We distinguish a subclass of stochastic point processes which we call "Giambelli compatible point processes". A striking feature of these processes is that the Giambelli formula for the Schur functions remains invariant under the averaging over random configurations.
We prove that orthogonal polynomial ensembles, z-measures on partitions, and the spectral measures of characters of generalized regular representations of S(\infty) induce point processes of that type.
We show that the Giambelli compatible point processes are characterized by certain identities which are reduced to formulae for averages of characteristic polynomials in the case of orthogonal polynomial ensembles. These identities imply the determinantal structure of the correlation functions. Based on these identities we provide the most straightforward (among those known so far) derivation of the associated correlation kernels.