Erdos L, Krüger T, Renfrew D (2023)

**Publication Type:** Journal article

**Publication year:** 2023

**Book Volume:** 33

**Pages Range:** 3098-3144

**Journal Issue:** 4

**DOI:** 10.1214/22-AAP1886

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 random connectivity matrix X. We rigorously prove and slightly correct the explicit 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.

Institute of Science and Technology Austria (IST Austria)
Austria (AT)
Binghamton University / State University of New York, Binghamton
United States (USA) (US)

**APA:**

Erdos, L., Krüger, T., & Renfrew, D. (2023). RANDOMLY COUPLED DIFFERENTIAL EQUATIONS WITH ELLIPTIC CORRELATIONS. *Annals of Applied Probability*, *33*(4), 3098-3144. https://dx.doi.org/10.1214/22-AAP1886

**MLA:**

Erdos, László, Torben Krüger, and David Renfrew. "RANDOMLY COUPLED DIFFERENTIAL EQUATIONS WITH ELLIPTIC CORRELATIONS." *Annals of Applied Probability* 33.4 (2023): 3098-3144.

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