On the correlation structure of some random point processes on the line

The correlation structure of some remarkable point processes on the one-dimensional real line is investigated. More specifically, focus is on translation invariant determinantal, permanental and/or renewal point processes. In some cases, anomalous (non-Poissonian) fluctuations for the number of points in a large window can be observed. This may be read from the total correlation function of the point process. We try to understand when and why this occurs and what are the anomalous behaviors to be expected.From examples, it is shown that determinantal (fermion) point processes can be super-homogeneous (the number variance grows slower than the number mean) and even hyper-uniform (when variance growth saturates).Renewal point processes with bounded spacings variance are essentially Poissonian (the number variance grows like the number mean as in Poisson models).Under certain conditions, permanental (boson) point processes can be sub-homogeneous or critical (in the sense that the number variance grows faster than the number mean).We give several detailed examples illustrating these properties of interest together with unexpected behaviors.     

Free energy and some sample path properties of a random walk with random potential

We study the asymptotic behavior of the free energy for a model (defined by Sinai) of one-dimensional random walk with random potential. In particular, we obtain a central limit theorem and a strong law of large numbers for this free energy. We use some results on the free energy to study some sample path properties of this random walk which are related respectively to its recurrence and localization. Some exponents describing the recurrence and localization are found.     

On some properties of axicons

The radial and axial pump power distribution in the line focus of a cone (as simplest example of an axicon) is discussed. It is also shown that four conical surfaces covered with antireflection coatings can make an efficient polarizer for radial and tangential polarization of a light beam. Glass cones can be used as efficient pump optics for laser-pumped lasers. The advantages of the novel pumping scheme are discussed in detail for the cases of dye-laser and x-ray laser pumping.     

Nondiffracting random intensity patterns

We report a simple approach for generating nondiffracting random intensity patterns that do not change with propagation. Experimental results are shown where the generating patterns are encoded onto a phase-only liquid-crystal display. We expect that these results will be useful for encryption or surveillance applications.     

On birth and death processes in symmetric random environment

We prove a limit theorem for a process in a random one-dimensional medium, which has been considered before as a model for hopping conduction in a disordered medium. To the edge between the two integersj and (j+ 1) a rate _j > 0 is attached. These _j :j integral are taken as independent, identically distributed random variables, and represent the medium. For given values _j, X(t) is a Markov chain in continuous time which jumps fromj to (j + 1) and from (j + 1) toj at the same rate _j. We show that in many cases there exists normalizing constants y(t) (which tend to oo witht) such that the distribution of X(t)/(t), or more generally of the whole processX(st)/(t) _S0, converges to a limit as t . The limit process is continuous and self-similar.     

On Pearson's random walk and some statistical properties of a quasiperiodic observable in a simple quantum model

The dynamics of the atomic inversion of a quantum model of field-atom interaction is studied from a statistical point of view. We determine its mean motion and its partial recurrence frequencies. We employed the mathematical analysis used by Lagrange, Wintner, and Weyl in their pioneering studies of the perturbed planetary motion and its connection with the studies of Pearson's random walks.Presented at the Symposium on Random Walks, Gaithersburg, MD, June 1982.Research supported in part by the U.S. Department of Energy.     

On some applications of diffusion processes for image processing

We propose a new algorithm inspired by the properties of diffusion processes for image filtering. We show that purely nonlinear diffusion processes ruled by Fisher equation allows contrast enhancement and noise filtering, but involves a blurry image. By contrast, anisotropic diffusion, described by Perona and Malik algorithm, allows noise filtering and preserves the edges. We show that combining the properties of anisotropic diffusion with those of nonlinear diffusion provides a better processing tool which enables noise filtering, contrast enhancement and edge preserving.     

Simulation of random envelope processes

Efficient and practical methods of simulating stationary and nonstationary random envelope processes are presented. The stationary envelope processes are simulated by using the fast Fourier transform while the nonstationary envelope processes are simulated as the square root of the sum of a series of cosine functions and a series of sine functions with random phase angles. Typical applications of the envelope simulation are the simulations of peaks and troughs which play an important role in the analyses of the first excursion probability, fatigue and crack propagation. In particular, applications to the crack propagation under random loadings are demonstrated in detail.     

A formulation of quantum stochastic processes and some of its properties

In an earlier paper by one of us [K.-E. Hellwig (1981)], elements of discrete quantum stochastic processes which arise when the classical probability space is replaced by quantum theory have been considered. In the present paper a general formulation is given and its properties are compared with those of classical stochastic processes. Especially, it is asked whether such processes can be Markovian. An example is given and similarities to methods in quantum statistical thermodynamics are pointed out.     

On the finite size corrections to some random matching problems

We get back to the computation of the leading finite size corrections to some random link matching problems, first adressed by Mézard and Parisi [J. Phys. France 48, 1451 (1987)]. In the so-called bipartite case, their result is in contradiction with subsequent works. We show that they made some mistakes, and correcting them, we get the expected result. In the non bipartite case, we agree with their result but push the analytical treatment further. Received 28 April 2002 Published online 14 October 2002 RID="a" ID="a"e-mail: giorgio.parisi@roma1.infn.it RID="b" ID="b"e-mail: matthieu.ratieville@roma1.infn.it