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

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

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Two-player Knock 'em Down

We analyze the two-player game of Knock 'em Down, asymptotically as the number of tokens to be knocked down becomes large. Optimal play requires mixed strategies with deviations of order sqrt(n) from the naive law-of-large numbers allocation. Upon rescaling by sqrt(n) and sending n to infinity, we show that optimal play's random deviations always have bounded support and have marginal distributions that are absolutely continuous with respect to Lebesgue measure.Comment: 15 pages, 1 figure. v2 has minor revision

Balanced Boolean functions that can be evaluated so that every input bit is unlikely to be read

A Boolean function of n bits is balanced if it takes the value 1 with probability 1/2. We exhibit a balanced Boolean function with a randomized evaluation procedure (with probability 0 of making a mistake) so that on uniformly random inputs, no input bit is read with probability more than Theta(n^{-1/2} sqrt{log n}). We give a balanced monotone Boolean function for which the corresponding probability is Theta(n^{-1/3} log n). We then show that for any randomized algorithm for evaluating a balanced Boolean function, when the input bits are uniformly random, there is some input bit that is read with probability at least Theta(n^{-1/2}). For balanced monotone Boolean functions, there is some input bit that is read with probability at least Theta(n^{-1/3}).Comment: 11 page