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Diffusive scaling of the spectral gap for the dilute Ising lattice-gas dynamics below the percolation threshold
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  • Published: August 2001

Diffusive scaling of the spectral gap for the dilute Ising lattice-gas dynamics below the percolation threshold

  • N. Cancrini1 &
  • F. Martinelli2 

Probability Theory and Related Fields volume 120, pages 497–534 (2001)Cite this article

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  • 5 Citations

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Abstract.

We consider a conservative stochastic lattice-gas dynamics reversible with respect to the canonical Gibbs measure of the bond dilute Ising model on ℤd at inverse temperature β. When the bond dilution density p is below the percolation threshold we prove that for any particle density and any β, with probability one, the spectral gap of the generator of the dyamics in a box of side L centered at the origin scales like L −2. Such an estimate is then used to prove a decay to equilibrium for local functions of the form where ε is positive and arbitrarily small and α = ½ for d = 1, α=1 for d≥2. In particular our result shows that, contrary to what happes for the Glauber dynamics, there is no dynamical phase transition when β crosses the critical value β c of the pure system.

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Authors and Affiliations

  1. INFM Unità di Roma “La Sapienza” and Dipartimento di Energetica, Università di L'Aquila, Località Monteluco, 67040 Roio Poggio, L'Aquila, Italy. e-mail: nicoletta.cancrini@roma1.infn.it, , , , , , IT

    N. Cancrini

  2. Dipartimento di Matematica, Università di Roma Tre, Largo S. Murialdo 1, 00146 Roma, Italy. e-mail: martin@mat.uniroma3.it, , , , , , IT

    F. Martinelli

Authors
  1. N. Cancrini
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  2. F. Martinelli
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Received: 10 April 2000 / Revised version: 23 October 2000 / Published online: 5 June 2001

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Cancrini, N., Martinelli, F. Diffusive scaling of the spectral gap for the dilute Ising lattice-gas dynamics below the percolation threshold. Probab Theory Relat Fields 120, 497–534 (2001). https://doi.org/10.1007/PL00008790

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  • Issue Date: August 2001

  • DOI: https://doi.org/10.1007/PL00008790

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  • Mathematics Subject Classification (2000): 82B44, 82C22, 82C44, 60K35
  • Key words or phrases: Kawasaki dynamics – Random ferromagnet – Spectral gap – Equivalence of ensembles
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