The Making of Modern Science: Science, Technology, Medicine, and Modernity, 1789-1914

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Schramm's proof of Watts' formula

G\'{e}rard Watts predicted a formula for the probability in percolation that there is both a left--right and an up--down crossing, which was later proved by Julien Dub\'{e}dat. Here we present a simpler proof due to Oded Schramm, which builds on Cardy's formula in a conceptually appealing way: the triple derivative of Cardy's formula is the sum of two multi-arm densities. The relative sizes of the two terms are computed with Girsanov conditioning. The triple integral of one of the terms is equivalent to Watts' formula. For the relevant calculations, we present and annotate Schramm's original (and remarkably elegant) Mathematica code.Comment: Published in at http://dx.doi.org/10.1214/11-AOP652 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A forward-backward single-source shortest paths algorithm

We describe a new forward-backward variant of Dijkstra's and Spira's Single-Source Shortest Paths (SSSP) algorithms. While essentially all SSSP algorithm only scan edges forward, the new algorithm scans some edges backward. The new algorithm assumes that edges in the outgoing and incoming adjacency lists of the vertices appear in non-decreasing order of weight. (Spira's algorithm makes the same assumption about the outgoing adjacency lists, but does not use incoming adjacency lists.) The running time of the algorithm on a complete directed graph on $n$ vertices with independent exponential edge weights is $O(n)$, with very high probability. This improves on the previously best result of $O(n\log n)$, which is best possible if only forward scans are allowed, exhibiting an interesting separation between forward-only and forward-backward SSSP algorithms. As a consequence, we also get a new all-pairs shortest paths algorithm. The expected running time of the algorithm on complete graphs with independent exponential edge weights is $O(n^2)$, matching a recent algorithm of Demetrescu and Italiano as analyzed by Peres et al. Furthermore, the probability that the new algorithm requires more than $O(n^2)$ time is exponentially small, improving on the $O(n^{-1/26})$ probability bound obtained by Peres et al

Excited Random Walk

A random walk on Z^d is excited if the first time it visits a vertex there is a bias in one direction, but on subsequent visits to that vertex the walker picks a neighbor uniformly at random. We show that excited random walk on Z^d, is transient iff d>1.Comment: 7 pages, v2 is journal versio

The looping rate and sandpile density of planar graphs

We give a simple formula for the looping rate of loop-erased random walk on a finite planar graph. The looping rate is closely related to the expected amount of sand in a recurrent sandpile on the graph. The looping rate formula is well-suited to taking limits where the graph tends to an infinite lattice, and we use it to give an elementary derivation of the (previously computed) looping rate and sandpile densities of the square, triangular, and honeycomb lattices, and compute (for the first time) the looping rate and sandpile densities of many other lattices, such as the kagome lattice, the dice lattice, and the truncated hexagonal lattice (for which the values are all rational), and the square-octagon lattice (for which it is transcendental)

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