We consider the static and the dynamic phases in a Rosenzweig-Porter (RP) random matrix ensemble with a distribution of off-diagonal matrix elements of the form of the large-deviation ansatz. We present a general theory of survival probability in such a random-matrix model and show that the {\it averaged} survival probability may decay with time as a simple exponent, as a stretch-exponent and as a power-law or slower. Correspondingly, we identify the exponential, the stretch-exponential and the frozen-dynamics phases. As an example, we consider the mapping of the Anderson localization model on Random Regular Graph onto the RP model and find exact values of the stretch-exponent $\kappa$ in the thermodynamic limit. As another example we consider the logarithmically-normal RP random matrix ensemble and find analytically its phase diagram and the exponent $\kappa$. Our theory allows to describe analytically the finite-size multifractality and to compute the critical length with the exponent $\nu_{MF}=1$ associated with it.

中文翻译：

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“多重分形” Rosenzweig-Porter 模型中的动态相

我们考虑具有大偏差 ansatz 形式的非对角矩阵元素分布的 Rosenzweig-Porter (RP) 随机矩阵系综中的静态和动态阶段。我们在这样的随机矩阵模型中提出了生存概率的一般理论，并表明 {\it averaged} 生存概率可能会随着时间的推移而衰减为简单指数、拉伸指数和幂律或更慢。相应地，我们确定了指数、拉伸指数和冻结动力学阶段。例如，我们考虑将随机正则图上的 Anderson 定位模型映射到 RP 模型，并在热力学极限中找到拉伸指数 $\kappa$ 的精确值。作为另一个例子，我们考虑对数正态 RP 随机矩阵系综，并通过分析找到它的相图和指数 $\kappa$。我们的理论允许分析地描述有限大小的多重分形并计算临界长度，其中指数 $\nu_{MF}=1$ 与之相关。