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A random phase property establishing a link between quasi-one-dimensional random Schroedinger operators and full random matrix theory is advocated. Briefly summarized it states that the random transfer matrices placed into a normal system…
We study the diffusion of complex Wishart matrices and derive a partial differential equation governing the behavior of the associated averaged characteristic polynomial. In the limit of large size matrices, the inverse Cole-Hopf transform…
In arXiv:1410.7268v3, the authors consider eigenvalues of overlapping Wishart matrices and prove that its fluctuations asymptotically convergence to the Gaussian free field. In this brief note, their result is extended to show that when the…
We introduce the generalized upper box dimension which is defined for any set, whether the set is bounded or unbounded. We study basic properties of the generalized upper box dimension. We prove that the generalized upper box and upper box…
The article reviews recent developments in the theory of fluctuations and correlations of energy levels and eigenfunction amplitudes in diffusive mesoscopic samples. Various spatial geometries are considered, with emphasis on…
Many fluctuating systems consist of macroscopic structures in addition to noisy signals. Thus, for this class of fluctuating systems, the scaling behaviors are very complicated. Such phenomena are quite commonly observed in Nature, ranging…
We show that self-similar sets arising from iterated function systems that satisfy the Moran open-set condition, a canonical class of fractal sets, are `equi-homogeneous'. This is a regularity property that, roughly speaking, means that at…
We study various models of random non-crossing configurations consisting of diagonals of convex polygons, and focus in particular on uniform dissections and non-crossing trees. For both these models, we prove convergence in distribution…
We present results on a meso-scale model for amorphous matter in athermal, quasi-static (a-AQS), steady state shear flow. In particular, we perform a careful analysis of the scaling with the lateral system size, $L$, of: i) statistics of…
A number of recent studies have estimated the inter-galactic void probability function and investigated its departure from various random models. We study a family of parametric statistical models based on gamma distributions, which do give…
We consider Galton--Watson trees conditioned on both the total number of vertices $n$ and the number of leaves $k$. The focus is on the case in which both $k$ and $n$ grow to infinity and $k = \alpha n + O(1)$, with $\alpha \in (0, 1)$.…
We show that when the standard techniques for calculating fractal dimensions in empirical data (such as the box counting) are applied on uniformly random structures, apparent fractal behavior is observed in a range between physically…
We propose experimentally feasible ways to probe universal features of absorbing phase transitions from two different approaches, both based on numerical validations. On one hand, we numerically study a probability distribution of…
The problem of the effects of compressibility and large-scale anisotropy on anomalous scaling behavior is considered for two models describing passive advection of scalar density and tracer fields. The advecting velocity field is Gaussian,…
The fractal dimension of domain walls produced by changing the boundary conditions from periodic to anti-periodic in one spatial direction is studied using both the strong-disorder renormalization group and the greedy algorithm for the…
We develop sharp bounds on the statistical distance between high-dimensional permutation mixtures and their i.i.d. counterparts. Our approach establishes a new geometric link between the spectrum of a complex channel overlap matrix and the…
Given any regularly varying dislocation measure, we identify a natural self-similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of…
We consider the multi-time correlation and covariance structure of a random surface growth with a wall introduced in arXiv:0904.2607. It is shown that the correlation functions associated with the model along space-like paths have…
In this paper, we consider the random plane forest uniformly drawn from all possible plane forests with a given degree sequence. Under suitable conditions on the degree sequences, we consider the limit of a sequence of such forests with the…
In the environmental modeling field, the exploratory analysis of responses often exhibits spatial correlation as well as some non-Gaussian attributes such as skewness and/or heavy-tailedness. Consequently, we propose a general spatial model…