Noise-stability and central limit theorems for effective resistance of random electric networks

We investigate the (generalized) Walsh decomposition of point-to-point effective resistances on countable random electric networks with i.i.d. resistances. We show that it is concentrated on low levels, and thus point-to-point effective resistances are uniformly stable to noise. For graphs that satisfy some homogeneity property, we show in addition that it is concentrated on sets of small diameter. As a consequence, we compute the right order of the variance and prove a central limit theorem for the effective resistance through the discrete torus of side length $n$ in $\mathbb {Z}^d$, when $n$ goes to infinity.Comment: Published at http://dx.doi.org/10.1214/14-AOP996 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Threshold for monotone symmetric properties through a logarithmic Sobolev inequality

Threshold phenomena are investigated using a general approach, following Talagrand [Ann. Probab. 22 (1994) 1576--1587] and Friedgut and Kalai [Proc. Amer. Math. Soc. 12 (1999) 1017--1054]. The general upper bound for the threshold width of symmetric monotone properties is improved. This follows from a new lower bound on the maximal influence of a variable on a Boolean function. The method of proof is based on a well-known logarithmic Sobolev inequality on $\{0,1\}^n$. This new bound is shown to be asymptotically optimal.Comment: Published at http://dx.doi.org/10.1214/009117906000000287 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation

We consider the standard first passage percolation model in $\mathbb{Z}^d$ for $d\geq 2$. We are interested in two quantities, the maximal flow $\tau$ between the lower half and the upper half of the box, and the maximal flow $\phi$ between the top and the bottom of the box. A standard subadditive argument yields the law of large numbers for $\tau$ in rational directions. Kesten and Zhang have proved the law of large numbers for $\tau$ and $\phi$ when the sides of the box are parallel to the coordinate hyperplanes: the two variables grow linearly with the surface $s$ of the basis of the box, with the same deterministic speed. We study the probabilities that the rescaled variables $\tau /s$ and $\phi /s$ are abnormally small. For $\tau$, the box can have any orientation, whereas for $\phi$, we require either that the box is sufficiently flat, or that its sides are parallel to the coordinate hyperplanes. We show that these probabilities decay exponentially fast with $s$, when $s$ grows to infinity. Moreover, we prove an associated large deviation principle of speed $s$ for $\tau /s$ and $\phi /s$, and we improve the conditions required to obtain the law of large numbers for these variables.Comment: 39 pages, 4 figures; improvement of the moment conditions and introduction of new results in the revised versio

Law of large numbers for the maximal flow through tilted cylinders in two-dimensional first passage percolation

Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random capacities. We prove a law of large numbers for the maximal flow crossing a rectangle in $\mathbb{R}^2$ when the side lengths of the rectangle go to infinity. The value of the limit depends on the asymptotic behaviour of the ratio of the height of the cylinder over the length of its basis. This law of large numbers extends the law of large numbers obtained by Grimmett and Kesten (1984) for rectangles of particular orientation.Comment: 27 pages, 4 figures; minor modification

Clarifying creative nonfiction through the personal essay

In a recent issue of TEXT, Matthew Ricketson sought to clarify the &lsquo;boundaries between fiction and nonfiction&rsquo;.&nbsp;In his capacity as a teacher of the creative nonfiction form he writes, &lsquo;I have lost count of the number of times, in classes and in submitted work, that students have described a piece of nonfiction as a novel&rsquo;. The confusion thus highlighted is not restricted to Ricketson&rsquo;s journalism students. In our own university&rsquo;s creative writing cohort, students also struggle with difficulties in melding the research methodology of the journalist with the language and form of creative writing required to produce nonfiction stories for a 21st century readership.Currently in Australia creative nonfiction is enthusiastically embraced by publishers and teaching institutions. Works of memoir proliferate in the lists of mainstream publishers, as do anthologies of the essay form. During a time of increasing competition and desire for differentiation between institutions, when graduate outcomes form a basis for marketing university degrees, it is hardly surprising that, increasingly, tertiary writing teachers focus on this genre in their writing programs. A second tension has arisen in higher education more generally, which affects our writing students&rsquo; approaches to tertiary study. The student writers of the 21st century emerge from a digitally literate and socially collaborative generation: the NetGen(eration). From a learner-centric viewpoint, they could be described as time-poor, and motivated by work-integrated learning with its perceived close links to workplace contexts and to writing genres. They seek just-in-time learning to meet their immediate employment needs, which inhibits the development of their capacity to adapt their researching and writing to various genres and audiences. This article examines issues related to moving these NetGen student writers into the demanding and rapidly expanding creative nonfiction market. It is form rather than genre that denotes creative nonfiction and, we argue, it is the unique features of the personal essay, based as it is on doubt, discovery and the writer&rsquo;s personal voice that can be instrumental in teaching creative nonfiction writing to our digitally and socially literate cohort of students.<br /

Is inequality harmful for the environment in a growing economy ?

In this paper we investigate the relationship between inequality and the environment in a growing economy from a political economy perspective. We consider an endogenous growth economy, where growth generates pollution and a deterioration of the environment. Public expenditures may either be devoted to supporting growth or abating pollution. The decision over the public programs is done in a direct democracy, with simple majority rule. We prove that the median voter is decisive and show that inequality is harmful for the environment : the poorer the median voter relative to the average individual, the less she will tax and devote resources to the environment, preferring to support growth.Inequality, growth, environmental policy, political economy.