Scaling limits for planar aggregation with subcritical fluctuations

We study scaling limits of a family of planar random growth processes in which clusters grow by the successive aggregation of small particles. In these models, clusters are encoded as a composition of conformal maps and the location of each successive particle is distributed according to the density of harmonic measure on the cluster boundary, raised to some power. We show that, when this power lies within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability. The methodology developed in this paper provides a blueprint for analysing more general random growth models, such as the Hastings-Levitov family.


Introduction
We study a family of planar random growth processes in which clusters grow by the successive aggregation of particles. Clusters are encoded as a composition of conformal maps, following an approach first introduced by Carleson and Makarov [3] and Hastings and Levitov [5]. The specific models that we study fall into the class of Laplacian growth models in which the growth rate of the cluster boundary is determined by the density of harmonic measure of the boundary as seen from infinity. In our case, the location of each successive particle is distributed according to the density of harmonic measure raised to some power. Our set-up is closely related to that of the Hastings-Levitov family of models, HL(α), α ∈ [0, ∞) [5], which includes off-lattice versions of the physically occurring dielectric-breakdown models [8], in particular the Eden model for biological growth [4] and diffusion-limited aggregation (DLA) [15]. Our family of models shares with the HL(0) model the unphysical feature that new particles are distorted by the conformal map which encodes the current cluster. However, in subsequent work [9], we show that these models share common behaviour with the HL(α) models when α = 0, so the present paper serves to develop methods applicable to these more physical models.
We establish scaling limits of the growth processes in the small-particle scaling regime where the size of each particle converges to zero as the number of particles becomes large. We show that, when the power of harmonic measure is chosen within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability. This phase transition in fluctuations can be interpreted as the beginnings of a macroscopic phase transition, from disks to non-disks.

Description of the model
Our clusters will grow from the unit disk by the aggregation of many small particles. Let We fix a non-empty subset P of D 0 and set We assume that P is chosen so that K is compact and simply connected. Then we call P a basic particle.
We will call a conformal map F , defined on D 0 and having values in D 0 , a basic map if it is univalent and satisfies F (∞) = ∞, F (∞) ∈ (1, ∞).
By the Riemann mapping theorem, there is a one-to-one correspondence between basic particles and basic maps given by For convenience, we will assume throughout that F has a continuous extension to the unit circle. It is well understood geometrically when this holds. The map F has the form for some c > 0 and sequence (a k : k 0) in C. The value e c is called the logarithmic capacity of the cluster K. We define the capacity of the particle P (or, interchangeably, of the map F ) by cap(P ) = log F (∞) = c.
For the purpose of this introduction, we will assume that we have chosen a family of basic particles (P (c) : c ∈ (0, ∞)), such that cap(P (c) ) = c. Figure 1 shows four representative particles from some families we have in mind. Write (F (c) : c ∈ (0, ∞)) for the family of associated basic maps. Given a sequence of attachment angles (Θ n : n 1) and capacities (c n : n 1), set F n (z) = e iΘn F (cn) (e −iΘn z).
We think of the compact set K n as a cluster, formed from the unit disk K 0 by the addition of n particles. By choosing the sequences (Θ n : n 1) and (c n : n 1) in different ways, we can obtain a wide variety of growth processes. In the aggregate Loewner evolution (ALE) model with parameters α ∈ R, η ∈ R, c ∈ (0, ∞) and σ ∈ [0, ∞), which was introduced in [14] for slit particles, and abbreviated as ALE(α, η), we set h n (θ) = |Φ n−1 (e σ+iθ )| −η Z n , Z n = 2π 0 |Φ n−1 (e σ+iθ )| −η dθ = 1 2πˆ2 π 0 |Φ n−1 (e σ+iθ )| −η dθ (3) and we take c n = c|Φ n−1 (e σ+iΘn )| −α with (Θ n : n 1) a sequence of random variables whose distribution given by where F n = σ(Θ 1 , . . . , Θ n ). In this paper, we will consider only the case where α = 0, which takes as data a single basic map F = F (c) and a choice of η ∈ R and σ ∈ [0, ∞). For simplicity, we refer to this model here as the ALE(η) model with basic map F and regularization parameter σ. If, on the other hand, we were to take η = σ = 0 and fix α ∈ [0, ∞), then we would obtain the HL(α) model considered by Hastings and Levitov [5]. The parameters α and η play a similar role in adjusting the 'local growth rate of capacity' as a function of the current cluster shape. Indeed, in the subsequent paper [9] we show that, modulo a deterministic timechange and under the same restrictions on the parameter σ as will be used in this paper, the scaling limit of ALE(α, η) depends primarily on the sum α + η provided that α + η 1. This means that ALE(η) and regularized HL(α) have qualitatively similar behaviour when α = η. Moreover, the range of the attachment densities considered in ALE(η) corresponds exactly to those used to define the dielectric-breakdown models, so the full family ALE(α, η) is of wider interest than HL(α) alone. See [14] for a comprehensive discussion of other models related to ALE.
One of the challenges of studying HL(α) when α = 0 is that the capacity of the cluster K n is random and could be quite badly behaved. It is therefore a priori unclear how to tune parameters in order to obtain non-trivial scaling limits. One way in which ALE(η) is simpler is that the capacity of the cluster K n is always cn, where c = log F (∞). Nevertheless, the models have much in common, and it has turned out that the framework developed here for ALE(η) provides useful ideas for the analysis of HL(α). In this paper we will focus on the case where η ∈ (−∞, 1]. We will establish scaling limits and fluctuations for ALE(η) in the small-particle regime, where simultaneously c → 0, σ → 0 and n → ∞ with n tuned so that nc → t, for some fixed t ∈ R, thereby giving clusters of macroscopic capacity.

Review of related work
Much effort has been devoted to the analysis of lattice-based random growth models. These are models in which, at each step, a lattice site adjacent to the current cluster is added, chosen according to a distribution determined by the current cluster. Examples include the Eden model [4], diffusion limited aggregation (DLA) [15] and the family of dielectric-breakdown models [8]. Around 20 years ago, Carleson and Makarov [3] and Hastings and Levitov [5] introduced an alternative approach in the planar case, which allows the formulation of a discrete particle model directly in the continuum by encoding clusters in terms of conformal maps, as described in the preceding subsection. In [3], the authors obtained a growth estimate for a deterministic analogue of DLA which is formulated in terms of the Loewner equation. In [5], the HL(α) model was studied numerically and experimental evidence was shown for a phase transition in behaviour at α = 1: when α < 1, clusters appeared to converge to disks; on the other hand, when α > 1, a turbulent growth regime emerged, in which clusters behaved randomly at large scale. Hastings and Levitov argued that HL(1) is a candidate for an off-lattice version of the Eden model, and HL(2) corresponds to DLA. Establishing the existence of this phase transition rigorously is one of the main open problems in this area.
In [14], Sola, Turner and Viklund showed the existence of a phase transition in the ALE(η) model. They showed that, for η > 1, if particles are taken to be slits, and the regularisation parameter σ is sufficiently small then, in the small-particle limit, the clusters themselves grow from the unit disk by the emergence of a radial slit, at a random angle.
This behaviour is qualitatively different to the known behaviour of ALE(0), that is to say HL(0), in the same scaling regime. In [10], Norris and Turner showed that the HL(0) clusters converge to disks with internal branching structure given by the Brownian web. More recently, Silvestri [13] analysed the fluctuations in HL(0) and showed that these converge to a log-correlated fractional Gaussian field. Several other papers consider modifications of the HL(0) model [6,7,12].
In this paper, we approach the question of the phase transition in ALE(η) at η = 1 from the opposite direction to that in [14] by showing convergence to a disk for ALE(η) for all η 1, provided that σ does not converge to zero too fast. Further, we prove convergence of the associated fluctuations to an explicit limit, which depends on η, and which would exhibit unstable behaviour if one took η > 1. Our results apply in a different regime to that considered in [14]. We require that the regularization parameter σ c 1/2 (and sometimes more), which enables us to show that, for each η 1, the disk limit and the fluctuations hold universally for a wide class of particle shapes. By contrast, in [14] the parameter σ c and the results rely heavily on the slit particle being non-differentiable at its tip.

Statement of results
Our main results will be proved under the technical assumption (4) below, which we will show in Appendix A to be satisfied for small particles of any given shape. This assumption expresses that the basic particle P is concentrated near the point 1 on the unit circle in a certain controlled way. Let F be a basic map of capacity c ∈ (0, 1], in the sense of Subsection 1.1, that is to say, a univalent conformal map from {|z| > 1} into {|z| > 1} such that .
Here and below we choose the branch of the logarithm so that log(F (z)/z) is continuous on {|z| > 1} with limit c at ∞. Our results will concern the limit c → 0 with Λ fixed, but are otherwise universal in the choice of particle. We will show that, for η ∈ (−∞, 1], in this limit, provided the regularisation parameter σ does not converge to 0 too fast, the cluster K n converges to a disk of radius e cn , and the fluctuations, suitably rescaled, converge to the solution of a certain stochastic partial differential equation. Theorem 1.1. Let η ∈ (−∞, 1], Λ ∈ [0, ∞) and ε ∈ (0, 1/2) be given. Let (Φ n : n 0) be an ALE(η) process with basic map F and regularization parameter σ. Assume that F has capacity c and regularity Λ, and that e σ 1 + c 1/2−ε . For all η ∈ (−∞, 1), m ∈ N and T ∈ [0, ∞), there is a constant C = C(η, ε, Λ, m, T ) < ∞ with the following property. There is an event Ω 1 of probability exceeding 1 − c m on which, for all n T /c and all |z| 1 + c 1/2−ε , Moreover, in the case where η = 1, provided ε ∈ (0, 1/5) and e σ 1 + c 1/5−ε , there is also a constant C = C(ε, Λ, m, T ) < ∞ with the following property. There is an event Ω 1 of probability exceeding 1 − c m on which, for all n T /c and all |z| 1 + c 1/5−ε , We remark that Theorem 1.1 can be recast in terms of a regularized particle P (σ) given by Note that P (σ) also has capacity c and is associated to the conformal map n : n 0) be an ALE process with basic map F (σ) and regularization parameter 0. Then Φ (σ) n (z) = e −σ Φ n (e σ z) for an ALE process (Φ n : n 0) with basic map F and regularization parameter σ. Hence, if we replace Φ n by Φ (σ) n in Theorem 1.1, then under the same restrictions on σ, the same estimates are valid but now for all |z| 1 and without regularization in the density of attachment angles.
The simulations on the left side of Figure 2 illustrate the conjectured phase transition in macroscopic shape from disks to non-disks at η = 1. The simulations on the right show the sensitivity of the fluctuations of the level lines θ → Φ n (re iθ ) in ALE(0) to taking r − 1 ≈ c 1/2 versus r − 1 c 1/2 . This provides evidence that the speed at which σ → 0 as c → 0 in ALE(η) significantly affects cluster behaviour.
We also establish the following characterization of the limiting fluctuations, which shows in particular that they are universal within the class of particles considered. Theorem 1.2. Let η ∈ (−∞, 1], Λ ∈ [0, ∞) and ε ∈ (0, 1/6) be given. Let (Φ n : n 0) be an ALE(η) process with basic map F and regularization parameter σ. Assume that F has capacity c and regularity Λ. Assume further that Set n(t) = t/c . Then, in the limit c → 0 with σ → 0, uniformly in F , where H is the set of analytic functions on {|z| > 1} vanishing at ∞, equipped with the metric of uniform convergence on compacts, and where F is given by the following stochastic PDE driven by the analytic extension ξ in D 0 of space-time white noise on the unit circle, The space H and the meaning of this PDE are discussed in more detail in Section 7. For η = 0 we recover the fluctuation result in [13]. The solution to the above stochastic PDE is an Ornstein-Uhlenbeck process in H. This process converges to equilibrium as t → ∞. When η < 1, the equilibrium distribution is given by the analytic extension in D 0 of a log-correlated Gaussian field defined on the unit circle. In the case η = 0, this is known as the augmented Gaussian Free Field. When η = 1, the equilibrium distribution is the analytic extension of complex white noise on the unit circle. The equation (6) can be interpreted as a family of independent equations for the Laurent coefficients of F(t, .), given in (50). These equations may be considered also for η > 1 but now the equation for the kth Laurent coefficient shows exponential growth of solutions at rate (η − 1)k, so there is no solution to (6) in H, indicating a destabilization of dynamics as η passes through 1.
Although we have stated our theorems above for η ∈ (−∞, 1], in many of our arguments we restrict to the case η ∈ [0, 1]. The proofs are largely similar when η < 0 except in the way that we decompose the operator in Section 4. We remark on the correct decomposition in the case η < 0 at the relevant point.

Remarks on context and scope of results
The process of conformal maps (Φ n : n 0) is Markov and takes values in an infinitedimensional vector space. In the limit considered, where c → 0, the jumps of this process become small, while we speed up the discrete time-scale to obtain a non-trivial limiting drift. So we are in the domain of fluid limits for Markov processes. The analysis of such limits, and of the renormalized fluctuations around them, is well understood in finite dimensions. However, while the formal lines of this analysis transfer readily to infinite dimensions, its detailed implementation is not so clear, not least because it is necessary to choose a norm, which should be well adapted to the dynamics, and the limiting drift will in general be a non-linear and unbounded operator.
In the case at hand, there are a number of special features which are important to the analysis. First, while the limiting dynamics is not in equilibrium, it is an explicit steady state, which allows us to handle convergence of the Markov process in terms of linearizations around this steady state: we find that the difference Φ n (z) − e cn z may usefully be expressed by an interpolation in time, in which each term describes the error introduced by a single added particle. Second, the map Φ n is determined by its restriction to the unit circle (Φ n (e iθ ) : θ ∈ [0, 2π)) and the action of each jump, besides being small, also becomes localized in θ in the limit c → 0. This is one of the features contributing to the explicit form found for the limiting fluctuations. Third, we have at our disposal, not only the usual tools of stochastic analysis, but also a range of tools from complex analysis, including distortion estimates, and L p -estimates for multiplier operators, which turn out to mesh well with L p -martingale inequalities.
We have tried to optimise, as far as our present techniques allow, the constraints in our results on the regularization parameter σ. In the case η < 1, we establish the disk limit for σ c 1/2 . Indeed, for η < 1, in the limit considered, we show that the derivative of the fluctuations at radius e σ , which controls the scale of h n (θ)−1, is at most of order c 1/2 /(e σ −1). Therefore, to leading order, the distribution of each attachment angle is approximately uniform and the bulk dynamics of our process resemble that of HL(0). As seen in Proposition A.2, the scale of individual particles is c 1/2 , so for σ ∼ c 1/2 the fluctuations of e −c f (e σ+iθ ) around 1 are scale-invariant. With that choice of σ we would expect to see macroscopic variations of h n (θ), so the attachment distributions would no longer be well approximated by the uniform distribution. We therefore believe our constraint on σ is close to optimal within this regime and it remains a challenging open problem to allow σ ∼ c 1/2 . When η = 1, on the other hand, we show that the derivative of the fluctuations at radius e σ is at most of order c 1/2 /(e σ − 1) 3/2 . The break-down in the uniform approximation may therefore well happen for larger σ than σ ∼ c 1/2 and the form of the fluctuations is suggestive of σ ∼ c 1/3 . Although we need a stronger regularization for the fluctuation result (cf. Theorem 1.2), we find that the fluctuations develop variations on all spatial scales, so the modification of dynamics from HL(0) to ALE(η), even with the averaging enforced by our choice of regularization, results in a feedback which affects the limiting evolution, and which identifies the case η = 1 as critical.

Organisation of the paper
The structure of the paper is as follows. In Section 2, we give a simplified proof of convergence to a disk in the case η = 0, corresponding to HL(0). This is followed by an overview of the proof when η = 0. In Section 3, we decompose the increment Φ n (z) − Φ n−1 (e c z) as a sum of martingale difference and drift terms, which we expand to leading order in c with error estimates. In Section 4 we obtain the evolution equation and decomposition for the fluctuations. The remainder of the paper analyses this equation. Specifically, in Section 5 we use the estimates from Section 3 to obtain bounds on the terms arising in the decomposition of the differentiated fluctuations. These bounds are then used in Section 6 to obtain our disk limit Theorem 1.1. Finally the fluctuation limit Theorem 1.2 is derived in Section 7.
Some necessary but technical estimates are deferred to appendices. In Appendix A we show that our main assumption (4) is satisfied for small particles of any given shape. Appendix B contains the estimates for multiplier operators used in the paper. In Appendices C and D we derive the specific estimates on ALE(η) used in our main results.
2 HL(0) and overview of the proof of Theorem 1.1 In this section we give a quick argument for the scaling limit of HL(0) (which is the same as ALE(0)), where the attachment angles (Θ n : n 1) are independent and uniformly distributed. Then we discuss the structure of the proof of Theorem 1.1, some aspects of which follow the argument used for HL(0).
For a measurable function f on {|z| > 1}, for p ∈ [1, ∞) and r > 1, we will write In the case where f is analytic and is bounded at ∞, we have, for ρ ∈ (1, r), The notation · p will be reserved for the L p (P)-norm on the probability space.

Disk limit for η = 0
We now show that HL(0) converges to a disk in the small-particle limit. A weaker form of this result was shown in [10] by fluid limit estimates on the Markov processes (Φ −1 n (z) : n 0). Here, we will use a new method, based on estimating directly the conformal maps Φ n . This both gives a simpler argument and leads to a stronger result.
Theorem 2.1. Let (Φ n : n 0) be an HL(0) process with basic map F . Assume that F has capacity c ∈ (0, 1] and regularity Λ ∈ [0, ∞). Then, for all p ∈ [2, ∞), there is a constant C = C(Λ, p) < ∞ such that, for all r > 1 and n 0, we have We remark that by taking p large enough it is possible to deduce that, for all ε ∈ (0, 1/2) and T 0, we have sup in probability as c → 0. As this is spelled out more generally in Section 6.2, we omit the details at this stage. Indeed, on applying Theorem 1.1 to HL(0), say with σ = 1, we obtain the stronger estimate sup with high probability as c → 0. This improvement can be traced to the iterative argument used in the proof of Proposition 6.1.
Proof of Theorem 2.1. It will suffice to consider the case where r 1 + √ c. Set Note that Φ n−1 (e c z) is the map we would obtain after n steps if we substituted F n (z) by e c z in (2). As we aim to show that Φ n (z) is close to e cn z, ∆ n (z) can be understood as the error due to the nth particle. We can write Φ n as a telescoping sum The functions F and Φ j−1 are analytic in {|z| > 1} and F (z)/z → e c as z → ∞, so the function is analytic in {0 < |w| < |z|} and extends analytically to {|w| < |z|}. Hence, almost surely, by Cauchy's theorem, Since Φ j−1 is univalent on {|z| > 1} and Φ j−1 (z)/z → e c(j−1) as z → ∞, by a standard distortion estimate, for all |z| = r > 1, Hence, for |z| = r > 1 + √ c/2, we have Burkholder's inequality (see Section B.1) applies to the sum of martingale differences (10) where we used an integral comparison for the last inequality. Set and write ρ = (r + 1)/2. Then, for and the claimed estimate follows.

Overview of the proof of Theorem 1.1
We now discuss how the above strategy can be adapted to the case where η ∈ (−∞, 1]. Write (9). We split ∆ j (z) as the sum of a martingale difference and a drift term (which vanished in the case η = 0) SetΦ n (z) = e −cn Φ n (z) − z as above. We start by identifying the leading term in the drift, showing that where R j (z) is small provided Φ j−1 ∞,e σ is sufficiently small. This gives the following de-compositionΦ where P is the operator which acts on analytic functions on {|z| > 1} by The reader is alerted to the fact that, while we used P to denote our basic particle in Sections 1 and Appendix A, in the rest of the paper, P will refer to this operator instead. Solving the recursion we end up with Note that for η = 0 the operator P has the simple form P f (z) = e −c f (e c z) and we recover (10).
We treat the general case η ∈ (−∞, 1] by observing that P acts diagonally on the Laurent coefficients, thus is a Fourier multiplier operator, which we can bound in · p,r -norm by means of the Marcinkiewicz multiplier theorem (see Appendix B.2). The proof strategy for the disk theorem then goes as follows. For δ = δ(c) small, to be specified, introduce the stopping time Then for all n N (δ) the angle density h n defined in (3) is approximately uniform. This, together with the multiplier theorem, can be used to bound both the martingale term (the first term in (15)) and the remainder term (the second term in (15)), thus leading to a bound for the mapΦ n . At this point it remains to show that we can pick δ 0 such that N (δ 0 ) T /c with high probability to conclude the proof. To this end, it turns out to be convenient to work instead with the differentiated dynamics Ψ n (z) = zΦ n (z) for which a decomposition similar to (15) holds (see (32) below). We use it to show that Ψ n 1 {n N 0 } p,r is small in L p (P) (see Proposition 6.1), where we have set N 0 = N (δ 0 ) to ease the notation slightly. The analyticity of Ψ n then allows us to make this bound into a high probability statement on the supremum norm of Ψ n 1 {n N 0 } , at the price of taking p large enough (see Proposition 6.2). By showing that this bound is smaller than δ for all n N 0 , we deduce that in fact we must have N 0 T /c , thus concluding the proof.

Choice of state variables
The sequence of conformal maps (Φ n ) n 0 is a Markov process. This allows an approach to the desired scaling limits using martingale estimates. Above, we introduced the analytic function Ψ n on {|z| > 1} given by where we set Df (z) = zf (z) andΦ n is the time-rescaled process of fluctuations given bỹ Then the process (Ψ n ) n 0 is also Markov and it proves more convenient to use this as our primary state variable. In doing this, we forget the limiting values (Φ n (∞)) n 0 , so we see the clusters only up to an unknown displacement. Otherwise, the use of (Ψ n ) n 0 may be considered as a particular choice of coordinates for the sequence of clusters. The function Φ n has a Laurent expansion in {|z| > 1} of the form In the final section of the paper, we will characterise the limit distribution of the rescaled fluctuations, by analysing the Laurent coefficients.

Expansions to first order and error estimates
In this section we identify the leading order behaviour of several quantities of interest and gather together bounds on the error terms which hold while the differentiated fluctuation process (Φ n ) n 0 is well-behaved. Our main objective is to justify (13). Fix δ 0 ∈ (0, 1/8] and consider the stopping time N 0 = N (δ 0 ) where N (δ) is defined in (16). Several of our estimates will be made under the assumption that n N 0 . In fact, in this section, we only use that |Φ j (e σ+iθ )| δ 0 1/8 when j = n − 1. However, we will need this to hold for all j n − 1 in the remainder of the paper and it simplifies notation to make the assumption here. This assumption guarantees that h n , defined in (3), can be bounded above and below by absolute constants. Bounding very crudely, A more refined analysis shows that, for all n N 0 , where C = C(η) is a constant depending only on the value of η. As the precise computation consists of elementary manipulations, it is deferred to Appendix C (see (65) and (66)).
Recall the definitions of ∆ n (θ, z) and ∆ n (z) from (9) and the definitions of A n (z) and B n (z) from (12) and (11). Then Furthermore, A n and B n are analytic in {|z| > 1} and, almost surely, As we showed in the proof of Theorem 2.1, by Cauchy's theorem, We now identify the leading order terms in ∆ n (z) and A n (z), in the limit c → 0. Where the computations add little to the intuition, these are also deferred to Appendix C.
Using (19), (18), (21) and that |β − 1| Λ √ c/2 (cf. Proposition A.1), the leading term of where the equality follows by Cauchy's integral formula. To be precise, set Then, by the argument in Appendix C, for n N 0 and |z| = r with r 1 + √ c, (27) and |R n (∞)| Cce cn δ 2 0 for some constant C = C(η, Λ) < ∞ (possibly different to the constant C obtained earlier). By the maximum principle, it follows that provided one takes δ 0 √ c/(e σ − 1) and r e σ 1 + √ c, From this bound, it can be easily seen that R n (z) is small if Φ n−1 ∞,e σ is sufficiently small, which is what we wanted to show. However, the assumption that r e σ is too restrictive for our needs, so in subsequent analysis we revert to the more general estimate (27).

Linear evolution equation for the fluctuations
In this section, our objective is to justify the expansion (15). In fact, we obtain an analogous expansion which makes it clearer which terms determine the leading order fluctuations.
In Section 1.1 we decomposed ∆ n (z) = Φ n (z) − Φ n−1 (e c z) as a sum of a martingale difference B n (z) and drift A n (z), and in the previous section we justified writing In view of (23), it is convenient to split the martingale difference B n as a sum of analytic functions We will see that M n is the main term: its explicit form allows for precise estimates, and it determines the Gaussian fluctuations. On the other hand, W n is accessible less directly, but is of smaller order, so can also be handled adequately. Then, using (26), so we obtain the linear evolution equatioñ where P is as in (14). Note that P acts diagonally on the Laurent coefficients, with multipliers In the case η ∈ [0, 1], we factorize P by writing It is straightforward to check then that, for all k, In order to adapt our argument to the case η ∈ (−∞, 0), we would modify the equation The subsequent argument is very similar so we will not give further details for this case. Write P 0 for the multiplier operator acting on analytic functions on {|z| > 1} by and note that We iterate (28) to obtainΦ Then, on differentiating, We will focus initially on bounding the terms in the decomposition (32) of the differentiated fluctuations Ψ n . We will refer to M n , W n and R n as the principal martingale term, the second martingale term and the remainder term respectively. Later, we will return also to the undifferentiated decomposition (31).

Norms
We conclude this section by describing the normed spaces on which we will obtain our bounds.
Recall from (7) the definition of f p,r for a measurable function f on {|z| > 1}. For a random such function Φ, we will write where · p denotes the L p (P)-norm on the probability space. Note that, for all n 0, the boundedness and monotonicity seen in (30) allows an application of the Marcinkiewicz multiplier theorem (see Appendix B.2), with m k = p 0 (k) n and M = 1 to see that for all p ∈ (1, ∞) and all r > 1, there is a constant C = C(p) < ∞ such that P n 0 f p,r C f p,r .
Some further operator estimates which will be used in the subsequent analysis are stated in Appendix B.2.

Estimation of terms in the decomposition of the differentiated fluctuations
In this section we collect estimates for the principal martingale term, the second martingale term and remainder term. We first estimate the principal martingale term M n (z) in the decomposition (32) of the differentiated fluctuation process, which is given by where r n = re c(1−η)n .
It follows that if r 1 + c 1/2−ε for some ε ∈ (0, 1/2), Proof. By Burkholder's inequality (cf. Theorem B.1), for all p ∈ [2, ∞), there is a constant So, on taking the · p/2,r -norm, Observe that Hence, almost surely, and, for |z| = r, For j N 0 , we have h j (θ) 3/2, so we obtain, for |z| = r, almost surely, Hence, for |z| = r, almost surely, Moreover, Finally, we take the . p/2,r -norm and substitute into (35) to obtain (34). Now suppose r 1 + c 1/2−ε for some ε ∈ (0, 2) and p 1 + 1/(2ε). If η < 1, by using an integral comparison in (34) we obtain where we used the assumption on p in the second inequality, and absorbed a factor of 2 + 1/(2 − 2η) in the final constant C. Hence If η = 1, we now have r n = r, so and then, using that p 2, We now state the estimate of the second martingale term W n (z) in the decomposition (32) of the differentiated fluctuation process, which is given by The proof is deferred to Appendix D.1.
where we have used the same notation as in Lemma 5.2. Now suppose that n T /c for some constant T > 0. Then there is a constant C = C(Λ, η, p, T ) < ∞ such that when η < 1 and when η = 1.

Convergence to a disk for ALE(η)
In this section we derive our main disk theorem. Recall that First we show that Ψ n 1 {n N 0 } p,r is small, provided δ 0 is appropriately chosen. Then we deduce estimates on the random norms Ψ n 1 {n N 0 } ∞,r , valid with high probability, and use them to dispense with the restriction that n N 0 . Finally, we apply these results to show that Φ n (z) is close to e cn z.

L p -estimates on the differentiated fluctuations
The proposition below shows that, for an appropriately chosen δ 0 , the · p,r norm of the differentiated fluctuation process Ψ n 1 {n N 0 } is of order √ c, with quantitative control of the singularity as r → 1 and the decay as r → ∞. The dependence of the estimate on σ is also explicit, allowing one to consider limits in which σ → 0 as c → 0. For small c, the estimates are strongest when ε and ν are taken to be small. A second argument, given in the next subsection, will show that the event {n N 0 } appearing in (40) and (41) is of high probability in the limit c → 0.
Proof. As before, constants referred to in the proof by the letter C may change from line to line and are all assumed to lie in [1, ∞). They may depend on Λ, η, T , ε, ν and p but they do not depend on c, n, σ and r.
We begin with a crude estimate which allows us to restrict further consideration to small values of c. The function e −cn Φ n (z) is univalent on {|z| > 1}, with e −cn Φ n (z) ∼ z as z → ∞. By same distortion estimate used in Section 2.1, for all |z| = r > 1, It is straightforward to check that this implies the claimed estimates in the case where c > 1/C, for any given constant C of the allowed dependence. Hence it will suffice to consider the case where c 1/C. Consider first the case where η < 1. Fix T , ε, p and ν as in the statement, and assume that c 1/e and r 1 + c 1/2−ε/2 and e σ 1 + c 1/2−ε and n T /c. Set ρ = (r + 1)/2. It will suffice to prove the result for p large enough, so assume p > 1 + 1/(2ε).
By the triangle inequality, where, by Lemmas 5.1, 5.2 and 5.3, On combining the estimates above and substituting the chosen value of δ 0 , we obtain, for all r 1 + c 1/2−ε/2 , Note that, for all r 1 + c 1/2−ε , we havē δ(r) Cc ε + Cc ε−ν (log(1/c)) 2 + C √ c log(1/c) c ε/2 1 for all sufficiently small c. Similarly, for r 1 + c 1/2−ε/2 , we haveδ(r) 1 for all sufficiently small c. As noted above, it suffices to deal with the case where c is sufficiently small. A complication in the analysis is that the right hand side of the inequality (43) requires estimates of Ψ j−1 (z) when |z| = ρ, but the left hand side only gives information about Ψ n (z) when |z| = r > ρ. Our approach is therefore to use the universal distortion estimate (42) to obtain an initial (very weak) bound and then recursively feed the bounds through the inequality. This generates stronger and stronger estimates, but at the cost of moving r further away from 1.

Spatially-uniform high-probability estimates on the differentiated fluctuations
We now use the results from the previous section to obtain uniform estimates on Ψ n (z).
Proof. We will give details for the case η ∈ [0, 1). The minor modifications needed for the case η = 1 are left to the reader. Fix η, ε, ν, m and T as in the statement. It will suffice to consider the case where e σ 1 + 2c 1/2−ε , and to find an event Ω 0 of probability exceeding 1 − c m on which (45) holds whenever r 1 + 2c 1/2−ε and n T /c. Set K = min{k 1 : 2 k c 1/2−ε 1}, N = T /c .

L p -estimates on the fluctuations
In this section we prove a result analogous to Proposition 6.1 for the undifferentiated dynamics. This allows us to prove Theorem 1.1.
Proof. Let us first consider η ∈ [0, 1). It suffices to prove the result for p > 1 + 1/(2ε). We modify the estimates leading to (34), by deleting D and k 2 and considering separately the constant term of the Laurent expansion, to obtain Similarly, for the second martingale term we obtain where we used the bound on Ψ n from Proposition 6.1. Finally, for the remainder term, we find On assembling these bounds, and simplifying using our constraints on r and σ, we obtain (46) and (47).
As in the proof of Proposition 6.1, in the case η = 1, we do not benefit from the push-out of r n = re c(1−η)n , and the bound on Ψ n is weaker. After some straightforward modifications, for p sufficiently large we obtain On assembling these bounds, and simplifying using our constraints on r and σ, we obtain (48). Similarly

giving (49).
Proof of Theorem 1.1. The argument is a variation of that for Proposition 6.2. We do it when η < 1; the η = 1 case is similar. Let Ω 0 , N , K, r(k), ρ(k) and λ be as in the proof of Proposition 6.2. Define and C is the larger of the constant in (46) and that in (47). Then P(Ω 1 ) 2c m and the desired uniform estimate on Φ n holds on Ω 1 , by the argument used in the proof of Proposition 6.2. In arriving at this estimate we use the fact that for r 1+c 1/2 we have (1+log(r/(r−1))) 1/2 c −ε for all sufficiently small c > 0, for all ε > 0.

Fluctuation scaling limit for ALE(η)
In this section, we show that the fluctuations of ALE(η) for η ∈ (−∞, 1] are of order √ c, and we determine the distribution of the rescaled fluctuations. Let (Φ n ) n 0 be an ALE(η) process with basic map F and regularization parameter σ. Assume that F has capacity c ∈ (0, 1] and regularity bound Λ ∈ [0, ∞). We consider the limit c → 0 with σ → 0, and will show weak limits which are otherwise uniform in F , subject to the given regularity bound. We embed in continuous time by setting n(t) = t/c and defining We will show that the process of analytic functions (Φ(t, .)/ √ c) t 0 converges weakly to a Gaussian limit.
Let us define the metric spaces our processes will live in. To start with, let D[0, ∞) denote the space of complex-valued càdlàg processes equipped with the Skorohod metric d.
To discuss weak convergence of sequences of Laurent coefficients, it is convenient to introduce the product space D[0, ∞) Z + of sequences of complex-valued càdlàg processes, with the metric of coordinate-wise convergence, given by d(a(k), b(k))) .
Finally, to talk about convergence of functions, let H denote the space of analytic functions on D 0 = {|z| > 1} with limits at ∞, equipped with the metric of uniform convergence on compacts in D 0 ∪ {∞}, given by We let D H [0, ∞) denote the space of H-valued càdlàg processes equipped with the associated Skorohod metric d H . Then all the above spaces are complete separable metric spaces [1] Here, on the left, we use the tensor product from R 2 . Thus (x + iy) ⊗ (x + iy ) = xx xy yx yy .
By standard estimates, the following series both converge almost surely, uniformly on compacts Hence F = (F(t, .) : t 0) and ξ = (ξ(t, .) : t 0) are continuous random processes in H. It is straightforward to check that and ξ is the analytic extension in D 0 of space-time white noise on the unit circle, so F satisfies the stochastic PDE (6). In this section we prove Theorem 1.2 by showing thatΦ/ √ c → F in distribution on D H [0, ∞).

Discarding lower order fluctuations
Our analysis is based on the decomposition (31), which we rewrite in continuous time, with obvious notation asΦ (t, z) =M(t, z) +W(t, z) +R(t, z).
In a first step, we will show thatM 0 is the only term that matters in the limiting fluctuations.
We first consider the case η ∈ (−∞, 1). Recall that in the proof of Proposition 6.3 we showed that, for all T ∈ [0, ∞), p > 1 + 1/(2ε) and r > 1, there is a constant C = C(Λ, η, T, ε, p, r) < ∞ such that for all c 1/C, e σ 1 + c 1/2−ε and n T /c, we have Here we have used that |β − 1| Λ √ c. Note that under the further restriction σ c 1/4−ε , By arguments from the proof of Proposition 6.2, it follows that in probability as c → 0, uniformly on compacts in (t, z) ∈ [0, T ] × (D 0 ∪ {∞}), and uniformly in σ and F subject to the given constraints. On the other hand, by Proposition 6.2, we know that P(N 0 < T /c) → 0 in the same limiting regime. The claim of the lemma follows. The case η = 1 is handled by the same argument with straightforward modifications.

Covariance structure
We now focus on the leading order fluctuations, coming from the martingale term where Let (Θ u n ) n 1 be a sequence of independent uniform random variables in [0, 2π). Define for |z| > 1 where F u n−1 is the σ-algebra generated by {Θ u k : k n − 1}. Expanding in Laurent series, we find Recalling that the operator P acts diagonally on Laurent coefficients, set and define for t 0Ã LetM u (t, z) be defined as in (51) with M 0 j replaced by M u j . Then we havẽ By an elementary calculation, we obtain Recall that for η ∈ [0, 1] p 0 (k) = e c(1+(1−η)k) p(k).
By some straightforward estimation, recalling that σ → 0, we have Note that if j t/c for some t > 0, and k is fixed, then the right hand side converges to 0 as c → 0. In the case η < 0, define p 0 (k) exactly as above (note that this differs from the definition in (29)). Provided c is taken sufficiently small that σ − c − c|η| > 0, we have 1 + cηke −(σ−c)(k+1) e cηk , and hence p 0 (k) 1. A straightforward estimation therefore gives Now, for any k, k 0 and s, t ∈ [0, ∞) with s t, the following limit holds in probability as c → 0, uniformly in σ and F , To see this, recall that by Proposition 6.1 for all m ∈ N there exists a constant C = C(Λ, η, ε, m, T ) < ∞ such that, for c 1/C and δ 0 defined as in the proof of Lemma 7.1, there exists an event Ω 0 of probability at least 1 − c m on which, for all n T /c and all θ ∈ [0, 2π), |Φ n (e σ+iθ )| δ 0 1, and hence, by (17), |h n (θ) − 1| 63δ 0 . Then, on Ω 0 , for c 1/C and t T , e −c(n(s)+n(t)−2j) .
Since cδ 0 n(s) → 0 as c → 0, this shows the claimed limit in probability.

Convergence of Laurent coefficients
We now show that the processes of rescaled Laurent coefficients (Ã(·, k)) k 0 ofM 0 (t, z) converge weakly to those of the limiting process F. Proof. It will suffice to show that the finite-dimensional distributions of (Ã(·, k)) k 0 converge to those of (A(·, k)) k 0 , and that for each fixed k the processesÃ(·, k) are tight in (D[0, ∞), d).
We start by proving convergence of finite-dimensional distributions. Fix positive integers K and m and pick arbitrary 0 t 1 < t 2 < · · · < t m . We aim to show the following convergence in distribution   Ã Write n i in place of n(t i ) for brevity. Fix real-linear maps α k,l : C → R, for k = 1, . . . , K and l = 1, . . . , m and consider the real-valued random variables given by It is readily verified that (X j,nm : j = 1, . . . , n m ) is a martingale difference sequence with respect to the filtration (F j : j = 1, . . . , n m ). Set We will use the following martingale central limit theorem [1, Theorem 18.1].
We can apply this theorem to the limit c → 0 and the martingale difference array (X j,nm : j = 1, . . . , n m ), with n m = n(t m ) = t m /c . We have in probability as c → 0 by (53) and (52), which proves (i). To see (ii) note that |a j,n (k)| 4 √ c for all k K, j n, from which, for arbitrary ε > 0 and a constant C allowed to depend on the constants α k,l , K and m, for all sufficiently small c, nm j=1 E(|X j,nm | 2 1 {|X j,nm |>ε} ) Cc nm j=1 P(|X j,nm | > ε) Ct m P max j nm |X j,nm | > ε = 0.
Since the linear maps α k,l were arbitrary, this shows convergence of the finite-dimensional distributions of (Ã(t, k)) k 0 to those of (A(t, k)) k 0 . It remains to prove tightness. We will show that, for all p ∈ [2, ∞), all k 0 and all T ∈ [0, ∞), there is a constant C = C(p, η, k, T ) < ∞ such that, for all s, t ∈ [0, T ], lim sup Since we may choose p > 2, this implies tightness, by a standard criterion. Recall thatÃ and that (M 0 j (k) : j 0) is a martingale difference sequence with |M 0 j (k)| 2ce cj . Also 0 p(k) 1 and, estimating as above, Fix s, t ∈ [0, T ] with s t and note that n(t) − n(s) 1 + (t − s)/c. Theñ and so, by Burkholder's inequality, for some constant C = C(p, η, k, T ) < ∞, The asymptotic Hölder condition (54) follows.

Convergence as an analytic function
In this section we deduce the convergence ofM 0 (t, z) from that of the Laurent coefficients, thus concluding the proof of Theorem 1.2. To this end, set Proof. Fix ε, T, r, δ as in the statement, and partition the interval [0, T ] into sub-intervals |Ã(t, k)|r −k and so Recall thatÃ which shows that the process (p(k) −n(t)Ã (t, k)) t 0 is a martingale for each k 0, with Doob's L 2 inequality then gives for some positive constant C, depending on T , changing from line to line. In the last inequality we have used that p(k) 1 and cn(lδ) T + 1. Noting that n(lδ) − n((l − 1)δ) 1 + δ/c, and that p(k) e −c(k+1) for η ∈ [0, 1], we find as K → ∞ since e −2δ r > 1. If η < 0, the result follows from the same argument using that, for c small enough that σ − c − c|η| > 0, we have

Appendices A Particle estimates
Let c ∈ (0, ∞) and Λ ∈ [0, ∞). Recall that we say a univalent function F from D 0 = {|z| > 1} into D 0 has capacity c and regularity Λ if it satisfies condition (4), that is to say, for all z ∈ D 0 , .
We show that this in fact implies a similar condition for F but with better decay as z → ∞. Then we will give some explicit examples of suitable maps F . Finally, we will show that (4) holds whenever the corresponding particle is not too flat. Only Subsection A.1 is used in the paper.
A.1 Precise form of the particle hypothesis Our particle hypothesis (4) can be reformulated more precisely in terms of the coefficient a 0 in the Laurent expansion (1).
On letting z → ∞, we see that Then h is analytic in D 0 and bounded at ∞. We have whenever |z| = 2. Then, by the maximum principle, for all |z| 2, we have |h(z)| 6Λc 3/2 and hence .
Note that (57) with |β − 1| Λ √ c/2 implies (4) with Λ replaced by 7Λ. Thus the two conditions are equivalent up to adjustment of the constant by a universal factor.

A.2 Spread out particles
Consider for γ ∈ C the map on D 0 given by It is straightforward to check that F c,γ is univalent into D 0 if and only if |γ| γ(c) = 1 + c + √ 2c + c 2 .
Then F c,γ has capacity c and, since log we see that F c,γ has regularity Λ = 2|γ − 1|/ √ c. The corresponding particles P c,γ are spread all around the unit circle, as illustrated in the rightmost particle in Figure 1. When γ = γ(c) we find F (1) = 0 so P c,γ(c) has the form of a cusp with endpoint F (1). Moreover, in the limit c → 0 with γ = γ(c), the regularity constant Λ stays bounded and log F (1) ∼ √ 2c, so the endpoint lies at distance F (1) − 1 ∼ √ 2c from the unit circle.

A.3 Small particles of a fixed shape
The following proposition shows that our condition (4) holds generically for particles attached near 1 which are not too flat. In particular, it shows that, for particles of a fixed shape, such as slits or disks, attached to the unit circle at 1, in the small diameter limit δ → 0, the capacity c → 0 while the regularity constant Λ stays bounded, which is the regime in which our limit theorems apply.
Proof. The bounds on c are well known. The lower bound relies on Beurling's projection theorem and a comparison with the case of a slit particle. The upper bound follows from a comparison with the case P δ = S δ ∩ D 0 , where S δ is the closed disk whose boundary intersects the unit circle orthogonally at e ±iθ δ with θ δ ∈ [0, π] is determined by |e iθ δ − 1| = δ. See Pommerenke [11].
The desired bound on Λ then follows from (59) and (60) and the lower bound on c.
We can write where u and v are harmonic functions in D with u(z) → c and v(z) → 0 as z → ∞. Since F maps into D 0 , we have u(e iθ ) 0 for all θ ∈ [0, 2π). We have to show that u(e iθ ) = 0 whenever |θ| ∈ [a, π]. Set where B is a complex Brownian motion. Consider the conformal map f of D 0 to the upper half-plane H given by Set b = f (e −iθ δ ) = sin θ δ /(1 + cos θ δ ). Since δ 1, we have θ δ δπ/3 and then b 2πδ/9. By conformal invariance, .

B.2 Operator estimates
We note some L p -estimates on operators which act on the set of analytic functions f on {|z| > 1} which are bounded at ∞, and hence have a Laurent expansion Firstly, for the operator Df (z) = zf (z), by a standard argument using Cauchy's integral formula, there is an absolute constant C < ∞ such that, for all p ∈ N and 1 < ρ < r, Secondly, let L be an operator which acts as multiplication by m k on the the kth Laurent coefficient. Thus Assume that there exists a finite constant M > 0 such that, for all k 0, and, for all integers K 0, The Marcinkiewicz multiplier theorem [16, Vol. II, Theorem 4.14] then asserts that, for all p ∈ (1, ∞), there is a constant C = C(p) < ∞ such that, for all r > 1, Lf p,r CM f p,r .
We will use also the following estimate. Write L p,ρ→r for the smallest constant K such that Then, for all r, ρ > 1, we have h p/2,r L 2 p,ρ→r g 2 p,ρ f 2 2,ρ .
Proof. We can write Then

C Computations of first order and error estimates
In this section, we provide the detailed calculations behind the estimates stated in Section 3. We only explicitly state estimates for η ∈ [0, 1], taking advantage of the fact that certain constants can be chosen uniformly over such values of η. Similar estimates hold for η ∈ (−∞, 0), and we leave the necessary adjustments to the reader. Furthermore, throughout this section we assume that c, σ 1. This assumption can be relaxed at the cost of the absolute constants.

C.3 A more refined decomposition
The estimate (24), is not sufficiently tight for all our needs. In this section, we give a decomposition of w n , which can be used for more refined estimates. Set l(z) = log F (z) z − c, q(z) = l(z) − 2cβ z − 1 .
We will use the following Taylor expansioñ In this section, we give the proof of Lemma 5.2, which bounds the second martingale term W n (z) in the decomposition (32) of the differentiated fluctuation process, which is given by Then, on taking the · p/2,r -norm, we deduce that While it is possible to use the estimate (24) to bound this expression, the bound is only sufficient to prove our final result for σ c 1/3 . In order to obtain a bound that works all the way down to σ c 1/2 , we need the refined decomposition (71), for some m ∈ N which we will choose later. Define Then W n = U 0 n + U 1 n + U 2 n so, with obvious notation, Q W j,n 3(Q 0 j,n + Q 1 j,n + Q 2 j,n ) and so Q W j,n p/2,r 3 Q 0 j,n p/2,r + Q 1 j,n p/2,r + Q 2 j,n p/2,r . We estimate the terms on the right. First, for j N 0 , we have Cc 3 e 2cj r 3 (r − 1) 5 .