Zeros and amoebas of partition functions

Singular sectors Z_sing (loci of zeros) for real-valued non-positively defined partition functions Z of n variables are studied. It is shown that Z_sing have a stratified structure where each stratum is a set of certain hypersurfaces in R^n. The concept of statistical amoeba is introduced and the properties of a family of statistical amoebas are studied. The relation with algebraic amoebas is discussed. Tropical limits of statistical amoebas are considered too. Applications of the concept of statistical amoeba to the analysis of singular sectors for integrable equations and properties of macroscopic systems with multiple equilibria, including frustrated systems, are discussed.