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Infinite Particle Systems: Complex Systems III (June 2007, Kazimierz Dolny, Poland)

In the years 2002-2005, a group of German and Polish mathematicians worked under a DFG research project No 436 POL 113/98/0-1 entitled "Methods of stochastic analysis in the theory of collective phenomena: Gibbs states and statistical hydrodynamics". The results of their study were summarized at the German-Polish conference, which took place in Poland in October 2005. The venue of the conference was Kazimierz Dolny upon Vistula - a lovely town and a popular place for various cultural, scientific, and even political events of an international significance. The conference was also attended by scientists from France, Italy, Portugal, UK, Ukraine, and USA, which predetermined its international character. Since that time, the conference, entitled "Infinite Particle Systems: Complex Systems" has become an annual international event, attended by leading scientists from Germany, Poland and many other countries. The present volume of the "Condensed Matter Physics" contains proceedings of the conference "Infinite Particle Systems: Complex Systems III", which took place in June 2007.

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Title: Continuous unitary transformation approach to pairing interactions in statistical physics
Author(s):

T.Domański (Institute of Physics, M. Curie Skłodowska University, 20-031 Lublin, Poland)

We apply the flow equation method to the study of the fermion systems with pairing interactions which lead to the BCS instability signalled by the appearance of the off-diagonal order parameter. For this purpose we rederive the continuous Bogoliubov transformation in a fashion of renormalization group procedure where the low and high energy sectors are treated subsequently. We further generalize this procedure to the case of fermions interacting with the discrete boson mode. Andreev-type interactions are responsible for developing a gap in the excitation spectrum. However, the long-range coherence is destroyed due to strong quantum fluctuations.

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Title: Random walks in random environment with Markov dependence on time
Author(s):

	C.Boldrighini (Dipartimento di Matematica, Università di Roma "La Sapienza", Piazzale Aldo Moro 2, 00185 Roma, Italy. Partially supported by INdAM (G.N.F.M.) and M.U.R.S.T. research founds) ,
	R.A.Minlos (Institute for Problems of Information Transmission, Russian Academy of Sciences, B. Karetnyi Per. 19, 127994, GSP-4, Moscow, Russia. Partially supported by RFBR grants 99-01-024, 97-01-00714 and CRDF research funds N RM1-2085) ,
	A.Pellegrinotti (Dipartimento di Matematica, Università di Roma Tre, Largo S. Leonardo Murialdo 1, 00146 Roma, Italy. Partially supported by INdAM (G.N.F.M.) and M.U.R.S.T. research founds)

We consider a simple model of discrete-time random walk on Ζ^ν, ν=1,2,... in a random environment independent in space and with Markov evolution in time. We focus on the application of methods based on the properties of the transfer matrix and on spectral analysis. In section 2 we give a new simple proof of the existence of invariant subspaces, with an explicit condition on the parameters. The remaining part is devoted to a review of the results obtained so far for the quenched random walk and the environment from the point of view of the random walk, with a brief discussion of the methods.

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Title: On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum
Author(s):

	E.Lytvynov (Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, U.K.) ,
	P.T.Polara (Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, U.K.)

We deal with the two following classes of equilibrium stochastic dynamics of infinite particle systems in continuum: hopping particles (also called Kawasaki dynamics), i.e., a dynamics where each particle randomly hops over the space, and birth-and-death process in continuum (or Glauber dynamics), i.e., a dynamics where there is no motion of particles, but rather particles die, or are born at random. We prove that a wide class of Glauber dynamics can be derived as a scaling limit of Kawasaki dynamics. More precisely, we prove the convergence of respective generators on a set of cylinder functions, in the L²-norm with respect to the invariant measure of the processes. The latter measure is supposed to be a Gibbs measure corresponding to a potential of pair interaction, in the low activity-high temperature regime. Our result generalizes that of [Random. Oper. Stoch. Equa., 2007, 15, 105], which was proved for a special Glauber (Kawasaki, respectively) dynamics.

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Title: Extension of explicit formulas in Poissonian white noise analysis using harmonic analysis on configuration spaces
Author(s):

	Yu.G.Kondratiev (Fakultät für Mathematik, Universität Bielefeld, D 33615 Bielefeld, Germany; Fakultät für Mathematik, Universität Bielefeld, D 33615 Bielefeld, Germany 1; National University "Kyiv-Mohyla Academy", Kiev, Ukraine) ,
	T.Kuna (Fakultät für Mathematik, Universität Bielefeld, D 33615 Bielefeld, Germany; Fakultät für Mathematik, Universität Bielefeld, D 33615 Bielefeld, Germany 1),
	M.J.Oliveira (Fakultät für Mathematik, Universität Bielefeld, D 33615 Bielefeld, Germany 1; Universidade Aberta, P 1269-001 Lisbon, Portugal; Universidade Aberta, P 1269-001 Lisbon, Portugal 1)

Harmonic analysis on configuration spaces is used in order to extend explicit expressions for the images of creation, annihilation, and second quantization operators in L²-spaces with respect to Poisson point processes to a set of functions larger than the space obtained by directly using chaos expansion. This permits, in particular, to derive an explicit expression for the generator of the second quantization of a sub-Markovian contraction semigroup on a set of functions which forms a core of the generator.

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Title: Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions
Author(s):

	M.Röckner (Department of Mathematics and BiBoS, Bielefeld University, Bielefeld, Germany; Department of Mathematics and BiBoS, Bielefeld University, Bielefeld, Germany 1) ,
	B.Schmuland (Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada) ,
	X.Zhang (Department of Statistics, School of Mathematics and Statistics, University of New South Wales, Sydney, Australia)

The purpose of this note is to give a complete and detailed proof of the fundamental Yamada-Watanabe Theorem on infinite dimensional spaces, more precisely in the framework of the variational approach to stochastic partial differential equations.

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Title: Equilibrium stochastic dynamics of Poisson cluster ensembles
Author(s):

	L.Bogachev (Department of Statistics, University of Leeds, Leeds LS2 9JT, UK) ,
	A.Daletskii (Department of Mathematics, University of York, York YO10 5DD, UK)

The distribution μ of a Poisson cluster process in Χ=R^d (with n-point clusters) is studied via the projection of an auxiliary Poisson measure in the space of configurations in Χⁿ, with the intensity measure being the convolution of the background intensity (of cluster centres) with the probability distribution of a generic cluster. We show that μ is quasi-invariant with respect to the group of compactly supported diffeomorphisms of Χ, and prove an integration by parts formula for μ. The corresponding equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms.

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Title: Invariance principle for diffusions in random environment
Author(s):

S.Struckmeier (Department of Mathematics, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, Germany)

We will show an invariance principle for the diffusive motion of a particle interacting with a random frozen configuration of infinitely many other particles in R^d. The interaction is described by a symmetric, translation invariant pair potential with repulsion at zero distance and proper decay at infinity.

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Title: Selection-mutation balance models with epistatic selection
Author(s):

	Yu.G.Kondratiev (Universität Bielefeld, Postfach 10 01 31, D-33501 Bielefeld, Germany; BiBoS, Univ. Bielefeld, Germany) ,
	T.Kuna (Universität Bielefeld, Postfach 10 01 31, D-33501 Bielefeld, Germany; BiBoS, Univ. Bielefeld, Germany; University of Reading, Department of Mathematics, Whiteknights, PO Box 220, Reading RG6 6AX, UK) ,
	N.Ohlerich (Universität Bielefeld, Postfach 10 01 31, D-33501 Bielefeld, Germany; BiBoS, Univ. Bielefeld, Germany)

We present an application of birth-and-death processes on configuration spaces to a generalized mutation-selection balance model. The model describes the aging of population as a process of accumulation of mutations in a genotype. A rigorous treatment demands that mutations correspond to points in abstract spaces. Our model describes an infinite-population, infinite-sites model in continuum. The dynamical equation which describes the system, is of Kimura-Maruyama type. The problem can be posed in terms of evolution of states (differential equation) or, equivalently, represented in terms of Feynman-Kac formula. The questions of interest are the existence of a solution, its asymptotic behavior, and properties of the limiting state. In the non-epistatic case the problem was posed and solved in [Steinsaltz D., Evans S.N., Wachter K.W., Adv. Appl. Math., 2005, 35(1)]. In our model we consider a topological space X as the space of positions of mutations and the influence of an epistatic potential on these mutations.

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Title: The Gibbs fields approach and related dynamics in image processing
Author(s):

	X.Descombes (Ariana, Joint group, CNRS/INRIA/UNSA, INRIA, 2004, route des Lucioles, BP93, 06902, Sophia-Antipolis Cedex, France) ,
	E.Zhizhina (Institute for Information Transmission Problems, Bolshoy Karetny per. 19, 127994 GPS-4, Moscow, Russia)

We give in the paper a brief overview of how the Gibbs fields and related dynamics approaches are applied in image processing. We discuss classical pixel-wise models as well as more recent spatial point process models in the framework of the Gibbs fields approach. We present a new multi-object adapted algorithm for object detection based on a spatial birth-and-death process and a discrete time approximation of this process.

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Title: Bassalygo-Dobrushin uniqueness for continuous spin systems on irregular graphs
Author(s):

	D.Kępa (Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, 20-031 Lublin, Poland) ,
	Yu.Kozitsky (Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, 20-031 Lublin, Poland)

An extension of the Bassalygo-Dobrushin technique of proving uniqueness of Gibbs fields on irregular graphs, developed in [Theory of Probab. Appl., 1986, 31, 572-589], to the case of continuous spins has been presented.

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Title: Analysis of urban complex networks
Author(s):

D.Volchenkov (Bielefeld Bonn Stochastic Research Center (BiBoS), University of Bielefeld, Postfach 100131, D-33501, Bielefeld, Germany)

We analyze the dual graph representation of urban textures by the methods of complex network theory and spectral graph theory. We present the empirical diagrams of distributions of the nearest and far-away neighbors in the several European compact urban patterns and the spectra of normalized Laplace operator defined on their dual graphs.

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Title: Modelling complex networks by random hierarchical graphs
Author(s):

M.Wróbel (Institute of Mathematics, Maria Curie-Skłodowska University, Lublin, Poland)

Numerous complex networks contain special patterns, called network motifs. These are specific subgraphs, which occur oftener than in randomized networks of Erdős-Rényi type. We choose one of them, the triangle, and build a family of random hierarchical graphs, being Sierpiński gasket-based graphs with random "decorations". We calculate the important characteristics of these graphs - average degree, average shortest path length, small-world graph family characteristics. They depend on probability of decorations. We analyze the Ising model on our graphs and describe its critical properties using a renormalization-group technique.

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Title: On the implementation of cryptoalgorithms based on algebraic graphs over some commutative rings
Author(s):

	J.S.Kotorowicz (University of Maria Curie-Skłodowska, Plac M.C. Skłodowkiej 1, 20-031 Lublin, Poland) ,
	V.A.Ustimenko (University of Maria Curie-Skłodowska, Plac M.C. Skłodowkiej 1, 20-031 Lublin, Poland)

The paper is devoted to computer implementation of some graph based stream ciphers. We compare the time performance of this new algorithm with fast, but no very secure RC4, and with DES. It turns out that some of new algorithms are faster than RC4. They satisfy the Madryga requirements, which is unusual for stream ciphers (like RC4). The software package with new encryption algorithms is ready for the demonstration.

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Title: Differential functional von Foerster equations with renewal
Author(s):

H.Leszczyński (Univ. Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland)

Natural iterative methods converge to the exact solution of a differential-functional von Foerster-type equation which describes a single population dependent on its past time and state densities as well as on its total size. On the lateral boundary we impose a renewal condition.

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Title: Almost sure functional central limit theorems for multiparameter stochastic processes
Author(s):

	E.B.Czerebak-Morozowicz (Department of Mathematics, Technical University of Rzeszów, ul. Wincentego Pola 2, 35-959 Rzeszów, Poland) ,
	Z.Rychlik (Institute of Mathematics, Maria Curie-Skłodowska University, pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland) ,
	M.Urbanek (Institute of Mathematics, Maria Curie-Skłodowska University, pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland)

We present almost sure central limit theorems for stochastic processes whose time parameter ranges over the d-dimensional unit cube. Our purpose here is to generalize the classic functional central limit theorem of Prokhorov (1956) for such processes. We prove multidimensional analogues of Glivenko-Cantelli type theorems.

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Title: Quantum codes from algebraic curves with automorphisms
Author(s):

T.Shaska (Science and Engineering Building, Department of Mathematics and Statistics, Oakland University, Rochester, MI, 48309 367 Science and Engineering Building, Department of Mathematics and Statistics, Oakland University, Rochester, MI, 48309; University of Maria Curie Sklodovska, Lublin, Poland University of Maria Curie Sklodovska, Lublin, Poland) ,

Let Χ be an algebraic curve of genus g ≥ 2 defined over a field F_q of characteristic p > 0. From Χ, under certain conditions, we can construct an algebraic geometry code C. If the code C is self-orthogonal under the symplectic product then we can construct a quantum code Q, called a QAG-code. In this paper we study the construction of such codes from curves with automorphisms and the relation between the automorphism group of the curve Χ and the codes C and Q.

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Erratum: Inelastic neutron scattering applied to the investigation of collective excitations in topologically disordered matter [Condens. Matter Phys., 2008, vol. 11, 1(53), 7]
Author(s):

J.-B.Suck

In the above mentioned publication the denominator of the prefactor of equation (31) was unfortunately wrongly given.