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Fredholm determinants, differential equations and matrix models. (English) Zbl 0813.35110
Orthogonal polynomial random matrices of N. N. Hermitian matrices lead to Fredholm determinants of integral operators with kernel $$(f(x)g(y) - g(x)f(y))/(x-y)$$. This paper shows that these Fredholm determinants can be expressed in terms of solutions of partial differential equations as long as $$f$$ and $$g$$ satisfy some kinds of differentiation formulae. The model applies to sine, Airy and Bessel kernels as well as Hermite, Laguerre and Jacobi ensembles. The analysis of these equations provides explicit solution in terms of Painlevé transcendents for the distribution functions of the largest and smallest eigenvalues in the finite Hermite and Laguerre ensembles, and for the distributions functions of the largest and smallest singular values of rectangular matrices of which the entries are independent identically distributed complex Gaussian variables.

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 33C10 Bessel and Airy functions, cylinder functions, $${}_0F_1$$ 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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