LABORATORY 4
Dobrushin Mathematics Laboratory

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.)	D. Akhiezer	Dr.Sc. (Math.)	M. Tsfasman
Dr.Sc. (Math.)	L. Bassalygo	Dr.	M. Boguslavskii
Dr.Sc. (Math.)	M. Blank	Dr.	S. Gelfand
Dr.Sc. (Math.)	V. Blinovsky	Dr.	G. Kabatyanskii
Dr.Sc. (Math.)	A. Kirillov	Dr.	A. Kuznetsov
Dr.Sc. (Math.)	M. Kontsevich	Dr.	V. Lebedev
Dr.Sc. (Math.)	G. Margulis	Dr.	D. Nogin
Dr.Sc. (Math.)	M. Men’shikov	Dr.	A. Okun’kov
Dr.Sc. (Math.)	N. Nadirashvili	Dr.	E. Pecherskii
Dr.Sc. (Math.)	G. Olshanski	Dr.	S. Popov
Dr.Sc. (Math.)	D. Panyushev	Dr.	A. Rybko
Dr.Sc. (Math.)	V. Prelov	Dr.	S. Vladuts
Dr.Sc. (Math.)	S. Shlosman	Dr.	S. Yashkov
Dr.Sc. (Math.)	V. Shehtman	Dr.	E. Zhizhina
Dr.Sc. (Math.)	Yu. Suhov	Dr.	Yu. Zhukov

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 asymptotic behavior of sufficiently long directed polimers in a random Markov medium is investigated. The central (integral and local) limit theorems for a distribution of a polimer edge under the fixed start point are obtained.

The Gibbs modification of -process with low decreasing pair potential is investigated. The existence and uniqueness of a corresponding limit measure.

The spectral properties of the generator of the Glauber dynamics for a 1-D disordered stochastic Ising model with random bounded couplings are studied. An asymptotic formula for the integrated density of states of the generator near the upper edge of the spectrum has been obtained. As a consequence the asymptotics for the average over the disorder of the time-autocorrelation function has been found.

Logarithmic asymptotics of probabilities of large delays are derived for the "last come-first served'' system and system with priorities. Trajectories that determine the mean dynamics of arrival flow under the condition of large delay are described.

Probabilities of large delay in several queueing systems are investigated. One-server systems, which obey various service rules, are considered. The concepts of the Cram\'er transform and the large deviations are discussed for a homogeneous random walk on the d-dimensional Lobachevsky, or hyperbolic, space , with the emphasis on geometric and algebraic aspects. In dimension d=3 the form of the Cramer transform is derived.

We present a unified approach to establishing the Gibbsian character of a wide class of non-Gibbsian states, arising in the Renormalisation Group theory.

The geometric solutions of the various extremal problems of statistical mechanics and combinatorics. Together with the Wulff construction, which predicts the shape of the crystals, the construction which exhibit the shape of a typical Young diagram and of a typical skyscraper, is discussed.

A new property of typical configurations of the low temperature pure phases, which we call the Amoeba Property is established. This property is less restrictive than the Path Large Deviation property. It allows us to extend the range of the validity of the Weakly Gibbs property of the random fields.

Statistical mechanics on non-amenable graphs are discussed, and the features of the phase transition, which are due to non-amenability are studied. For the Ising model on the usual lattice it is known that fluctuations of magnetization are much less likely in the states with non-zero magnetic field than in the pure states with zero field. It is shown that on the Cayley tree the corresponding fluctuations have the same order.

The three-parameter family of point stochastic processes on the 1-D lattice arising in the infinite symmetric group representation theory is investigated. It is proved that correlation functions of these processes are given by a determinant function with Gauss hypergeometric function.

The asymptotic of Plancherel measures on large Young diagrams are considered. It is shown that the local structure of typical stochastic Young diagrams "inside" of the limit curve converges to a point stochastic process with determinant correlation functions.

Examples of non-negatives harmonic functions on the Young lattice are constructed.

The maximum cardinality of binary codes (linear and nonlinear) is estimated under arbitrary restrictions are imposed on the set of distances between codewords. The focus was on the case when a single distance is forbidden. A series of asymptotic results about the minimal number of weighting for the problem of false coin search is obtained. The positive result is obtained for a mathematical problem in copyright for the digital form of the representation of a subject.

The problem of the calculation of information rate in stationary memoryless channels with an additive noise and a slowly varying input signal is considered. Under the assumption that the input signal is a stationary Markov chain with rare transitions, it is shown that the information rate is asymptotically equivalent to the entropy of the chain and, therefore, the main term of its asymptotics does not depend on the channel noise.

The problem of filtering for a singular stationary stochastic process under nonstationary distortions is considered. For certain models of nonstationary distortions, sufficient conditions are obtained under which error-free filtering of such processes is possible.

By using the theory of large deviations for sums of independent random variables the logarithmic asymptotics of entropy of ellipsoids in the Hamming space is investigated.

The two-parameter family of unitarily invariant probability measures on a space of infinite Hermitian matrices is studied. It is shown that an expansion of each such measure into ergodic components is described by a determinant point process on a one-dimensional space. Its correlation kernel is computed.

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 2000

Boldrighini C., Minlos R., Pellegrinotti A. Random walk in random environment with Markov evoluthion. In: AMS. Math Soc. transl. (2) vol 198, 2000, pp.13-35.

Lorenzi J., Minlos R.A. Gibbs distibution for path measure with help two point interaction, submitted to Comm. Math. Phys.

Zhizhina E. The Lifshitz tail and relaxation to equilibrium in the one-dimensional disordered Ising model // J. Stat. Phys. 2000. V. 98. No 3/4. P. 701-721.

Pechersky E., Suhov Yu., Vvedenskaya N.. Large deviations for some queuing systems // Probl. Peredachi Inf. 2000. V. 36. No. 1. P. 48-59 (in Russian).

Karpelevich F.I., Pechersky E.A., Suhov Yu.M. The Cramеr Transform And Large Deviations On The 3d Lobachevsky Space.

Maes Ch., Redig F., Shlosman S., van Moffaert A. Percolation, Path Large Deviations and Weakly Gibbs States // Comm. Math. Phys. 2000. No. 209. P. 517-545.

Chayes L., Shlosman S, Zagrebnov V. Discontinuity of the Magnetization in Diluted O(n) Models // J. Statist. Phys. 2000. No. 98 P. 537-549.

Shlosman S. Geometric variational problems of statistical mechanics and of combinatorics. Probabilistic techniques in equilibrium and nonequilibrium statistical physics // J. Math. Phys. 2000. No. 41. P. 1364-1370.

Shlosman S. Path Large Deviation and Other Typical Properties of the Low-Temperature Models, with Applications to the Weakly Gibbs States // Markov Processes and Related Fields. 2000. No. 6, P. 121-134.

Shlosman S., Tsfasman M. Random Lattices and Random Sphere Packings: Typical Properties, arXiv.org e-Print archive, math-ph/0011040, accepted by Moscow Math. Journal.

S. Shlosman. Wulff construction in statistical mechanics and in combinatorics, arXiv.org e-Print archive, math-ph/0010039, accepted by Russian Math. Surveys.

Bleher P., Ruiz J., Schonmann R.H., Shlosman S., Zagrebnov V. Rigidity of the critical phases on a Cayley tree, http://rene.ma.utexas.edu/mp\_arc/, \# 00-418, submitted to Comm. Math. Phys.

Pinsker M.S., Prelov V.V., van der Meulen E. Information Rate of Slowly Varying Input Signals over Discrete Memoryless Stationary Channels // Probl. Peredachi Inf. 2000. V. 36. No. 3. P. 29-38.

Pinsker M.S., Prelov V.V. The problem of filtering for a singular stationary stochastic process under nonstationary distortions // Probl. Peredachi Inf. 2000. V. 36. No. 4. P. 3-10.

Pinsker M.S., Prelov V.V., van der Meulen E.C. Information Transmission of Slowly Varying Input Signals over Discrete Memoryless Stationary Channels // Proc. 21-th Symp. Inform. Theory in the Benelux. Wassenaar, May 25-26, 2000. P. 277-283.

Prelov V.V. On the Entropy of Ellipsoids in the Hamming Space // Proc. Seventh Intern. Workshop Algebraic and Combinatorial Coding Theory, Bansko, Bulgaria, June 18-24, 2000, P. 269-272.

Pinsker M. S., Prelov V. V. On Error-Free Filtering under Dependent Distortions // Proc. IEEE Intern. Symp. Inform. Theory. Sorrento, Italy, June 25--30, 2000. P. 359.

Pinsker M.S., Prelov V.V., van der Meulen E.C. Transmission of a Slowly Varying Markov Signal over Memoryless Channels // Proc. IEEE Intern. Symp. Inform. Theory. Sorrento, Italy, June 25-30, 2000. P. 488.

Barg A., Kabatyansky G. Codes with identifiable parent property: the case of multiple parents // Proc. of the 7th International Workshop on Algebraic and Combinatorial Coding Theory, Bansko, Bulgaria, June 18-24, 2000, p. 68-71.

Barg A., Nogin D. Bounds on packings of spheres in the Grassmann manifolds // DIMACS Technical Report 2000-19, DIMACS, New Jersey.

Ashikhmin A., Litsyn S., Tsfasman M. Asymptotically Good Quantum Codes // Phys. Rev. A (in print).

Shlosman S., Tsfasman M. "Random Lattices and Random Sphere Packings: Typical Properties", Moscow Math. J. (in print).

Bassalygo L.A., Pinsker M.S. Вычисление асимптотики суммарной пропускной способности М-частотного бесшумного канала с множественным доступом для Т пользователей // Probl. Peredachi Inf. 2000. V. 36. No. 2. P. 3-9.

Bassalygo L., Cohen G., Zemor G. Codes with forbidden distances // Discrete Mathematics. 2000. V. 213. P. 3-11.

Kabatyansky G., Lebedev V., Thorpe J. The Mastermind game and the rigidity of Hamming spaces // Proc. IEEE on Information Theory, 2000, Sorento, Italy.

Borodin A., Olshanski G. Distributions on partitions, point processes, and the hypergeometric kernel // Comm. Math. Phys. 2000. No. 211. P. 335-358.

Borodin A, Okounkov A., Olshanski G. Asymptotics of Plancherel measures for symmetric groups // J. Amer. Math. Soc. 2000. No. 13. P. 481-51.

Borodin A., Olshanski G. Harmonic functions on multiplicative graphs and interpolation polynomials. // Electronic J. Comb. 2000. No. 7, paper 38.

Borodin A., Olshanski G. Infinite random matrices and ergodic measures // Preprint, 2000, 39 pp., math-ph/0010015.

Olshanski G., Regev A., Vershik A. Frobenius – Schur functions: summary of results // Preprint, 2000, 12 pp., math.CO/0003031.

Blank M. Variational principles in the analysis of traffic flows (Why it is worth to go against the flow.) // Markov Processes and Related Fields. 2000. V. 6. No. 3. P. 287-304.

Blank M. Exact analysis of dynamical systems arising in models of transport flows // Uspehi Math. Nauk. 2000. V. 50. No. 3. P. 167-168.