Trees, functional inversion, and the virial expansion
Trees are ubiquitous. Probabilists may think of branching processes and ask about extinction or survival. The recursive structure of trees leads to functional equations for generating functions, of interest in analytic combinatorics. Trees also help organize power series expansions in various areas of analysis and mathematical physics, from numerics (Butcher trees) to renormalization (Gallavotti-Niccolo trees). The talk presents yet another application, namely inverse function theorems for functionals in measure spaces for which Banach inversion is not possible. Combined with cluster expansions from equilibrium statistical mechanics, the theorem allows for a rigorous derivation, in a restricted parameter regime, of density functionals used in analytic models of materials. The talk is based on joint work with Tobias Kuna and Dimitrios Tsagkarogiannis (arXiv:1906.02322 [math-ph]) and considerably improves earlier results based on Lagrange-Good inversion (J., Tate, Tsagkarogiannis, Ueltschi, CMP 2014).