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

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