Anomalous heat-kernel decay for random walk among bounded random conductances

We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$, driven by a field of bounded random conductances $\omega_{xy}\in[0,1]$. The conductance law is i.i.d. subject to the condition that the probability of $\omega_{xy}>0$ exceeds the threshold for bond percolation on $\Z^d$. For environments in which the origin is connected to infinity by bonds with positive conductances, we study the decay of the $2n$-step return probability $P_\omega^{2n}(0,0)$. We prove that $P_\omega^{2n}(0,0)$ is bounded by a random constant times $n^{-d/2}$ in $d=2,3$, while it is $o(n^{-2})$ in $d\ge5$ and $O(n^{-2}\log n)$ in $d=4$. By producing examples with anomalous heat-kernel decay approaching $1/n^2$ we prove that the $o(n^{-2})$ bound in $d\ge5$ is the best possible. We also construct natural $n$-dependent environments that exhibit the extra $\log n$ factor in $d=4$. See also math.PR/0701248.Comment: 22 pages. Includes a self-contained proof of isoperimetric inequality for supercritical percolation clusters. Version to appear in AIHP + additional correction

Hebb and the art of spine remodeling

The notion that synaptic remodeling underlies certain forms of learning is one of the main tenets of Hebb's inspiring theories dating from the 1940s. Until recently, however, direct evidence for tight relationships between synaptic remodeling and behavior has been scarce. Fascinating data from recent studies on the remodeling of postsynaptic structures known as dendritic spines indicates that such relationships might be more complex than initially expected

Strengthening Online Dispute Resolution Justice

This paper adopts a systems-design approach to focus courts and lawyers on the unexamined: how involving lawyers in the design, development and implementation of court-annexed online dispute resolution (ODR) programs, will strengthen their justice outcomes. The phrase “ODR programs” refers to the new menu of processes for dispute resolution and litigation offered online by courts

Peer-to-Peer File Sharing and Cultural Trade Protectionism

We examine the Internet’s impact on the cross-border distribution of cultural goods and assess its implications for cultural policy and cultural diversity. We present a stylized model of a two-country economy where governments are endowed with political preferences over the consumption of domestic content and enact import barriers and subsidies to protect it. We introduce peer-to-peer file sharing as a distinct distribution channel enabled by the Internet that provides access to all media products at a low cost. We report two main findings. First, the Internet renders legacy cultural policy inefficient, and the elimination of import barriers and the reduction of subsidized production can be desirable even when governments exhibit paternalistic preferences favoring the consumption of domestic content. And second, even though the Internet increases cultural diversity within countries, it can also reduce diversity across them

Can citizens guess how other citizens voted based on demographic characteristics?

How well do citizens understand the associations between social groups and political divisions in their societies? Previous research has indicated systematic biases in how the demographic composition of party supporters are perceived, but this need not imply that citizens misperceive the likely voting behavior of specific individuals. We report results from two experiments where subjects were provided with randomly selected demographic profiles of respondents to the 2017 British Election Study (BES) and then asked to assess either (1) which party that individual was likely to have voted for in the 2017 UK election or (2) whether that individual was likely to have voted Leave or Remain in the 2016 UK referendum on EU membership. We find that, despite substantial overconfidence in individual responses, on average citizens’ guesses broadly reflect the actual distribution of groups supporting the parties and referendum positions

What Dinosaurs Can Teach Lawyers About How to Avoid Extinction in the ODR Evolution

This paper is a wake-up call for the legal profession: Heed the justice changes that are upon us or risk extinction. Online dispute resolution (hereinafter ODR) is currently being incorporated into U.S and international court systems, re-shaping and re-defining justice as we know it today. Courts and clients, two stakeholders in our justice system, are increasingly receptive to ODR as a viable option to help provide and access justice efficiently and affordably. The legal profession, the third stakeholder in our justice system, however, has been slower to react. As ODR plays an increasingly prominent role in the court system, it will eliminate some of the justice roles currently reserved for lawyers, diminish others, and create new areas of practice. We highlight ODR innovations already in the justice system and project the paths of ODR’s likely expansion. This paper alerts the legal profession and legal education community to take heed of these developments and become active contributors in shaping these justice innovations. Viewing ODR’s entry into the court as an evolution of the justice system, we identify six adaptive skills that will redefine “thinking like a lawyer” and help the legal profession avoid extinction and remain relevant. Some of these are currently marginally addressed in the law school curriculum, others are entirely absent. Law schools, the primary disseminators of legal education, must re align their curriculum with the skills that practice-competent lawyers require to succeed in the ODR-infused justice system

Why a Single-Star Model Cannot Explain the Bipolar Nebula of Eta Carinae

I examine the angular momentum evolution during the 1837-1856 Great Eruption of the massive star Eta Carinae. I find that the new estimate of the mass blown during that eruption implies that the envelope of Eta Car substantially spun-down during the 20 years eruption. Single-star models, most of which require the envelope to rotate close to the break-up velocity, cannot account for the bipolar nebula (the Homunculus) formed from matter expelled in that eruption. The kinetic energy and momentum of the Homunculus further constrains single-star models. I discuss how Eta Car can fit into a unified model for the formation of bipolar lobes where two oppositely ejected jets inflate two lobes (or bubbles). These jets are blown by an accretion disk, which requires stellar companions in the case of bipolar nebulae around stellar objects.Comment: ApJ, in press. New references and segments were adde

Solar-Like Cycle in Asymptotic Giant Branch Stars

I propose that the mechanism behind the formation of concentric semi-periodic shells found in several planetary nebulae (PNs) and proto-PNs, and around one asymptotic giant branch (AGB) star, is a solar-like magnetic activity cycle in the progenitor AGB stars. The time intervals between consecutive ejection events is about 200-1,000 years, which is assumed to be the cycle period (the full magnetic cycle can be twice as long, as is the 22-year period in the sun). The magnetic field has no dynamical effects; it regulates the mass loss rate by the formation of magnetic cool spots. The enhanced magnetic activity at the cycle maximum results in more magnetic cool spots, which facilitate the formation of dust, hence increasing the mass loss rate. The strong magnetic activity implies that the AGB star is spun up by a companion, via a tidal or common envelope interaction. The strong interaction with a stellar companion explains the observations that the concentric semi-periodic shells are found mainly in bipolar PNs.Comment: 10 pages, submitted to Ap

Curves of every genus with many points, II: Asymptotically good families

We resolve a 1983 question of Serre by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power q there is a positive constant c_q with the following property: for every non-negative integer g, there is a genus-g curve over F_q with at least c_q * g rational points over F_q. Moreover, we show that there exists a positive constant d such that for every q we can choose c_q = d * (log q). We show also that there is a constant c > 0 such that for every q and every n > 0, and for every sufficiently large g, there is a genus-g curve over F_q that has at least c*g/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to (Z/nZ)^r for some r > c*g/n.Comment: LaTeX, 18 page