具有椭圆相关的随机耦合微分方程
Randomly coupled differential equations with elliptic correlations
摘要：
研究了一类随机系数$N$线性微分方程大系统的长时间渐近性态。我们允许系数之间存在一般的椭圆相关结构，因此我们大大推广了我们之前仅限于独立情况的工作[14]。特别地，我们分析了神经网络理论中的一个最近的模型[27]，该模型特别关注随机连接矩阵$X$中分布不对称性的影响。我们严格证明并稍微修正了[28]中关于时间衰减作为不对称参数的函数的显式公式。我们的主要工具是一个渐近精确的公式规范化的轨迹$f(X) g(X^*)$，在大$N$限制，其中$f$和$g$是解析函数。
英文摘要：
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 [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.
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