Georgia Institute of Technology College of Sciences

We study the number of random permutations needed to invariably generate the symmetric group, S_n, when the distribution of cycle counts has the strong \alpha-logarithmic property. The canonical example is the Ewens sampling formula, for which the number of k-cycles relates to a conditioned Poisson random variable with mean \alpha/k. The special case \alpha=1 corresponds to uniformly random permutations, for which it was recently shown that exactly four are needed.For strong \alpha-logarithmic measures, and almost every \alpha, we show that precisely $\lceil( 1- \alpha \log 2 )^{-1} \rceil$ permutations are needed to invariably generate S_n. A corollary is that for many other probability measures on S_n no bounded number of permutations will invariably generate S_n with positive probability. Along the way we generalize classic theorems of Erdos, Tehran, Pyber, Luczak and Bovey to permutations obtained from the Ewens sampling formula.

For a fixed graph $G$, let $\mathcal{L}_G$ denote the family of Lipschitz functions $f:V(G) \rightarrow \mathbb{R}$ such that $0 = \sum_u f(u)$. The \emph{spread} of $G$ is denoted $c(G) := \frac{1}{|V(G)|} \max_{f \in \mathcal{L}_G} \sum_u f(u)^2$ and the subgaussian constant is $e^{\sigma_G^2} := \sup_{t > 0} \max_{f \in \mathcal{L}_G} \left( \frac{1}{|V(G)|} \sum_u e^{t f(u)} \right)^{2/t^2}$. Motivation of these parameters comes from their relationship with the isoperimetric number of a graph (given a number $t$, find a set $W \subset V(G)$ such that $2|W| \geq |V(G)|$ that minimizes $i(G,t) := |\{u : d(u, W) \leq t \}|$). While the connection to the isoperimetric number is interesting, the spread and subgaussian constant have not been any easier to understand. In this talk, we will present results that describe the functions $f$ achieving the optimal values. As a corollary to these results, we will resolve two conjectures (one false, one true) about these parameters. The conjectures that we resolve are the following. We denote the Cartesian product of $G$ with itself $d$ times as $G^d$. Alon, Boppana, and Spencer proved that the set $\{u: f(u) < k\}$ for extremal function $f$ for the spread of $G^d$ gives a value that is asymptotically close to the isoperimetric number when $d, t$ grow at specific rates and $k=0$; and they conjectured that the value is exactly correct for large $d$ and $k,t$ in ``appropriate ranges.'' The conjecture was proven true for hypercubes by Harper and the discrete torus of even order by Bollob\'{a}s and Leader. Bobkov, Houdr\'{e}, and Tetali constructed a function over a cycle that they conjectured to be optimal for the subgaussian constant, and it was proven correct for cycles of even length by Sammer and Tetali. This work appears in the manuscript https://arxiv.org/abs/1705.09725 .

Official School Holiday: Thanksgiving Break

We study an online algorithm for making a well—equidistributed random set of points in an interval, in the spirit of "power of choice" methods. Suppose finitely many distinct points are placed on an interval in any arbitrary configuration. This configuration of points subdivides the circle into a finite number of intervals. At each time step, two points are sampled uniformly from the interval. Each of these points lands within some pair of intervals formed by the previous configuration. Add the point that falls in the larger interval to the existing configuration of points, discard the other, and then repeat this process. We then study this point configuration in the sense of its largest interval, and discuss other "power of choice" type modifications. Joint work with Pascal Maillard.