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 Teor. Veroyatnost. i Primenen., 2004, Volume 49, Issue 1, Pages 164–171 (Mi tvp242)

Short Communications

On the lower bound of the spectrum of some mean-field models

B. L. Granovskiia, A. I. Zeifmanb

a Technion – Israel Institute of Technology
b Vologda State Pedagogical University

Abstract: We find the lower bound of the spectrum of the $q$-matrix for a variety of mean-field models, as the number of interacting sites goes to infinity. We also make a comparative study of the asymptotic behavior of the lower bound and the spectral gap and establish a characterization of a class of mean-field models for which both bounds of the spectrum attain their extremal values. The results are obtained with the help of the method suggested by the second author in the late 1980s.

Keywords: mean-field models, birth-death processes, random walks on graphs, spectrum of the generator, maximal and minimal rates of exponential convergence, spectral gap.

DOI: https://doi.org/10.4213/tvp242

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English version:
Theory of Probability and its Applications, 2005, 49:1, 148–155

Bibliographic databases:

Revised: 30.06.2003

Citation: B. L. Granovskii, A. I. Zeifman, “On the lower bound of the spectrum of some mean-field models”, Teor. Veroyatnost. i Primenen., 49:1 (2004), 164–171; Theory Probab. Appl., 49:1 (2005), 148–155

Citation in format AMSBIB
\Bibitem{GraZei04} \by B.~L.~Granovskii, A.~I.~Zeifman \paper On the lower bound of the spectrum of some mean-field models \jour Teor. Veroyatnost. i Primenen. \yr 2004 \vol 49 \issue 1 \pages 164--171 \mathnet{http://mi.mathnet.ru/tvp242} \crossref{https://doi.org/10.4213/tvp242} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2141336} \zmath{https://zbmath.org/?q=an:1094.60066} \transl \jour Theory Probab. Appl. \yr 2005 \vol 49 \issue 1 \pages 148--155 \crossref{https://doi.org/10.1137/S0040585X97980920} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000228185300011} 

• http://mi.mathnet.ru/eng/tvp242
• https://doi.org/10.4213/tvp242
• http://mi.mathnet.ru/eng/tvp/v49/i1/p164

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Zeifman A., Chegodaev A., Shilova G., “Almost absorbing continuous-time Markov's chains on finite state space”, International Conference on Modelling of Business, Industrial and Transport Systems, 2008, 213–217
2. E. van Doorn, A. I. Zeifman, T. L. Panfilova, “Bounds and Asymptotics for the Rate of Convergence of Birth-Death Processes”, Theory Probab. Appl., 54:1 (2010), 97–113
3. van Doorn E.A., van Foreest N.D., Zeifman A.I., “Representations for the extreme zeros of orthogonal polynomials”, J. Comput. Appl. Math., 233:3 (2009), 847–851
4. Ya. A. Satin, A. I. Zeifman, A. V. Korotysheva, S. Ya. Shorgin, “Ob odnom klasse markovskikh sistem obsluzhivaniya”, Inform. i ee primen., 5:4 (2011), 18–24
5. Zeifman A.I. Korolev V.Yu., “Two-Sided Bounds on the Rate of Convergence For Continuous-Time Finite Inhomogeneous Markov Chains”, Stat. Probab. Lett., 103 (2015), 30–36
6. Zeifman A.I., Korolev V.Yu., Satin Ya.A., Kiseleva K.M., “Lower Bounds For the Rate of Convergence For Continuous-Time Inhomogeneous Markov Chains With a Finite State Space”, Stat. Probab. Lett., 137 (2018), 84–90
7. Zeifman I A., Satin Y.A., Kiseleva K.M., “On Obtaining Sharp Bounds of the Rate of Convergence For a Class of Continuous-Time Markov Chains”, Stat. Probab. Lett., 161 (2020), 108730
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