The algebraic method in tree percolation

TL;DR

This paper uses algebraic methods to analyze percolation on complete k-ary trees, providing explicit formulas for algebraic invariants that describe percolation behavior and its asymptotics.

Contribution

It introduces a novel algebraic framework for studying percolation on trees, including explicit recursive formulas for Betti numbers and Hilbert series of associated monomial ideals.

Findings

Explicit recursive formulas for Betti numbers of the ideals

Analysis of bounds and asymptotic behavior of percolation

Application of algebraic techniques to percolation models

Abstract

We apply the methods of algebraic reliability to the study of percolation on trees. To a complete $k$-ary tree $T_{k,n}$ of depth $n$ we assign a monomial ideal $I_{k,n}$ on $\sum_{i=1}^n k^i$ variables and $k^n$ minimal monomial generators. We give explicit recursive formulae for the Betti numbers of $I_{k,n}$ and their Hilbert series, which allow us to study explicitly percolation on $T_{k,n}$. We study bounds on this percolation and study its asymptotical behavior with the mentioned commutative algebra techniques.