Slow Convergence in Bootstrap Percolation

In the bootstrap percolation model, sites in an L by L square are initially infected independently with probability p. At subsequent steps, a healthy site becomes infected if it has at least 2 infected neighbours. As (L,p)->(infinity,0), the probability that the entire square is eventually infected is known to undergo a phase transition in the parameter p log L, occurring asymptotically at lambda = pi^2/18. We prove that the discrepancy between the critical parameter and its limit lambda is at least Omega((log L)^(-1/2)). In contrast, the critical window has width only Theta((log L)^(-1)). For the so-called modified model, we prove rigorous explicit bounds which imply for example that the relative discrepancy is at least 1% even when L = 10^3000. Our results shed some light on the observed differences between simulations and rigorous asymptotics.Comment: 22 pages, 3 figure

A sharper threshold for bootstrap percolation in two dimensions

Two-dimensional bootstrap percolation is a cellular automaton in which sites become 'infected' by contact with two or more already infected nearest neighbors. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n x n square, with sites initially infected independently with probability p. The critical probability p_c is the smallest p for which the probability that the entire square is eventually infected exceeds 1/2. Holroyd determined the sharp first-order approximation: p_c \sim \pi^2/(18 log n) as n \to \infty. Here we sharpen this result, proving that the second term in the expansion is -(log n)^{-3/2+ o(1)}, and moreover determining it up to a poly(log log n)-factor. The exponent -3/2 corrects numerical predictions from the physics literature.Comment: 21 page

Good jobs, scam jobs: Detecting, normalizing, and internalizing online job scams during the COVID-19 pandemic

Good jobs that allow remote work have enabled white-collar professionals to stay home during COVID-19, but for precarious workers, online advertisements for work-from-home employment are often scams. In this article, based on in-depth interviews conducted between April and July 2020 with nearly 200 precarious workers, we find that precarious workers regularly encountered fraudulent job advertisements via digital media. Drawing on Swidler's concepts of the cultural tool kit and cultural logic, we find that in this time of uncertainty, workers defaulted to the focus on personal responsibility that is inherent in insecurity culture. Following the cultural logic of personal responsibility, job seekers did not place blame on job search websites for allowing the scams to be posted, but normalized the situation, deploying a scam detection repertoire in response. In addition, the discovery that advertised "good jobs" are often scams affecting workers' desire to continue job hunting and perceptions of potential future success

The Side Hustle Safety Net: Precarious Workers and Gig Work during COVID-19

While social distancing measures are essential in limiting the impact of a pandemic, such measures are often less feasible for low-income groups such as precarious workers who continue to travel on public transit and are less able to practice social distancing measures. In this paper, based on in-depth remote interviews conducted from April 2020 through June 2020, with more than 130 gig and precarious workers in New York City, we find that precarious workers experience three main hurdles in regard to accessing unemployment assistance that can be broadly categorized as knowledge, sociological, and temporal/financial barriers. Drawing on worker interview responses, we have named these responses: (1) Didn’t Know, (2) Didn’t Want, and (3) Can’t Wait. These challenges have led workers to turn to gig and precarious work, further highlighting the inequities of the pandemic. As a result, for some workers, so-called “side hustles” have become their primary social safety net

Phase Fluctuations and Pseudogap Properties: Influence of Nonmagnetic Impurities

The presence of nonmagnetic impurities in a 2D ``bad metal'' depresses the superconducting Berezinskii-Kosterlitz-Thouless transition temperature, while leaving the pairing energy scale unchanged. Thus the region of the pseudogap non-superconducting phase, where the modulus of the order parameter is non-zero but its phase is random, and which opens at the pairing temperature is substantially bigger than for the clean system. This supports the premise that fluctuations in the phase of the order parameter can in principle describe the pseudogap phenomena in high-$T_c$ materials over a rather wide range of temperatures and carrier densities. The temperature dependence of the bare superfluid density is also discussed.Comment: 11 pages, LaTeX, 1 EPS figure; final version to appear in Low.Temp.Phy