# Transversal Fluctuations of the ASEP, Stochastic Six Vertex Model, and Hall-Littlewood Gibbsian Line Ensembles

@article{Corwin2017TransversalFO,
title={Transversal Fluctuations of the ASEP, Stochastic Six Vertex Model, and Hall-Littlewood Gibbsian Line Ensembles},
author={Ivan Corwin and Evgeni Dimitrov},
journal={Communications in Mathematical Physics},
year={2017},
volume={363},
pages={435-501}
}
• Published 21 March 2017
• Mathematics
• Communications in Mathematical Physics
We consider the ASEP and the stochastic six vertex model started with step initial data. After a long time, T, it is known that the one-point height function fluctuations for these systems are of order T1/3. We prove the KPZ prediction of T2/3 scaling in space. Namely, we prove tightness (and Brownian absolute continuity of all subsequential limits) as T goes to infinity of the height function with spatial coordinate scaled by T2/3 and fluctuations scaled by T1/3. The starting point for proving…
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## References

SHOWING 1-10 OF 87 REFERENCES
KPZ and Airy limits of Hall-Littlewood random plane partitions
In this paper we consider a probability distribution on plane partitions, which arises as a one-parameter generalization of the q^{volume} measure. This generalization is closely related to the
Current fluctuations of the stationary ASEP and six-vertex model
Our results in this paper are two-fold. First, we consider current fluctuations of the stationary asymmetric simple exclusion process (ASEP), run for some long time $T$, and show that they are of
The ASEP and Determinantal Point Processes
• Mathematics
• 2016
We introduce a family of discrete determinantal point processes related to orthogonal polynomials on the real line, with correlation kernels defined via spectral projections for the associated Jacobi
Brownian Gibbs property for Airy line ensembles
• Mathematics
• 2011
Consider a collection of N Brownian bridges $B_{i}:[-N,N] \to \mathbb{R}$, Bi(−N)=Bi(N)=0, 1≤i≤N, conditioned not to intersect. The edge-scaling limit of this system is obtained by taking a weak
The Kardar-Parisi-Zhang equation and universality class
Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or
Between the stochastic six vertex model and Hall-Littlewood processes
• Mathematics
• 2016
We prove that the joint distribution of the values of the height function for the stochastic six vertex model in a quadrant along a down-right path coincides with that for the lengths of the first
Transversal fluctuations for increasing subsequences on the plane
Abstract. Consider a realization of a Poisson process in ℝ2 with intensity 1 and take a maximal up/right path from the origin to (N, N) consisting of line segments between the points, where maximal
The KPZ fixed point
• Mathematics
Acta Mathematica
• 2021
An explicit Fredholm determinant formula is derived for the multipoint distribution of the height function of the totally asymmetric simple exclusion process with arbitrary initial condition. The
Scale Invariance of the PNG Droplet and the Airy Process
• Mathematics
• 2001
We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process A(y). The Airy
Limit Processes for TASEP with Shocks and Rarefaction Fans
• Mathematics
• 2010
We consider the totally asymmetric simple exclusion process (TASEP) with two-sided Bernoulli initial condition, i.e., with left density ρ− and right density ρ+. We study the associated height