A note on exponential decay in the random field Ising model

For the two-dimensional random field Ising model (RFIM) with bimodal (i.e., two-valued) external field, we prove exponential decay of correlations either (1) when the temperature is larger than the critical temperature of the Ising model without external field and the magnetic field strength is small or (2) at any temperature when the magnetic field strength is sufficiently large. Unlike previous work on exponential decay, our approach is not based on cluster expansions but rather on arguably simpler methods; these combine an analysis of the Kert\'{e}sz line and a coupling of Ising measures (and also their random cluster representations) with different boundary conditions. We also show similar but weaker results for the RFIM with a general field distribution and in any dimension.Comment: 16 pages, minor revisio

Assessing the likelihood of having false positives caused by population stratification

Population stratification is always a concern in association analysis. There is a debate on the extent of the problem in less extreme situations (Thomas and Witte [1], Wacholder et al. [2]). Wacholder et al.[3] and Ardlie et al. [4] showed that hidden population structure is not a serious threat to case-control designs. We propose a method of assessing the seriousness of the population stratification before designing association studies. If population stratification is not a serious problem, one may consider using case-control study instead of family-based design to get more power. In a case-control design, we compare chi-square statistics from a structured population (a union of two subpopulations) and a homogeneous population with the same prevalence and allele frequencies. We provide an explicit formula to calculate the chi-square statistics from 17 parameters, such as proportions of subpopulation, allele frequencies in subpopulations, etc. We choose these factors because they have potential to cause false associations. Each parameter takes a random value in a chosen range. We then calculate the likelihood of getting opposite conclusions in the structured and the homogeneous populations. This is the likelihood of having false positives caused by population stratification. The advantage of this method is to provide a cost effective way to choose between using case-control data and using family data before actually collecting those data. We conclude that sample sizes have a significant effect on the likelihood of false positive caused by population stratification. The larger the sample size is, the more likely to have false positive if the population structure is ignored. If the sample size will be smaller than 200 by budget constraints, then case-control study may be a better choice because of its power

Weighted selective collapsing strategy for detecting rare and common variants in genetic association study

<p>Abstract</p> <p>Background</p> <p>Genome-wide association studies (GWAS) have been used successfully in detecting associations between common genetic variants and complex diseases. However, common SNPs detected by current GWAS only explain a small proportion of heritable variability. With the development of next-generation sequencing technologies, researchers find more and more evidence to support the role played by rare variants in heritable variability. However, rare and common variants are often studied separately. The objective of this paper is to develop a robust strategy to analyze association between complex traits and genetic regions using both common and rare variants.</p> <p>Results</p> <p>We propose a weighted selective collapsing strategy for both candidate gene studies and genome-wide association scans. The strategy considers genetic information from both common and rare variants, selectively collapses all variants in a given region by a forward selection procedure, and uses an adaptive weight to favor more likely causal rare variants. Under this strategy, two tests are proposed. One test denoted by <it>B_wSC</it>is sensitive to the directions of genetic effects, and it separates the deleterious and protective effects into two components. Another denoted by <it>B_wSCd</it>is robust in the directions of genetic effects, and it considers the difference of the two components. In our simulation studies, <it>B_wSC</it>achieves a higher power when the casual variants have the same genetic effect, while <it>B_wSCd</it>is as powerful as several existing tests when a mixed genetic effect exists. Both of the proposed tests work well with and without the existence of genetic effects from common variants.</p> <p>Conclusions</p> <p>Two tests using a weighted selective collapsing strategy provide potentially powerful methods for association studies of sequencing data. The tests have a higher power when both common and rare variants contribute to the heritable variability and the effect of common variants is not strong enough to be detected by traditional methods. Our simulation studies have demonstrated a substantially higher power for both tests in all scenarios regardless whether the common SNPs are associated with the trait or not.</p

Market Manipulation: A Comprehensive Study of Stock Pools

Using a hand collected new data set, this paper examines in detail a classic account of stock market manipulation—the “stock pools” of the 1920s, which prompted the current anti-manipulation rules in the United States. We examine abnormal turnover and returns and the relationship between them, as well as the long-term performance of the selected stocks. We conclude that the evidence suggests informed trading rather than manipulation. Our findings have implications for regulatory policy as well as the investigation and prosecution of manipulation cases

Motion of Lee-Yang zeros

We consider the zeros of the partition function of the Ising model with ferromagnetic pair interactions and complex external field. Under the assumption that the graph with strictly positive interactions is connected, we vary the interaction (denoted by $t$) at a fixed edge. It is already known that each zero is monotonic (either increasing or decreasing) in $t$; we prove that its motion is local: the entire trajectories of any two distinct zeros are disjoint. If the underlying graph is a complete graph and all interactions take the same value $t\geq 0$ (i.e., the Curie-Weiss model), we prove that all the principal zeros (those in $i[0,\pi/2)$) decrease strictly in $t$.Comment: 16 pages, 1 figur