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Painleve II in random matrix theory and related fields

We review some occurrences of Painlevé II transcendents in the study of two-dimensional Yang-Mills theory, fluctuation formulas for growth models, and as distribution functions within random matrix theory. We first discuss settings in which the parameter $α$ in the Painlevé equation is zero, and the boundary condition is that of the Hasting-MacLeod solution. As well as expressions involving the Painlevé transcendent itself, one encounters the sigma form of the Painlevé II equation, and Lax pair equations in which the Painlevé transcendent occurs as coefficients. We then consider settings which give rise to general $α$ Painlevé II transcendents. In a particular random matrix setting, new results for the corresponding boundary conditions in the cases $α= \pm 1/2$, 1 and 2 are presented.