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Register a new userThe Clairvoyant Demon Has a Hard Task
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Peter Gacs
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For some m ge 4, let us color each column of the integer lattice L = Z^2 independently and uniformly into one of m colors. We do the same for the rows, independently from the columns. A point of L will be called blocked if its row and column have the same color. 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, and avoiding the blocked points. As a problem arising in distributed computing, it has been conjectured that for m ge 4, the configuration percolates with positive probability. This has now been proved (in a later paper), for large m. Here, 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.
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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< n_{i} - n_{i-1} < m, w(i) = v(n_i) for all i > 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.
The main result of this paper is that almost every realization of the sine-process with one particle removed is a uniqueness set for the Paley-Wiener space; with two particles removed, a zero set for the Paley-Wiener space.
In this paper, we show that the first passage time in the frog model on $Z^d$ with $dgeq 2$ has a sublinear variance. This implies that the central limit theorem does not holds at least with the standard diffusive scaling. The proof is based on the method introduced in cite{BRo, DHS} combining with a control of the maximal weight of paths in locally dependent site-percolation. We also apply this method to get the linearity of the lengths of optimal paths..
We study the large-$n$ limit of the probability $p_{2n,2k}$ that a random $2ntimes 2n$ matrix sampled from the real Ginibre ensemble has $2k$ real eigenvalues. We prove that, $$lim_{nrightarrow infty}frac {1}{sqrt{2n}} log p_{2n,2k}=lim_{nrightarrow infty}frac {1}{sqrt{2n}} log p_{2n,0}= -frac{1}{sqrt{2pi}}zetaleft(frac{3}{2}right),$$ where $zeta$ is the Riemann zeta-function. Moreover, for any sequence of non-negative integers $(k_n)_{ngeq 1}$, $$lim_{nrightarrow infty}frac {1}{sqrt{2n}} log p_{2n,2k_n}=-frac{1}{sqrt{2pi}}zetaleft(frac{3}{2}right),$$ provided $lim_{nrightarrow infty} left(n^{-1/2}log(n)right) k_{n}=0$.
Local perturbations of a Brownian motion are considered. As a limit we obtain a non-Markov process that behaves as a reflected Brownian motion on the positive half line until its local time at zero reaches some exponential level, then changes a sign and behaves as a reflected Brownian motion on the negative half line until some stopping time, etc.
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