A diagrammatic equation for oriented planar graphs

In this paper we derive a diagrammatic equation for the planar sector of square non-Hermitian random matrix models. Our fundamental equation is first obtained by a graph counting argument (inspired by the Polchinski equation in quantum field theory) and subsequently derived independently by a precise saddle point analysis of the corresponding random matrix integral. We solve the equation perturbatively for a generic model and conclude by exhibiting two duality properties of the perturbative solution.     

Percolation threshold on planar Euclidean Gabriel graphs

In the present article, numerical simulations have been performed to find the bond andsite percolation thresholds on two-dimensional Gabriel graphs (GG) for Poisson pointprocesses. GGs belong to the family of “proximity graphs” and are discussed, e.g., incontext of the construction of backbones for wireless ad-hoc networks. Finite-size scalinganalyses have been performed to find the critical points and critical exponentsν,β andγ. Thecritical exponents obtained this way verify that the associated universality class is thatof standard 2Dpercolation.     

Symmetric vertex models on planar random graphs

We solve a 4-(bond)-vertex model on an ensemble of 3-regular (Φ³) planar random graphs, which has the effect of coupling the vertex model to 2D quantum gravity. The method of solution, by mapping onto an Ising model in field, is inspired by the solution by Wu et.al. of the regular lattice equivalent – a symmetric 8-vertex model on the honeycomb lattice, and also applies to higher valency bond vertex models on random graphs when the vertex weights depend only on bond numbers and not cyclic ordering (the so-called symmetric vertex models).The relations between the vertex weights and Ising model parameters in the 4-vertex model on Φ³ graphs turn out to be identical to those of the honeycomb lattice model, as is the form of the equation of the Ising critical locus for the vertex weights. A symmetry of the partition function under transformations of the vertex weights, which is fundamental to the solution in both cases, can be understood in the random graph case as a change of integration variable in the matrix integral used to define the model.Finally, we note that vertex models, such as that discussed in this paper, may have a role to play in the discretisation of Lorentzian metric quantum gravity in two dimensions.     

The quantum mechanical planar propagator: From vector models to planar graphs

In the framework of a systematic investigation of planar field theory, we study the planar two-point Green function. We define and evaluate an expansion which preserves the formal properties of the theory and compute the three lowest lying poles of the propagator in a one-dimensional space-time, finding excellent agreement with known results.     

Random graphs as a model for pregeometry

A toy model for pregeometry based on random graphs is proposed, and it is shown how it is related to other models in the literature. A prediction of the dimensionality of space is given and we obtain bounds on the Euler number.     

Efficiency of star-like graphs and the Atlanta subway network

The distance d(i,j)$d(i,j)$ between any two vertices i$i$ and j$j$ in a graph is the number of edges in a shortest path between i$i$ and j$j$. If there is no path connecting i$i$ and j$j$, then d(i,j)=∞$d(i,j)=\infty$. In 2001, Latora and Marchiori introduced the measure of efficiency between vertices in a graph (Latora and Marchiori, 2001) [1]. The efficiency between two vertices i$i$ and j$j$ is defined to be _∈i,j=j${\in }_{i,j}=j$. In this paper, we investigate the efficiency of star-like networks, and show that networks of this type have a high level of efficiency. We apply these ideas to an analysis of the Metropolitan Atlanta Rapid Transit Authority (MARTA) Subway system, and show this network is 82% as efficient as a network where there is a direct line between every pair of stations.     

Random removal of edges from scale free graphs

It has been discovered that many naturally occurring networks (the internet, the power grid of the western US, various biological networks, etc.) satisfy a power-law degree distribution. Such scale-free networks have many interesting properties, one of which is robustness to random damage. This problem has been analyzed from the point of view of node deletion and connectedness. Recently, it has also been considered from the point of view of node deletion and scale preservation. In this paper we consider the problem from the point of view of edge deletion and scale preservation. In agreement with the work on node deletion and scale preservation, we show that a scale-free graph should not be expected to remain scale free when edges are removed at random.     

Non-affinity: The emergence of networks from amorphous planar graphs

The primary research of physics is to reveal the underlying law of the physical world. However, for a complex system consisting of multiple components, even if we know every details of each component, we still cannot predict the collective behavior due to the emergence phenomenon. The amorphous networks of mass points hinged by springs belong to such a complex system.Owing to the non-affinity, one of the emergence phenomena of the network, the displacements field of the internal mass points unde...     

The network of commuters in London

We have studied the directed and weighted network in which the wards of London are vertices and two vertices are connected whenever there is at least one person commuting to work from one ward to another. Remarkably the in-strength and in-degree distribution tail is a power law with exponent around −2, while the out-strength and out-degree distribution tail is exponential. We propose a simple square lattice model to explain the observed empirical behaviour.     

Random Sierpinski network with scale-free small-world and modular structure

In this paper, we define a stochastic Sierpinski gasket, on the basis of which we construct a network called random Sierpinski network (RSN). We investigate analytically or numerically the statistical characteristics of RSN. The obtained results reveal that the properties of RSN is particularly rich, it is simultaneously scale-free, small-world, uncorrelated, modular, and maximal planar. All obtained analytical predictions are successfully contrasted with extensive numerical simulations. Our network representation method could be applied to study the complexity of some real systems in biological and information fields.     

Yang–Lee zeros of the Ising model on random graphs of non planar topology

We obtain in a closed form the 1/N² contribution to the free energy of the two Hermitian N×N random matrix model with nonsymmetric quartic potential. From this result, we calculate numerically the Yang–Lee zeros of the 2D Ising model on dynamical random graphs with the topology of a torus up to n=16 vertices. They are found to be located on the unit circle on the complex fugacity plane. In order to include contributions of even higher topologies we calculated analytically the nonperturbative (sum over all genus) partition function of the model for the special cases of N=1,2 and graphs with n≤20 vertices. Once again the Yang–Lee zeros are shown numerically to lie on the unit circle on the complex fugacity plane. Our results thus generalize previous numerical results on random graphs by going beyond the planar approximation and strongly indicate that there might be a generalization of the Lee–Yang circle theorem for dynamical random graphs.     

Random resistor network model of minimal conductivity in graphene

Transport in undoped graphene is related to percolating current patterns in the networks of n- and p-type regions reflecting the strong bipolar charge density fluctuations. Finite transparency of the p-n junctions is vital in establishing the macroscopic conductivity. We propose a random resistor network model to analyze scaling dependencies of the conductance on the doping and disorder, the quantum magnetoresistance and the corresponding dephasing rate.     

切比雪夫馈电网络开口谐振环天线阵列

提出了一种基于切比雪夫馈电网络的13×14开口谐振环天线阵列。该天线阵列由天线单元和切比雪夫馈电网络组成。根据八木天线原理,该天线阵列设计的辐射贴片作为天线单元的引向器,金属地面作为反射器;辐射贴片由开口谐振环结构以及I型谐振结构组成,增强天线单元的辐射能力并提高增益;天线单元的馈电端由圆弧单极子组成,提高调节天线单元阻抗匹配的灵活性;电流矩阵指导非均匀电流分布切比雪夫馈电网络的设计,减小天线阵列的旁瓣;天线单元竖直插入馈电网络的基板,减少带有馈电网络阵列的口径大小。通过实物加工后,测试结果表明,该天线阵列在工作频率点10.1 GHz处实现了22.3 dBi的增益,E面和H面旁瓣电平分别为-16 dB和-17.66 dB,具有高增益、低副瓣的特点。     

Random resistivity network calculations for cuprate superconductors with an electronic phase separation transition

The resistivity as a function of temperature for high temperature superconductors is very unusual and, despite its importance, lacks a unified theoretical explanation. It is linear with the temperature for overdoped compounds but it falls more quickly as the doping level decreases. The resistivity of underdoped cuprates increases like that of an insulator below a characteristic temperature where it shows a minimum. We show that this overall behavior can be explained by calculations using an electronic phase segregation into two main component phases with low and high electronic densities. The total resistance is calculated from the various contributions through several processes of random picking of the local resistivities and using a common statistical random resistor network approach.     

Random Walk on the Range of Random Walk

We study the random walk X on the range of a simple random walk on ℤ ^d in dimensions d≥4. When d≥5 we establish quenched and annealed scaling limits for the process X, which show that the intersections of the original simple random walk path are essentially unimportant. For d=4 our results are less precise, but we are able to show that any scaling limit for X will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when d=4 similar logarithmic corrections are necessary in describing the asymptotic behavior of the return probability of X to the origin.     

Ghosts and renormalization in the planar gauge

We make two points about the planar gauge in non-abelian gauge theories: (a) Although ghosts are absent from S-matrix elements they ar necessary to formulate Slavnov identities. (b) The Slavnov identities by themselves are insufficient to control the field renormalization-a supplementary argument is necessary.     

QCD box graphs and the quark-antiquark potential

The $t\bar t$ system allows a truly perturbative treatment of the potential. Completing previous computations, we calculate the contributions of QCD box graph corrections, which make the “relativistic”O(α ⁴) corrections in the non-Abelian case differ from the well known corrections of the same order in QED.     

Polymers and random graphs

We establish a precise connection between gelation of polymers in Lushnikov's model and the emergence of the giant component in random graph theory. This is achieved by defining a modified version of the Erdös-Rényi process; when contracting to a polymer state space, this process becomes a discrete-time Markov chain embedded in Lushnikov's process. The asymptotic distribution of the number of transitions in Lushnikov's model is studied. A criterion for a general Markov chain to retain the Markov property under the grouping of states is derived. We obtain a noncombinatorial proof of a theorem of Erdös-Rényi type.     

Non-Hermitian Tridiagonal Random Matrices and Returns to the Origin of a Random Walk

We study a class of tridiagonal matrix models, the q-roots of unity models, which includes the sign (q=2) and the clock (q=) models by Feinberg and Zee. We find that the eigenvalue densities are bounded by and have the symmetries of the regular polygon with 2q sides, in the complex plane. Furthermore, the averaged traces of M ^k are integers that count closed random walks on the line such that each site is visited a number of times multiple of q. We obtain an explicit evaluation for them.