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Spin glass analysis of the invariant distribution of a Lotka-Volterra SDE with a large random interaction matrix

Mohammed Younes Gueddari, Walid Hachem

TL;DR

The paper analyzes the invariant distribution of a generalized Lotka-Volterra SDE with a large random interaction matrix drawn from a deformed GOE ensemble. It shows existence and uniqueness for the deterministic LV dynamics and derives a rigorous Parisi-type variational formula for the limiting free energy of the random Gibbs measure, accounting for the non-compact support of the ambient space. The main result characterizes the large-n free energy via a Parisi functional P_a(ζ,h,γ) minimized over a Parisi measure ζ and auxiliary parameters, with the optimization domain depending on the sign of the deformation parameter α. The approach blends Guerra interpolation, RPC representations, and spin-glass techniques to rigorize previous replica-based predictions and to handle the non-compact Gibbs measure setting. This provides a mathematically rigorous bridge between theoretical ecology models with random interactions and disordered systems methods, opening the way to analyses under non-compact supports and broader random-matrix ecosystems.

Abstract

The generalized Lotka-Volterra stochastic differential equation with a symmetric food interaction matrix is frequently used to model the dynamics of the abundances of the species living within an ecosystem when these interactions are mutualistic or competitive. In the relevant cases of interest, the Markov process described by this equation has an unique invariant distribution which has a Hamiltonian structure. Following an important trend in theoretical ecology, the interaction matrix is considered in this paper as a large random matrix. In this situation, the (conditional) invariant distribution takes the form of a random Gibbs measure that can be studied rigorously with the help of spin glass techniques issued from the field of physics of disordered systems. Considering that the interaction matrix is an additively deformed GOE matrix, which is a well-known model for this matrix in theoretical ecology, the free energy of the model is derived in the limit of the large number $n$ of species, making rigorous some recent results from the literature. The free energy analysis made in this paper could be adapted to other situations where the Gibbs measure is non compactly supported.

Spin glass analysis of the invariant distribution of a Lotka-Volterra SDE with a large random interaction matrix

TL;DR

The paper analyzes the invariant distribution of a generalized Lotka-Volterra SDE with a large random interaction matrix drawn from a deformed GOE ensemble. It shows existence and uniqueness for the deterministic LV dynamics and derives a rigorous Parisi-type variational formula for the limiting free energy of the random Gibbs measure, accounting for the non-compact support of the ambient space. The main result characterizes the large-n free energy via a Parisi functional P_a(ζ,h,γ) minimized over a Parisi measure ζ and auxiliary parameters, with the optimization domain depending on the sign of the deformation parameter α. The approach blends Guerra interpolation, RPC representations, and spin-glass techniques to rigorize previous replica-based predictions and to handle the non-compact Gibbs measure setting. This provides a mathematically rigorous bridge between theoretical ecology models with random interactions and disordered systems methods, opening the way to analyses under non-compact supports and broader random-matrix ecosystems.

Abstract

The generalized Lotka-Volterra stochastic differential equation with a symmetric food interaction matrix is frequently used to model the dynamics of the abundances of the species living within an ecosystem when these interactions are mutualistic or competitive. In the relevant cases of interest, the Markov process described by this equation has an unique invariant distribution which has a Hamiltonian structure. Following an important trend in theoretical ecology, the interaction matrix is considered in this paper as a large random matrix. In this situation, the (conditional) invariant distribution takes the form of a random Gibbs measure that can be studied rigorously with the help of spin glass techniques issued from the field of physics of disordered systems. Considering that the interaction matrix is an additively deformed GOE matrix, which is a well-known model for this matrix in theoretical ecology, the free energy of the model is derived in the limit of the large number of species, making rigorous some recent results from the literature. The free energy analysis made in this paper could be adapted to other situations where the Gibbs measure is non compactly supported.
Paper Structure (17 sections, 19 theorems, 206 equations, 1 figure)

This paper contains 17 sections, 19 theorems, 206 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The LV SDE analysis
  3. Realizability bound for deformed GOE interaction matrix
  4. Asymptotics of the free energy
  5. Proof of Theorem \ref{['F->P']}: some preparation
  6. Ruelle probability cascades
  7. Useful bounds
  8. Proof of Theorem \ref{['F->P']}: upper bound for $\alpha\leq 0$
  9. Proof of Theorem \ref{['F->P']}: lower bound
  10. Proof of Theorem \ref{['F->P']} for $\alpha > 0$.
  11. Lower bound
  12. Upper bound
  13. Proof of Proposition \ref{['prop-eds']}
  14. Proof of Lemma \ref{['F<F^A']}
  15. Proof of Lemma \ref{['F>Fa']}
  16. ...and 2 more sections

Key Result

Proposition 1

Assume that Condition l+ is satisfied. Then, for each initial probability measure $\mu$ such that $x_0 \sim \mu$, the SDE eds admits an unique strong solution on ${{\mathbb R}}_+$.

Figures (1)

  • Figure 1: The curve $\boldsymbol\lambda_+(\alpha,\kappa) = 1$

Theorems & Definitions (27)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • proof : Proof of Proposition \ref{['erg']}
  • Proposition 4
  • Remark 1
  • Lemma 5
  • Lemma 6
  • proof
  • Theorem 7
  • ...and 17 more