19 research outputs found
The clairvoyant demon has a hard task
Consider the integer lattice L = ℤ2. For some m [ges ] 4, let us colour each column of this lattice independently and uniformly with one of m colours. We do the same for the rows, independently of the columns. A point of L will be called blocked if its row and column have the same colour. We say that this random configuration percolates if there is a path in L starting at the origin, consisting of rightward and upward unit steps, avoiding the blocked points. As a problem arising in distributed computing, it has been conjectured that for m [ges ] 4 the configuration percolates with positive probability. This question remains open, but we prove that the probability that there is percolation to distance n but not to infinity is not exponentially small in n. This narrows the range of methods available for proving the conjecture
Clairvoyant embedding in one dimension
Let v, w be infinite 0-1 sequences, and m a positive integer. We say that w
is m-embeddable in v, if there exists an increasing sequence n_{i} of integers
with n_{0}=0, such that 0 0.
Let X and Y be independent coin-tossing sequences. We will show that there is
an m with the property that Y is m-embeddable into X with positive probability.
This answers a question that was open for a while. The proof generalizes
somewhat the hierarchical method of an earlier paper of the author on dependent
percolation.Comment: 49 pages. Some errors corrected. arXiv admin note: substantial text
overlap with arXiv:math/010915
Avoidance Coupling
We examine the question of whether a collection of random walks on a graph
can be coupled so that they never collide. In particular, we show that on the
complete graph on n vertices, with or without loops, there is a Markovian
coupling keeping apart Omega(n/log n) random walks, taking turns to move in
discrete time.Comment: 13 pages, 3 figure