Percolation on dual lattices with k-fold symmetry

Zhang found a simple, elegant argument deducing the non-existence of an infinite open cluster in certain lattice percolation models (for example, p=1/2 bond percolation on the square lattice) from general results on the uniqueness of an infinite open cluster when it exists; this argument requires some symmetry. Here we show that a simple modification of Zhang's argument requires only 2-fold (or 3-fold) symmetry, proving that the critical probabilities for percolation on dual planar lattices with such symmetry sum to 1. Like Zhang's argument, our extension applies in many contexts; in particular, it enables us to answer a question of Grimmett concerning the anisotropic random cluster model on the triangular lattice.Comment: 11 pages, 1 figure. Revised with applications added; to appear in Random Structures and Algorithm

A derivative formula for the free energy function

We consider bond percolation on the ${\bf Z}^d$ lattice. Let $M_n$ be the number of open clusters in $B(n)=[-n, n]^d$. It is well known that $E_pM_n / (2n+1)^d$ converges to the free energy function $\kappa(p)$ at the zero field. In this paper, we show that $\sigma^2_p(M_n)/(2n+1)^d$ converges to $-(p^2(1-p)+p(1-p)^2)\kappa'(p)$.Comment: 8 pages 1 figur

Uniqueness and multiplicity of infinite clusters

The Burton--Keane theorem for the almost-sure uniqueness of infinite clusters is a landmark of stochastic geometry. Let $\mu$ be a translation-invariant probability measure with the finite-energy property on the edge-set of a $d$-dimensional lattice. The theorem states that the number $I$ of infinite components satisfies $\mu(I\in\{0,1\})=1$. The proof is an elegant and minimalist combination of zero--one arguments in the presence of amenability. The method may be extended (not without difficulty) to other problems including rigidity and entanglement percolation, as well as to the Gibbs theory of random-cluster measures, and to the central limit theorem for random walks in random reflecting labyrinths. It is a key assumption on the underlying graph that the boundary/volume ratio tends to zero for large boxes, and the picture for non-amenable graphs is quite different.Comment: Published at http://dx.doi.org/10.1214/074921706000000040 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

World Trade Organization (WTO) Decisions and Their Effect in U.S. Law

[Excerpt] Congress has comprehensively dealt with the legal effect of World Trade Organization (WTO) agreements and dispute settlement results in the United States in the Uruguay Round Agreements Act (URAA), P.L. 103-465. The act provides that domestic law prevails over conflicting provisions of WTO agreements and prohibits private remedies based on alleged violations of these agreements. As a result, provisions of WTO agreements and WTO panel and Appellate Body reports adopted by the WTO Members that are in conflict with federal law do not have domestic legal effect unless and until Congress or the executive branch, as the case may be, takes action to modify or remove the conflicting statute, regulation, or regulatory action. Violative state laws may be withdrawn by the state or, in rare circumstances, invalidated through legal action by the federal government. The URAA also contains requirements for agencies to follow where a change in a regulation or the issuance of a new agency determination in a trade remedy proceeding is needed to comply with a WTO decision and existing law may be sufficient to carry out the action. While the URAA prohibits private rights of action based on Uruguay Round agreements, plaintiffs, in cases brought under other statutes, have argued that the agency actions they are challenging in court are inconsistent with a WTO agreement or a WTO decision and should conform with U.S. WTO obligations. Although courts have deemed WTO decisions to be persuasive, they have also held that they are not binding on the United States, U.S. agencies, or the judiciary, leaving the issue of whether and how the United States complies in a particular WTO proceeding to the executive branch. Legislation introduced in recent Congresses generally reflected congressional concerns that the WTO Appellate Body had interpreted WTO agreements in an overly broad manner to the detriment of the United States and that the executive branch had in some cases too readily used existing statutory authorities to comply with these decisions, particularly where U.S. trade remedies were involved. Legislation particularly focused on WTO decisions finding the U.S. use of “zeroing” in antidumping proceedings to be in violation of the WTO Antidumping Agreement and an administrative modification instituted by the Department of Commerce in original anti-dumping investigations in response to one of the earliest of these decisions. Under the practice, the department calculates dumping margins by taking into account only sales below fair market value—generally the price in the exporting country—and assigns a zero value to sales at or above this price. While it is argued that zeroing improperly creates or inflates dumping margins, U.S. courts have consistently upheld the department’s use of the practice as valid under U.S. antidumping law