The trunks of CLE(4) explorations

Abstract

The family of SLE4⟨μ⟩(−2) exploration processes with parameter μ∈R forms a natural class of conformally invariant ways for discovering the loops of a conformal loop ensemble CLE4. Such an exploration consists of one simple continuous path called the trunk of the exploration that discovers CLE4 loops along the way. The parameter μ appears in the Loewner chain description of the path that traces the trunk and all CLE4 loops encountered by the trunk in chronological order. These explorations can also be interpreted in terms of level lines of a Gaussian free field. It has been shown by Miller, Sheffield and Werner that the trunk of such an exploration is an SLE4(ρ,−2−ρ) process for some (unknown) value of ρ∈(−2,0). The main result of the present paper is to establish the relation between μ and ρ, more specifically to show that μ=−πcot(πρ/2).