Local behavior and hitting probabilities of the \text{ Airy}_1 process

We obtain a formula for the $n$ -dimensional distributions of the $\text{ Airy}_1$ process in terms of a Fredholm determinant on $L^2(\mathbb{R })$ , as opposed to the standard formula which involves extended kernels, on $L^2(\{1,\dots ,n\}\times \mathbb{R })$ . The formula is analogous to an earlier formula of Prähofer and Spohn (J Stat Phys 108(5–6):1071–1106, 2002) for the $\text{ Airy}_2$ process. Using this formula we are able to prove that the $\text{ Airy}_1$ process is Hölder continuous with exponent $\frac{1}{2}$ —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 $\text{ Airy}_2$ process. As a consequence of these two results, we derive a formula for the continuum statistics of the $\text{ Airy}_1$ process, analogous to that obtained in Corwin et al. (Commun Math Phys 2011, to appear) for the $\text{ Airy}_2$ process.