately with tight linear relations that are statistically robust against the uncertainty in the isophotal shape measurements. There is also a host of significant correlations between the disky fraction and indicators of nuclear activity (both in the optical and in the radio) and environment (soft X-rays, group mass, group hierarchy). Our analysis shows however that these correlations can be accurately matched by assuming that the disky fraction depends only on galaxy luminosity or mass. We therefore conclude that neither the level of activity, nor group mass or group hierarchy help in better predicting the isophotal shape of early-type galaxies.Comment: 31 pages, 10 figures, accepted for publication in Ap

Polymer-Chain Adsorption Transition at a Cylindrical Boundary

In a recent letter, a simple method was proposed to generate solvable models that predict the critical properties of statistical systems in hyperspherical geometries. To that end, it was shown how to reduce a random walk in $D$ dimensions to an anisotropic one-dimensional random walk on concentric hyperspheres. Here, I construct such a random walk to model the adsorption-desorption transition of polymer chains growing near an attractive cylindrical boundary such as that of a cell membrane. I find that the fraction of adsorbed monomers on the boundary vanishes exponentially when the adsorption energy decreases towards its critical value. When the adsorption energy rises beyond a certain value above the critical point whose scale is set by the radius of the cell, the adsorption fraction exhibits a crossover to a linear increase characteristic to polymers growing near planar boundaries.Comment: latex, 12 pages, 3 ps-figures, uuencode

Correlation energies by the generator coordinate method: computational aspects for quadrupolar deformations

We investigate truncation schemes to reduce the computational cost of calculating correlations by the generator coordinate method based on mean-field wave functions. As our test nuclei, we take examples for which accurate calculations are available. These include a strongly deformed nucleus, 156Sm, a nucleus with strong pairing, 120Sn, the krypton isotope chain which contains examples of soft deformations, and the lead isotope chain which includes the doubly magic 208Pb. We find that the Gaussian overlap approximation for angular momentum projection is effective and reduces the computational cost by an order of magnitude. Cost savings in the deformation degrees of freedom are harder to realize. A straightforward Gaussian overlap approximation can be applied rather reliably to angular-momentum projected states based on configuration sets having the same sign deformation (prolate or oblate), but matrix elements between prolate and oblate deformations must be treated with more care. We propose a two-dimensional GOA using a triangulation procedure to treat the general case with both kinds of deformation. With the computational gains from these approximations, it should be feasible to carry out a systematic calculation of correlation energies for the nuclear mass table.Comment: 11 pages revtex, 9 eps figure