We show that the supremum of the average of the Airy process and its time reversal minus a parabola is distributed as the maximum of two independent GUE Tracy-Widom random variables. The proof is obtained by considering a directed last passage percolation model with a rotational symmetry in two different ways. We also review other known identities between the Airy process and the Tracy-Widom distributions.
"On the average of the Airy process and its time reversal." Electron. Commun. Probab. 18 1 - 10, 2013. https://doi.org/10.1214/ECP.v18-2952