We prove a novel inversion theorem for functionals given as power series in infinite-dimensional spaces. This provides a rigorous framework to prove convergence of density functionals for inhomogeneous systems with applications in classical density function theory, liquid crystals, molecules with various shapes or other internal degrees of freedom. The key technical tool is the representation of the inverse via a fixed point equation and a combinatorial identity for trees, which allows us to obtain convergence estimates in situations where Banach inversion fails.

We prove a new convergence condition for the activity expansion of correlation functions in equilibrium statistical mechanics with possibly negative pair potentials. For non-negative pair potentials, the criterion is an if and only if condition. The condition is formulated with a sign-flipped Kirkwood-Salsburg operator and known conditions such as Kotecky-Preiss and Fernandez-Procacci are easily recovered. In addition, we deduce new sufficient convergence conditions for hard-core systems in R-d and Z(d) as well as for abstract polymer systems. The latter improves on the Fernandez-Procacci criterion.

"The survey is dedicated to a celebrated series of quantitave results, developed by the Lithuanian school of probability, on the normal approximation for a real-valued random variable. The key ingredient is a bound on cumulants of the type vertical bar kappa(j)(X)vertical bar <= j!(1+gamma)/Delta(j-2), which is weaker than Cramer's condition of finite exponential moments. We give a self-contained proof of some of the ""main lemmas"" in a book by Saulis and StatuleviCius (1989), and an accessible introduction to the Cramer-Petrov series. In addition, we explain relations with heavy-tailed Weibull variables, moderate deviations, and mod-phi convergence. We discuss some methods for bounding cumulants such as summability of mixed cumulants and dependency graphs, and briefly review a few recent applications of the method of cumulants for the normal approximation."

We prove a multivariate Lagrange-Good formula for functionals of uncountably many variables and investigate its relation with inversion formulas using trees. We clarify the cancellations that take place between the two aforementioned formulas and draw connections with similar approaches in a range of applications.

A generalized version of Groeneveld's convergence criterion for the virial expansion and generating functionals for weighted two-connected graphs is proven. This criterion works for inhomogeneous systems and yields bounds for the density expansions of the correlation functions rho (s) (a.k.a. distribution functions or factorial moment measures) of grand-canonical Gibbs measures with pairwise interactions. The proof is based on recurrence relations for graph weights related to the Kirkwood-Salsburg integral equation for correlation functions. The proof does not use an inversion of the density-activity expansion,. however, a Mobius inversion on the lattice of set partitions enters the derivation of the recurrence relations.

We investigate a toy model for phase transitions in mixtures of incompressible droplets. The model consists of non-overlapping hypercubes in Z(d) of sidelengths 2(j), j is an element of N-0. Cubes belong to an admissible set B such that if two cubes overlap, then one is contained in the other. Cubes of sidelength 2(j) have activity z(j) and density rho(j). We prove explicit formulas for the pressure and entropy, prove a van-der-Waals type equation of state, and invert the density-activity relations. In addition we explore phase transitions for parameter-dependent activities z j (mu) = exp(2(dj)mu - E-j). We prove a sufficient criterion for absence of phase transition, show that constant energies E-j equivalent to lambda lead to a continuous phase transition, and prove a necessary and sufficient condition for the existence of a first-order phase transition.

We consider a binary system of small and large objects in the continuous space interacting via a nonnegative potential. By integrating over the small objects, the effective interaction between the large ones becomes multi-body. We prove convergence of the cluster expansion for the grand canonical ensemble of the effective system of large objects. To perform the combinatorial estimate of hypergraphs (due to the multi-body origin of the interaction), we exploit the underlying structure of the original binary system. Moreover, we obtain a sufficient condition for convergence which involves the surface of the large objects rather than their volume. This amounts to a significant improvement in comparison to a direct application of the known cluster expansion theorems. Our result is valid for the particular case of hard spheres (colloids) for which we rigorously treat the depletion interaction.