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Limit processes for TASEP with shocks and rarefaction fans
Mathematical Physics · Submitted 18 Feb 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We characterize the large time scaling limit of the fluctuations as a function of the densities rho_{+/-} and of the different macroscopic regions.
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The abstract, sentence by sentence
We consider the totally asymmetric simple exclusion process (TASEP) with two-sided Bernoulli initial condition, i.e., with left density rho_- and right density rho_+. We consider the associated height function, whose discrete gradient is given by the particle occurrences. Macroscopically one has a deterministic limit shape with a shock or a rarefaction fan depending on the values of rho_{+/-}. We characterize the large time scaling limit of the fluctuations as a function of the densities rho_{+/-} and of the different macroscopic regions. Moreover, using a slow decorrelation phenomena, the results are extended from fixed time to the whole space-time, except along the some directions (the characteristic solutions of the related Burgers equation) where the problem is still open. On the way to proving the results for TASEP, we obtain the limit processes for the fluctuations in a class of corner growth processes with external sources, of equivalently for the last passage time in a directed percolation model with two-sided boundary conditions. Additionally, we provide analogous results for eigenvalues of perturbed complex Wishart (sample covariance) matrices.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider the totally asymmetric simple exclusion process (TASEP) with two-sided Bernoulli initial condition, i.e., with left density rho_- and right density rho_+. | background 0.67 | 0.49 | 0.03 | asserted 0.89 |
| s1 | We consider the associated height function, whose discrete gradient is given by the particle occurrences. | background 0.43 | 0.36 | 0.02 | asserted 0.98 |
| s2 | Macroscopically one has a deterministic limit shape with a shock or a rarefaction fan depending on the values of rho_{+/-}. | background 0.84 | 0.14 | 0.12 | asserted 1.00 |
| s3 | We characterize the large time scaling limit of the fluctuations as a function of the densities rho_{+/-} and of the different macroscopic regions. | result 0.78 | 0.94 | 0.54 | asserted 1.00 |
| s4 | Moreover, using a slow decorrelation phenomena, the results are extended from fixed time to the whole space-time, except along the some directions (the characteristic solutions of the related Burgers equation) where the problem is still open. | limitation 0.58 | 0.93 | 0.18 | asserted 1.00 |
| s5 | On the way to proving the results for TASEP, we obtain the limit processes for the fluctuations in a class of corner growth processes with external sources, of equivalently for the last passage time in a directed percolation model with two-sided boundary conditions. | result 0.51 | 0.95 | 0.10 | asserted 1.00 |
| s6 | Additionally, we provide analogous results for eigenvalues of perturbed complex Wishart (sample covariance) matrices. | result 0.61 | 0.93 | 0.05 | asserted 1.00 |
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The paper
Original abstract
We consider the totally asymmetric simple exclusion process (TASEP) with two-sided Bernoulli initial condition, i.e., with left density rho_- and right density rho_+. We consider the associated height function, whose discrete gradient is given by the particle occurrences. Macroscopically one has a deterministic limit shape with a shock or a rarefaction fan depending on the values of rho_{+/-}. We characterize the large time scaling limit of the fluctuations as a function of the densities rho_{+/-} and of the different macroscopic regions. Moreover, using a slow decorrelation phenomena, the results are extended from fixed time to the whole space-time, except along the some directions (the characteristic solutions of the related Burgers equation) where the problem is still open. On the way to proving the results for TASEP, we obtain the limit processes for the fluctuations in a class of corner growth processes with external sources, of equivalently for the last passage time in a directed percolation model with two-sided boundary conditions. Additionally, we provide analogous results for eigenvalues of perturbed complex Wishart (sample covariance) matrices.
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References
How the literature uses it
5 arXiv papers cite this in 7 sentences
Papers citing it · 6 sentences without a confident label
Cites it as background 1 paper
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Much recent work on TASEP has focused on the joint distributions of the particle current [ref] .
Formulas for Joint Probabilities for the Asymmetric Simple Exclusion Process · Introduction · 0.90
What it cites · 21 sentences without a confident label
Builds on 2 papers
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For presentation simplicity in [ref] only the fixed-time result was stated, but slow decorrelation allow immediately to extend it as in Theorem [ref].
Limit process of stationary TASEP near the characteristic line · Continuous time TASEP · 0.90
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They provide generalizations of Lemma 4.1 and Lemma 4.2 of [ref] from one-point functions to n-point functions.
Current fluctuations for TASEP: A proof of the Pr\"{a}hofer--Spohn conjecture · Coupling Lemmas · 0.97
Uses the method or tool of 3 papers
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In order to bootstrap the one-sided process result to the full two-sided case we appeal to a coupling method introduced in [ref] .
Current fluctuations for TASEP: A proof of the Pr\"{a}hofer--Spohn conjecture · Introduction · 0.99
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The first probabilistic method we use is slow decorrelation [ref] (given as Proposition [ref]).
Universality of slow decorrelation in KPZ growth · Introduction · 1.00
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Using again Appendix A in [ref] [equation] The exponential decay (as in ([ref])) for large positive $s_i$ follows from the fact that the $w-$contour (resp. $z$-contour) lies to the left (resp. right) of $w_c$ and with a distance at least $κ'/T^1/3.$ Again one shall choose $κ'≥max_i |τ_i|+κ.$
Limit process of stationary TASEP near the characteristic line · The case where $η=γ/1+γ$ · 0.91
Cites as background 3 papers
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The definition of the process for stationary TASEP, $𝒜_stat$, is quite intricate [ref] .
Limit process of stationary TASEP near the characteristic line · Limit processes: definitions · 0.93
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Very recently, due to the efforts of Tracy and Widom [ref] , Derrida and Gerschenfeld [ref] , Balázs and Seppäläinen [ref] , Quastel and Valkó [ref] , Mountford and Guiol [ref] significant progress has been made in answering this question in the general PASEP.
math.PR/0608400 · Introduction · 0.96
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Now we briefly expose the ideas of the asymptotics since such arguments have already been developed many times (see e.g. [ref] or at the beginning of the proof of Lemma 6.1 in [ref] , where the steps are explained).
Anisotropic growth of random surfaces in 2+1 dimensions · The case where $η> γ/1+γ$ · 0.87
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