Definition

2.5 Disc SLE

2.5.1 Definition

convergence of LERW to SLE2( see Chapter 4). It is not known now whether this is true for other κ.

Proof of Proposition 2.5.1. For fixed r ∈ (−∞, a), let ϕ^r_t, r ≤ t < a, be the solution of

∂tϕ^r_t(z) =ϕ^r_t(z)S_|t|(ϕ^r_t(z)/e^iξ(t)), ϕ^r_r(z) =z. (2.5.2) For r ≤ t < a, let K_t^r be the set of z ∈ A_|r| such that ϕ^r_s(z) blows up at some time s ∈ [r, t]. Then s 7→ K_r+s^r , 0 ≤ s < a−r, is a Loewner chain in A_|r| on C₀, and ϕ^r_t maps A_|r|\K_t^r conformally onto A_|t| with ϕ^r_t(C_|r|) = C_|t|. By the uniqueness of the solution of ODE, ift₁ ≤t₂ ≤t₃ < a, thenϕ^t_t²₃◦ϕ^t_t¹₂(z) = ϕ^t_t¹₃(z), forz ∈A_|t1|\K_t^t₃¹. For t <0, define R_t(z) = e^t/z. ThenR_t maps A_|t| conformally onto itself, and exchanges its two boundary components. Defineϕb^r_t =R_t◦ϕ^r_t◦R_r, andKb_t^r =R_r(K_t^r). Then Kb_t^r is a hull inA_|r|onC_|r|, andϕb^r_t mapsA_|r|\Kb_t^r conformally ontoA_|t|withϕb^r_t(C₀) =C₀. We also have ϕb^t_t²₃ ◦ϕb^t_t¹₂(z) = ϕb^t_t¹₃(z), for z ∈ A_|t1|\Kb_t^t₃¹, if t₁ ≤ t₂ ≤ t₃ < a. And ϕb^r_t satisfies

∂_tϕb^r_t(z) =ϕb^r_t(z)Sb_|t|(ϕb^r_t(z)/e^−iξ(t)), ϕb^r_r(z) =z, where bS_p(z) := 1−S_p(e^−p/z) forp > 0. A simple computation gives:

|Sb_p(z)| ≤8e^−p/|z|, if 4e^−p ≤ |z| ≤1.

We then have

|ϕb^r_t(z)−z| ≤8e^t, if r≤t <0, and 12e^t≤ |z| ≤1. (2.5.3)

Now let ψb_t^r be the inverse of ϕb^r_t. If t₁ ≤ t₂ ≤ t₃ < a, then ψb_t^t1₂ ◦ψb_t^t₃²(z) = ψb_t^t₃¹(z), for any z ∈A_|t3|. For fixed t ∈(−∞, a), {ψb^r_t :r ∈ (−∞, t]} is a family of uniformly bounded conformal maps on A_|t|, so is a normal family. This implies that we can find a sequence r_n → −∞ such that for any m ∈ N, {ψb_−m^rn } converges to some ψb_−m, uniformly on each compact subset of A_m. Let β_n = ψb^r_−mⁿ (C_m/2). Then β_n is a Jordan curve in A|rn|\Kb_−m^rn that separates the two boundary components. So 0 is contained in the Jordan domain determined by β_n. Note that {ψb_−m^rn } maps A_m/2 onto the domain bounded by βn and C0, whose modulus has to be m/2. So βn is

not contained in B(0;e^−m/2). This implies that the diameter of βn is not less than e^−m/2. Soψb_−mcan’t be a constant. By Lemma 2.5.1,ψb_−m mapsA_m conformally onto some domain D−m, and ψb_−m^rn (Am)→D−m. Sinceψb^r_−mⁿ (Am) = A_|rn|\Kb_−m^rn ⊂D\ {0}, D_−m ⊂ D\ {0}. Since M(A_|rn|\Kb_−m^rn ) = m, there is some a_m ∈ (0,1) such that B(0;e^rn)∪ Kb_−m^rn ⊂ B(0;e^−am) for all rn. So Aam contains no boundary points of A_|rn| \ Kb_−m^rn = ψb^r_−mⁿ (A_m). Since these domains converge to D_−m as n → ∞, so Aam contains no boundary points of D−m, which means that either Aam ⊂ D−m or A_am ∩D_−m = ∅. Now let γ_n = ψb^r_−mⁿ (C_am/2). For the same reason as β_n, we have γn 6⊂ B(0;e^−am/2). So there is zn ∈ C_am/2 such that |ψb_−m^rn (zn)| ≥ e^−am/2. Let z0 be any subsequential limit of {z_n}, then z₀ ∈ C_am/2 ⊂ A_m and |ψb_−m(z₀)| ≥ e^−am/2, so ψb−m(z0) ∈ Aam. Thus D−m ∩Aam 6= ∅, and so Aam ⊂ D−m. Hence D−m has one boundary component C₀. Using similar arguments, we have ψb_t(C₀) =C₀.

If rn < −m1 < −m2, then ψb^r_−mⁿ ₁ ◦ψb_−m^−m1₂ = ψb^r_−mⁿ ₂, which implies ψb−m1 ◦ψb_−m^−m₂¹ = ψb_−m2. For t ∈ (−∞,0), choose m ∈ N with −m ≤ t, define ψb_t = ψb_−m ◦ψb_t^−m and D_t=ψb_t(A_|t|). It is easy to check that the definition ofψb_tis independent of the choice ofm, and the following properties hold. For allt∈(−∞,0),D_tis a doubly connected subdomain of D\ {0} that has one boundary component C₀, and ψb_t(C₀) = C₀; ψb_t^rn converges to ψb_t, uniformly on each compact subset of A_|t|. If r < t < 0, then ψb_t =ψb_r◦ψb_t^r; D_t$D_r, and D_r\D_t =ψb_r(Kb_t^r).

Let ϕb_t on D_t be the inverse of ψb_t. By Lemma 2.5.1, ϕb^r_tⁿ converges to ϕb_t as n → ∞, uniformly on each compact subset of D_t. Thus from formula (2.5.3), we have |ϕb_t(z)−z| ≤ 8e^t, if 12e^t ≤ |z| < 1. It follows that lim_t→−∞ϕb_t(z) = z, for any z ∈ D\ {0}. We also have ϕb_t(z) = ϕb^−m_t ◦ϕb_−m(z), if −m ≤ t < 0 and z ∈ D_t. Let ϕ_t=R_t◦ϕb_t onD_t. Then ϕ_t maps D_t conformally onto A_|t|, takes C₀ toC_|t|, and

t→−∞lim e^t/ϕt(z) = lim

t→−∞ϕbt(z) =z, for any z ∈D\ {0}.

If −m ≤t, then ϕ_t(z) = ϕ^−m_t ◦R_−m◦ϕb_−m(z),∀z ∈D_t. By formula (2.5.2), we have

∂_tϕ_t(z) = ϕ_t(z)S_|t|(ϕ_t(z)/e^iξ(t)), −m≤t <0.

Since we may choosem ∈N arbitrarily, formula (2.5.1) holds.

Let K_t = D\D_t. Since D_t is a doubly connected subdomain of D\ {0} with a boundary component C0, Kt is an interior hull in D and 0∈Kt. The fact M(Dt) =

|t| → ∞ as t → −∞ implies that the diameter of K_t tends to 0 as t → −∞. So {0}=∩Kt. If t1 < t2 < a, then Kt1 $Kt2, asDt1 %Dt2. Fix any r∈ (−∞, a). For t ∈ [r, a), K_t\K_r = D_r \D_t = ψb_r(Kb_t^r). From conformal invariance, s → ψb_r(Kb_r+s^r ), 0≤s < a−r, is a Loewner chain in Dr on∂Kr. Thust 7→Kt is an interior Loewner chain in D started from 0.

For any t∈(−∞, a) and ε∈(0, a−t), we have

ϕ_t(K_t+ε\K_t) =ϕ_t(ψb_t(Kb_t+ε^t )) =R_t◦ϕb_t◦ψb_t◦R_t(K_t+ε^t ) =K_t+ε^t .

Since (K_t+ε^t ,0 ≤ ε < a− t) is a family of standard modulus p annulus LE hulls driven by ξ(t+·), so we have {ξ(t)} =∩_ε>0K_t+ε^t , from which follows that {ξ(t)} =

∩ε>0ϕt(Kt+ε\Kt).

Suppose t 7→ K_t^∗, −∞ < t < a, is an interior Loewner chain in D started from 0, and ϕ^∗_t, −∞ < t < a, is a family of maps such that for each t, ϕ^∗_t maps D\K_t^∗ conformally onto A_|t| and formula (2.5.1) holds with ϕ_t replaced by ϕ^∗_t. By the uniqueness of the solution of ODE, we have ϕ^∗_t =ϕ^r_t ◦ϕ^∗_r, if r≤t < 0. So Rt◦ϕ^∗_t =

b

ϕ^r_t ◦R_r◦ϕ^∗_r. Now choose r = r_n and let n → ∞. Since R_rn ◦ϕ^∗_rn → id by formula (2.5.1), and ϕb^r_tⁿ → ϕbt, so Rt◦ϕ^∗_t = ϕbt, from which follows that ϕ^∗_t = Rt◦ϕbt = ϕt

and K_t^∗ =K_t. 2

Proposition 2.5.2 Suppose t 7→K_t, −∞ < t < a, is an interior Loewner chain in D started from 0 such that M(D\K_t) =|t| for each t. Then (K_t, −∞< t < a, is a family of standard disc LE interior hulls. And if we don’t assume thatM(D\K_t) = |t|

for each t, then after a time-change, we can make (K_t)to be a family of standard disc interior LE hulls.

Proof. We only need to consider the case thatM(D\K_t) = |t|, for all−∞< t < a.

For each t ∈ (−∞, a), choose g_t^∗ which maps D\K_t conformally onto A_|t| so that

g_t^∗(1) = 1. Let ϕ^∗_t =Rt◦g_t^∗, where Rt(z) =e^t/z. Then ϕ^∗_t maps D\Kt conformally ontoA_|t|withϕ^∗_t(C₀) =C_|t|andϕ^∗_t(1) =e^t. For anyr ≤t <0, letK_r,t^∗ =ϕ^∗_r(K_t\K_r).

Then for fixed r < a, s 7→ K_r,r+s^∗ , 0 ≤ s < a−r), is a Loewner chain in A_|r| onC0. Nowϕ^∗_t◦(ϕ^∗_r)⁻¹mapsA_|r|\K_r,t^∗ conformally ontoA_|t|, and satisfiesϕ^∗_t◦(ϕ^∗_r)⁻¹(e^r) = e^t. From the proof of Proposition 2.4.1, there exists some continuousξ_r^∗ : [r,0)→Rsuch that for r≤t <0,

∂tϕ^∗_t ◦(ϕ^∗_r)⁻¹(w) =ϕ^∗_t ◦(ϕ^∗_r)⁻¹(w)[S|t|(ϕ^∗_t ◦(ϕ^∗_r)⁻¹(w)/e^iξr∗(t))−iImS|t|(e^t/e^iξ∗r(t))].

It then follows that

∂_tϕ^∗_t(z) =ϕ^∗_t(z)[S_|t|(ϕ^∗_t(z)/e^iξ∗r(t))−iImS_|t|(e^t/e^iξ∗r(t))], r≤t <0.

Soe^iξ∗r1(t)=e^iξ∗r2(t)ifr₁, r₂ ≤t. We then can construct a continuousξ^∗ : (−∞, a)→R, such that

∂tϕ^∗_t(z) = ϕ^∗_t(z)[S_|t|(ϕ^∗_t(z)/e^iξ∗(t))−iImS_|t|(e^t/e^iξ∗(t))], − ∞ ≤t < a.

Consequently,

∂tg^∗_t(z) = g_t^∗(z)[Sb|t|(ϕ^∗_t(z)/e^−iξ∗(t))−iImbS|t|(e^iξ∗(t))], − ∞ ≤t < a.

Since |Sb_|t|(z)| ≤ 8e^t when 4e^t ≤ |z| ≤ 1, |ImbS_|t|(e^iξ∗(t))| decays exponentially as t→ −∞. Let θ(t) =R_t

−∞ImSb_|s|(e^iξ∗(s))ds, g_t(z) = e^iθ(t)g_t^∗(z), andξ(t) =ξ^∗(t)−θ(t).

Then g_t maps D\K_t conformally onto A_|t| with g_t(C₀) =C₀, and

∂_tlng_t(z) = ∂_tlng^∗_t(z) +iθ⁰(t) =Sb_|t|(g^∗_t/e^−iξ∗(t)) = Sb_|t|(ϕ_t/e^−iξ(t)).

Thus∂_tg_t(z) =g_t(z)Sb_|t|(g_t(z)/e^−iξ(t)). From the estimation ofSb_|t|, we have

|g_t(z)−g_r(z)| ≤8e^t, if 12e^t≤ |g_r(z)| ≤1, and r ≤t <0.

Since Kt contains 0 and M(D\Kt) = |t|, the diameter of Kt tends to zero as t→ −∞. LetD_t=D\K_t. Then for any sequencet_n → −∞, we haveD_tn →D\ {0}.

Sincegtn is uniformly bounded, there is a subsequence that converges to some function g onD\ {0}uniformly on each compact subset of D\ {0}. By checking the image of C1 under gtn similarly as in the proof of Proposition 2.5.1, we see that g cannot be constant. So by Lemma 2.5.1,gmapsD\{0}conformally onto some domainD₀ which is a subsequential limit ofA_|tn|=gtn(Dtn). Sincetn→ −∞,D0 has to beD\ {0}and so g(z) =χz for some χ ∈C₀. Now this χ may depend on the subsequence of {t_n}.

But we always have limt→−∞|gt(z)| =|z| for anyz ∈ D\ {0}. Now fix z ∈ D\ {0}, there is s(z)<0 such that when r ≤t < s(z), we have 12e^t≤ |g_r(z)| ≤1. Therefore

|gt(z)−gr(z)| ≤8e^tforr ≤t < s(z). Thus limt→−∞gt(z) exists for every z ∈D\ {0}.

Since we have a sequence t_n → −∞ such that {g_tn} converges pointwise to z 7→ χz on D\ {0} for some χ ∈C0, so limt→−∞ϕt(z) = χz, for all z ∈D\ {0}. Finally, let ϕ_t(z) =R_t◦g_t(z/χ). Then ϕ_t maps D\K_t conformally onto A_|t|, takes C₀ to C_|t|, and satisfies (disc LC1). 2

We still use B(t) to denote a standard Brownian motion. Let xbe some uniform random point on [0,2π), independent of B(t). For κ > 0 and −∞ < t < 0, write ξ_κ(t) =x+√

κB(|t|). The process (e^iξ(t)) is determined by the following properties: for any fixedr <0, (e^iξ(t)/e^iξ(r), r ≤t <0) has the same law as (e^{iB(κ(t−r))}, r ≤t <0) and is independent frome^iξ(r). IfK_tandϕ_t,−∞< t <0, are the standard disc interior LE hulls and maps, respectively, driven by ξ_κ, then we call the law of (K_t) the standard disc SLE_κ. Suppose D is a simply connected domain and p∈D. Let W map (D; 0) conformally onto (D;p) and W⁰(0) >0. Then the disc SLE_κ(D;p→ ∂D) is defined as the image of the standard disc SLE_κ under the mapW. The existence of standard annulus SLE_κ trace then implies the a.s. existence of standard disc SLE_κ trace, which is a curve β : [−∞,0) → D such that γ(−∞) = 0, and for each t ∈ (−∞,0), K_t is the complement of the unbounded component of C\γ[−∞, t]. If κ ≤4, the trace is a simple curve; otherwise, it is not simple.