Fluctuations of Rectangular Young Diagrams of Interlacing Wigner Eigenvalues

We prove a new CLT for the difference of linear eigenvalue statistics of a Wigner random matrix $H$ and its minor $\hat H$ and find that the fluctuation is much smaller than the fluctuations of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of $H$ and $\hat H$. In particular our theorem identifies the fluctuation of Kerov's rectangular Young diagrams, defined by the interlacing eigenvalues of $H$ and $\hat H$, around their asymptotic shape, the Vershik-Kerov-Logan-Shepp curve. This result demonstrates yet another aspect of the close connection between random matrix theory and Young diagrams equipped with the Plancherel measure known from representation theory. For the latter a CLT has been obtained in [18] which is structurally similar to our result but the variance is different, indicating that the analogy between the two models has its limitations. Moreover, our theorem shows that Borodin's result [7] on the convergence of the spectral distribution of Wigner matrices to a Gaussian free field also holds in derivative sense.Comment: New citations and appendix added. 24 pages, 2 figures. Updated numbering to match the published versio

Phase Transition in the Density of States of Quantum Spin Glasses

We prove that the empirical density of states of quantum spin glasses on arbitrary graphs converges to a normal distribution as long as the maximal degree is negligible compared with the total number of edges. This extends the recent results of [6] that were proved for graphs with bounded chromatic number and with symmetric coupling distribution. Furthermore, we generalise the result to arbitrary hypergraphs. We test the optimality of our condition on the maximal degree for $p$-uniform hypergraphs that correspond to $p$-spin glass Hamiltonians acting on $n$ distinguishable spin-$1/2$ particles. At the critical threshold $p=n^{1/2}$ we find a sharp classical-quantum phase transition between the normal distribution and the Wigner semicircle law. The former is characteristic to classical systems with commuting variables, while the latter is a signature of noncommutative random matrix theory.Comment: 21 pages, 2 figure

The Linear Boltzmann Equation as the Low Density Limit of a Random Schrodinger Equation

We study the evolution of a quantum particle interacting with a random potential in the low density limit (Boltzmann-Grad). The phase space density of the quantum evolution defined through the Husimi function converges weakly to a linear Boltzmann equation with collision kernel given by the full quantum scattering cross section.Comment: 74 pages, 4 figures, (Final version -- typos corrected

Bounds on the norm of Wigner-type random matrices

We consider a Wigner-type ensemble, i.e. large hermitian $N\times N$ random matrices $H=H^*$ with centered independent entries and with a general matrix of variances $S_{xy}=\mathbb E|H_{xy}|^2$. The norm of $H$ is asymptotically given by the maximum of the support of the self-consistent density of states. We establish a bound on this maximum in terms of norms of powers of $S$ that substantially improves the earlier bound $2\| S\|^{1/2}_\infty$ given in [arXiv:1506.05098]. The key element of the proof is an effective Markov chain approximation for the contributions of the weighted Dyck paths appearing in the iterative solution of the corresponding Dyson equation.Comment: 25 pages, 8 figure

Complex Analyses of Spatial Vegetation Gradients and Boundaries on the Example of the Villány Mts and Mt Tubes

12

Random Matrices with Slow Correlation Decay

We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of [arXiv:1604.08188] to allow slow correlation decay and arbitrary expectation. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion.Comment: 41 pages, 1 figure. We corrected a typo in (4.1b

Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case

For complex Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner-Dyson-Mehta universality conjecture for the last remaining universality type in the complex Hermitian class. Our analysis holds not only for exact cusps, but approximate cusps as well, where an extended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp for both symmetry classes. This result is also used in the companion paper [arXiv:1811.04055] where the cusp universality for real symmetric Wigner-type matrices is proven.Comment: 58 pages, 2 figures. Updated introduction and reference

On the terms related to spatial ecological gradients and boundaries

Ecological gradients and boundaries are currently in the focus of research interest. A widely accepted terminology, however, is still lacking, thus the use of the terms related to gradients and boundaries continues to be confusing. In this paper, we provide new more elaborated definition of the spatial boundary. We distinguish between the gradient (transition) and the space-segment (transitional zone). Our paper identifies the main difference between the two types of gradients: cline and tone. We discuss the meanings of the synonyms boundary line, boundary zone, edge, margin and border. We review the importance of scale and organizational levels in the field of gradients and boundaries. The article also enlightens the difficulties of vegetation mapping associated with boundaries. At last, we identify some important research topics for the future, where intensive studies are needed