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

Directions of activity:

· spectral analysis of generators in stochastic model of mathematical physics;

· Gibbs random fields and Markov chains with local interactions;

· mean-field models of queueing systems;

· large deviations and its applications;

· queueing systems;

· combinatorial and probabilistic problems for information transmission and

protection in modern communication systems;

· algebraic geometry and number theory;

· combinatorial and probabilistic aspects of representation theory;

· modal logics.

MAIN RESULTS

For a stochastic model (birth-annihilation model) the one-particle subspace in a configuration space of continuous gas is constructed.

Two leader invariant subspaces for a generator of a Blume-Capel stochastic model (model with nearest neighborhood interaction and three-value spin) are constructed in a high-temperature region. It is shown that on each these subspaces the generator is unitarily equivalent to the operator of production on a bounded function.

For a weakly bounded coupling of a free general quantum lattice system it is proved that if the free system has a non-degenerated bound state and a spectral gap, then the coupled system also has a non-degenerated bound state and a spectral gap.

The central limit theorem is proved for a time-distribution of the end-point of a random walk for arbitrary values of the coupling parameter.

For M/G/1 foreground-background processor-sharing queue with minimal attained service time (a simplified version of TCP/IP protocol) the time-dependent distribution of attained service times is found.

For asymptotic dynamic of infinite-particle system on one-dimensional lattice under the action of a constant force and a friction force it is proved the existence of two different ergodic phases corresponding to two critical values of a particle density and the structure of these phases is studied. The existence of phase transition with an hysteresys is proved and explicit expressions for the life-time of a single particle cluster and the average motion speed are obtained.

Pseudobilliard systems with a special reflection law are introduced and studied. It is shown that such systems may have chaotic, stable and neutral behavior that may coexist in one system.

Iterative stochastic algorithms are proposed for image processing. These algorithms are based on properties of diffusive dynamics (anniling procedure). A number of approximation schemes for numerical solutions of the recovery problem are considered. The convergence of Markov chains to a continuous process is investigated, conditions providing the ergodicity of the Euler approximation scheme are stated.

Queuing systems with two servers and dynamic routing are investigated. Numerical experiments showing a good coincidence of results (obtained with help of large deviations and numerical calculations) are carried out.

A model of magnetostriction is developed. This phenomenon signifies that some substances can discontinuously change their form and size depending on exterior parameters – temperature or magnetic field.

The existence of a first order phase transition is strictly proved for some statistical models with continuous symmetry. The existence of a discontinuous transition is proved for some kind of models with a large entropy.

The problem of 3D-crystall growth is discussed. A hypothesis about a discontinuous growth is stated. The hypothesis is proved for a model “solid-on-solid”.

The validity of the Poisson hypothesis (about the asymptotic behavior of large queuing systems) is established in a naturally general class of time-service distributions. Frameworks of the hypothesis (outside of which it is broken) are indicated,

It is proved that certain difference analogs of the Painleve equations, which have numerous applications in mathematical physics, can be interpreted as isomorphisms of modules of d-connections on the projective line with given singularities. The most general difference Painleve equation known so far is derived; it degenerates to both difference Painleve V and classical (differential) Painleve VI equations.

Odd analogs of the classical and quantum family algebras are introduced and studied. As an example, the structure of a g-module on the exterior power of the adjoint representation for the Lie algebras A_3, A_4, G_2 is explicitly determined.

A new method for computing correlations functions of random point processes was found. This method allows one to obtain simpler proofs of two known results, the Eynard-Mehta theorem and the Okounkov-Reshetikhin formula for the correlation kernels of Schur processes. The new method also make it possible to obtain Pfaffian analogs of these results.

A family of Markov processes on the set of Young diagrams is constructed. It is proved that the correlation functions of these processes have determinantal structure, and the corresponding correlation kernels are explicitly found. The results show that the recently discovered surprising analogy between asymptotic properties of random partitions arising in representation theory and those of spectra of random matrices extends to the associated time-dependent models. A connection with the classical results of Karlin and McGregor is also established (those results, published in the fifties, concern the birth-death processes associated to various systems of orthogonal polynomials).

Two survey papers, based on talks delivered at the 4th European Congress of Mathematics (Stockholm, 2004), were prepared. Their subjects are: (1) connection between enumerative geometry of algebraic curves and random surfaces, and (2) application of random point processes in representation theory.

Linear codes correcting errors associated with multi-dimensional projective manifolds on a finite field are considered. Codes on Shubert submanifolds in Grassman manifolds (Shubert codes) are investigated. Explicit formulas for the distance and dimension of a arbitrary Shubert code are obtained. The hypothesis about the minimal distance is verified for the case of codes associated with Shubert divisors.

Complex spaces with action of compact Lie group (extending spherical algebraical manifolds) are investigated. For such spaces the simplicity of representation spectrum of a given group in linear fibering cut is proved.

A ring of algebraical cobordisms of Pfister quadric is calculated. It is show that this ring has no the rotating.

The irreducibility of commutator manifolds for some simple Lie algebras involutions (i.e. for some symmetric spaces) is proved. The case for symmetric spaces of rang 1 is completely investigated.

A series of papers is devoted to the investigation of ideals in borel subalgebra of simple Lie algebras. Obtained results are related to combinatorial aspects of the theory. The relation of ideal with properties of Weyl affine group elements is investigated.

New results about finite-range approximability of productions of modal logics are obtained. A new method of the proof for the finite-range approximability is constructed. A series of open problems about finite-range approximability of many-dimensional modal logics is solved as well as about the theory of binary relations with Boolean operations and compositions with fixed relations and their inverse ones.

The second order asymptotic for the mutual information between inlet and outlet signal in the case when the ratio signal/noise tends to zero is found for the first time and for a very general model of the communication channel.

The asymptotic of a ellipsoid epsilon-entropy in the Euclidean space is found under the assumption that the dimension of the space tends to infinity. The asymptotic depends only on the volume of subellipsoid with axes having the length greater than two epsilon.

"Good" codes (i.e. with asymptotically non-zero rate) with polynomial (on code length) complexity "parents identification" are constructed. A new solution for the problem of “digital imprints” is proposed for the case of two users coalition.

Researchers of the laboratory are teaching in institutes of Moscow (in particular, Moscow State University and Independent Moscow University). Under their leadership 4 post-graduate students are working. One Ph.D. thesis is defended.

A prize of the European mathematical society is awarded to A. Yu. Okunkov for results in asymptotical combinatorial calculus with applications to the topology of module spaces, the ergodic theory, the random surfaces theory and the algebraic geometry.

GRANTS FROM:

· Grant of the President of Russia for support of leading scientific schools (No. NS-934.2003.1): school by R. A. Minlos.

· Russian Foundation of Basic Research (No. 02-01-00444): "Gibbs States and Dynamic Systems". Project leader R. А. Minlos.

· Russian Foundation of Basic Research (No. 03-01-00098): "Combinatorial and Probabilistic Problems of Information Transmission and Protection for Modern Communication Systems". Project leader L. А. Bassalygo.

Publications in 2004

Books

1. Blank M.L. Lectures on ergodic theory of irreversible transformations. Moscow, MCCME, 2004.

2. Kirillov A. A. Lectures on fractals. Moscow, MCCME, 2004.

3. Kirillov A. A. Lectures on the orbit method. Graduate Studies in Mathematics, vol. 64. Providence, RI: American Mathematical Society, Providence, RI, 2004.

Articles

1. Akhiezer D., Heinzner P. Spherical Stein spaces, Manuscripta math., 2004, 114, pp. 327-334.

2. Boguslavsky M., Boguslavskaya E. Arbitrage under Power. Risk, June 2004, pp. 69-73.

3. Boldrighini C., Minlos R., Pellegrinotti A. Random walks in quenched i.i.d. space-time random envoriment are always a.s. diffusive. Probab. Theory and Relat. Fields, 2004, vol. 129, pp. 133-156.

4. Comets F., Popov S. A note on quenched moderate deviations for Sinai's random walk in random environment. ESAIM: Probability and Statistics, 2004, vol. 8, pp. 6-65.

5. Decombes X., Zhizhina E.A. Method of Gibbs random fields in image processes problems. Information Trasmition Problems, 2004, Vol. 40 (3), pp. 108-125.

6. Duffy K., Malone D., Pechersky E.A., Suhov Yu.M., Vvedenskaya N.D. Large deviations provide good approximation to queueing system i-th dynamic routing. Preprint. No. DIAS-STP-04-15, Dublin, 2004.

7. Dumer I., Pinsker M.S., Prelov V.V. On Covering of Ellipsoids in Euclidean Spaces. IEEE Trans. Inform. Theory, 2004, vol. 50, no. 10, pp. 2348-2356.

8. Kim H.K., Lebedev V.S. On the Optimality of Trivial (w,r)-Cover-Free Codes. Problemy Peredachi Informatsii. 2004, vol. 40, no. 3, pp. 13-20.

9. Kurkova I., Popov S., Vachkovskaia M. On infection spreading and competition between independent random walks. Electronic Journal of Probability, 2004, vol. 9, no. 11, pp. 293-315.

10. Lakshtanov E.A., Minlos R.A. The spectrum of Two-Particle Bound States for the Transfer Matrices of Gibbs Fields (an Isolated Bound State). Functional Analysis and Its Applications, 2004, vol. 38, no. 3, pp. 202-214.

11. MacPhee I.M., Menshikov M.V. Critical random walks on two-dimensional complexes with applications to polling systems. Ann. Appl. Probab., 2003, vol. 13, no. 4, pp.1399-1422.

12. Menshikov M. V., Popov S. Yu., Vachkovskaia, M. On a multiscale continuous percolation model with unbounded defects. Sixth Brazilian School in Probability (Ubatuba, 2002). Bull. Braz. Math. Soc. (N.S.), 34 (2003), no. 3, 417-435.

13. Menshikov M., Petritis D., Popov S. Bindweeds or random walks in random environments on multiplexed trees and their asymptotics. Discrete Math. Theor. Comput. Sci. Proc., AC, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2003.

14. Menshikov M.V., Popov S.Yu., Sisko V., Vachkovskaia, M. On a many-dimensional random walk in a rarefied random environment. Markov Processes and Related Fields, 2004, vol. 10, no. 1, pp. 137-160.

15. Minlos R.A., Kondratiev Yu., Zhizhina E. One-particle subspace of the Glauber dynamics generator for continuous particle systems. Reviews in Math. Phys, 2004, 16 (9), pp. 1-42.

16. Okounkov A., Pandharipande R. Hodge integrals and invariants of the unknot. Geometry and Topology, 2004, 8, pp. 675-699.

17. Olshanski G., Kerov S., Vershik A. Harmonic analysis on the infinite symmetric group. Inventiones Mathematicae, 2004, 158, no. 3, pp. 551-642.

18. Panyushev D.I. Ad-nilpotent ideals of a Borel subalgebra: generators and duality. J. Algebra, 2004, 274, pp. 822-846.

19. Panyushev D.I. Irreducibility of commutator manifolds related with involutions of simple Lie algebra. Funct.Analys. and Its Appl., 2004, 38, Vol. 1, стр. 47-55.

20. Panyushev D.I. Long Abelian ideals. Adv. in Math., 2004, 186, pp. 307-316.

21. Panyushev D.I. Regions in the dominant chamber and nilpotent orbits. Bull. Sci. Math., 2004, 128, pp. 1-6.

22. Panyushev D.I. Short antichains in root systems, semi-Catalan arrangements, and $B$-stable subspaces. Europ. J. Combinatorics, 2004, 25, pp. 93-112.

23. Panyushev D.I. Weight multiplicity free representations, $\mathfrak g$-endomorphism algebras, and Dynkin polynomials. J. London Math. Soc., 2004, 69, Part 2, pp. 273-290.

24. Prelov V.V., Verdu S. Second-Order Asymptotics of Mutual Information. IEEE Trans. Inform. Theory, 2004, vol. 50, no. 8, pp. 1567-1580.

25. Rovinsky M. Admissible semi-linear representations. Preprint. Max-Planck-Inst. Math. 2004.

26. Shlosman S.B., Zagrebnov V.A. Magnetostriction Transition, Journal of Statistical Physics, 2004, vol. 114, no. 3/4, pp. 563-574.

27. Tsfasman M.A. Algebraic geometry, theory of number and dense packing. In: Globus. Math. seminar of NMU. Vol. 1. Moscow, MCCME, 2004, pp. 66-77.

28. Tsfasman M.A. Bourguignon J.-P., Broue M., Digne F., Henniart G., Ilyashenko Yu., Keller B., Kosmann-Schwarzbach Y., Rosso M., Sossinsky A. Pierre Cartier. Moscow Mathematical Journal, 2004, vol. 4, no. 1, pp. 3-4.

29. Tsfasman M.A. Preface to Russian traduction. In.: E.Artin. Galois theory. Moscow, MCCME, 2003, p. 3.

30. Tsfasman M.A. Preface to. In: Глобус. Общематематический семинар Независимого Московского университета. Globus. Math. seminar of NMU. Vol. 1. Moscow, MCCME, 2004, pp. 3-7.

31. Tsfasman M.A., Beilinson A., Belavin A., Drinfeld V., Finkelberg M., Frenkel, Fuchs D., Ilyashenko Yu., Lando S., Sossinsky A., Vassiliev V., Zelevinsky A. Boris Feigin. Moscow Mathematical Journal, 2004, vol. 4, no. 3, pp. 537-538.

32. Tsfasman M.A., Brin M., Hasselblat B., Ilyashenko Yu., Kushnirenko A., Pesin Ya., Sossinsky A. Anatole Katok. Moscow Mathematical Journal, 2004, vol.4, no. 4, pp. 977-979.

33. Tsfasman M.A., Feigin B., Ilyashenko Yu., Manin Yu., Shlosman S. Serge Vladuts. Moscow Mathematical Journal, 2004, vol. 4, no. 2, pp. 531-532.

34. Vershik A., Okunkov A. New approach to the representation theory of symmetric groups II, Proceed. Of Sci. seminars of St.Peterburg Math.Instit., 2004, Vol. 307, стр. 57-98.

35. Yarotsky D.A. Perturbations of ground states in weakly interacting quantum spin systems. J. Math. Phys. 2004, 45, pp. 2134-2152.

36. Yarotsky D.A. Scattering of quasi-particle excitations in weakly coupled stochastic lattice spin systems. Comm. Math. Phys, 2004, 249, pp. 449-474.

37. Yashkov S.F., Yashkova A.S. The time-dependent solution of the M/G/1-FBPS queue. Information Processes, 2004, vol. 4, no. 2, p.175-187.

38. Zhizhina E. Convergence properties of quasi-particles of various species in the stochastic Blume-Capel model. Markov Processes and Related Fields, 2004, vol. 10 (2), pp. 307-326.

In print

1. Adler M., Borodin A., van Moerbeke A. Expectations of hook products on large partitions, 31 pp.; arxiv: math.PR/0409554

2. Arinkin D., Borodin A. Moduli spaces of d-connections and difference Painleve equations, 30 pp.; arXiv: math.AG/0411584.

3. Barg A., Kabatiansky G. Class of i.p.p codes with effective tracing algorithm. Journal of Complexity, 2004.

4. Bassalygo L.A., Zinoviev V.A. About special polynomials on finite fields of odd characteristic and reaching Weyl bound. Matem. Zametki.

5. Blank M. Hysteresis phenomenon in deterministic traffic flows. J. Stat. Phys.

6. Blank M., Bunimovich M. Switched flow systems: pseudo billiard dynamics. Dynamical Systems: An International Journal.

7. Bodineau T., Schonmann R.H., Shlosman S. 3D crystal: how flat its flat facets are? Communications in Math. Physics.

8. Boldrighini C., Minlos R., Pellegrinotti A. Diffusive behavior of directed polymers for small stochastics. Markov Process and Rel. Fields.

9. Borodin A. Isomonodromy transformations of linear systems of difference equations, Annals of Mathematics, 160 (2004), no. 3.

10. Borodin A., Olshanski G. Markov processes on partitions, 73 pp.; arXiv: math-ph/0409075.

11. Borodin A., Olshanski G. Stochastic dynamics related to Plancherel measure on partitions. In a volume dedicated to A. M. Vershik, Amer. Math. Soc., 13 pp.; arXiv: math-ph/0402064.

12. Borodin A., Rains E. Eynard-Mehta theorem, Schur process, and their pfaffian analogs, 17 pp.; arXiv: math-ph/0409059.

13. Borodin A., Strahov E. Averages of characteristic polynomials in random matrix theory. Communications in Pure and Applied Mathematics. http://arxiv.org/abs/math-ph/0407065

14. Golyshev V.V. Modularity of equations D3 and Iskovskih classification. Doklady Akademii Nauk.

15. Kirillov A. A., Rybnikov L. G. Odd family algebras. In a volume dedicated to A. Joseph.

16. Lakshtanov E.A., Minlos R.A. Two-particle spectrum of transfer-matrix of Gibbs fields (2D-case). Functional analysis.

17. Lebensztayn E., Machado F.P., Popov S. An improved upper bound for the critical probability of the frog model on homogeneous trees. Journal of Statistical Physics.

18. Maulik D., Nekrasov N., Okounkov A., Pandharipande R. Gromov-Witten theory and Donaldson-Thomas theory, II, 28 pp.; arXiv: math.AG/0406092

19. Menshikov M., Petritis D., Popov S. Matrix valued multiplicative chaos and bindweeds. Markov Processes and Related Fields.

20. Nogin D.Yu. Weight functions and extended weight of linear codes. Information Transmission Problems.

21. Okounkov A., Olshanski G. Limits of $BC$-type orthogonal polynomials as the number of variables goes to infinity. Hall-Littlewood and Macdonald polynomials, Contemporary Mathematics series, Amer. Math. Soc.

22. Okounkov A., Pandharipande R. Quantum cohomology of the Hilbert scheme of points in the plane, 46 pp.; arXiv: math.AG/0411210.

23. Popov S. Random Bulgarian solitaire. Random Structures and Algorithms.

24. Rovinsky M. Motives and admissible representations of automorphism groups of fields. Math.Zeit. Published online 9 July 2004.

25. Rovinsky M. Semi-linear representations of $PGL$. math.RT/0306333 v3, http://arXiv.org/.

26. Rybko A. N., Shlosman S.B. Poisson Hypothesis for Information Networks II. Violation and Phase Transitions. Probability theory and Related fields.

27. Rybko A.N., Shlosman S.B. Poisson Hypothesis for Information Networks (A study in non-linear Markov processes) I. Domain of Validity. Probability theory and Related fields, 2004.

28. Tsfasman M.A., Ghorpade S.R. Schubert varieties, linear codes and enumerative combinatorics. Finite Fields and Appl., 2004.

29. van Enter A.C.D., Shlosman S.B. Provable first-order transitions for liquid crystal and lattice gauge models with continuous symmetries. Comm. Math. Phys.

30. Vishik A. On the Chow groups of quadratic Grassmannians. Dokumenta Mathematica, 2004.

31. Yarotsky D.A. Uniqueness of the ground state in weak perturbations of non-interacting gapped quantum lattice systems. J. Stat. Phys., 2005, 118, pp. 119-144.

32. Рыбко А.Н., Шлосман С.Б. Пуассоновская гипотеза: комбинаторные аспекты. Теория информации и ее приложения.

Theses

1. Blakley G.R., Kabatiansky G. Random coding technique for digital fingerprinting codes: fighting two pirates revisited. Proc. IEEE Symp. Information Theory. Chicago, 2004, p. 203.

2. Borodin A., Olshanski G. Representation theory and random point processes. Proceedings 4th European Congress of Mathematics. Stockholm: 2004.

3. Kabatiansky G., Good ternary 2-tracebility codes exist. Proc. IEEE Symp. Information Theory, 2004, p. 204.

4. Kabatiansky G., Tavernier C. List decoding of Reed-Muller codes. 9th Intern. Workshop "Algebraic and Combinatorial Coding Theory", Kranevo: 2004, p. 230-235.

5. Kim H., Lebedev V. Uniqueness of some optimal superimposed codes. 9th Intern. Workshop "Algebraic and Combinatorial Coding Theory", Kranevo: 2004, p. 241-246.

6. Okounkov A. Random surfaces enumerating algebraic curves. Proceedings 4th European Congress of Mathematics. Stockholm: 2004.

7. Pechersky E.A., Suhov Y.M., Vvedenskaya N.D. Large deviation in two-server system with dynamic routing. IEEE 2004 ISIT Abstracts. Chicago, 2004.

8. Prelov V. On the entropy and optimal covering of ellipsoids. Proc. 9th Intern. Workshop "Algebraic and Combinatorial Coding Theory", Kranevo: 2004, p. 338-344.

9. Shehtman V.B. A new version of the filtration method. Advances in Modal Logic. Proceedings of the International Conference, Manchetster, 2004, pp. 344-356.

10. Veretennikov A.Yu., Zhizhina E.A. On stochastic algorithms for the global optimization problem on a compact. Proceedingd of Leeds Annual Statistics Research (Bioinformatics, Images and Wavelets). Leeds, 2004.

11. Vladuts S.G. On the order of points on elliptic curves over a finite field. The Finite Fields Oberwolfach workshop, Oberwolfach: 2004.

12. Yashkov S.F., Yashkova A.S. Time-dependent distribution of attained service times for the M/G/1 foreground-background processor-sharing queue. Transactions of the XXIV Int. Seminar on Stability Problems for Stochastic Models. Yurmala: 2004, pp. 156-162.