A. Daletskii

The aim of this paper is to give a review of recent results of Yu. Kondratiev, Yu. Kozitsky, T. Pasurek and myself on the multiplicity of Gibbs states (phase transitions) in infinite spin systems on random configurations, and provide a `pedestrian' route following Georgii–Haggstrom approach to (closely related to phase transitions) percolation problems for a class of random point processes.

We study a class of measures on the space $\Gamma _{X}$ of locally finiteconfi\-gurations in $X=\mathbb{R}^{d}$, obtained as images of ''lattice'' Gibbs measures on $X^{\mathbb{Z}^{d}}$ with respect to an embedding $\mathbb{Z}^{d}\subset \mathbb{R}^{d}$. For these measures, we prove the integration by parts formula andlog-Sobolev inequality.

We prove that the natural action of permutations in a tensor product of type $\mathrm{II}$ factors is free, and compute the von Neumann trace of the projection onto the space of symmetric and antisymmetric elements respectively. We apply this result to computation of von Neumann dimensions of the spaces of square-integrable harmonic forms ($L^{2}$-Betti numbers) of $N$-point configurations in Riemannian manifolds with infinite discrete groups of isometries.