Abstract
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 , in the large N limit, where f and g are analytic functions.
Funding Statement
LE was partially supported by ERC Advanced Grant RANMAT No. 338804 and RMTBeyond No. 101020331.
TK was partially supported by VILLUM FONDEN research grant no. 29369.
DR was partially supported by Austrian Science Fund (FWF): M2080-N35.
Acknowledgments
DR would like to thank Nicolas Brunel and Johnatan Aljadeff for fruitful discussions as well as sharing unpublished notes. The authors would like to thank the anonymous referees for their helpful comments.
Citation
László Erdős. Torben Krüger. David Renfrew. "Randomly coupled differential equations with elliptic correlations." Ann. Appl. Probab. 33 (4) 3098 - 3144, August 2023. https://doi.org/10.1214/22-AAP1886
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