One-dimensional scaling limits in a planar Laplacian random growth model

We consider a family of growth models defined using conformal maps in which the local growth rate is determined by $|\Phi_n'|^{-\eta}$, where $\Phi_n$ is the aggregate map for $n$ particles. We establish a scaling limit result in which strong feedback in the growth rule leads to one-dimensional limits in the form of straight slits. More precisely, we exhibit a phase transition in the ancestral structure of the growing clusters: for $\eta>1$, aggregating particles attach to their immediate predecessors with high probability, while for $\eta<1$ almost surely this does not happen.


Conformal aggregation processes
Laplacian growth models describe processes where the local growth rate of a piece of the boundary of a growing compact cluster is determined by the Green's function of the exterior of the cluster. Such growth processes can be used to model a range of physical phenomena, including ones involving aggregates of diffusing particles. Discrete versions can be formulated on a lattice in all dimensions: some famous examples of this type of growth process include diffusion-limited aggregation (DLA) [30], the Eden model [4], or the more general dielectric breakdown model (DBM) [24]. Despite considerable numerical evidence suggesting that the clusters that arise in these processes exhibit fractal features, very few rigorous results are known (for DLA, see [16]) and it remains a formidable challenge to rigorously analyze long-term behavior such as sharp growth rates of the clusters.
One objection that can be leveled at lattice-based models is that the underlying discrete spacial structure could potentially introduce anisotropies in the growing clusters that are not present in the physical setting of the plane or three-space. Indeed, large-scale simulations in two dimensions demonstrate anisotropy along the coordinate axes [5]. This fact provides one motivation for the study of off-lattice versions of aggregation processes. In the plane, such off-lattice models can be formulated in terms of iterated conformal mappings, providing access to complex analytic machinery. Clusters produced by these conformal growth processes are initially isotropic by construction, but simulations suggest that in many instances, anisotropic structures appear on timescales where the number of aggregated particles become large compared to the size of the individual constituent particles. Nevertheless, proving the existence of such small-particle limits, whether anisotropic or not, has proved elusive, similarly to the case of lattice-based models.
A fascinating feature of Laplacian growth models is competition between concentration and dispersion of particle arrivals on the cluster boundary. Protruding structures ("branches") and their endpoints ("tips") tend to attract relatively many arrivals, but they compete with each other as well as the remainder of the boundary. (Kesten's discrete Beurling estimate gives an upper bound on the tip concentration in the case of DLA.) The degree to which tips are favored is determined by the exact choice of growth rule, and several models contain one or more parameters that affect concentration, dispersion, and competition [24,8,2,18].
Previous work on small-particle limits of conformal aggregation models [25,14,29] has yielded growing discs, that is, smooth and isotropic shapes; the dispersion effect "wins" in the limit. In this paper, we study a particular instance of a conformal growth model, focusing instead on the concentration aspect of Laplacian growth and showing that anisotropic scaling limits arise in the presence of strong feedback in the growth rule. The scaling limits we exhibit are highly degenerate in the sense that growth, which is initially spread out, favors tips very strongly, and eventually collapses onto a single growing slit.
To state our results, we first describe the general class of processes our object of study fits into. Let c > 0, and let f c denote the unique conformal map having f c (z) = e c z + O(1) at infinity, and sending the exterior disk ∆ to the complement of the closed unit disk with a slit of length d = d(c) attached to the unit circle T at the point 1. The capacity increment c and the length d of the slit satisfy e c = 1 + d 2 4(1 + d) ; (1) in particular, d ≍ c 1/2 as c → 0. In terms of aggregation, the closed unit disk can be viewed as a seed, while the slit represents an attached particle. Typically, we think of the particle as being small compared to the seed. A general two-parameter framework to model random or deterministic aggregation, based on conformal maps, is given by the following construction. Pick a sequence {θ k } ∞ k=1 in [−π, π), and let {d k } ∞ k=1 , or, equivalently, {c k } ∞ k=1 , be a sequence of non-negative numbers connected via (1). From the two numerical sequences {θ k } and {c k }, we obtain a sequence {f k } ∞ k=1 of rotated and rescaled conformal maps, referred to as building blocks, via Finally, we set Φ n (z) = f 1 • · · · • f n (z), n = 1, 2, . . . .
Each Φ n is itself a conformal map sending the exterior disk onto the complement of a compact set K n ⊂ C, that is, Φ n : ∆ → C \ K n .
The sets {K n } ∞ n=1 are called clusters. They satisfy K n−1 ⊂ K n , and model a growing twodimensional aggregate formed of n particles. At infinity, we have is the total capacity of the nth cluster. When modeling random aggregates formed via diffusion, one chooses the angles {θ k } to be iid, and uniform in [−π, π). Due to the conformal invariance of harmonic measure, this has the effect of attaching the nth particle at a point chosen according to harmonic measure (seen from infinity) on the boundary of K n−1 . This type of setup has been considered in a number of papers, see for instance [8,19,1,21,28,10,25,13,14,29]; we shall only briefly mention models that are particularly pertinent to our study.

ALE: Aggregate Loewner Evolution
The main object of study in the present paper is a model we refer to as aggregate Loewner evolution, abbreviated ALE(α, η), with parameters α ∈ R and η ∈ R. In ALE(α, η), conformal maps Φ n are defined as in (2) as follows.
Initialize by setting Φ 0 (z) = z and letting F 0 be the trivial σ-algebra.
• Next, we define a sequence of capacity increments for k = 1, 2, 3, . . . by taking We note that ALE(α, 0) is the same model as the Hastings-Levitov HL(α) model studied in [8,3,28,14], and in particular ALE(0, 0) coincides with the HL(0) model studied in depth in [25,29]. The Hastings-Levitov model was introduced as a conformal mapping model of dielectric breakdown (DBM) [24], a discrete model in which vertices are added to a growing cluster by drawing bonds from among the neighboring lattice points. At stage n of DBM(η), a point is added to the cluster K n by including a neighbor of (j, k) ∈ K n with probability Here, summation is over lattice neighbors of K n and the function φ n is discrete harmonic, that is ∆φ n = 0 on Z 2 \ K n , and has φ n = 0 on K n and φ n = 1 on some large external circle.
Off-lattice versions of DBM involving non-uniform angle choices determined by the derivative of a conformal map have been considered by several authors. Hastings [6], and subsequently Mathiesen and Jensen [21], study a model that essentially corresponds to ALE(2, η) modulo a slightly different parametrization in η. (In fact, an alternative name for the growth model in this paper could have been DBM(α, η) or HL(α, η), but we have opted for a different terminology to avoid confusion with lattice models, and also to emphasize connections with the Loewner equation, see below.) Hastings argues that large enough exponents, more precisely, for η 3 in our parametrization, the corresponding clusters become one-dimensional; he also points out that the behavior of the models depends strongly on the choice of regularization.
Another model that fits into this general framework is the Quantum Loewner Evolution model (QLE(γ, η)) of Miller and Sheffield [22,23] which is proposed as a scaling limit of DBM(η) on a γ-Liouville quantum gravity surface. In the QLE model, particles are attached according to a distribution which depends on the power of the derivative of the cluster map, as in (4), but with an additional term involving the Gaussian Free Field due to the presence of Liouville quantum gravity. In the construction of QLE, capacity increments are kept constant, as for ALE(0, η). However, each particle in QLE is constructed as an SLE curve, rather than the straight slits used in ALE.
Common to all conformal mapping models of Laplacian growth is the difficulty that derivatives of conformal mappings do not remain bounded away from 0 or ∞ as they approach the boundary and therefore the map θ → |Φ ′ n (e iθ )| −1 can be very badly behaved. For instance, even when n = 1, |Φ ′ n (e iθ )| −η is not integrable over T for certain values of η and hence the ALE(α, η) model would not be well defined if we were to use |Φ ′ n (e iθ )| −η as angle density. As mentioned above, for this reason we define the model via the regularization parameter σ as in (4), and then let σ → 0 together with the (pre-image) particle size. A similar difficulty arises from the dependence of the particle sizes on the derivatives of the conformal mappings. Although in this case the model is well-defined without the need for a regularization parameter in (5), it is no longer possible to guarantee that the resulting clusters have total capacity bounded above and below. Indeed, even with the presence of a regularization parameter, it is not clear that the total capacity remains bounded as σ → 0. The exception is the ALE(0, η) model: in light of (3), taking n ≍ c −1 is a natural choice of time-scaling in ALE(0, η) as with this choice the resulting clusters have total capacity bounded above and below. This in turn means that the total diameter of the clusters K n remains bounded as a consequence of Koebe's 1/4-theorem, see [27]. The fact that we have some a priori control over the global size of clusters is our main motivation for moving from studying HL(α) with α large to ALE(0, η) with η large. Simulations suggest that one-dimensional limits are present also in HL(α) for large α but showing that this is the case seems technically more difficult.
In this paper, we mainly focus on ALE(0, η) for η > 1, and show that the conformal maps Φ n converge to a randomly oriented single-slit map in the regime where n ≍ c −1 . This can be viewed as a rigorous version of Hastings' investigation [6] of ALE(2, η) for the ALE(0, η) model. To obtain our convergence results, we exploit what is in a way the most extreme mechanism that could lead to a single-slit limit, namely that of aggregated particles becoming attached to their immediate predecessors. The main difficulties in the proof are that the angle densitites induced by slit maps have maxima and minima of different orders, even in the presence of regularization, making it hard to show convergence to a point mass. Furthermore, the feedback mechanism in (4) is sensitive so that a single "bad" angle can destroy the genealogical structure of the growing slit by leading to the creation of a new, competing tip further down the slit, which could lead to a splitting of growth into two branches.

Overview of results
Clusters that are formed by successively composing slit maps come with a natural notion of ancestry for their constituent particles. We say that a particle j has parent 0 if it attaches directly to the unit disk and that the particle j has parent k if the jth particle is directly attached to the kth particle. More precisely, suppose that β c ∈ (0, π) is defined by |θ| < β c } so e ±iβc is mapped by the basic slit map to the base point of the slit i.e. f c (e ±iβc ) = 1. Therefore particle j has parent 0 if |Φ j (e i(θ j ±βc) )| = 1 and particle j has parent k 1 if where In the ALE(0, η) model, each successive particle chooses its attachment point on the cluster according to the relative density of harmonic measure (as seen from infinity) raised to the power η. As the highest concentration of harmonic measure occurs at the tips of slits, intuitively one would expect that for sufficiently large values of η each particle is likely to attach near the tip of the previous particle. In this paper we show that this indeed happens, and we identify the values of η for which the above event occurs with high probability in the limit as c → 0. Figure 1 displays ALE(0, η) clusters for different values of η.
The limiting behavior of the model is quite sensitive to the rate at which σ → 0 as c → 0. Figure  2 show how the angle sequences {θ k } in ALE(0, 4) are affected by the choice of exponent γ when regularizing by σ = c γ ; this phenomenon is also observed by Hastings in [6] for a related model. In [14], which deals with slow-decaying σ scaling limits in a strongly regularized version of HL(α), it is shown that the scaling limits of the clusters are disks. By choosing σ to decay sufficiently slowly compared to c, once can prove that the corresponding scaling limits in ALE(0, η) are again disks, this is the topic of forthcoming work of Norris, Silvestri, and Turner. As we seek results which do not strongly depend on the choice of regularisation parameter, part of our objective is to identify the minimal value of η for which there exists some σ 0 (dependent on c and η) such that, provided σ < σ 0 , with high probability each particle lands on the tip of the previous particle.
The following is the main result of the paper and shows that the ALE(0, η) model exhibits a phase transition at η = 1 in the genealogy of the growing cluster in the small-particle limit. See Theorem 16 for a complete statement and proof, in particular we give sufficient conditions on γ.  In the case when η > 1 and σ < σ 0 , it follows that, for any r > 1, and the cluster K n(t) converges in the Hausdorff topology to a slit of capacity t at position z = e iθ 1 .

A related Markovian model
Observe that, for each k, we are free to specify the interval of length 2π in which to sample θ k , and this choice does not have any effect on the maps Φ n . It is convenient to choose to sample θ k from the interval [θ k−1 − π, θ k−1 + π). In this case, we can express the event as (By definition, β c ∈ (0, π) and e i±βc is mapped by the basic slit map to the base point of the slit i.e. f c (e ±iβc ) = 1.) One of the main difficulties in analysing this event is that the distribution of θ k conditional on F k−1 (as defined in (4)), depends non-trivially on the entire sequence θ 1 , . . . , θ k−1 . In this subsection, we introduce an auxiliary model for random growth in the exterior unit disk in which the sequence of attachment angles is Markovian. The auxiliary model is relatively straightforward to analyse and we show below that it exhibits an analogous phase transition to that described above. The remainder of the paper is concerned with examining how ALE(0, η) and the auxiliary model relate to each other. Set Φ * 0 (z) = z and let {Φ * n } be conformal maps obtained through composing where each f * k is a building block with c k = c, and rotation angle θ * k having conditional distribution with density Here, we have set and suppressed the dependence on c, σ and η to ease notation. In order for the measure to be well-defined when η 1, we require σ > 0. In words, the density of the kth angle distribution in this model is obtained by replacing the complicated (k − 1)th cluster map of ALE by a simple slit map "centered" at θ * k−1 , and with deterministic capacity c(k − 1). For this model we obtain the following theorem: we again set n(t) = ⌊t/c⌋.
Furthermore, when η > 1 and σ < σ 0 , for any r > 1, and the cluster K n(t) converges in the Hausdorff topology to a slit of capacity t at position z = e iθ * 1 .
Proof. Since we can always rotate the clusters K n by a fixed angle, without loss of generality, we assume that the initial angle θ * 1 = 0. As above, we choose to sample θ * k from the interval [θ * k−1 − π, θ * k−1 + π). This means that we can write θ * n = u 2 + · · · + u n where the u k are independent [−π, π)-valued random variables and u k = θ * k − θ * k−1 has symmetric distribution h * k (θ|0). First suppose η > 1. Then by (19) and Lemma 8 below there exists some constant A (which may change from line to line), depending only on T and η, such that A −1 (kc) 1/2 < β kc < A(kc) 1/2 , Hence, for η > 1, as c → 0. Now suppose that η < 1. Then, using Lemmas 8 and 9 and letting c → 0, we get To show convergence of Φ * n(t) (z) to f t (z) for t < T when η > 1 and σ < σ 0 , by Proposition 3 it is enough to show that sup n N |θ * n | → 0 with high probability as c → 0. To do this, we write and note that M * n = n k=2 u k 1 {|u k |<βc/2} is a martingale. Since θ * n = M * n with high probability, convergence of sup n N |θ * n | to 0 follows from moment bounds in Lemma 9 together with standard martingale arguments.

Overview of the proof of Theorem 1 and organization of the paper
The main idea for the proof is to show that the Markovian model of the previous section is a good approximation of the ALE(0, η) process. In order to do this one approach would be to try to argue that |Φ ′ n (e σ+iθ )| can be globally well approximated by |(f θn nc ) ′ (e σ+iθ )|. However, this seems difficult to make work to sufficient precision when evaluating the maps close to the boundary. Specifically, the map Φ ′ n (z) has zeros (respectively singularities) at each of the points on the boundary of the unit disk which are mapped to the tip (respectively to the base) of one of the slits corresponding to an individual particle. In contrast, for the map (f θn nc ) ′ (z), the points corresponding to tips and bases of successive particles coincide and therefore the singularities and zeros corresponding to intermediate particles cancel each other out, leaving only a zero at the point mapped to the tip of the last particle and singularities at the two points which are mapped the base of the first particle (see Figure 3).
Interactions between nearby tips can be subtle and are in general hard to analyze [2]. Our strategy is instead to establish two properties of the distribution function h n (θ).
• The first is to show that near the tip of the last particle to arrive the derivatives are in fact very close and so for very small values of θ, h n (θ + θ n−1 ) can be well approximated by h * n (θ|0).
• The second property is to show that h n (θ) concentrates the measure so close to θ n−1 that even though the probability of attaching to earlier particles is higher than for the Markovian model, Ω N still occurs with high probability, provided we now require when regularizing by σ < σ 0 = c γ ; see Figure 4 for plots of the lower bounds on γ and γ * .
We now give a brief overview of the structure of the paper. In Section 3 we provide some background information on the Loewner differential equation, which allows us to represent the aggregate maps Φ n as solutions corresponding to a [−π, π)-valued driving process with equally spaced jump times and positions given by the random angles (4). In particular, we explain how convergence of an angle sequence {θ k } allows us to deduce convergence of the corresponding conformal maps Φ n . Figure 3: Diagram illustrating the presence of zeros and singularities in the derivative at each successive particle tip and base in Φ n (z) (left). These zeros and singularities are absent in f nc (z) except at the tip of the final particle and base of the first particle (right).
In Section 4 we obtain estimates on the slit map used to construct ALE, as well as estimates on its derivatives. These latter estimates lead to moment bounds for [−π, π)-valued random variables constructed from slit map derivatives. The arguments used are elementary in nature, and heavily use the explicit form of the slit map.
Section 5 contains most of the technical machinery needed for the proof. In this section, we obtain estimates on the distance between two solutions to the Loewner equation in terms of the distance between their respective driving functions in the case where we know that one of the solutions is a slit map. These estimates, which we believe may be of independent interest, enable us to obtain much more precise estimates than exist for generic solutions. In particular, our estimates give very good approximations when the conformal mappings are quite close to the boundary, whereas generic estimates blow up in this region. We perform this analysis by splitting the Loewner equation into radial and angular parts, and linearizing the resulting differential equations.
In Section 6, we use our estimates on Loewner derivatives at the tip and away from the approximate slit to show that h n (θ), the density function for the nth angle θ n , has the required behaviour. Then, similar arguments to those used in the proof of Theorem 2 are used to establish Theorem 1, but since {θ k } does not have a Markovian structure, there are further terms to control. Finally, we discuss some extensions of our results, valid for certain instances of the ALE(α, η) model as well as related models.

Notation
Many of the estimates presented in this paper, especially in Section 5, are more precise than what is strictly needed for the proof of our main theorem, in that we frequently keep track of the dependence of constants on parameters, and similar. We have opted to record detailed versions to enable potential further applications where such dependencies may be important. Generic constants, which may change from line to line, will mainly be denoted by the capital letters A and B.

Loewner flows
We shall make extensive use of Loewner techniques in this paper. Loewner equations describe the flow of families {Ψ t } t 0 of conformal maps of a reference domain in C ∪ {∞} onto evolving domains in the plane in terms of measures on the boundary. We only give a very brief overview here, and refer the reader to [17] and the references therein for a discussion of Loewner theory.

Loewner's equation
Let {µ t } t>0 be a family of probability measures on the unit circle T, in this context referred to as driving measures. Then the Loewner partial differential equation for the exterior disk, with initial condition Ψ 0 (z) = z admits a unique solution {Ψ t } t 0 called a Loewner chain. Each Ψ t (z) is a conformal map of the exterior disk onto a simply connected domain, and at ∞ we have the power series expansion Ψ t (z) = e t z + O(1). The growing compact sets {K t } t 0 are called hulls, satisfy K s K t for s < t, and have cap(K t ) = e t for t 0, where cap(K) denotes the logarithmic capacity of a compact set K ⊂ C.
The limit functions appearing in Theorem 1 can be realized in terms of Loewner chains, and in fact have a very simple Loewner representation.
Example 1 (Growing a slit). Let µ t = δ 1 , a point mass at ζ = 1. Then (7) reads With initial condition f 0 (z) = z, the solution has the explicit representation (viz. [20, p. 772]) The solution precisely consists of the slit maps f t : This means that the growing hulls are K t = D ∪ (1, 1 + d(t)], the closed unit disk plus a radial slit emanating from ζ = 1.
In this paper, we are mainly concerned with the case µ t = δ e iξ t for some function ξ t : (0, T ] → R and in that setting, we refer to ξ t as a driving term. The conformal maps arising in ALE(α, η) have the following simple Loewner representation. We first solve the Loewner equation with driving measure µ t = δ e iξ t , where with C k = k j=1 c k , and the angles {θ k } and capacity increments {c k } given by (4) and (5), respectively. Explicitly then, the Loewner problem associated with ALE(α, η) reads To obtain the composite ALE(α, η)-maps Φ n described in Section 1, we evaluate the solution to (11) at t = cn; thus Φ n = Ψ cn , n = 1, 2, . . . .
The random driving function ξ t can be viewed as a càdlàg jump process exhibiting a complicated dependence structure encoded through angles and capacity increments. When α = 0, the dependence structure is only present in the distribution of the increments, as the jump times are deterministic, and equal to c · k for k = 1, 2, . . .. We emphasize that this is the main technical reason why the ALE(0, η) is easier to analyze then the general ALE(α, η) model or the Hastings-Levitov model HL(α). Another version of Loewner's equation arises in the setting of the upper half-plane this is usually referred to as the chordal version of the Loewner equation. While the ALE model is defined in ∆, the half-plane formulas are simpler and are used in our analysis of the tip behavior, and so we give a quick overview here. The chordal version of Loewner's equation with driving and we again consider the initial conditionΨ 0 (z) = z. Solving (12), we obtain a family of conformal mapsΨ t : H → H \K t , mapping the half-plane to the half-plane minus a compact set and having expansionΨ t (z) = z − hcap(K t )/z + O(1/z 2 ) at infinity. The quantity hcap(K t ) = 2t is referred to as the half-plane capacity.
Example 2 (Growing a slit in the half-plane). In the case whereξ t = 0, the chordal equation becomes and we again obtain slit map solutions. In the half-plane, the corresponding closed formula is given by a substantially simpler expression than for ∆, namely

Reverse-time Loewner flow
The Loewner equation (11) is a first-order partial differential equation, and in the ALE(α, η) model, it gives rise to a non-linear PDE problem since the driving measures depend on the maps f t via their derivatives. As is common in Loewner theory, we shall analyze solutions by passing to the backwards flow associated with (11): this essentially entails employing the method of characteristics to obtain an ordinary differential equation that describes the evolution at hand. See [17,1] for detailed derivations and discussions. Let T > 0 be fixed. The equation for the backward or reverse-time flow in the exterior disk is where we define Then, setting u 0 (z) = z, we obtain (see [17,Chapter 4]) where f t denotes the solution to the forward equation (11) with driving function ξ t . Note that this holds in general only at the special time T . In the half-plane, the corresponding reverse flow is governed by the equation with driving functionΞ t =ξ T −t and initial condition h 0 (z) = z. For each T > 0, we again have whereΨ T (z) is the solution to the Loewner PDE (12) for the upper half-place with driving functioñ ξ t , evaluated at time T .
The main advantage of the backward flow is the fact that, for each z, (13) and (14) are now formally ODEs, simplifying the problems of analyzing and estimating solutions to the corresponding flow problems. Such analyses are carried out in Section 5 and will be crucial in the proof of Theorem 1.

Convergence of Loewner chains
Our strategy will be to argue that the driving functions arising in the ALE process are close, in the regime where n ≍ c −1 , to the constant driving function ξ t = θ 1 . We would then like to argue that the resulting conformal maps are close. These kinds of continuity results have been established in several settings, see for instance [10, Proposition 3.1] and [13, Proposition 1], and [9] for a more systematic discussion.
Since the ALE driving processes exhibit synchronous jumps, it is natural to measure distances between them in the uniform norm · ∞ . For T > 0, we denote the space of piecewise continuous functions ξ : [0, T ) → R endowed with this norm by D T . We consider the space Σ consisting of conformal maps f (z) = Cz + O(1), with C > 0 uniformly bounded, and we endow Σ with the topology of uniform convergence on compact subsets of ∆. We then view the conformal maps Ψ t , and hence the aggregate maps Φ n , as random elements of Σ.
The following result is well-known, but we give a proof for completeness. (With additional work, one could obtain estimates on rates of convergence, but we do not pursue this direction here.) be the solution to the Loewner equation (7) with driving term ξ Proof. Fix s ∈ [0, T ] and consider the reverse-time Loewner equation (13). We let u (j) t be the reverse flow driven by ξ (j) Taking the difference and differentiating H = u (1) − u (2) with respect to t giveṡ .
Since the flows move away from the unit circle, these expressions show that there is a constant A depending only on T such that if |z| 1 + ǫ then for all 0 t s T , Since H(0) = 0, Hence we can take δ < e −A/ǫ 2 ǫ 3 /A and this is clearly uniform in 0 t T .
Thus, we obtain convergence in law of conformal maps provided we can show convergence in law of driving processes. Note that in our main result we have convergence to a degenerate deterministic limit (modulo rotation). As is explained in [10, Section 4.2], we can strengthen the convergence that follows from Proposition 3 in this instance, and obtain convergence of K n with respect to the Hausdorff metric in ∆.

Analysis of the slit map
In our arguments, we shall need effective bounds on the building blocks f k making up the aggregated map Φ n = f 1 • · · · • f n , as well as on the derivatives f ′ k , in order to estimate moments of angle sequences, among other things. In this section, we present both global and local estimates for the slit map.
An explicit formula for the slit map f t : ∆ → ∆ \ (1, 1 + d(t)] was given in (8), while the length d(t) of the growing slit is given by (9). When performing estimates, it is sometimes helpful to view f t as a composition of Möbius maps with a slit map in the upper half-plane.
Define conformal maps m H : ∆ → H = {z ∈ C : Imz > 0} and m ∆ : H → ∆ by setting and For d > 0, we check thatf maps the upper half-plane conformally onto the upper half-plane minus the slit 0, i d d+2 . Note that (17) is a rescaling of the half-plane slit map from Section 3.
By following the image of the unit circle T under composition, and checking the trajectories of the point at infinity and the boundary point 1 ∈ ∆, we verify that where d(t) is the explicit formula in (9). We note that f t (1) = 1 + d(t), and that one can compute that f t (e iβt ) = f t (e −iβt ) = 1 for We shall refer to exp(iβ t ) and exp(−iβ t ) as the base points of the slit. In our scaling limit results, we will make use of the facts that β t d(t) → 1 and d(t) 2t 1/2 → 1, as t → 0.

The half-plane
We start by analyzing the half-plane slit map F t (z) = √ z 2 − 4t in detail near the tip. We note that Thus, F ′ t has a zero at z = 0, which gets mapped to the tip, and singularities at ±2t 1/2 , the points that are mapped to the base of the slit.
Lemma 4 (Near the tip, half-plane version). Let z = x + iy, where 0 y t 1/2 and |x| 6 5 and Proof. By symmetry, it suffices to establish these bounds for x > 0. The upper bound in (21)

The exterior disk
We now transfer back to the exterior disk, the setting of the ALE model, using the decomposition To do this, we reparametrize and rescale the half-plane slit map to obtaiñ We recall from (19) that ρ(t)/t 1/2 → 1 as t → 0. First, we describe the radial and angular effect of the building block map f t near the tip for small values of t.
Using the fact that the real part is maximized on the unit circle, at arg z = β t /2, together with the identities sin(arctan(x)) = x √ 1+x 2 and cos(arctan(x)) = 1 √ 1+x 2 , we find that for 1 < r < 1 + 1 4 t 1/2 and |θ| β t /2, We also have and by a similar argument to the one used to estimate the real part, Thus, we can apply Lemma 4 to obtain To finish the proof, note that if w = x + iy ∈ H, For y > x, which is satisfied when We also have The assertion of the Lemma now follows from the bounds in (22) and the fact that 2Im[f d(t) (m H (z)] 5 2 ρ(t) 3 4 , by our assumption on t.
We now examine the derivative of the slit map in ∆.
Lemma 6. For 0 < t 1 and 1 < |z| 2, we have where H t (z) is holomorphic in z, has lim z→∞ H t (z) = 1, and satisfies for an absolute constant A > 0.
Proof. Since the slit map f t (z) solves the Loewner equation Differentiating the explicit expression (8) with respect to t, we find that Inserting this into (24), we obtain It remains to show that H t is bounded above and below. But this follows immediately upon writing H t (z) = z −1 e −t f t (z), where f t is the slit map itself. Finally, we verify that z t = e iβt solves (z + 1) 2 − 4e −t z = 0, and this leads to the factorization (z + 1) 2 − 4e −t z = (z − e iβt )(z − e −iβt ). Our analysis of the ALE model will require local estimates on the derivative of the slit map. Representative graphs of how θ → |f ′ t (e σ+iθ )| varies with t and σ are shown in Figure 5. Lemma 7. Let 0 < t 1 and suppose |z| − 1 d(t). Then the derivative of the slit map admits the following estimates: 1. (Near the tip) For | arg z| < 1 2 β t ,
Near the base, the same reasoning as before shows that |z − 1| ≍ d(t). On the other hand, where the lower bound is attained when arg(z) = β t . Combining these bounds leads to the claimed estimates for | arg z ± β t | βt 2 . On each fixed radius, the function v : arg(z) → with v(π) = (e log |z| + 1)/((e log |z| + cos β t ) 2 + sin 2 β t ) 1/2 1. So in order to obtain the last set of estimates, it suffices to note that v remains bounded above and below as arg(z) → 3 2 β t , by the same arguments as before.

Moment computations
We now return to random growth models and present the moment bounds used in Section 2. As before, σ > 0 is our regularization parameter, while η > 0 is a model parameter.
For t > 0, define the normalization factor Lemma 8. Let 0 < t < 1 and suppose σ t 1/2 . The total mass Z * t satisfies the following.
• (η < 1) There are constants A 1 and A 2 such that In particular, Z * t remains finite as σ → 0.
We turn to the case η > 1. Since the integral |f ′ t (e is )| −η ds now diverges due to the singularity at s = 0, it again suffices to estimate the contribution coming from |θ| < β t /2 in order to establish (27). We have after a change of variables. Since ∞ 0 (1 + u 2 ) −η/2 du is now finite, the upper bound follows. Similar reasoning together with the assumption that σ t 1/2 yields the lower bound on the integral. The estimates on the normalized derivative for η > 1 follow upon dividing through in Lemma 7 by the total mass Z * t .
We now turn to moment bounds for η > 1.

Lemma 9.
Under the hypotheses of Lemma 8, we have, for each η > 0, Suppose x ∈ (σ, βt 2 ). Then, for 1 < η < 3, we have and for η = 3, we have For η > 3, we have Proof. The statement that θ|f ′ t (e σ+iθ )| −η dθ = 0 follows immediately from symmetry of the function θ → |f ′ t (e σ+iθ )| for each σ and t. We turn to second moments, and deal with the parameter range 1 < η 3 first. By Lemma 8, Performing a change of variables, and assuming η < 3, we obtain the integral as claimed. An obvious modification of the argument leads to bounds for η = 3. Finally, we treat the case η > 3 and show that the second moment decays like σ 2 independently of η. It now suffices to examine The integral on the right now converges since η > 3, and in fact .
To get the lower bound, we use the assumption 1 < x/σ to bound the integral from below. The second assertion of the Lemma follows.

Estimates on conformal maps via Loewner's equation
We now obtain refined estimates on the distance between solutions to the Loewner equation in terms of the distance between their driving functions, in the special case when the driving functions are close to constant. Generic estimates between conformal maps tend to blow up close to the boundary (as seen in, for example, Proposition 3).
As we wish to compare |Φ ′ n (e σ+iθ )| to |(f θn nc ) ′ (e σ+iθ )| when σ is typically much smaller than the difference between the respective driving functions, we need bespoke estimates which behave well close to the boundary.
Suppose Ψ t (z) is the solution to the Loewner equation (11). In this section, we compare Ψ t (z) and Ψ ′ t (z) to f t (z) and f ′ t (z) in the case when ξ, the driving function of Ψ, is close to zero. For fixed T > 0, let ξ T = sup Writing u ξ t = r ξ t e iϑ ξ t , substituting into (13) and separating Re[(u ξ t + 1)/(u ξ t − 1)] and Im[(u ξ t + 1)/(u ξ t − 1)] we obtain the two differential equations and In the special case when ξ ≡ 0 it is straightforward to see that if |z| > 1 and arg z = 0, then ϑ 0 t = 0 for all t, whereas if |z| = 1 and | arg z| > 0, then r 0 t = 1 for all t < inf{s > 0 : ϑ 0 s = 0}. Therefore, in these two cases, solving the pair of differential equations above reduces to solving a single ordinary differential equation, and we are able to obtain explicit solutions. We establish our bespoke estimates by linearising the differential equations around these explicit solutions.
In Section 5.1 we perform the analysis in the case when arg z is close to 0, and in Section 5.2 we carry out the case when |z| is close to 1. We also obtain cruder estimates which apply in the intermediate regime between these two cases and will be used in the next section to "glue" the two results together.

Near the tip
In this subsection, we analyse Ψ t (z) in the case when arg z is close to zero. We begin by computing an explicit expression for f t (r) when r > 1. Consider the reverse-time Loewner equations above for z = r, with ξ t = 0 for all t. Since there is no dependency on T , we have that f t (r) = u 0 t for all t > 0. From (28) it is immediate that ϑ 0 t = 0 for all t > 0. Substituting this into (28) we get Solving this gives log (r 0 t + 1) 2 r r 0 t (r + 1) 2 = t or Observe that if r = 1, then r 0 t = d(t) + 1. Now consider Ψ t having general driving function ξ t which is assumed to be small. Proposition 10. There exists some absolute constant A such that, for all 1 < |z| < 2 and T > 0 and where r 0 t is given by (31) with r = |z|. Hence A can be chosen so that Proof. First suppose that z = r 0 ∈ R and let u ξ t = r ξ t e iϑ ξ t be the solution to (13) starting from z. We begin by showing that |ϑ ξ t | ξ T for all t T . Suppose that there exists some t 1 T such that ϑ ξ t 1 > ξ T . Since ϑ ξ 0 = 0 and t → ϑ ξ t is continuous, there exists s 1 < t 1 such that ϑ ξ s 1 = ξ T and ϑ ξ t > ξ T for all t ∈ (s 1 , t 1 ] (take s 1 = sup{t t 1 : ϑ ξ t < ξ T }). Then since ξ T −t ξ T for all t T , sin(ϑ ξ t − ξ T −t ) > 0 for all t ∈ (s 1 , t 1 ]. Therefore, by (29), Hence ϑ ξ t 1 < ϑ ξ s 1 = ξ T which contradicts our assumption that ϑ ξ t 1 > ξ T . By symmetry, there is also no t T for which ϑ ξ t < − ξ T and hence |ϑ ξ t | < ξ T for all t T . We now turn to r ξ t . Note that by (28), r 0 r ξ t r 0 t . Set δ r (t) = r 0 t − r ξ t and let Suppose that t T ξ . Set Here we have used Then, using (30), d dt and hence By Taylor's Theorem, for some a t ∈ (0, 1). Now We now bound these functions for r > 1, seeking to reduce the singularities in the denominators as much as possible by exploiting simultaneous vanishing in the numerators. By using the fact that 2r(1 − cos x) r 2 − 2r cos x + 1, we have Observe that since ξ T A −1 (r 0 − 1) A −1 (r 0 t − 1), provided A 2, we obtain the crude estimates (r 0 Therefore, using the fact that t T ξ , there exists some absolute constant A ′ such that Substituting this bound into (34), we get Here we have used that r 0 t + 1 5e t and that x log(1/x) is bounded by 1. Therefore, taking A > 10A ′ , we have δ r (t) ξ T and hence T ξ = T . It follows immediately that | arg Ψ T (z)| ξ T and 0 r 0 Now suppose that arg z = θ = 0. It is straightforward to show that where ψ T (z) solves the Loewner equation with driving function ξ t − θ. It follows that |Ψ T (z)| = |ψ T (|z|)| and arg Ψ T (z) = arg ψ T (|z|) + θ.
The result follows by replacing Ψ T (z) by ψ T (|z|) and ξ t by ξ t − θ in (36). Finally, for the last part of the theorem, note that, if ξ t ≡ 0, then clearly the conditions of the theorem are satisfied. Therefore, |f T (z)| − 1 and arg f T (z) also satisfy the above bounds from which the second statement of the theorem follows.
We now turn to comparing the derivatives Ψ ′ t (z) and f ′ t (z).
Proposition 11. There exists some absolute constant A (taken to be at least as large as the absolute constant in Proposition 10), such that for all 1 < |z| < 2 and T > 0 satisfying ξ T + | arg z| Proof. Recall that Ψ T (z) = u ξ T (z) where u ξ t (z) is the solution to (13) starting from u ξ 0 (z) = z. Hence Ψ ′ T (z) = ∂ z u ξ T (z). Set y ξ (t) = ∂ z u ξ t (z). Differentiating both sides of (13) gives Hence, using y ξ (0) = 1, we get and so

Away from the slits
(Note that δ r (t) and T ξ as defined here are not the same as the δ r (t) and T ξ defined in Section 5.1). We begin by obtaining some crude bounds on δ r (t) and |δ θ (t) − ξ T −t | that hold under the assumption that t T ξ . In what follows, let A 12. Firstly, note that by standard trigonometric identities, and using the explicit value of ϑ 0 t from (40), Hence, using the fact that ϑ 0 t is decreasing in t, and Then, for any a ∈ (0, 1), and We now turn to (28) and (29). Define h 1 (r, x) as in (32) and We first analyse δ r (t). Set Here we have used the expressions for the partial derivatives of h 1 (r, x) in (33). By computing the integrating factor corresponding to this expression, we obtain Hence, using that δ r (0) = |z| − 1 and ϑ 0 0 = θ, log δ r (t) tan By Taylor's Theorem, for some a t ∈ (0, 1). Using (35), together with the fact that (r − 1) 2 r 2 − 2r cos x + 1, for r ∈ (1, 3/2) and x ∈ (0, π) we have that Hence, using that t T ξ and the crude bounds (42) -(45), for some absolute constant A ′ , where we used the assumption on T to show that cot(θ/2) 1/ √ e T − 1.
Substituting this bound into (47), we get Here we have used the condition on T to deduce that √ e T − 1 tan(θ/2). Finally, observe that, provided A A ′ ∨ A ′′ , this bound and the bound on δ r (t) together imply that T = T ξ . Hence Finally, for the last part of the theorem, note that, if ξ t ≡ 0, then clearly the conditions of the theorem are satisfied. Therefore, |f T (z)| − 1 and arg f T (z) also satisfy the above bounds from which the second statement of the theorem follows.
We now turn to comparing the derivatives Ψ ′ t (z) and f ′ t (z).
Hence, using the crude estimates (42) -(45), Using exactly the same argument as in Proposition 12, there exists some absolute constant A such that The same bound holds for |f ′ T (e σ+iθ )| and hence Finally, we give a lower bound on the modulus of the derivative of a map associated with a general Loewner driving function.

Proposition 14.
There exists some absolute constant B such that, for all T > 0, Proof. We first obtain a generic lower bound on |Ψ ′ T (z)|, without making any assumptions on the driving function ξ or initial value z. Using the notation of the previous proof, recall that By (28), Therefore, using the fact that |Ψ T (z)| |z|, Suppose T satisfies the conditions of Proposition 13. Then |Ψ ′ T (z)| e −1 tan(arg(z)/2) cot(ϑ 0 t /2) 1 3 and hence, since T log 2 1+cos arg z implies that so the required result holds provided B 6. If T does not satisfy the conditions from Proposition 13, then there exists some 0 < S 1 < T such that . We can write Ψ T (z) = Ψ T −S 1 (ψ S 1 (z)) where ψ S 1 is the solution to the Loewner equation for some driving function which is bounded by ξ T . Using the generic estimate above, the results of Propositions 12 and 13 applied to ψ S 1 (z), the identity in (41), and that |Ψ T (z)| − 1 d(T ) 4e T , Taking the absolute constant B = 36A, gives the required result.

Ancestral lines and convergence for ALE
We now return to the ALE(0, η) process and show how the bounds obtained above allow us to prove the analogue of Theorem 2 for the Φ n maps that generate ALE(0, η) clusters.

Combined derivative estimates
We first begin by using the results of Section 5 to estimate how well |Φ ′ n (e σ+iθ )| can be approximated by |(f θn nc ) ′ (e σ+iθ )|. In Section 2, we discussed how the intermediate particles are visible in the derivative of Φ n (z) in a way they are not in f θn nc (z) (see Figure 3). The estimates below capture this discrepancy.
(i) There exists some constant A, depending only on T , such that if |θ − θ n | < c 1/2 and ǫ n < (ii) There exists a constant B only dependent on T , such that Proof. (i) By the chain rule, .
Hence, by Proposition 11, there exists some constant A, depending only on T , such that Note how this argument uses that Φ n evolves in discrete steps, allowing us to invoke Lemma 5.

The ancestral lines and convergence theorem
We now use the Lemma above to prove our main result. Fix T > 0 and set N = ⌊T /c⌋. Recall the definition of Ω N from Section 2.
Furthermore, when η > 1 and σ < σ 0 , for any r > 1, and hence the cluster K n(t) converges in the Hausdorff topology to a slit of capacity t at position 1.
Proof. Fix η > 1 and let We shall first show that P(N T = N ) → 1 as c → 0. Suppose that n < N T . Using the fact that σ < σ 0 , we have Hence, using the fact that γ > λ + 1/2, there exists some c 1 > 0, dependent only on T and η, such that if c < c 1 , then ǫ n satisfies the conditions of Lemma 15. From now on assume that c < c 1 . Then, by Lemma 15, there exists A n such that, if |θ − θ n | c 1/2 and furthermore A n = A η σ 2 c −1 n 2λ (log c −1 ) 12λ for A η that depends only on η and T . Also, for some B ′ that depends only on η and T . We begin by getting estimates on Z n . Using the notation of Section 2, recall from Lemma 8 that there exist A ′ , A ′′ depending only on η and T such that Hence, for some B η that depends only on η and T . Set B n = B η σ η−1 n λη c −(2η−1)/2 (log c −1 ) 6λη .
Observe that the choice of γ ensures that, provided σ < σ 0 , we have c −1/2 A n → 0 and N B N → 0. We shall see that these conditions are sufficient to prove our result. Now Similarly, we can show that Z n (1 − A n − B n )Z * nc . Since A n + B n → 0 as c → 0 there exists 0 < c 2 c 1 , depending only on T and η, such that A n + B n < 1/2 provided c < c 2 . Assume from now on that c < c 2 . Hence, if |θ − θ n | < c 1/2 then, where α n = 7(A n + B n ). Equivalently As in Section 2, we choose to sample θ k from the interval [θ k−1 − π, θ k−1 + π) and so we can write θ n = u 2 + · · · + u n where the u k are [−π, π)-valued random variables and, conditional on F k−1 , u k = θ k − θ k−1 has distribution function h k (θ + θ k−1 ). We write where We first show M n is small with high probability. By Lemma 9, for some constant A depending only on T and η. Hence M n is a martingale with quadratic variation By Bernstein's inequality, we obtain that P |M n | > σn λ (log c −1 ) 6λ /2 for all n N T AN exp −A(log c −1 ) 4λ → 0 as c → 0 as desired. We next turn to the second term in (51). We use that by the symmetry of h * k (θ|0). Hence, by a similar computation to that in Lemma 9, for some constant A depending only on T and η. Therefore, if 1 < η < 2, σn λ (log c −1 ) 6λ An 1−λ α n (log c −1 ) −2(3λ−1) .
By our choice of γ, there exists 0 < c 3 c 2 , depending only on T and η, such that n k=1 E u k 1 {|u k | k λ σ(log c −1 ) 2λ } F k−1 < σn λ (log c −1 ) 6λ /2 provided c < c 3 . From now on assume that c < c 3 . Finally, we deal with the last term in (51). The same computation as used to bound Z n can be used to show that P(|u k | c 1/2 ; k N T ) B k .

Modifications of the model
One criticism that can be levelled at the ALE(0, η) model, from the point of view of modelling physical phenomena, is that the conformal mappings distort the sizes of particles as they are added to the growing cluster. Using the result proved above that the scaling limit of the ALE(0, η) cluster is a growing slit, it can be shown that the size of the nth particle is approximately equal to d(cn) − d(c(n − 1)). Using the expression for d(t) in (9), we obtain d(cn) − d(c(n − 1)) ≍ 2c 1/2 n 1/2 +(n−1) 1/2 if cn ≪ 1; 2ce cn if cn ≫ 1.
In particular, the first particle is of size approximately 2c 1/2 , whereas all subsequent particles are strictly smaller. A number of modifications to the model are possible which result in clusters where all of the particles are roughly the same size. The simplest modification is to recursively choose a deterministic sequence of capacities with c 1 = c and c n satisfying Another modification (see [6,21]) is to take the capacity of the nth particle to be c n = c|Φ ′ n−1 (eσ +iθn )| −2 for some regularization parameterσ > 0, not necessarily equal to the angular regularization parameter σ. Closely related (see [1,28]), is to choose capacity c n corresponding to slit length d n = inf{d > 0 : d|Φ ′ n−1 ((1 + d)e iθn )| = d(c)}.
In each of these modified models, the total capacity of the cluster no longer grows linearly in the number of particles and is potentially random. It is therefore necessary to modify the timescale in which to obtain scaling results. More precisely, given some fixed T > 0, let n(t) = sup{n : C n < t} for t T, and set N = n(T ). The event Ω N can then be defined as before.
It is relatively straightforward to verify that the proof and conclusion of Theorem 16 still hold for these modified models (and further generalisations). We only state the modified result for η > 1, as the case η < 1 is identical to that for the Markov model, for any choice of capacity sequence.
Provided there exists some constant A > 0, depending only on T and η, such that P(c k Ac for all k = 1, . . . N ) → 1 as c → 0, it holds that P(Ω N ) → 1 as c → 0. Furthermore, such a constant A exists for the three modifications defined above as well as for ALE(α, η) for any α > 0.
In this case, for any r > 1, sup t T sup |z|>r |Φ n(t) (z) − f t (z)| → 0 in probability as c → 0, and hence the cluster K n(t) converges in the Hausdorff topology to a slit of capacity t at position 1.
Proof. The proof consists of checking step by step that each inequality in the proofs of Lemma 15 and Theorem 16 still holds (possibly with new constants). The only changes are that we compare Φ n to f θn Cn = f θn c 1 • · · · • f θn cn instead of f θn cn and we need to define N T = inf k 1 : |θ k | > σk λ (log c −1 ) 6λ or c k < Ac ∧ N and then use the additional assumption in the statement of the corollary to show that N T = N with high probability.
To show that the additional assumption holds for the modified models defined above, it is enough to show that, so long as n N T , there exists some constant A (depending only on T and η), such that |Φ ′ n−1 (eσ +iθn )| −1 > A. But this follows by using the (analogous) estimates in Lemma 15 for the modified model and observing that there exists some constant A ′ (depending only on T ) such that whenever | arg(z)| β t /2 and t T .