scholarly journals The speed of the tagged particle in the exclusion process on Galton–Watson trees

2020 ◽  
Vol 25 (0) ◽  
Author(s):  
Nina Gantert ◽  
Dominik Schmid
Keyword(s):  
Exclusion Process ◽  
Tagged Particle

First displacement time of a tagged particle in a stochastic cluster in a simple exclusion process with random slow bonds

2019 ◽  
Vol 51 (03) ◽  
pp. 717-744
Author(s):  
Adriana Uquillas ◽  
Adilson Simonis
Keyword(s):  
Poisson Process ◽  
Exclusion Process ◽  
Upper And Lower Bounds ◽  
Density Profiles ◽  
Tagged Particle ◽  
Simple Exclusion Process ◽  
Discrete Torus ◽  
Simple Exclusion ◽  
The One ◽  
Initial Measure

AbstractWe consider the nearest-neighbour simple exclusion process on the one-dimensional discrete torus $\mathbb{T}_N=\mathbb{Z}/N\mathbb{Z}$ , with random rates $c_N=\{c_{x,N}\colon x \in \mathbb{T}_N\}$ defined in terms of a homogeneous Poisson process on $\mathbb{R}$ with intensity $\lambda$ . Given a realization of the Poisson process, the jump rate along the edge $\{x,x+1\}$ is 1 if there is not any Poisson mark in $ (x,x+1) $ ; otherwise, it is $\lambda/N,\, \lambda \in( 0,1]$ . The density profile of this process with initial measure associated to an initial profile $\rho_0\colon \mathbb{R} \rightarrow [0,1]$ , evolves as the solution of a bounded diffusion random equation. This result follows from an appropriate quenched hydrodynamic limit. If $\lambda=1$ then $\rho$ is discontinuous at each Poisson mark with passage through the slow bonds, otherwise the conductance at the slow bonds decreases meaning no passage through the slow bonds in the continuum. The main results are concerned with upper and lower quenched and annealed bounds of $T_j$ , where $T_j$ is the first displacement time of a tagged particle in a stochastic cluster of size j (the cluster is defined via specific macroscopic density profiles). It is possible to observe that when time t grows, then $\mathbb{P}\{T_j \geq t\}$ decays quadratically in both the upper and lower bounds, and falls as slow as the presence of more Poisson marks neighbouring the tagged particle, as expected.

Poissonian approximation for the tagged particle in asymmetric simple exclusion

1996 ◽  
Vol 33 (02) ◽  
pp. 411-419
Author(s):  
P. A. Ferrari ◽  
L. R. G. Fontes
Keyword(s):  
Nearest Neighbor ◽  
Exclusion Process ◽  
Initial Distribution ◽  
Zero Range Process ◽  
Tagged Particle ◽  
In Series ◽  
Asymmetric Simple Exclusion ◽  
Net Flux ◽  
Simple Exclusion ◽  
The One

We consider the position of a tagged particle in the one-dimensional asymmetric nearest-neighbor simple exclusion process. Each particle attempts to jump to the site to its right at rate p and to the site to its left at rate q. The jump is realized if the destination site is empty. We assume p > q. The initial distribution is the product measure with density λ, conditioned to have a particle at the origin. We call X, the position at time t of this particle. Using a result recently proved by the authors for a semi-infinite zero-range process, it is shown that for all t ≧ 0, Xt = Nt − Bt + B 0 , where {N t} is a Poisson process of parameter (p – q)(1– λ) and {Bt } is a stationary process satisfying E exp (θ | B, |) < ∞ for some θ > 0. As a corollary we obtain that, properly centered and rescaled, the process {Xt } converges to Brownian motion. A previous result says that in the scale t 1/2, the position Xt is given by the initial number of empty sites in the interval (0, λt) divided by λ. We use this to compute the asymptotic covariance at time t of two tagged particles initially at sites 0 and rt. The results also hold for the net flux between two queues in a system of infinitely many queues in series.

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