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Second time scale of the metastability of reversible inclusion processes

A Correction to this article was published on 20 May 2023

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Abstract

We investigate the second time scale of the metastable behavior of the reversible inclusion process in an extension of the study by Bianchi et al. (Electron J Probab 22:1–34, 2017), which presented the first time scale of the same model and conjectured the scheme of multiple time scales. We show that \(N/d_{N}^{2}\) is indeed the correct second time scale for the most general class of reversible inclusion processes, and thus prove the first conjecture of the foresaid study. Here, N denotes the number of particles, and \(d_{N}\) denotes the small scale of randomness of the system. The main obstacles of this research arise in calculating the sharp asymptotics for the capacities, and in the fact that the methods employed in the former study are not directly applicable due to the complex geometry of particle configurations. To overcome these problems, we first thoroughly examine the landscape of the transition rates to obtain a proper test function of the equilibrium potential, which provides the upper bound for the capacities. Then, we modify the induced test flow and precisely estimate the equilibrium potential near the metastable valleys to obtain the correct lower bound for the capacities.

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Notes

  1. Bosonic particle systems represent dynamics in which particles tend to attract each other. They are mostly used to represent dynamical systems in low temperatures.

  2. We take \(1/\infty \) to be 0 in the following.

  3. For two subsets A and B of S, the graph distance between A and B is defined as \(\min \{n\ge 0:\exists x_{0},\ldots ,x_{n}\in S\text { such that }x_{0}\in A,\;x_{n}\in B,\text { and }r(x_{i},x_{i+1})>0\text { for }0\le i\le n-1\}\).

References

  1. Armendáriz, I., Grosskinsky, S., Loulakis, M.: Metastability in a condensing zero-range process in the thermodynamic limit. Probab. Theory Relat. Fields 169, 105–175 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ayala, M.; Carinci, G.; Redig, F.: Condensation of SIP particles and sticky Brownian motion. arXiv:1906.09887v1 [math.PR] (2019)

  3. Beltrán, J., Landim, C.: Metastability of reversible condensed zero range processes on a finite set. Probab. Theory Relat. Fields 152, 781–807 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beltrán, J., Landim, C.: Metastability of reversible finite state Markov processes. Stoch. Proc. Appl. 121, 1633–1677 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beltrán, J., Landim, C.: Tunneling and metastability of continuous time Markov chains. J. Stat. Phys. 140, 1065–1114 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Beltrán, J., Landim, C.: Tunneling and metastability of continuous time Markov chains II, the nonreversible case. J. Stat. Phys. 149, 598–618 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bianchi, A., Dommers, S., Giardinà, C.: Metastability in the reversible inclusion process. Electron. J. Probab. 22, 1–34 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bovier, A., den Hollander, F.: Metastabillity: A Potential-Theoretic Approach. Grundlehren der mathematischen Wissenschaften. Springer (2015)

  9. Cao, J., Chleboun, P., Grosskinsky, S.: Dynamics of condensation in the totally asymmetric inclusion process. J. Stat. Phys. 155, 523–543 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chleboun, P.: Large deviations and metastability in condensing stochastic particle systems. PhD thesis, The University of Warwick (2011)

  11. Chleboun, P., Grosskinsky, S., Jatuviriyapornchai, W.: Structure of the condensed phase in the inclusion process. J. Math. Phys. 178, 682–710 (2020)

    MathSciNet  MATH  Google Scholar 

  12. Gaudillière, A., Landim, C.: A Dirichlet principle for non reversible Markov chains and some recurrence theorems. Probab. Theory Relat. Fields 158, 55–89 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Giardinà, C., Kurchan, J., Redig, F.: Duality and exact correlations for a model of heat conduction. J. Math. Phys. 48, 033301 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Giardinà, C., Kurchan, J., Redig, F., Vafayi, K.: Duality and hidden symmetries in interacting particle systems. J. Stat. Phys. 135, 25–55 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Giardinà, C., Redig, F., Vafayi, K.: Correlation inequalities for interacting particle systems with duality. J. Stat. Phys. 141, 242–263 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grosskinsky, S., Redig, F., Vafayi, K.: Condensation in the inclusion process and related models. J. Stat. Phys. 142, 952–974 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Grosskinsky, S., Redig, F., Vafayi, K.: Dynamics of condensation in the symmetric inclusion process. Electron. J. Probab. 18, 1–23 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kuoch, K., Redig, F.: Ergodic theory of the symmetric inclusion process. Stoch. Proc. Appl. 126, 3480–3498 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kim, S., Seo, I.: Condensation and metastable behavior of non-reversible inclusion processes. Commun. Math. Phys. (2021). https://doi.org/10.1007/s00220-021-04016-y

    Article  MathSciNet  MATH  Google Scholar 

  20. Landim, C.: A topology for limits of Markov chains. Stoch. Proc. Appl. 125, 1058–1088 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Landim, C.: Metastability for a non-reversible dynamics: the evolution of the condensate in totally asymmetric zero range processes. Commun. Math. Phys. 330, 1–32 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Landim, C., Loulakis, M., Mourragui, M.: Metastable Markov chains: from the convergence of the trace to the convergence of the finite-dimensional distributions. Electron. J. Probab. 23, 1–34 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Opoku, A., Redig, F.: Coupling and hydrodynamic limit for the inclusion process. J. Stat. Phys. 160, 532–547 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Seo, I.: Condensation of non-reversible zero-range processes. Commun. Math. Phys. 366, 781–839 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  25. Slowik, M.: A note on variational representations of capacities for reversible and nonreversible Markov chains. Unpublished, Technische Universität Berlin (2013)

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Acknowledgements

We appreciate the reviewers and editors for providing numerous valuable comments that were greatly helpful to improve the performance of the article.

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Correspondence to Seonwoo Kim.

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S. Kim received support from the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2018R1C1B6006896 and NRF-2019-Global Ph.D. Fellowship Program).

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Kim, S. Second time scale of the metastability of reversible inclusion processes. Probab. Theory Relat. Fields 180, 1135–1187 (2021). https://doi.org/10.1007/s00440-021-01036-6

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  • DOI: https://doi.org/10.1007/s00440-021-01036-6

Keywords

  • Metastability
  • Multiple time scales
  • Interacting particle systems
  • Inclusion process

Mathematics Subject Classification

  • 60J28
  • 60K35
  • 82C22