The trunks of CLE(4) explorations

Abstract

The family of ${SLE}_4^{⟨\mathit{\mu }⟩}(-2)$ exploration processes with parameter $\mathit{\mu }\in \mathbb{R}$ forms a natural class of conformally invariant ways for discovering the loops of a conformal loop ensemble ${CLE}_4$. Such an exploration consists of one simple continuous path called the trunk of the exploration that discovers ${CLE}_4$ loops along the way. The parameter μ appears in the Loewner chain description of the path that traces the trunk and all ${CLE}_4$ 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 ${SLE}_4(\mathit{\rho },-2-\mathit{\rho })$ process for some (unknown) value of $\mathit{\rho }\in (-2,0)$. The main result of the present paper is to establish the relation between μ and ρ, more specifically to show that $\mathit{\mu }=-\mathit{\pi }cot(\mathit{\pi }\mathit{\rho }/2)$.