We consider problems in two areas at the intersection of discrete probability and statistical physics: the chemical distance in planar percolation models and phase transitions of ballistic annihilation. Although the two models are quite different, the techniques and results involved are often of combinatorial interest. We first discuss the chemical distance in planar Bernoulli percolation and random cluster models in the critical regime. By extending Damron--Hanson--Sosoe’s results, we give upper bounds on the expected number of edges to cross a square box from left to right, as well as from the origin to the boundary, in critical percolation clusters. Along the way, we also extend two classical results to new settings. In the second part of the thesis, we consider a one-dimensional annihilating particle system called ballistic annihilation. We introduce two variants of the symmetric three-velocity ballistic annihilation, for which we prove the existence of phase transitions and compute the critical density using an exactly solvable approach pioneered by Haslegrave--Sidoravicius--Tournier. These variants have considerably more complicated dynamics and require tools that observe broader symmetry.