Generating-functional analysis of random Lotka-Volterra systems: A step-by-step guide

TL;DR

This paper offers a detailed, step-by-step guide to applying generating-functional analysis to random Lotka-Volterra systems, serving as an educational resource for understanding complex disordered systems.

Contribution

It provides a comprehensive, beginner-friendly tutorial on the generating-functional method applied to Lotka-Volterra equations, filling a gap in accessible educational materials.

Findings

Clarifies the step-by-step derivation process

Connects Lotka-Volterra analysis with disordered systems

Serves as a foundational resource for further research

Abstract

This paper provides what is hopefully a self-contained set of notes describing the detailed steps of a generating-functional analysis of systems of generalised Lotka-Volterra equations with random interaction coefficients. Nothing in these notes is original, instead the generating-functional method (also known as the Martin-Siggia-Rose-DeDominic-Janssen formalism) and the resulting dynamic mean field theories have been used for the study of disordered systems and spin glasses for decades. But it is hard to find unifying sources which would allow a beginner to learn step-by-step how these methods can be used. My aim is to provide such a source. Most of the calculations are specific to generalised Lotka-Volterra systems, but much can be transferred to disordered systems in more general.