Localization-delocalization phenomena for random interfaces

We consider d-dimensional random surface models which for d=1 are the standard (tied-down) random walks (considered as a random ``string''). In higher dimensions, the one-dimensional (discrete) time parameter of the random walk is replaced by the d-dimensional lattice \Z^d, or a finite subset of it. The random surface is represented by real-valued random variables \phi_i, where i is in \Z^d. A class of natural generalizations of the standard random walk are gradient models whose laws are (formally) expressed as P(d\phi) = 1/Z \exp[-\sum_{|i-j|=1}V(\phi_i-\phi_j)] \prod_i d\phi_i, V:\R -> R^+ convex, and with some growth conditions. Such surfaces have been introduced in theoretical physics as (simplified) models for random interfaces separating different phases. Of particular interest are localization-delocalization phenomena, for instance for a surface interacting with a wall by attracting or repulsive interactions, or both together. Another example are so-called heteropolymers which have a noise-induced interaction. Recently, there had been developments of new probabilistic tools for such problems. Among them are: o Random walk representations of Helffer-Sj\"ostrand type, o Multiscale analysis, o Connections with random trapping problems and large deviations We give a survey of some of these developments

Prohibited Floor Trading Activities Under the Commodity Exchange Act

In algorithmic graph theory, a classic open question is to determine the complexity of the Maximum Independent Set problem on Pt -free graphs, that is, on graphs not containing any induced path on t vertices. So far, polynomial-time algorithms are known only for t≤5 (Lokshtanov et al., in: Proceedings of the twenty-fifth annual ACM-SIAM symposium on discrete algorithms, SODA 2014, Portland, OR, USA, January 5–7, 2014, pp 570–581, 2014), and an algorithm for t=6 announced recently (Grzesik et al. in Polynomial-time algorithm for maximum weight independent set on P6 -free graphs. CoRR, arXiv:1707.05491, 2017). Here we study the existence of subexponential-time algorithms for the problem: we show that for any t≥1 , there is an algorithm for Maximum Independent Set on Pt -free graphs whose running time is subexponential in the number of vertices. Even for the weighted version MWIS, the problem is solvable in 2O(tnlogn√) time on Pt -free graphs. For approximation of MIS in broom-free graphs, a similar time bound is proved. Scattered Set is the generalization of Maximum Independent Set where the vertices of the solution are required to be at distance at least d from each other. We give a complete characterization of those graphs H for which d-Scattered Set on H-free graphs can be solved in time subexponential in the size of the input (that is, in the number of vertices plus the number of edges): If every component of H is a path, then d-Scattered Set on H-free graphs with n vertices and m edges can be solved in time 2O(|V(H)|n+m√log(n+m)) , even if d is part of the input. Otherwise, assuming the Exponential-Time Hypothesis (ETH), there is no 2o(n+m) -time algorithm for d-Scattered Set for any fixed d≥3 on H-free graphs with n-vertices and m-edges

Localization and delocalization of random interfaces

The probabilistic study of effective interface models has been quite active in recent years, with a particular emphasis on the effect of various external potentials (wall, pinning potential, ...) leading to localization/delocalization transitions. I review some of the results that have been obtained. In particular, I discuss pinning by a local potential, entropic repulsion and the (pre)wetting transition, both for models with continuous and discrete heights.Comment: Published at http://dx.doi.org/10.1214/154957806000000050 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Quantum corrections and wall-crossing via Lagrangian intersections

This article introduces the past and ongoing works on quantum corrections in SYZ from the author’s perspective. It emphasizes on a method of gluing local pieces of mirrors using isomorphisms between immersed Lagrangians, which is an ongoing joint work with Cho and Hong. It gives a canonical construction of mirrors and generalizes the SYZ setting

The impact of Facebook use on micro-level social capital: a synthesis

The relationship between Facebook use and micro-level social capital has received substantial scholarly attention over the past decade. This attention has resulted in a large body of empirical work that gives insight into the nature of Facebook as a social networking site and how it influences the social benefits that people gather from having social relationships. Although the extant research provides a solid basis for future research into this area, a number of issues remain underexplored. The aim of the current article is twofold. First, it seeks to synthesize what is already known about the relationship between Facebook use and micro-level social capital. Second, it seeks to advance future research by identifying and analyzing relevant theoretical, analytical and methodological issues. To address the first research aim, we first present an overview and analysis of current research findings on Facebook use and social capital, in which we focus on what we know about (1) the relationship between Facebook use in general and the different subtypes of social capital; (2) the relationships between different types of Facebook interactions and social capital; and (3) the impact of self-esteem on the relationship between Facebook use and social capital. Based on this analysis, we subsequently identify three theoretical issues, two analytical issues and four methodological issues in the extant body of research, and discuss the implications of these issues for Facebook and social capital researchers