Randomly coupled differential equations with elliptic correlations (Q6165249)

In order to describe the evolution of a network of \(N\) fully connected neurons, one can consider the following system of linear differential equations: \[ \partial_t u(t) = -u(t) + g X u(t), \] where \(u\) has values in \(\mathbb{C}^N\), \(X\in\mathbb{C}^{N\times N}\) is a random matrix, and \(g>0\) is a coupling parameter. In the simplest setting the entries of the matrix \(X\) would be independent or even i.i.d.\ random variables. However, recent experimental data suggests that in reality \(X_{ij}\) and \(X_{ij}\) are not independent. One could therefore assume that for every \(i\neq j\), \[ \mathbb{E} X_{ij} X_{ji} = \rho \mathbb{E} |X_{ij}|^2 = \frac{\rho}{N}, \] where the complex number \(\rho\) is the correlation coefficient (the pairs \((X_{ij}, X_{ji})\) are still assumed to be independent for different index pairs \(\{i,j\}\)). Note that \(\rho=0\) in the independent case and \(\rho = 1\) if \(X=X^*\) is Hermitian. Depending on the value of the coupling parameter \(g\) the solution \(u\) typically grows or decays exponentially, but for a critical value it exhibits a power law decay with an exponent which depends on the symmetry properties of \(X\). The extreme cases \(\rho=0\) and \(\rho = 1\) were studied in a previous article of the authors [SIAM J. Math. Anal. 50, No. 3, 3271--3290 (2018; Zbl 1392.60011)]. In the paper under review, they focus on the remaining case \(0<|\rho| <1\) (some mathematically non-rigorous analysis of this intermediate case appears in the literature; here the authors present complete proofs). The results about the long time asymptotic behaviour of the above system of \(N\) linear differential equations are presented under a general assumption that \(X\) is an elliptic or elliptic-type random matrix. One of the main tools in the proofs is an asymptotically precise formula for \(\frac{1}{N}\mathbb{E} \operatorname{Tr}(f(X) g(X^*))\), where \(f\) and \(g\) are analytic functions. The paper contains 4 figures.