Volume 1 (2001), Number 3. Abstracts

S. Albeverio, Yu. Kondratiev, T. Pasurek, and M. Röckner. Euclidean Gibbs States of Quantum Crystals [PDF]

We prove existence and uniform a priori estimates for Euclidean Gibbs states corresponding to quantum anharmonic crystals. Our method is based on a characterization of Gibbs measures in terms of their Radon—Nikodym derivatives with respect to local shifts of the configuration space and corresponding integration by parts formulas.

Keywords. Quantum crystals, Euclidean Gibbs states, existence problem

2000 Mathematics Subject Classification. 82B10

M. Blank. Perron—Frobenius Spectrum for Random Maps and its Approximation [PDF]

To study the convergence to equilibrium in random maps, we develop the spectral theory of the corresponding transfer (Perron—Frobenius) operators acting in a certain Banach space of generalized functions (distributions). The random maps under study in a sense fill the gap between expanding and hyperbolic systems, since among their (deterministic) components there are both expanding and contracting ones. We prove the stochastic stability of the Perron—Frobenius spectrum and develop its finite rank operator approximations by means of a `stochastically smoothed' Ulam approximation scheme. A counterexample to the original Ulam conjecture about the approximation of the SBR measure and the discussion of the instability of spectral approximations by means of the original Ulam scheme are presented as well.

Keywords.Perron—Frobenius operator, invariant measure, spectrum, random map, mixing, finite rank approximation

2000 Mathematics Subject Classification. 37A30, 37A25, 37H10

P. Bleher, J. Ruiz, R. Schonmann, S. Shlosman, and V. Zagrebnov. Rigidity of the Critical Phases on a Cayley Tree [PDF]

We discuss statistical mechanics on nonamenable graphs, and we study the features of the phase transition, which are due to nonamenability. For the Ising model on the usual lattice it is known that below the critical temperature fluctuations of magnetization are much less likely in the states with nonzero magnetic field than in the pure states with zero field. We show that on the Cayley tree the corresponding fluctuations have the same order.

Keywords. Tree, nonamenable graph, Ising model, large deviations, droplet

2000 Mathematics Subject Classification. 60F10, 82B20

C. Boldrighini and A. Pellegrinotti. T^-1/4-Noise for Random Walks in Dynamic Environment on Z [PDF]

We consider a discrete-time random walk $X_t$ on $\Z$ with transition probabilities $P(X_{t+1} = x+u \mid X_t=x, \xi) = P_0(u) + c(u; \xi(t,x))$, depending on a random field $\xi =\{\xi(t,x)\colon (t,x)\in \Z\times \Z\}$. The variables $\xi(t,x)$ take finitely many values, are i.i.d.\ and $c(u;\cdot\,)$ has zero average. Previous results show that for small stochastic term the CLT holds almost surely, with dispersion independent of the field. Here we prove that the first correction in the CLT asymptotics is a term of order $T^{-1/4}$ depending on the field, with asymptotically gaussian distribution as $T\to \infty$.

Keywords. Random walk, random environment, Central Limit Theorem

2000 Mathematics Subject Classification. 60J15, 60F05, 60G60, 82B41

E. Dinaburg and Ya. Sinai. A Quasilinear Approximation for the Three-Dimensional Navier—Stokes System [PDF]

In this paper a modification of the 3-dimensional Navier—Stokes system which defines some system of quasilinear equations in Fourier space is considered. Properties of the obtained system and its finite-dimensional approximations are studied.

Keywords. Three-dimensional Navier—Stokes system, quasilinear system, characteristics

2000 Mathematics Subject Classification. 76D05

W. Faris. Ornstein—Uhlenbeck and Renormalization Semigroups [PDF]

The Ornstein—Uhlenbeck semigroup combines Gaussian diffusion with the flow of a linear vector field. In infinite-dimensional settings there can be non-Gaussian invariant measures. This gives a context for one version of the renormalization group. The adjoint of the Ornstein—Uhlenbeck semigroup with respect to an invariant measure need not be an Ornstein—Uhlenbeck semigroup. This adjoint is the appropriate semigroup to analyze the local stability of the invariant measure under the renormalization group.

Keywords. Ornstein—Uhlenbeck semigroup, Mehler semigroup, random field, renormalization group, invariant measure

2000 Mathematics Subject Classification. Primary 81T17, 82B28, 60G60, 47D06; Secondary 60J60

K. Khanin, D. Khmelev, A. Rybko, and A. Vladimirov. Steady Solutions for FIFO Networks [PDF]

We consider the fluid model of a reentrant line with FIFO discipline and look for solutions with constant flows (steady solutions). In the case of constant viscosities we prove the uniqueness of such a solution. If viscosities are different, we present an example with multiple steady solutions. We also prove that for some classes of reentrant lines uniqueness holds even if the viscosities are different.

Keywords. Kelly networks, fluid models, uniqueness of steady solution, fixed points

2000 Mathematics Subject Classification. 90B10, 94C99, 37Lxx

C. Maes, F. Redig, and M. Verschuere. From Global to Local Fluctuation Theorems [PDF]

The Gallavotti—Cohen fluctuation theorem suggests a general symmetry in the fluctuations of the entropy production, a basic concept in the theory of irreversible processes, based on results in the theory of strongly chaotic maps. We study this symmetry for some standard models of nonequilibrium steady states. We give a general strategy to derive a local fluctuation theorem exploiting the Gibbsian features of the stationary space-time distribution. This is applied to spin flip processes and to the asymmetric exclusion process.

Keywords. Entropy production, nonequilibrium ensembles, fluctuation symmetry

2000 Mathematics Subject Classification. 60K35, 82C05, 82C22

V. Malyshev, A. Yambartsev, A. Zamyatin. Two-Dimensional Lorentzian Models [PDF]

The goal of this paper is to present rigorous mathematical formulations and results for Lorentzian models, introduced in physical papers. Lorentzian models represent two dimensional models, where instead of a two-dimensional lattice one considers an ensemble of triangulations of a cylinder, and natural probability measure (Gibbs family) on this ensemble. It appears that correlation functions of this model can be found explicitly. Such models can be considered as an example of a new approach to quantum gravity, based on the notion of a causal set. Causal set is a partially ordered set, thus having a causal structure, similar to Minkowski space. We consider subcritical, critical and supercritical cases. In the critical case the scaling limit of the light cone can be restored.

Keywords. Gibbs families, transfer matrix, triangulation, random walk, continuous limit

2000 Mathematics Subject Classification. 83C27, 82B41, 37K99

A. Vershik and Yu. Yakubovich. The Limit Shape and Fluctuations of Random Partitions of Naturals with Fixed Number of Summands [PDF]

We consider the uniform distribution on the set of partitions of integer $n$ with $c \sqrt n$ numbers of summands, $c>0$ is a positive constant. We calculate the limit shape of such partitions, assuming $c$ is constant and $n$ tends to infinity. If $c \to \infty$ then the limit shape tends to known limit shape for unrestricted number of summands (see references). If the growth is slower than $\sqrt n$ then the limit shape is universal ($e^{-t}$). We prove the invariance principle (central limit theorem for fluctuations around the limit shape) and find precise expression for correlation functions. These results can be interpreted in terms of statistical physics of ideal gas, from this point of view the limit shape is a limit distribution of the energy of two dimensional ideal gas with respect to the energy of particles. The proof of the limit theorem uses partially inversed Fourier transformation of the characteristic function and refines the methods of the previous papers of authors (see references).

Keywords. Young diagram, partition of integer, limit shape, fluctuations

2000 Mathematics Subject Classification. 05A17, 11P82, 82B05

Moscow Mathematical Journal
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