The problem of describing a point process from its low order correlation functions is well-known in statistical mechanics as realizability problem, but naturally occurs also in several other areas dealing with the analysis of complex systems. In this talk we first give an introduction to this problem outlining its interpretation as an infinite dimensional version of the classical moment problem. Then we focus on a specific instance of realizability problem, namely, on the question of establishing whether two given functions non-negative and symmetric on Z^d are the first two correlation functions of a translation invariant point process on Z^d (here d is a positive integer). Explicit constructions of such realizing processes for d=1 are available in literature under some natural assumptions relevant for applications in fluid theory and material science. We will review such one-dimensional constructions and then present an explicit construction recently developed in a joint work with E. Caglioti and T. Kuna for any d>=2. Our construction is the first to produce a lower bound for the maximal realizable density which improves the general lower bounds already known in literature and so provides a better approximation of the feasibility region for this class of realizability problems.
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