Equilibrium size distribution and phase separation of multivalent, molecular assemblies in dilute solution

Equilibrium self-assembly, gelation, and phase separation of multivalent molecules in dilute solutions analyzed using statistics of lattice animals depicted here.

Expansion in High Dimension for the Growth Constants of Lattice Trees and Lattice Animals

We compute the first three terms of the 1/d expansions for the growth constants and one-point functions of nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice $\mathbb{Z}^d$, with rigorous error estimates. The proof uses the lace expansion, together with a new expansion for the one-point functions based on inclusion–exclusion.