论文标题
在随机常规图上的定位过渡为对数正常rosenzweig-porter随机矩阵集合集合中的不稳定三级点
Localization transition on the Random Regular Graph as an unstable tricritical point in a log-normal Rosenzweig-Porter random matrix ensemble
论文作者
论文摘要
高斯罗森兹维格 - 波特（GRP）随机矩阵集合是唯一一个稳健多型相和厄尔及性跃迁具有数学定理状态的矩阵。然而，GRP模型中的这个阶段过于简化：分形维度的光谱是退化的，并且局部频谱中的迷你频段不是多重分数。在本文中，我们建议通过采用对数正常（LN）分布的分子基质元素来扩展GRP模型。这样的LN-RP模型的家族由对称参数$ p $参数化，并且它在$ p \ rightarrow 0 $的GRP之间进行了插值，而Levy emembles则以$ p \ rightarrow \ rightarrow \ infty $。特殊点$ p = 1 $表明是随机常规图上安德森本地化模型的最简单近似值。我们详细研究LN-RP模型的相图，并表明$ p = 1 $是多型相位首先崩溃的三级点。相对于对数正态分布的截断，这种崩溃表明是不稳定的。我们建议一个非共性阶段稳定性的新标准，并证明LN-RP模型中的Anderson过渡在所有$ p> 0 $上都是不连续的。
Gaussian Rosenzweig-Porter (GRP) random matrix ensemble is the only one in which the robust multifractal phase and ergodic transition have a status of a mathematical theorem. Yet, this phase in GRP model is oversimplified: the spectrum of fractal dimensions is degenerate and the mini-band in the local spectrum is not multifractal. In this paper we suggest an extension of the GRP model by adopting a logarithmically-normal (LN) distribution of off-diagonal matrix elements. A family of such LN-RP models is parametrized by a symmetry parameter $p$ and it interpolates between the GRP at $p\rightarrow 0$ and Levy ensembles at $p\rightarrow\infty$. A special point $p=1$ is shown to be the simplest approximation to the Anderson localization model on a random regular graph.We study in detail the phase diagram of LN-RP model and show that $p=1$ is a tricritical point where the multifractal phase first collapses. This collapse is shown to be unstable with respect to the truncation of the log-normal distribution. We suggest a new criteria of stability of the non-ergodic phases and prove that the Anderson transition in LN-RP model is discontinuous at all $p>0$.
