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We consider the noncolliding Brownian motion (BM) with $N$ particles startingfrom the eigenvalue distribution of Gaussian unitary ensemble (GUE) of $N\times N$ Hermitian random matrices with variance $\sigma^2$. We prove thatthis process is equivalent with the time shift $t \to t+\sigma^2$ of thenoncolliding BM starting from the configuration in which all $N$ particles areput at the origin. In order to demonstrate nontriviality of such equivalencefor determinantal processes, we show that, even from its special consequence,determinantal expressions are derived for the ensemble averages of products ofcharacteristic polynomials of random matrices in GUE. Another determinantalprocess, noncolliding squared Bessel process with index $\nu >-1$, is alsostudied in parallel with the noncolliding BM and corresponding results forcharacteristic polynomials are given for random matrices in the chiral GUE aswell as in the Gaussian ensembles of class C and class D.

Characteristic Polynomials of Random Matrices and Noncolliding Diffusion Processes