Percolation of discrete GFF in dimension two I. Arm events in the random walk loop soup

In this work, which is the first part of a series of two papers, we study the random walk loop soup in dimension two. More specifically, we estimate the probability that two large connected components of loops come close to each other, in the subcritical and critical regimes. The associated four-arm event can be estimated in terms of exponents computed in the Brownian loop soup, relying on the connection between this continuous process and conformal loop ensembles (with parameter $\kappa \in (8/3,4]$). Along the way, we need to develop several useful tools for the loop soup, based on separation for random walks and surgery for loops, such as a "locality" property and quasi-multiplicativity. The results established here then play a key role in a second paper, in particular to study the connectivity properties of level sets in the random walk loop soup and in the discrete Gaussian free field.