Scaling window for mean-field percolation of averages

For a complete graph of size $n$, assign each edge an i.i.d. exponential variable with mean $n$. For $\lambda>0$, consider the length of the longest path whose average weight is at most $\lambda$. It was shown by Aldous (1998) that the length is of order $\log n$ for $\lambda < 1/\mathrm{e}$ and of order $n$ for $\lambda > 1/\mathrm{e}$. Aldous (2003) posed the question on detailed behavior at and near criticality $1/\mathrm{e}$. In particular, Aldous asked whether there exist scaling exponents $\mu, \nu$ such that for $\lambda$ within $1/\mathrm{e}$ of order $n^{-\mu}$, the length for the longest path of average weight at most $\lambda$ has order $n^\nu$. We answer this question by showing that the critical behavior is far richer: For $\lambda$ around $1/\mathrm{e}$ within a window of $\alpha(\log n)^{-2}$ with a small absolute constant $\alpha>0$, the longest path is of order $(\log n)^3$. Furthermore, for $\lambda \geq 1/\mathrm{e} + \beta (\log n)^{-2}$ with $\beta$ a large absolute constant, the longest path is at least of length a polynomial in $n$. An interesting consequence of our result is the existence of a second transition point in $1/\mathrm{e} + [\alpha (\log n)^{-2}, \beta (\log n)^{-2}]$. In addition, we demonstrate a smooth transition from subcritical to critical regime. Our results were not known before even in a heuristic sense.Comment: 17pages. Minor revision upon previous version. To appear in Annals of Probabilit

Mixing under monotone censoring

We initiate the study of mixing times of Markov chain under monotone censoring. Suppose we have some Markov Chain $M$ on a state space $\Omega$ with stationary distribution $\pi$ and a monotone set $A \subset \Omega$. We consider the chain $M'$ which is the same as the chain $M$ started at some $x \in A$ except that moves of $M$ of the form $x \to y$ where $x \in A$ and $y \notin A$ are {\em censored} and replaced by the move $x \to x$. If $M$ is ergodic and $A$ is connected, the new chain converges to $\pi$ conditional on $A$. In this paper we are interested in the mixing time of the chain $M'$ in terms of properties of $M$ and $A$. Our results are based on new connections with the field of property testing. A number of open problems are presented.Comment: 6 page