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}
}
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… 
Spatial tightness at the edge of Gibbsian line ensembles
Consider a sequence of Gibbsian line ensemble whose lowest labeled curve (i.e., the edge) has tight one-point marginals. Then, given certain technical assumptions on the nature of the Gibbs property
Stochastic six-vertex model in a half-quadrant and half-line open ASEP
Abstract. We consider the asymmetric simple exclusion process (ASEP) on the positive integers with an open boundary condition. We show that, when starting devoid of particles and for a certain
Brownian structure in the KPZ fixed point
Many models of one-dimensional local random growth are expected to lie in the Kardar-Parisi-Zhang (KPZ) universality class. For such a model, the interface profile at advanced time may be viewed in
KPZ equation correlations in time
We consider the narrow wedge solution to the Kardar-Parisi-Zhang stochastic PDE under the characteristic $3:2:1$ scaling of time, space and fluctuations. We study the correlation of fluctuations at
TIGHTNESS AND LOCAL FLUCTUATION ESTIMATES FOR THE KPZ LINE ENSEMBLE
Abstract. In this paper we study the KPZ line ensemble H = {Hn}n∈N under the t 1/3 vertical and t horizontal scaling. We prove quantitative (uniformly in t) local fluctuation estimates on curves in H
Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation
  • A. Hammond
  • Mathematics
    Memoirs of the American Mathematical Society
  • 2022
The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the
Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process
We consider the asymmetric simple exclusion process (ASEP) on the positive integers with an open boundary condition. We show that, when starting devoid of particles and for a certain boundary
Limit Shapes and Local Statistics for the Stochastic Six-Vertex Model
  • A. Aggarwal
  • Mathematics
    Communications in Mathematical Physics
  • 2019
In this paper we consider the stochastic six-vertex model on a cylinder with arbitrary initial data. First, we show that it exhibits a limit shape in the thermodynamic limit, whose density profile is
Time evolution of the Kardar-Parisi-Zhang equation
The use of the non-linear SPDEs are inevitable in both physics and applied mathematics since many of the physical phenomena in nature can be effectively modeled in random and non-linear way. The
Limits and fluctuations of p-adic random matrix products
We show that singular numbers (also known as invariant factors or Smith normal forms) of products and corners of random matrices over $\mathbb{Q}_p$ are governed by the Hall-Littlewood polynomials,
...
...

References

SHOWING 1-10 OF 96 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
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
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
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
Limit Processes for TASEP with Shocks and Rarefaction Fans
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
Lectures on Integrable Probability: stochastic vertex models and symmetric functions
We consider a homogeneous stochastic higher spin six vertex model in a quadrant. For this model we derive concise integral representations for multi-point q-moments of the height function and for the
Variants of geometric RSK, geometric PNG and the multipoint distribution of the log-gamma polymer
We show that the reformulation of the geometric Robinson-Schensted-Knuth (gRSK) correspondence via local moves, introduced in \cite{OSZ14} can be extended to cases where the input matrix is replaced
...
...