Combinatorial Theory Seminar

Consider a tail trivial, positively associated site percolation process such that the set of open vertices is stochastically dominated by the set of closed ones. We show that, for any planar graph $G$, such a process must contain zero or infinitely many infinite connected components. The assumptions cover Bernoulli site percolation at parameter $p$ less than or equal to one half, resolving a conjecture of Benjamini and Schramm. As a corollary, we prove that $p_c$ is greater than or equal to $1/2$ for any unimodular, invariantly amenable planar graphs.

We will then apply this percolation statement to the loop $O(n)$ model on the hexagonal lattice, and show that, whenever $n$ is between $1$ and $2$ and $x$ is between $1/\sqrt{2}$ and $1$, the model exhibits infinitely many loops surrounding every face of the lattice, giving strong evidence for conformally invariant behavior in the scaling limit (as conjectured by Nienhuis).

This is joint work with Alexander Glazman (University of Innsbruck) and Nathan Zelesko (Northeastern University).

What do the roots of random polynomials look like? Classical works of Erdős-Turán and others show that most roots are near the unit circle and they are approximately rotationally equidistributed. We will begin with an understanding of why this happens and see how ideas from extremal combinatorics can mix with analytic and probabilistic arguments to show this. Another main feature of random polynomials is that their roots tend to "repel" each other. We will see various quantitative statements that make this rigorous. In particular, we will study the smallest separation $m_n$ between pairs of roots and show that typically $m_n$ is on the order of $n^{-5/4}$. We will see why this reflects repulsion between roots and discuss where this repulsion comes from. This is based on joint work with Oren Yakir.

Given a finite poset $P$ we ask how small a family of subsets of $[n]$ can be such that it does not contain an induced copy of the poset, but adding any other subset creates such a copy. This number is called the saturation number of $P$, denoted by $\operatorname{sat}^*(n,P)$. Despite the apparent similarity to the saturation for graphs, this notion is vastly different. For example, it has been shown that the saturation numbers exhibit a dichotomy: for any poset, the saturation number is either bounded, or at least $2 n^{1/2}$. In fact, it is believed that the saturation number is always bounded or exactly linear. In this talk we will be discussing the most recent advances in this field, with the focus on the diamond poset, whose saturation number was unknown until recently.

Joint with Sean Jaffe.

How many vectors are needed to simultaneously generate $m$ complete flags in $\mathbb{R}^d$, in the worst-case scenario?  A classical linear algebra fact, essentially equivalent to the Bruhat cell decomposition for $\text{GL}_d$, says that the answer is $d$ when $m=2$.  We obtain a precise answer for all values of $m$ and $d$.  Joint work with Federico Glaudo and Chayim Lowen.