Zeros and amoebas of partition functions

TL;DR

This paper investigates the structure of zeros in partition functions, introduces statistical amoebas, and explores their properties, stratification, and relation to algebraic amoebas, including tropical limits.

Contribution

It introduces the concept of statistical amoebas, analyzes their stratified structure, and discusses their relation to algebraic amoebas and tropical limits.

Findings

Zeros form stratified hypersurfaces in ^n

Statistical amoebas exhibit specific geometric properties

Tropical limits of statistical amoebas are characterized

Abstract

Singular sectors $\mathcal{Z}_{\mathrm{sing}}$ (loci of zeros) for real-valued non-positively defined partition functions $\mathcal{Z}$ of $n$ variables are studied. It is shown that $\mathcal{Z}_{\mathrm{sing}}$ have a stratified structure and each stratum is a set of certain hypersurfaces in $\mathbb{R}^n$. The concept of statistical amoebas is introduced and their properties are studied. Relation with algebraic amoebas is discussed. Tropical limit of statistical amoebas is considered too.