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

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

Evaluating the efficiency of commercial banks: does our view of what banks do matter?

Bank management

Why do banks disappear? The determinants of U.S. bank failures and acquisitions

This paper examines the determinants of individual bank failures and acquisitions in the United States during 1984-1993. We use bank-specific information suggested by examiner CAMEL-rating categories to estimate competing-risks hazard models with time-varying covariates. We focus especially on the role of management quality, as reflected in alternative measures of x-efficiency and find the inefficiency increases the risk of failure, while reducing the probability of a bank's being acquired. Finally, we show that the closer to insolvency a bank is, as reflected by a low equity-to-assets ratio, the more likely its acquisition.Bank failures

New evidence on returns to scale and product mix among U.S. commercial banks

Numerous studies have found that banks exhaust scale economies at low levels of output, but most are based on the estimation of parametric cost functions which misrepresent bank cost. Here we avoid specification error by using nonparametric kernal regression techniques. We modify measures of scale and product mix economies introduced by Berger et al. (1987) to accommodate the nonparametric estimation approach, and estimate robust confidence intervals to assess the statistical significance of returns to scale. We find that banks experience increasing returns to scale up to approximately $500 million of assets, and essentially constant returns thereafter. We also find that minimum efficient scale has increased since 1985.Banks and banking ; Banks and banking - Costs ; Economies of scale

END-USER GAINS FROM INPUT CHARACTERISTICS IMPROVEMENT

Production Economics,