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Correlation functions, cluster functions, and spacing distributions for random matrices. (English) Zbl 0942.60099
The $$n$$-point distribution function for orthogonal and symplectic matrix ensembles is proportional to the multidimensional integral $\int dx_{n+ 1}\cdots \int dx_N \prod^N_{j= 1}w(x_j) \prod_{1\leq j< k\leq N}|x_k- x_j|^\beta$ for $$\beta= 1$$ and 4, respectively. Following on from pioneering work of F. J. Dyson and M. L. Mehta, it has been shown by a number of authors that the multidimensional integral in both the orthogonal and symplectic cases can be expressed as a quaternion determinant with elements involving skew orthogonal polynomials associated with the appropriate skew inner product. (A quaternion determinant is essentially a Pfaffian, while a skew inner product is characterized by the property $$\langle f,g\rangle= -\langle g,f\rangle$$.) Here a new approach is given to obtain the Pfaffian formulas which avoids all reference to the theory of quaternion determinants and skew orthogonal polynomials. Crucial use is made of the identity $$\text{det}(1+ AB)= \text{det}(1+ BA)$$.

##### MSC:
 60K40 Other physical applications of random processes 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics 15B52 Random matrices (algebraic aspects)
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