UCLA

This thesis consists of six chapters based on papers on probabilities events in percolationtheory. • Chapter 1 is based on paper [GP24] written with Igor Pak. There we study colored percolation, a generalization of the classical percolation model. • Chapter 2 is based on paper [GZ24] written with Aleksandr Zimin. There we show that bond percolation does not simulate site percolation. • Chapter 3 is based on paper [G24]. There we study percolation inequalities and decision trees. • Chapter 4 is based on paper [GPZ24] with Igor Pak and Aleksandr Zimin. There we disprove the bunkbed conjecture. The appendix proves the bunkbed conjecture for the case of 2 transversal vertices and wasn’t published before. • Chapter 5 is based on an unpublished manuscript written with Aleksandr Zimin. There we prove multiple inequalities of the form similar to the Harris–Kleitman inequality. This work generalizes and supercedes a previous paper [G24b] by the author. • Chapter 6 contains an unpublished result related to the main body of work in the thesis