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

[2510.15754] Spin glass analysis of the invariant distribution of a Lotka-Volterra SDE with a large random interaction matrix Skip to main content Search Submit Donate Log in Mathematics > Probability arXiv:2510.15754 (math) [Submitted on 17 Oct 2025 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: Spin glass analysis of the invariant distribution of a Lotka-Volterra SDE with a large random interaction matrix Authors: Mohammed Younes Gueddari , Walid Hachem View PDF HTML (experimental) 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. Subjects: Probability (math.PR) ; Mathematical Physics (math-ph) Cite as: arXiv:2510.15754 [math.PR] (or arXiv:2510.15754v2 [math.PR] for this version) https://doi.org/10.48550/arXiv.2510.15754 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Mohammed-Younes Gueddari [ view email ] [v1] Fri, 17 Oct 2025 15:38:48 UTC (104 KB) [v2] Thu, 27 Aug 2026 13:36:21 UTC (105 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2025-10 Change to browse by: math math-ph math.MP References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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