# Brownian Gibbs property for Airy line ensembles

@article{Corwin2011BrownianGP,
title={Brownian Gibbs property for Airy line ensembles},
author={Ivan Corwin and Alan Hammond},
journal={Inventiones mathematicae},
year={2011},
volume={195},
pages={441-508}
}
• Published 10 August 2011
• Mathematics
• Inventiones mathematicae
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 limit as N→∞ of the collection of curves scaled so that the point (0,21/2N) is fixed and space is squeezed, horizontally by a factor of N2/3 and vertically by N1/3. If a parabola is added to each of the curves of this scaling limit, an x-translation invariant process sometimes called the multi-line…
162 Citations
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
On the limiting law of line ensembles of Brownian polymers with geometric area tilts
• Mathematics
• 2022
. We study the line ensembles of non-crossing Brownian bridges above a hard wall, each tilted by the area of the region below it with geometrically growing pre-factors. This model, which mimics the
Maximum of an Airy process plus Brownian motion and memory in KPZ growth
We obtain several exact results for universal distributions involving the maximum of the Airy$_2$ process minus a parabola and plus a Brownian motion, with applications to the 1D Kardar-Parisi-Zhang
On the probability of several near geodesics with shared endpoints in Brownian last passage percolation, and Brownian bridge regularity for the Airy line ensemble
The Airy line ensemble is a positive-integer indexed system of continuous random curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the
Reunion Probability of N Vicious Walkers: Typical and Large Fluctuations for Large N
• Mathematics
• 2013
We consider three different models of N non-intersecting Brownian motions on a line segment [0,L] with absorbing (model A), periodic (model B) and reflecting (model C) boundary conditions. In these
Extreme statistics of non-intersecting Brownian paths
• Mathematics
• 2016
We consider finite collections of $N$ non-intersecting Brownian paths on the line and on the half-line with both absorbing and reflecting boundary conditions (corresponding to Brownian excursions and
Extremes of N Vicious Walkers for Large N: Application to the Directed Polymer and KPZ Interfaces
We compute the joint probability density function (jpdf) PN(M,τM) of the maximum M and its position τM for N non-intersecting Brownian excursions, on the unit time interval, in the large N limit. For
Local behavior and hitting probabilities of the Airy1 process
• Mathematics
• 2012
We obtain a formula for the $n$-dimensional distributions of the Airy$_1$ process in terms of a Fredholm determinant on $L^2(\rr)$, as opposed to the standard formula which involves extended kernels,
Edge scaling limit of Dyson Brownian motion at equilibrium with general β ľ 1
Abstract: For general β ľ 1, we consider Dyson Brownian motion at equilibrium and prove convergence of the extremal particles to an ensemble of continuous sample paths in the limit N Ñ 8. For each
Universality in models of local random growth via Gibbs
• Mathematics
• 2019
An important technique for understanding a random system is to find a higher dimensional random system that enjoys an attractive and tractable structure and that has the system of interest as a

## References

SHOWING 1-10 OF 97 REFERENCES
Airy processes with wanderers and new universality classes.
• Mathematics
• 2010
Consider n + m nonintersecting Brownian bridges, with n of them leaving from 0 at time t = -1 and returning to 0 at time t = -1, while the m remaining ones (wanderers) go from m points a(i) to m
Extremes of N Vicious Walkers for Large N: Application to the Directed Polymer and KPZ Interfaces
We compute the joint probability density function (jpdf) PN(M,τM) of the maximum M and its position τM for N non-intersecting Brownian excursions, on the unit time interval, in the large N limit. For
Nonintersecting Brownian Motions on the Half-Line and Discrete Gaussian Orthogonal Polynomials
We study the distribution of the maximal height of the outermost path in the model of N nonintersecting Brownian motions on the half-line as N→∞, showing that it converges in the proper scaling to
Nonintersecting Brownian interfaces and Wishart random matrices.
• Mathematics
Physical review. E, Statistical, nonlinear, and soft matter physics
• 2009
It is shown that, for a large system and with an appropriate choice of the external confining potential, the joint distribution of the heights of the N nonintersecting interfaces at a fixed point on the substrate can be mapped to the joint Distribution of the eigenvalues of a Wishart matrix of size N with complex entries, thus providing a physical realization of the Wishart Matrix.
Markov processes of infinitely many nonintersecting random walks
• Mathematics
• 2011
Consider an N-dimensional Markov chain obtained from N one-dimensional random walks by Doob h-transform with the q-Vandermonde determinant. We prove that as N becomes large, these Markov chains
Nonintersecting Brownian Excursions
• Mathematics
• 2006
Author(s): Tracy, Craig A.; Widom, Harold | Abstract: We consider the process of $n$ Brownian excursions conditioned to be nonintersecting. We show the distribution functions for the top curve and
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
PDEs for the joint distributions of the Dyson, Airy and Sine processes
• Mathematics
• 2005
In a celebrated paper, Dyson shows that the spectrum of an n x n random Hermitian matrix, diffusing according to an Ornstein-Uhlenbeck process, evolves as n noncolliding Brownian motions held