RANDOMLY COUPLED DIFFERENTIAL EQUATIONS WITH ELLIPTIC CORRELATIONS

• 成果类型: Article
• 署名作者: Erdos, Laszlo; Krueger, Torben; Enfrew, David
• 署名单位: University of Erlangen Nuremberg; State University of New York (SUNY) System; Binghamton University, SUNY
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1886
• 发表日期: 2023
• 页码: 3098-3144
• 关键词: random matrices; eigenvectors; statistics; network

摘要

We consider the long time asymptotic behavior of a large system of N linear differential equations with random coefficients. We allow for general elliptic correlation structures among the coefficients, thus we substantially generalize our previous work (SIAM J. Math. Anal. 50 (2018) 3271-3290) that was restricted to the independent case. In particular, we analyze a recent model in the theory of neural networks (Phys. Rev. E 97 (2018) 062314) that specifically focused on the effect of the distributional asymmetry in the ran-dom connectivity matrix X. We rigorously prove and slightly correct the ex-plicit formula from (J. Math. Phys. 41 (2000) 3233-3256) on the time decay as a function of the asymmetry parameter. Our main tool is an asymptotically precise formula for the normalized trace of f (X)g(X*), in the large N limit, where f and g are analytic functions.