Conditioned Galton-Watson trees do not grow

An example is given which shows that, in general, conditioned Galton-Watson trees cannot be obtained by adding vertices one by one, as has been shown in a special case by Luczak and Winkler.Comment: 5 pages, 2 figure

The evolution of the cover time

The cover time of a graph is a celebrated example of a parameter that is easy to approximate using a randomized algorithm, but for which no constant factor deterministic polynomial time approximation is known. A breakthrough due to Kahn, Kim, Lovasz and Vu yielded a (log log n)^2 polynomial time approximation. We refine this upper bound, and show that the resulting bound is sharp and explicitly computable in random graphs. Cooper and Frieze showed that the cover time of the largest component of the Erdos-Renyi random graph G(n,c/n) in the supercritical regime with c>1 fixed, is asymptotic to f(c) n \log^2 n, where f(c) tends to 1 as c tends to 1. However, our new bound implies that the cover time for the critical Erdos-Renyi random graph G(n,1/n) has order n, and shows how the cover time evolves from the critical window to the supercritical phase. Our general estimate also yields the order of the cover time for a variety of other concrete graphs, including critical percolation clusters on the Hamming hypercube {0,1}^n, on high-girth expanders, and on tori Z_n^d for fixed large d. For the graphs we consider, our results show that the blanket time, introduced by Winkler and Zuckerman, is within a constant factor of the cover time. Finally, we prove that for any connected graph, adding an edge can increase the cover time by at most a factor of 4.Comment: 14 pages, to appear in CP

A Markov growth process for Macdonald's distribution on reduced words

We give an algorithmic-bijective proof of Macdonald's reduced word identity in the theory of Schubert polynomials, in the special case where the permutation is dominant. Our bijection uses a novel application of David Little's generalized bumping algorithm. We also describe a Markov growth process for an associated probability distribution on reduced words. Our growth process can be implemented efficiently on a computer and allows for fast sampling of reduced words. We also discuss various partial generalizations and links to Little's work on the RSK algorithm.Comment: 16 pages, 5 figure

Stochastic Ordering of Infinite Binomial Galton-Watson Trees

We consider Galton-Watson trees with ${\rm Bin}(d,p)$ offspring distribution. We let $T_{\infty}(p)$ denote such a tree conditioned on being infinite. For $d=2,3$ and any $1/d\leq p_1 <p_2 \leq 1$, we show that there exists a coupling between $T_{\infty}(p_1)$ and $T_{\infty}(p_2)$ such that ${\mathbb P}(T_{\infty}(p_1) \subseteq T_{\infty}(p_2))=1.$Comment: 19 page