The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion

For independent nearest-neighbour bond percolation on Z^d with d >> 6, we prove that the incipient infinite cluster's two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit. The proof is based on an extension of the new expansion for percolation derived in a previous paper, and involves treating the magnetic field as a complex variable. A special case of our result for the two-point function implies that the probability that the cluster of the origin consists of n sites, at the critical point, is given by a multiple of n^{-3/2}, plus an error term of order n^{-3/2-\epsilon} with \epsilon >0. This is a strong statement that the critical exponent delta is given by delta =2.Comment: 56 pages, 3 Postscript figures, in AMS-LaTeX, with graphicx, epic, and xr package

The Alexander-Orbach conjecture holds in high dimensions

We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior has been established, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous diffusion with the spectral dimension d_s=4/3, that is, p_t(x,x)= t^{-2/3+o(1)}. This establishes a conjecture of Alexander and Orbach. En route we calculate the one-arm exponent with respect to the intrinsic distance.Comment: 25 pages, 2 figures. To appear in Inventiones Mathematica

A preferential attachment model with random initial degrees

In this paper, a random graph process ${G(t)}_{t\geq 1}$ is studied and its degree sequence is analyzed. Let $(W_t)_{t\geq 1}$ be an i.i.d. sequence. The graph process is defined so that, at each integer time $t$, a new vertex, with $W_t$ edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on $G(t-1)$, the probability that a given edge is connected to vertex i is proportional to $d_i(t-1)+\delta$, where $d_i(t-1)$ is the degree of vertex $i$ at time $t-1$, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent $\tau=\min\{\tau_{W}, \tau_{P}\}$, where $\tau_{W}$ is the power-law exponent of the initial degrees $(W_t)_{t\geq 1}$ and $\tau_{P}$ the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze, which is surveyed.Comment: In the published form of the paper, the proof of Proposition 2.1 is incomplete. This version contains the complete proo

Diffusion and drift of graphene flake on graphite surface

Diffusion and drift of a graphene flake on a graphite surface are analyzed. A potential energy relief of the graphene flake is computed using ab initio and empirical calculations. Based on the analysis of this relief, different mechanisms of diffusion and drift of the graphene flake on the graphite surface are considered. A new mechanism of diffusion and drift of the flake is proposed. According to the proposed mechanism, rotational transition of the flake from commensurate to incommensurate state takes place with subsequent simultaneous rotation and translational motion until a commensurate state is reached again, and so on. Analytic expressions for the diffusion coefficient and mobility of the flake corresponding to different mechanisms are derived in wide ranges of temperatures and sizes of the flake. The molecular dynamics simulations and estimates based on ab initio and empirical calculations demonstrate that the proposed mechanism can be dominant under certain conditions. The influence of structural defects on the diffusion of the flake is examined on the basis of calculations of the potential energy relief and molecular dynamics simulations. The methods of control over the diffusion and drift of graphene components in nanoelectromechanical systems are discussed. The possibility to experimentally determine the barriers to relative motion of graphene layers based on the study of diffusion of a graphene flake is considered. The results obtained can be also applied to polycyclic aromatic molecules on graphene and should be qualitatively valid for a set of commensurate adsorbate-adsorbent systems.Comment: 42 pages, 7 figure

Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk

We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if $d>2(\alpha\wedge2)$ for self-avoiding walk and the Ising model, and $d>3(\alpha\wedge2)$ for percolation, where $d$ denotes the dimension and $\alpha$ the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (2007)Comment: 43 pages, many figures. Version v2 with various (minor) changes (in particular in Sections 1.4 and A.1), and Sect. 4 is shortened. Journal of Statistical Physics (to appear

High-Dimensional Incipient Infinite Clusters Revisited

The incipient infinite cluster (IIC) measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite. Using the lace expansion, we construct the IIC measure for high-dimensional percolation models in three different ways, extending previous work by the second-named author and Járai. We show that each construction yields the same measure, indicating that the IIC is a robust object. Furthermore, our constructions apply to spread-out versions of both finite-range and long-range percolation models. We also get estimates on structural properties of the IIC, such as the volume of the intersection between the IIC and Euclidean balls. Keywords: Percolation; Incipient infinite cluster; Lace expansion; Critical behavio

P3HT-Based Solar Cells: Structural Properties and Photovoltaic Performance

Each year we are bombarded with B.Sc. and Ph.D. applications from students that want to improve the world. They have learned that their future depends on changing the type of fuel we use and that solar energy is our future. The hope and energy of these young people will transform future energy technologies, but it will not happen quickly. Organic photovoltaic devices are easy to sketch, but the materials, processing steps, and ways of measuring the properties of the materials are very complicated. It is not trivial to make a systematic measurement that will change the way other research groups think or practice. In approaching this chapter, we thought about what a new researcher would need to know about organic photovoltaic devices and materials in order to have a good start in the subject. Then, we simplified that to focus on what a new researcher would need to know about poly-3-hexylthiophene:phenyl-C61-butyric acid methyl ester blends (P3HT: PCBM) to make research progress with these materials. This chapter is by no means authoritative or a compendium of all things on P3HT:PCBM. We have selected to explain how the sample fabrication techniques lead to control of morphology and structural features and how these morphological features have specific optical and electronic consequences for organic photovoltaic device applications