arXiv subjects
Publications and source records attributed to Matheus Secco.
In this paper we construct a tropical moduli space parametrizing roots of divisors on tropical curves. We study the relation between this space and the skeleton of Jarvis moduli space of nets of limit roots on stable curves. We show that the combinatorics of the moduli space of tropical roots is governed by the poset of flows, a poset parametrizing certain flows on graphs.
Let the random variable $X\, :=\, e(\mathcal{H}[B])$ count the number of edges of a hypergraph $\mathcal{H}$ induced by a random $m$-element subset $B$ of its vertex set. Focussing on the case that the degrees of vertices in $\mathcal{H}$ vary significantly we prove bounds on the probability that $X$ is far from its mean. It is possible to apply these results to discrete structures such as the set of $k$-term arithmetic progressions in the $\{1,\dots, N\}$. Furthermore, our main theorem allows us to deduce results for the case $B\sim B_p$ is generated by including each vertex independently with probability $p$. In this setting our result on arithmetic progressions extends a result of Bhattacharya, Ganguly, Shao and Zhao \cite{BGSZ}. We also mention connections to related central limit theorems.
Let the random variable $X\, :=\, e(\mathcal{H}[B])$ count the number of edges of a hypergraph $\mathcal{H}$ induced by a random $m$ element subset $B$ of its vertex set. Focussing on the case that $\mathcal{H}$ satisfies some regularity condition we prove bounds on the probability that $X$ is far from its mean. It is possible to apply these results to discrete structures such as the set of $k$-term arithmetic progressions in the cyclic group $\mathbb{Z}_N$. Furthermore, we show that our main theorem is essentially best possible and we deduce results for the case $B\sim B_p$ is generated by including each vertex independently with probability $p$.