Signatures of partition functions and their complexity reduction through the KP II equation

TL;DR

This paper explores the signatures of partition functions related to KP II soliton solutions, revealing how certain sign choices preserve determinantal structures and satisfy the KP hierarchy, with implications in geometry and information theory.

Contribution

It characterizes signatures that maintain determinantal form and KP compatibility, linking sign choices to the KP hierarchy and exploring their geometric and informational properties.

Findings

Signatures preserving determinantal form are characterized by sign choices in coefficient matrices.

Signatures compatible with KP II solutions satisfy the entire KP hierarchy.

Connections are made between signatures, tropical limits, and geometric properties.

Abstract

A statistical amoeba arises from a real-valued partition function when the positivity condition for pre-exponential terms is relaxed, and families of signatures are taken into account. This notion lets us explore special types of constraints when we focus on those signatures that preserve particular properties. Specifically, we look at sums of determinantal type, and main attention is paid to a distinguished class of soliton solutions of the Kadomtsev-Petviashvili (KP) II equation. A characterization of the signatures preserving the determinantal form, as well as the signatures compatible with the KP II equation, is provided: both of them are reduced to choices of signs for columns and rows of a coefficient matrix, and they satisfy the whole KP hierarchy. Interpretations in term of information-theoretic properties, geometric characteristics, and the relation with tropical limits are discussed.