LABORATORY 4

Head of Laboratory – Dr.Sc. (Mathematics) **Robert Minlos**

Tel.: (095) 299-83-54; E-mail: __minl@iitp.ru__

The leading researchers of the laboratory include:

Dr.Sc. (Math.) |
L. Bassalygo |
Dr.Sc. (Math.) |
V. Prelov |

Dr.Sc. (Math.) |
M. Blank |
Dr.Sc. (Math.) |
S. Shlosman |

Dr.Sc. (Math.) |
V. Blinovsky |
Dr.Sc. (Math.) |
Yu. Suhov |

Dr.Sc. (Math.) |
A. Kirillov |
Dr.Sc. (Math.) |
M. Tsfasman |

Dr.Sc. (Math.) |
G. Margulis |
Dr. |
A. Rybko |

Dr.Sc. (Math.) |
N. Nadirashvili |
Dr. |
V. Shehtman |

Dr.Sc. (Math.) |
G. Olshanski |
Dr. |
S. Vladuts |

Directions of activity:

- the Gibbs random fields and Markov chains with local interactions;
- mean-field models of queuing systems;
- fluid models of queuing networks;
- large deviations and its applications;
- queuing systems;
- systems of information transmission, information channels and coding theory;
- algebraic geometry and number theory;
- combinatorial and probabilistic aspects of representation theory;
- modal logics.

MAIN RESULTS

The uniqueness of Gibbs measure (in a bounded range of temperatures) in the space of trajectories induced by the Feynmann-Kac representation is proved for a quantum system of angarmonic oscillators on the lattice.

The limiting Hamiltonian describing elementary excitations of bound states for a weak-coupled system of lattice planar rotators is constructed. Also one-particle invariant subspaces of this Hamiltonian are found. The limiting Hamiltonian and its one-particle subspaces are found for the case of a lattice quantum system of 3-D rotators (a quantum Heisenberg model).

The spectral structure (point spectrum with the localization of eigenvectors) and its position have been found for a generator of stochastic dynamic of 1-D Ising model with random interaction.

A cluster expansion for a system of quantum oscillator with multidimensional spin space is constructed.

A central limit theorem is proved for a random walk of a particle in a random environment which is a Markov random field with short memory.

The uniqueness of a Gibbs measure on trajectories induced my Feynmann-Kac representation is proved for a weak-coupled system of lattice spin from a compact manifold.

Spectral properties of a 1-D stochastic Ising model with random bounded couplings are studied. It is proved that the integrated density of states for the generator of the corresponding dynamics near the upper spectrum edge has the form of the Lifshitz' tail. The asymptotic formula for the relaxation to equilibrium in average over the disorder for the system is obtained.

For a classical gas of particles in R^d interacting via a pair potential a region of parameters (inverse temperature and chemical potential) for that the set of Gibbs states is unique or it does not exist is investigated.

Ergodic properties of random maps from a compact set into itself are studied. Under some assumptions the quasi-compactness of a corresponding Perron-Frobenius operator is proved and its spectrum is analyzed. In this case it is shown that its spectrum constist of not more than a countable number of isolated eigenvalues of finite multiplicity.

It is proved that Markov processes semigroups describing finite closed networks converge to a limiting determinant dynamical system. The attractors of the limiting dynamical system for a class of closed networks is described.

Ergodic properties for open networks (with increasing to the infinity trajectory of corresponding hydrodynamical models) are investigated. A non-ergodicity of the original random process is proved under the assumption of the existence of a stable trajectory.

A Poisson hypothesis for general closed symmetric networks in the thermodynamical limit on finite time intervals is proved.

A stationary channel with a random parameter which is a completely singular stationary process independent of an input signal is considered. It is shown that under rather weak additional conditions, the information rate between the input signal

and output signal of such a channel coincides with the conditional information rate.

The capacity of non-binary constant-weight codes of weight one correcting a single localized error was find. For nonbinary codes it is proved that the Hamming bound

is asymptotically sharp in some range of the code rate.

The proofs of several theorems on codes are clarified using the Radon transformation for the suitable pair of dual homogeneous spaces. The new lattice invariant is introduced, namely, the formal power series which generalises the theta function and which is close to the McWilliams weight enumerator in coding theory.

For sequences of number fields with growing discriminant we prove generalizations of the Odlyzko – Serre bounds and of the Brauer – Siegel theorem, taking into account archimedean places.

The probability that the group of points of an elliptic curve over a fixed finite field is cyclic is investigated. An asymptotic formula for this probability (the cardinality of the field tending to infinity) is given. A description is given for the finite fields over which the group of all elliptic curves is cyclic.

It is shown that the minimum r-weight d_r of an anticode can be expressed in terms of the maximum r-weight of the corresponding code. As examples, we consider anticodes from homogeneous hypersurfaces (quadrics and Hermitian varieties). In a number of cases, all differences (except for one) of the weight hierarchy of such an anticode meet an analog of the generalized Griesmer bound.

It is proved that the weight function of a linear code, that is, an integer function defined on the vector space of messages, uniquely determines the code up to equivalence.

An axiomatisation of modal logic of connected topological spaces with universal and local modal operators is proposed. Completeness and the finite model property of this logic is proved. Studies of compactness property of modal and intermediate logics with respect to topological semantics are continued. It is proved that every modal logic with a single transitive modality is strongly complete (compact) in topological semantics if it is complete in relational semantics. We give counterexample to this claim in the bimodal case.

For three-paramereters family of stochastic point processes on 1-D lattice (arising in the representation theory) it is proved that correlation functions of these processes are given by a determinant formula with a kernel expressed in terms of Gauss hypergeometric function.

It is shown that a serie of known results about the relation of asymptotic combinatoric promblems and point ensemble, arising from random matrices, can be obtained as degenerations of a model, related to representations of an infinite simmetric group.

For an asymptotic of Plansherel measures on big Young diagrams it is shown that the local structure of typical random diagrams "inside" of the limit curve converges to a point random process with determinant correlation functions.

A new deduction of results about correlation functions for point processes with different families of partitions is proposed.

A version of Bike-Deift-Johanson hypothesis is proved. This hypothesis associate the asymptotic of the string length of Young random diagram with the asymptotic of maximum eigenvalues of random matrices.

CRANTS FROM:

**Russian Foundation of Basic Research (No. 98-01-00303):**"Infinite-dimensional groups and quantum algebras" (head – Grigirii Ol'shanskii).**Russian Foundation of Basic Research (No. 99-01-00828):**"Information transmission and its protection in arbitrarily varying channels" (head – L. Bassalygo).**Russian Foundation of Basic Research (No. 99-01-00284):**"Gibbs states and dynamic systems" (head – R. Minlos).**Russian Foundation of Basic Research (No. 99-01-00003):**"Asymptotic methods of analysis of multicomponent systems: statistical mechanics and queuing networks" (head – F.Karpelevich).**Russian Foundation of Basic Research (No. 99-01-01204):**"Algebraic-geometric and number-theoretic methods in noise stable coding" (head – M. Tsfasman).

Publications in 1999

- Albeverio S., Kondratiev Yu., Rebenko A., Minlos R. Small mass behavior of quantum Gibbs states for lattice model with unbounded spin // J. Stat. Phys. 1999. V. 92. No. 5/6. P. 1153-1172.
- Kondratiev Yu.G., Minlos R.A., Zhizhina E.A. Lower branches of spectrum of infinite-particle Hamiltonian with compact spin space // Proceed. of Moscow Math. Soc. 1999. V. 60, P. 259-302.
- Albeverio S., Minlos R., Scacciatelli E., Zhizhina E. Spectral analysis of the disorder stochastic 1-D Ising operator // Comm. Math. Phys. 1999.
- Minlos R., Faris W. A quantum crystal with multidimensional anharmonico-scillators // J. Stat. Phys. 1999.
- Boldrigini C., Minlos R., Pellegrinotti A. Random walks in a fluctuating random environment with Markov evolution // Submitted to Adv. Sov. Math.
- Albeverio S., Minlos R. A., Scacciatelli E., Zhizhina E. Spectral analysis of the disordered stochastic 1-D Ising model // Commun. Math. Phys. 1999. No. 204. P. 651-668.
- Zhizhina E. The Lifshitz tail and relaxation to equilibrium in the 1-D disordered Ising model // J. Stat. Phys. (to appear).
- Kondratiev Yu., Minlos R., Rockner M., Schepanuk G. Exponential mixing for classical continuous systems // Letter of Math. Phys. (in print).
- Minlos R. One- and two-particle branches of the spectrum of "many-component" operators // A Collection of Papers Dedicated to the 60
^{th}Anniversary of of S. Albeverio (accepted for publication). - Minlos R.A. The corpuscular structure of the spectra of operators describing large system // A Collection of Papers Published by the London Mathematical Society (accepted for publication).
- Angelesku N., Minlos R., Zagrebnov V. The lower branch of the spectrum of the generator of the stochastic dynamics for the classical Heisenberg model // Advanced Sov. Math. (submitted).
- Angelesku N., Minlos R., Zagrebnov V. One-particle branch of spectrum of Hamiltonian of lattice weak-coupling field with values on two-dimensional sphere // J. Math. Phys. (submitted).
- Albeverio S., Kondratiev Yu., Minlos R., Schepanuk G. Uniqueness problem for quantum lattice system with compact spin // Phys. Letters. (submitted).
- Pechersky E.A., Zhukov Yu. Uniqueness of Gibbs state for Nonideal Gas in R^d: the case of pair potentials // J. Stat. Phys. 1999. V. 97. No. 1/2. P. 145-172.
- Karpelevich F.I., Rybko A.N. Thermodynamical limit for symmetrical closed queueing network // In memory of R. L. Dobrushin (AMS, Providence).
- Karpelevich F.I., Rybko A.N. Thermodynamical limit for the Markov processes describing some closed queueing networks // Markov Pocesses and Relat. Topics. 1999. No. 5.
- Blank M.L. Ruelle resonances for random maps. In: “International Conference on Differential Equations and Functional Differential Equations” // Steklov Inst. of Mathematics, MAI, MMS, August 16-21, 1999.
- Shehtman V.B. Everywhere and Here // J. Appplied Non-Classical Logic. V. 9, No. 2-3. P. 369-380.
- Shehtman V.B. On strong neighbourhood completeness of modal and intermediate propositional logics, part 2. In: JFAK. Essays dedicated to Johan Van Benthem on the occasion of his 50th birthday. University of Amsterdam.
- Pinsker M. S., Prelov V. V., van der Meulen E. C.. Stationary Channels with a Random Parameter which is a Completely Singular Process // Probl. Peredachi Inf. 1999. V. 35. No. 1. P. 3-12.
- Pinsker M.S., Prelov V.V., van der Meulen E.C. On Certain Channels with a Random Parameter // Proc. 20-th Symp. Inform. Theory in the Benelux. Haasrode, May 27-28, 1999. P. 165-172.
- Pinsker M.S., Prelov V.V., van der Meulen E.C. Information Rates and its Asymptotics for a General Class of Channels with a Random Parameter // Proc. Intern. Conf. Computer Science and Inform. Technologies. Yerevan, August 17-22, 1999. P. 146-148.
- Lebedev V.S. Correcting a Singl Localized Error by Non-Binary Constant-Weight Codes of Weight One // Probl. Peredachi Inf. 1999. V. 35. No. 1. P. 38-43.
- Bassalygo L.A., Pinsker M.S.. Centered error-correcting codes // Probl. Peredachi Inf. 1999. V. 35. No. 1. P. 30-37.
- Ahlswede R., Bassalygo L.A., Pinsker M.S. On the Hamming bound for nonbinary localized-error-correcting codes // Probl. Peredachi Inf. 1999. V. 35. No. 2. P. 29-37.
- Boguslavsky M. Lattices, Codes, and Radon Transforms // Proceedings of the INRIA Workshop on Coding and Cryptography. Paris, January, 1999.
- Boguslavski M. Radon Transforms and Packings // Discrete Mathematics and Applications (to appear).
- Nogin D. Higher weights of anticodes and the generalized Griesmer bound // Finite Fields and Their Applications. 1999. No. 5. P. 409-423.
- Nogin D. Generalized Hamming Weight as a Weight Function // INRIA Research Report No.3762, September 1999, 24 p.
- Tsfasman M.A., Vladuts S.G. Asymptotic Properties of Global Fields and Generalised Brauer – Siegel theorem // Pretirage IML (Marseille). No. 99-30/98-35, 63 p.
- Vladuts S.G. Cyclicity statistics for elliptic curves over a finite field // Finite Fields and their Applications. 1999. 5:1, P. 13-25.
- Kirillov A.A. Merits and demerits of the orbit method // Bull. Amer. Math. Soc. (N.S.) 1999. V. 36. No. 4. P. 433--488.
- Borodin A., Olshanski G. Measures on partitions, Robinson-Schensted-Knuth correspondence, and beta=2 random matrix ensembles // Proc. of the MSRI Workshop on Random Matrix Theory (to appear).
- Borodin A., Okounkov A.A Fredholm determinant formula for Toeplitz determinants // Integr. Equat. Operator Theory (to appear).
- Borodin A, Okounkov A., Olshanski G. Asymptotics of Plancherel measures for symmetric groups // Preprint math. CO/9905032.
- Borodin A., Olshanski G. Distributions on partitions, point processes, and the ypergeometric kernel // Comm. Math. Phys. (to appear).
- Okounkov A. Infinite wedge and measures on partitions // Preprint math. RT/9907127.
- Okounkov A. Random Matrices and Random Permutations // Preprint math. CO/9903176.