随机关联势能量模型多重分形指数的预冻结

非厄米准晶中的对数关联势场与局域化

Logarithmically Correlated Landscapes and Localization in Non-Hermitian Quasicrystals

该研究分析了经零准周期调制的非互易Hatano-Nelson环，推导其势场特性，揭示了刚度阈值决定局域化与临界多重分形相的转变，且机制可应用于非互易拓扑电路。

Dynamical phases in a "multifractal" Rosenzweig-Porter model

We consider the static and dynamic phases in a Rosenzweig-Porter (RP) random matrix ensemble with the tailed 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 the simple exponent, as the 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 model on Ran…

冻结转变与极值：随机矩阵理论、$ζ(1/2+it)$与无序景观

Freezing Transitions and Extreme Values: Random Matrix Theory, $ζ(1/2+it)$, and Disordered Landscapes

我们认为，在1/f噪声随机能量模型的统计力学中，先前被证明存在的冻结跃迁情形，经过重新解释后，支配着N个大的\times N个随机酉（CUE）矩阵的特征多项式p_N（θ）的模的最大值的值分布，即当N个\rightarrow为\infty时p_N（θ）的极值统计.此外，我们还论证了它在区间的总长度μ_N（x）上导致了类似多重分形的行为，其中|p_N（θ）|>N^x，x>0，在相同的极限内。我们推测，我们的结果可以推广到黎曼ζ函数ε（s）在临界线s=1/2+it上的大值，并给出ε（1/2+it）的大值的数值计算结果。我们的主要目的是提请注意这些不同的极值问题之间的意想不到的联系。

高维随机景观中单个粒子的统计力学

On statistical mechanics of a single particle in high-dimensional random landscapes

我们讨论了最近的结果的副本方法的统计力学的一个单一的经典粒子放置在一个随机的N（>>1）维高斯景观。特别注意的是支付的情况下，景观的相关性不断增长，并在最近的概括。这些景观产生了丰富的多重分形空间结构的相关玻尔兹曼-吉布斯措施。我们还简要地提到了相关的结果计数的平稳点的随机高斯曲面，以及正在进行的研究统计力学在随机景观构建本地添加许多平方高斯分布的条款。

Localization transition on the Random Regular Graph as an unstable tricritical point in a log-normal Rosenzweig-Porter random matrix ensemble

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\rightarr…

Log-correlated Random Energy Models with extensive free energy fluctuations: pathologies caused by rare events as signatures of phase transitions

We address systematically an apparent non-physical behavior of the free energy moment generating function for several instances of the logarithmically correlated models: the Fractional Brownian Motion with Hurst index $H = 0$ (fBm0) (and its bridge version), a 1D model appearing in decaying Burgers turbulence with log-correlated initial conditions, and finally, the two-dimensional logREM introduced in [Cao et al., Phys.Rev.Lett.,118,090601] based on the 2D Gaussian free field (GFF) with background charges and directly related to the Liouville field theory. All these models share anomalously la…

One step replica symmetry breaking and extreme order statistics of logarithmic REMs

Building upon the one-step replica symmetry breaking formalism, duly understood and ramified, we show that the sequence of ordered extreme values of a general class of Euclidean-space logarithmically correlated random energy models (logREMs) behave in the thermodynamic limit as a randomly shifted decorated exponential Poisson point process. The distribution of the random shift is determined solely by the large-distance ("infra-red", IR) limit of the model, and is equal to the free energy distribution at the critical temperature up to a translation. the decoration process is determined solely b…