SLE background and prior convergence results

We now give a brief definition of (chordal) SLE() for>0. See also the surveys [W], [KN], [L4], [Ca] or [L3]. The discussion below along with further discussion of the special properties of SLE(4) appears in another paper by the current authors [SS]. That paper shows that SLE(4) is the scaling limit of a random interface called the harmonic explorer (designed in part to be a toy model for the DGFF contour line addressed here).

LetT >0. Suppose thatγ: [0, T]!His a continuous simple path in the closed up-per half-plane H which satisfiesγ[0, T]∩R={γ(0)}={0}. For every t∈[0, T], there is a unique conformal homeomorphismg_t:H\γ[0, t] which satisfies the so-calledhydrodynamic normalization at infinity

zlim!∞(gt(z)−z) = 0.

The limit

cap_∞(γ[0, t]) := lim

z!∞

z(gt(z)−z) 2

is real and monotone increasing int. It is called the (half-plane)capacity ofγ[0, t] from

∞, or justcapacity, for short. Since cap_∞(γ[0, t]) is also continuous int, it is natural to reparameterizeγso that cap_∞(γ[0, t])=t. Loewner’s theorem states that in this case the mapsgt satisfy his differential equation

∂tgt(z) = 2 gt(z)−Wt

, g0(z) =z, (1.3)

where Wt=gt(γ(t)). (Since γ(t) is not in the domain of definition ofgt, the expression gt(γ(t)) should be interpreted as a limit ofgt(z) asz!γ(t) insideH\γ[0, t]. This limit does exist.) The functiont7!Wt is continuous int, and is called the driving parameter forγ.

One may also try to reverse the above procedure. Consider the Loewner evolution defined by the ordinary differential equation (ODE) (1.3), where Wt is a continuous, real-valued function. For a fixedz, the evolution defines gt(z) as long as |gt(z)−Wt| is bounded away from zero. For z∈Hletτz be the first time t>0 in whichgt(z) andWt

collide, or set τz=∞ if they never collide. Thengt(z) is well defined on {z∈H:τz>t}.

The setKt:={z∈H:τz6t} is sometimes called theevolving hull of the evolution. In the case discussed above where the evolution is generated by a simple pathγ parameterized by capacity and satisfyingγ(t)∈Hfort>0, we haveK_t=γ[0, t].

The path of the evolution is defined asγ(t)=lim_z!Wtg⁻¹_t (z), wherez tends toW_t from within the upper half-plane H, provided that the limit exists and is continuous.

However, this is not always the case. The process (chordal) SLE() in the upper half-plane, beginning at 0 and ending at ∞, is the path γ(t) when W_t is B_t√

, where B_t=B(t) is a standard 1-dimensional Brownian motion. (Standard meansB(0)=0 and E[B(t)²]=t, t>0. Since (B_t√

k:t>0) has the same distribution as (B_t:t>0), taking W_t=B_t is equivalent.) In this case, almost surely γ(t) does exist and is a continuous path. See [RS] (6=8) and [LSW4] (=8).

We now define the processes SLE(;%₁, %₂). Given a Loewner evolution defined by a continuous Wt, we let xt and yt be defined by xt:=sup{gt(x):x<0 andx /∈Kt} and yt:=inf{gt(x):x>0 andx /∈Kt}. When the Loewner evolution is generated by a simple pathγ(t) satisfyingγ(t)∈Hfort>0, these pointsxtandytcan be thought of as the two images of 0 undergt. Note that, by (1.3), noting that existence and uniqueness of solutions to this SDE (at least from the initial timeruntil the firsts>rfor which eitherx_s=W_sorW_s=y_s) follow easily from standard results in [RY]. (The x_t and y_t are called force points because they apply a “force”

affecting the drift of the processW_tby an amount inversely proportional to their distance fromW_t.)

Some subtlety is involved in extending the definition of SLE(;%1, %2) beyond times whenWthits the force points, and in starting the process from the natural initial values x0=W0=y0=0. This is closely related to the issues which come up when defining the Bessel processes of dimension less than 2 and will be discussed in more detail in §4.

Although many random self-avoiding lattice paths from the statistical physics litera-ture are conjeclitera-tured to have forms of SLE as scaling limits, rigorous proofs have thus far appeared only for a few cases: site percolation cluster boundaries on the hexagonal lattice (SLE(6), [Sm]; see also [CN]), branches (loop-erased random walk) and outer boundaries (random Peano curves) of uniform spanning trees (forms of SLE(2) and SLE(8), respec-tively, [LSW4]), the harmonic explorer (SLE(4), [SS]), and boundaries of simple random walks (forms of SLE ⁸₃

, [LSW3]).

In the latter case, conformal invariance properties follow almost immediately from the conformal invariance of 2-dimensional Brownian motion. In each of the other cases listed above, the initial step of the proof is to show that a certain function of the partially generated pathsγ([0, t]), which is a martingale intwhenγis SLE() for the appropriate , has a discrete analog which is (approximately or exactly) a martingale for the dis-crete paths and is approximately equivalent to the continuous version in the fine mesh limit. For loop-erased random walk, harmonic explorer, and uniform spanning tree Peano curves, this initial step is the easy part of the argument; it follows almost immediately from the fact that simple random walk converges to Brownian motion. The analogous step for site percolation on the hexagonal lattice, as given by [Sm], is an ingenious but nonetheless short and simple argument.

By contrast, the analogous step in this paper (which requires the proof of the height gap lemma, as given in §3) is quite involved; it is the most technically challenging part of the current work and includes many new techniques and lemmas about the geometry of DGFF contours that we hope are interesting for their own sake.

Another way in which the DGFF differs from percolation, the harmonic explorer, and the uniform spanning tree is that it has a natural continuum analog (the GFF) which can be easily rigorously constructed without any reference to SLE, and which is itself (like Brownian motion) an object of great significance. It becomes natural to ask whether the DGFF results enable us to define the “contour lines” of the continuum GFF in a canonical way; we plan to answer this question (affirmatively) in a subsequent work (see§1.7).

A final difference is that, for the DGFF, there is a continuum of choices for left and right boundary conditions (a and b) which are equally natural a priori, so we are led to consider a family of paths SLE(4;a/λ−1, b/λ−1) instead of simply SLE(4). (The case a=b=0 is particularly natural; see Figure 1.4.) In these processes, the driving

parameters Wt are generally no longer Brownian motions (rather, they are continuous semimartingales with constant quadratic variation and a drift term that can become singular on a fractal set). Proving driving parameter convergence to these processes requires some rather general convergence infrastructure (§4.4), which we hope will be useful in other settings as well.