scholarly journals Cutoff profile of ASEP on a segment

2022 ◽  
Author(s):  
Alexey Bufetov ◽  
Peter Nejjar
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Total Variation ◽  
Exclusion Process ◽  
Hecke Algebra ◽  
Initial Condition ◽  
Simple Exclusion Process ◽  
Mixing Behavior ◽  
Asymmetric Simple Exclusion ◽  
Simple Exclusion

AbstractThis paper studies the mixing behavior of the Asymmetric Simple Exclusion Process (ASEP) on a segment of length N. Our main result is that for particle densities in (0, 1),  the total-variation cutoff window of ASEP is $$N^{1/3}$$ N 1 / 3 and the cutoff profile is $$1-F_{\mathrm {GUE}},$$ 1 - F GUE , where $$F_{\mathrm {GUE}}$$ F GUE is the Tracy-Widom distribution function. This also gives a new proof of the cutoff itself, shown earlier by Labbé and Lacoin. Our proof combines coupling arguments, the result of Tracy–Widom about fluctuations of ASEP started from the step initial condition, and exact algebraic identities coming from interpreting the multi-species ASEP as a random walk on a Hecke algebra.

scholarly journals Mapping TASEP back in time

2021 ◽  
Author(s):  
Leonid Petrov ◽  
Axel Saenz
Keyword(s):  
Markov Process ◽  
Continuous Time ◽  
Exclusion Process ◽  
Discrete Space ◽  
Baxter Equation ◽  
Simple Exclusion Process ◽  
Open Questions ◽  
Asymmetric Simple Exclusion ◽  
Simple Exclusion ◽  
Hammersley Process

AbstractWe obtain a new relation between the distributions $$\upmu _t$$ μ t at different times $$t\ge 0$$ t ≥ 0 of the continuous-time totally asymmetric simple exclusion process (TASEP) started from the step initial configuration. Namely, we present a continuous-time Markov process with local interactions and particle-dependent rates which maps the TASEP distributions $$\upmu _t$$ μ t backwards in time. Under the backwards process, particles jump to the left, and the dynamics can be viewed as a version of the discrete-space Hammersley process. Combined with the forward TASEP evolution, this leads to a stationary Markov dynamics preserving $$\upmu _t$$ μ t which in turn brings new identities for expectations with respect to $$\upmu _t$$ μ t . The construction of the backwards dynamics is based on Markov maps interchanging parameters of Schur processes, and is motivated by bijectivizations of the Yang–Baxter equation. We also present a number of corollaries, extensions, and open questions arising from our constructions.

A stochastic airplane boarding model in a framework of ASEP with distinguishable particles

2018 ◽  
Vol 29 (10) ◽  
pp. 1850093
Author(s):  
ShengJie Qiang ◽  
Bin Jia ◽  
QingXia Huang
Keyword(s):  
Removal Rate ◽  
Exclusion Process ◽  
System Size ◽  
Nonequilibrium Systems ◽  
One Dimensional ◽  
Simple Exclusion Process ◽  
Maximum Current ◽  
Asymmetric Simple Exclusion ◽  
Simple Exclusion ◽  
The Difference

The asymmetric simple exclusion process (ASEP) is a paradigmatic model for nonequilibrium systems and has been used in many applications. Airplane boarding provides another interesting example where this framework can be applied. We propose a simple model for boarding process, in which a particle moves along a one-dimensional aisle after being injected, and finally is removed at a reserved site. Different from the typical ASEP model, particles are removed in a disorderly or a parallel way. Detailed calculations and discussions of some related characteristics, such as mean boarding time and parallelism indicator, are provided based on Monte-Carlo simulations. Results show that three phases exist in the boarding process: free-flow, jamming and maximum current. Transitions between these phases are governed by the difference between the injection and removal rate. Further analysis shows how the scaling behavior depends on the system size and the boarding conditions. Those results emphasize the importance of utilizing the whole length of the aisle to reduce the boarding time when designing an efficient boarding strategy.

scholarly journals Corners in Tree-Like Tableaux

10.37236/5712 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Paweł Hitczenko ◽  
Amanda Lohss
Keyword(s):  
Exclusion Process ◽  
Limiting Distribution ◽  
Tree Structure ◽  
Random Tree ◽  
Type B ◽  
Combinatorial Objects ◽  
Simple Exclusion Process ◽  
Asymmetric Simple Exclusion ◽  
Simple Exclusion ◽  
Symmetric Tree

In this paper, we study tree–like tableaux, combinatorial objects which exhibit a natural tree structure and are connected to the partially asymmetric simple exclusion process (PASEP). There was a conjecture made on the total number of corners in tree–like tableaux and the total number of corners in symmetric tree–like tableaux. In this paper, we prove both conjectures. Our proofs are based off of the bijection with permutation tableaux or type–B permutation tableaux and consequently, we also prove results for these tableaux. In addition, we derive the limiting distribution of the number of occupied corners in random tree–like tableaux and random symmetric tree–like tableaux.

Stochastic comparisons of density profiles for the road-hog process

1991 ◽  
Vol 28 (04) ◽  
pp. 852-863
Author(s):  
Rengarajan Srinivasan
Keyword(s):  
Exclusion Process ◽  
Stochastic Ordering ◽  
Asymmetric Simple Exclusion Process ◽  
The Other ◽  
Density Profiles ◽  
Poisson Input ◽  
Simple Exclusion Process ◽  
Asymmetric Simple Exclusion ◽  
Simple Exclusion ◽  
The Right

We consider the asymmetric simple exclusion process which starts from a product measure such that all the sites to the left of zero (including zero) are occupied and the right of 0 (excluding 0) are empty. We label the particle initially at 0 as the leading particle. We study the long-term behaviour of this process near large sites when the leading particle's holding time is different from that of the other particles. In particular, we assume that the leading particle moves at a slower rate than the other particles. We call this modified asymmetric simple exclusion process the road-hog process. Coupling and stochastic ordering techniques are used to derive the density profile of this process. Road-hog processes are useful in modelling series of exponential queues with Poisson and non-Poisson input process. The density profiles dramatically illustrate the flow of customers through the queues.

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