  The tacnode process is a universal determinantal point process arising from
non-intersecting particle systems and tiling problems. It is the aim of this
work to explore the integrable structure and large gap asymptotics for the gap
probability of the thinned/unthinned tacnode process over $(-s,s)$. We
establish an integral representation of the gap probability in terms of the
Hamiltonian associated with a system of differential equations. With the aids
of some remarkable differential identities for the Hamiltonian, we also compute
large gap asymptotics, up to and including the constant term in the thinned
case. As direct applications, we obtain expectation, variance and a central
limit theorem for the associated counting function.
