# SLE scaling limits for a Laplacian random growth model

Frankie Higgs<EMAIL_ADDRESS>
(Department of Mathematics and Statistics, Lancaster University, Lancaster LA1
4YF, UK.
)

###### Abstract

We consider a model of planar random aggregation from the ALE$(\alpha,\eta)$
family where particles are attached preferentially in areas of low harmonic
measure. We find that the model undergoes a phase transition in negative
$\eta$, where for sufficiently large values the attachment distribution of
each particle becomes atomic in the small particle limit, with each particle
attaching to one of the two points at the base of the previous particle. This
complements the result of Sola, Turner and Viklund for large positive $\eta$,
where the attachment distribution condenses to a single atom at the tip of the
previous particle.

As a result of this condensation of the attachment distributions we deduce
that in the limit as the particle size tends to zero the ALE cluster converges
to a Schramm-Loewner evolution with parameter $\kappa=4$ (SLE4).

Keywords: scaling limits; Schramm-Loewner evolution; conformal aggregation;
harmonic measure.

Classifiers: 60F99, 60D05, 30C45.

###### Contents

  1. 1 Introduction
    1. 1.1 Conformal aggregation
    2. 1.2 Aggregate Loewner evolution
    3. 1.3 Our results
    4. 1.4 Loewner’s equation and the Schramm-Loewner evolution
    5. 1.5 Structure of paper
    6. 1.6 Table of notation
  2. 2 Spatial distortion of points
  3. 3 The newest basepoints
    1. 3.1 A lower bound on the normalising factor
    2. 3.2 Concentration about each basepoint
    3. 3.3 Symmetry of the two most recent basepoints
  4. 4 Analysis of the density away from the main basepoints
    1. 4.1 Regular points
    2. 4.2 Old singular points
  5. 5 Proof of main results

##  1 Introduction

###  1.1 Conformal aggregation

There has been a great deal of research into models of random aggregation,
where particles are added at each time step to the existing cluster at random
locations. These models are perhaps most easily defined on the lattice
$\mathbb{Z}^{d}$, where each particle is one vertex, for example diffusion-
limited aggregation (DLA) [17] or the Eden model [4]. However, the underlying
anisotropy of $\mathbb{Z}^{d}$ may be retained by the cluster on large scales,
making these models a poor approximation of reality under some conditions [5]
[2].

To resolve this problem in two dimensions we may consider models of _conformal
growth_ in which $\mathbb{Z}^{2}$ is replaced by the complex plane
$\mathbb{C}$. In this paper we will study the aggregate Loewner evolution
(ALE($\alpha,\eta$)) model introduced in [16], which is a generalisation of
the Hastings-Levitov model (HL($\alpha$)) [6].

In a conformal aggregation model, we add particles to our cluster by composing
conformal maps from a fixed reference domain to smaller domains. Our initial
cluster will be the closed unit disc
$K_{0}=\overline{\mathbb{D}}=\\{z\in\mathbb{C}:|z|\leq 1\\}$. We attach a
particle to $K_{0}$ by applying a map from
$\Delta:=\mathbb{C}_{\infty}\setminus\overline{\mathbb{D}}$ to a smaller
domain, and then the new cluster will be the complement of the image of
$\Delta$. We will use particles of the form $(1,1+d]$ for $d>0$.

######  Definition 1.

For any $d>0$, by the Riemann mapping theorem there exists a unique bijective
conformal map

$f^{d}\colon\Delta\to\Delta\setminus(1,1+d]$

such that $f^{d}(z)=e^{\mathbf{c}}z+O(1)$ near $\infty$, for some
$\mathbf{c}=\mathbf{c}(d)\in\mathbb{R}$.

One advantage of the slit particles we use in this paper over more general
particle shapes is that we have an explicit expression for $f^{d}(z)$ [12]. We
call $\mathbf{c}>0$ the _(logarithmic) capacity_ of the particle. As the name
suggests, we can view $\mathbf{c}$ as measuring the “size” of a set in a
certain sense. As we consider the “small-particle limit” we will parameterise
the model by the particle capacity $\mathbf{c}$ (equivalently by $d$).

######  Definition 2.

The preimage of the particle $(1,1+d]$ under $f$ is
$\\{e^{i\theta}:-\beta\leq\theta\leq\beta\\}$, where $0<\beta(\mathbf{c})<\pi$
is uniquely determined by $f(e^{i\beta})=1$.

We can explicitly relate the quantities $\mathbf{c}$, $\beta$ and $d$ using
two equations found in [12] and [16]: $4e^{\mathbf{c}}=(d+2)^{2}/(d+1)$ and
$e^{i\beta}=2e^{-\mathbf{c}}-1+2ie^{-\mathbf{c}}\sqrt{e^{\mathbf{c}}-1}$.
Asymptotically, as $\mathbf{c}\to 0$, these give us $\beta(\mathbf{c})\sim
d(\mathbf{c})\sim 2\mathbf{c}^{1/2}$.

We have maps which can attach one particle, so now we want to be able to build
a cluster with multiple particles by composing maps which attach particles in
different positions. For $\theta\in\mathbb{R}$ and $\mathbf{c}>0$, define the
rotated map

$\displaystyle f^{\theta,\mathbf{c}}\colon\Delta\to\Delta\setminus
e^{i\theta}(1,1+d(\mathbf{c})],$ $\displaystyle
f^{\theta,\mathbf{c}}(z)=e^{i\theta}f^{d(\mathbf{c})}(e^{-i\theta}z),$

and note that it has the same behaviour
$f^{\theta,\mathbf{c}}(z)=e^{\mathbf{c}}z+O(1)$ near $\infty$ as does
$f^{d(\mathbf{c})}$.

Now we want to attach multiple particles.

######  Definition 3.

Given a sequence of angles $(\theta_{n})_{n\in\mathbb{N}}$ and of capacities
$(c_{n})_{n\in\mathbb{N}}$, if we write $f_{j}=f^{\theta_{j},c_{j}}$ then we
can define

$\Phi_{n}=f_{1}\circ f_{2}\circ\cdots\circ f_{n},$ (1)

and define the $n$th cluster $K_{n}$ as the complement of $\Phi_{n}(\Delta)$,
so

$\Phi_{n}\colon\Delta\to\mathbb{C}_{\infty}\setminus K_{n}.$

Note that the total capacity is $\mathbf{c}(K_{n})=\sum_{k=1}^{n}c_{k}$, i.e.
$\Phi_{n}(z)=e^{\sum_{k=1}^{n}c_{k}}z+O(1)$ near $\infty$.

We can now use this setup to construct various models of _random_ growth, by
choosing the angles $(\theta_{n})_{n\in\mathbb{N}}$ and capacities
$(c_{n})_{n\in\mathbb{N}}$ according to a stochastic process.

###  1.2 Aggregate Loewner evolution

The _aggregate Loewner evolution_ model introduced in [16] is a conformal
aggregation model as in Section 1.1, where for the $(n+1)$th particle the
distribution of its attachment angle $\theta_{n+1}$ and its capacity
$c_{n+1}=\mathbf{c}(P_{n+1})$ are functions of the density of _harmonic
measure_ on the boundary of $K_{n}$. The conditional distribution of
$\theta_{n+1}$ and the way we obtain $c_{n+1}$ are respectively controlled by
the two parameters $\eta$ and $\alpha$.

######  Definition 4.

Inductively, we choose $\theta_{n+1}$ for $n\geq 1$ conditionally on
$\theta_{1},\cdots,\theta_{n}$ according to the probability density function

$h_{n+1}(\theta)=\frac{1}{Z_{n}}\left|\Phi_{n}^{\prime}\left(e^{\sigma+i\theta}\right)\right|^{-\eta},\,\theta\in(-\pi,\pi],$
(2)

where
$Z_{n}=\int_{\mathbb{T}}|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|^{-\eta}\,\mathrm{d}\theta$
is a normalising factor. We have introduced a $\sigma=\sigma(\mathbf{c})>0$ as
the poles and zeroes of $\Phi_{n}^{\prime}$ on the boundary mean the measure
$h_{n+1}$ is not necessarily well-defined if $\sigma=0$, but we take
$\sigma\to 0$ as $\mathbf{c}\to 0$.

After choosing $\theta_{n+1}$, we choose the capacity of the $(n+1)$th
particle to be

$c_{n+1}=\mathbf{c}|\Phi_{n}^{\prime}(e^{\sigma+i\theta_{n+1}})|^{-\alpha}$
(3)

where $\mathbf{c}$ is a capacity parameter and $c_{1}=\mathbf{c}$, and we will
later consider the limit shape of the cluster as $\mathbf{c}\to 0$.

Figure 1: The final particle (the rightmost, in orange) of the cluster $K_{n}$
is highly distorted by the application of the first $n-1$ maps
$f_{n-1},f_{n-2},\cdots,f_{1}$. The distortion is much greater near the base
of the particle: we have had to fill in a guess (the red dashed line) for the
behaviour of the particle deep into the cluster, as the distortion is so large
there that we are unable to find the exact location of enough points to draw a
sensible diagram. In fact, the red dashed section corresponds to only 1/500
000th the length of the original, undistorted slit.

###  1.3 Our results

Figure 2:  One cluster of the ALE$(0,-\infty)$ process, with 3000 particles
each of capacity $\mathbf{c}=0.0001$.

In this paper, we study the ALE model defined in Section 1.2 with $\alpha=0$
and large negative values of the parameter $\eta$, which controls the
influence of harmonic measure on our attachment locations.

The case $\alpha=2$ often gives a model in which each particle is
approximately the same size. In this paper we take $\alpha=0$, where the model
can be easier to analyse as the capacities are deterministic. In this case,
the distortion of particles can lead to physically unrealistic outcomes, as in
[11] where the distorted size of the final particle in the cluster does not
disappear in the limit. For the model we are considering here, Figure 1 shows
that the distortion affects the shape as well as the size of the particles.

For $\eta>0$ the density $h_{n+1}$ in (2) is an exaggeration of harmonic
measure, and in [16] the authors find that for $\eta>1$ the attachment
distribution is concentrated around the point of highest harmonic measure,
converging to a single atom as $\mathbf{c}\to 0$. For a slit particle the
point of highest harmonic measure is the tip (see Figure 3), so this
corresponds to the growth of a straight line.

In this paper, we find the equivalent phase transition in negative $\eta$: for
$\eta<-2$ the attachment distribution is concentrated around the points of
lowest harmonic measure. For a slit particle the two points of lowest harmonic
measure are either side of the base (see Figure 3 again), and so
$\theta_{2}\approx\theta_{1}\pm\beta$ with the probability of each tending to
$1/2$ as $\mathbf{c}\to 0$. We go on to find that for all $n$ the distribution
of $\theta_{n+1}$ is concentrated around $\theta_{n}\pm\beta$, and so the
angle sequence approximates a random walk of step length $\beta\sim
2\mathbf{c}^{1/2}$.

This gives us the following statement about the _driving function_ generating
the cluster (see Section 1.4):

######  Proposition 5.

Fix some $T>0$. For $\eta<-2$ and if
$\sigma(\mathbf{c})\leq\mathbf{c}^{2^{2^{1/\mathbf{c}}}}$ for all
$\mathbf{c}<1$ let $(\theta_{n}^{\mathbf{c}})_{n\geq 1}$ be the sequence of
angles we obtain from the $\mathrm{ALE}(0,\eta)$ process with capacity
parameter $\mathbf{c}$. Let $\xi_{t}^{\mathbf{c}}=\theta_{\left\lfloor
t/\mathbf{c}\right\rfloor+1}^{\mathbf{c}}$ for all $0\leq t\leq T$. Then

$(\xi_{t}^{\mathbf{c}})_{t\in[0,T]}\to(2B_{t})_{t\in[0,T]}\text{ in
distribution as }\mathbf{c}\to 0,$

as a random variable in the Skorokhod space $D[0,T]$.

We explain in Section 1.4 that by using Loewner’s equation we can immediately
turn a result about convergence of such a driving function into a result about
convergence of clusters in an appropriate space $\mathcal{K}$. The main
theorem of the paper therefore follows immediately from the proposition:

######  Theorem 6.

For $\eta,\sigma$ as in Proposition 5, let the corresponding
$\mathrm{ALE}(0,\eta)$ cluster with $N=\left\lfloor T/\mathbf{c}\right\rfloor$
particles each of capacity $\mathbf{c}$ be $K_{N}^{\mathbf{c}}$. Then as
$\mathbf{c}\to 0$, $K_{N}^{\mathbf{c}}$ converges in distribution as a random
variable in $\mathcal{K}$ to a radial $\mathrm{SLE}_{4}$ curve of capacity
$T$.

We can see in Figure 2 a cluster corresponding to a random walk, which despite
being visibly composed of slits resembles an SLE4 curve.

######  Remark.

We can give $\eta$ a physical interpretation if we think of growth in which
access to environmental resources (proportional to harmonic measure) affects
the growth rate in a _non-linear_ manner. For negative $\eta$ we could also
interpret ALE$(\alpha,\eta)$ as modelling a cluster in an environment which
inhibits growth, so growth is concentrated in areas with the least exposure to
the environment.

In any case, the remarkable thing about the $\eta<-2$ case is that it is drawn
from a family of models which naturally extend DLA-type growth, and we obtain
an SLE4 scaling-limit for a whole range of parameters.

######  Remark.

The convergence of attachment distributions to atomic measures for $\eta<-2$
complements the phase transition result of [16] in which it is shown that the
limiting attachment measures are atomic for $\eta>1$. For $-2<\eta<1$ the
distribution $h_{2}$ of the second particle is supported on all of
$\mathbb{T}$ even in the limit $\mathbf{c}\to 0$, showing that we do indeed
have three qualitative phases: for extreme values of $\eta$ the attachment
measures are degenerate, but this is not the case for $-2<\eta<1$.

###  1.4 Loewner’s equation and the Schramm-Loewner evolution

We obtain a _Schramm-Loewner evolution_ (SLE) cluster as the scaling limit of
our model, so we will give a brief overview here of what SLE is and a few
useful facts from Loewner theory which we use to establish our scaling limit.
For a more detailed treatment, see [1], [8] and [3].

Firstly, we look at _Loewner’s equation_ , which encodes our growing cluster
by a “driving function” taking values on the circle.

######  Definition 7.

Let $\xi:[0,T]\to\mathbb{R}$ be a càdlàg function. Then there is a unique
solution to _Loewner’s equation_

$\varphi_{0}(z)=z,\quad\frac{\partial}{\partial
t}\varphi_{t}(z)=\varphi_{t}^{\prime}(z)z\frac{z+e^{i\xi_{t}}}{z-e^{i\xi_{t}}},\quad
z\in\Delta,$ (4)

corresponding to a growing cluster via
$\varphi_{t}(\Delta)=\mathbb{C}_{\infty}\setminus K_{t}$.

For a sequence of angles $(\theta_{n})_{n\geq 1}$ and a capacity $\mathbf{c}$,
a growth model constructed as in Section 1.1 corresponds to the cluster
obtained by solving Loewner’s equation with the driving function

$\xi_{t}=\theta_{\left\lfloor n/\mathbf{c}\right\rfloor+1}.$

######  Definition 8.

If $(B_{t})_{t\in[0,T]}$ is a standard Brownian motion, then the _Schramm-
Loewner evolution_ with parameter $\kappa>0$ (SLEκ) is the random cluster
obtained by solving Loewner’s equation with the driving function given by
$\xi_{t}=\sqrt{\kappa}B_{t}$.

######  Remark.

One very useful property of Loewner’s equation for this paper is that the map
$D[0,T]\to\mathcal{K}$ given by $\xi\mapsto K_{T}$ is continuous [7], where
$D[0,T]$ is the usual Skorokhod space and $\mathcal{K}$ is the set of compact
subsets of $\mathbb{C}$ containing $0$, equipped with the Carathéodory
topology described in [3].

This property of Loewner’s equation means we can deduce convergence of an
ALE(0,$\eta$) cluster to an SLE4 for $\eta<-2$ by showing that the cluster is
corresponds to a driving function converging to $2B$ for a standard Brownian
motion $B$ as $\mathbf{c}\to 0$.

Schramm-Loewner evolutions arise as both the scaling limit of discrete models
such as the loop-erased random walk, which converges to an SLE2 curve [9], or
the percolation interface and SLE6 [15]; and there are many links between the
SLE and other continuous random objects, such as the links between the paths
of Brownian motion and SLE6 [10], and between the Gaussian free field and SLE4
[14].

###  1.5 Structure of paper

Our proof of Proposition 5 will involve showing that the distribution of
$\theta_{n+1}$ conditional on the previous angles
$(\theta_{1},\cdots,\theta_{n})$ converges to
$\frac{1}{2}(\delta_{\theta_{n}+\beta}+\delta_{\theta_{n}-\beta})$, and so the
whole path $\xi^{\mathbf{c}}$ converges to the same limit as a simple random
walk with step length $\beta\sim 2\mathbf{c}^{1/2}$.

We can use a heuristic approach to see why we might expect this to be the
case. If we formally take $\eta=-\infty$ and $\sigma=0$, so the $n$th
attachment point $\theta_{n+1}$ is chosen uniformly from the finite set
$\\{\theta:\liminf_{\sigma\to
0}\inf_{w\in\mathbb{T}}|\Phi_{n}^{\prime}(e^{\sigma}e^{i\theta})|/|\Phi_{n}^{\prime}(e^{\sigma}w)|>0\\}$
(i.e. among the “strongest poles” of $\Phi_{n}^{\prime}$), and let
$\tau=\inf\\{n:|\theta_{n}-\theta_{n-1}|\not=\beta\\}$, then we can calculate
that for $N=\left\lfloor T/\mathbf{c}\right\rfloor$ in the limit
$\mathbf{c}\to 0$ we have $\mathbb{P}[\tau\leq N]\to 0$ as $\mathbf{c}\to 0$.
In other words, at each step the distribution $h_{n+1}$ is equal to
$\frac{1}{2}(\delta_{\theta_{n}-\beta}+\delta_{\theta_{n}+\beta})$.

Our approach for finite $\eta<-2$ will therefore be to find a small upper
bound on $h_{n+1}(\theta)$ for $\theta$ away from the poles of
$\Phi_{n}^{\prime}$ to deduce that $h_{n+1}$ is an approximation to a sum of
atoms at the poles. Then we show separately that the contribution to
$Z_{n}=\int_{\mathbb{T}}h_{n+1}(\theta)\,\mathrm{d}\theta$ from poles other
than $e^{i(\theta_{n}\pm\beta)}$ is small.

Figure 3:  Left: the one-slit cluster of our process with 1000 points in red
sampled according to harmonic measure on the boundary. Right: the three-slit
cluster of the process with 10 000 points sampled according to harmonic
measure. Note in the second image that there are almost no points landing near
the base of the most recent (longest) particle.

In the actual model with $-\infty<\eta<-2$, we can only show that $h_{n+1}$
_approximates_
$\frac{1}{2}(\delta_{\theta_{n}-\beta}+\delta_{\theta_{n}+\beta})$ as
$\mathbf{c}\to 0$. However, weak convergence of these measures is not enough
to prove Proposition 5, so we will need to introduce some extra notation to
describe the possible behaviour of the process $(\theta_{n})_{n\geq 1}$, and
make precise the way in which its steps converge to the SSRW steps as above.

######  Definition 9.

For a small $D=D(\mathbf{c})$ (which we will specify later), define the
stopping time

$\tau_{D}:=\inf\\{n\geq
2:\min(|\theta_{n}-(\theta_{n-1}+\beta)|,|\theta_{n}-(\theta_{n-1}-\beta)|)>D\\}.$

######  Remark.

Given that $n<\tau_{D}$, we have a lot of information about the angle sequence
$(\theta_{1},\cdots,\theta_{n})$, and so can say quite a lot about the
conditional distribution of $\theta_{n+1}$. In particular, we can say that the
probability that $n+1=\tau_{D}$ is very low, and that the distribution of
$\theta_{n+1}-\theta_{n}$ is (approximately) symmetric. The results of all the
following sections will be used to establish these two facts.

######  Theorem 10.

Suppose that $\nu>2$. There exists a constant $A>0$ depending only on $\nu$
and $T$ such that when $\sigma\leq\mathbf{c}^{2^{2^{1/\mathbf{c}}}}$, then for
$D=\mathbf{c}^{9/2}\sigma^{1/2}$, whenever $n<N\wedge\tau_{D}$ and
$\mathbf{c}$ is sufficiently small,

$\int_{F_{n}}h_{n+1}(\theta)\,\mathrm{d}\theta\leq A\mathbf{c}^{4}$ (5)

with probability $1$, where
$F_{n}=\\{\theta\in\mathbb{T}:|\theta-(\theta_{n}+\beta)|\geq D\text{ and
}|\theta-(\theta_{n}-\beta)|\geq D\\}$, and with probability $1$

$\left|\int_{\theta_{n}+\beta-D}^{\theta_{n}+\beta+D}h_{n+1}(\theta)\,\mathrm{d}\theta-\int_{\theta_{n}-\beta-D}^{\theta_{n}-\beta+D}h_{n+1}(\theta)\,\mathrm{d}\theta\right|\leq
A\mathbf{c}^{11/4}.$ (6)

In Section 2 we prove a number of technical results about the positions of the
images and preimages of points $w\in\Delta$ under the maps $f_{j}$,
$\Phi_{n}=f_{1}\circ\cdots\circ f_{n}$, and
$\Phi_{j,n}=\Phi_{j}^{-1}\circ\Phi_{n}$ when $w$ is close to the poles of
$\Phi_{n}^{\prime}$. When dealing with points away from these poles, we make
extensive use of results from [16]. Our estimates for the positions of these
images will be useful when we find upper bounds on the derivative
$|\Phi_{n}^{\prime}(w)|=|f_{n}^{\prime}(w)|\times|f_{n-1}^{\prime}(\Phi_{n-1,n}(w))|\times\cdots\times|f_{1}^{\prime}(\Phi_{1,n}(w))|$,
using lower bounds on the distance between $\Phi_{j,n}(w)$ and the poles of
$f_{j}^{\prime}$.

In Section 3.1 we integrate the pre-normalised density
$|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|^{\nu}$ over the regions around
$\theta_{n}\pm\beta$, and so obtain a lower bound on

$Z_{n}=\int_{\mathbb{T}}|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|^{\nu}\,\mathrm{d}\theta.$

In Section 3.2 and Section 4 we find upper bounds on
$|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|$ for $\theta\in F_{n}$, and so using
the lower bound on $Z_{n}$ we can establish the bound (5).

In Section 3.3 we establish the technical results needed to prove (6).

######  Remark.

In our proof of Theorem 10, the convergence of $h_{n+1}$ to
$\frac{1}{2}(\delta_{\theta_{n}+\beta}+\delta_{\theta_{n}-\beta})$ does not
rely on the convergence of $h_{1},\cdots,h_{n}$ to these _symmetric_ discrete
measures, only that $n<\tau_{D}$. If we were to use the fact that the angle
sequence up until time $n$ is very close to a simple symmetric random walk,
then some properties (such as the fact that the longest interval on which a
SSRW is monotone has length of order $O(\log n)$) would allow us to optimise
our choice of $\sigma$ further than we have. However, for the convergence of
our cluster to an SLE4 curve, we do require a $\sigma$ which decays at least
as quickly as $\mathbf{c}^{1/\mathbf{c}}$, which is already much faster than
the fixed power of $\mathbf{c}$ used in [16] and elsewhere, so we have not
attempted to optimise our choice of
$\sigma\leq\mathbf{c}^{2^{2^{1/\mathbf{c}}}}$.

###  1.6 Table of notation

As we introduce a lot of notation in this paper, we will give a list here so
that it is possible to look up any notation appearing in any section without
searching for where it was introduced.

Subsets of the complex plane

$\mathbb{C}_{\infty}$

The Riemann sphere, $\mathbb{C}\cup\\{\infty\\}$

$\mathbb{D}$

The open unit disc $\\{z\in\mathbb{C}:|z|<1\\}$.

$\overline{\mathbb{D}}$

The closed unit disc $\\{z\in\mathbb{C}:|z|\leq 1\\}$.

$\Delta$

The exterior disc $\mathbb{C}_{\infty}\setminus\overline{\mathbb{D}}$.

$\mathbb{T}$

The unit circle
$\partial\Delta=\\{z\in\mathbb{C}:|z|=1\\}=\\{e^{i\theta}:\theta\in\mathbb{R}\\}$.
We will often abuse notation and identify $\mathbb{T}$ with
$\mathbb{R}/2\pi\mathbb{Z}$.

$\partial U$

The boundary of a set $U\subseteq\mathbb{C}_{\infty}$, defined as $\partial
U=\overline{U}\setminus U^{\circ}$.

Conformal maps

$f$

The conformal map
$f_{\mathbf{c}}\colon\Delta\to\Delta\setminus(1,1+d(\mathbf{c})]$ which we say
attaches a particle to the unit circle at the point 1.

$f_{j}$

Given a sequence of angles $(\theta_{j})_{j\geq 1}$, $f_{j}$ attaches a
particle to the unit circle at the point $e^{i\theta_{j}}$, so
$f_{j}(z):=e^{i\theta_{j}}f(e^{-i\theta_{j}}z)$.

$\beta$

The distance from $1$ of the points which are sent to the base of the particle
by $f$. Defined uniquely as the $\beta=\beta(\mathbf{c})\in(0,\pi)$ such that
$f_{\mathbf{c}}(e^{\pm i\beta})=1$, and obeys $\beta\sim 2\mathbf{c}^{1/2}$ as
$\mathbf{c}\to 0$.

$d$

The length of the particle attached by $f$, defined by
$f_{\mathbf{c}}(1)=1+d(\mathbf{c})$. Obeys $d\sim\beta\sim 2\mathbf{c}^{1/2}$
as $\mathbf{c}\to 0$.

$\Phi_{n}$

The conformal map which attaches the entire cluster of $n$ particles to the
unit circle at the point $1$. Constructed as $f_{1}\circ f_{2}\circ\cdots\circ
f_{n}$.

$\Phi_{j,n}$

The conformal map which attaches only the most recent $n-j$ particles to the
unit circle. Given by $\Phi_{j,n}=\Phi_{j}^{-1}\circ\Phi_{n}$.

Model parameters

$\eta$

The parameter controlling the relationship between our attachment
distributions and the harmonic measure on the boundary of the cluster.
Throughout this paper we take $\eta<-2$.

$\nu$

We write $\nu=-\eta$. Note that $\nu>2$ throughout.

$T$

The total capacity of our cluster, fixed throughout.

$\mathbf{c}$

The capacity of each individual particle attached to the cluster. We consider
in this paper the limit $\mathbf{c}\to 0$, so all the following parameters are
functions of $\mathbf{c}$.

$\sigma$

A regularisation parameter, used so that we do not evaluate our conformal maps
$\Phi_{n}^{\prime}$ at their poles on $\mathbb{T}$, instead evaluating
everything on $e^{\sigma}\mathbb{T}$. We take $\sigma$ to be a function of
$\mathbf{c}$, decaying very rapidly as $\mathbf{c}\to 0$:
$\sigma\leq\mathbf{c}^{2^{2^{1/\mathbf{c}}}}$

$L$

The maximum distance of $z$ from $e^{i(\theta_{n}\pm\beta)}$ at which we rely
on the estimates for $|\Phi_{j,n}(z)-e^{i\theta_{j+1}}|$ we obtain in the
proof of Theorem 17. We take $L$ to be a function of $\mathbf{c}$ which does
not decay as rapidly as $\sigma$: $L=\mathbf{c}^{2^{N+1}}$.

$D$

A bound on $\min_{\pm}|\theta_{n+1}-(\theta_{n}\pm\beta)|$ which holds with
high probability. If this distance exceeds $D$, we stop the process. We can
take $D=\mathbf{c}^{9/2}\sigma^{1/2}$.

Points in $\mathbb{T}$

$\theta^{\top}_{j}$

The point in $\mathbb{T}$ which $\theta_{j}$ was “supposed to” attach nearby
to, i.e. the unique choice of $\theta_{j-1}\pm\beta$ which $\theta_{j}$ is
within $D$ of (if $\theta_{j}$ is not within $D$ of either, we will have
stopped the process at time $\tau_{D}\leq j$).

$\theta^{\bot}_{j}$

The choice of $\theta_{j-1}\pm\beta$ which isn’t $\theta_{j}^{\top}$.

$\widehat{z}^{n}_{j}$

The point on $\mathbb{T}$ corresponding to the base of the $j$th particle in
the cluster $K_{n}$, for $1\leq j\leq n-1$. Given by
$\widehat{z}^{n}_{j}:=\Phi_{j,n}^{-1}(e^{i\theta^{\bot}_{j+1}})$. See Figure 5
for an illustration. We refer to the points on $\mathbb{T}$ close to
$\widehat{z}_{j}^{n}$ for some $j$ as _singular points_ for $h_{n+1}$, and
points away from all $\widehat{z}_{j}^{n}$ as _regular points_.

Probabilistic objects

$h_{n+1}$

The density of the distribution on $\mathbb{T}$ of $\theta_{n+1}$, conditional
on $\theta_{1},\cdots,\theta_{n}$. Given by
$h_{n+1}(\theta)\propto|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|^{\nu}$.

$Z_{n}$

The normalising factor for $h_{n+1}$. Given by
$Z_{n}:=\int_{\mathbb{T}}|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|^{\nu}\,\mathrm{d}\theta$.

$\mathbb{P}$

The law of $(\theta_{n})_{n\in\mathbb{N}}$. Implicitly depends on $\mathbf{c}$
and $\sigma$.

$\tau_{D}$

The first time at which some $\theta_{n+1}$ is further than $D$ from both of
$\theta_{n}\pm\beta$. We stop the process when this happens, but show in
Section 3 and Section 4 that with high probability $\tau_{D}>N:=\left\lfloor
T/\mathbf{c}\right\rfloor$.

Approximations and bounds
We will use the following notation when we have two functions depending on a
parameter $x$ which is converging to some
$x_{0}\in\mathbb{R}\cup\\{\pm\infty\\}$, and we want to say the two functions
are similar in some way, or that one bounds the other.

$f(x)\sim g(x)$

The ratio $\frac{f(x)}{g(x)}\to 1$ as $x\to x_{0}$.

$f(x)=O(g(x))$

The ratio $\left|\frac{f(x)}{g(x)}\right|$ is bounded above as $x\to x_{0}$,
so there exists a constant $C>0$ such that $|f(x)|\leq C|g(x)|$ in a
neighbourhood of $x_{0}$. The constant $C$ should not depend on any other
parameter or variable. If the value of $C$ does depend on a parameter $\rho$,
we will write $f(x)=O_{\rho}(g(x))$. Throughout this paper we hold $T$ and
$\nu=-\eta$ fixed, so we may occasionally omit these as subscripts when the
constant depends on them.

$f(x)=o(g(x))$

The ratio $\left|\frac{f(x)}{g(x)}\right|\to 0$ as $x\to x_{0}$.

When $f$ and $g$ are non-negative (particularly when they are probabilities or
densities), we may use the following alternative notations.

$f(x)\lesssim g(x)$

The same as $f(x)=O(g(x))$, i.e. there exists a constant $C>0$ such that
$f(x)\leq Cg(x)$ in a neighbourhood of $x_{0}$.

$f(x)\ll g(x)$

The same as $f(x)=o(g(x))$, i.e. $f(x)/g(x)\to 0$ as $x\to x_{0}$.

$f(x)\asymp g(x)$

Both $f(x)=O(g(x))$ and $g(x)=O(f(x))$, i.e. there exists constants
$C_{1},C_{2}>0$ such that
$C_{1}g(x)\leq f(x)\leq C_{2}g(x)$ in a neighbourhood of $x_{0}$.

Finally, we may write $f(x)\approx g(x)$, but this will only be used
informally to mean that $f$ and $g$ behave similarly in some sense.

##  2 Spatial distortion of points

There are several steps we need to establish our upper bound on $\int
h_{n+1}(\theta)\,\mathrm{d}\theta$ in (5), including precise estimates for
$|\Phi_{n}^{\prime}|$ near its poles. We can decompose the derivative

$\Phi_{n}^{\prime}(w)=\prod_{j=0}^{n-1}f_{n-j}^{\prime}(\Phi_{n-j,n}(w))$ (7)

where

$\Phi_{k,n}:=\Phi_{k}^{-1}\circ\Phi_{n}=f_{k+1}\circ f_{k+2}\circ\cdots\circ
f_{n}.$ (8)

Then we have precise estimates on $|f^{\prime}|$ near to its poles $e^{\pm
i\beta}$, and upper bounds away from these poles, and so we write

$|\Phi_{n}^{\prime}(w)|=\prod_{j=0}^{n-1}\left|f^{\prime}\left(e^{-i\theta_{n-j}}\Phi_{n-j,n}(w)\right)\right|.$
(9)

We will show that if $w$ is close to one of $e^{i(\theta_{n}\pm\beta)}$, then
for each $j$, the point $e^{-i\theta_{n-j}}\Phi_{n-j,n}(w)$ is close to a pole
of $|f^{\prime}|$, and we will derive specific estimates on the distance in
terms of the distance $|w-e^{i(\theta_{n}\pm\beta)}|$. Conversely, we will
show that the only way for _every_ image $e^{-i\theta_{n-j}}\Phi_{n-j,n}(w)$
to be close to a pole is for $w$ to be close to $e^{i(\theta_{n}\pm\beta)}$,
and so the measure $\mathrm{d}h_{n+1}$ is concentrated around
$\theta_{n}+\beta$ and $\theta_{n}-\beta$.

Firstly, we will establish an estimate for $|f^{\prime}|$ close to its poles
$e^{\pm i\beta}$, and a universal upper bound away from these two points.

######  Lemma 11.

There are universal constants $A_{1},A_{2}>0$ such that for all
$\mathbf{c}<1$, for $w\in\Delta$, if $|w-e^{i\beta}|\leq\frac{3}{4}\beta$,
then

$A_{1}\frac{\beta^{1/2}}{|w-e^{i\beta}|^{1/2}}\leq|f_{\mathbf{c}}^{\prime}(w)|\leq
A_{2}\frac{\beta^{1/2}}{|w-e^{i\beta}|^{1/2}},$ (10)

and similarly if $|w-e^{-i\beta}|\leq\frac{3}{4}\beta$.

Moreover, there is a third constant $A_{3}$ such that if
$\min\\{|w-e^{i\beta}|,|w-e^{-i\beta}|\\}>\frac{3}{4}\beta$, then

$|f_{\mathbf{c}}^{\prime}(w)|\leq A_{3}.$

###### Proof.

See Lemma 5 of [16]. ∎

This lemma tells us that $|\Phi_{n}^{\prime}(w)|$ will be large only when many
of the points $e^{-i\theta_{n-j}}\Phi_{n-j,n}(w)$ in (9) are close to one of
the poles $e^{\pm i\beta}$. We will next introduce some technical estimates
which will allow us to determine for which points $w$ this is true.

######  Remark.

If we imagine an idealised path in which $|\theta_{i+1}-\theta_{i}|=\beta$ for
all $i$, then $f_{n}(e^{i(\theta_{n}\pm\beta)})=e^{i\theta_{n-1}}$, and
$f_{n-1}(e^{i\theta_{n-1}})=e^{i\theta_{n-2}}$, and so on. Hence
$\Phi_{n-j,n}(e^{i(\theta_{n}\pm\beta)})=e^{i\theta_{n-j+1}}=e^{i(\theta_{n-j}+s_{n-j}\beta)}$,
where $s_{n-j}\in\\{\pm 1\\}$. So if a point $w$ is close to one of
$e^{i(\theta_{n}\pm\beta)}$ then, as $f$ is continuous when extended to
$\overline{\Delta}$, each of the points in (9) is close to
$e^{is_{n-j}\beta}$, but continuity alone does not allow us to make precise
what we mean by “$w$ is close to $e^{i(\theta_{n}\pm\beta)}$”, so to estimate
the size of $|\Phi_{n}^{\prime}(w)|$, we need a precise estimate for
$|f(w)-f(e^{i\beta})|$ in terms of $|w-e^{i\beta}|$.

######  Lemma 12.

For $w\in\Delta$, for all $\mathbf{c}<1$, if $|w-e^{i\beta}|\leq\beta/2$, then

$\displaystyle|f_{\mathbf{c}}(w)-1|=$ $\displaystyle
2(e^{\mathbf{c}}-1)^{1/4}|w-e^{i\beta}|^{1/2}$ (11)
$\displaystyle\times\left(1+O\left[\frac{|w-e^{i\beta}|}{\mathbf{c}^{1/2}}\vee\mathbf{c}^{1/4}|w-e^{i\beta}|^{1/2}\right]\right).$

###### Proof.

We will work with the half-plane slit map
$\widetilde{f}_{\mathbf{c}}\colon\mathbb{H}\to\mathbb{H}\setminus(0,i\sqrt{1-e^{-\mathbf{c}}}\,]$
by conjugating $f$ with the Möbius map
$m_{\mathbb{H}}\colon\Delta\to\mathbb{H}$ given by

$m_{\mathbb{H}}(w)=i\frac{w-1}{w+1},$ (12)

and its inverse

$m_{\Delta}(z):=m_{\mathbb{H}}^{-1}(z)=\frac{1-iz}{1+iz}.$ (13)

The benefit of this is that $\widetilde{f}_{\mathbf{c}}$ has a simple explicit
form:

$\widetilde{f}_{\mathbf{c}}(\zeta)=e^{-\mathbf{c}/2}\sqrt{\zeta^{2}-(e^{\mathbf{c}}-1)}$
(14)

where the branch of the square root is given by
$\arg\colon\mathbb{C}\setminus[0,\infty)\to(0,2\pi)$, so we write

$f_{\mathbf{c}}=m_{\Delta}\circ\widetilde{f}_{\mathbf{c}}\circ m_{\mathbb{H}}$

and will derive a separate estimate for each of the three maps.

As $w$ is close to
$e^{i\beta}=2e^{-\mathbf{c}}-1+2ie^{-\mathbf{c}}\sqrt{e^{\mathbf{c}}-1}$, we
will expand each map about the images (given by a simple calculation)
$m_{\mathbb{H}}(e^{i\beta})=-\sqrt{e^{\mathbf{c}}-1}$,
$\widetilde{f}_{\mathbf{c}}(-\sqrt{e^{\mathbf{c}}-1})=0$, and
$m_{\Delta}(0)=1$. Our calculations will show that $m_{\Delta}$ and
$m_{\mathbb{H}}$ behave like scaling by a constant close to the relevant
points, and that the behaviour of $f_{\mathbf{c}}$ seen in (11) is due to the
behaviour of $\widetilde{f}_{\mathbf{c}}$ close to
$\pm\sqrt{e^{\mathbf{c}}-1}$.

First, when $w=e^{i\beta}+\delta$,

$\displaystyle|m_{\mathbb{H}}(w)-m_{\mathbb{H}}(e^{i\beta})|$
$\displaystyle=\left|\frac{2\delta}{(e^{i\beta}+1+\delta)(e^{i\beta}+1)}\right|$
$\displaystyle=\frac{1}{2}e^{\mathbf{c}}|\delta|(1+O(|\delta|))$ (15)

since a simple calculation shows that $|e^{i\beta}+1|^{2}=4e^{-\mathbf{c}}$.

Next, we will evaluate $\widetilde{f}_{\mathbf{c}}$ at a point close to one of
the two preimages of $0$, $\pm\sqrt{e^{\mathbf{c}}-1}$:

$\displaystyle\left|\widetilde{f}_{\mathbf{c}}(\pm\sqrt{e^{\mathbf{c}}-1}+\lambda)\right|$
$\displaystyle=e^{-\mathbf{c}/2}\left|\sqrt{\pm
2\sqrt{e^{\mathbf{c}}-1}\lambda+\lambda^{2}}\right|$
$\displaystyle=\sqrt{2}e^{-\mathbf{c}/2}(e^{\mathbf{c}}-1)^{1/4}|\lambda|^{1/2}\left(1+O\left(\frac{|\lambda|}{\mathbf{c}^{1/2}}\right)\right).$
(16)

Finally, for a small $z\in\mathbb{H}$,

$\displaystyle|m_{\Delta}(z)-1|=\left|\frac{1-iz}{1+iz}-1\right|=\left|\frac{-2iz}{1+iz}\right|=2|z|(1+O(|z|)).$
(17)

Then for $w$ close to $e^{i\beta}$, applying (15), (16) and (17) in turn, we
obtain

$\displaystyle|f(w)-1|=\ $ $\displaystyle
2(e^{\mathbf{c}}-1)^{1/4}|w-e^{i\beta}|^{1/2}$ $\displaystyle\times$
$\displaystyle\left(1+O\left(\frac{|w-e^{i\beta}|}{\mathbf{c}^{1/2}}\right)\right)\left(1+O\left(\mathbf{c}^{1/4}|w-e^{i\beta}|^{1/2}\right)\right).$

Then for $\mathbf{c}^{3/2}\leq|w-e^{i\beta}|\leq\beta/2$, we have the estimate
(11) with error term of order $\mathbf{c}^{-1/2}|w-e^{i\beta}|$, and for
$|w-e^{i\beta}|\leq\mathbf{c}^{3/2}$ the error term has order
$\mathbf{c}^{1/4}|w-e^{i\beta}|^{1/2}$. ∎

######  Remark.

Unlike most results in this section, we will not use the following lemma in
Section 3, but it will be very useful in Section 4.2. We include it here and
omit the proof as it is very similar to Lemma 12.

######  Lemma 13.

For all $\mathbf{c}<1$, if $z\in\Delta\setminus(1,1+d(\mathbf{c})]$ has
$|z-1|\leq\mathbf{c}$, then

$\min_{\pm}|f^{-1}(z)-e^{\pm
i\beta}|=\frac{|z-1|^{2}}{4(e^{\mathbf{c}}-1)^{1/2}}\left(1+O\left(|z-1|\right)\right).$

Now we have all the technical results we need in order to prove our lower
bound on $|\Phi_{n}^{\prime}(w)|$ when $w$ is close to one of the two “most
recent basepoints” $e^{i(\theta_{n}\pm\beta)}$. We will derive the bound
itself in Section 3.1, and here we will show that each of the points
$\Phi_{n-j,n}(w)$ in (9) is close to $e^{i\theta_{n-j+1}}$.

######  Proposition 14.

Let $L=L(\mathbf{c},N)=\mathbf{c}^{2^{N+1}}$, and let $n<N\wedge\tau_{D}$. If
$\delta:=\min|w-e^{i(\theta_{n}\pm\beta)}|\leq L$, and $|w|\geq e^{\sigma}$,
then for all $1\leq j\leq n$,

$\left|\Phi_{n-j,n}(w)-e^{i\theta^{\top}_{n-j+1}}\right|=\left[2(e^{\mathbf{c}}-1)^{\frac{1}{4}}\right]^{2(1-2^{-j})}\delta^{2^{-j}}(1+O(\mathbf{c}^{4})).$
(18)

Before we begin the proof we will introduce some notation in order to make the
argument easier to follow.

######  Definition 15.

By definition of $\tau_{D}$, for each $n<\tau_{D}$ one of the two angles
$\theta_{n-1}\pm\beta$ is within distance $D$ of $\theta_{n}$. We will call
the closer of the two angles $\theta^{\top}_{n}$, and the other angle
$\theta^{\bot}_{n}$.

###### Proof of Proposition 14.

We will proceed by induction on $j$. For $j=1$, the estimate (18) follows
directly from Lemma 12. For a given $1\leq j\leq n-1$, assume that

$|\Phi_{n-j,n}(w)-e^{i\theta^{\top}_{n-j+1}}|=\left[2(e^{\mathbf{c}}-1)^{\frac{1}{4}}\right]^{2(1-2^{-j})}\delta^{2^{-j}}(1+O(\mathbf{c}^{4})),$

(as $|\theta_{n}-\theta^{\top}_{n}|<D\ll\mathbf{c}^{4}$, this certainly holds
for $j=1$) and then by the triangle inequality, since
$|e^{i\theta_{n-j}}-e^{i\theta^{\top}_{n-j}}|\leq|\theta_{n-j}-\theta^{\top}_{n-j}|<D$,
we have

$\displaystyle|\Phi_{n-j-1,n}(w)-e^{i\theta_{n-j}}|-D\leq|$
$\displaystyle\Phi_{n-j-1,n}(w)-e^{i\theta^{\top}_{n-j}}|\leq|\Phi_{n-j-1,n}(w)-e^{i\theta_{n-j}}|+D.$

Now by Lemma 12,

$\displaystyle|$
$\displaystyle\Phi_{n-j-1,n}(w)-e^{i\theta_{n-j}}|=|f_{n-j}(\Phi_{n-j,n}(w))-f_{n-j}(e^{i\theta^{\top}_{n-j+1}})|$
$\displaystyle=|f(e^{-i\theta^{\top}_{n-j+1}}\Phi_{n-j,n}(w))-1|$
$\displaystyle=2(e^{\mathbf{c}}-1)^{\frac{1}{4}}|e^{-i\theta^{\top}_{n-j+1}}\Phi_{n-j,n}(w)-1|^{1/2}(1+O(\mathbf{c}^{1/4}|e^{-i\theta^{\top}_{n-j+1}}\Phi_{n-j,n}(w)-1|^{1/2}))$
$\displaystyle=\left[2(e^{\mathbf{c}}-1)^{\frac{1}{4}}\right]^{1+(1-2^{-j})}\delta^{2^{-(j+1)}}(1+O(\mathbf{c}^{4}))(1+O(\mathbf{c}^{3/8}\delta^{2^{-(j+1)}}))$
$\displaystyle=\left[2(e^{\mathbf{c}}-1)^{\frac{1}{4}}\right]^{2(1-2^{-(j+1)})}\delta^{2^{-(j+1)}}(1+O(\mathbf{c}^{4}))$

and the second error term is absorbed since $\delta^{2^{-(j+1)}}\leq
L^{2^{-(j+1)}}\leq\mathbf{c}^{4}$.

Now as $\delta=|w-e^{i(\theta_{n}\pm\beta)}|\geq|w|-1\geq\sigma$, and
$D\sim\mathbf{c}^{9/2}\sigma^{1/2}$ (see Section 1.6), we have

$\displaystyle|\Phi_{n-j-1,n}(w)-e^{i\theta^{\top}_{n-j}}|$
$\displaystyle=|\Phi_{n-j-1,n}(w)-e^{i\theta_{n-j}}|\left(1+O\left(\frac{D}{\mathbf{c}^{\frac{1}{2}(1-2^{-(j+1)})}\delta^{2^{-(j+1)}}}\right)\right)$
$\displaystyle=|\Phi_{n-j-1,n}(w)-e^{i\theta_{n-j}}|\left(1+O\left(\mathbf{c}^{4}\sigma^{1/4}\right)\right),$

and hence our result holds for all $1\leq j\leq n$ by induction. ∎

##  3 The newest basepoints

###  3.1 A lower bound on the normalising factor

We defined in (2) the density function $h_{n+1}(\theta)$ and the $n$th
normalising factor

$Z_{n}=\int_{\mathbb{T}}|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|^{-\eta}\,\mathrm{d}\theta.$
(19)

If we are going to find upper bounds on $h_{n+1}$ by bounding
$|\Phi_{n}^{\prime}|$, then we will need to have some lower bound on the
normalising factor $Z_{n}$. In this section, we will obtain a lower bound on
$Z_{n}$, and it will give us our upper bound on $h_{n+1}$ in Section 4.2.
First, we will need a good estimate for $|\Phi_{n}^{\prime}|$ around the poles
$e^{i(\theta_{n}\pm\beta)}$.

######  Lemma 16.

Let $n<\left\lfloor T/\mathbf{c}\right\rfloor\wedge\tau_{D}$. There are
constants $A_{1},A_{2}>0$ such that for any $\mathbf{c}<1$, whenever
$|\varphi|<L$,

$A_{1}^{n}\frac{\mathbf{c}^{\frac{1}{2}(1-2^{-n})}}{(\sigma^{2}+\varphi^{2})^{\frac{1}{2}(1-2^{-n})}}\leq\left|\Phi_{n}^{\prime}\left(e^{\sigma+i(\theta_{n}\pm\beta+\varphi)}\right)\right|\leq
A_{2}^{n}\frac{\mathbf{c}^{\frac{1}{2}(1-2^{-n})}}{(\sigma^{2}+\varphi^{2})^{\frac{1}{2}(1-2^{-n})}}$

provided that
$\sigma=\sigma(\mathbf{c})\leq\mathbf{c}^{2^{2^{1/\mathbf{c}}}}$.

###### Proof.

For $|\varphi|<L$, without loss of generality take
$\theta=\theta_{n}+\beta+\varphi$. Since $\Phi_{n}=f_{1}\circ\cdots\circ
f_{n}$, by the chain rule,

$|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|=\prod_{j=0}^{n-1}\left|f^{\prime}\left(e^{-i\theta_{n-j}}\Phi_{n-j,n}(e^{\sigma+i\theta})\right)\right|,$

where $\Phi_{k,n}=\Phi_{k}^{-1}\circ\Phi_{n}=f_{k+1}\circ
f_{k+2}\circ\cdots\circ f_{n}$.

By Proposition 14, if
$\delta:=|e^{\sigma+i\theta}-e^{i(\theta_{n}+\beta)}|<L$, then for all $1\leq
j\leq n-1$,
$|\Phi_{n-j,n}(e^{\sigma+i\theta})-e^{i\theta^{\top}_{n-j+1}}|=[2(e^{\mathbf{c}}-1)^{\frac{1}{4}}]^{2(1-2^{-j})}\delta^{2^{-j}}(1+O(\mathbf{c}^{4}))$,
and so by Lemma 11 (the above estimate shows that
$e^{-i\theta_{n-j}}\Phi_{n-j,n}(e^{\sigma+i\theta})$ is close enough to one of
$e^{\pm i\beta}$ to apply this lemma),

$\displaystyle\left|f^{\prime}\left(e^{-\theta_{n-j}}\Phi_{n-j,n}(e^{\sigma+i\theta})\right)\right|$
$\displaystyle\asymp\beta^{1/2}|\Phi_{n-j,n}(e^{\sigma+i\theta})-e^{i\theta^{\top}_{n-j+1}}|^{-1/2}$
$\displaystyle=\beta^{1/2}[2(e^{\mathbf{c}}-1)^{\frac{1}{4}}]^{-(1-2^{-j})}\delta^{-2^{-j-1}}(1+O(\mathbf{c}^{4}))$
$\displaystyle\asymp\mathbf{c}^{2^{-(j+2)}}\delta^{-2^{-(j+1)}}.$

For $j=0$, as $\Phi_{n,n}$ is the identity map,

$\displaystyle|f^{\prime}(e^{-i\theta_{n}}\Phi_{n,n}(e^{\sigma+i\theta}))|=|f^{\prime}(e^{\sigma+i(\theta-\theta_{n})})|$
$\displaystyle\asymp A_{1}\beta^{1/2}\delta^{-1/2}$ $\displaystyle\asymp
A\mathbf{c}^{1/4}\delta^{-1/2}.$

Now if we combine the bounds for each term in the above product for
$|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|$, we have

$\displaystyle|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|$
$\displaystyle\geq\prod_{j=0}^{n-1}\left(A_{1}\mathbf{c}^{2^{-(j+2)}}\delta^{-2^{-(j+1)}}\right)$
$\displaystyle=A_{1}^{n}\mathbf{c}^{\frac{1}{2}(1-2^{-n})}\delta^{-(1-2^{-n})}.$

and a similar upper bound. Finally, $\delta$ is given by

$\displaystyle\delta$
$\displaystyle=|e^{\sigma+i\theta}-e^{i(\theta_{n}+\beta)}|$
$\displaystyle=|e^{\sigma+i\varphi}-1|$
$\displaystyle=(\sigma^{2}+\varphi^{2})^{1/2}(1+O(\sigma+|\varphi|))$
$\displaystyle\asymp(\sigma^{2}+\varphi^{2})^{1/2},$

and so, modifying the constants as necessary, we have our result. ∎

We can now obtain our lower bound on the normalising factor.

######  Proposition 17.

If $\nu>2$, then there exists a constant $A$ depending only on $\nu$ such that
for any fixed $T>0$, for sufficiently small $\mathbf{c}$ and for
$n<\left\lfloor T/\mathbf{c}\right\rfloor\wedge\tau_{D}$,

$Z_{n}\geq
A^{n}\mathbf{c}^{\frac{\nu}{2}(1-2^{-n})}\sigma^{-[\nu(1-2^{-n})-1]}$ (20)

provided that
$\sigma=\sigma(\mathbf{c})\leq\mathbf{c}^{2^{2^{1/\mathbf{c}}}}$.

###### Proof.

The normalising factor $Z_{n}$ is given by the integral
$\int_{\mathbb{T}}|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|^{\nu}\,\mathrm{d}\theta$,
and Lemma 16 gives us a lower bound on the integrand for $\theta$ close to
$\theta_{n}+\beta$:

$|\Phi_{n}^{\prime}(e^{\sigma+i(\theta_{n}+\beta+\varphi)})|^{\nu}\geq
A^{n}\mathbf{c}^{\frac{\nu}{2}(1-2^{-n})}(\sigma^{2}+\varphi^{2})^{-\frac{\nu}{2}(1-2^{-n})}$

when $|\varphi|<L$.

We will now integrate our lower bound over the interval
$(\theta_{n}+\beta-L,\theta_{n}+\beta+L)$. First, note that

$\displaystyle\int_{-L}^{L}(\sigma^{2}+\varphi^{2})^{-\frac{\nu}{2}(1-2^{-n})}\,\mathrm{d}\varphi$
$\displaystyle=\int_{-L/\sigma}^{L/\sigma}(\sigma^{2}+\sigma^{2}x^{2})^{-\frac{\nu}{2}(1-2^{-n})}\,\sigma\mathrm{d}x$
$\displaystyle=\sigma^{1-\nu(1-2^{-n})}\int_{-L/\sigma}^{L/\sigma}\frac{\mathrm{d}x}{(1+x^{2})^{\frac{\nu}{2}(1-2^{-n})}}$
$\displaystyle\geq A^{\prime}\sigma^{1-\nu(1-2^{-n})}$

for a constant $A^{\prime}$, since the integral term on the right hand side is
increasing as $\mathbf{c}\to 0$ because $\sigma\ll L$. Note that this all
remains true for any $\eta<0$, and the fact that $\eta<-2$ will only be
necessary in Section 3.2.

Finally, we can put together our bounds (and modify our constant $A$) to get

$\displaystyle\int_{\theta_{n}+\beta-L}^{\theta_{n}+\beta+L}|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|^{\nu}\,\mathrm{d}\theta$
$\displaystyle\geq
A^{n}\mathbf{c}^{\frac{\nu}{2}(1-2^{-n})}\int_{-L}^{L}(\sigma^{2}+\varphi^{2})^{\frac{\nu}{2}(1-2^{-n})}\,\mathrm{d}\varphi$
$\displaystyle\geq
A^{n}\mathbf{c}^{\frac{\nu}{2}(1-2^{-n})}\sigma^{1-\nu(1-2^{-n})}$

as required. ∎

###  3.2 Concentration about each basepoint

Most of our upper bounds on $|\Phi_{n}^{\prime}|$ will be established in
Section 4, but we will find one here as it uses the estimates from the
previous section. Using the terminology we introduce in Section 4 and
illustrate in Figure 4, in this section we look at _singular points_ which are
within $L$ of one of the “main” poles $e^{i(\theta_{n}\pm\beta)}$ so the
estimate of Lemma 16 is valid, but are not within $D$ of these poles.

######  Proposition 18.

Let $n<\left\lfloor T/\mathbf{c}\right\rfloor\wedge\tau_{D}$. For
$\sigma(\mathbf{c})\leq\mathbf{c}^{2^{2^{1/\mathbf{c}}}}$, then with
$L=\mathbf{c}^{2^{N+1}}$ and $D=\mathbf{c}^{9/2}\sigma^{1/2}\ll L$,

$\frac{1}{Z_{n}}\int_{[-L,L]\setminus[-D,D]}|\Phi_{n}^{\prime}(e^{\sigma+i(\theta_{n}\pm\beta+\varphi)})|^{\nu}\,\mathrm{d}\varphi=o(\mathbf{c}^{4})$

as $\mathbf{c}\to 0$.

###### Proof.

Using the symmetry of our upper bound in Lemma 16, it will be enough to find
the upper bound
$\int_{D}^{L}|\Phi_{n}^{\prime}(e^{\sigma+i(\theta_{n}+\beta+\varphi)})|^{\nu}\,\mathrm{d}\varphi\ll\mathbf{c}^{2}Z_{n}$.
We have, modifying the constant $A_{2}$ where necessary,

$\displaystyle\int_{D}^{L}|\Phi_{n}^{\prime}(e^{\sigma+i(\theta_{n}+\beta+\varphi)})|^{\nu}\,\mathrm{d}\varphi$
$\displaystyle\leq
A_{2}^{n}\mathbf{c}^{\frac{\nu}{2}(1-2^{-n})}\int_{D}^{L}(\sigma^{2}+\varphi^{2})^{-\frac{\nu}{2}(1-2^{-n})}\,\mathrm{d}\varphi$
$\displaystyle=A_{2}^{n}\frac{\mathbf{c}^{\frac{\nu}{2}(1-2^{-n})}}{\sigma^{\nu(1-2^{-n})-1}}\int_{D/\sigma}^{L/\sigma}(1+x^{2})^{-\frac{\nu}{2}(1-2^{-n})}\,\mathrm{d}x$
$\displaystyle\leq
A_{2}^{n}\frac{\mathbf{c}^{\frac{\nu}{2}(1-2^{-n})}}{\sigma^{\nu(1-2^{-n})-1}}\int_{D/\sigma}^{L/\sigma}x^{-\nu(1-2^{-n})}\,\mathrm{d}x$
$\displaystyle\leq
A_{2}^{n}\frac{\mathbf{c}^{\frac{\nu}{2}(1-2^{-n})}}{D^{\nu(1-2^{-n})-1}},$

and so, using our lower bound on $Z_{n}$,

$\displaystyle\frac{\int_{D}^{L}|\Phi_{n}^{\prime}(e^{\sigma+i(\theta_{n}+\beta+\varphi)})|^{\nu}\,\mathrm{d}\varphi}{Z_{n}}$
$\displaystyle\leq(A_{2}/A)^{n}\left(\frac{\sigma}{D}\right)^{\nu(1-2^{-n})-1}$
$\displaystyle=(A_{2}/A)^{n}\left(\mathbf{c}^{-9/2}\sigma^{1/2}\right)^{\nu(1-2^{-n})-1}$

which, since $\nu(1-2^{-n})-1\geq\frac{1}{2}\nu-1>0$, decays faster than any
power of $\mathbf{c}$ as $\mathbf{c}\to 0$. ∎

Note that the above proof is the only place in which we use that $\eta<-2$. If
$-2\leq\eta<0$, then early on in the process we will attach the $n$th particle
where $n=\tau_{D}$, and the inductive arguments we used in the proofs of
Proposition 14 and Lemma 16 will fail. It then becomes extremely difficult to
say how the process behaves, but the scaling limit as $\mathbf{c}\to 0$ is
unlikely to be described by the Schramm-Loewner evolution.

###  3.3 Symmetry of the two most recent basepoints

There are two parts to the statement in Theorem 10 about convergence of
$h_{n+1}$ to the discrete measure
$\frac{1}{2}(\delta_{\theta_{n}-\beta}+\delta_{\theta_{n}+\beta})$: the
previous two sections and Section 4 establish that $h_{n+1}$ is concentrated
very tightly around $\theta_{n}\pm\beta$, and we will show here that the
weight given to each of these two points is approximately equal.

######  Remark.

Unlike the results from the previous two sections, the following proposition
is not inductive, i.e. as long as $n<\left\lfloor
T/\mathbf{c}\right\rfloor\wedge\tau_{D}$, the previous choices of angle do not
have to have been symmetric, so it would still apply in the extreme case where
$(\theta_{n})_{n\in\mathbb{N}}$ is close to an arithmetic progression:
$\theta_{2}\approx\theta_{1}+\beta$,
$\theta_{3}\approx\theta_{2}+\beta,\cdots,\theta_{n}\approx\theta_{n-1}+\beta$.

######  Proposition 19.

Let $n<\left\lfloor T/\mathbf{c}\right\rfloor\wedge\tau_{D}$. Then

$\sup_{|\varphi|<D}\left|\log\left(\frac{\left|\Phi_{n}^{\prime}\left(e^{\sigma+i(\theta_{n}+\beta+\varphi)}\right)\right|}{\left|\Phi_{n}^{\prime}\left(e^{\sigma+i(\theta_{n}-\beta-\varphi)}\right)\right|}\right)\right|\leq
A\mathbf{c}^{11/4}$

for some constant $A$ depending only on $T$.

###### Proof.

Let
$z_{\pm}=\exp\left(\sigma+i\left[\theta_{n}\pm(\beta+\varphi)\right]\right)$
for $|\varphi|<D$, and write
$\lambda_{\pm}=z_{\pm}-e^{i(\theta_{n}\pm\beta)}$. We can then write

$\log\left(\frac{|\Phi_{n}^{\prime}(z_{+})|}{|\Phi_{n}^{\prime}(z_{-})|}\right)=\sum_{j=0}^{n-1}\log\left(\frac{|f_{n-j}^{\prime}(\Phi_{n-j,n}(z_{+}))|}{|f_{n-j}^{\prime}(\Phi_{n-j,n}(z_{-}))|}\right)$
(21)

and so we can estimate each term in (21) separately.

The $j=0$ term is exactly 0, by the symmetry of $|f_{n}^{\prime}|$ about
$\theta_{n}$.

For $1\leq j\leq n-1$, we will use Lemma 4 of [16], which states that
$f^{\prime}(z)=\frac{f(z)}{z}\frac{z-1}{(z-e^{i\beta})^{1/2}(z-e^{-i\beta})^{1/2}}$,
to compare the two derivatives in the $j$th term of (21). Write
$z_{\pm}^{j}=\Phi_{n-j,n}(z_{\pm})$, then the $j$th term in (21) is

$|f_{n-j}^{\prime}(z_{\pm}^{j})|=\frac{|z_{\pm}^{j+1}|}{|z_{\pm}^{j}|}\frac{|z_{\pm}^{j}-e^{i\theta_{n-j}}|}{|z_{\pm}^{j}-e^{i\theta^{\bot}_{n-j+1}}|^{1/2}|z_{\pm}^{j}-e^{i\theta^{\top}_{n-j+1}}|^{1/2}}$
(22)

There will be some telescoping in the product which allows us to find

$\prod_{j=1}^{n-1}\frac{|z_{\pm}^{j+1}|}{|z_{\pm}^{j}|}=\frac{|z_{\pm}^{n}|}{|z_{\pm}^{1}|}.$

Then recall that in Section 2 we derived estimates for the distance of
$z_{\pm}^{n}$ from $e^{i\theta^{\top}_{n-j+1}}$ in terms of $|\lambda_{\pm}|$.
So by Proposition 14, as $e^{i\theta^{\top}_{1}}=1$,

$|z_{\pm}^{n}-1|=\left[2(e^{\mathbf{c}}-1)^{\frac{1}{4}}\right]^{2(1-2^{-n})}|\lambda_{\pm}|^{2^{-n}}(1+O(\mathbf{c}^{4}))=O(\mathbf{c}^{17/4})$

since $|\lambda_{\pm}|^{2^{-n}}\lesssim D^{2^{-n}}\ll
L^{2^{-n}}\leq\mathbf{c}^{4}$. Therefore
$|z_{\pm}^{n}|=1+O(\mathbf{c}^{17/4})$, and similarly
$|z_{\pm}^{1}|=1+O(\mathbf{c}^{17/4})$.

Having dealt with the first fraction in all derivatives (22) at once, we will
tackle the remaining terms individually for each $1\leq j\leq n-1$.

First note that by definition of $\theta^{\top}_{-}$,
$|e^{i\theta^{\top}_{n-j+1}}-e^{i\theta_{n-j}}|=|e^{i\beta}-1|$. Hence, using
Proposition 14 again,

$\displaystyle|z_{\pm}^{j}-e^{i\theta_{n-j}}|$
$\displaystyle=|e^{i\theta^{\top}_{n-j+1}}-e^{i\theta_{n-j}}|\left[1+O\left(\frac{|z_{\pm}^{j}-e^{i\theta^{\top}_{n-j+1}}|}{|e^{i\theta^{\top}_{n-j+1}}-e^{i\theta_{n-j}}|}\right)\right]$
$\displaystyle=|e^{i\beta}-1|\left[1+O\left(\mathbf{c}^{-2^{-(j+1)}}|\lambda_{\pm}|^{2^{-j}}\right)\right]$
$\displaystyle=|e^{i\beta}-1|\left[1+O\left(\mathbf{c}^{15/4}\right)\right]$

since $|\lambda_{\pm}|^{2^{-j}}\ll L^{2^{-(n-1)}}\leq\mathbf{c}^{4}$.

Similarly,

$\displaystyle|z_{\pm}^{j}-e^{i\theta^{\bot}_{n-j+1}}|$
$\displaystyle=|e^{2i\beta}-1|(1+O(\mathbf{c}^{15/4})),$

and finally, directly from Proposition 14,

$|z_{\pm}^{j}-e^{i\theta^{\top}_{n-j+1}}|=\left[2(e^{\mathbf{c}}-1)^{\frac{1}{4}}\right]^{2(1-2^{-j})}|\lambda_{\pm}|^{2^{-j}}(1+O(\mathbf{c}^{4})).$

Note that for the three estimates we just found, the only part which depends
on the choice of $\pm$ is the error term (as $|\lambda_{+}|=|\lambda_{-}|$).
Hence the part of the ratio of $|f_{n-j}^{\prime}(z_{+}^{j})|$ to
$|f_{n-j}^{\prime}(z_{-}^{j})|$ which comes from the second fraction in (22)
is just $1+O(\mathbf{c}^{15/4})$.

We can therefore find a constant $A$ (which does not depend on $n$ or
$\varphi$) such that for each $1\leq j\leq n-1$,
$\left|\log\left(\frac{|f_{n-j}^{\prime}(z_{+}^{j})|}{|f_{n-j}^{\prime}(z_{-}^{j})|}\right)\right|\leq
A\mathbf{c}^{15/4}$. As there are $O_{T}(\mathbf{c}^{-1})$ such terms in the
product (21), we have

$\left|\log\left(\frac{|\Phi_{n}^{\prime}(z_{+})|}{|\Phi_{n}^{\prime}(z_{-})|}\right)\right|=O_{T}(\mathbf{c}^{11/4})$

as we wanted. ∎

Now we can deduce that $h_{n+1}$ gives (asymptotically) the same measure to
the sets $(\theta_{n}+\beta-D,\theta_{n}+\beta+D)$ and
$(\theta_{n}-\beta-D,\theta_{n}-\beta+D)$.

######  Remark.

Recall that earlier we used the heuristic argument that if $\eta=-\infty$ (so
we choose from points with the highest-order pole), then we attach the
$(n+1)$th particle to one of $\theta_{n}\pm\beta$, with equal probability.
With finite $\eta<-2$, the derivative $|\Phi_{n}^{\prime}|$ in fact differs
slightly at each of $e^{\sigma+i(\theta_{n}+\beta)}$ and
$e^{\sigma+i(\theta_{n}-\beta)}$, and so choosing to attach a particle at
$e^{i\theta}$ for $\theta$ maximising
$|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|$ leads to a deterministic process
rather than our SLE4 limit.

However, when we have a finite $\eta<-2$, integrating over the range $(-D,D)$
around each $\theta_{n}\pm\beta$ means that only the asymptotic behaviour of
$|\Phi_{n}^{\prime}|$ needs to be the same in order to have symmetry between
the two points $\theta_{n}\pm\beta$.

######  Corollary 20.

For $n<\left\lfloor T/\mathbf{c}\right\rfloor\wedge\tau_{D}$,

$\left|\int_{-D}^{D}h_{n+1}(\theta_{n}+\beta+\varphi)\,\mathrm{d}\varphi-\int_{-D}^{D}h_{n+1}(\theta_{n}-\beta-\varphi)\,\mathrm{d}\varphi\right|=O_{T}(\mathbf{c}^{11/4}).$
(23)

###### Proof.

From Proposition 19, we have

$\displaystyle\int_{-D}^{D}$ $\displaystyle
h_{n+1}(\theta_{n}+\beta+\varphi)\,\mathrm{d}\varphi-\int_{-D}^{D}h_{n+1}(\theta_{n}-\beta-\varphi)\,\mathrm{d}\varphi$
$\displaystyle=\frac{1}{Z_{n}}\int_{-D}^{D}\left(|\Phi_{n}^{\prime}(e^{\sigma+i(\theta_{n}+\beta+\varphi)})|^{\nu}-|\Phi_{n}^{\prime}(e^{\sigma+i(\theta_{n}-\beta-\varphi)})|^{\nu}\right)\mathrm{d}\varphi$
$\displaystyle=\frac{1}{Z_{n}}\int_{-D}^{D}\left(|\Phi_{n}^{\prime}(e^{\sigma+i(\theta_{n}+\beta+\varphi)})|^{\nu}-e^{O_{T}(\mathbf{c}^{11/4})}|\Phi_{n}^{\prime}(e^{\sigma+i(\theta_{n}+\beta+\varphi)})|^{\nu}\right)\mathrm{d}\varphi$
$\displaystyle=O_{T}\left(\mathbf{c}^{11/4}\frac{\int_{-D}^{D}|\Phi_{n}^{\prime}(e^{\sigma+i(\theta_{n}+\beta+\varphi)})|^{\nu}\,\mathrm{d}\varphi}{Z_{n}}\right)$

which is just $O_{T}(\mathbf{c}^{11/4})$ by definition of $Z_{n}$. ∎

##  4 Analysis of the density away from the main basepoints

In this section, we will classify the points $\theta\in\mathbb{T}$ with
$|\theta-(\theta_{n}\pm\beta)|\geq D$ (i.e. the set $F_{n}$ from Theorem 10)
into _regular points_ $R_{n}$ where $h_{n+1}(\theta)\ll 1$, and _singular
points_ $S_{n}$ where $h_{n+1}(\theta)\gtrsim 1$. We make this classification
based on how close the image $\Phi_{n}(e^{\sigma+i\theta})$ is to the common
basepoint of the cluster, which is the image of all the poles of
$\Phi_{n}^{\prime}$, as we can see in Figure 4.

In Section 4.1 we make this classification explicit and establish a bound on
$h_{n+1}$ for the regular points. In Section 4.2 we analyse the singular
points more carefully and establish an upper bound on
$\int_{S_{n}}h_{n+1}(\theta)\,\mathrm{d}\theta$ using similar techniques as in
Section 3.1.

$\Phi_{3}$ Figure 4:  We can see on the left the three types of points in
$e^{\sigma}\mathbb{T}$ for the three-slit cluster: we have the _singular
points_ in red and orange and the _regular points_ in yellow. The right hand
side of the diagram shows that a point on $e^{\sigma}\mathbb{T}$ is classified
as regular if its image under $\Phi_{n}$ is far from the common basepoint
(Proposition 21 in Section 4.1 shows that this implies $h_{n+1}\ll 1$), and
the singular points are further classified into the two main (red) arcs
containing $e^{i(\theta_{n}\pm\beta)}$, and the other (orange) singular
points. We have $h_{n+1}\gtrsim 1$ for all singular points, but we obtained a
lower bound on the integral of $|\Phi_{n}^{\prime}|$ over the red regions in
Section 3.1, and we will find an upper bound on the integral of this
derivative over the orange regions in Section 4.2. Note that the choice of
$\sigma$ we have used for this diagram is around $\mathbf{c}^{2}$ rather than
the much smaller $\mathbf{c}^{2^{1/\mathbf{c}}}$, which is necessary to make
the envelope $\Phi_{3}(e^{\sigma}\mathbb{T})$ clear, but does mean that some
“regular” points are closer to the common basepoints than the red “singular”
points. With a sufficiently small $\sigma$ this isn’t the case.

###  4.1 Regular points

In this section, we will establish a criterion for $\theta\in\mathbb{T}$ to be
in our set of _regular_ points for which $h_{n+1}(\theta)\ll 1$, based on the
position of $\Phi_{n}(e^{\sigma+i\theta})$, as shown in Figure 4.

We will first derive an upper bound on $|\Phi_{n}^{\prime}(w)|$ in terms of
$|\Phi_{n}(w)-1|$, so we can classify $w\in\Delta$ as a regular point using
the distance of its image $\Phi_{n}(w)$ from $1$.

######  Proposition 21.

Let $n<N(\mathbf{c})\wedge\tau_{D}$. For $\theta\in\mathbb{R}$, let
$w=\exp(\sigma+i\theta)$.

For any function $a:\mathbb{R}_{+}\to\mathbb{R}_{+}$ with
$D^{2^{-N}}/\beta\leq a(\mathbf{c})\leq\mathbf{c}^{3/2}$ for all
$0<\mathbf{c}<1$, if

$|\Phi_{n}(w)-1|\geq\beta a(\mathbf{c})$ (24)

then, for sufficiently small $\mathbf{c}$,

$|\Phi_{n}^{\prime}(w)|\leq
A^{n}\beta^{n/2}\left(\frac{a(\mathbf{c})}{8}\right)^{-\frac{1}{2}(2^{n}-1)}$
(25)

where $A$ is a universal constant independent of $a$.

###### Proof.

We will use the estimate (11) from Lemma 12. For convenience, let
$z=\Phi_{n}(w)$, and we will estimate
$|\Phi_{n}^{\prime}(w)|=|(\Phi_{n}^{-1})^{\prime}(z)|^{-1}$ by using (9) and
estimating each term separately, using Lemma 12 to obtain estimates on
$\Phi_{n-j,n}(w)=\Phi_{n-j}^{-1}(z)$ by induction on $j$.

First we claim that for $A(\mathbf{c})\leq\mathbf{c}^{1/2}$, and
$\zeta\in\Delta\setminus(1,1+d(\mathbf{c})]$, if we have $|\zeta-1|\geq\beta
A(\mathbf{c})$, then

$\min_{\pm}(|f^{-1}(\zeta)-e^{\pm i\beta}|)\geq\frac{1}{4}\beta
A(\mathbf{c})^{2}$ (26)

for all $\mathbf{c}<c_{0}$, where $c_{0}>0$ is a universal constant which
doesn’t depend on $A$.

To see this, suppose that $|f^{-1}(\zeta)-e^{i\beta}|<\frac{1}{4}\beta
A(\mathbf{c})^{2}$. Then by Lemma 12, setting $\varepsilon=2^{1/4}-1>0$, for
sufficiently small $\mathbf{c}$,

$\displaystyle|\zeta-1|$ $\displaystyle=|f(f^{-1}(\zeta))-f(e^{i\beta})|$
$\displaystyle=2(e^{\mathbf{c}}-1)^{1/4}|f^{-1}(\zeta)-e^{i\beta}|^{1/2}(1+O\left(A(\mathbf{c})^{2}\vee\mathbf{c}^{1/2}A(\mathbf{c})\right))$
$\displaystyle<2(\beta/2)^{1/2}(1+\varepsilon)\frac{1}{2}\beta^{1/2}A(\mathbf{c})(1+\varepsilon)$
$\displaystyle=\beta A(\mathbf{c}),$

so we have shown the contrapositive for our claim.

The derivative $|\Phi_{n}^{\prime}(w)|$ is decomposed in (9) into the product
of $n$ terms $\left|f^{\prime}(e^{-i\theta_{k}}\Phi_{k,n}(w))\right|$, and so
we can find an upper bound on $|\Phi_{n}^{\prime}(w)|$ by obtaining lower
bounds on each
$|\Phi_{k,n}(w)-e^{i(\theta_{k}\pm\beta)}|=|\Phi_{k}^{-1}(z)-e^{i(\theta_{k}\pm\beta)}|$
for $0\leq k\leq n-1$ and applying Lemma 11.

We claim that, for each $0\leq k\leq n-1$,

$|\Phi_{k}^{-1}(z)-e^{i\theta_{k+1}}|\geq\beta\times
8\left(\frac{a(\mathbf{c})}{8}\right)^{2^{k}}$ (27)

and we will show this using induction. For $k=0$, (27) is exactly the
assumption (24) of this proposition. For $k\geq 1$, we assume as the induction
step that

$|\Phi_{k-1}^{-1}(z)-e^{i\theta_{k}}|\geq\beta\times
8\left(\frac{a(\mathbf{c})}{8}\right)^{2^{k-1}}$

and aim to obtain (27) by applying (26).

Taking $A(\mathbf{c})=8\left(\frac{a(\mathbf{c})}{8}\right)^{2^{k-1}}$ in (26)
gives us

$|\Phi_{k}^{-1}(z)-e^{i\theta^{\top}_{k+1}}|\geq\beta\times
16\left(\frac{a(\mathbf{c})}{8}\right)^{2^{k}},$

and so since $8\beta\left(\frac{a(\mathbf{c})}{8}\right)^{2^{k}}\geq 2D$ when
$k\leq N\wedge\tau_{D}$ (for $\mathbf{c}$ sufficiently small),

$\displaystyle|\Phi_{k}^{-1}(z)-e^{i\theta_{k+1}}|$
$\displaystyle\geq|\Phi_{k}^{-1}(z)-e^{i\theta^{\top}_{k+1}}|-|e^{i\theta_{k+1}}-e^{i\theta^{\top}_{k+1}}|$
$\displaystyle\geq 16\beta\left(\frac{a(\mathbf{c})}{8}\right)^{2^{k}}-2D$
$\displaystyle\geq 8\beta\left(\frac{a(\mathbf{c})}{8}\right)^{2^{k}},$

verifying (27).

Then (27) tells us, using (26), that for each $0\leq k\leq n-1$,

$|\Phi_{k}^{-1}(z)-e^{i(\theta_{k}\pm\beta)}|\geq\beta\times
16\left(\frac{a(\mathbf{c})}{8}\right)^{2^{k}},$ (28)

and so, by Lemma 11, for $\mathbf{c}$ sufficiently small,

$\displaystyle|\Phi_{n}^{\prime}(w)|$
$\displaystyle=\prod_{k=0}^{n-1}|f_{k+1}^{\prime}(\Phi_{k}^{-1}(z))|$
$\displaystyle\leq
A^{n}\beta^{n/2}\prod_{k=1}^{n-1}\left(\beta^{1/2}\left[\beta\times
16\left(\frac{a(\mathbf{c})}{8}\right)^{2^{k}}\right]^{-1/2}\right)$
$\displaystyle=(A/4)^{n}\beta^{n/2}\left(\frac{a(\mathbf{c})}{8}\right)^{-\frac{1}{2}(2^{n}-1)}$

for a universal constant $A$. ∎

In the next section we will use these results with $a(\mathbf{c})$ equal to
$\frac{L}{4\beta}$. We can easily check now that if we use this choice of $a$
in Proposition 21 then, comparing (25) with (20), if $\sigma$ decays as fast
as $\mathbf{c}^{2^{2^{N}}}$ then $|\Phi_{n}^{\prime}(z)|^{\nu}$ is far smaller
than $\mathbf{c}Z_{n}$, for $z$ away from the preimages of $e^{i\theta_{1}}$,
and so if we classify our regular points as those $\theta$ for which
$|\Phi_{n}(e^{\sigma+i\theta})-1|\geq\frac{L}{4}$ then we do have
$\sup\limits_{\theta\in R_{n}}h_{n+1}(\theta)\ll 1$.

###  4.2 Old singular points

In Section 3, we established a lower bound on the $n$th normalising factor
$Z_{n}$. So to show that the probability is low that the $(n+1)$th particle is
attached at a point in $E\subseteq\mathbb{T}$, we need to find an upper bound
on $\int_{E}|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|^{\nu}\,\mathrm{d}\theta$.

We did this over certain regions in Section 4.1 by finding a bound
$|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|^{\nu}\ll\mathbf{c}Z_{n}$. In this
section we will consider _singular_ points where we can have
$|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|^{\nu}\gg Z_{n}$. However, if we look
at Figure 4 we can see that not all singular points are close to the preimages
$\theta_{n}\pm\beta$ of the base of the most recent particle; there are
singular points at the preimages of the base of each particle. We will
therefore need to estimate the integrand $|\Phi_{n}^{\prime}|^{\nu}$ more
carefully, and show that when integrated over the singular points around these
old bases and normalised by $Z_{n}$, the resulting probability is small.

The first thing we need to do is to describe precisely which points we are
integrating over. We have previously classified our points into regular points
$R_{n}$ and singular points $S_{n}$ by looking at the distance
$|\Phi_{n}(w)-1|$. Points are singular when $|\Phi_{n}(w)-1|<\beta
a(\mathbf{c})$ (for an $a(\mathbf{c})$ we will specify later), and we will
find a way of differentiating between the “new” singular points around the
preimages of the $n$th particle’s base and the “older” singular points around
the preimages of the other particles’ bases. To make this clear, we will first
give names to all of these preimages.

$f_{3}$$f_{2}$$f_{1}$$\Phi_{3}$$\widehat{z}_{-}^{3}$$\widehat{z}_{+}^{3}$$\widehat{z}_{2}^{3}$$\widehat{z}_{1}^{3}$$e^{i\theta^{\top}_{3}}$$e^{i\theta^{\bot}_{3}}$$\widehat{z}_{1}^{2}$$e^{i\theta^{\top}_{2}}$$e^{i\theta^{\bot}_{2}}$$e^{i\theta_{1}}$
Figure 5:  The construction of a cluster with three particles by composing the
three maps $f_{3}$, $f_{2}$ and $f_{1}$. The top left diagram has labelled the
four poles $\widehat{z}^{3}_{\pm}$, $\widehat{z}_{2}^{3}$ and
$\widehat{z}_{1}^{3}$ of $\Phi_{3}^{\prime}$ with text, and the markers $+$,
$\times$ and $\circ$ have been used to track the images of
$e^{\sigma}\widehat{z}$ for each pole $\widehat{z}$. By following the
preimages of each point in the upper-right diagram through each map $f_{1}$,
$f_{2}$ and $f_{3}$, we can see how we defined the “lesser” poles
$\widehat{z}_{2}^{3}$ and $\widehat{z}_{1}^{3}$: for example, in the lower-
right diagram $e^{i\theta^{\bot}_{2}}$ is a pole of $f_{1}^{\prime}$, its
preimage under $f_{2}$ is $\widehat{z}_{1}^{2}$, and the preimage of
$\widehat{z}_{1}^{2}$ under $f_{3}$ is $\widehat{z}_{1}^{3}$.

Firstly, we have the two “most attractive” points: the preimages of the base
of the most recent ($n$th) slit. We will call these two points
$\widehat{z}^{n}_{\pm}=e^{i(\theta_{n}\pm\beta)}$. Now the other points
correspond to the bases of the $n-1$ other slits in the cluster, and we will
denote them by $\widehat{z}^{n}_{j}$ for $1\leq j\leq n-1$. The base of the
first slit is the image under $f_{1}$ of the choice of
$e^{i(\theta_{2}\pm\beta)}$ which is _not_ close to $e^{i\theta_{2}}$. We
defined this in Definition 15 to be $e^{i\theta^{\bot}_{2}}$, and so the point
sent to the base of the first slit by $\Phi_{n}$ is the preimage under
$f_{2}\circ\cdots\circ f_{n}=\Phi_{1,n}$ of $e^{i\theta^{\bot}_{2}}$, so set
$\widehat{z}^{n}_{1}=\Phi_{1,n}^{-1}(e^{i\theta^{\bot}_{2}})$.

In general, when the $j$th slit is attached to the cluster by $f_{j}$, there
are two points which are mapped to the base of the slit:
$e^{i\theta^{\top}_{j+1}}$ (where the later slits are also attached), and
$e^{i\theta^{\bot}_{j+1}}$, which has nothing else attached to it. Therefore,
the point sent to the base of the $j$th slit by $\Phi_{n}$ is the preimage of
$e^{i\theta^{\bot}_{j+1}}$ under $f_{j+1}\circ\cdots\circ f_{n}$. We can see
this illustrated in Figure 5.

######  Definition 22.

The base of the $j$th slit for $1\leq j\leq n-1$ is the image of

$\widehat{z}^{n}_{j}:=\Phi_{j,n}^{-1}\left(e^{i\theta^{\bot}_{j+1}}\right)$
(29)

under $\Phi_{n}$.

Note that for all $n<N\wedge\tau_{D}$ and $1\leq j\leq n-1$,

$f_{n}(\widehat{z}^{n}_{j})=\widehat{z}^{n-1}_{j},$ (30)

where we adopt the convention that
$\widehat{z}^{n-1}_{n-1}=e^{i\theta^{\bot}_{n}}$.

######  Remark.

We will bound $|\Phi_{n}^{\prime}(w)|$ above when $w$ is close to
$\widehat{z}_{j}^{n}$, so first we will have to show that these points
$\widehat{z}_{j}^{n}$ for $1\leq j\leq n-1$ are not close to the points
$e^{i(\theta_{n}\pm\beta)}$ where we have already shown $|\Phi_{n}^{\prime}|$
is large.

######  Lemma 23.

For $n<N\wedge\tau_{D}$ and $1\leq j\leq n-1$,

$|e^{i(\theta_{n}\pm\beta)}-\widehat{z}_{j}^{n}|\geq\mathbf{c}^{2^{n-j}},$

when $\mathbf{c}$ is sufficiently small.

###### Proof.

Assume for contradiction that
$|e^{i(\theta_{n}+\beta)}-\widehat{z}_{j}^{n}|<\mathbf{c}^{2^{n-j}}$. By Lemma
12,

$\displaystyle|e^{i\theta_{n}}-\widehat{z}_{j}^{n-1}|$
$\displaystyle=|f_{n}(e^{i(\theta_{n}+\beta)})-f_{n}(\widehat{z}_{j}^{n})|$
$\displaystyle=2(e^{\mathbf{c}}-1)^{1/4}\mathbf{c}^{2^{n-j-1}}\left(1+O\left(\mathbf{c}^{1/4}\mathbf{c}^{2^{n-j-1}}\right)\right)$
$\displaystyle<\frac{1}{2}\mathbf{c}^{2^{n-j-1}}$

for $\mathbf{c}$ smaller than some universal $c_{0}$ (with
$(c_{0}-1)^{1/4}<1/4$, and small enough to make the error term irrelevant),
and so

$\displaystyle|e^{i\theta^{\top}_{n}}-\widehat{z}_{j}^{n-1}|\leq|e^{i\theta^{\top}_{n}}-e^{i\theta_{n}}|+|e^{i\theta_{n}}-\widehat{z}_{j}^{n-1}|<\mathbf{c}^{2^{n-j-1}},$
(31)

since $|e^{i\theta^{\top}_{n}}-e^{i\theta_{n}}|\lesssim
D\ll\mathbf{c}^{2^{n-j-1}}$. Then, as $\theta^{\top}_{n}=\theta_{n-1}\pm\beta$
for some choice of $\pm$, we can apply this argument repeatedly until we
arrive at
$|e^{i\theta^{\top}_{j+1}}-\widehat{z}_{j}^{j}|<\mathbf{c}^{2^{j-j}}=\mathbf{c}$.
But as we noted after (30), $\widehat{z}_{j}^{j}=e^{i\theta^{\bot}_{j+1}}$,
and $|e^{i\theta^{\top}_{j+1}}-e^{i\theta^{\bot}_{j+1}}|\sim
4\mathbf{c}^{1/2}\gg\mathbf{c}$, and so we have our contradiction. ∎

######  Remark.

In fact the lower bound in Lemma 23 is fairly generous; it would take only a
small amount of extra work in the proof above to get a tighter bound of
$\mathbf{c}^{2^{n-j-1}}$, and we could improve this even further as we used
the weak bound $(e^{\mathbf{c}}-1)^{1/4}<\frac{1}{4}$ in the initial
calculation. However, all we need from Lemma 23 is a bound which decays more
slowly than $L=\mathbf{c}^{2^{N+1}}$, and so we have chosen the bound which
leads to the simplest possible proof.

######  Remark.

The following corollary (which we will not prove) is not used in the proof of
our main results, but does answer a question we may worry about: if we know
that $w$ is within $L$ of some $\widehat{z}_{j}^{n}$, then is that $j$
uniquely determined?

######  Corollary 24.

For $n<N\wedge\tau_{D}$, if $1\leq j<k\leq n-1$, then

$|\widehat{z}_{j}^{n}-\widehat{z}_{k}^{n}|\geq\mathbf{c}^{2^{n-j}}$

for sufficiently small $\mathbf{c}$.

######  Remark.

The next result will be useful in telling us for which points
$\theta\in\mathbb{T}$ we can bound $|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|$
above using Proposition 21, and will later help us locate those points for
which Proposition 21 does not provide an upper bound.

######  Lemma 25.

Suppose that $n<N\wedge\tau_{D}$, and let $w\in\Delta$. For all $\mathbf{c}$
sufficiently small, if $|\Phi_{n}(w)-1|\leq\frac{L}{4}$, then either
$\min\limits_{\pm}|w-e^{i(\theta_{n}\pm\beta)}|\leq L$, or there exists some
$1\leq j\leq n-1$ such that

$|\Phi_{j,n}(w)-e^{i\theta^{\bot}_{j+1}}|\leq\frac{\beta}{4}\left(\frac{L}{\beta}\right)^{2^{j}}.$

###### Proof.

Suppose that there is no such $j$. We will show that
$\min_{\pm}|w-e^{i(\theta_{n}\pm\beta)}|\leq L$. We now claim that
$|\Phi_{j,n}(w)-e^{i\theta^{\top}_{j+1}}|\leq\frac{\beta}{4}\left(\frac{L}{\beta}\right)^{2^{j}}$
for all $0\leq j\leq n-1$ (where $\Phi_{0,n}=\Phi_{n}$ and
$\theta^{\top}_{1}=\theta_{1}=0$). For $j=0$ the claim is the true by
assumption, and if the claim is true for $0\leq j<n-1$, then by Lemma 13, as
$|\Phi_{j,n}-e^{i\theta_{j+1}}|\leq\frac{\beta}{4}\left(\frac{L}{\beta}\right)^{2^{j}}+|e^{i\theta^{\top}_{j+1}}-e^{i\theta_{j+1}}|\leq\frac{\beta}{2}\left(\frac{L}{\beta}\right)^{2^{j}}$,
for sufficiently small $\mathbf{c}$,

$\displaystyle\min(|\Phi_{j+1,n}(w)-e^{i\theta^{\top}_{j+2}}|,|\Phi_{j+1,n}(w)-e^{i\theta^{\bot}_{j+2}}|)$
$\displaystyle\leq\frac{\frac{1}{4}\beta^{2}\left(\frac{L}{\beta}\right)^{2^{j+1}}}{4(e^{\mathbf{c}}-1)^{1/2}}\left(1+\frac{1}{2}\right)$
$\displaystyle=\frac{3\beta/2}{4(e^{\mathbf{c}}-1)^{1/2}}\times\frac{\beta}{4}\left(\frac{L}{\beta}\right)^{2^{j}}$
$\displaystyle\leq\frac{\beta}{4}\left(\frac{L}{\beta}\right)^{2^{j}}$

since $\beta\sim 2(e^{\mathbf{c}}-1)^{1/2}$ for small $\mathbf{c}$. But we
supposed initially that
$|\Phi_{j+1,n}(w)-e^{i\theta^{\bot}_{j+2}}|>\frac{\beta}{4}\left(\frac{L}{\beta}\right)^{2^{j+1}}$,
and so the above shows that
$|\Phi_{j+1,n}(w)-e^{i\theta^{\top}_{j+2}}|\leq\frac{\beta}{4}\left(\frac{L}{\beta}\right)^{2^{j+1}}$,
and by induction our claim holds. Finally, one more application of Lemma 13
after the $j=n-1$ case of our claim,
$|\Phi_{n-1,n}(w)-e^{i\theta_{n}}|\leq\frac{\beta}{2}\left(\frac{L}{\beta}\right)^{2^{n-1}}$,
tells us that
$\min\limits_{\pm}|w-e^{i(\theta_{n}\pm\beta)}|\leq\frac{3\beta/2}{16(e^{\mathbf{c}}-1)^{1/2}}\beta\left(\frac{L}{\beta}\right)^{2^{n}}\ll
L$, as required. ∎

######  Remark.

We intend to use this lemma to find a precise expression for our set $S_{n}$
of singular points and then we can make a precise estimate on the size of
$|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|$ for $\theta\in S_{n}$ as we did in
Lemma 16. For a singular point $w$, Lemma 25 tells us that for some $j$,
$\Phi_{j,n}(w)$ is close to $e^{i\theta^{\bot}_{j+1}}$, and we now need to
turn that into an estimate for the distance between $w$ and
$\Phi_{j,n}^{-1}(e^{i\theta^{\bot}_{j+1}})=\widehat{z}_{j}^{n}$.

######  Corollary 26.

Suppose that $n<N\wedge\tau_{D}$, and let $w\in\Delta$. For all $\mathbf{c}$
sufficiently small, if $|\Phi_{n}(w)-1|\leq\frac{L}{4}$ then either
$\min_{\pm}|w-e^{i(\theta_{n}\pm\beta)}|\leq L$ or there exists some $1\leq
j\leq n-1$ such that

$|w-\widehat{z}_{j}^{n}|\leq
A^{n-j}\frac{\beta}{4}\left(\frac{L}{\beta}\right)^{2^{j}},$

where $A$ is some universal constant.

###### Proof.

To deduce this from Lemma 25, we need only show that there is some constant
$A$ such that
$|\Phi_{j,n}(w)-e^{i\theta^{\bot}_{j+1}}|\leq\frac{\beta}{4}\left(\frac{L}{\beta}\right)^{2^{j}}\implies|w-\widehat{z}_{j}^{n}|\leq
A^{n-j}\frac{\beta}{4}\left(\frac{L}{\beta}\right)^{2^{j}}$. Fix some $1\leq
j\leq n-1$. We will show that for $j\leq k\leq n-1$,
$|\Phi_{k+1,n}(w)-\widehat{z}_{j}^{k+1}|\leq
A|\Phi_{k,n}(w)-\widehat{z}_{j}^{k}|$.

Fix a path $\gamma:(0,1]\to\Delta$ with $\lim_{\varepsilon\downarrow
0}\gamma(\varepsilon)=\widehat{z}_{j}^{k}$, $\gamma(1)=\Phi_{k,n}(w)$, and
$|\gamma(t)-\widehat{z}_{j}^{k}|\leq|\Phi_{k,n}(w)-\widehat{z}_{j}^{k}|$ for
all $t\in(0,1]$. We can also choose $\gamma$ in such a way that it has arc
length $\ell:=\int_{\gamma}|\mathrm{d}z|\leq
2|\Phi_{k,n}(w)-\widehat{z}_{j}^{k}|$. By the fundamental theorem of calculus,

$\displaystyle|\Phi_{k+1,n}(w)-\widehat{z}_{j}^{k+1}|$
$\displaystyle=|f_{k+1}^{-1}(\Phi_{k,n}(w))-f_{k+1}^{-1}(\widehat{z}_{j}^{k})|$
$\displaystyle=\left|\int_{\gamma}(f_{k+1}^{-1})^{\prime}(\zeta)\,\mathrm{d}\zeta\right|$
$\displaystyle\leq\ell\times\sup_{\zeta\in\gamma(0,1]}|(f_{k+1}^{-1})^{\prime}(\zeta)|$
$\displaystyle=\frac{\ell}{\inf_{\omega\in
f_{k+1}^{-1}(\gamma(0,1])}|f_{k+1}^{\prime}(\omega)|}.$

Now there must be some constant $M\geq 1$ such that
$|\omega-e^{i\theta_{k+1}}|\geq\beta/M$ for all $\omega\in
f_{k+1}^{-1}(\gamma(0,1])$ (otherwise, if
$|\omega-e^{i\theta_{k+1}}|<\beta/M$, then it is easy to check using the
explicit form of $f_{\mathbf{c}}$ from [12] that
$|f_{k+1}(\omega)-e^{i\theta_{k+1}}(1+d)|=O(\beta/M^{2})$, and so
$|\widehat{z}_{j}^{k}-e^{i\theta_{k+1}}(1+d)|\leq|f_{k+1}(\omega)-e^{i\theta_{k+1}}(1+d)|+|f_{k+1}(\omega)-\widehat{z}_{j}^{k}|\leq\frac{1}{2}d$
for $M$ sufficiently large, contradicting $\widehat{z}_{j}^{k}\in\mathbb{T}$).
There is therefore, by Lemma 11, some constant $A$ such that
$\inf\limits_{\omega\in
f_{k+1}^{-1}(\gamma(0,1])}|f_{k+1}^{\prime}(\omega)|\geq 2A^{-1}$.

We therefore obtain

$|\Phi_{k+1,n}(w)-\widehat{z}_{j}^{k+1}|\leq
A|\Phi_{k,n}(w)-\widehat{z}_{j}^{k}|$ (32)

for all $j\leq k\leq n-1$, and so

$|w-\widehat{z}_{j}^{n}|=|\Phi_{n,n}(w)-\widehat{z}_{j}^{n}|\leq
A^{n-j}|\Phi_{j,n}(w)-\widehat{z}_{j}^{j}|\leq
A^{n-j}\frac{\beta}{4}\left(\frac{L}{\beta}\right)^{2^{j}},$

as required. ∎

If we let $L_{j}^{n}$ be the upper bound in Corollary 26, then we can now
classify $\theta\in\mathbb{T}$ as regular or singular based only on its
location: if $e^{\sigma+i\theta}\in\Delta$ is within $L_{j}^{n}$ of
$\widehat{z}_{j}^{n}$ for some $j$, then $\theta$ is _singular_ , and
otherwise it is _regular_. Now that we know where the singular points are, we
can find a precise estimate for $|\Phi_{n}^{\prime}|$ on $S_{n}$ as we did in
Lemma 16. The proof for this estimate will also be similar to the proof of
Lemma 16.

######  Lemma 27.

Let $n<N\wedge\tau_{D}$, and $1\leq j\leq n-1$. If $\mathbf{c}$ is
sufficiently small, then for all $w\in\Delta$ with $|w|=e^{\sigma}$ and
$|w-\widehat{z}_{j}^{n}|\leq
A^{n-j}\frac{\beta}{4}\left(\frac{L}{\beta}\right)^{2^{j}}$, for $A$ as in
Corollary 26, we have

$|\Phi_{n}^{\prime}(w)|\leq
B^{n}\mathbf{c}^{\frac{n-j}{4}+1}\mathbf{c}^{\frac{1}{2}(1-2^{-j})}\frac{1}{\mathbf{c}^{2^{n-j}}}|w-\widehat{z}_{j}^{n}|^{-(1-2^{-j})}$

where $B$ is a universal constant.

###### Proof.

We will complete the proof by finding bounds on
$|\Phi_{j,n}(w)-e^{i\theta^{\bot}_{j+1}}|$; an upper bound to show
$|\Phi_{j,n}^{\prime}(w)|$ is small, and a lower bound to show
$|\Phi_{j}^{\prime}(\Phi_{j,n}(w))|$ is small. The rest of the proof will be
similar to the way we deduced Lemma 16 from Proposition 14.

First, we will estimate the positions of
$\Phi_{n-1,n}(w),\Phi_{n-2,n}(w),\cdots,\Phi_{j,n}(w)$. As in the proof of
Corollary 26, for $j+1\leq k\leq n$,

$\displaystyle|\Phi_{k-1,n}(w)-\widehat{z}_{j}^{k-1}|$
$\displaystyle=|f_{k}(\Phi_{k,n}(w))-f_{k}(\widehat{z}_{j}^{k})|$
$\displaystyle\leq
2|\Phi_{k,n}(w)-\widehat{z}_{j}^{k}|\times\sup_{|\zeta-\widehat{z}_{j}^{k}|\leq|\Phi_{k,n}(w)-\widehat{z}_{j}^{k}|}|f_{k}^{\prime}(\zeta)|,$
(33)

so we need only bound $|f_{k}^{\prime}(\zeta)|$ for $\zeta$ close to
$\widehat{z}_{j}^{k}$. We will also need inductively that
$|\Phi_{k,n}(w)-\widehat{z}_{j}^{k}|$ is small in order to say that $\zeta$
_is_ close to $\widehat{z}_{j}^{k}$.

######  Claim.

For $j+1\leq k\leq n$, $|\Phi_{k,n}(w)-\widehat{z}_{j}^{k}|\leq
A^{n-j}\mathbf{c}^{3\times 2^{n}}$ for sufficiently small $\mathbf{c}$.

The claim is true for $k=n$, as $|w-\widehat{z}_{j}^{n}|\leq
A^{n-j}\mathbf{c}^{1/2}\left(\frac{1}{2}\mathbf{c}^{2^{n+1}-1/2}\right)^{2^{j}}\leq
A^{n-j}\mathbf{c}^{2^{n+j+1}-2^{j-1}}\leq A^{n-j}\mathbf{c}^{2^{n+2}-2^{n}}$.
Then, if the claim holds for all $l\geq k$, we have

$\displaystyle|\Phi_{l,n}(w)-\widehat{z}_{j}^{l}|$ $\displaystyle\leq
A^{n-j}\mathbf{c}^{3\times 2^{n}}\leq\frac{1}{2}\mathbf{c}^{2^{l-j}}$

for all sufficiently small $\mathbf{c}$, and so, by Lemma 23 and the triangle
inequality, for all $\zeta$ such that
$|\zeta-\widehat{z}_{j}^{l}|\leq|\Phi_{l,n}(w)-\widehat{z}_{j}^{l}|$, we have
$\min_{\pm}|\zeta-e^{i(\theta_{l}\pm\beta)}|\geq\frac{1}{2}\mathbf{c}^{2^{l-j}}$.
Hence by Lemma 11,

$|f_{k}^{\prime}(\zeta)|\leq
A_{2}\frac{\mathbf{c}^{1/2}}{\mathbf{c}^{2^{l-j-1}}}$

Therefore, by (33),

$\displaystyle|\Phi_{k-1,n}(w)-\widehat{z}_{j}^{k-1}|$ $\displaystyle\leq
2^{n-k+1}|\Phi_{n,n}(w)-\widehat{z}_{j}^{n}|\times\prod_{l=k}^{n}\left(A_{2}\mathbf{c}^{\frac{1}{2}-2^{l-j-1}}\right)$
$\displaystyle\leq(2A_{2})^{n-k+1}A^{n-j}\frac{\beta}{4}\left(\frac{L}{\beta}\right)^{2^{j}}\mathbf{c}^{\frac{n-k+1}{4}}\mathbf{c}^{-\sum_{l=k-j-1}^{n-j-1}2^{l}}$
$\displaystyle\leq\left[(2A_{2})^{n-k+1}\mathbf{c}^{\frac{n-k+1}{4}}\right]A^{n-j}\left(\mathbf{c}^{2^{n+1}-\frac{1}{2}}\right)^{2^{j}}\mathbf{c}^{-(2^{n-j}-2^{k-j-1})}$
$\displaystyle\leq A^{n-j}\mathbf{c}^{2^{n+j+1}-2^{j-1}-2^{n-j}+2^{k-j-1}}$
$\displaystyle\leq A^{n-j}\mathbf{c}^{2^{n+2}-2^{n-1}-2^{n-1}}$
$\displaystyle=A^{n-j}\mathbf{c}^{3\times 2^{n}},$

and so our claim holds by induction.

We can also see, from the same computation, that

$|\Phi_{j,n}(w)-e^{i\theta^{\bot}_{j+1}}|=|\Phi_{j,n}(w)-\widehat{z}_{j}^{j}|\leq\mathbf{c}^{3\times
2^{n}}.$ (34)

Then for each $j+1\leq k\leq n$, as $\mathbf{c}^{3\times
2^{n}}\leq\frac{1}{2}\mathbf{c}^{2^{k-j}}$, we have by the triangle inequality
and Lemma 23 that
$|\Phi_{k,n}(w)-e^{i(\theta_{k}\pm\beta)}|\geq\frac{1}{2}\mathbf{c}^{2^{k-j}}$,
and so by Lemma 11,

$\displaystyle|\Phi_{j,n}^{\prime}(w)|$
$\displaystyle=\prod_{k=j+1}^{n}|f_{k}^{\prime}(\Phi_{k,n}(w))|$
$\displaystyle\leq\prod_{k=j+1}^{n}A_{2}\frac{\beta^{1/2}}{(\frac{1}{2}\mathbf{c}^{2^{k-j}})^{1/2}}$
$\displaystyle\leq(2A_{2})^{n-j}\mathbf{c}^{\frac{n-j}{4}-\sum_{k=0}^{n-j-1}2^{k}}$
$\displaystyle=(2A_{2})^{n-j}\mathbf{c}^{\frac{n-j}{4}-2^{n-j}+1}$ (35)

for sufficiently small $\mathbf{c}$.

We will next establish an upper bound on $|\Phi_{j}^{\prime}(\Phi_{j,n}(w))|$.
By the arguments used to prove Corollary 26, we have a lower bound on
$|\Phi_{j,n}(w)-e^{i\theta^{\bot}_{j+1}}|$ as well as the upper bound we just
established:

$|\Phi_{j,n}(w)-e^{i\theta^{\bot}_{j+1}}|\geq
A^{-(n-j)}|w-\widehat{z}_{j}^{n}|,$ (36)

where $A$ is a constant. The upper bound in (34) is less than
$\mathbf{c}^{2^{n+1}}$, and so we can apply (the proof of) Lemma 16 to say

$\displaystyle|\Phi_{j}^{\prime}(\Phi_{j,n}(w))|\leq\frac{(A^{\prime})^{j}}{A^{\frac{n-j}{2}}}\frac{\mathbf{c}^{\frac{1}{2}(1-2^{-j})}}{|w-\widehat{z}_{j}^{n}|^{1-2^{-j}}},$
(37)

and so we can combine (35) and (37) to obtain

$\displaystyle|\Phi_{n}^{\prime}(w)|$
$\displaystyle=|\Phi_{j,n}^{\prime}(w)|\times|\Phi_{j}^{\prime}(\Phi_{j,n}(w))|$
$\displaystyle\leq\left(\frac{2A_{2}}{\sqrt{A}}\right)^{n-j}(A^{\prime})^{j}\mathbf{c}^{\frac{n-j}{4}-2^{n-j}+1}\mathbf{c}^{\frac{1}{2}(1-2^{-j})}|w-\widehat{z}_{j}^{n}|^{-(1-2^{-j})}$
$\displaystyle\leq(A^{\prime\prime})^{n}\mathbf{c}^{\frac{n-j}{4}+1}\mathbf{c}^{\frac{1}{2}(1-2^{-j})}\frac{1}{\mathbf{c}^{2^{n-j}}}|w-\widehat{z}_{j}^{n}|^{-(1-2^{-j})}$

where $A^{\prime\prime}=\max(\frac{2A_{2}}{\sqrt{A}},A^{\prime})$ is a
constant. ∎

######  Corollary 28.

Let $n<N\wedge\tau_{D}$, and $1\leq j\leq n-1$. Then for
$L_{j}^{n}=A^{n-j}\frac{\beta}{4}\left(\frac{L}{\beta}\right)^{2^{j}}$, we
have

$\displaystyle\int_{-L_{j}^{n}}^{L_{j}^{n}}|\Phi_{n}^{\prime}(\widehat{z}_{j}^{n}e^{\sigma+i\varphi})|^{\nu}\,\mathrm{d}\varphi\leq
B_{\nu}^{n}\frac{\mathbf{c}^{\nu(\frac{n-j}{4}+1)}}{\mathbf{c}^{\nu
2^{n-j}}}\mathbf{c}^{\frac{\nu}{2}(1-2^{-j})}\sigma^{-\left[\nu(1-2^{-j})-1\right]}$
(38)

where $B_{\nu}$ is a constant depending only on $\nu$.

###### Proof.

As
$|\widehat{z}_{j}^{n}e^{\sigma+i\varphi}-\widehat{z}_{j}^{n}|\asymp(\sigma^{2}+\varphi^{2})^{1/2}$,
the bound follows immediately from Lemma 27 (in the same way as we obtained
Proposition 17 from Lemma 16). ∎

##  5 Proof of main results

With the results of the previous sections, we are finally ready to prove our
main scaling limit result, that the cluster $K_{N}^{\mathbf{c}}$ converges in
distribution, as $\mathbf{c}\to 0$, to an SLE4 cluster. To help picture the
sets $S_{n,j}$ and $R_{n}$, it may be useful to refer to Figure 4.

###### Proof of Theorem 10.

We want to show that
$h_{n+1}(F_{n})=\int_{F_{n}}h_{n+1}(\theta)\,\mathrm{d}\theta$ is small, and
so we will decompose $F_{n}$ into several sets.

Let
$R_{n}=\\{\theta\in\mathbb{T}:|\Phi_{n}(e^{\sigma+i\theta})-1|>\frac{L}{4}\\}$,
$S_{n}=F_{n}\setminus R_{n}$. We will further decompose $S_{n}$: first define

$T_{n}=\\{\theta\in
S_{n}:D<\min_{\pm}|e^{\sigma+i\theta}-e^{i(\theta_{n}\pm\beta)}|\leq L\\},$

and for $1\leq j\leq n-1$ define

$S_{n,j}=\\{\theta\in S_{n}:|e^{\sigma+i\theta}-\widehat{z}_{j}^{n}|\leq
L_{j}^{n}\\},$

where $L_{j}^{n}$ is the bound appearing in Corollary 26, then Corollary 26
tells us that $S_{n}=T_{n}\cup\left(\bigcup_{j=1}^{n-1}S_{n,j}\right)$. We can
then split the integral as

$h_{n+1}(F_{n})\leq
h_{n+1}(R_{n})+h_{n+1}(T_{n})+\sum_{j=1}^{n-1}h_{n+1}(S_{n,j}).$ (39)

We showed in Section 3.2 that $h_{n+1}(T_{n})=o(\mathbf{c}^{4})$, and so we
only need to bound $h_{n+1}(R_{n})$ and each $h_{n+1}(S_{n,j})$. Bounding
$h_{n+1}(R_{n})$ is simple using Proposition 21, as for any $\theta\in R_{n}$,
we have

$|\Phi_{n}^{\prime}(e^{\sigma+i\theta})|\leq
A^{n}\beta^{n/2}\left(\frac{L}{32\beta}\right)^{-\frac{1}{2}(2^{n}-1)}\ll\mathbf{c}^{4}Z_{n},$

and so $h_{n+1}(R_{n})=o(\mathbf{c}^{4})$. Finally, we will bound
$h_{n+1}(S_{n,j})$. Using the bounds from Proposition 17 and Corollary 28, we
have

$\displaystyle h_{n+1}(S_{n,j})$
$\displaystyle\asymp\frac{1}{Z_{n}}\int_{-L_{j}^{n}}^{L_{j}^{n}}|\Phi_{n}^{\prime}(\widehat{z}_{j}^{n}e^{\sigma+i\varphi})|^{\nu}\,\mathrm{d}\varphi$
$\displaystyle\leq\frac{B_{\nu}^{n}\frac{\mathbf{c}^{\nu(\frac{n-j}{4}+1)}}{\mathbf{c}^{\nu
2^{n-j}}}\mathbf{c}^{\frac{\nu}{2}(1-2^{-j})}\sigma^{-\left[\nu(1-2^{-j})-1\right]}}{A^{n}\mathbf{c}^{\frac{\nu}{2}(1-2^{-n})}\sigma^{-[\nu(1-2^{-n})-1]}}$
$\displaystyle=\left(\frac{B_{\nu}}{A}\right)^{\mspace{-5.0mu}n}\underbrace{\mathbf{c}^{\nu(\frac{n-j}{4}+1)}\mathbf{c}^{-\frac{\nu}{2}(2^{-j}-2^{-n})}}_{o(\mathbf{c}^{5})}\mathbf{c}^{-\nu
2^{n-j}}\sigma^{\nu(2^{-j}-2^{-n})}$
$\displaystyle\ll\mathbf{c}^{5}\left(\frac{B_{\nu}}{A}\right)^{\mspace{-5.0mu}n}\mathbf{c}^{-\nu
2^{n-j}}\sigma^{\nu 2^{-n}},$

then as $\sigma\leq\mathbf{c}^{2^{2^{1/\mathbf{c}}}}$, we have
$\mathbf{c}^{-\nu 2^{n-j}}\sigma^{\nu
2^{-n}}\leq\mathbf{c}^{\nu\left(2^{2^{1/\mathbf{c}}-n}-2^{n-j}\right)}\leq\mathbf{c}^{\nu\left(2^{2^{1/\mathbf{c}}-N}-2^{N}\right)}$
which decays faster than exponentially in $N$. Therefore
$h_{n+1}(S_{n,j})=o_{T}(\mathbf{c}^{5})$, and so
$\sum_{j=1}^{n-1}h_{n+1}(S_{n,j})=o_{T}(\mathbf{c}^{4})$, establishing (5).
The second bound, (6), comes immediately from Corollary 20. ∎

######  Remark.

We have now seen that $(\theta^{\mathbf{c}}_{n})_{n\leq\left\lfloor
T/\mathbf{c}\right\rfloor}$ is very close to a simple symmetric random walk
with step length $\beta\sim 2\mathbf{c}^{1/2}$, and so we expect
$(\xi^{\mathbf{c}}_{t})_{t\in[0,T]}=(\theta_{\left\lfloor
t/\mathbf{c}\right\rfloor})_{t\in[0,T]}$ will converge in distribution to
$(2B_{t})_{t\in[0,T]}$, where $B$ is a standard Brownian motion. We can use a
result on diffusive limits of near-martingales to establish this rigorously.

###### Proof of Corollary 5.

Corollary 3.8 of [13] gives us three conditions to check in order to obtain
that as $\mathbf{c}\to 0$, $\xi^{\mathbf{c}}\to 2B$ in distribution with
respect to the topology of $D[0,T]$; for any $t\in[0,T]$ we need all of the
following to converge in probability:

$\displaystyle\sum_{j=1}^{\left\lfloor
t/\mathbf{c}\right\rfloor}\int_{-\pi}^{\pi}\varphi^{2}h_{j+1}(\theta_{j}+\varphi)1[|\varphi|>\varepsilon]\,\mathrm{d}\varphi$
$\displaystyle\to 0\text{ for any $\varepsilon>0$;}$ (40)
$\displaystyle\sum_{j=1}^{\left\lfloor
t/\mathbf{c}\right\rfloor}\int_{-\pi}^{\pi}\varphi^{2}h_{j+1}(\theta_{j}+\varphi)\,\mathrm{d}\varphi$
$\displaystyle\to 4t;$ (41) $\displaystyle\sum_{j=1}^{\left\lfloor
t/\mathbf{c}\right\rfloor}\left|\int_{-\pi}^{\pi}\varphi
h_{j+1}(\theta_{j}+\varphi)\,\mathrm{d}\varphi\right|$ $\displaystyle\to 0.$
(42)

The $j$th term in the sum (40) is bounded by
$\pi^{2}\mathbb{P}(|\theta_{j+1}-\theta_{j}|>\varepsilon|(\theta_{1},\cdots,\theta_{j}))$,
which, once $\mathbf{c}$ is small enough that $\beta+D<\varepsilon$, is
bounded by $\pi^{2}\mathbb{P}[\tau_{D}\leq j+1]$. Then (5) tells us that for
all $1\leq j\leq\left\lfloor T/\mathbf{c}\right\rfloor$,
$\mathbb{P}[\tau_{D}\leq j+1]\leq A\mathbf{c}^{4}+\mathbb{P}[\tau_{D}\leq j]$
almost surely, and so $\mathbb{P}[\tau_{D}\leq j+1]\leq
A(j+1)\mathbf{c}^{4}\leq AT\mathbf{c}^{3}$, which clearly converges to $0$ in
probability as $\mathbf{c}\to 0$.

Next, in (41), we know that $h_{j+1}$ approximates
$\frac{1}{2}(\delta_{\theta_{j}-\beta}+\delta_{\theta_{j}+\beta})$, and so we
can write

$\displaystyle\int_{-\pi}^{\pi}\varphi^{2}h_{j+1}(\theta_{j}+\varphi)\,\mathrm{d}\varphi$
$\displaystyle=\int_{\beta-D}^{\beta+D}\varphi^{2}h_{j+1}(\theta_{j}+\varphi)\,\mathrm{d}\varphi+\int_{-\beta-D}^{-\beta+D}\varphi^{2}h_{j+1}(\theta_{j}+\varphi)\,\mathrm{d}\varphi+E_{j}$
$\displaystyle=(\beta+O(D))^{2}\int_{\mathbb{T}\setminus
F_{j}}h_{j+1}(\theta)\,\mathrm{d}\theta+E_{j}$
$\displaystyle=\beta^{2}+O(\beta D)+E_{j}^{\prime},$

where $E_{j}^{\prime}$ is the sum of
$\int_{F_{j}}\theta^{2}h_{j+1}(\theta)\,\mathrm{d}\theta\leq\pi^{2}\mathbb{P}[\tau_{D}\leq
j+1]+\pi^{2}A\mathbf{c}^{4}=O_{T}(\mathbf{c}^{3})$ and
$\int_{F_{j}}h_{j+1}(\theta)\,\mathrm{d}\theta=O_{T}(\mathbf{c}^{3})$, and so

$\displaystyle\sum_{j=1}^{\left\lfloor
t/\mathbf{c}\right\rfloor}\int_{-\pi}^{\pi}\varphi^{2}h_{j+1}(\theta_{j}+\varphi)\,\mathrm{d}\varphi$
$\displaystyle=\left\lfloor
t/\mathbf{c}\right\rfloor\beta^{2}+O\left(\frac{\beta
D}{\mathbf{c}}\right)+O(\mathbf{c}^{2})$ $\displaystyle=\left\lfloor
t/\mathbf{c}\right\rfloor
4\mathbf{c}(1+O(\mathbf{c}^{1/2}))+O\left(\frac{\beta
D}{\mathbf{c}}\right)+O(\mathbf{c}^{2})$ $\displaystyle\to 4t$

in probability as $\mathbf{c}\to 0$.

Finally, for the symmetry condition (42), combining (5) and (6) gives us

$\displaystyle\left|\int_{-\pi}^{\pi}\varphi
h_{j+1}(\theta_{j}+\varphi)\,\mathrm{d}\varphi\right|$
$\displaystyle\leq\left|\int_{\beta-D}^{\beta+D}\varphi
h_{j+1}(\theta_{j}+\varphi)\,\mathrm{d}\varphi+\int_{-\beta-D}^{-\beta+D}\varphi
h_{j+1}(\theta_{j}+\varphi)\,\mathrm{d}\varphi\right|+E_{j}$
$\displaystyle=(\beta+O(D))\left|\int_{\theta_{n}+\beta-D}^{\theta_{n}+\beta+D}h_{j+1}(\theta)\,\mathrm{d}\theta-\int_{\theta_{n}-\beta-D}^{\theta_{n}-\beta+D}h_{j+1}(\theta)\,\mathrm{d}\theta\right|+E_{j}$
$\displaystyle=O(\mathbf{c}^{11/4}),$

and so we have a bound

$\sum_{j=1}^{\left\lfloor
t/\mathbf{c}\right\rfloor}\left|\int_{-\pi}^{\pi}\varphi
h_{j+1}(\theta_{j}+\varphi)\,\mathrm{d}\varphi\right|=O_{T}(\mathbf{c}^{7/4})$

and so (42) tends to zero in probability as $\mathbf{c}\to 0$. ∎

## Acknowledgements

The author would like to thank Amanda Turner for her guidance throughout the
project, and Vincent Beffara for his very useful comments on an early version
of the paper about the $\eta=-\infty$ case.

## References

  * [1] Lennart Carleson and Nikolai Makarov.  Aggregation in the plane and Loewner’s equation.  Communications in Mathematical Physics, 216(3):583–607, 2001.  doi:10.1007/s002200000340.
  * [2] Olivier Couronné, Nathanaël Enriquez, and Lucas Gerin.  Construction of a short path in high-dimensional first passage percolation.  Electron. Commun. Probab., 16:22–28, 2011.  doi:10.1214/ECP.v16-1595.
  * [3] Peter L. Duren.  Univalent functions, volume 259.  Springer Science & Business Media, 2001.
  * [4] Murray Eden.  A two-dimensional growth process.  Dynamics of fractal surfaces, 4:223–239, 1961.
  * [5] Denis S. Grebenkov and Dmitry Beliaev.  How anisotropy beats fractality in two-dimensional on-lattice diffusion-limited-aggregation growth.  Phys. Rev. E, 96:042159, Oct 2017.  doi:10.1103/PhysRevE.96.042159.
  * [6] M. B. Hastings and L. S. Levitov.  Laplacian growth as one-dimensional turbulence.  Physica D: Nonlinear Phenomena, 116:244–252, 08 1996.  doi:10.1016/S0167-2789(97)00244-3.
  * [7] Fredrik Johansson and Alan Sola.  Rescaled Lévy-Loewner hulls and random growth.  Bulletin des Sciences Mathématiques, 133(3):238 – 256, 2009\.  doi:https://doi.org/10.1016/j.bulsci.2008.12.006.
  * [8] Gregory F. Lawler.  Conformally Invariant Processes in the Plane.  Mathematical surveys and monographs. American Mathematical Society, 2008\.
  * [9] Gregory F. Lawler, Oded Schramm, and Wendelin Werner.  Conformal invariance of planar loop-erased random walks and uniform spanning trees.  In Selected Works of Oded Schramm, pages 931–987. Springer, 2011\.  doi:10.1007/978-1-4419-9675-6_30.
  * [10] Gregory F. Lawler, Oded Schramm, and Wendelin Werner.  Values of Brownian intersection exponents, I: Half-plane exponents.  In Selected Works of Oded Schramm, pages 859–895. Springer, 2011\.  doi:10.1007/BF02392618.
  * [11] George Liddle and Amanda Turner.  Scaling limits and fluctuations for random growth under capacity rescaling.  arXiv preprint, 2019.  URL: https://arxiv.org/abs/1907.09321.
  * [12] Donald E. Marshall and Steffen Rohde.  The Loewner differential equation and slit mappings.  Journal of the American Mathematical Society, 18(4):763–778, 2005\.  doi:10.1090/S0894-0347-05-00492-3.
  * [13] D. L. McLeish.  Dependent central limit theorems and invariance principles.  Ann. Probab., 2(4):620–628, 08 1974.  doi:10.1214/aop/1176996608.
  * [14] Oded Schramm and Scott Sheffield.  A contour line of the continuum Gaussian free field.  Probability Theory and Related Fields, 157(1-2):47–80, 2013.  doi:10.1007/s00440-012-0449-9.
  * [15] Stanislav Smirnov.  Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits.  Comptes Rendus de l’Académie des Sciences - Series I - Mathematics, 333(3):239 – 244, 2001.  doi:10.1016/S0764-4442(01)01991-7.
  * [16] Alan Sola, Amanda Turner, and Fredrik Viklund.  One-dimensional scaling limits in a planar Laplacian random growth model.  Communications in Mathematical Physics, 371(1):285–329, Oct 2019\.  doi:10.1007/s00220-019-03460-1.
  * [17] Thomas A. Witten Jr. and Leonard M. Sander.  Diffusion-limited aggregation, a kinetic critical phenomenon.  Physical review letters, 47(19):1400, 1981.  doi:10.1103/PhysRevLett.47.1400.