Annals of Applied Probability

A version of Aldous’ spectral-gap conjecture for the zero range process

Jonathan Hermon and Justin Salez

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Abstract

We show that the spectral gap of a general zero range process can be controlled in terms of the spectral gap for a single particle. This is in the spirit of Aldous’ famous spectral-gap conjecture for the interchange process, now resolved by Caputo et al. Our main inequality decouples the role of the geometry (defined by the jump matrix) from that of the kinetics (specified by the exit rates). Among other consequences, the various spectral gap estimates that were so far only available on the complete graph or the $d$-dimensional torus now extend effortlessly to arbitrary geometries. As an illustration, we determine the exact order of magnitude of the spectral gap of the rate-one zero-range process on any regular graph and, more generally, for any doubly stochastic jump matrix.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 4 (2019), 2217-2229.

Dates
Received: October 2018
First available in Project Euclid: 23 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1563869041

Digital Object Identifier
doi:10.1214/18-AAP1449

Mathematical Reviews number (MathSciNet)
MR3984254

Zentralblatt MATH identifier
07120707

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Comparison Dirichlet form spectral gap mixing time zero range process particle system expanders

Citation

Hermon, Jonathan; Salez, Justin. A version of Aldous’ spectral-gap conjecture for the zero range process. Ann. Appl. Probab. 29 (2019), no. 4, 2217--2229. doi:10.1214/18-AAP1449. https://projecteuclid.org/euclid.aoap/1563869041


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References

  • [1] Aldous, D. and Fill, J. (2002). Reversible Markov chains and random walks on graphs. Unfinished manuscript. Available at http://www.stat.berkeley.edu/~aldous/RWG/book.html.
  • [2] Alon, G. and Kozma, G. (2018). Comparing with octopi. Preprint. Available at arXiv:1811.10537.
  • [3] Boudou, A.-S., Caputo, P., Dai Pra, P. and Posta, G. (2006). Spectral gap estimates for interacting particle systems via a Bochner-type identity. J. Funct. Anal. 232 222–258.
  • [4] Caputo, P. (2004). Spectral gap inequalities in product spaces with conservation laws. In Stochastic Analysis on Large Scale Interacting Systems. Adv. Stud. Pure Math. 39 53–88. Math. Soc. Japan, Tokyo.
  • [5] Caputo, P., Liggett, T. M. and Richthammer, T. (2010). Proof of Aldous’ spectral gap conjecture. J. Amer. Math. Soc. 23 831–851.
  • [6] Caputo, P. and Posta, G. (2007). Entropy dissipation estimates in a zero-range dynamics. Probab. Theory Related Fields 139 65–87.
  • [7] Dai Pra, P. and Posta, G. (2005). Logarithmic Sobolev inequality for zero-range dynamics. Ann. Probab. 33 2355–2401.
  • [8] Diaconis, P. and Saloff-Coste, L. (1993). Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 696–730.
  • [9] Diaconis, P. and Shahshahani, M. (1981). Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57 159–179.
  • [10] Diaconis, P. and Stroock, D. (1991). Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1 36–61.
  • [11] Janvresse, E., Landim, C., Quastel, J. and Yau, H. T. (1999). Relaxation to equilibrium of conservative dynamics. I. Zero-range processes. Ann. Probab. 27 325–360.
  • [12] Jerrum, M. and Sinclair, A. (1989). Approximating the permanent. SIAM J. Comput. 18 1149–1178.
  • [13] Jonasson, J. (2012). Mixing times for the interchange process. ALEA Lat. Am. J. Probab. Math. Stat. 9 667–683.
  • [14] Lacoin, H. (2016). Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion. Ann. Probab. 44 1426–1487.
  • [15] Landim, C., Sethuraman, S. and Varadhan, S. (1996). Spectral gap for zero-range dynamics. Ann. Probab. 24 1871–1902.
  • [16] Levin, D. A. and Peres, Y. (2017). Markov Chains and Mixing Times, 2nd ed. Amer. Math. Soc., Providence, RI.
  • [17] Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 324. Springer, Berlin.
  • [18] Liggett, T. M. (2005). Interacting Particle Systems. Classics in Mathematics. Springer, Berlin. Reprint of the 1985 original.
  • [19] Merle, M. and Salez, J. (2018). Cutoff for the mean-field zero-range process. Preprint. Available at arXiv:1804.04608.
  • [20] Montenegro, R. and Tetali, P. (2006). Mathematical aspects of mixing times in Markov chains. Found. Trends Theor. Comput. Sci. 1 x+121.
  • [21] Morris, B. (2006). Spectral gap for the zero range process with constant rate. Ann. Probab. 34 1645–1664.
  • [22] Oliveira, R. I. (2013). Mixing of the symmetric exclusion processes in terms of the corresponding single-particle random walk. Ann. Probab. 41 871–913.
  • [23] Spitzer, F. (1970). Interaction of Markov processes. Adv. Math. 5 246–290.