Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs

We show how to compute the probabilities of various connection topologies for uniformly random spanning trees on graphs embedded in surfaces. As an application, we show how to compute the "intensity" of the loop-erased random walk in ${\mathbb Z}^2$, that is, the probability that the walk from (0,0) to infinity passes through a given vertex or edge. For example, the probability that it passes through (1,0) is 5/16; this confirms a conjecture from 1994 about the stationary sandpile density on ${\mathbb Z}^2$. We do the analogous computation for the triangular lattice, honeycomb lattice and ${\mathbb Z} \times {\mathbb R}$, for which the probabilities are 5/18, 13/36, and $1/4-1/\pi^2$ respectively.Comment: 45 pages, many figures. v2 has an expanded introduction, a revised section on the LERW intensity, and an expanded appendix on the annular matri

Approximation Techniques for Planar Periodic Structures

The rigorous calculation of electromagnetic properties of periodic meshes using moment methods requires considerable algebraic work and computer resources. In this paper, a number of easy to use approximation techniques for analyzing thin structures with square, rectangular, and circular holes are presented. Formulas for the effective impedante of these meshes are described which can easily take into account oblique incidence and the presence of a dielectric substrate. In addition, techniques for analyzing more complex-shaped apertures such as a cross are discussed. These methods are more accurate than existing approximation techniques and can be applied to a wide range of situations that could not be handled before

Boundary Partitions in Trees and Dimers

Given a finite planar graph, a grove is a spanning forest in which every component tree contains one or more of a specified set of vertices (called nodes) on the outer face. For the uniform measure on groves, we compute the probabilities of the different possible node connections in a grove. These probabilities only depend on boundary measurements of the graph and not on the actual graph structure, i.e., the probabilities can be expressed as functions of the pairwise electrical resistances between the nodes, or equivalently, as functions of the Dirichlet-to-Neumann operator (or response matrix) on the nodes. These formulae can be likened to generalizations (for spanning forests) of Cardy's percolation crossing probabilities, and generalize Kirchhoff's formula for the electrical resistance. Remarkably, when appropriately normalized, the connection probabilities are in fact integer-coefficient polynomials in the matrix entries, where the coefficients have a natural algebraic interpretation and can be computed combinatorially. A similar phenomenon holds in the so-called double-dimer model: connection probabilities of boundary nodes are polynomial functions of certain boundary measurements, and as formal polynomials, they are specializations of the grove polynomials. Upon taking scaling limits, we show that the double-dimer connection probabilities coincide with those of the contour lines in the Gaussian free field with certain natural boundary conditions. These results have direct application to connection probabilities for multiple-strand SLE_2, SLE_8, and SLE_4.Comment: 46 pages, 12 figures. v4 has additional diagrams and other minor change

What Have Two Decades of British Economic Reform Delivered?

Beginning in 1979 with the newly electted Thatcher Government and continuing under successive Conservative and Labour Governments, the United Kingdom has embarked on a two-decade-long experiment in economic reform. We present evidence that the reform process has succeeded in making the UK more market-friendly than its European competitors. In fact, by the 1990s Britain ranked near the top of the league tables for freedom of markets, in some cases even ahead of the United States. To evaluate the effects of these reforms we compare trends in macroeconomic outcomes in the UK relative to the US, Germany, and France. During the 1980s and 1990s Britain halted the relative declines in GDP per capita and labour productivity that had characterized earlier decades, and partially closed the gap in income per capita with France and Germany. These gains were mainly attributable to relative rises in employment and hours. Unlike its EU competitors, Britain was able to achieve high employment-population rates with rising real wages for workers. The case that the change in economic performance can be credited to market-oriented reforms is harder to prove. Nevertheless, based on our own macro-level analyses, and micro-level evidence from several companion studies, we conclude that economic reforms contributed to halting the nearly century-long trend in relative economic decline of the UK relative to its historic competitors, Germany and France.