Stability of regularized Hastings-Levitov aggregation in the subcritical regime

We prove bulk scaling limits and fluctuation scaling limits for a two-parameter class ALE$(\alpha,\eta)$ of continuum planar aggregation models. The class includes regularized versions of the Hastings--Levitov family HL$(\alpha)$ and continuum versions of the family of dielectric breakdown models, where the local attachment intensity for new particles is specified as a negative power $-\eta$ of the density of arc length with respect to harmonic measure. The limit dynamics follow solutions of a certain Loewner--Kufarev equation, where the driving measure is made to depend on the solution and on the parameter $\zeta=\alpha+\eta$. Our results are subject to a subcriticality condition $\zeta\le1$: this includes HL$(\alpha)$ for $\alpha\le1$ and also the case $\alpha=2,\eta=-1$ corresponding to a continuum Eden model. Hastings and Levitov predicted a change in behaviour for HL$(\alpha)$ at $\alpha=1$, consistent with our results. In the regularized regime considered, the fluctuations around the scaling limit are shown to be Gaussian, with independent Ornstein--Uhlenbeck processes driving each Fourier mode, which are seen to be stable if and only if $\zeta\le1$.


Hastings-Levitov aggregation
In many physical contexts there appear clusters whose shape is complex, formed apparently by some mechanism of random growth. It has long been a challenge to account for the observed variety of complex cluster shapes, starting from plausible physical principles governing the aggregation of individual microscopic particles. For clusters which are essentially twodimensional, there is an approach introduced by Carleson and Makarov [2] and Hastings and Levitov [4], in which clusters are encoded as a composition of conformal maps, one for each particle. In this approach, a growing cluster is modelled by an increasing sequence of compact sets K n ⊆ C which are assumed to be simply connected. We will take the initial set K 0 to be the closed unit disk {|z| 1}. The increments K n \ K n−1 are then thought of as a sequence of particles added to the cluster. The idea is to study the clusters K n via the conformal isomorphisms Φ n : D 0 → D n where D n is the complementary domain C \ K n and Φ n is normalized by Φ n (∞) = ∞ and Φ n (∞) > 0. Then Φ 0 (z) = z for all z and K n has logarithmic capacity Φ n (∞) > 1 for all n 1. This formulation is convenient because the harmonic measure from ∞ on the boundary ∂D n , which provides a natural way to choose the location of the next particle, is then simply the image under Φ n of the uniform distribution on ∂D 0 = {|z| = 1}. Having chosen a random angle Θ n+1 to locate the next particle, and a model particle P n+1 attached to K 0 at e iΘ n+1 , for example a small disk tangent to K 0 , the cluster map is updated to where F n+1 is the conformal isomorphism D 0 → D 0 \ P n+1 , normalized similarly to Φ n . Then Φ n+1 encodes the cluster K n+1 = K n ∪ Φ n (P n+1 ).
Thus, once we specify distributions for the angles Θ n and model particles P n , we have specified a mechanism to grow a random cluster. We will write cap(K n ) = log Φ (∞), c n = log F n (∞) and we will refer to cap(K n ) as the capacity 4 of K n and c n as the capacity of P n . Then cap(K n ) = c 1 + · · · + c n .
We will be looking for scaling limits where the particle capacities c n and the associated particles P n become small, but where n is chosen sufficiently large that the cluster capacities cap(K n ) grow macroscopically. A simple case is to choose Θ n+1 uniformly distributed on the unit circle and to take P n+1 = e iΘ n+1 P , where P is a small disk tangent to the unit disk at 1, of radius r(c), chosen so that P has capacity c. Then in fact r(c)/ √ c has a positive limit as c → 0. The location of the new particle Φ n (P n+1 ) is then distributed according to harmonic measure on ∂K n . However, if we assume that ∂K n is approximately linear on the scale of P , then we would have Φ n (P n+1 ) ≈ Φ n (e iΘ n+1 ) + Φ n (e iΘ n+1 )P (1) so we would add an approximate disk of diameter proportional to √ c|Φ n (e iΘ n+1 )|. In order to compensate for this distortion, Hastings and Levitov proposed the HL(α) family of models where, once Θ n+1 is chosen, we choose P n+1 to be a particle of capacity c n+1 = |Φ n (e iΘ n+1 )| −α c.
Then, in the case α = 2, the particles added to the cluster would be approximately of constant size. The approximation (1) is in fact misleading, at least on a microscopic level, because ∂K n develops inhomogeneities on the scale of the particles. Nevertheless, HL (2) has been considered as a variant of diffusion-limited aggregation (DLA) [17], with some justification, see [4], derived from numerical experiments.
In general, the HL(α) model offers a convenient mechanism for such experiments, and moves away from the lattice formulation of [17] which has been shown to lead to unphysical effects on large scales. See for example [?]. Moreover, it might be hoped that an evolving family of conformal maps would present a more tractable framework for the analysis of scaling limits than other growth models, while potentially sharing the same bulk scaling limit and fluctuation universality class. That is the direction explored in this paper.
Besides the mechanism of diffusive aggregation, based on harmonic measure, there is another one-parameter family of models, conceived originally in the lattice case, called dielectric breakdown models [9], which interpolates between DLA and the Eden model [3]. In the Eden model, each boundary site is chosen with equal probability. In the continuum setting, for an Eden-type model we would choose an attachment point on the boundary according to normalized arc length, which has density proportional to |Φ n (e iθ )| with respect to harmonic measure. We can widen our family of models to include a continuum analogue of dielectric breakdown models by choosing P(Θ n+1 ∈ dθ|Φ n ) ∝ |Φ n (e iθ )| −η dθ.
The case η = −1 then provides a continuum variant of the Eden model.
In a law-of-large-numbers regime, it might be guessed that bulk characteristics of the cluster for the model incorporating both the α and η modifications would depend only on their sum ζ = α + η since, once this is fixed, up to a global time-scaling, the growth rate of capacity due to particles attached near e iθ does not depend further on α or η. We will show, in the regime which we can address, that this is indeed true.
In this paper we investigate the two-parameter family of models just described, but modified by the introduction of a regularization parameter σ > 0, which controls the minimum length scale over which feedback occurs through c n+1 and Θ n+1 . We will require throughout that σ √ c (and sometimes σ c 1/4 , or more) and we will restrict attention to the subcritical regime ζ 1. This includes the Eden case (α = 2, η = −1) but excludes continuum DLA (α = 2, η = 0). In the regularized models, we will show fluctuation behaviour which is universal over all choices of particle family. Our first main result shows that, in this regime, in the limit c → 0, disks are stable, that is, an initial disk cluster remains close to a disk as particles are added and its capacity becomes large. Our second main result is to prove convergence of the normalized fluctuations of the cluster around its deterministic limit, to an explicit Gaussian process. The constraint ζ 1 appears sharp for this behaviour: we see an explicit dependence of the fluctuations on α and η and, in particular, an exponential instability of rate (ζ − 1)k in the kth Fourier mode if we formally take ζ > 1.

Statement of results
In this section, we define the continuous-time ALE(α, η) model, which is our object of study, and we specify our standing assumptions for individual particles. We then state our main results. Our model is constructed as a composition of univalent functions on the exterior unit disk D 0 = {|z| > 1}. Each of these functions corresponds to a choice of attachment angle θ ∈ [0, 2π) and a basic particle P . Recall that K 0 = {|z| 1}. By a basic particle P we mean a non-empty subset of D 0 such that K 0 ∪ P is compact and simply connected. Set D = D 0 \ P . By the Riemann mapping theorem, there is a c ∈ (0, ∞) and a conformal isomorphism F : D 0 → D with Laurent expansion of the form Then F is uniquely determined by P , and P has capacity c. Our model depends on three parameters α, η ∈ R and σ ∈ (0, ∞), along with the choice of a family of basic particles (P (c) : c ∈ (0, ∞)) with P (c) of capacity c. The associated maps F c : D 0 → D (c) then have the form (2) with a k = a k (c) for all k. We assume throughout that F c extends continuously to {|z| 1}. We require that our particle family is nested and satisfies, for some Λ ∈ [1, ∞), δ(c) Λr 0 (c) for all c where r 0 (c) = sup{|z| − 1 : z ∈ P (c) }, δ(c) = sup{|z − 1| : z ∈ P (c) }. where in each case δ is a suitable increasing homeomorphism of (0, ∞). It will be convenient to place our aggregation models from the outset in continuous time. By a (continuous-time) aggregate Loewner evolution of parameters α, η ∈ R, or ALE(α, η), we mean a finite-rate, continuous-time Markov chain (Φ t ) t 0 taking values in the set of univalent functions D 0 → D 0 , starting from Φ 0 (z) = z, which, when in state φ, jumps to φ • F c(θ,φ),θ at rate λ(θ, φ)dθ/(2π), where Since σ > 0, the rate λ(θ, φ) is continuous in θ, so the total jump rate is finite. The model may be thought of equivalently in term of the random process of compact sets (K t ) t 0 given by The effect of the jump just described is then to add to the current cluster the set φ(e iθ P (c(θ,φ)) ) thereby increasing its capacity by c(θ, φ). In the case where (Φ t ) t 0 takes exactly n jumps by time t, we have where C n is the capacity of the nth particle and Θ n is its attachment angle, as in Figure 1. Moreover, the capacity T t of the cluster K t is then given by For certain parameter values, the process (Φ t ) t 0 may explode, that is, may take infinitely many jumps in a finite time interval. In fact this can happen only if T t → ∞ at the same time, and this possibility is excluded (with high probability) over the relevant time interval in the conclusions of our main results. So we make no attempt to define Φ t beyond explosion. 5 The discrete-time process (Φ n ) n 0 in the introductory discussion is given by the Markov chain formed of the sequence of distinct values taken by (Φ t ) t 0 . We denote this process from now on by (Φ disc n ) n 0 for clarity. Prior work on ALE models [10,16] was framed in terms of this discrete-time process. The continuous-time framework allows a more local specification of the dynamics, without the need to normalise the distribution of attachment angles. It further allows us to organise the computation of martingales in terms of a standard calculus for Poisson random measures.
We can now state our first main result. Define Note that τ t → ∞ as t → t ζ for all ζ. The result identifies the small-particle scaling limit of K t in the case ζ 1 as a disk of radius e τt , with quantified error estimates. It is proved in Propositions 5.7 and 5.8. The range of parameter values to which the result applies is indicated by the region shaded red in Figure 2, with diagonal lines showing parameter pairs (α, η) sharing a common bulk scaling limit. Recall that T t = log Φ t (∞), which is the capacity of K t , and setΦ t (z) = Φ t (z)/e Tt . 5 The total jump rate λ(φ) at a state φ is given by Similarly, there is a constant C(α, σ) < ∞ such that the next jump in capacity ∆τ satisfies These estimates imply by standard arguments that, almost surely, explosion occurs if and only if both η < 0 and ζ = α + η < 0, and only if T t → ∞ at the same time. For all α, η ∈ R with ζ = α + η 1, for all ε ∈ (0, 1/2] and ν ∈ (0, ε/4], for all m ∈ N and T ∈ [0, t ζ ), there is a constant C = C(α, η, Λ, ε, ν, m, T ) < ∞ with the following property. In the case ζ < 1, for all c 1/C and all σ c 1/2−ε , with probability exceeding 1 − c m , for all t T , and, for all |z| = r 1 + c 1/2−ε , Moreover, in the case ζ = 1 with ε ∈ (0, 1/5], for all c 1/C and all σ c 1/5−ε , with probability exceeding 1 − c m , for all t T , and, for all |z| = r 1 + c 1/5−ε , We will show a similar result for the discrete-time process (Φ disc n ) n 0 . Set The following result is proved at the end of Section 5.2. The case α = 0 is Theorem 1.1 in [10].
Theorem 1.2. For all α, η ∈ R with ζ = α + η 1, for all ε ∈ (0, 1/2] and ν ∈ (0, ε/4], for all m ∈ N and N ∈ [0, n α ), not necessarily an integer, there is a constant C = C(α, η, Λ, ε, ν, m, N ) < ∞ with the following property. In the case ζ < 1, for all c 1/C and all σ c 1/2−ε , with probability exceeding 1 − c m , for all n N/c, and, for all |z| = r 1 + c 1/2−ε , Moreover, in the case ζ = 1 with ε ∈ (0, 1/5], for all c 1/C and all σ c 1/5−ε , with probability exceeding 1 − c m , for all n N/c, e σ e σ − 1 3 and, for all |z| = r 1 + c 1/5−ε , We turn to our second main result, which describes the limiting fluctuations of ALE(α, η). Denote by H the set of all holomorphic functions on D 0 = {|z| > 1} which are bounded at ∞. We equip H with the the topology of uniform convergence on {|z| r} for all r > 1. Define for t < t ζΨ Let (B t ) t 0 be a (real) Brownian motion. Let (B t (k)) t 0 for k 0 be a sequence of independent complex Brownian motions, independent of (B t ) t 0 . We can define continuous Gaussian processes (Γ t (k)) t<t ζ and (Γ cap t ) t<t ζ by the following Ornstein-Uhlenbeck-type stochastic differential equations We show in Section 6.2 that the following series converges in H, uniformly on compacts in [0, t ζ ), almost surelyΓ In fact (Γ t ) t<t ζ satisfies the following stochastic differential equation in H The following two results are proved in Section 6.
and ε > 0 and consider the limit c → 0 with σ → 0 subject to the constraint Then As in the bulk scaling limit, we can deduce an analogous discrete-time fluctuation theorem. The case α = 0 recovers Theorem 1.2 in [10]. Define for t 0 We have seen already in Theorem 1.2, for N < n α , that (T disc n − τ disc n ) n N/c does not fluctuate at scale √ c. We can define a continuous Gaussian process (Γ disc t ) t<nα in H by , not necessarily an integer, and fix ε > 0. In the limit c → 0 with σ → 0 considered in Theorem 1.3, we have

Commentary and review of related work
Hastings and Levitov [4] introduced the family of planar aggregation models HL(α), which are the cases η = σ = 0 of our ALE(α, η) model. They discovered by numerical experiments that, for small particles, the models underwent a transition at α = 1: for α 1 the cluster grows like a disk, while for α > 1 it exhibits fractal properties. The HL(0) model was subsequently investigated rigorously in a series of works [13], [11] (bulk scaling limit), [5], [15] (fluctuation scaling limit). The σ-regularized variant of HL(α) was proposed in [6], where it was shown for slit maps that, if σ (log(1/c)) −1/2 , there is disk-like behaviour for all α 0: it appeared that the observed fractal properties of HL(α) for α > 1 were suppressed by strong regularization. In contrast, for the weaker regularization used in the present paper, the phase transition at α = 1 (or ζ = 1) becomes visible. The method of [6] used a comparison with an HL(0)-type model which breaks down for smaller values of σ. The regularized ALE(α, η) model appeared in [16], where it was shown that, for slit maps and for η > 1, a σ-regularized ALE(0, η) grows as a line for sufficiently small σ c γ , for some γ < ∞ depending on η.
A new approach was begun in [10], treating regularized ALE(0, η) as a Markov chain in univalent functions by martingale arguments: a bulk scaling limit and fluctuation scaling limit were shown, subject to the constraint η 1 and to restrictions on σ as a fractional power of c.
These limits turn out not to depend on the details of individual particle shapes. In this paper, we extend the analysis of [10] to ALE(α, η), subject now to the constraint ζ = α + η 1. Thus we now include regularized HL(α) for α 1. Hastings and Levitov had argued that there should be a trade-off between α and η, with only ζ affecting the bulk scaling limit, and on this basis proposed HL(1), that is ALE(1, 0), as a continuum variant of the Eden model. A more direct continuum analogue of the Eden model is ALE(2, −1). Our results, in the regularized case, both justify the trade-off argument and show a disk scaling limit whenever ζ 1. On the other hand, we show that ALE(1, 0) and ALE(2, −1) have different fluctuation behaviour. As in [10], the behaviour of fluctuations as a function of ζ is consistent with the conjectured transition in behaviour at ζ = 1.
Hastings and Levitov [4] identify a Loewner-Kufarev-type equation, which they propose as governing the small-particle limit of HL(α), citing a discussion of Shraiman and Bensimon [14] for the Hele-Shaw flow, where α is taken to be 2. This is the LK(α) equation, which is the subject of the next section. As noted by Sola in a contribution to [7], there is a lack of mathematical theory for the LK(α) equation, except in the case α = 2 when some special techniques become available. In this paper, since our focus is on clusters initiated as a disk, we are able to use an explicit solution of the equation, along with its linearization around that solution, so we do not rely on a general theory. However, the particle interpretation established here offers some evidence that for α 1, the LK(α) equation may have a suitable existence, uniqueness and stability theory, and that it may be possible to derive the equation as a limit of particle models. McEnteggart [8] has shown short-time existence and uniqueness for holomorphic initial data, by adapting a classical argument for the case α = 2.
Our results depend on constraints on the regularization parameter σ, though substantially weaker ones than those used in [6]. These constraints limit the interactions of individual particles and place us in the simplest case of Gaussian fluctuations. At a technical level, for Theorem 1.1, these constraints come from the need to haveδ(e σ ) c ε in Proposition 5.3, while for Theorem 1.3 they are needed to show that the Poisson integral process (Π t ) t 0 is a good approximation to the fluctuations in Proposition 5.7. In the case ζ = 1, the regularizing operator Q obtained by linearization of the LK(ζ) equation collapses from a fixed multiple of the Cauchy operator to σ times the second derivative. In general, for scaling regimes where σ → 0 faster than our fluctuation results allow, it remains possible that ALE(α, η) has different universal fluctuation behaviour, such as KPZ, as has been conjectured for the lattice Eden model.

Structure of the paper
In the next section, we discuss the Loewner-Kufarev equation for the limit dynamics. Then, in Section 3, we derive an interpolation formula between ALE(α, η) and solutions of the limit equation. The terms in this formula are estimated in Section 4. Equipped with these estimates, we show the bulk scaling limit in Section 5 and the fluctuation scaling limit in Section 6. We collect in an Appendix 7 some further estimates needed in the course of the paper, including estimates on the conformal maps which encode single particles and particle families.

Loewner-Kufarev equation
Let S denote the set of univalent holomorphic functions φ on {|z| > 1} with φ(∞) = ∞ and φ (∞) ∈ (0, ∞). Then each φ ∈ S has the form for some c ∈ R and some sequence (a k : k 0) in C. Fix parameters ζ ∈ R and σ 0. Given φ 0 ∈ S, consider the following Cauchy problem for where The case σ = 0 of this equation is the equation proposed by Hastings and Levitov as scaling limit for HL(ζ), which we will call the LK(ζ) equation. When ζ = 0, the value of σ is immaterial and there is a unique solution given by φ t (z) = φ 0 (e t z).
Where ζ = 2 and σ = 0, (7) is the Loewner-Kufarev equation associated to the Hele-Shaw flow. For σ > 0, we will refer to (7) as the σ-regularized LK(ζ) equation. We will be interested in the subcritical case ζ ∈ (−∞, 1]. The general form of the Loewner-Kufarev equation is given bẏ with (µ t : t 0) a given family of measures on [0, 2π). Thus the σ-regularized LK(ζ) equation is obtained by requiring that the driving measures are given by Note that the density of these driving measures is the product of the density of the local attachment rate and the local particle capacity (5) for ALE(α, η). By the Loewner-Kufarev theory, for any solution (φ t ) t 0 of (7), the sets form an increasing family of simply-connected compacts, with capacities given by

Linearization
We compute the linearization of (7) around a solution (φ t ) t 0 . For ψ holomorphic in {|z| > 1}, we have where and, setting ρ = ψ /φ , Note that first-order variations in S have the form The process of first-order variations (ψ t ) t 0 around a solution (φ t ) t 0 can be expected to satisfy the linearized equationψ

Linear stability of disk solutions in the subcritical case
For all σ 0, a trial solution φ t (z) = e τt z for (7) leads to the equationτ t = e −ζτt . Let us consider first the case where ζ 0. We can solve to obtain We thus find solutions (φ t ) t 0 to (7) in which the sets K t form a growing family of disks. For such disk solutions, we have φ t (z) = e τt for all z, so we can evaluate the integral (8) to obtain Here and below, we write Dψ(z) for zψ (z). Note that Q acts as a multiplier on the Laurent coefficients. For ψ(z) = ∞ k=−1 ψ k z −k , we have We split Q as a sum of multiplier operators Q 0 + Q 1 with multipliers given by Define for δ 0 P (δ) = e −δQ , P 0 (δ) = e −δQ 0 , P 1 (δ) = e −δQ 1 .
We turn to the case where ζ < 0. The differential equationτ t = e −ζτt has now only a local solution, given by τ t = z −1 log(e ζτ 0 + ζt), t < t ζ = e ζτ 0 /|ζ| with τ t → ∞ as t → t ζ . It is convenient in this case to split Q differently, setting Then, making similar definitions in all other respects, we have and ψ t remains holomorphic for all t < t ζ , as in the case ζ ∈ [0, 1]. Define for r > 1 The following inequality will be used in Section 4.
Lemma 2.1. For all p ∈ (1, ∞), there is a constant C(p) < ∞ such that, for all ζ ∈ R and all σ 0, for all holomorphic functions ψ on {|z| > 1} bounded at ∞, all t 0 and all r > 1, we have Proof. Consider first the case ζ 0. The operator P 1 (t) acts as multiplication by Hence the conditions of the Marcinkiewicz multiplier theorem (as recalled in Section 7.2) hold for P 1 (t) with A = 1. The desired estimate follows.
In the case ζ < 0, we modified the split so thatq 1 (k) 0, so 0 p 1 (k, t) 1 for all k. Nowq 1 (k) is no longer increasing but is unimodal in k, sop 1 (k, t) is also unimodal in k, and so ∞ k=0 Hence the Marcinkiewicz theorem applies with A = 2 and we can conclude as before.

Transformation to (Schlicht function, capacity) coordinates
Write S 1 for the set of 'Schlicht functions at ∞' on {|z| > 1}, given by It will be convenient to use coordinates (φ, τ ) on S, given bŷ Thenφ ∈ S 1 and τ ∈ R. It is straightforward to show that, for a solution (φ t ) t 0 to (7), the transformed variables (φ t , τ t ) t 0 satisfy whereb On linearizing (14) around a solution (φ t , τ t ) t 0 , we obtain equations for first-order variations (ψ t , ψ cap t ) t 0 in the new coordinates, where nowψ t is bounded at ∞ for all t, reflecting the normalization ofφ t . These are then related to the first-order variations (ψ t ) t 0 in the old coordinates by . The equations for first-order variations are then given bẏ

Interpolation formula for Markov chain fluid limits
We use an interpolation formula between continuous-time Markov chains and differential equations, which we first review briefly in a general setting. This formula is then applied to an ALE(α, η) aggregation process (Φ t ) t 0 with capacity parameter c, regularization parameter σ and particle family (P (c) : c ∈ (0, ∞)), taking as limit equation the σ-regularized LK(ζ) equation with ζ = α +η. We use (Schlicht function, capacity) coordinates for both the process and the limit equation.

General form of the interpolation formula
Let (X t ) t 0 be a continuous-time Markov chain with state-space E and transition rate kernel q, starting from x 0 say. Suppose for this general discussion that .
Then Z 0 = x t and Z t = X t and, on computing the martingale decomposition of (Z s ) s t , we obtain the interpolation formula where and where β is the drift of (X t ) t 0 , given by We will use this formula in a case where the state-space E is infinite-dimensional. Rather than justify its validity generally in such a context, in the next section, we will prove directly the special case of the formula which we require. Note that the integrands in M t and A t depend on t. Nevertheless, we will call M t the martingale term and A t the drift term.

Proof of the formula for ALE(α, η)
Let (Φ t ) t 0 be an ALE(α, η) aggregation process with capacity parameter c, regularization parameter σ and particle family (P (c) : c ∈ (0, ∞)). See Section 1.2 and (5) for the specification of this process. We use (Schlicht function, capacity) coordinates, as in Section 2.3, to obtain a continuous-time Markov chain and We can and do assume that the process is constructed from a Poisson random measure µ on [0, 2π) × [0, ∞) × (0, ∞) of intensity (2π) −1 dθdvdt by the following stochastic differential equation: and We use the vector field b = (b, b cap ) of the σ-regularized LK(ζ) equation (14), written in (Schlicht function, capacity) coordinates. We consider the disk solution (x t ) t 0 = (φ t , τ t ) t<t ζ with initial capacity τ 0 = 0, which is given bŷ We will compute the form of the interpolation formula in this case and then prove directly and, for y = (ψ, ψ cap ), ∇b(x t )y = −e −ζτt ((Q + 1)ψ, ζψ cap ) and the first-order variation at time t due to a variation y at time s t is given by Writeμ for the compensated Poisson random measurẽ µ(dθ, dv, ds) = µ(dθ, dv, ds) − (dθ/2π)dvds.
Fix t 0 and set δ s = τ t − τ s . We alert the reader to the concealed dependence of δ s on t.
The martingale term M t = (M t , M cap t ) in the interpolation formula may then be written The following interpolation identities may then be obtained formally by splitting equation (15) into its Schlicht function and capacity components.
We will show that, for all x ∈ [0, t] and all |z| > 1, The case x = t gives the claimed identities. In the case x = 0, all terms are 0. The left-hand and right-hand sides are piecewise continuously differentiable in x, except for finitely many jumps, at the jump times of (Φ x ) 0 x t , which occur when µ has an atom at (θ, v, x) with v Λ x (θ). It will suffice to check that the jumps and derivatives agree. NowÂ x,t (z) and A cap x,t are continuous in x and, at the jump times of Φ x , the jumps inΨ x,t (z) and Ψ cap x,t are given by So it remains to check the derivatives. We will use a spectral calculation for the semigroup of multiplier operators P (τ ) = e −τ Q , whose justification is straightforward.
So, between the jump times, we have Hence, between the jump times, as required.

Estimation of terms in the interpolation formula
We obtain some estimates on the terms in the interpolation formula (17) for ALE(α, η) when it is close to the disk solution (16) We estimate first the martingale term and then the drift term.

Estimates for the martingale terms
Recall that the martingale term (M t , M cap t ) in the interpolation formula is given bŷ Consider the following approximations toM t (z) and M cap t , which are obtained by replacinĝ Φ s− byφ s , T s− by τ s and e −c F c (θ, z) − z by 2cz/(e −iθ z − 1). (Under our assumptions on the particle family, the last approximation becomes good in the limit c → 0. See Section 7.1 and in particular equation (102).) Definê where Lemma 4.1. For all α, η ∈ R, all p 2 and all T < t ζ , there is a constant C(α, η, p, T ) < ∞, such that, for all c ∈ (0, 1], all σ 0 and all δ 0 Proof. Write T 0 for T 0 (δ 0 ) throughout the proofs. Consider the martingale (M t ) t<t ζ given by By an inequality of Burkholder, for all p 2, there is a constant C(p) < ∞ such that, for all t 0, We write here M * t for sup s t |M s | and similarly for other processes. See [1, Theorem 21.1] for the discrete-time case. The continuous-time case follows by a standard limit argument. Now For all t T 0 ∧ T and all θ ∈ [0, 2π), we have Here and below, we write C for a finite constant of the dependence allowed in the statement. The value of C may vary from one instance to the next. We remind the reader that C t (θ) and Λ t (θ) are defined at (18). Hence Since M cap t = e −ζτt M t for all t T 0 , the first claimed estimate follows. For the second estimate, we use instead the martingale (M t ) t 0 given by Then M t Ccδ 0 and (∆M ) t Cc. Hence, by Burkholder's inequality, Note that, sinceΦ t takes values in S 1 ,Ψ t (z) =Φ t (z) − z is bounded at ∞ and hence has a limiting valueΨ t (∞). The same is true for the termsM t andÂ t in the interpolation formula. Instead of estimating these terms directly, we estimate first their values at ∞ and then their derivatives DM t and DÂ t , since this gives the best control of the derivative ofΦ t near the unit circle, which drives the dynamics of the process. For all α, η ∈ R with ζ = α + η 1, all p 2 and all T < t ζ , there is a constant C(α, η, Λ, p, T ) < ∞, such that, for all c ∈ (0, 1], all σ 0, all δ 0 ∈ (0, 1/2] and all t T , Proof. By considering the Laurent expansions of F c andφ, we have Consider the martingale (M t ) t<t ζ given by and, for p 2, since The first claimed estimate then follows from Burkholder's inequality (22). For the second estimate, we consider instead the martingale (M t ) t<t ζ given by ThenM t (∞)−Π t (∞) = e −τt M t for all t T 0 . By Proposition 7.4, we have |a 0 (c)−2c| Cc 3/2 . We combine this with (23) and (24) to see that The second estimate then follows by Burkholder's inequality as above.
Recall that, for p ∈ [1, ∞) and r > 1, we set For a measurable function Ψ on Ω × {|z| > 1}, we set and while in the case ζ = 1 the same bounds hold with an additional factor r r−1 1/2 on the righthand side.
Proof. We restrict our account to the case ζ ∈ [0, 1], omitting the minor modifications needed when ζ < 0. The case ζ < 0 proceeds just as for ζ ∈ [0, 1) but only for T < t ζ and using the alternative split Q =Q 0 +Q 1 and taking r s = e δs r.
Fix t T < ∞ and consider for |z| > 1, the martingale (M x (z)) 0 x t given by where ρ s = (r s + 1)/2 and r s = e (1−ζ)δs r. Here we used Proposition 2.1 and the inequality (111). By Burkholder's inequality, for p 2 and all |z| > 1, For t T 0 , we have DM t (z) = M t (z) so, on taking the . p,r -norm in (29), we obtain Now Also We have Λ s (θ) C/c for all s T 0 and θ ∈ [0, 2π). Hence We will split the jump ∆(θ, z, c,φ) as the sum of several terms, and thereby split H s (θ, z) and hence M t also as a sum of terms. For each of these terms, we will use one of the inequalities (32), (33), (34) and one of (31), (35) to obtain a suitable upper bound for the right-side of (30). These bounds will combine to prove the first claimed estimate. Recall thatφ(z) = z +ψ(z), so where We further split the second term by expanding in Taylor series, using an interpolation from z to F c (θ, z). For u ∈ [0, 1], define Then F c,0 (θ, z) = z and F c,1 (θ, z) = F c (θ, z). Fix c, θ and z and set Set m = 1/(8ε) and recall that our constants C are allowed to depend on ε. Then where, for k = 1, . . . , m,

By Proposition 7.4, we have
Hence, for s T 0 , where we have used the fact that |F c,u (θ, z)| |z| to see that Hence we obtain for ζ = 1 while, using Lemma 7.7, for ζ < 1 we have Here we have used our choice of m 1/(8ε) and the assumption r 1 + c 1/2−ε to see that The bound (46) remains valid with M m+1 in place of M k . Hence for ζ < 1 and for ζ = 1 Now and we have shown that all terms on the right-hand side can be bounded by the right-hand side in (27), so this first estimate is now proved. It remains to show the second estimate. Fix t 0 and consider, for |z| > 1, the martingale (Π x (z)) x 0 given by e −(τt−τs) P (τ t − τ s )DH(θ, z)2c s 1 {v λs, s T 0 }μ (dθ, dv, ds).
For all but the first term on the right, the bounds (43) Otherwise we can proceed as for M 0,0 to arrive as the following estimates, which suffice for (28). For ζ < 1, we have
The claimed estimate follows on integrating (51) in θ.

Bulk scaling limit for ALE(α, η)
Recall that we write our ALE(α, η) process (Φ t ) t 0 in (Schlicht function, capacity) coordinates (Φ t , T t ), and that we setΨ where (φ t , τ t ) t<t ζ is the disk solution to the LK(ζ) equation with initial capacity τ 0 = 0. We obtained the following interpolation formula (17) and have estimated the terms on the right-hand sides in the preceding section. We now put these estimates together to obtain first L p -estimates and then pointwise high-probability estimates which allow us to prove Theorems 1.1 and 1.2.

L p -estimates
Recall that Proposition 5.1. For all α, η ∈ R, all p 2 and all T < t ζ , there is a constant C(α, η, p, T ) < ∞ such that, for all c ∈ (0, 1] and all δ 0 ∈ (0, 1/2], and sup and sup Proof. The first two estimates follow immediately from Lemmas 4.1 and 4.6. From Lemmas 4.2 and 4.7, we obtain, for all t T , from which the third estimate follows by Gronwall's lemma. The fourth estimate follows from the third, together with Lemmas 4.2 and 4.7.
The following estimates follow immediately from Lemmas 4.3 and 4.7.
Proposition 5.5. For all α, η ∈ R, all p 2 and all T < t ζ , there is a constant C(α, η, p, T ) < ∞ such that, for all c ∈ (0, 1] and all δ 0 ∈ (0, 1/2], We can also improve on the estimate of T t by τ t in Proposition 5.1. Define, for cV t < n α , where τ disc n = α −1 log(1 + αcn) as at (6). We leave any modifications needed for the case α = 0 to the reader. By allowingT t to depend on the random time-scale of particle arrivals, we remove the main source of error when estimating T t by τ t . Proposition 5.6. For all α, η ∈ R, all p 2, all T < t ζ and all N < n α , there is a constant C(α, η, p, T, N ) < ∞ such that, for all c 1/C and all δ 0 ∈ (0, 1/2], We have, for cV t N , and, for t T 0 (δ 0 ), as in the proof of Lemma 4.4, and, using (53),

Spatially-uniform high-probability estimates
We now pass from the L p -estimates of the preceding section to pointwise estimates which hold with high probability on the functionΨ t (z) =Φ t (z) − z, uniformly in t ∈ [0, T ] and |z| r(c) as c → 0, for a suitable function r(c), which is specified in the next result, and tends to 1 as c → 0. In order to show the desired uniformity, we combine the usual L p -tail estimate with suitable dissections of [0, T ] and {|z| r(c)}, choosing p large to deal with an increasing number of terms as c → 0. We see at the same time that the event {T 0 (δ 0 ) > T }, to which our previous estimates were restricted, is in fact an event of high probability as c → 0, thus closing the argument for convergence to a disk.
Fix α, η, ε, ν, m and T as in the statement. By adjusting the value of ε, it will suffice to consider the case where e σ 1 + 2c 1/2−ε , and to find an event Ω 0 ⊆ {T 0 (δ 0 ) > T }, of probability exceeding 1 − c m , on which (76), (77) and (78) hold whenever r 1 + 2c 1/2−ε and t T . There is a constant C < ∞ of the desired dependence, such that δ 0 1/2 whenever c 1/C. We restrict to such c. Set Recall that V t denotes the number of particles added to the cluster by time t. Consider the event Note that, on Ω 1 , for all t T 0 (δ 0 ) ∧ T , there exists n ∈ {1, . . . , N 0 } such thatΨ t =Ψ t(n) . Since δ 0 1/2, there is a constant C < ∞ of the desired dependence such that the process (V t ) t T 0 (δ 0 ) is a thinning of a Poisson process of rate C/c. Hence and hence P(Ω c 1 ) c m /3 for all c 1/(3C). We restrict to such c. Fix an integer p 2, to be chosen later, depending on m and ν. By Proposition 5.1, there is a constant C < ∞ of the desired dependence such that, for Set λ 0 = (6c −m ) 1/p and consider the event Then P(Ω c 2 ) 2λ −p 0 = c m /3. We choose p m/ν. Then, since e σ 1 + 2c 1/2−ε and ν ε, there is a constant C < ∞ of the desired dependence such that, for c 1/C, on the event Ω 2 , for all t T 0 (δ 0 ) ∧ T , We restrict to such c. Set K = min{k 1 : 2 k c 1/2−ε 1}.
We deduce (76) using the identity On the event Ω 2 , for all t T 0 (δ 0 ) ∧ T , On the other hand, Ω 0 ⊆ Ω 2 and on Ω 0 we have T 0 (δ 0 ) > T and, using (77), for t T and Since r 1 + c 1/2 , the log factor can be absorbed in c 1/2−ν by adjustment of ν. Then, on combining the last two estimates, we obtain (76).
It is now straightforward to deduce the following high-probability estimates from Proposition 5.1 using L p -tail estimates and the fact that P(Ω 0 ) 1 − c m from Proposition 5.7. The details are left to the reader. Propositions 5.7 and 5.8 together imply Theorem 1.1.
Proposition 5.8. For all α, η ∈ R with ζ = α + η 1, all ε ∈ (0, 1/2] and all ν ∈ (0, ε/4], all m ∈ N and all T < t ζ , there is a constant C(α, η, Λ, ε, ν, m, T ) < ∞ with the following property. In the case ζ < 1, for all c 1/C, for e σ 1 + c 1/2−ε , with probability exceeding 1 − c m , for all t T , (81) Moreover, in the case ζ = 1 with ε ∈ (0, 1/5], for all c 1/C, for e σ 1 + c 1/5−ε , with probability exceeding 1 − c m , for all t T , (82) Proof of Theorem 1.2. We will write the argument for the case ζ < 1, omitting the modifications needed for ζ = 1, which are left to the reader. Since N < n α , we can choose δ = δ(α, η, N ) > 0 and T < t ζ such that ν T = N + δ. Choose δ 0 and Ω 0 as in Proposition 5.7, with the choice of T just made. Write C for the constant appearing in Proposition 5.7 and set Then, for all |z| 1 + c 1/2−ε and all t T , on the event Ω 0 , we have |Φ t (z) − z| ∆. Then, by Propositions 5.5 and 5.6, choosing δ 0 as in Proposition 5.7 and using an L p -tail estimate for suitably large p, there is an event Ω 1 ⊆ Ω 0 , of probability exceeding 1 − 2c m , on which, for all t T , both |cV t − ν t | ∆ and, provided cV t N , also We can choose C so that, for c 1/C, we have ∆ δ, so cV T N + δ − ∆ N always on Ω 1 . Now, for all n N/c, we have V t = n for some t T with cV t N , so on Ω 1 , for all e σ e σ − 1 2 .

Fluctuation scaling limit for ALE(α, η)
Given an ALE(α, η) process (Φ t ) t 0 , recall that The fluctuations in these coordinates are given bŷ Recall that we write H for the set of holomorphic functions on {|z| > 1} which are bounded at ∞, and we use on H the topology of uniform convergence on {|z| r} for all r > 1. In this section we prove Theorem 1.3 and then, at the end, we deduce Theorem 1.4.

Reduction to Poisson integrals
Our starting point is the interpolation formula (17) As a first step, we study the approximationsΠ t (z) and Π cap t toM t (z) and M cap t which have a simple form and which prove to be the dominant terms in the considered limit. Set Recall the multiplier operator P (δ) defined at (11). Then Recall that c t = ce −ατt and λ t = c −1 e −ητt , and that we define for |z| > 1 The following result allows us to deduce the weak limit of the normalized fluctuations from that of the Poisson integrals (Π t , Π cap t ) t 0 .
Note that 0 e κτs − e κ 0 τs (κ − κ 0 )τ s e κτs so, by a similar argument, For Π 0 s (k), we used the estimate (85) with κ replaced by κ 0 , which is the special case σ = 0. A similar argument holds in the case ζ < 0, with the roles of κ and κ 0 interchanged, which leads to the same estimate. It remains to show the third estimate, which we will do for general σ 0. We have e κτs e i(k+1)θ 2c s 1 {v λs}μ (dθ, dv, ds).
Cch. On the other hand, since T t, The claimed estimate follows.

Gaussian limit process
By Proposition 6.1, in order to compute the weak limit of c −1/2 (Ψ t , Ψ cap t ) t<t ζ , it suffices to compute the weak limit of c −1/2 (Π t , Π cap t ) t<t ζ . This process is a deterministic linear function of the compensated Poisson random measureμ. We are guided to find the weak limit process by replacingμ in (83) and (84) by a Gaussian white noise on [0, 2π) × [0, ∞) × (0, ∞) of the same intensity. At the same time, we set σ = 0 in the limit 8 , replacing the multiplier operator P (δ) by P 0 (δ). Then, using the scaling properties of white noise, we arrive at candidate limit processes (Γ t (z)) t<t ζ and (Γ cap t ) t<t ζ which are defined as follows. Let W be a Gaussian white noise on [0, 2π) × (0, ∞) of intensity (2π) −1 dθdt. Define for each |z| > 1 and t ∈ [0, t ζ ) Γ t (z) = 2ˆt 0ˆ2 π 0 e −(τt−τs) P 0 (τ t − τ s )H(θ, z)e −(α+η/2)τs W (dθ, ds), where these Gaussian integrals are understood by the usual L 2 isometry. Define for t 0 and k 0 π 0 W (dθ, ds). 8 It is not necessary to pass to the limit σ → 0. Indeed, the best Gaussian approximation for given σ > 0 would be obtained using P instead of P 0 . The limit c → 0 with σ fixed then holds uniformly in σ, subject to the restrictions stated in Theorem 1.3, and the limit processes for σ fixed converge weakly to the case σ = 0. We have stated only the joint limit, since this seems to us of main interest, and since the limit fluctuations have in this case a slightly simpler form.
We can and do choose versions of (B t (k)) t 0 and (B t ) t 0 which are continuous in t. Then (B t (k)) t 0 is a complex Brownian motion for all k, (B t ) t 0 is a real Brownian motion, and all these processes are independent. Note that, almost surely, for all t < t ζ , Define for t ∈ [0, t ζ ) and k 0 The following estimate may be obtained by (a simpler version of) the argument used for Proposition 6.2.
Proposition 6.3. For all α, η ∈ R with α + η = ζ 1, and all t < t ζ , there is a constant C(α, η, t) < ∞ such that, for all k 0, The following identity holds in L 2 for all |z| > 1 and t < t ζ By Proposition 6.3, almost surely, the right-hand side in (86) converges uniformly on compacts in [0, t ζ ), uniformly on {|z| r}, for all r > 1. So we can and do use (86) to choose a version ofΓ t (z) for each t < t ζ and |z| > 1 such that (Γ t ) t<t ζ is a continuous process in H and (86) holds for all ω.

Convergence
Given Proposition 6.1, the following result will complete the proof of Theorem 1.3.
Proof of Theorem 1.4. We will argue via the Skorokhod representation theorem. It will suffice to show the claimed convergence for all sequences c k → 0 and σ k → 0 subject to the constraint assumed in Theorem 1.3. Given N < n α , choose δ > 0 and T < t ζ such that ν T = N + δ, as in the proof of Theorem 1. But (Γ t(ν) ) ν<nα has the same distribution as (Γ disc ν ) ν<nα . Hence weakly in D([0, N ], H).
We assume throughout that δ 1. We use the following well known estimates on the capacity c. There is an absolute constant C < ∞ such that 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 [12]. Write log F (z) z = u(z) + iv(z) where we understand the argument to be determined for each z ∈ D 0 so that the left-hand side is holomorphic in D 0 and such that v(z) → 0 as z → ∞. Then u and v are bounded and harmonic in D 0 , with continuous extensions to {|z| 1}, and u(z) → c as z → ∞. Note also that 0 u(e iθ ) log(1 + r 0 ) r 0 for all θ. 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, .
Since v is continuous and is non-decreasing on the complementary interval, this inequality then holds for all θ. Now v is bounded and harmonic in D 0 with limit 0 at ∞, sô 2π 0 v(e iθ )dθ = 0.
Hence, for |z| > 1 with |z − 1| 2a, We can extend F to a holomorphic function in {|z − 1| > a} by setting F (z −1 ) = F (z) We sometimes use exponentiated versions of the inequalities just proved, which are straightforward to deduce and are noted here for easy reference. There is an absolute constant C < ∞ with the following properties. Suppose that δ 1/C. Then, for all |z| > 1, Proposition 7.3. There is an absolute constant C < ∞ with the following properties. Let P 1 , P 2 be basic particles with P 1 ⊆ P 2 . For i = 1, 2, write F i for the associated conformal map D 0 → D 0 \ P i and write c i for the capacity of P i . Set δ i = δ(P i ) and a 0,i = a 0 (P i ) and set Assume that δ 2 1/C. Then and, for all |z| > 1 with |z − 1| Cδ 2 , and Proof. The inequalities (104) and (105) follow from (103) by the same argument used to deduce (99) and (100) from (98). SetP = F −1 1 (P 2 \ P 1 ). WriteF for the associated conformal map D 0 → D 0 \P and writec for the capacity ofP . Then Note that, for z ∈P , we have F 1 (z) ∈ P 2 , so |F 1 (z) − 1| δ 2 . But |e −c 1 F 1 (z) − z| Cδ 1 for all |z| > 1 and c 1 Cδ 2 1 . Hence |z − 1| Cδ 2 for all z ∈P and sõ δ = δ(P ) Cδ 2 .
Hence, for C sufficiently large and δ 2 1/C, for all |z| > 1 with |z − 1| Cδ 2 , and in particular Set z t = z exp(t log(F (z)/z)) and f (t) = log(F 1 (z t )/F 1 (z)). Then Now | log(F (z)/z)| Cδ 2 , so |z t − z| Cδ 2 for all t. Hence, for C sufficiently large and |z − 1| Cδ 2 , we have |z t − 1| C 0 δ 1 for all t, where C 0 is the constant from Proposition 7.2. Then where we used Cauchy's integral formula for the second inequality, adjusting the value of C if necessary. On combining these estimates with (106) and (107), we see that as claimed.