Level lines of Gaussian Free Field I: Zero-boundary GFF

Abstract

We study level lines of Gaussian Free Field h emanating from boundary points. The article has two parts. In the first part, we show that the level lines are random continuous curves which are variants of SLE 4 path. We show that the level lines with different heights satisfy the same monotonicity behavior as the level lines of smooth functions. We prove that the time-reversal of the level line coincides with the level line of −h. This implies that the time-reversal of SLE 4 (ρ) process is still an SLE 4 (ρ) process. We prove that the level lines satisfy "target-independent" property. In the second part, we discuss the relation between Gaussian Free Field and Conformal Loop Ensemble (CLE). A CLE is a collection of disjoint SLE-loops. Since the level lines of GFF are SLE 4 paths, the collection of level loops of GFF corresponds to CLE 4 . We study the coupling between GFF and CLE 4 with time parameter which sheds lights on the conformal invariant metric on CLE 4 . Primary 60G60; secondary 60J67.

Two-valued local sets of the 2D continuum Gaussian free field: connectivity, labels, and induced metrics

“…1 Here, and elsewhere this means that the boundary conditions are given by a piecewise constant function that changes only finitely many times 6 For convenience we also use the notion of a (−a, −a + 2λ)-level line of Γ + u: it is a generalised level line of Γ + a − λ + u and has boundary conditions −a, −a + 2λ with respect to the field Γ + u. Moreover, it is known that when u = 0 this level line has the law of a SLE 4 (−a/λ, a/λ − 2) process, see Theorem 1.1.1 of [WW16].…”

“…One of the simplest families of BTLS are the generalised level lines, first described in [SS13]. We recall here some of their properties, see [WW16,ASW17] for a more thorough treatment of the subject. To simplify our statements take D := H. Furthermore, let u be a harmonic function in D. We say that (η(t)) t≥0 , a curve parametrised by half-plane capacity, is the generalised level line for the GFF Γ + u in D up to a stopping time τ if for all t ≥ 0: ( * ): The set η([0, t ∧ τ ]) is a BTLS of the GFF Γ, with harmonic function h t := h η([0,t∧τ ]) satisfying the following properties: h t + u is a harmonic function in D\η([0, min(t, τ )]) with boundary values −λ on the left-hand side of η, +λ on the right side of η, and with the same boundary values as u on ∂D.…”

“…In this paper, we are just going to work with piecewise constant boundary conditions 1 . A careful treatment of level lines in this regime is done in [WW16]. We are now going to state Theorem 1.1.1 of [WW16].…”

“…A careful treatment of level lines in this regime is done in [WW16]. We are now going to state Theorem 1.1.1 of [WW16]. Let u be a bounded harmonic function with piecewise constant boundary data such that u(0 − ) < λ and u(0 + ) > −λ.…”

“…Thus, it suffices to prove that the loops of the first type are connected via a finite path only using loops of the first type, i.e., loops that will also belong to G s . However, this fact follows from the fact that the level line is a.s. non-self-crossing, continuous up to its target point, and attains its target point almost surely ( [WW16]). Now, notice that the rest follows inductively: indeed, any loop of A −a,−a+2λ that was not present at A n , but is present at A n+1 is side-connected to a loop that appears at level A n .…”