Statistics Seminar

Brownian Regularity for the Airy Line Ensemble
Monday, September 26, 2016 - 4:30pm
Sequoia Hall, room 200
Alan Hammond (UC Berkeley)
Abstract / Description: 

The Airy line ensemble is a positive-integer indexed ordered system of continuous random curves on the real line whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the Airy2 process, describes after the subtraction of a parabola the limiting law of the scaled weight of a geodesic running from the origin to a variable point on an anti-diagonal line in such problems as Poissonian last passage percolation. The Airy line ensemble enjoys a simple and explicit spatial Markov property, the Brownian Gibbs property.

In this talk, I will discuss how this resampling property may be used to analyse the Airy line ensemble. Arising results include a close comparison between the ensemble's curves after affine shift and Brownian bridge. The Brownian Gibbs technique is also used to compute the value of a natural exponent describing the decay in probability for the existence of several near geodesics with common endpoints in Brownian last passage percolation, where the notion of "near" refers to a small deficit in scaled geodesic weight, with the parameter specifying this nearness tending to zero.