Competing first passage percolation on random regular graphs

We consider two competing first passage percolation processes started from uniformly chosen subsets of a random regular graph on N vertices. The processes are allowed to spread with different rates, start from vertex subsets of different sizes or at different times. We obtain tight results regarding the sizes of the vertex sets occupied by each process, showing that in the generic situation one process will occupy vertices, for some . The value of α is calculated in terms of the relative rates of the processes, as well as the sizes of the initial vertex sets and the possible time advantage of one process. The motivation for this work comes from the study of viral marketing on social networks. The described processes can be viewed as two competing products spreading through a social network (random regular graph). Considering the processes which grow at different rates (corresponding to different attraction levels of the two products) or starting at different times (the first to market advantage) allows to model aspects of real competition. The results obtained can be interpreted as one of the two products taking the lion share of the market. We compare these results to the same process run on d dimensional grids where we show that in the generic situation the two products will have a linear fraction of the market each. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 534–583, 2017     

Surface‐plasmon wavefront and spectral shaping by near‐field holography

Surface‐plasmon‐polariton waves are two‐dimensional electromagnetic surface waves that propagate at the interface between a metal and a dielectric. These waves exhibit unusual and attractive properties, such as high spatial confinement and enhancement of the optical field, and are widely used in a variety of applications, such as sensing and subwavelength optics. The ability to precisely control the spatial and spectral properties of the surface‐plasmon wave is required in order to support the growing interest in both research and applications of plasmonic waves, and to bring it to the next level. Here, we review the challenges and methods for shaping the wavefront and spectrum of plasmonic waves. In particular, we present the recent advances in plasmonic spatial and spectral shaping, which are based on the realization of plasmonic holograms for the optical nearfield.

     

Can Extra Updates Delay Mixing?

We consider Glauber dynamics (starting from an extremal configuration) in a monotone spin system, and show that interjecting extra updates cannot increase the expected Hamming distance or the total variation distance to the stationary distribution. We deduce that for monotone Markov random fields, when block dynamics contracts a Hamming metric, single-site dynamics mixes in O(n log n) steps on an n-vertex graph. In particular, our result completes work of Kenyon, Mossel and Peres concerning Glauber dynamics for the Ising model on trees. Our approach also shows that on bipartite graphs, alternating updates systematically between odd and even vertices cannot improve the mixing time by more than a factor of log n compared to updates at uniform random locations on an n-vertex graph. Our result is especially effective in comparing block and single-site dynamics; it has already been used in works of Martinelli, Toninelli, Sinclair, Mossel, Sly, Ding, Lubetzky, and Peres in various combinations.     

Percolation of electrical conductivity in solution-cast blends containing polyaniline

The potential offered by intrinsically conductive polymers is limited by their poor mechanical properties. Blending with common thermoplastics can improve processability and mechanical properties and still preserve the electrical conductivity. In such blends, the morphology determines the mechanical and electrical properties. In this research, blends of polyaniline (PANI)-dodecyl benzene sulfonic acid (DBSA) with either polystyrene (PS) in xylene or polyvinylchloride (PVC) in bromobenzene were solution cast. The morphologies of the blends were characterized by optical microscopy (OM), electron microscopy, and small-angle X-ray scattering (SAXS). Electrical conductivity was measured for various compositions. The formation of a continuous network was strongly associated with percolation and conductivity. The morphologies of the two blends are significantly different. This difference arises from the different solvents used and their ability to swell the PAN1 aggregates and to promote their disintegration into small particles. Molecular calculations show that, indeed, bromobenzene, used in the PVC system, is able to penetrate the PANI-DBSA aggregates, while xylene, used in the PS system, cannot. Nevertheless, the small PANI-DBSA particles in the PVC matrix form a conductive network only at a relatively high content, while the coarse aggregates in the PS matrix form conductive paths at a relatively low content. These results are discussed in terms of the formation and stability of the PANI-DBSA dispersion.     

Large deviations for random walks on Galton–Watson trees: averaging and uncertainty

 In the study of large deviations for random walks in random environment, a key distinction has emerged between quenched asymptotics, conditional on the environment, and annealed asymptotics, obtained from averaging over environments. In this paper we consider a simple random walk {X _n } on a Galton–Watson tree T, i.e., on the family tree arising from a supercritical branching process. Denote by |X _n | the distance between the node X _n and the root of T. Our main result is the almost sure equality of the large deviation rate function for |X _n |/n under the “quenched measure” (conditional upon T), and the rate function for the same ratio under the “annealed measure” (averaging on T according to the Galton–Watson distribution). This equality hinges on a concentration of measure phenomenon for the momentum of the walk. (The momentum at level n, for a specific tree T, is the average, over random walk paths, of the forward drift at the hitting point of that level). This concentration, or certainty, is a consequence of the uncertainty in the location of the hitting point. We also obtain similar results when {X _n } is a λ-biased walk on a Galton–Watson tree, even though in that case there is no known formula for the asymptotic speed. Our arguments rely at several points on a “ubiquity” lemma for Galton–Watson trees, due to Grimmett and Kesten (1984). Received: 15 November 2000 / Revised version: 27 February 2001 / Published online: 19 December 2001     

Fluctuation of a planar Brownian loop capturing a large area

We consider a planar Brownian loop that is run for a time and conditioned on the event that its range encloses the unusually high area of , with being large. The conditioned process, denoted by , was proposed by Senya Shlosman as a model for the fluctuation of a phase boundary. We study the deviation of the range of from a circle of radius . This deviation is measured by the inradius and outradius , which are the maximal radius of a disk enclosed by the range of , and the minimal radius of a disk that contains this range. We prove that, in a typical realization of the conditioned measure, each of these quantities differs from by at most .

     

Universal finitary codes with exponential tails

In 1977, Keane and Smorodinsky showed that there exists a finitaryhomomorphism from any finite-alphabet Bernoulli process to anyother finite-alphabet Bernoulli process of strictly lower entropy.In 1996, Serafin proved the existence of a finitary homomorphismwith finite expected coding length. In this paper, we constructsuch a homomorphism in which the coding length has exponentialtails. Our construction is source-universal, in the sense thatit does not use any information on the source distribution otherthan the alphabet size and a bound on the entropy gap betweenthe source and target distributions. We also indicate how ourmethods can be extended to prove a source-specific version ofthe result for Markov chains.     

Polymer and biopolymer mediated self-assembly of gold nanoparticles

Gold nanoparticle-polymer composites are versatile and diverse functional materials, with applications in optical, electronic and sensing devices. This tutorial review focuses on the use of polymers to control the assembly of gold nanoparticles. Examples of synthetic polymers and biopolymers are provided, as well as applications of the composite materials in sensing and memory devices.     

An LIL for cover times of disks by planar random walk and Wiener sausage

Let be the radius of the largest disk covered after steps of a simple random walk. We prove that almost surely

where denotes 3 iterations of the function. This is motivated by a question of Erdos and Taylor. We also obtain the analogous result for the Wiener sausage, refining a result of Meyre and Werner.