Alan Hammond

TL;DR: In this paper, it was shown that the top line of the Airy line ensemble without a parabola attains its maximum at a unique point, which is the case of the non-intersecting version of the multi-line Airy process.

TL;DR: In this article, the authors constructed an ensemble of random continuous curves with three properties: 1) the diffusion of O'Connell and Yor, 2) the convergence result of Moreno Flores et al. (in preparation) of the lowest indexed curve of that diffusion to the solution of the KPZ equation with narrow wedge initial data, and 3) the one-point distribution formula proved by Amir et al (Commun Pure Appl Math 64:466-537, 2011).

TL;DR: In this article, the authors constructed an ensemble of random continuous curves with three properties: 1. The lowest index curve is distributed as the time $t$ Hopf-Cole solution to the Kardar-Parisi-Zhang (KPZ) stochastic PDE with narrow wedge initial data; 2. The entire ensemble satisfies a resampling invariance which is called the $\mathbf{H}$-Brownian Gibbs property (with $\mathBF{H(x)=e^{x}$); 3. Concentration in the $t^{2

TL;DR: In this paper, the authors employ the Brownian Gibbs property to make a close comparison between the Airy line ensemble's curves after affine shift and Brownian bridge, proving the finiteness of a superpolynomially growing moment bound on Radon-Nikodym derivatives.

TL;DR: In this article, a biased random walk Xn on a Galton-Watson tree with leaves in the sub-ballistic regime was considered, and it was shown that Δn/n1/γ does not converge in law.