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 [14] that was restricted to the independent case. In
particular, we analyze a recent model in the theory of neural networks [27]
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 [28] 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.Comment: 46 pages, 4 figures. Paper has been reorganized. Examples have been
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