GOE and Airy2→1 Marginal Distribution via Symplectic Schur Functions

We derive Sasamoto’s Fredholm determinant formula for the Tracy-Widom GOE distribution, as well as the one-point marginal distribution of the Airy2→1 process, originally derived by Borodin-Ferrari-Sasamoto, as scaling limits of point-to-line and point-to-half-line directed last passage percolation with exponentially distributed waiting times. The asymptotic analysis goes through new expressions for the last passage times in terms of integrals of (the continuous analog of) symplectic and classical Schur functions, obtained recently in [6].