Large deviations of radial SLE$_{\infty}$

We derive the large deviation principle for radial SLE$_\kappa$ on the unit disk as $\kappa \rightarrow \infty$. Restricting to the time interval $[0,1]$, the good rate function is finite only on a certain family of Loewner chains driven by absolutely continuous probability measures $\{\phi_t^2 (\zeta)\, d\zeta\}_{t \in [0,1]}$ on the unit circle and equals $\int_0^1 \int_{S^1} |\phi_t'|^2/2\,d\zeta \,dt$. Our proof relies on the large deviation principle for the long-time average of the Brownian occupation measure by Donsker and Varadhan.


Introduction
The Schramm-Loewner evolution is a one parameter family of random fractal curves (denoted as SLE κ with parameter κ > 0). It was introduced by Oded Schramm [Sch00] and has been a central topic in the two dimensional random conformal geometry. A version of such curves starting from a fixed boundary point to a fixed interior point on some two-dimensional simply connected domain D are called radial SLEs. Let us recall briefly the definition. The radial SLE κ on the unit disk D = {ζ ∈ C : |ζ| = 1} targeted at 0 is the random curve associated to the radial Loewner chain with driving function t → ζ t given by a Brownian motion on the unit circle S 1 = {ζ ∈ C : |ζ| = 1} with variance κ, i.e.
with the initial condition f 0 (z) = g 0 (z) = z. For a given t > 0, f t is a conformal map from D onto a simply connected domain D t ⊂ D (and s → g s (z) is a well-defined solution of (1.2) up to time t if and only if z ∈ D t ) such that f t (0) = 0 and f t (0) = e −t . The family of conformal maps {f t } t≥0 is called the capacity parametrized radial Loewner chain or normalized subordination chain driven by t → ζ t and the SLE κ is a curve t → γ t and can be defined as lim r→1− f t (rζ t ) = γ t , see [RS05]. In particular, the curve starts at γ 0 = 1. The radial SLE κ on an arbitrary simply connected domain D is defined via the unique conformal map from D to D respecting the starting and target points. It is well-known that SLE κ exhibits phase transitions as κ varies. Larger values of κ correspond in some sense to "wilder" SLE κ curves; in the κ ≥ 8 regime the curve is space-filling.
In this work, we study the κ → ∞ asymptotic behavior of radial SLE. To simplify notation we consider SLE κ run on the time interval [0, 1] throughout the paper. Our results are easily generalized to arbitrary bounded time intervals. We let {·} denote the family {·} t∈ [0,1] .
Our first result (Proposition 1.1) roughly says that as κ → ∞, the time-evolution of the SLE κ hulls, i.e. t → γ [0,t] , converges to a deterministic limit t → D\e −t D. We argue heuristically as follows. We view the time-dependent vector field {−z(z + ζ t )/(z − ζ t )} which generates the flow {g t } as { S 1 −z(z + ζ)/(z − ζ)δ B κ t (ζ)}, where δ B κ t is the Dirac mass at B κ t . During a short interval where the flow is well-defined for the point z, we have g t (z) ≈ g t+∆t (z) and hence where L κ t is the occupation measure (or local time) on S 1 of B κ up to time t. We show that as κ → ∞, the driving function oscillates so quickly that its local time in [t, t + ∆t] is almost uniform on S 1 , so in the limit we get a measure-driven Loewner chain with driving measure uniform on S 1 . That is, where dζ denotes the Lebesgue measure. This implies ∂ t g t (z) = g t (z), that is, g t (z) = e t z or equivalently f t (z) = e −t z. See Section 2 for more details on the measure-driven Loewner chain. We show in Section 3.2: converges to {z → e −t z} almost surely, with respect to the uniform Carathéodory topology.
We shall mention that Loewner chains are also used in the study of the Hastings-Levitov model of randomly aggregating particles and similar small-particle limits have been studied, see [JVST12] and references therein.
The heuristic argument above suggests that the large deviations of SLE κ boil down to the large deviations of the Brownian occupation measure, which we now describe.
For any metric space X, let M 1 (X) denote the set of Borel probability measures equipped with the Prokhorov topology. Let The condition imposed here means we can write ρ ∈ N as a disintegration {ρ t } over the time interval [0, 1] (with ρ t ∈ M 1 (S 1 ) for a.e. t); see (2.1). We identify ρ and the time-indexed family {ρ t }. The second result we show is: Theorem 1.2. The process of measures {δ B κ t } ∈ N satisfies the large deviation principle with good rate function E(ρ) : where {ρ t } is the driving measure whose Loewner transform is {K t }.
Let us conclude the introduction with two comments.
The study of large deviations of SLE, while of inherent interest, is also motivated by problems from complex analysis and geometric function theory. In particular, in a forthcoming work [VW], Viklund  It is also natural to consider the large deviations of chordal SLE ∞ (say, in H targeted at ∞). However, in contrast with the radial case, the family indexed by κ of random measures {δ Wκt } on R × [0, 1] is not tight and the corresponding Loewner flow converges to the identity map for any fixed time t. To obtain a non-trivial limit, one needs to renormalize appropriately (see e.g., Beffara's thesis [Bef03, Sec.5.2] for a non-conformal normalization) and consider generalized Stieltjes transformation of measures for the large deviations. Therefore, for simplicity we choose to study the radial case and suggest the large deviations of chordal SLE ∞ as an interesting question. We will show a simulation of a large κ chordal SLE and discuss some other questions at the end of the paper.
The paper is organized as follows: In Section 2, we explain the measure-driven radial Loewner evolution. In Section 3 we prove the main results of our paper. In Section 4 we present some comments, observations and questions.

Measure-driven radial Loewner transform
In this section we collect some known facts on the measure-driven Loewner transform that are essential to our proofs. Recall that From the disintegration theorem, for each measure ρ ∈ N there exists a Borel measurable map t → ρ t (sending [0, 1] → M 1 (S 1 )) such that for every measurable function ϕ : The Loewner chain driven by a measure ρ ∈ N is defined similarly to (1.2). For z ∈ D, consider the Loewner ODE with the initial condition g 0 (z) = z. Let T z be the supremum of all t such that the solution is well-defined up to time t with g t (z) ∈ D, and set D t := {z ∈ D : T z > t}. We define the hull K t := D \ D t associated with the Loewner chain; this is a compact subset of D with simply connected complement. Note that when κ ≥ 8, the family {γ[0, t]} of radial SLE κ is exactly the family of hulls {K t } driven by the measure {δ B κ t }. The function g t defined above is the unique conformal map of D t onto D such that g t (0) = 0 and g t (0) > 0; moreover g t (0) = e t (i.e. D t has conformal radius e −t seen from 0). Indeed (see e.g. [Law05,Thm. 4.13]), If g t is the solution of a Loewner ODE then its inverse f t = g −1 t satisfies the Loewner PDE: In fact, L is a bijection; this is an easy consequence of the following theorem.
• and there is a (t-a.e. unique) function h(z, t) that is analytic in z, measurable in t with h(0, t) = 1 and Re h(z, t) > 0, so that for t-a.e. we have From the Herglotz representation of h(·, t), there exists a unique ρ t ∈ M 1 (S 1 ) such that Therefore {f t } satisfies the Loewner PDE driven by the (a.e. uniquely determined) measurable function t → ρ t . This concludes the proof that L is a bijection.
We now equip S with a topology. View S as the set of normalized chains of subordinations {f t } on [0, 1], and change notation by writing f (z, t) = f t (z). We endow S with the topology of uniform convergence of f on compact subsets of D × [0, 1]. (Equivalently, if we view S as the set of processes of hulls {K t }, this is the topology of uniform Carathéodory convergence.) The continuity of L has been, e.g., derived in [MS16b, Proposition 6.1] (see also [JVST12]).

Proofs of the main results
In this section, we study {δ B κ t } ∈ N ; this is a random probability measure on S 1 × [0, 1] having time marginal Leb [0,1] .
In Section 3.1 we approximate N by spaces of time-averaged measures. In Section 3.2 we verify that {δ B κ t } ∈ N converges almost surely as κ → ∞ to the uniform measure on S 1 ×[0, 1]; this yields Proposition 1.1. In Section 3.3, we review the large deviation principle for the circular Brownian motion occupation measure, which is a special case of seminal work of Donsker and Varadhan [DV75]. Finally, in Section 3.4 we prove Theorem 1.2, the large deviation principle for {δ B κ t } ∈ N . In this section, we discuss approximations of measures ρ ∈ N and express N as a projective limit of such approximations. We emphasize that the results of this section are wholly deterministic.

Time-discretized approximations of measures
For n ≥ 0, let I n := {0, 1, 2, · · · , 2 n − 1} be an index set, and define We note that Y n is endowed with the product topology. For each i ∈ I n we define a function P i n : N → M 1 (S 1 ) via where here {ρ t } is a disintegration of ρ with respect to t, as in (2.1). We define also the map P n : N → Y n via P n = (P i n ) i∈In . That is, P n averages ρ along each 2 −n -time interval, and outputs the 2 n -tuple of these 2 n time-averages.
We consider Y n to be the space of time-discretized approximations of N , in the following sense. Define a map F n : Y n → N via Then one can view F n (P n (ρ)) as a "level n approximation" of ρ, in the sense that F n (P n (ρ)) converges in the Prokhorov topology to ρ as n → ∞.
We have provided a way of projecting an element of N to the space of level n approximations Y n . Now we write down a map P n,n+1 : Y n+1 → Y n which takes in a finer approximation and outputs a coarser approximation: That is, we average pairs of components of Y n+1 . It is clear that Finally, as n → ∞, the space of approximations Y n converges in some sense to N . We rigorously state this in Lemma 3.1.
That is, as topological spaces, N is the projective (inverse) limit of Y n as n → ∞.
Remark. We explain the intuitive meaning of the above statement. The convergence of P n (ρ j ) for the functions f which are "piecewise constant in time for each time interval (i/2 n , (i + 1)/2 n )". For each fixed n this is a coarser topology than that of N , but taking n → ∞, these "piecewise constant in time" functions can approximate arbitrarily closely any element in C(S 1 × [0, 1]). Lemma 3.1 is a formal way of saying that the n → ∞ topology agrees with that of N .
Proof. Let Y = lim ← − Y n ; this is the subset of ∞ j=0 Y j comprising elements (y 0 , y 1 , . . . ) such that P n,n+1 (y n+1 ) = y n for all n ≥ 0. The topology on Y is inherited from ∞ j=0 Y j . We will construct a homeomorphism from N to Y.
We now show that P is continuous, that P is a bijection, and that P −1 is continuous, then we are done.
Showing that P is continuous. Since the topology on Y is inherited from the product topology on ∞ n=0 Y j , it suffices to show that the map P : N → ∞ n=0 Y n is continuous, i.e. P n : N → Y n is continuous for each n. But this is clear: if two measures in N are close in the Prokhorov topology, then so is the time-average of these measures on a time interval.
Now we show that P is a bijection. By Clearly T y maps nonnegative functions to nonnegative reals, so the Riesz-Markov-Kakutani representation theorem tells us there is a unique 1 Borel measure ρ on S 1 × [0, 1] such that ρ(f ) = T y (f ) for all f ∈ C 1 (S 1 × [0, 1]); it is easy to check that ρ ∈ N . Thus, for each y ∈ Y the equation P (ρ) = y has a unique solution in N , so P is a bijection.
Showing that P −1 : Y → N is continuous. This is equivalent to the statement that for any sequence (y k ) k≥0 ⊂ Y converging to y ∞ and any continuous function f ∈ C(S 1 × [0, 1]), we have lim Fix ε > 0. By (3.3), we see that for all large n we have Since lim k→∞ y n k = y n ∞ , for all large k we have |(F n (y n k ))(f ) − (F n (y n ∞ ))(f )| < ε, and hence |T y k (f ) − T y∞ (f )| < 3ε. Thus P −1 is continuous.

Almost sure limit of SLE driving measures
Consider a Brownian motion B κ t on the unit circle S 1 = {ζ ∈ C : |ζ| = 1} started at 0 with variance or diffusion rate κ, that is where W t is a standard linear Brownian motion. We study B κ t as the driving function of the radial Loewner equation (1.2) defining SLE κ .
Let L κ t = t −1 L κ t be the average occupation measure of B κ at time t (its normalization gives L κ t ∈ M 1 (S 1 )). An easy consequence of the ergodic theorem is the following almost sure t → ∞ limit of L 1 t ; we include the proof for completeness.

Lemma 3.2. Almost surely, as
Proof. It suffices to show that for any continuous function f : S 1 → R we have almost surely Given this, by choosing a suitable countable collection of functions, we obtain the lemma.
Let (Ω, P) be a probability space (we suppress the σ-algebra) such that {B 1 t (ω)} is Brownian motion started at 1. Consider the expanded probability space given by (Ω × S 1 , P ⊗ (2π) −1 Leb S 1 ), and let (ω, e iθ ) correspond to the random path {e iθ B 1 t (ω)}. That is, after sampling an instance of Brownian motion B 1 t (ω) started at B 1 0 (ω) = 1, we apply an independent uniform rotation to the circle so the Brownian motion starts at e iθ instead. A consequence of Birkhoff's ergodic theorem is that for a.e. (ω, e iθ ) ∈ Ω × S 1 , we have Equivalently, for a.e. e iθ ∈ S 1 , we have (3.6) for a.e. ω. Taking a countable sequence of e iθ converging to 1 and using the uniform continuity of f , we obtain (3.5). This concludes the proof of Lemma 3.2.
Now we justify the heuristic argument in the introduction, which said that as κ → ∞, the Brownian motion B κ t moves so quickly that the driving measure converges to Proof. Lemma 3.1 states that N is the projective limit of the spaces Y n defined in Section 3.1, with projection map from N to Y n given by (P i n ) i∈In . It thus suffices to show that as κ → ∞, the random measure P i n {δ B κ t } a.s. converges to P i n ((2π) −1 Leb S 1 ⊗ Leb [0,1] ) = (2π) −1 Leb S 1 in the Prokhorov topology. That is, almost surely This is true since Lemma 3.2 tells us that almost surely, in the Prokhorov topology we have Hence Lemma 3.3 holds.
We thus obtain Proposition 1.1, an almost sure description of the κ → ∞ limit of SLE κ .
Proof of Proposition 1.1. The proof follows immediately from Theorem 2.2 and Lemma 3.3.
The rest of this section is devoted to the proof of Theorem 1.2.

Large deviation principle of occupation measures
In this section, we discuss the large deviation principle of Brownian motion occupation measures on S 1 as κ → ∞. This is a special case of [DV75].
Recall that L κ t = t −1 L κ t is the average occupation measure of B κ at time t. By Brownian scaling we have (recall that the upper index is diffusivity and the lower index is time) where L(u) = u /2 is the infinitesimal generator of the Brownian motion on S 1 . The average occupation measure L κ 1 admits a large deviation principle as κ → ∞, with rate functionĨ. That is, for any closed set C ⊂ M 1 (S 1 ), and for any open set G ∈ M 1 (S 1 ), andĨ is lower-semicontinuous.
Note that the lower-semicontinuity follows from the definition ofĨ. In fact, let µ n be a sequence converging weakly to µ. We have We verify two immediate properties ofĨ that will be useful later.
Lemma 3.5. The rate functionĨ is convex and good, i.e. the sub-level sets Proof. The convexity follows from the fact thatĨ(·) is the supremum of the linear (hence convex) functionals where the supremum is taken over all positive u ∈ C 2 (S 1 ). SinceĨ is lower-semicontinuous, the sets {µ ∈ M 1 (S 1 ) :Ĩ(µ) ≤ c} are closed in M 1 (S 1 ) which is compact itself.
For the convenience of those readers who may not be so familiar with the statement of Theorem 3.4, let us provide an outline of the proof of the upper bound (3.8) in order to explain where this rate function comes from.
Let P ζ denote the law of a Brownian motion B on S 1 (with diffusivity 1) starting from ζ ∈ S 1 and Q ζ,t the law of the average occupation measure L 1 t under P ζ . Fix a small number h > 0, and let π h (ζ, dξ) be the law of B h under P ζ . We consider the Markov chain X n := B nh , so that π h is the transition kernel of X. We write E for the expectation with respect to P 1 . Now let u ∈ C 2 (S 1 ) such that u > 0. From the Markov property, we inductively get Since the Brownian motion is a Feller process with infinitesimal generator L, we have Therefore, where n is chosen to be the integer part of t/h. Hence, for some M (u) < ∞ depending only on the function u > 0. For any measurable set C ⊂ M 1 (S 1 ), since for arbitrary u, we have lim sup When C is closed (hence compact), some topological considerations allow us to swap the inf and sup in the above expression, and we obtain inf u>0,u∈C 2 which is the upper bound (3.8). As it is often the case in the derivation of large deviation principles, the lower bound turns out to be trickier, and uses approximation by discrete time Markov chains and a change of measure argument. We refer to the original paper [DV75] for more details.
The rate functionĨ of Theorem 3.4 is somewhat unwieldy but can be simplified for Brownian motion as noted in [DV75]. We provide here an alternative elementary proof.
In the definition (3.7), take u = e λh where h is smooth and λ is a real number. This gives

Comments
Let us make further comments and list a few questions in addition to those in the introduction.
1. As we have discussed in the introduction, one may wonder what the limit and large deviations of chordal SLE ∞ are.  The simulation of these counterflow lines is done by imaginary geometry as described in [MS16a], and are approximated via linear interpolation of an 800 × 800 discrete Gaussian free field with suitable boundary conditions. The color represents the time (capacity) parametrization of the SLE curve.
2. Corollary 1.3 shows that SLE ∞ concentrates around the family of Loewner chains driven by an absolutely continuous measure ρ with E(ρ) < ∞. In [VW] we characterize geometrically the Loewner chains driven by such measures. Note that the answer to the same question for the large deviation rate function of SLE 0+ , namely the family of Jordan curves of finite Loewner energy, is well-understood. That family has been shown to be exactly the family of Weil-Petersson quasicircles [Wan19b], which has far-reaching connections to different areas of mathematics and mathematical physics.
3. The rate function (1.5) for the Brownian occupation measure coincides with the rate function of the square of the Brownian bridge (or Gaussian free field) on S 1 . Is there a deep reason or is this merely a coincidence? One could attempt to relate the large deviations of the Brownian occupation measure to the large deviations of the occupation measure of a Brownian loop soup on S 1 .
4. The fluctuations of the circular Brownian occupation measure L κ t were investigated by Bolthausen. We express this result in terms of the local time t : S 1 → [0, ∞), defined via L 1 t = t (ζ)dζ. Note that t is a.s. a random continuous function.
We wonder whether there are interesting consequences to the fluctuations of SLE ∞ .