Uniform existence of the integrated density of states for random Schr\"odinger operators on metric graphs over Zd\mathbb{Z}^d

We consider ergodic random magnetic Schr\"odinger operators on the metric graph $\mathbb{Z}^d$ with random potentials and random boundary conditions taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable. This limit, the integrated density of states, can be expressed by a closed Shubin-Pastur type trace formula. It supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. Among other examples we discuss percolation models.Comment: 17 pages; typos removed, references updated, definition of subgraph densities explaine

New spectral bounds on the chromatic number encompassing all eigenvalues of the adjacency matrix

The purpose of this article is to improve existing lower bounds on the chromatic number chi. Let mu_1,...,mu_n be the eigenvalues of the adjacency matrix sorted in non-increasing order. First, we prove the lower bound chi >= 1 + max_m {sum_{i=1}^m mu_i / - sum_{i=1}^m mu_{n-i+1}} for m=1,...,n-1. This generalizes the Hoffman lower bound which only involves the maximum and minimum eigenvalues, i.e., the case $m=1$. We provide several examples for which the new bound exceeds the {\sc Hoffman} lower bound. Second, we conjecture the lower bound chi >= 1 + S^+ / S^-, where S^+ and S^- are the sums of the squares of positive and negative eigenvalues, respectively. To corroborate this conjecture, we prove the weaker bound chi >= S^+/S^-. We show that the conjectured lower bound is tight for several families of graphs. We also performed various searches for a counter-example, but none was found. Our proofs rely on a new technique of converting the adjacency matrix into the zero matrix by conjugating with unitary matrices and use majorization of spectra of self-adjoint matrices. We also show that the above bounds are actually lower bounds on the normalized orthogonal rank of a graph, which is always less than or equal to the chromatic number. The normalized orthogonal rank is the minimum dimension making it possible to assign vectors with entries of modulus one to the vertices such that two such vectors are orthogonal if the corresponding vertices are connected. All these bounds are also valid when we replace the adjacency matrix A by W * A where W is an arbitrary self-adjoint matrix and * denotes the Schur product, that is, entrywise product of W and A

LpL^p-approximation of the integrated density of states for Schr\"odinger operators with finite local complexity

We study spectral properties of Schr\"odinger operators on \RR^d. The electromagnetic potential is assumed to be determined locally by a colouring of the lattice points in \ZZ^d, with the property that frequencies of finite patterns are well defined. We prove that the integrated density of states (spectral distribution function) is approximated by its finite volume analogues, i.e.the normalised eigenvalue counting functions. The convergence holds in the space $L^p(I)$ where $I$ is any finite energy interval and $1\leq p< \infty$ is arbitrary.Comment: 15 pages; v2 has minor fixe

Probing turbulent superstructures in Rayleigh-B\'{e}nard convection by Lagrangian trajectory clusters

We analyze large-scale patterns in three-dimensional turbulent convection in a horizontally extended square convection cell by Lagrangian particle trajectories calculated in direct numerical simulations. A simulation run at a Prandtl number Pr $=0.7$, a Rayleigh number Ra $=10^5$, and an aspect ratio $\Gamma=16$ is therefore considered. These large-scale structures, which are denoted as turbulent superstructures of convection, are detected by the spectrum of the graph Laplacian matrix. Our investigation, which follows Hadjighasem {\it et al.}, Phys. Rev. E {\bf 93}, 063107 (2016), builds a weighted and undirected graph from the trajectory points of Lagrangian particles. Weights at the edges of the graph are determined by a mean dynamical distance between different particle trajectories. It is demonstrated that the resulting trajectory clusters, which are obtained by a subsequent $k$-means clustering, coincide with the superstructures in the Eulerian frame of reference. Furthermore, the characteristic times $\tau^L$ and lengths $\lambda_U^L$ of the superstructures in the Lagrangian frame of reference agree very well with their Eulerian counterparts, $\tau$ and $\lambda_U$, respectively. This trajectory-based clustering is found to work for times $t\lesssim \tau\approx\tau^L$. Longer time periods $t\gtrsim \tau^L$ require a change of the analysis method to a density-based trajectory clustering by means of time-averaged Lagrangian pseudo-trajectories, which is applied in this context for the first time. A small coherent subset of the pseudo-trajectories is obtained in this way consisting of those Lagrangian particles that are trapped for long times in the core of the superstructure circulation rolls and are thus not subject to ongoing turbulent dispersion.Comment: 12 pages, 7 downsized figures, to appear in Phys. Rev. Fluid

Coherent structure colouring: identification of coherent structures from sparse data using graph theory

We present a frame-invariant method for detecting coherent structures from Lagrangian flow trajectories that can be sparse in number, as is the case in many fluid mechanics applications of practical interest. The method, based on principles used in graph colouring and spectral graph drawing algorithms, examines a measure of the kinematic dissimilarity of all pairs of fluid trajectories, measured either experimentally, e.g. using particle tracking velocimetry, or numerically, by advecting fluid particles in the Eulerian velocity field. Coherence is assigned to groups of particles whose kinematics remain similar throughout the time interval for which trajectory data are available, regardless of their physical proximity to one another. Through the use of several analytical and experimental validation cases, this algorithm is shown to robustly detect coherent structures using significantly less flow data than are required by existing spectral graph theory methods