Scaling Limit for the Space-Time Covariance of the Stationary Totally Asymmetric Simple Exclusion Process

By P. Ferrari, Herbert Spohn. 2005

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The totally asymmetric simple exclusion process (TASEP) on the one-dimensional lattice with the Bernoulli ρ measure as initial conditions, 0<ρ<1, is stationary in space and time. Let Nt(j) be the number of particles which have crossed the bond from j to j+1 during the time span [0,t]. For we prove that the fluctuations of Nt(j) for large t are of order t1/3 and we determine the limiting distribution function , which is a generalization of the GUE Tracy-Widom distribution. The family of distribution functions have been obtained before by Baik and Rains in the context of the PNG model with boundary sources, which requires the asymptotics of a Riemann-Hilbert problem. In our work we arrive at through the asymptotics of a Fredholm determinant. is simply related to the scaling function for the space-time covariance of the stationary TASEP, equivalently to the asymptotic transition probability of a single second class particle.

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/7255cb5f4434123dd1dcb8fabe827eeb134f8221

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Citations in the corpus

Nonlinear Fluctuating Hydrodynamics for Anharmonic Chains (2013, methodology): “As a supporting approach, discrete models have been studied, like the PNG model [46, 14] and the TASEP [47].”
Introduction to KPZ (2011, background): “The marginal Astat(0) has the Baik-Rains distribution F0 [9], [55].”
Height Fluctuations for the Stationary KPZ Equation (2014, methodology): “17, which demonstrates a universality result, namely that in the large time limit we recover the T 1/3 fluctuation scaling and the one-point probability distribution function previously obtained previously for stationary PNG and TASEP [9, 47, 57, 69].”
The KPZ fixed point (2016, methodology): “2 xu+ ˙@ x˘ satisﬁed by u= @ xhfrom (1.2). Dynamic renormalization was performed by [FNS77] leading to the dynamic scaling exponent 3=2. The equilibrium space-time covariance function was computed in [FS06] by taking a limit from TASEP: With = = 1=4 and ˙= 1, E[u(t;x)u(0;0)] = 2 7=3t 2=3g00 sc(2 1t ”
The Kardar-Parisi-Zhang equation and universality class (2011, unknown): “= −X X<−1 T1+X2 2 X∈[−1,0] T/2 X>0 h¯(T,TX) = −X X<−1 T1+X2 2 X∈[−1,0] T/2 X>0 h¯(T,TX) = T/2 Fluctuations •One pt: FGUE [9, 93] •Multi pt: Airy2 [137, 94, 29] •One pt: F0 [11, 69] •Multi pt: Airystat [10] •One pt: FGOE [12, 13, 70, 146] •Multi pt: Airy1 [31, 32] •One pt: (FGOE)2 [”
Scaling for a one-dimensional directed polymer with boundary conditions (2009, background): “6) with exponential or geometric weights ω(i, j), not only have the predicted exponents been confirmed but also limiting Tracy-Widom fluctuations for Gm,n have been proved [3, 4, 10, 13, 16, 17].”
Tropical Combinatorics and Whittaker functions (2011, background): “s as in equation (5.9). For (n,N) = tC~ and suitably chosen coeﬃcients c1 = c1(µ,ν) and c2 = c2(µ,ν): (5.13) lim t→∞ P nlogz N,1(n) −tc1 c2t1/3 ≤s o = F0(s) where F0(s) is the Baik-Rains distribution [9, 39]. One can also state similar conjectures for other boundary conditions which should limit to ”
A pedestrian's view on interacting particle systems, KPZ universality and random matrices (2008, background): “A proof of the Prähofer–Spohn conjecture for the TASEP was recently presented by Ben Arous and Corwin [4] (see also [5, 48, 96] for earlier partial results).”
Fluctuation Properties of the TASEP with Periodic Initial Configuration (2006, background): “However, the limiting distribution differ: as we will see, in the first case it is the GOE Tracy-Widom of random matrices [41] which differs from the stationary case [12].”
Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials (2009, background): “This connection between the TASEP, Young diagrams, and random matrices has been extremely fruitful, see [2, 4, 16 , 30].”
Evidence for Geometry-Dependent Universal Fluctuations of the Kardar-Parisi-Zhang Interfaces in Liquid-Crystal Turbulence (2012, background): “For example, stationary interfaces are not realized in the liquid-crystal turbulence yet, which constitute an as important situation as the growing interfaces and are theoretically predicted to have another nontrivial distribution called the F0 distribution [5, 19, 27, 42, 49, 72].”
An appetizer to modern developments on the Kardar–Parisi–Zhang universality class (2017, background): “11(b)), and the Baik-Rains distribution was identified [84].”