Central Limit Theorems for Infinite Series of Queues and Applications to Simple Exclusion

Kipnis, Claude

doi:10.1214/aop/1176992523

Cited by 97 publications

(72 citation statements)

References 11 publications

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“…This paper was largely motivated by Srinivasan (1989), who applied results of Rost (1981), Andjel (1982), Andjel and Kipnis (1984), Kipnis (1986), and Benassi and Fouque (1987) about interacting particle systems (in particular, the zero-range process and the asymmetric simple exclusion process) to describe the hydrodynamic limit for our model in the special case of exponential service times (stil with mean 1 It is easy to apply Srinivasan's hydrodynamic limit in the saturated case (with i.i.d.…”

Section: D(kmentioning

confidence: 99%

Departures from Many Queues in Series

Glynn¹,

Whitt²

1991

Ann. Appl. Probab.

138

111

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Section: D(kmentioning

confidence: 99%

Departures from Many Queues in Series

Glynn¹,

Whitt²

1991

Ann. Appl. Probab.

138

111

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“…Other ordered particle systems derived from independent driftless Brownian motions were studied by Arratia in [1], and by Sznitman in [38]. Several other papers study connections between systems of queues and one-dimensional interacting particle systems: [24,14,15,34]. Links to the directed percolation and the directed polymer models, as well as the GUE random matrix ensemble, can be found in [26].…”

Section: Introductionmentioning

confidence: 99%

Infinite systems of competing Brownian particles

Sarantsev¹

2017

Ann. Inst. H. Poincaré Probab. Statist.

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Abstract. Consider a system of infinitely many Brownian particles on the real line. At any moment, these particles can be ranked from the bottom upward. Each particle moves as a Brownian motion with drift and diffusion coefficients depending on its current rank. The gaps between consecutive particles form the (infinite-dimensional) gap process. We find a stationary distribution for the gap process. We also show that if the initial value of the gap process is stochastically larger than this stationary distribution, this process converges back to this distribution as time goes to infinity. This continues the work by Pal and Pitman (2008). Also, this includes infinite systems with asymmetric collisions, similar to the finite ones from Karatzas, Pal and Shkolnikov (2016).

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“…It was found that it depends on the initial conditions. In a steady state with a given density, the position fluctuation of a tagged particle grows as t 1/2 [9]. The exponent 1/2 is the same as that of the Brownian particle.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a Tagged Particle in the Asymmetric Exclusion Process with the Step Initial Condition

Imamura

Sasamoto

2007

J Stat Phys

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The one-dimensional totally asymmetric simple exclusion process (TASEP) is considered. We study the time evolution property of a tagged particle in the TASEP with the step initial condition. Calculated is the multi-time joint distribution function of its position. Using the relation of the dynamics of the TASEP to the Schur process, we show that the function is represented as the Fredholm determinant. We also study the scaling limit. The universality of the largest eigenvalue in the random matrix theory is realized in the limit. When the hopping rates of all particles are the same, it is found that the joint distribution function converges to that of the Airy process after the time at which the particle begins to move. On the other hand, when there are several particles with small hopping rate in front of a tagged particle, the limiting process changes at a certain time from the Airy process to the process of the largest eigenvalue in the Hermitian multi-matrix model with external sources.

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Central Limit Theorems for Infinite Series of Queues and Applications to Simple Exclusion

Cited by 97 publications

References 11 publications

Departures from Many Queues in Series

Departures from Many Queues in Series

Infinite systems of competing Brownian particles

Dynamics of a Tagged Particle in the Asymmetric Exclusion Process with the Step Initial Condition

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