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Condensation of Non-reversible Zero-Range Processes

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Abstract

In this article, we investigate the condensation phenomena for a class of non-reversible zero-range processes on a fixed finite set. By establishing a novel inequality bounding the capacity between two sets, and by developing a robust framework to perform quantitative analysis on the metastability of non-reversible processes, we prove that the condensed site of the corresponding zero-range processes approximately behaves as a Markov chain on the underlying graph whose jump rate is proportional to the capacity with respect to the underlying random walk. The results presented in the current paper complete the generalization of the work of Beltran and Landim (Probab Theory Relat Fields 152:781–807,2012) on reversible zero-range processes, and that of Landim (Commun Math Phys 330:1–32,2014) on totally asymmetric zero-range processes on a one-dimensional discrete torus.

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Acknowledgements

I. Seo was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (Nos. 2018R1C1B6006896 and 2017R1A5A1015626). I. Seo wishes to thank Claudio Landim, and Fraydoun Rezakhanlou for providing valuable ideas through numerous discussions. Part of this work was done during the author’s stay at the IMPA for the conference “XXI Escola Brasileira de Probabilidade”. The author thanks IMPA for the hospitality and support for this visit.

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  1. Department of Mathematical Sciences and RIMS, Seoul National University, 27-212, Gwanak-Ro 1, Gwanak-Gu, Seoul, 08826, Republic of Korea

    Insuk Seo

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Correspondence to Insuk Seo.

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Communicated by H. Spohn

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Seo, I. Condensation of Non-reversible Zero-Range Processes. Commun. Math. Phys. 366, 781–839 (2019). https://doi.org/10.1007/s00220-019-03346-2

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