Infinitely divisible distribution functions of class L and the Lee-Yang theorem

It is shown that the free-energy density of a large class of ferromagnets satisfying the Lee-Yang property is to be connected with the limit characteristic function of a suitably renormalized sum of independent and non-identically distributed random variables. Using the canonical representation formulae of such characteristic functions, various chains of inequalities are derived for the Ursell functions.     

Incoherent convergence of diffraction free fields

We describe a method for the synthesis of diffraction free fields by means of an ensemble of optical fields mutually incoherent. The constituent optical fields are generated by controlling the angular correlation function between two points distributed on a circle in the frequency space. The angular position and the separation between the points are considered as random variables. The implicit probability density function allows us to generate diffraction free beams with easily tunable profiles. Inverse problems are also analyzed, which consists in finding the joint probability density function for a known irradiance distribution. Experimental and computational results are shown.     

Multibeam scintillation cumulative distribution function

The gamma-gamma probability density function is commonly used to model the scintillation of a single laser beam propagating through atmospheric turbulence. One method proposed to reduce scintillation at the receiver plane involves the use of multiple channels propagating through independent paths, resulting in a sum of independent gamma-gamma random variables. Recently, a novel approach for an accurate, closed-form approximation for the sum of independent, identically distributed gamma-gamma random variables was introduced by Chatzidiamantis et al. [GLOBECOM 2009--2009 IEEE Global Telecommunications Conference (2009)]. Using this approximation, we present the first analytic model for the distribution of irradiance due to propagating multiple independent beams. This model compares favorably to wave-optics simulations.     

Grassmann phase space methods for fermions. I. Mode theory

In both quantum optics and cold atom physics, the behaviour of bosonic photons and atoms is often treated using phase space methods, where mode annihilation and creation operators are represented by c-number phase space variables, with the density operator equivalent to a distribution function of these variables. The anti-commutation rules for fermion annihilation, creation operators suggest the possibility of using anti-commuting Grassmann variables to represent these operators. However, in spite of the seminal work by Cahill and Glauber and a few applications, the use of Grassmann phase space methods in quantum–atom optics to treat fermionic systems is rather rare, though fermion coherent states using Grassmann variables are widely used in particle physics.     

An Application of the Renormalization Group Method to Stable Limit Laws

Let \((X_n)_{n=1}^{\infty }\) be a sequence of independent identically distributed random variables. We study the normalized partial sums and the corresponding renormalization group flow in the space of probability densities. We prove the convergence to stable limit laws under suitable assumptions on the initial density.     

On the Fluctuations about the Vlasov Limit for <Emphasis Type="Italic">N</Emphasis>-particle Systems with Mean-Field Interactions

Whereas the Vlasov (a.k.a. “mean-field”) limit for N-particle systems with sufficiently smooth potentials has been the subject of many studies, the literature on the dynamics of the fluctuations around the limit is sparse and somewhat incomplete. The present work fulfills two goals: 1) to provide a complete, simple proof of a general theorem describing the evolution of a given initial fluctuation field for the particle density in phase space, and 2) to characterize the most general class of initial symmetric probability measures that lead (in the infinite-particle limit) to the same Gaussian random field that arises when the initial phase space coordinates of the particles are assumed to be i.i.d. random variables (so that the standard central limit theorem applies). The strategy of the proof of the fluctuation evolution result is to show first that the deviations from mean-field converge for each individual system, in a purely deterministic context. Then, one obtains the corresponding probabilistic result by a modification of the continuous mapping theorem. The characterization of the initial probability measures is in terms of a higher-order chaoticity condition (a.k.a. “Boltzmann property”).     

Integro-differential equation for joint probability density in phase space associated with continuous-time random walk

We derive an integro-differential equation for the joint probability density function in phase space associated with the continuous-time random walk, with generic waiting time probability density function and external force. This equation permits us to investigate whole diffusion processes covering initial-, intermediate-, and long-time ranges, which can distinguish the evolution details for systems having the same behavior in the long-time limit with different initial- and intermediate-time behaviors. Moreover, we obtained analytic solutions for probability density functions both in velocity and phase spaces, and interesting dynamic behaviors are discovered.     

Asymptotic probability density of nonlinear phase noise

The asymptotic probability density of nonlinear phase noise, often called the Gordon-Mollenauer effect, is derived analytically when the number of fiber spans is large. Nonlinear phase noise is the summation of infinitely many independently distributed noncentral chi2 random variables with two degrees of freedom. The mean and the standard deviation of those random variables are both proportional to the square of the reciprocal of all odd natural numbers. Nonlinear phase noise can also be accurately modeled as the summation of a noncentral chi2 random variable with two degrees of freedom and a Gaussian random variable.     

A random necklace model

We consider a Laplace operator on a random graph consisting of infinitely many loops joined symmetrically by intervals of unit length. The arc lengths of the loops are considered to be independent, identically distributed random variables. The integrated density of states of this Laplace operator is shown to have discontinuities provided that the distribution of arc lengths of the loops has a nontrivial pure point part. Some numerical illustrations are also presented.     

A random necklace model

We consider a Laplace operator on a random graph consisting of infinitely many loops joined symmetrically by intervals of unit length. The arc lengths of the loops are considered to be independent, identically distributed random variables. The integrated density of states of this Laplace operator is shown to have discontinuities provided that the distribution of arc lengths of the loops has a nontrivial pure point part. Some numerical illustrations are also presented.     

On critical phenomena in interacting growth systems. Part I: General

Components which are placed in a finite or infinite space have integer numbers as possible states. They interact in a discrete time in a local deterministic way, in addition to which all the components' states are incremented at every time step by independent identically distributed random variables. We assume that the deterministic interaction function is translation-invariant and monotonic and that its values are between the minimum and the maximum of its arguments. Theorems 1 and 2 (based on propositions which we give in a separate Part II), give sufficient conditions for a system to have an invariant distribution or a bounded mean. Other statements, proved herein, provide background for them by giving conditions when a system has no invariant distribution or the mean of its components' states tends to infinity. All our main results use one and the same geometrical criterion.     

Bosonization and quantum hydrodynamics

It is shown that it is possible to bosonize fermions in any number of dimensions using the hydrodynamic variables, namely the velocity potential and density. The slow part of the Fermi field is defined irrespective of dimensionality and the commutators of this field with currents and densities are exponentiated using the velocity potential as conjugate to the density. An action in terms of these canonical bosonic variables is proposed that reproduces the correct current and density correlations. This formalism in one dimension is shown to be equivalent to the Tomonaga-Luttinger approach as it leads to the same propagator and exponents. We compute the one-particle properties of a spinless homogeneous Fermi system in two spatial dimensions with long-range gauge interactions and highlight the metal-insulator transition in the system. A general formula for the generating function of density correlations is derived that is valid beyond the random phase approximation. Finally, we write down a formula for the annihilation operator in momentum space directly in terms of number conserving products of Fermi fields.     

Density of near-extreme events

We provide a quantitative analysis of the phenomenon of crowding of near-extreme events by computing exactly the density of states (DOS) near the maximum of a set of independent and identically distributed random variables. We show that the mean DOS converges to three different limiting forms depending on whether the tail of the distribution of the random variables decays slower than pure exponential, faster than pure exponential, or as a pure exponential function. We argue that some of these results would remain valid even for certain correlated cases and verify it for power-law correlated stationary Gaussian sequences. Satisfactory agreement is found between the near-maximum crowding in the summer temperature reconstruction data of western Siberia and the theoretical prediction.     

Rigorous mean-field model for coherent-potential approximation: Anderson model with free random variables

A model of a randomly disordered system with site-diagonal random energy fluctuations is introduced. It is an extension of Wegner'sn-orbital model to arbitrary eigenvalue distribution in the electronic level space. The new feature is that the random energy values are not assumed to be independent at different sites, but free. Freeness of random variables is an analog of the concept of independence for noncommuting random operators. A possible realization is the ensemble of randomly rotated matrices at different lattice sites. The one- and two-particle Green functions of the proposed Hamiltonian are calculated exactly. The eigenstates are extended and the conductivity is novanishing everywhere inside the band. The long-range behavior and the zero-frequency limit of the two-particle Green function are universal with respect to the eigenvalue distribution in the electronic level space. The solutions solve the CPA equation for the one- and two-particle Green function of the corresponding Anderson model. Thus our (multisite) model is a rigorous mean-field model for the (single-site) CPA. We show how the Lloyd model is included in our model and treat various kinds of noises.     

Correlations of record events as a test for heavy-tailed distributions

A record is an entry in a time series that is larger or smaller than all previous entries. If the time series consists of independent, identically distributed random variables with a superimposed linear trend, record events are positively (negatively) correlated when the tail of the distribution is heavier (lighter) than exponential. Here we use these correlations to detect heavy-tailed behavior in small sets of independent random variables. The method consists of converting random subsets of the data into time series with a tunable linear drift and computing the resulting record correlations.     

Thermodynamic limit for the one-dimensional Ising systems with random interaction of finite range

The existence of the thermodynamic limit is proved for the random one-dimensional Ising systems under the assumption that the interaction energies are random variables taking on continuous values and the distribution of the random variables is given by a continuous function. It is assumed that the total number of possible configurations for each lattice site is finite and the range of interaction is finite.     

Look at Tempered Subdiffusion in a Conjugate Map: Desire for the Confinement

The Laplace distribution of random processes was observed in numerous situations that include glasses, colloidal suspensions, live cells, and firm growth. Its origin is not so trivial as in the case of Gaussian distribution, supported by the central limit theorem. Sums of Laplace distributed random variables are not Laplace distributed. We discovered a new mechanism leading to the Laplace distribution of observable values. This mechanism changes the contribution ratio between a jump and a continuous parts of random processes. Our concept uses properties of Bernstein functions and subordinators connected with them.     

Representations of Canonical Commutation Relations Describing Infinite Coherent States

We investigate the infinite volume limit of quantized photon fields in multimode coherent states. We show that for states containing a continuum of coherent modes, it is mathematically and physically natural to consider their phases to be random and identically distributed. The infinite volume states give rise to Hilbert space representations of the canonical commutation relations which we construct concretely. In the case of random phases, the representations are random as well and can be expressed with the help of Itô stochastic integrals. We analyze the dynamics of the infinite state alone and the open system dynamics of small systems coupled to it. We show that under the free field dynamics, initial phase distributions are driven to the uniform distribution. We demonstrate that coherences in small quantum systems, interacting with the infinite coherent state, exhibit Gaussian time decay. The decoherence is qualitatively faster than the one caused by infinite thermal states, which is known to be exponentially rapid only. This emphasizes the classical character of coherent states.     

Synchronization in a power-driven moving agent network

Motivated by the fact that couplings between individual units of many real-world complex systems are relevant to energy, we propose a power-driven moving agent network model as a simple representation. The presented network exhibits a directed and time-varying topological structure, where each agent associated with a chaotic oscillator is depicted as a random walker in a planar space, and interactions among agents are established via communication by assigning different emission powers to them. To investigate the effect of power distribution, synchronization is further explored for the power-driven moving agent network. Under the constraint of fast-switching, we theoretically show that synchronization of the agent network is determined by the power density which is independent of both the power distribution and the size of network. Several numerical simulations are given to validate the acquired results.