Bond percolation on the square lattice - the Harris-Kesten Theorem

The thesis studies the percolation phase transition in a 2-dimensional lattice with random i.i.d. nearest neighbor connections. Percolation theory focuses on the size of the largest set of connected nodes. The main goal of the thesis is to give a proof of the Harris-Kesten Theorem, a fundamental result in percolation theory, that describes how the probability of an infinite cluster changes when the probability of connection between nearest neighbor nodes increases from zero to one. Such a probability - i.e. the probability that there exists an infinite set of connected nodes - shows a phase transition as it abruptly passes from zero to a strictly positive value when the probability of connection between nearest neighbor nodes crosses a critical value, called the percolation threshold.