For first passage percolation on @math with i.i.d. bounded edge weights, we consider the upper tail large deviation event; i.e., the rare situation where the first passage time between two points at distance @math , is macroscopically larger than typical. It was shown by Kesten (1986) that the probability of this event decays as @math . However the question of existence of the rate function i.e., whether the log-probability normalized by @math tends to a limit, had remained open. We show that under some additional mild regularity assumption on the passage time distribution, the rate function for upper tail large deviation indeed exists. Our proof can be generalized to work in higher dimensions and for the corresponding problem in last passage percolation as well. The key intuition behind the proof is that a limiting metric structure which is atypical causes the upper tail large deviation event. The formal argument then relies on an approximate version of the above which allows us to dilate the large deviation environment to compare the upper tail probabilities for various values of @math
Kesten @cite_18 considered both upper and lower tail large deviations for first passage percolation. Let @math (throughout this section for brevity we will use @math to denote the passage time between @math and @math in @math although it was initially defined only for @math ) denote the lower tail large deviation event. Using a subadditive argument, Kesten showed that for @math ,
@cite_18
1981891714 2136645553 1559523984 2127129887
Let @math and @math be standard normal random variables with covariance matrices @math and @math , respectively. Slepian's lemma says that if @math for @math , the lower bound @math is at least @math . In this paper an upper bound is given. The usefulness of the upper bound is justified with three concrete applications: (i) the new law of the iterated logarithm of Erdős and Revesz, (ii) the probability that a random polynomial does not have a real zero and (iii) the random pursuit problem for fractional Brownian particles. In particular, a conjecture of Kesten (1992) on the random pursuit problem for Brownian particles is confirmed, which leads to estimates of principal eigenvalues. We prove upper and lower bounds and give an approximation algorithm for the cover time of the random walk on a graph. We introduce a parameter M motivated by the well-known Matthews bounds (P. Matthews, 1988) on the cover time, C, and prove that M 2<C= O(M(lnlnn) sup 2 ). We give a deterministic-polynomial time algorithm to approximate M within a factor of 2; this then approximates C within a factor of O((lnlnn) sup 2 ), improving the previous bound O(lnn) due to Matthews. The blanket time B was introduced by P. Winkler and D. Zuckerman (1996): it is the expectation of the first time when all vertices are visited within a constant factor of the number of times suggested by the stationary distribution. Obviously C spl les B. Winkler and Zuckerman conjectured B=O(C) and proved B=O(Clnn). Our bounds above are also valid for the blanket time, and so it follows that B=O(C(lnlnn) sup 2 ). A natural problem in extremal combinatorics is to maximize the number of distinct subsequences for any length- @math string over a finite alphabet @math ; this value grows exponentially, but slower than @math . We use the probabilistic method to determine the maximizing string, which is a cyclically repeating string. The number of distinct subsequences is exactly enumerated by a generating function, from which we also derive asymptotic estimates. For the alphabet @math , @math has the maximum number of distinct subsequences, namely @math . We also consider the same problem with sub strings in lieu of sub sequences . Here, we show that an appropriately truncated de Bruijn word attains the maximum. For both problems, we compare the performance of random strings with that of the optimal ones. Given a branching random walk, let @math be the minimum position of any member of the @math th generation. We calculate @math to within O(1) and prove exponential tail bounds for @math , under quite general conditions on the branching random walk. In particular, together with work by Bramson [Z. Wahrsch. Verw. Gebiete 45 (1978) 89―108], our results fully characterize the possible behavior of @math when the branching random walk has bounded branching and step size.