17,574 research outputs found
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 vertices with independent exponential edge
weights is , with very high probability. This improves on the previously
best result of , 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 , matching
a recent algorithm of Demetrescu and Italiano as analyzed by Peres et al.
Furthermore, the probability that the new algorithm requires more than
time is exponentially small, improving on the 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 , 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 . We do the analogous
computation for the triangular lattice, honeycomb lattice and , for which the probabilities are 5/18, 13/36, and
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
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