A Riemann-Hilbert problem for equations of Painlev\'e type in the one matrix model with semi-classical potential

We study the hermitian one matrix model with semi-classical potential. This is a general unitary invariant random matrix ensemble in which the potential has a derivative that is a rational function and the measure is supported on some collection of disjoint closed intervals. Such models have attracted much interest both due to their physical applications and relations to integrable systems. An object of central interest in random matrix theory is the correlation kernel, as this encodes the eigenvalue correlation functions. In recent years many results have been obtained proving that the correlation kernel near special points in the spectrum can be expressed in terms of Painlev\'{e} transcendents and their associated Riemann-Hilbert problems. In the present work we build on this success by proposing a model problem that is general enough to describe the limiting kernel at any point in the spectrum. In the most general situation this would include cases of logarithmic singularities and essential singularities in the weight colliding with soft or hard edges, the bulk of the spectrum or even births of a cut.