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     

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.

     

The method of stress variation applied to minimal design for multiple loading

The method of stress variation, previously developed for the optimal design of axisymmetric sandwich plates obeying the Tresca criterion for a single loading condition[1–3], is extended to multiple loading conditions. The method consists of a systematic reduction in the design parameter until the optimum is reached. For the class of loadings treated in this paper, it is shown that repeated application of only a single stress variation is necessary to produce the optimum design for a simply supported plate. The method is such that it can easily be adapted to automatic computation, and the computer time required is simply proportional to the number of separate loads to be considered. An example of a simply supported plate for three separate loads is presented.     

What do your footsteps sound like? An investigation on interactive footstep sounds adjustment

This paper presents an experiment where participants were asked to adjust, while walking, the spectral content and the amplitude of synthetic footstep sounds in order to match the sounds of their own footsteps. The sounds were interactively generated by means of a shoe-based system capable of tracking footfalls and delivering real-time auditory feedback via headphones. Results allowed identification of the mean value and the range of variation of spectral centroid and peak level of footstep sounds simulating various combinations of shoe type and ground material. Results showed that the effect of ground material on centroid and peak level depended on the type of shoe. Similarly, the effect of shoe type on the two variables depended on the type of ground material. In particular, participants produced greater amplitudes for hard sole shoes than for soft sole shoes in presence of solid surfaces, while similar amplitudes for both types of shoes were found for aggregate, hybrids, and liquids. No significant correlations were found between each of the two acoustic features and participants’ body size. This result might be explained by the fact that while adjusting the sounds participants did not primarily focus on the acoustic rendering of their body. In addition, no significant differences were found between the values of the two acoustic features selected by the experimenters and those adjusted by participants. This result can therefore be considered as a measure of the goodness of the design choices to synthesize the involved footstep sounds for a generic walker. More importantly, this study showed that the relationships between the ground-shoes combinations are not changed when participants are actively walking. This represents the first active listening confirmation of this result, which had previously only been shown in passive listening studies. The results of this research can be used to design ecologically-valid auditory rendering of foot-floor interactions in virtual environments.     

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     

Methods for calculating tracer distribution in multiphase plug flow systems

A novel continuous time and space Monte Carlo simulation technique is presented for solving equations describing tracer distribution in multiphase inhomogeneous plug-flow systems. Comparison is made with the commonly employed method of characteristics. This comparison indicates that the method of characteristics is unreliable for systems with spacially varying parameters. Further, results indicate that the suggested Monte Carlo technique is more efficient in the use of computer time than the method of characteristics. Examples are given.     

A stability result for balanced dictatorships in Sn

We prove that a balanced Boolean function on S_n whose Fourier transform is highly concentrated on the first two irreducible representations of S_n, is close in structure to a dictatorship, a function which is determined by the image or pre‐image of a single element. As a corollary, we obtain a stability result concerning extremal isoperimetric sets in the Cayley graph on S_n generated by the transpositions. Our proof works in the case where the expectation of the function is bounded away from 0 and 1. In contrast, [6] deals with Boolean functions of expectation O(1/ n) whose Fourier transform is highly concentrated on the first two irreducible representations of S_n. These need not be close to dictatorships; rather, they must be close to a union of a constant number of cosets of point‐stabilizers. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 494–530, 2015