On some properties of the Gauss ensemble of random matrices: Integrable system
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説明
The reduced density of p points xi,. . . , xP on the real line W of the Gaussian ensemble is given by the following integral: for n = N - p, where dTN,p denotes dxp+l A . ~dx,. It is called “orthogonal,” “unitary,” and “symplectic” according as X = 1, 2, and 4. Mehta and others have investigated in detail these three cases and gave an expression of it in determinant forms by means of Hermite polynomials (see [M2]). At least when X is equal to a positive integer, their method may be applied, but it seems to be difficult to get a simple and clear-cut formula. Here we consider it for general X. Our approach is completely different in the sense of using basic results about the twisted de Rham cohomology attached to (1.1) developed in [A31 and to give a complete system of linear differential equations (so-called “com- pletely integrable” or “holonomic” system) in the variables x1, . . . , xP satisfied by FN,p. In fact, in our case, the following identification makes us possible to use the de Rham theory for the Gibbs integral of type (1.1): Gibbs states = elements of the twisted de Rham n-cohomology. 147
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- Advances in Applied Mathematics
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Advances in Applied Mathematics 8 147-153, 1987-06-01
Elsevier BV