As one of the most significant models, the uniform recursive tree (URT) has found many applications in a variety of fields. In this paper, we study rigorously the structural features and spectral properties of the adjacency matrix for a family of det
erministic uniform recursive trees (DURTs) that are determinist
The explicit determinations of the mean first-passage time (MFPT) for trapping problem are limited to some simple structure, e.g., regular lattices and regular geometrical fractals, and determining MFPT for random walks on other media, especially com
plex real networks, is a theoretical challenge. In this paper, we investigate a simple random walk on the the pseudofractal scale-free web (PSFW) with a perfect trap located at a node with the highest degree, which simultaneously exhibits the remarkable scale-free and small-world properties observed in real networks. We obtain the exact solution for the MFPT that is calculated through the recurrence relations derived from the structure of PSFW. The rigorous solution exhibits that the MFPT approximately increases as a power-law function of the number of nodes, with the exponent less than 1. We confirm the closed-form solution by direct numerical calculations. We show that the structure of PSFW can improve the efficiency of transport by diffusion, compared with some other structure, such as regular lattices, Sierpinski fractals, and T-graph. The analytical method can be applied to other deterministic networks, making the accurate computation of MFPT possible.
The uniform recursive tree (URT) is one of the most important models and has been successfully applied to many fields. Here we study exactly the topological characteristics and spectral properties of the Laplacian matrix of a deterministic uniform re
cursive tree, which is a deterministic version of URT. Firstly, from the perspective of complex networks, we determine the main structural characteristics of the deterministic tree. The obtained vigorous results show that the network has an exponential degree distribution, small average path length, power-law distribution of node betweenness, and positive degree-degree correlations. Then we determine the complete Laplacian spectra (eigenvalues) and their corresponding eigenvectors of the considered graph. Interestingly, all the Laplacian eigenvalues are distinct.
Recently, significant interest has emerged in fabricated systems that mimic the behavior of geometrically-frustrated materials. We present the full realization of such an artificial spin ice system on a two-dimensional kagome lattice and demonstrate
rigid adherence to the local ice rule by directly counting individual pseudo-spins. The resulting spin configurations show not only local ice rules and long-range disorder, but also correlations consistent with spin ice Monte Carlo calculations. Our results suggest that dipolar corrections are significant in this system, as in pyrochlore spin ice, and they open a door to further studies of frustration in general.
The progress of semiconductor electronics toward ever-smaller length scales and associated higher power densities brings a need for new high-resolution thermal microscopy techniques. Traditional thermal microscopy is performed by detecting infrared r
adiation with far-field optics, where the resolution is limited by the wavelength of the light. By adopting a serial, local-probe approach, near-field and scanned-probe microscopies can surpass this limit but sacrifice imaging speed. In the same way that electron microscopy was invented to overcome the resolution limits of light microscopy, we here demonstrate a thermal imaging technique that uses an electron microscope to overcome the limits of infrared thermal microscopy, without compromising imaging speed. With this new technique, which we call electron thermal microscopy, temperature is resolved by detecting the liquid-solid transition of arrays of nanoscale islands, producing thermal maps in real-time (30 thermal images per second over a 16um^2 field-of-view). The experimental demonstration is supported by combined electrical and thermal modeling.
