Characterization of equilibrium measures for critical reversible Nearest Particle Systems

Thomas Mountford; Li Wu

Open Mathematics (2008)

  • Volume: 6, Issue: 2, page 237-261
  • ISSN: 2391-5455

Abstract

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We show that for critical reversible attractive Nearest Particle Systems all equilibrium measures are convex combinations of the upper invariant equilibrium measure and the point mass at all zeros, provided the underlying renewal sequence possesses moments of order strictly greater than
and obeys some natural regularity conditions.

How to cite

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Thomas Mountford, and Li Wu. "Characterization of equilibrium measures for critical reversible Nearest Particle Systems." Open Mathematics 6.2 (2008): 237-261. <http://eudml.org/doc/269317>.

@article{ThomasMountford2008,
abstract = {We show that for critical reversible attractive Nearest Particle Systems all equilibrium measures are convex combinations of the upper invariant equilibrium measure and the point mass at all zeros, provided the underlying renewal sequence possesses moments of order strictly greater than \[ \frac\{\{7 + \sqrt\{41\} \}\}\{2\} \] and obeys some natural regularity conditions.},
author = {Thomas Mountford, Li Wu},
journal = {Open Mathematics},
keywords = {particle system; attractiveness; equilibrium measure},
language = {eng},
number = {2},
pages = {237-261},
title = {Characterization of equilibrium measures for critical reversible Nearest Particle Systems},
url = {http://eudml.org/doc/269317},
volume = {6},
year = {2008},
}

TY - JOUR
AU - Thomas Mountford
AU - Li Wu
TI - Characterization of equilibrium measures for critical reversible Nearest Particle Systems
JO - Open Mathematics
PY - 2008
VL - 6
IS - 2
SP - 237
EP - 261
AB - We show that for critical reversible attractive Nearest Particle Systems all equilibrium measures are convex combinations of the upper invariant equilibrium measure and the point mass at all zeros, provided the underlying renewal sequence possesses moments of order strictly greater than \[ \frac{{7 + \sqrt{41} }}{2} \] and obeys some natural regularity conditions.
LA - eng
KW - particle system; attractiveness; equilibrium measure
UR - http://eudml.org/doc/269317
ER -

References

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  1. [1] Andjel E., Liggett T.M., Mountford T., Clustering in one dimensional threshold voter models, Stochastic Process. Appl., 1992, 42, 73–90 http://dx.doi.org/10.1016/0304-4149(92)90027-N Zbl0752.60086
  2. [2] Bezuidenhout C., Grimmett G., The critical contact process dies out, Ann. Probab., 1990, 18, 1462–1482 http://dx.doi.org/10.1214/aop/1176990627 Zbl0718.60109
  3. [3] Diaconis P., Stroock D., Genmetric bounds for eigenvalues of Markov chains, Ann. Appl. Probab., 1991, 1, 36–61 http://dx.doi.org/10.1214/aoap/1177005980 Zbl0731.60061
  4. [4] Griffeath D., Liggett T.M., Critical phenomena for Spitzer’s reversible nearest particle systems, Ann. Probab., 1982, 10, 881–895 http://dx.doi.org/10.1214/aop/1176993711 Zbl0498.60090
  5. [5] Liggett T.M., Interacting particle systems, Springer-Verlag, New York, 1985 Zbl0559.60078
  6. [6] Liggett T.M., L 2 rates of convergence for attractive reversible nearest particle systems: the critical case, Ann. Probab., 1991, 19, 935–959 http://dx.doi.org/10.1214/aop/1176990330 Zbl0737.60092
  7. [7] Liggett T.M., Branching random walks and contact processes on homogenous trees, Probab. Theory Related Fields, 1996, 106, 495–519 http://dx.doi.org/10.1007/s004400050073 
  8. [8] Mountford T., A complete convergence theorem for attractive reversible nearest particle systems, Canad. J. Math., 1997, 49, 321–337 Zbl0891.60092
  9. [9] Mountford T., A convergence result for critical reversible nearest particle systems, Ann. Probab., 2002, 30, 1–61 Zbl1061.60108
  10. [10] Mountford T., Sweet T., Finite approximations to the critical reversible nearest particle system, Ann. Probab., 1998, 26, 1751–1780 http://dx.doi.org/10.1214/aop/1022855881 Zbl0966.82013
  11. [11] Mountford T., Wu L.C., The time for a critical nearest particle system to reach equilibrium starting with a large gap, Electron. J. Probab., 2005, 10, 436–498 Zbl1111.60078
  12. [12] Schinazi R., Brownian fluctuations of the edge for critical reversible nearest-particle systems, Ann. Probab., 1992, 20, 194–205 http://dx.doi.org/10.1214/aop/1176989924 Zbl0742.60108
  13. [13] Sinclair A., Jerrum M., Approximate counting uniform generation and rapidly mixing Markov chains, Inform. and Comput., 1989, 82, 93–133 http://dx.doi.org/10.1016/0890-5401(89)90067-9 Zbl0668.05060
  14. [14] Spitzer F., Stochastic time evolution of one dimensional infinite particle systems, Bull. Amer. Math. Soc., 1977, 83, 880–890 http://dx.doi.org/10.1090/S0002-9904-1977-14322-X Zbl0372.60149
  15. [15] Sweet T.D., One dimensional spin systems, PhD thesis, University of California, Los Angeles, USA, 1997 

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