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## Ergodicity of a collective random walk on a circle

Nonlinearity. 2014. Vol. 27. No. 5. P. 953-971.

We discuss conditions for unique ergodicity of a collective random walk on a continuous circle. Individual particles in this collective motion perform independent (and different in general) random walks conditioned by the assumption that the particles cannot overrun each other. Additionally to sufficient conditions for the unique ergodicity we discover a new and unexpected way for its violation due to excessively large local jumps. Necessary and sufficient conditions for the unique ergodicity of the deterministic version of this system are obtained as well. Technically our approach is based on the interlacing property of the spin function which describes states of pairs of particles in coupled processes under study.

Blank M., Доклады Академии наук 2013 Т. 448 № 6 С. 629-632

We give conditions for unique ergodicity for a discrete time collective
random walk on a continuous circle. Individual particles in this collective
motion perform independent (and different) random walks conditioned
by the assumption that the particles cannot overrun each other.
Deterministic version of this system is studied as well. ...

Added: November 25, 2014

Tkachenko A., Multichannel queuing systems with balking and regenerative input flow / Высшая школа экономики. Series WP BRP "Science, Technology and Innovation". 2013. No. 14.

Motivated by the application to telephone call centers this paper is focused on the multichannel queueing system with heterogeneous servers, regenerative input flow and balking. Servers times are random variables but not necessary exponential. If a new customer encountering j other customers in the system then it stays for service with probability fj and gets ...

Added: August 1, 2013

Galkin S., Belmans P., Mukhopadhyay S., Graph potentials and moduli spaces of rank two bundles on a curve / Cornell University. Series math "arxiv.org". 2020. No. 2009.05568.

We introduce graph potentials, which are Laurent polynomials associated to (colored) trivalent graphs. These graphs encode degenerations of curves to rational curves, and graph potentials encode degenerations of the moduli space of rank 2 bundles with fixed determinant. We show that the birational type of the graph potential only depends on the homotopy type of ...

Added: April 15, 2021

Veretennikov A., Veretennikova M., On convergence rate for homogeneous Markov chains / Cornell University. Series "Working papers by Cornell University". 2019.

New convergence rate asymptotic bound for a class of homogeneous Markov chains is established. ...

Added: November 14, 2019

Ткаченко А.В., Вестник Московского университета. Серия 1: Математика. Механика 2014 № 1 С. 53-57

This paper is focused on multichannel queueing system with heterogeneous servers and regenerative input flow in a random environment. The environment can destroy all the system and then system is reconstructed. Ergodicity condition of the system is obtained. ...

Added: May 11, 2013

Tkachenko A., Esaulov D., Applied Mathematical Modelling 2016

This paper is focused on a multichannel queueing system with heterogeneous servers, regenerative input flow, and balking. Service times are random variables but not necessary exponential. If a new customer encounters j other customers in the system, then it joins the queue with probability fj and leaves with probability 1-fj . For this system the ...

Added: October 19, 2016

Afanasyeva L. G., Tkachenko A., Theory of Probability and Its Applications 2014 Vol. 58 No. 2 P. 174-192

We consider the multichannel queueing system with nonidentical servers and regenerative input flow. The necessary and sufficient condition for ergodicity is established, and functional limit theorems for high and ultra-high load are proved. As a corollary, the ergodicity condition for queues with unreliable servers is obtained. Suggested approaches are used to prove the ergodic theorem ...

Added: August 20, 2014

Tkachenko A., Moscow University Mathematics Bulletin 2014 Vol. 69 No. 1 P. 37-40

This paper is focused on a multichannel queueing system with heterogeneous servers and regenerative input flow operating in a random environment. The environment can destroy the whole system and the system is reconstructed after that. The necessary and sufficient ergodicity condition is obtained for the system. ...

Added: August 20, 2014

Kleptsyn V., Alvarez S., Malicet D. et al., Groups with infinitely many ends acting analytically on the circle / Cornell University. Series math "arxiv.org". 2015.

Added: June 22, 2016

А. В. Ткаченко, Вестник Московского университета. Серия 1: Математика. Механика 2013 № 2 С. 12-17

This paper is devoted to $M|GI|1|\infty$ queueing system with unreliable server and customer service times depending on the system state. Condition of ergodicity and generating function are found in the stationary state. ...

Added: March 27, 2013

Volk D., Kleptsyn V., Moscow Mathematical Journal 2014 Vol. 14 No. 2 P. 339-365

A one-dimensional confined nonlinear random walk is a tuple of N diffeomorphisms of the unit interval driven by a probabilistic Markov chain. For generic such walks, we obtain a geometric characterization of their ergodic stationary measures and prove that all of them have negative Lyapunov exponents. These measures appear to be probabilistic manifestations of physical measures for ...

Added: December 30, 2015

Shchegolev A., A new rate of convergence estimate for homogeneous discrete-time nonlinear Markov chains / Cornell University. Серия math "arxiv.org". 2021.

In the paper, we study a new rate of convergence estimate for homogeneous discrete-time nonlinear Markov chains based on the Markov-Dobrushin condition. This result generalizes the convergence estimates for any positive number of transition steps. An example of a class such a process provided indicates that such types of estimates considering several transition steps may ...

Added: October 22, 2021

Blank M., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2015 Т. 461 № 2 С. 1-5

We study the functional properties of the concept of interlacing introduced by I.M. Gelfand and show that in the context of collective random walks, this property leads to synchronization. ...

Added: March 20, 2015

Veretennikov A., Veretennikova M., Известия РАН. Серия математическая 2022 Т. 86 № 1 С. 98-133

We continue the work of improving the rate of convergence of ergodic homogeneous Markov chains. The setting is more general than in previous papers: we are able to get rid of the assumption about a common dominating measure and consider the case of inhomogeneous Markov chains as well as more general state spaces. We give examples ...

Added: March 14, 2022

Tkachenko A., Moscow University Mathematics Bulletin 2013 Vol. 68 No. 2 P. 93-97

This paper is devoted to $M|GI|1|\infty$ queueing system with unreliable server and customer service times depending on the system state. Condition of ergodicity and generating function are found in the stationary state. ...

Added: March 13, 2014

Tumanov M., Абдуллин С., Вестник Московского государственного университета леса - Лесной вестник 2015 № 3 С. 173-177

Discusses frequency methods to ensure the accuracy of the control system with model lag and the connection with ergodic theory (the theory of quasi-periodic functions) in the case where the delay model differs from the true. ...

Added: August 27, 2015

Valba O. V., Nechaev S., Tamm M., Химическая физика 2012 Т. 31 С. 23

В работе определяется свободная энергия связывания двух молекул РНК и обсуждаются такие ста тистические свойства, как флуктуации средней энергии связывания двух молекул РНК и распреде ление длин петель в образованной структуре. Анализ зависимости удельной свободной энергии комплекса двух длинных случайных молекул РНК от числа с типов нуклеотидов позволил выдви нуть гипотезу о выделенной роли используемого ...

Added: November 19, 2013

Veretennikov A., Veretennikova M., Doklady Mathematics 2020 Vol. 101 No. 1 P. 12-15

Изучаются улучшенные оценки скорости сходимости для эргодических однородных цепей Маркова. Даны примеры сравнения с классическими оценками. ...

Added: October 29, 2019

Skripchenko A., Dynnikov I., Journal of Modern Dynamics 2017 Vol. 11 P. 219-248

It is known since a 40-year-old paper by M.Keane that minimality is a generic (i.e., holding with probability one) property of an irreducible interval exchange transformation. If one puts some integral linear restrictions on the parameters of the interval exchange transformation, then minimality may become an "exotic" property. We conjecture in this paper that this ...

Added: April 20, 2018

Tamm M., Majumdar S., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2012 Vol. 86 P. 021135

We compute analytically the mean number of common sites, WN(t), visited by N independent random walkers each of length t and all starting at the origin at t=0 in d dimensions. We show that in the (N−d) plane, there are three distinct regimes for the asymptotic large-t growth of WN(t). These three regimes are separated by two critical lines d=2 and d=dc(N)=2N/(N−1) in the (N-d) plane. For d<2, WN(t)∼td/2 for large t (the N dependence is ...

Added: November 18, 2013

Боровков А. А., Cambridge University Press, 2020

This is a companion book to Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions by A.A. Borovkov and K.A. Borovkov. Its self-contained systematic exposition provides a highly useful resource for academic researchers and professionals interested in applications of probability in statistics, ruin theory, and queuing theory. The large deviation principle for random walks was first established ...

Added: March 30, 2021

Blank M., Problems of Information Transmission 2014 Vol. 50 No. 4 P. 350-363

We study functional consequences of the interlacing property consisting in
that a new configuration of “particles” occurs in gaps between elements of
a previous configuration. This property was introduced by I.M. Gelfand in
terms of spectra of sequences of matrices of increasing dimensions and turned
out to be highly needed in many areas of modern mathematics. We examine
conditions under ...

Added: March 20, 2015

Афанасьева Л. Г., Ткаченко А.В., Теория вероятностей и ее применения 2013 Т. 58 № 2 С. 210-234

We consider a multichannel queuing system with heterogeneous servers and regenerative input flow. The necessary and sufficient condition is established and functional limit theorems are proved in overloaded and critically loaded regimes. The ergodicity condition is obtained for multichannel system with unreliable servers. Some approaches for ergodicity of systems with abandonment are discussed. ...

Added: March 31, 2013

Davydov Y., Konakov V., Random walks in non homogeneous Poissonian environment / Cornell University. Series math "arxiv.org". 2016. No. 1609.07066.

We consider the moving particle process in Rd which is defined in the following way. There are two independent sequences (Tk) and (dk) of random variables. The variables Tk are non negative and form an increasing sequence, while variables dk form an i.i.d sequence with common distribution concentrated on the unit sphere. The values dk ...

Added: September 23, 2016