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 , indexed by a real α≥0. We compute exactly the mean number of real roots 〈N _n 〉 for large n. As α is varied, one finds three different phases. First, for 0≤α<1, one finds that . For 1<α<2, there is an intermediate phase where 〈N _n 〉 grows algebraically with a continuously varying exponent, . And finally for α>2, one finds a third phase where 〈N _n 〉∼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 _n 〉/n are real. This condensation occurs via a localization of the real roots around the values , 1≪k≤n.     

Weakly Non-Ergodic Statistical Physics

For weakly non ergodic systems, the probability density function of a time average observable is where is the value of the observable when the system is in state j=1,…L. p _j ^eq is the probability that a member of an ensemble of systems occupies state j in equilibrium. For a particle undergoing a fractional diffusion process in a binding force field, with thermal detailed balance conditions, p _j ^eq is Boltzmann’s canonical probability. Within the unbiased sub-diffusive continuous time random walk model, the exponent 0<α<1 is the anomalous diffusion exponent 〈x ²〉∼t ^α found for free boundary conditions. When α→1 ergodic statistical mechanics is recovered . We briefly discuss possible physical applications in single particle experiments.     

Spherical-separability of Non-Hermitian Hamiltonians and Pseudo- PT\mathcal{PT} -symmetry

Non-Hermitian but -symmetrized spherically-separable Dirac and Schr?dinger Hamiltonians are considered. It is observed that the descendant Hamiltonians H _r , H _θ , and H _φ play essential roles and offer some “user-feriendly” options as to which one (or ones) of them is (or are) non-Hermitian. Considering a -symmetrized H _φ , we have shown that the conventional Dirac (relativistic) and Schr?dinger (non-relativistic) energy eigenvalues are recoverable. We have also witnessed an unavoidable change in the azimuthal part of the general wavefunction. Moreover, setting a possible interaction V(θ)≠0 in the descendant Hamiltonian H _θ would manifest a change in the angular θ-dependent part of the general solution too. Whilst some -symmetrized H _φ Hamiltonians are considered, a recipe to keep the regular magnetic quantum number m, as defined in the regular traditional Hermitian settings, is suggested. Hamiltonians possess properties similar to the -symmetric ones (here the non-Hermitian -symmetric Hamiltonians) are nicknamed as pseudo- -symmetric.     

Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions

We consider the point process of zeroes of certain Gaussian analytic functions and find the asymptotics for the probability that there are more than m points of the process in a fixed disk of radius r, as . For the planar Gaussian analytic function, , we show that this probability is asymptotic to . For the hyperbolic Gaussian analytic functions, , we show that this probability decays like .In the planar case, we also consider the problem posed by Mikhail Sodin² on moderate and very large deviations in a disk of radius r, as . We partially solve the problem by showing that there is a qualitative change in the asymptotics of the probability as we move from the large deviation regime to the moderate.Research supported by NSF grant #DMS-0104073 and NSF-FRG grant #DMS-0244479.     

NLO contributions to <Emphasis Type="Italic">B</Emphasis>→<Emphasis Type="Italic">KK</Emphasis><Superscript>*</Superscript> decays in the pQCD approach

We calculate the important next-to-leading-order (NLO) contributions to the B→KK ^* decays from the vertex corrections, the quark loops, and the magnetic penguins in the perturbative QCD (pQCD) factorization approach. The pQCD predictions for the CP-averaged branching ratios are , , and Br(B ⁰→K ⁺ K ^*−+K ⁻ K ^*+)≈1.3×10⁻⁷, which agree well with both the experimental upper limits and the predictions based on the QCD factorization approach. Furthermore, the CP violating asymmetries of the considered decay modes are also evaluated. The NLO pQCD predictions for and decays are and .     

Detailed Examination of Transport Coefficients in Cubic-Plus-Quartic Oscillator Chains

We examine the thermal conductivity and bulk viscosity of a one-dimensional (1D) chain of particles with cubic-plus-quartic interparticle potentials and no on-site potentials. This system is equivalent to the FPU-α β system in a subset of its parameter space. We identify three distinct frequency regimes which we call the hydrodynamic regime, the perturbative regime and the collisionless regime. In the lowest frequency regime (the hydrodynamic regime) heat is transported ballistically by long wavelength sound modes. The model that we use to describe this behaviour predicts that as ω→0 the frequency dependent bulk viscosity, , and the frequency dependent thermal conductivity, , should diverge with the same power law dependence on ω. Thus, we can define the bulk Prandtl number, , where m is the particle mass and k _B is Boltzmann’s constant. This dimensionless ratio should approach a constant value as ω→0. We use mode-coupling theory to predict the ω→0 limit of Pr _ζ . Values of Pr _ζ obtained from simulations are in agreement with these predictions over a wide range of system parameters. In the middle frequency regime, which we call the perturbative regime, heat is transported by sound modes which are damped by four-phonon processes. This regime is characterized by an intermediate-frequency plateau in the value of . We find that the value of in this plateau region is proportional to T ⁻² where T is the temperature; this is in agreement with the expected result of a four-phonon Boltzmann-Peierls equation calculation. The Boltzmann-Peierls approach fails, however, to give a nonvanishing bulk viscosity for all FPU-α β chains. We call the highest frequency regime the collisionless regime since at these frequencies the observing times are much shorter than the characteristic relaxation times of phonons.     

Determination of the η and η<Superscript>′</Superscript> mixing angle from the pseudoscalar transition form factors

The possible range of the η– mixing angle is determined from the transition form factors F_ηγ(Q²) and with the help of up-to-date experimental data. For this purpose, the quark-flavor mixing scheme is adopted and the pseudoscalar transition form factors are calculated in the framework of light-cone pQCD, in which the transverse-momentum corrections and the contributions beyond the leading Fock state have been carefully taken into consideration. We construct a phenomenological expression to estimate the contributions to the form factors beyond the leading Fock state, based on their asymptotic behavior at Q²→0 and . By taking the quark-flavor mixing scheme, our results lead to , where the first error comes from the experimental uncertainty and the second error from the uncertainties of the parameters of the wavefunction. The possible intrinsic charm component in η and is discussed, and our present analysis also disfavors a large intrinsic charm component in η and , e.g. . PACS 13.40.Gp; 12.38.Bx; 14.40.Aq     

Parking on a Random Tree

Consider an infinite tree with random degrees, i.i.d. over the sites, with a prescribed probability distribution with generating function G(s). We consider the following variation of Rényi’s parking problem, alternatively called blocking RSA (random sequential adsorption): at every vertex of the tree a particle (or “car”) arrives with rate one. The particle sticks to the vertex whenever the vertex and all of its nearest neighbors are not occupied yet. We provide an explicit expression for the so-called parking constant in terms of the generating function. That is, the occupation probability, averaged over dynamics and the probability distribution of the random trees converges in the large-time limit to (1−α ²)/2 with .     

No Phase Transition for Gaussian Fields with Bounded Spins

Let a<b, and H be the (formal) Hamiltonian defined on Ω by

Quantum Mechanics and Imprecise Probability

An extension of the Born rule, the quantum typicality rule, has recently been proposed [B. Galvan in Found. Phys. 37:1540–1562 (2007)]. Roughly speaking, this rule states that if the wave function of a particle is split into non-overlapping wave packets, the particle stays approximately inside the support of one of the wave packets, without jumping to the others. In this paper a formal definition of this rule is given in terms of imprecise probability. An imprecise probability space is a measurable space endowed with a set of probability measures ℘. The quantum formalism and the quantum typicality rule allow us to define a set of probabilities on (X ^T ,ℱ), where X is the configuration space of a quantum system, T is a time interval and ℱ is the σ-algebra generated by the cylinder sets. Thus, it is proposed that a quantum system can be represented as the imprecise stochastic process , which is a canonical stochastic process in which the single probability measure is replaced by a set of measures. It is argued that this mathematical model, when used to represent macroscopic systems, has sufficient predictive power to explain both the results of the statistical experiments and the quasi-classical structure of the macroscopic evolution.     

Asymptotic Behavior of Inflated Lattice Polygons

We study the inflated phase of two dimensional lattice polygons with fixed perimeter N and variable area, associating a weight exp [pA−Jb] to a polygon with area A and b bends. For convex and column-convex polygons, we calculate the average area for positive values of the pressure. For large pressures, the area has the asymptotic behaviour , where , and ρ<1. The constant K(J) is found to be the same for both types of polygons. We argue that self-avoiding polygons should exhibit the same asymptotic behavior. For self-avoiding polygons, our predictions are in good agreement with exact enumeration data for J=0 and Monte Carlo simulations for J≠0. We also study polygons where self-intersections are allowed, verifying numerically that the asymptotic behavior described above continues to hold.     

Lower Bound of Minimal Time Evolution in Quantum Mechanics

We show that the total time of evolution from the initial quantum state to final quantum state and then back to the initial state, i.e., making a round trip along the great circle over S ², must have a lower bound in quantum mechanics, if the difference between two eigenstates of the 2×2 Hamiltonian is kept fixed. Even the non-hermitian quantum mechanics can not reduce it to arbitrarily small value. In fact, we show that whether one uses a hermitian Hamiltonian or a non-hermitian, the required minimal total time of evolution is same. It is argued that in hermitian quantum mechanics the condition for minimal time evolution can be understood as a constraint coming from the orthogonality of the polarization vector P of the evolving quantum state with the vector of the 2×2 hermitian Hamiltonians and it is shown that the Hamiltonian H can be parameterized by two independent parameters and Θ.     

Towards a measurement of the two-photon decay width of the Standard Model Higgs boson at a Photon Collider

A study of the measurement of the two photon decay width times the branching ratio of the Standard Model Higgs boson with a mass of 120 GeV in photon–photon collisions is presented, assuming a γ γ integrated luminosity of 80 fb⁻¹ in the high energy part of the spectrum. The analysis is based on the reconstruction of the Higgs events produced in the γ γ→H process, followed by the decay of the Higgs into a pair. A statistical error of the measurement of the two-photon width, Γ(H→γ γ), times the branching ratio of the Higgs boson, BR is found to be 2.1% for one year of data taking.     

Analysis of Ωb? (bss) and Ωc0 (css) with QCD sum rules

We calculate the masses and the pole residues of the heavy baryons Ω _c ⁰(css) and Ω _b ⁻(bss) with the QCD sum rules. The numerical values  GeV (or  GeV) and  GeV (or  GeV) are in good agreement with the experimental data.     

Random Walk Weakly Attracted to a Wall

We consider a random walk X _n in ℤ₊, starting at X ₀=x≥0, with transition probabilities

<Emphasis Type="Italic">X</Emphasis>(3872) and other possible heavy molecular states

We perform a systematic study of the possible molecular states composed of a pair of heavy mesons such as , , in the framework of the meson exchange model. The exchanged mesons include the pseudoscalar, scalar and vector mesons. Through our investigation, we find the following results. (1) The structure X(3764) is not a molecular state. (2) There exists strong attraction in the range r<1 fm for the system with J=0,1. If future experiments confirm Z ⁺(4051) as a loosely bound molecular state, its quantum number is probably J ^P =0⁺. Its partner state Φ ^**0 may be searched for in the π ⁰ χ _c1 channel. (3) Vector meson exchange provides strong attraction in the channel together with pion exchange. A bound state solution may exist with a reasonable cutoff parameter Λ∼1.4 GeV. X(3872) may be accommodated as a molecular state dynamically although drawing a very definite conclusion needs further investigation. (4) The molecular state may exist.     

Moment Bounds for the Smoluchowski Equation and their Consequences

We prove bounds on moments of the Smoluchowski coagulation equations with diffusion, in any dimension d ≥ 1. If the collision propensities α(n, m) of mass n and mass m particles grow more slowly than , and the diffusion rate is non-increasing and satisfies for some b ₁ and b ₂ satisfying 0 ≤ b ₂ < b ₁ < ∞, then any weak solution satisfies for every and T ∈(0, ∞), (provided that certain moments of the initial data are finite). As a consequence, we infer that these conditions are sufficient to ensure uniqueness of a weak solution and its conservation of mass. This work was performed while A.H. held a postdoctoral fellowship in the Department of Mathematics at U.B.C. This work is supported in part by NSF grant DMS0307021.     

Combinatorial entropies and statistics

We examine the combinatorial or probabilistic definition (“Boltzmann’s principle”) of the entropy or cross-entropy function H ∝ or D ∝ - , where is the statistical weight and the probability of a given realization of a system. Extremisation of H or D, subject to any constraints, thus selects the “most probable” (MaxProb) realization. If the system is multinomial, D converges asymptotically (for number of entities N ↦∞) to the Kullback-Leibler cross-entropy D_KL; for equiprobable categories in a system, H converges to the Shannon entropy H_Sh. However, in many cases or is not multinomial and/or does not satisfy an asymptotic limit. Such systems cannot meaningfully be analysed with D_KL or H_Sh, but can be analysed directly by MaxProb. This study reviews several examples, including (a) non-asymptotic systems; (b) systems with indistinguishable entities (quantum statistics); (c) systems with indistinguishable categories; (d) systems represented by urn models, such as “neither independent nor identically distributed” (ninid) sampling; and (e) systems representable in graphical form, such as decision trees and networks. Boltzmann’s combinatorial definition of entropy is shown to be of greater importance for “probabilistic inference” than the axiomatic definition used in information theory.