Zeros of Airy Function and Relaxation Process

Zeros of Airy Function and Relaxation Process

One-dimensional system of Brownian motions called Dyson’s model is the particle system with long-range repulsive forces acting between any pair of particles, where the strength of force is β/2 times the inverse of particle distance. When β = 2, it is realized as the Brownian motions in one dimension conditioned never to collide with each other. For any initial configuration, it is proved that D...

Philippe Flajolet and the Airy Function

1. Historical backgrounds: the Airy function in Physics 2 2. The area-Airy distributions: Brownian motion, linear probing hashing, additive parameters in grammars 4 2.1. Area under a Brownian excursion 4 2.2. On the analysis of linear probing hashing 4 2.3. Analytic variations on the Airy distribution 5 2.4. Hachage, arbres, chemins & graphes 6 3. Random matrices, Airy kernel and the Tracy–Wido...

On Asymptotics for the Airy Process

The Airy process t → A(t), introduced by Prähofer and Spohn, is the limiting stationary process for a polynuclear growth model. Adler and van Moerbeke found a PDE in the variables s1, s2 and t for the probability Pr (A(0) ≤ s1, A(t) ≤ s2). Using this they were able, assuming the truth of a certain conjecture and appropriate uniformity, to obtain the first few terms of an asymptotic expansion fo...

Absence of zeros and asymptotic error estimates for Airy and parabolic cylinder functions

Scale Invariance of the PNG Droplet and the Airy Process

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 process is stationary, it has continuous sample paths, its single “time” (fixed y) distribution is the Tracy-Widom distribution of the largest eigenvalue of a GUE random matrix, and the Airy process has ...