Asymptotics of the Tacnode process: a transition between the gap probabilities from the Tacnode to the Airy process

We study the gap probabilities of the single-time Tacnode process. Through steepest descent analysis of a suitable Riemann-Hilbert problem, we show that under appropriate scaling regimes the gap probability of the Tacnode process degenerates into a product of two independent gap probabilities of the Airy processes.