Size of the Clusters Under Low Density Zero-Range Invariant Measures

Jeon, Intae

doi:10.4134/ckms.2005.20.4.813

Cited by 24 publications

(54 citation statements)

References 7 publications

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“…This is to be contrasted with the size of the largest component in the case below criticality, which is of order log(L) [12]. The comparison gives a precise meaning to the phase transition experienced by the system, and is reminiscent of the Erdös-Renyi results on the largest cluster of a random graph.…”

Section: Notation and Resultsmentioning

confidence: 96%

“…Furthermore, it has been proved [10][11][12] that when the density is supercritical a condensation phenomenon emerges. Precisely, if ρ > ρ c and ε > 0 then…”

Section: Notation and Resultsmentioning

confidence: 99%

“…In this case none of the grandcanonical measures corresponds to a particle density higher than the critical ρ c , and the system undergoes a phase transition [11,12] from a fluid to a condensed phase, in a sense to be made precise later.…”

Section: Notation and Resultsmentioning

confidence: 99%

“…It is known [11,12] that when the density of particles exceeds a critical value ρ c , the invariant measures of the process concentrate on configurations where a macroscopic proportion of the total number of particles forms a randomly located cluster. In this article we analyse the thermodynamic limit of the invariant measures of the process conditioned to having a supercritical density, that is we let the number of sites L and the number of particles N grow to infinity in such a way that N /L → ρ > ρ c .…”

Section: Introductionmentioning

confidence: 99%

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Thermodynamic limit for the invariant measures in supercritical zero range processes

Armendáriz

Loulakis

2008

Probab. Theory Relat. Fields

112

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We prove a strong form of the equivalence of ensembles for the invariant measures of zero range processes conditioned to a supercritical density of particles. It is known that in this case there is a single site that accomodates a macroscopically large number of the particles in the system. We show that in the thermodynamic limit the rest of the sites have joint distribution equal to the grand canonical measure at the critical density. This improves the result of Großkinsky, Schütz and Spohn, where convergence is obtained for the finite dimensional marginals. We obtain as corollaries limit theorems for the order statistics of the components and for the fluctuations of the bulk.

show abstract

Section: Notation and Resultsmentioning

confidence: 96%

“…Furthermore, it has been proved [10][11][12] that when the density is supercritical a condensation phenomenon emerges. Precisely, if ρ > ρ c and ε > 0 then…”

Section: Notation and Resultsmentioning

confidence: 99%

Section: Notation and Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Thermodynamic limit for the invariant measures in supercritical zero range processes

Armendáriz

Loulakis

2008

Probab. Theory Relat. Fields

112

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“…Let the condensate be the site with the maximal occupancy. Precise estimates on the number of particles at the condensated, as well as its fluctuations, have been obtained in [9,8,5]. The equivalence of ensembles has been proved by Großkinsky, Schütz and Spohn [8].…”

Section: Introductionmentioning

confidence: 99%

Metastability of reversible condensed zero range processes on a finite set

Beltrán

Landim

2011

Probab. Theory Relat. Fields

160

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Abstract. Let r : S × S → R + be the jump rates of an irreducible random walk on a finite set S, reversible with respect to some probability measure m.Consider a zero range process on S in which a particle jumps from a site x, occupied by k particles, to a site y at rate g(k)r(x, y). Let N stand for the total number of particles. In the stationary state, as N ↑ ∞, all particles but a finite number accumulate on one single site. We show in this article that in the time scale N 1+α the site which concentrates almost all particles evolves as a random walk on S whose transition rates are proportional to the capacities of the underlying random walk.

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Metastability for a Non-reversible Dynamics: The Evolution of the Condensate in Totally Asymmetric Zero Range Processes

Landim

2014

Commun. Math. Phys.

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Let T L = Z/LZ be the one-dimensional torus with L points. For α > 0, let g : N → R + be given by g(0) = 0, g(1) = 1, g(k) = [k/(k − 1)] α , k ≥ 2. Consider the totally asymmetric zero range process on T L in which a particle jumps from a site x, occupied by k particles, to the site x + 1 at rate g(k). Let N stand for the total number of particles. In the stationary state, if α > 1, as N ↑ ∞, all particles but a finite number accumulate on one single site. We show in this article that in the time scale N 1+α the site which concentrates almost all particles evolves as a random walk on T L whose transition rates are proportional to the capacities of the underlying random walk, extending to the asymmetric case the results obtained in [5] for reversible zero-range processes on finite sets.

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Size of the Clusters Under Low Density Zero-Range Invariant Measures

Cited by 24 publications

References 7 publications

Thermodynamic limit for the invariant measures in supercritical zero range processes

Thermodynamic limit for the invariant measures in supercritical zero range processes

Metastability of reversible condensed zero range processes on a finite set

Metastability for a Non-reversible Dynamics: The Evolution of the Condensate in Totally Asymmetric Zero Range Processes

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