Local behavior and hitting probabilities of the Airy1 process

Details Local behavior and hitting probabilities of the Airy1 process Local behavior and hitting probabilities of the Airy1 process

2012-01-23

arxiv.org/abs/1201.4709v3

Mathematics

Probability

Expanded introduction, added Theorem 3, changed title from "Regularity and continuum statistics of the Airy1 process"

Scientific paper

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, on $L^2(\{1,...,n\}\times\rr)$. The formula is analogous to an earlier formula of [PS02] for the Airy$_2$ process. Using this formula we are able to prove that the Airy$_1$ process is H\"older continuous with exponent $\frac12-$ and that it fluctuates locally like a Brownian motion. We also explain how the same methods can be used to obtain the analogous results for the Airy$_2$ process. As a consequence of these two results, we derive a formula for the continuum statistics of the Airy$_1$ process, analogous to that obtained in [CQR11] for the Airy$_2$ process.

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