Invited Speaker
Mahshid Atapour
An animal in the simple cubic lattice is a finite connected subgraph of $\mathbb{Z}^3$. Let $a_n$ be the number (up to translation) of $n$-edge animals in $\mathbb{Z}^3$. In 1967, Klarner proved that $a_n$ grows exponentially. Let $e_n$ be the number (up to translation) of all $n$-edge linked clusters, i.e. subgraphs of $\mathbb{Z}^3$ in which the connected components (animals) are (topologically) non-splittable. In this presentation, I will briefly explain how it can be proved that $e_n$ also has a finite exponential growth rate. I will then mention some of the important consequences of this result in entanglement percolation.
This is a joint work with my postdoc supervisor N. Madras.