论文标题
通过随机矩阵相互作用的扩散：通过随机泰勒扩展的通用性
Diffusions interacting through a random matrix: universality via stochastic Taylor expansion
论文作者
论文摘要
考虑$（x_ {i}（t））$求解通过随机矩阵$ \ mathbf j =（j_ {ij}）$与独立（不一定要分布的）随机系数相互作用的$ n $随机微分方程的系统。我们表明，从$ \ Mathbf j $独立于$μ$初始化的$（x_i（t））$的平均观测值的轨迹是通用的，即仅取决于分发$ \ Mathbf {J j} $的选择。我们采用一般的组合方法来证明具有随机系数的动态系统的通用性，将随机的泰勒扩展与匹配类型的参数相结合。我们的结果意味着普遍性的具体设置包括球形SK旋转玻璃中的衰老，Langevin Dynamics和梯度流以对称和非对称Hopfield网络。
Consider $(X_{i}(t))$ solving a system of $N$ stochastic differential equations interacting through a random matrix $\mathbf J = (J_{ij})$ with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of $(X_i(t))$, initialized from some $μ$ independent of $\mathbf J$, are universal, i.e., only depend on the choice of the distribution $\mathbf{J}$ through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks.
