Up-to-constants estimates on four-arm events for simple conformal loop ensemble

We prove up-to-constants estimates for a general class of four-arm events in simple conformal loop ensembles, i.e. CLE$_\kappa$ for $\kappa\in (8/3,4]$. The four-arm events that we consider can be created by either one or two loops, with no constraint on the topology of the crossings. Our result is a key input in our series of works arxiv:2409.16230 and arxiv:2409.16273 on percolation of the two-sided level sets in the discrete Gaussian free field (and level sets in the occupation field of the random walk loop soup). In order to get rid of all constraints on the topology of the crossings, we rely on the Brownian loop-soup representation of simple CLE [Ann. Math. 176 (2012) 1827-1917], and a "cluster version" of a separation lemma for the Brownian loop soup. As a corollary, we also obtain up-to-constants estimates for a general version of four-arm events for SLE$_\kappa$ for $\kappa\in (8/3,4]$. This fixes (in the case of four arms and $\kappa\in(8/3,4]$) an essential gap in [Ann. Probab. 46 (2018) 2863-2907] and improves some estimates therein.