Percolation thresholds on planar Euclidean relative-neighborhood graphs

Indication of long-range correlations governing city size

City systems are characterized by the functional organization of cities on a regional or country scale. While there is a relatively good empirical and theoretical understanding of city size distributions, insights about their spatial organization remain on a conceptual level. Here, we analyze empirically the correlations between the sizes of cities (in terms of area) across long distances. Therefore, we (i) define city clusters, (ii) obtain the neighborhood network from Voronoi cells, and (iii) apply a fluctuation analysis along all shortest paths. We find that most European countries exhibit long-range correlations but in several cases these are anti-correlations. In an analogous way, we study a model inspired by Central Places Theory and find that it leads to positive long-range correlations, unless there is strong additional spatial disorder-contrary to intuition. We conclude that the interactions between cities extend over large distances reaching the country scale. Our findings have policy relevance as urban development or decline can affect cities at a considerable distance.

Ising Ferromagnets on Proximity Graphs with Varying Disorder of the Node Placement

We perform Monte Carlo simulations to determine the critical temperatures of Ising Ferromagnets (IFM) on different types of two-dimensional proximity graphs, in which the distribution of their underlying node sets has been changed systematically by means of a parameter σ. This allows us to interpolate between regular grids and proximity graphs based on complete random placement of nodes. Each edge of the planar proximity graphs carries a weighted ferromagnetic coupling. The coupling strengths are determined via the Euclidean distances between coupled spins. The simulations are carried out on graphs with N = 16² to N = 128² nodes utilising the Wolff cluster algorithm and parallel tempering method in a wide temperature range around the critical point to measure the Binder cumulant in order to obtain the critical temperature for different values of σ. Interestingly, the critical temperatures depend partially non-monotonously on the disorder parameter σ, corresponding to a non-monotonous change of the graph structure. For completeness, we further verify using finite-size scaling methods that the IFM on proximity graphs is for all values of the disorder in the same universality class as the IFM on the two-dimensional square lattice.

Fragmentation properties of two-dimensional proximity graphs considering random failures and targeted attacks

The pivotal quality of proximity graphs is connectivity, i.e., all nodes in the graph are connected to one another either directly or via intermediate nodes. These types of graphs are often robust, i.e., they are able to function well even if they are subject to limited removal of elementary building blocks, as may occur for random failures or targeted attacks. Here, we study how the structure of these graphs is affected when nodes get removed successively until an extensive fraction is removed such that the graphs fragment. We study different types of proximity graphs for various node-removal strategies. We use different types of observables to monitor the fragmentation process, simple ones like the number and sizes of connected components and more complex ones like the hop diameter and the backup capacity, which is needed to make a network N-1 resilient. The actual fragmentation turns out to be described by a second-order phase transition. Using finite-size scaling analyses we numerically assess the threshold fraction of removed nodes, which is characteristic for the particular graph type and node deletion scheme; this suffices to decompose the underlying graphs.

The effect of rod orientation on electrical anisotropy in silver nanowire networks for ultra-transparent electrodes

Two-dimensional networks made of metal nanowires are excellent paradigms for the experimental observation of electrical percolation caused by continuous jackstraw-like physical pathways. Such systems became very interesting as alternative material in transparent electrodes, which are fundamental components in display devices. This work presents the experimental characterization of low-haze and ultra-transparent electrodes based on silver nanowires. The films are created by dip-coating, a feasible and scalable liquid film coating technique. We have found dominant alignment of the silver nanowires in withdrawal direction. The impact of this structural anisotropy on electrical anisotropy becomes more pronounced for low area coverage. The rod alignment does not influence the technical usability of the films as significant electrical anisotropy occurs only at optical transmission higher than 99 %. For films with lower transmission, electrical anisotropy becomes negligible. In addition to the experimental work, we have carried out computational studies in order to explain our findings further and compare them to our experiments and previous literature. This paper presents the first experimental observation of electrical anisotropy in two-dimensional silver nanowire networks close at the percolation threshold.

Dimer covering and percolation frustration

Covering a graph or a lattice with nonoverlapping dimers is a problem that has received considerable interest in areas, such as discrete mathematics, statistical physics, chemistry, and materials science. Yet, the problem of percolation on dimer-covered lattices has received little attention. In particular, percolation on lattices that are fully covered by nonoverlapping dimers has not evidently been considered. Here, we propose a procedure for generating random dimer coverings of a given lattice. We then compute the bond percolation threshold on random and ordered coverings of the square and the triangular lattices on the remaining bonds connecting the dimers. We obtain p_{c}=0.367713(2) and p_{c}=0.235340(1) for random coverings of the square and the triangular lattices, respectively. We observe that the percolation frustration induced as a result of dimer covering is larger in the low-coordination-number square lattice. There is also no relationship between the existence of long-range order in a covering of the square lattice and its percolation threshold. In particular, an ordered covering of the square lattice, denoted by shifted covering in this paper, has an unusually low percolation threshold and is topologically identical to the triangular lattice. This is in contrast to the other ordered dimer coverings considered in this paper, which have higher percolation thresholds than the random covering. In the case of the triangular lattice, the percolation thresholds of the ordered and random coverings are very close, suggesting the lack of sensitivity of the percolation threshold to microscopic details of the covering in highly coordinated networks.

Efficient Cluster Algorithm for Spin Glasses in Any Space Dimension

Spin systems with frustration and disorder are notoriously difficult to study, both analytically and numerically. While the simulation of ferromagnetic statistical mechanical models benefits greatly from cluster algorithms, these accelerated dynamics methods remain elusive for generic spin-glass-like systems. Here, we present a cluster algorithm for Ising spin glasses that works in any space dimension and speeds up thermalization by at least one order of magnitude at temperatures where thermalization is typically difficult. Our isoenergetic cluster moves are based on the Houdayer cluster algorithm for two-dimensional spin glasses and lead to a speedup over conventional state-of-the-art methods that increases with the system size. We illustrate the benefits of the isoenergetic cluster moves in two and three space dimensions, as well as the nonplanar chimera topology found in the D-Wave Inc. quantum annealing machine.

Electrical percolation thresholds of semiconducting single-walled carbon nanotube networks in field-effect transistors

We related the network density of semiconducting single-walled carbon nanotube networks at the percolation threshold to their electrical characteristics.