# A Jointly Learned Deep Architecture for Facial Attribute Analysis and Face
Detection in the Wild

Keke He, Yanwei Fu, Xiangyang Xue
Fudan University
{kkhe15, yanweifu<EMAIL_ADDRESS>
###### Abstract

Facial attribute analysis in the real world scenario is very challenging
mainly because of complex face variations. Existing works of analyzing face
attributes are mostly based on the cropped and aligned face images. However,
this result in the capability of attribute prediction heavily relies on the
preprocessing of face detector. To address this problem, we present a novel
jointly learned deep architecture for both facial attribute analysis and face
detection. Our framework can process the natural images in the wild and our
experiments on CelebA and LFWA datasets clearly show that the state-of-the-art
performance is obtained.

##  1 Introduction

The problem of automatically analyzing the facial attributes received
increasing attention recently [24, 15, 37]. Face attributes may potentially
benefit a number of real-world applications, such as face alignment[22, 26,
36, 38], head pose estimation [40, 32] and face verification [28, 27].
Analyzing facial attributes still remains challenging in real-world scenarios.
For example, most existing works [24] predict the facial attributes on well-
cropped face images. In an extension to real-world scenarios, those works have
to utilize the face detector to localize the bounding box of human faces
before facial attribute analysis.

Such a pipeline of face detection followed by facial attribute prediction is
nevertheless undesirable in real-world facial attribute prediction due to two
reasons. Firstly, with the preprocessing of face detector, the capability of
facial attribute prediction has to heavily relies on the results of face
detection. Unfortunately in an uncontrolled setting [15], the face is likely
to have large out-of-plane tilting, occlusion and illumination variations,
which may affect the algorithms of face detection and facial attribute
analysis simultaneously. For example, the Viola-Jones face detector [29] works
well for near-frontal faces, but less effective for faces in the pose far from
frontal views. Second, face detection is heterogeneous but subtly correlated
with facial attribute prediction, and vice versa [33]. Efficiently and
effectively exploiting this correlation may help in both detecting faces and
predicting facial attributes.

We propose an end-to-end deep architecture to _jointly_ learning to detect
faces and analyze facial attributes. Essentially, the two tasks share the same
bottom layers (feature map in Fig. 1) in the architecture to alleviate the
problems of face variations aforementioned. For instance, one face region of
one image is failed to be detected as a face due to the partial occlusion, and
yet the network can detect the existence of some facial attributes such as
“eyeglasses”, and “smiling”. The face region can still be used to optimize our
network and in turn help supervise the learning of the face detection part, to
improve the performance of face detection part.

In this work, the deep learning architecture is proposed for both face
detection and facial attribute analysis. The network structure is illustrated
in Fig. 1. Given an entire image, our architecture will firstly pass the image
with convolutional layers (_conv1_ – _conv5_) and max pooling layers to
produce the _conv_ feature map for face region proposals. The region of
interest (RoI) pooling layer and face region proposal layer are two layers
introduced to facilitate the jointly learning of our architecture. A region of
interest (RoI) pooling layer pools each face proposal on the _conv5_ feature
map into a fixed-length feature vector which is further processed by two fully
connected layers (_fc6_ and _fc7_). Built on the _fc7_ layer, each individual
task of facial attribute analysis and face detection are jointly optimized. By
virtue of the face detection subnet, our architecture can directly predict the
facial attributes from the whole images, rather than using the well-cropped
images as previous work [20, 24, 31]. Extensive experiments on benchmark
CelebA [15] and LFWA [9, 15] datasets demonstrate that our method outperforms
state-of-the-art alternatives.

Contribution. Our main contribution is to propose a novel deep architecture of
jointly learning of face detection and facial attribute tasks, which is
capable of utilizing both tasks to better optimize the shared network and thus
improve the performance of both tasks. Two layers – region of interest (RoI)
pooling layer and face region proposal layer are introduced to enable learning
these two tasks simultaneously. More importantly, we can predict the facial
attributes in the wild and the input images do not need to be cropped and
aligned as the standard practice in [24, 31]. Finally, our joint deep
architecture achieves state-of-the-art result on the biggest benchmark dataset
for facial attribute analysis– CelebA dataset [15] and LFWA [15, 9] datasets.

##  2 Related Work

Figure 1: Overview of the proposed architecture.

Facial attribute analysis. It was first studied by Kumar et al. [13]. In term
of different visual features and distinctive learning paradigm, the facial
attribute analysis has been developed into three categories: (1) the methods
[13] of using hand-crafted visual features, such as SIFT [16] and LBP [18];
(2) the methods of utilizing the recent deep features [15, 37, 31]; and (3)
multi-task methods of learning facial attribute [1, 24]. We here highlight the
differences between our architecture and these previous works. Liu _et al._
[15] cascaded three deep networks pre-trained for facial attribute prediction.
In contrast, we show that the tasks of face detection and facial attribute
prediction are highly correlated and our jointly learning architecture can
improve both tasks. Rudd _et al._ [24] introduced a mixed objective
optimization network which utilizes distribution of attribute labels to learn
each task. Abdulnabi _et al._ [1] proposed a multi-task CNN model sharing of
visual knowledge between tasks for facial attribute analysis. Comparing with
[24, 1], we focus on jointly learning the face detection and facial attribute
analysis; and our model can predict facial attributes on the images in the
wild [24, 31]_i.e._ without cropped and aligned, as illustrated in Fig. 1 and
Fig. 5.

Face detection. The work of Viola and Jones [29, 30] made face detection
usable in many real world applications. The cascaded classifiers were built on
Harr-like features to detect human faces. The deformable part model (DPM) [6]
on top of HOG feature is a general object detector and can also be used for
face detection [4]. Recent advances of deep learning architectures also
inspire another category of methods for face detection. Yang [34] used the
fully convolutional networks (FCN) to generate the heat map of facial parts
for producing face proposals. In contrast to these works, our face detection
task gets benefit from not only the well-designed architecture (in Fig. 1),
but also the jointly learning process with facial attribute prediction.
Further inspired by Fast-RCNN [7, 23], we take the face detection as a special
case of the general semi-rigid object detection. More specifically, given an
image, our face detection will try to answer two questions: (1) whether this
patch contains faces or not? _i.e_. face score task in Fig. 1. (2) can we
detect the bounding box of faces if there is any face? _i.e_. face bounding
box task in Fig. 1. The jointly optimizing these two sub-tasks will better
solve face detection.

Multi-task learning. Our framework can be categorized as multi-task learning
[35, 19], which shares the information and explores the similarity of related
tasks on the same data. The multi-task learning can facilitate a wide range of
tasks and applications, including but not limited to action recognition [39],
information retrieval [14], facial landmark detection [36, 20], and facial
attribute prediction [25, 5, 1, 21, 17]. The recent technical report [20] also
combine face detection with the tasks of locating face landmarks and
recognizing gender. However, unlike our facial attribute prediction on the
images in the wild, their work still has to firstly detect face regions for
the facial landmarks and predicting gender.

##  3 Our Deep Architecture

In this section, we firstly overview our network in Sec. 3.1, and then the
tasks are defined in Sec. 3.2. The Face region proposal and RoI pooling layers
are explained accordingly in Sec. 3.3 and Sec. 3.4. Finally, we utilize the
network to solve the facial attribute prediction and face detection in Sec.
3.5.

###  3.1 Overview

Figure 1 shows our framework for jointly face detection and attribute
analysis. Our architecture takes an entire image as input and a set of face
bounding boxes as labels for training. The whole network firstly processes the
image with several convolutional layers (_conv1_ – _conv5_), and max pooling
layers to produce a _conv_ feature map for face region proposal. For feature
map of each proposed face, we employ a region of interest (RoI) pooling layer
to pool it into a fixed-length feature vector. Each feature vector is further
processed by two fully connected layers (_fc6_ and _fc7_) and thus used for
the tasks of facial attribute analysis and face detection. On top layers, our
architecture has _face detection branch_ and _facial attribute branch_.

Our network follows the art and design of VGG-16 [3]. Particularly, the kernel
size, stride and the number of filters in convolutional layers (_conv1_ –
_conv5_) and the two fully connected layers (_fc6_ and _fc7_) are exactly the
same as the corresponding layers in VGG-16 architecture.

###  3.2 Task formulation

Suppose we have the labelled source training dataset
$\mathcal{D}_{s}=\left\\{\mathbf{I},\mathbf{a},\mathbf{L}\right\\}$ with $N$
training instances and $M$ attributes. $\mathbf{I}$ denotes the patches of
training images and $\mathbf{L}$ denotes the labels. We use the $\mathbf{L}$
matrix to both denote whether an image patch contains the face, and whether a
facial attribute exists in the image patch. Particularly, for the $i$-th image
patch $\mathbf{I}_{i}\>(i=1,\cdots,N)$, we use
$\mathbf{L}_{i\star}=\mathbf{0}$ to indicate that this image patch does not
contain a human face. If $\mathbf{I}_{i}$ is a face image patch, we use
$\mathbf{L}_{ij}=+1$ to denote the existence of $j-$th facial attribute
$\mathbf{a}_{j}\,(j=1,\cdots,M)$; $\mathbf{L}_{ij}=-1$ otherwise.

As illustrated in Fig. 1, our network extracts the image patch to a $4096-dim$
feature vector, and it is denoted as $f\left(\mathbf{I};\Theta\right)$, where
$\Theta$ is the parameter set of the deep architecture. We have the prediction
tasks of facial attribute, face score and face bounding box. Each task has
their own parameters on the last layer.

Facial attribute prediction. To predict the attributes of the image
$\mathbf{I}^{*}$, we need to learn a function
$\mathbf{L}_{i\mathbf{a}}^{*}=\Psi\left(\mathbf{I}^{*}\right)$ to predict the
facial attribute $\mathbf{a}$. We thus have the predicting function
$\Psi=\left[\psi_{i}\right]_{i=1,\cdots,M}$, and
$\psi_{i}\left(\mathbf{I}\right):\mathbb{R}^{4096}\rightarrow\left\\{+1,-1\right\\}$;
specifically, we consider $\Psi\left(x\right)=W_{a}^{T}x$, where
$W_{a}\subseteq\mathbb{R}^{4096\times 40}$ for all $40$ attributes.

Face score. This task aims to predict the score whether an image patch is
face. Essentially, we need to learn
$\Phi\left(x\right):\mathbb{R}^{4096}\rightarrow\left\\{+1,-1\right\\}$, and
$\Phi\left(x\right)=W_{s}^{T}x$, where $W_{a}\subseteq\mathbb{R}^{4096\times
2}$ for binary prediction task.

Face bounding box. We regress the bounding box of faces by
$\Omega\left(x\right):\mathbb{R}^{4096}\rightarrow\mathbb{R}^{4}$, and we
configure the form as $\Omega\left(x\right)=W_{b}^{T}x$, and
$W_{b}\subseteq\mathbb{R}^{4096\times 4}$ for the new bounding box.

###  3.3 Face region proposal layer

Inspired by the work of object detection [23], our network has the branch of
face region proposal. Specifically, this branch takes an image of any size as
input and outputs a set of rectangular face proposals. To generate the facial
region proposal, we slide the branch over the convolutional feature map after
the five convolutional layers (_conv1_ – _conv5_) in Fig. 1. The extracted
_conv5_ features are further fed into two sibling fully connected layers, i.e.
_face bounding box_ (bbox) and _face score_ layer. Face bounding box layer
employs the smooth $L_{1}$ loss (defined in [7]) enables a regressor to
predict the facial bounding box. Face score layer utilizes the softmax loss to
indicate whether the bounding box is a face. For each image, we selected
top-300 face region proposals in term of the face scores computed.

###  3.4 The RoI pooling layer

The RoI pooling layer can convert the feature maps of face region proposal
(with the size of $h\times w$) to a fixed spatial extent of $H\times W$, which
facilitates the further process. Here, $h,w,H,W$ are heights and widths of
each rectangle region of feature map in Fig. 1. For the varying size of input
feature patches, we vary the size of filters of pooling layer with the sub-
window of approximate size $h/H\times w/W$. The RoI pooling technique is
adopted from the work of object detection [7] which inspired by the SPPnets
[8].

###  3.5 Facial attribute analysis and face detection

The fixed-length feature vector extracted from RoI Pooling layer is further
fed into two fully connected layers (_fc6_ and _fc7_) in order to enable the
tasks of facial attribute analysis. Particularly, built on the layers of _fc6_
and _fc7,_ the multi-task network structure is employed to analyze each
attribute individually. This can be modeled as the minimization of the
expected loss over all the training instances which is

$\displaystyle\left\\{\Theta,W_{a},W_{s},W_{b}\right\\}$
$\displaystyle=\underset{}{\mathrm{argmin}}\>\mathcal{L}\left(\mathbf{I};\Theta,W_{a},W_{s},W_{b}\right)$
(1)

where $\mathcal{L}\left(\mathbf{I};\Theta,W_{a},W_{s},W_{b}\right)$ is the
loss function of jointly learning; and we have

$\displaystyle\mathcal{L}\left(\mathbf{I};\Theta,W_{a},W_{s},W_{b}\right)$
$\displaystyle=\lambda_{1}\mathcal{L}_{a}\left(\Phi\left(f\left(\mathbf{I}\right)\right)\right)+\lambda_{2}\mathcal{L}_{s}\left(\Psi\left(f\left(\mathbf{I}\right)\right)\right)$
(2)
$\displaystyle+\lambda_{3}\mathcal{L}_{b}\left(\Omega\left(f\left(\mathbf{I}\right)\right)\right)$

here the loss function
$\mathcal{L}_{a}\left(\Phi\left(f\left(\mathbf{I}\right)\right)\right)$ on
_facial attribute prediction_ is the mean square error loss of all attributes;
we use softmax loss
$\mathcal{L}_{s}\left(\Psi\left(f\left(\mathbf{I}\right)\right)\right)$ for
_face score task,_ and finally the smooth $L_{1}-$ loss
$\mathcal{L}_{b}\left(\Omega\left(f\left(\mathbf{I}\right)\right)\right)$ [7]
is employed to regress the _face bounding box_.

We denote the shared parameters of deep architectures as $\Theta$ which are
jointly optimized by all these three tasks. Since these three tasks are highly
related, learning in such a way can not only greatly reduce the prediction
error of each individual task, but also accelerate the convergence rate of
learning the whole network.

For the testing image $\mathbf{I}_{k}$, the predicted result
$\hat{\mathbf{L}}_{kj}$ of the $j-$th facial attribute
$\mathbf{a}_{j}\,(j=1,\cdots,M)$ is thresholded by

$\hat{\mathbf{L}}_{kj}=\begin{cases}1&\psi\left(\mathbf{I}_{k}\right)>\tau\\\
-1&\psi\left(\mathbf{I}_{k}\right)\leq\tau\end{cases}$ (3)

where $\tau$ is the threshold parameter.

Figure 2: Performance comparison with state-of-the-art methods on CelebA on
all 40 attributes. X-axis indicates each attribute, while y-axis is the error
rate of attribute prediction (the lower value, the better performance of one
method). The averaged error rate of FaceTracer, LNets+ANet, Walk and Learn,
Moon, Baseline (Cropped images), Baseline (Aligned images), and ours are
$18.88\%$, $12.70\%$, $11.35\%$, $9.06\%$, $9.95\%$, $9.19\%$ and 8.41%
respectively.

Figure 3: Performance comparison with state-of-the-art methods on LFWA on all
40 attributes. The averaged error rate of FaceTracer, LNets+ANet, Walk and
Learn, Baseline (Cropped images), Baseline (Aligned images), and ours are
$26.07\%$, $16.15\%$, $13.40\%$, $15.99\%$, $15.06\%$ and 13.13% respectively.

##  4 Experiments

###  4.1 Datasets and settings

Figure 4:  The first row is aligned images provided by the CelebA. The second
row is CelebA raw images. We train our end-to-end model on raw images.

We conduct the experiments on the CelebA dataset [15] and LFWA dataset [9,
15].

CelebA [15] contains approximately 200k images of 10k identities. Each image
is annotated with $5$ landmarks (two eyes, the nose tips, the mouth corners)
and binary labels of $40$ attributes. To make a fair comparison with the other
facial attribute methods, the standard split is used here: the first $160k$
images are used for training, $20k$ images for validation and remaining $20k$
for testing. CelebA provides two types of training images, _i.e._ , aligned
and cropped face images and raw images as shown in the first and second row of
Fig. 4 respectively. We use the raw images for training our joint deep
architecture. Since there is no ground-truth bounding box label for faces in
raw images, the face detector [4, 11] is employed here to help generate the
face bounding box for training images. We use the implementation of dlib
toolbox [11, 4] for generating labels. Note that face bounding box generation
is only required in the training stages for synthesizing labels. At testing
phrase, raw images are directly input for both facial attribute analysis and
face detection.

LFWA [15] is constructed based on face datasets LFW [9]. It contains
approximately $13143$ images of $10k$ identities. Following [15], $50\%$ of
the images for training, and the other $50\%$ are used for testing. LFWA has
$40$ binary facial attributes, the same as CelebA. We also generate face
bounding box for LFWA to train our joint learning network.

Evaluation metrics. We take the attribute prediction as classification tasks
and thus mean accuracy can be computed. Particularly, we evaluate the
performance as comparable to [24, 15] by the mean error which is defined as
$mean\>error=1-mean\,accuracy$.

Implementation and Parameter settings. The $\tau$ is set as $0$ in Eq (3); and
square error loss is used in Eq (1). We empirically set
$\lambda_{1}=\lambda_{2}=1,\lambda_{3}=2$ in Eq (2). The convolutional layers
and fully connected layers are initialized by the 16 layer VGG network [3]
individually. We use the open source deep learning framework Caffe [10] to
implement our structure. A single end-to-end model is used for all the
testing. We employ the stochastic gradient descent to train our network.
Dropout is used for fully connected layers and the ratio is set to $0.5$. For
training CelebA dataset, with initial learning rate $0.001$, and gradually
decreased by $1/10$ at $100k$, $150k$ iterations, the total training
iterations are $180k$. For training LFWA dataset, as there are only limited
training images - $6263$, We fine-tune from pre-trained CelebA model, with
initial learning rate $0.0001$, and decreased by $1/10$ at $40k$ iterations,
the total training iterations is $60k$. Once trained, our framework can
predict the facial attributes and detect faces on the images in the wild for
any testing images.

Running cost. Our facial model get converged with $180k$ iterations and it
takes $29$ hours on CelebA with one NVIDIA TITANX GPU. On LFWA dataset, Our
facial model get converged with $60k$ iterations and takes $9$ hours. For
training all the model, and it takes around $4$ GB GPU memory.

###  4.2 Competitors.

Our model is compared against state-of-the-art methods and several baselines.
Particularly, (1) FaceTracer [12] is one of the best methods with the hand-
crafted features. The features used including HOG [2] and color histograms of
facial regions of interested to train SVM classifier for predicting facial
attributes. (2) LNets+ANet [15] migrates two deep CNN face localization
networks to one deep CNN network for facial attribute classification. (3) Walk
and Learn [31] learns good representations for facial attributes by exploiting
videos and contextual data (geo-location and weather) as the person walks. (4)
Moon [24] is a mixed objective optimization multi-task network to learn all
facial attributes, achieves the best result on CelebA dataset. (5) Cropped
model is a variant of our model without using the face detection branch. The
raw image is also used as the input and we crop the faces from the images by
Dlib toolkit to train facial attribute models. The processed face images are
utilized to train the model. (6) Aligned model uses the aligned and cropped
images provided by CelebA and LFWA to train facial attribute model, which is
used by most of the state-of-the-art methods. To make a fair comparison, these
two baseline models are initialized by 16 layer VGG network; same as the
proposed structure.

Methods | CelebA (%) | LFWA (%)
---|---|---
FaceTracer [12] | 18.88 | 26.07
LNets+ANet [15] | 12.70 | 16.15
Walk-and-Learn [31] | 11.35 | 13.40
Moon [25] | 9.06 | -
Baseline: Cropped Model | 9.95 | 15.99
Baseline: Aligned Model | 9.19 | 15.06
Ours | 8.41 | 13.13
Table 1:  Comparison of mean error on CelebA and LFWA datasets. Figure 5:  The
visualization of feature map for _conv5_ layer. The images are from CelebA
test set. The different types of images are shown in the first three columns.
These images have large pose variation of heads and high occlusion. Some
failed examples are shown in the last column.

###  4.3 Comparison with the State of the Art

We compared our model with state-of-the-art methods: FaceTracer [12],
LNets+ANet [15], Walk and Learn [31], Moon [24] and two baselines. The results
on the testing split of CelebA and LFWA are reported in Tab. 1, Fig. 2 and
Fig. 3, respectively. Note that please refer to the supplementary material for
the full comparison results on each attribute. Comparing with all the
competitors, we draw the following conclusions.

Our model beats all the other methods by the mean error on CelebA and LFWA
dataset. As we can see from Fig. 2, our approach obtains the mean error 8.41%
which is the lowest among all the competitors, and it outperforms all the
other methods on CelebA dataset. And on LFWA dataset as shown in Fig. 3, our
mean error is only 13.13% which is the lowest among all the competitors. This
validates the efficacy of our joint learning architecture. Particularly, We
compare the classification error in each individual attribute in Fig. 2 and
Fig. 3. We note that on more than half among the total 40 attributes, our
framework is significantly better than the other competitors, since our
jointly learning architecture can efficiently leverage the information between
face detection and facial attribute prediction. Particularly, comparing with
the other works, all our attribute prediction tasks share the same deep
learning architecture _, i.e. conv1_ -_conv5_ , feature map and _fc6_ , _fc7_
layers as illustrated in Fig. 1). These shared structures implicitly model the
correlations between each attribute task. Furthermore, on gender attribute
prediction, our framework can achieve $1.7\%$ error on CelebA; in contrast,
the error of HyperFace [20] is $3.0\%$. That indicates that our results can
get $1.3\%$ improvement over that of HyperFace [20].

Our model can process the images in the wild. Our face region proposal and RoI
pooling layers are flexible enough to directly process the images in the wild.
This thus better demonstrates the effectiveness of our models. Specifically,
our model not only achieves the best performance on the benchmark dataset –
CelebA and LFWA, but also our tasks of facial attribute prediction do not need
to align and crop the facial images as have done in many previous work [24,
12, 15]. Additionally, unlike the work of Walk and Learn [31], our model is
not trained by the external data; and yet still obtains better results than
Walk and Learn [31].

Our model is very efficient in term of jointly learning to detect faces and
predict facial attributes. Our joint learning results have greatly improved
over those of two baseline models – Cropped, Aligned models. These two methods
are yet another two naive baselines of directly learning the facial attribute
tasks. Particularly, on CelebA dataset, our model beats the two methods on the
classification error of $27$ attributes (totally $40$ facial attributes) as
compared in Fig. 2. Figure 3 shows that our joint model hit better results
than the two baseline models on all the $40$ attributes. This reveals
attribute prediction tasks get benefit from the face detection task. Also note
that unlike LNets+ANet [15] using two networks to localize the face and
another one network to extract features, we can train our model by a single
end-to-end network here since the shared architectures (_conv1 – conv5_) can
explicitly model the important facial parts, again thanks to the face
detection subnet.

Qualitative results. The important facial parts extracted by the shared
feature map are visualized in Fig. 5. In particular, four groups of images are
shown; and the feature map has higher activation on the regions of human
faces. The first 3 columns are success cases, even thought the images have
large pose (Column 2) or very high occlusion (Column 3). We also list some
failure examples in Column 4, which are some extreme cases. Some of them are
caused by ambiguous or too small view (e.g. the cat face is also similar to
human’s). The feature map reveals our model pay attention on the important
facial parts which helps to analyze facial attributes.

###  4.4 Results of Face Detection

Our face detection is compared against the face detector of dlib toolbox [11],
which is an implementation of the generic object detector [4] on face images.
On CelebA dataset, there are only 5 landmarks of faces, but no labeled ground-
truth bounding box of faces for raw images. We generate the ground truth
bounding box by 5 landmark coordinates for the test set. If the IOU
(Intersection over Union) of predicted bounding box and the ground truth
bounding box is larger than 0.5, we assume the prediction is correct. We
compare the two detectors on the testing split of CelebA.

The results are shown in Fig. 6. We compare the Precision-Recall curve for two
methods. We find that both the Dlib detector and our face detector can achieve
very high face detection on the CelebA test set as shown in the left subfigure
of Fig. 6, which shows the efficacy of both detectors of solving the task of
face detection. Nevertheless, our face detector still beats the Dlib detector
by a relatively large margin. The AUC values of Dlib detector and ours are
$0.938$ and $0.982$ respectively. To better show the difference, we highlight
the up-right corner of PR curve in the right subfigure as shown in Fig. 6.
This result further proves that our jointly learned architecture can make the
face detection and face attribute tasks help each other via sharing the same
parameters of the deep network. This shows that attribute learning does help
for face detection also it provides more detailed information about the face.

Figure 6: Precision-Recall (PR) results of face detection.

##  5 Conclusion

In this paper, we propose a novel joint deep architecture for facial attribute
prediction and face detection. Different from the previous pipeline of face
detection followed by facial attribute prediction, our architecture takes an
entire image as input, enables both face detection and facial attribute
analysis. The proposed architecture can not only exploit the correlation of
face detection and face attribute prediction, but also boost both tasks. The
experimental results on CelebA and LFWA datasets show the efficacy of proposed
methods over the other state-of-the-art methods.

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# On the density of 2D critical percolation gaskets and anchored clusters

Federico Camia  Division of Science, NYU Abu Dhabi, Saadiyat Island, Abu
Dhabi, UAE & Courant Institute of Mathematical Sciences, New York University,
251 Mercer st, New York, NY 10012, USA.
ORCiD: 0000-0002-3510-8535<EMAIL_ADDRESS>
###### Abstract.

We prove a formula, first obtained by Kleban, Simmons and Ziff using conformal
field theory methods, for the (renormalized) density of a critical percolation
cluster in the upper half-plane “anchored” to a point on the real line. The
proof is inspired by the method of images. We also show that more general
bulk-boundary connection probabilities have well-defined, scale-covariant
scaling limits, and prove a formula for the scaling limit of the
(renormalized) density of the critical percolation gasket in any domain
conformally equivalent to the unit disk.

###### Key words and phrases:

Critical percolation, connection probabilities, continuum scaling limit, upper
half-plane.

###### 2010 Mathematics Subject Classification:

Primary: 60K35, 82B43, 82B27. Secondary: 82B31, 60J67, 81T27

##  1\. Introduction

Despite being one of the first models for which the emergence of conformal
invariance in the scaling limit was verified rigorously, with Smirnov’s proof
[13] of Cardy’s formula [7] and the subsequent proofs of convergence of
interfaces to SLE6 [13, 4] and CLE6 [3, 5], critical percolation does not fit
as easily as other models into the conformal field theory (CFT) framework, due
to the lack of an obvious local field such as the spin (magnetization) field
in the Ising model.

In the physics literature, direct reference to local percolation fields is
often avoided by deriving results for the $q$-state Potts model and then
extrapolating them to percolation by taking the limit $q\to 1$. This is the
case, for example, in [7] and [9]. Working with the Potts model, which has a
well-defined magnetization field, makes it possible to use CFT tools that are
not directly available for percolation. However, the limit $q\to 1$ hides a
remarkable amount of subtlety (see, e.g., [8]) and, at the moment, has no
rigorous justification.

From a mathematical perspective, the rigorous derivation of conformal
covariance for observables that can be interpreted as percolation $n$-point
correlation functions in the bulk and for a local percolation field has been
achieved only very recently [1].

Given this situation, from a mathematical physics perspective, it is
worthwhile to try to match CFT derivations with rigorous results. This note
does that by combining the recent results and methods of [1] with insight and
ideas from [12].

In [12], among other results, P. Kleban, J. J. H. Simmons and R. M. Ziff
derive a formula for what they call the density of a critical percolation
cluster in the upper half-plane “anchored” to the origin. Their derivation
uses non-rigorous conformal field theory methods and relies on the
interpretation of the desired quantity as the two-point function, in the upper
half-plane, of two conformal primary fields. Using a method of images, this
two-point function in the upper half-plane can be written as a three-point
function on the full plane, which is fixed by conformal covariance up to a
multiplicative constant.

In this note, we show how the results and methods of [1] can be used to obtain
the formula of Kleban, Simmons and Ziff rigorously as the scaling limit of the
probability, appropriately rescaled, of the event that a critical percolation
cluster restricted to the upper half-plane and touching the origin of the
plane contains a point $z=x+iy$ with $y>0$. The proof of the formula proceeds
by considering two “specular” copies of the same event, one in the upper half-
plane and one in the lower half-plane, and is reminiscent of the method of
images used in electrostatics and field theory.

The results and methods of [1] also allow us to show that other bulk-boundary
connection probabilities have well-defined, scale-covariant scaling limits,
and to derive a formula for the scaling limit of the probability,
appropriately rescaled, of the event that a point in a domain belongs to the
critical percolation gasket of that domain. (The percolation gasket of a
domain with, say, open boundary condition is the set of all open vertices in
the domain that belong to the open cluster touching the boundary).

##  2\. Scaling limit of gaskets and anchored clusters

We consider critical site percolation on $a\mathbb{T}$, the triangular lattice
$\mathbb{T}$ scaled by a factor $a>0$. We embed $\mathbb{T}$ in
$\mathbb{R}^{2}$ as in Figure 1 and in such a way that one of its vertices
coincides with the origin of $\mathbb{R}^{2}$. We denote this vertex by $0$
and call it the _origin_ of $\mathbb{T}$. With $z^{a}$ we denote a vertex of
$a\mathbb{T}$ as well as, with a slight abuse of notation, the elementary
hexagon of the dual hexagonal lattice $a\mathbb{H}$ of which $z^{a}$ is the
center (see Figure 1). Each vertex of $a\mathbb{T}$ (or hexagonal cell of
$a\mathbb{H}$) is declared _open_ or _closed_ with equal probability,
independently of all other vertices. We let $\mathbb{P}^{a}$ denote the
corresponding probability measure on configurations of open and closed
vertices.

Figure 1. Embedding of the triangular and hexagonal lattices in
${\mathbb{R}}^{2}$.

Let $\pi_{a}$ denote the probability of a _bulk one-arm event_ , that is, the
probability that a vertex $z^{a}$ belongs to an open cluster that is not fully
contained in the disk $B_{1}(z^{a})$ of radius $1$ centered at $z^{a}$.
Analogously, let $\overline{\pi}_{a}$ denote the probability of a _boundary
one-arm event_ , that is, the probability that $0$ belongs to an open cluster
whose restriction to the upper half-plane is not fully contained in the disk
$B_{1}(0)$ of radius $1$ centered at $0$.

We write
$\pi_{a}(\varepsilon)=\mathbb{P}^{a}\big{(}z^{a}\longleftrightarrow\partial
B_{\varepsilon}(z^{a})\big{)}$, where $z^{a}\longleftrightarrow\partial
B_{\varepsilon}(z^{a})$ denotes the event that $z^{a}$ belongs to an open
cluster that is not fully contained in the disk $B_{\varepsilon}(z^{a})$ of
radius $\varepsilon$ centered at $z^{a}$. More generally, for a planar domain
$D$ containing $z^{a}$, we let $z^{a}\longleftrightarrow\partial D$ denote the
event that $z^{a}$ belongs to an open cluster that is not fully contained
inside $D$.

Note that $\pi_{a}=\pi_{a}(1)$. Moreover, one can prove [11] that, for any
$\varepsilon>0$,

(2.1) $\displaystyle\lim_{a\to 0}\pi_{a}^{-1}\pi_{a}(\varepsilon)=\lim_{a\to
0}\pi_{a}^{-1}\mathbb{P}^{a}\big{(}z^{a}\longleftrightarrow\partial
B_{\varepsilon}(z^{a})\big{)}=\varepsilon^{-5/48}$

and

(2.2) $\displaystyle\lim_{a\to
0}\overline{\pi}_{a}^{-1}\overline{\pi}_{a}(\varepsilon)=\varepsilon^{-1/3}.$

In the rest of the paper, it is convenient to identify $\mathbb{R}^{2}$ with
the complex plane $\mathbb{C}$. Our first result concerns the density $g_{D}$
of the percolation “gasket” in a planar domain $D$ (an open subset of
$\mathbb{C}$), that is, the scaling limit of the probability that a vertex
contained in $D$ belongs to an open cluster that reaches the boundary of $D$.

######  Theorem 2.1.

Let $D$ be a domain conformally equivalent to the upper half-plane. For any
$z\in D$, let $z^{a}\in a\mathbb{T}$ be chosen in such a way that $z^{a}\to
z$, as $a\to 0$. Then there exists a constant $C_{g}\in(0,\infty)$ such that

(2.3) $\displaystyle g_{D}(z):=\lim_{a\to
0}\pi_{a}^{-1}\mathbb{P}^{a}(z^{a}\longleftrightarrow\partial
D)=C_{g}\,\text{\emph{rad}}(z,D)^{-5/48},$

where $\text{\emph{rad}}(z,D)$ denotes the conformal radius of $D$ from $z$.

###### Proof.

By standard RSW arguments (see, e.g., the proofs of Lemmas 2.1 and 2.2 of
[6]), there are constants $0<K_{1}<K_{2}<\infty$, independent of $a$, such
that

(2.4) $\displaystyle
K_{1}\pi_{a}\leq\mathbb{P}^{a}(z^{a}\longleftrightarrow\partial D)\leq
K_{2}\pi_{a},$

which shows that $\pi_{a}^{-1}\mathbb{P}^{a}(z^{a}\longleftrightarrow\partial
D)$ stays bounded away from zero and infinity as $a\to 0$.

Take $\varepsilon>0$ such that $z$ is at distance greater than $\varepsilon$
from $\partial D$. Then, for all $a$ sufficiently small, $z^{a}$ is at
distance greater than $\varepsilon$ from $\partial D$ and the event
$z^{a}\longleftrightarrow\partial D$ implies the events
$z^{a}\longleftrightarrow\partial B_{\varepsilon}(z^{a})$. Therefore, using
(2.1), we have

(2.5) $\displaystyle\begin{split}&\lim_{a\to
0}\pi_{a}^{-1}\mathbb{P}^{a}\big{(}z^{a}\longleftrightarrow\partial
D\big{)}\\\ &\quad=\lim_{a\to
0}\pi_{a}^{-1}\mathbb{P}^{a}\big{(}z^{a}\longleftrightarrow\partial
D|z^{a}\longleftrightarrow\partial
B_{\varepsilon}(z^{a})\big{)}\mathbb{P}^{a}\big{(}z^{a}\longleftrightarrow\partial
B_{\varepsilon}(z^{a})\big{)}\\\ &\quad=\varepsilon^{-5/48}\lim_{a\to
0}\mathbb{P}^{a}\big{(}z^{a}\longleftrightarrow\partial
D|z^{a}\longleftrightarrow\partial B_{\varepsilon}(z^{a})\big{)}.\end{split}$

The last conditional probability is similar to that appearing in the first
line of equation (2.23) of [1], it is the probability of a similar
connectivity event and the conditioning is of the same type. This implies that
the arguments used to show the existence of the limit in the first line of
(2.23) of [1] can be used to show the existence of the limit of the
conditional probability in the last line of (2.5), so we can define

(2.6)
$\displaystyle\begin{split}&\mathbb{P}\big{(}z\longleftrightarrow\partial
D|z\longleftrightarrow\partial B_{\varepsilon}(z)\big{)}:=\lim_{a\to
0}\mathbb{P}^{a}\big{(}z^{a}\longleftrightarrow\partial
D|z^{a}\longleftrightarrow\partial B_{\varepsilon}(z^{a})\big{)}\end{split}$

and

(2.7) $\displaystyle\begin{split}&g_{D}(z):=\lim_{a\to
0}\pi_{a}^{-1}\mathbb{P}^{a}\big{(}z^{a}\longleftrightarrow\partial
D\big{)}=\varepsilon^{-5/48}\mathbb{P}\big{(}z\longleftrightarrow\partial
D|z\longleftrightarrow\partial B_{\varepsilon}(z)\big{)},\end{split}$

where the last equality is valid for all $\varepsilon>0$ sufficiently small.

The rest of the proof proceeds like the proof of Theorem 1.4 of [1]; we
provide a sketch of the argument for completeness. The conformal invariance
properties of the scaling limit of percolation [3, 5] imply that, for any
domain $D^{\prime}$ conformally equivalent to $D$, given a conformal map
$\phi:D\to D^{\prime}$, for all $\varepsilon>0$ sufficiently small,

(2.8)
$\displaystyle\begin{split}&g_{D^{\prime}}(\phi(z))=\varepsilon^{-5/48}\mathbb{P}\big{(}\phi(z)\longleftrightarrow\partial
D^{\prime}|\phi(z)\longleftrightarrow\partial
B_{\varepsilon}(\phi(z))\big{)}\\\
&\qquad\qquad=\varepsilon^{-5/48}\mathbb{P}\big{(}z\longleftrightarrow\partial
D|z\longleftrightarrow\phi^{-1}\big{(}\partial
B_{\varepsilon}(\phi(z))\big{)}\big{)},\end{split}$

where

(2.9) $\displaystyle\begin{split}\mathbb{P}\big{(}z\longleftrightarrow\partial
D|z\longleftrightarrow\phi^{-1}\big{(}\partial
B_{\varepsilon}(\phi(z))\big{)}\big{)}=\lim_{a\to
0}\mathbb{P}^{a}\big{(}z^{a}\longleftrightarrow\partial
D|\phi^{-1}\big{(}\partial B_{\varepsilon}(\phi(z))\big{)}\big{)}.\end{split}$

Now let $s=\phi^{\prime}(z)$ and let $A_{r,R}(z)=B_{R}(z)\setminus B_{r}(z)$
denote the thinnest annulus centered at $z$ containing the symmetric
difference of $\phi^{-1}(B_{\varepsilon}(\phi(z)))$ and
$B_{\varepsilon/s}(z)$. Since $\phi^{-1}$ is analytic and
$(\phi^{-1})^{\prime}(\phi(z))=1/s$, for every $w\in\partial
B_{\varepsilon}(\phi(z))$,
$|z-\phi^{-1}(w)|=\varepsilon/s+O(\varepsilon^{2})$, which implies that

(2.10) $\displaystyle\lim_{\varepsilon\to
0}\frac{r}{\varepsilon}=\lim_{\varepsilon\to
0}\frac{R}{\varepsilon}=\frac{1}{s}.$

Using (LABEL:eq:g_Dprime) and an analog of equation (2.32) of [1], we have
that

(2.11)
$\displaystyle\begin{split}&\Big{(}\frac{\varepsilon}{r}\Big{)}^{-5/48}r^{-5/48}\,\mathbb{P}\big{(}z\longleftrightarrow\partial
D\,|\,z\longleftrightarrow\partial B_{r}(z)\big{)}\\\
&\quad\leq\varepsilon^{-5/48}\mathbb{P}\big{(}z\longleftrightarrow\partial
D\,|\,z\longleftrightarrow\partial\phi^{-1}\big{(}\partial
B_{\varepsilon}(z)\big{)}\big{)}\\\
&\qquad\leq\Big{(}\frac{\varepsilon}{R}\Big{)}^{-5/48}R^{-5/48}\,\mathbb{P}\big{(}z\longleftrightarrow\partial
D\,|\,z\longleftrightarrow\partial B_{R}(z)\big{)}.\end{split}$

Combining this with (LABEL:def:g_D) and (LABEL:eq:g_Dprime) and taking
$\varepsilon$ (and hence $r$ and $R$) sufficiently small, we obtain

(2.12)
$\displaystyle\begin{split}\Big{(}\frac{\varepsilon}{r}\Big{)}^{-5/48}\,g_{D}(z)\leq
g_{D^{\prime}}(\phi(z))\leq\Big{(}\frac{\varepsilon}{R}\Big{)}^{-5/48}\,g_{D}(z).\end{split}$

Sending $\varepsilon\to 0$ and using (2.10) gives

(2.13) $\displaystyle g_{D^{\prime}}(\phi(z))=s^{-5/48}g_{D}(z).$

To conclude the proof, apply (2.13) to the unit disk, $D^{\prime}=\mathbb{D}$,
choose $\phi:D\to\mathbb{D}$ to be a conformal map such that $\phi(z)=0$, and
let $C_{g}=g_{\mathbb{D}}(0)$. ∎

The next theorem is the main result of this paper, it provides a rigorous
derivation of a formula, first obtained by Kleban, Simmons and Ziff [12] using
conformal field theory methods, for the (renormalized) density of a critical
percolation cluster in the upper half-plane “anchored” to a point on the real
line. We will approximate the upper half-plane $\mathbb{H}$ by the discrete
domain depicted in Fig. 2, with origin $0$, which we will refer to as the
_discrete upper half-plane_. (In the proof of the theorem, we will also use a
similar discrete approximation of the lower half-plane, with origin
$0^{\prime}$, as in Fig. 2.) For any hexagon $z^{a}$ in the discrete upper
half-plane, we will write
$z^{a}\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}0$ to denote
the event that $z^{a}$ and $0$ belong to the same open cluster and, moreover,
there is an open path (i.e., a sequence of open, nearest-neighbor hexagons)
from $z^{a}$ to $0$ contained in the discrete upper half-plane. Similarly,
$0\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}\partial
B_{\varepsilon}(0)$ will denote the event that there is an open path in the
discrete upper half-plane from $0$ to the (upper half of the) boundary of the
disk $B_{\varepsilon}(0)$.

Figure 2. Discrete approximations of the upper and lower half-planes with the
respective origins, $O$ and $O^{\prime}$.

######  Theorem 2.2.

Consider critical site percolation on the triangular lattice. Given
$z=x+iy=re^{i\theta}\in\mathbb{C}$ with $y>0$, for any sequence of vertices
$z^{a}\in a\mathbb{T}$ in the discrete upper half-plane chosen so that
$z^{a}\to z$ as $a\to 0$, there is a constant $C_{\mathbb{H}}\in(0,\infty)$
such that

(2.14) $\displaystyle\lim_{a\to
0}\pi_{a}^{-1}\overline{\pi}_{a}^{-1}\mathbb{P}^{a}\big{(}z^{a}\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}0\big{)}=C_{\mathbb{H}}\,\frac{y^{11/48}}{|z|^{2/3}}=C_{\mathbb{H}}\,\frac{(\sin\theta)^{11/48}}{r^{7/16}}.$

###### Proof.

By standard RSW arguments (see, e.g., the proofs of Lemmas 2.1 and 2.2 of
[6]), there are constants $0<K_{1}<K_{2}<\infty$, independent of $a$, such
that

(2.15) $\displaystyle
K_{1}\pi_{a}\overline{\pi}_{a}\leq\mathbb{P}^{a}\big{(}z^{a}\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}0\big{)}\leq
K_{2}\pi_{a}\overline{\pi}_{a},$

which shows that
$\pi_{a}^{-1}\overline{\pi}_{a}^{-1}\mathbb{P}^{a}\big{(}z^{a}\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}0\big{)}$
stays bounded away from zero and infinity as $a\to 0$.

Take $\varepsilon>0$ such that $z$ is at distance greater than $2\varepsilon$
from the real line. Then, for all $a$ sufficiently small, $z^{a}$ is at
distance greater than $2\varepsilon$ from the real line and the event
$z^{a}\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}0$ implies the
independent events $z^{a}\longleftrightarrow\partial B_{\varepsilon}(z^{a})$
and $0\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}\partial
B_{\varepsilon}(0)$. Therefore, using (2.1) and (2.2), we have

(2.16) $\displaystyle\begin{split}&\lim_{a\to
0}\pi_{a}^{-1}\overline{\pi}_{a}^{-1}\mathbb{P}^{a}\big{(}z^{a}\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}0\big{)}\\\
&\quad=\lim_{a\to
0}\pi_{a}^{-1}\overline{\pi}_{a}^{-1}\mathbb{P}^{a}\big{(}z^{a}\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}0|\\{z^{a}\longleftrightarrow\partial
B_{\varepsilon}(z^{a})\\}\cap\\{x^{a}\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}\partial
B_{\varepsilon}(0)\\}\big{)}\\\
&\qquad\quad\mathbb{P}^{a}\big{(}z^{a}\longleftrightarrow\partial
B_{\varepsilon}(z^{a})\big{)}\mathbb{P}^{a}\big{(}0\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}\partial
B_{\varepsilon}(0)\big{)}\\\ &\quad=\varepsilon^{-5/48-1/3}\lim_{a\to
0}\mathbb{P}^{a}\big{(}z^{a}\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}0|\\{z^{a}\longleftrightarrow\partial
B_{\varepsilon}(z^{a})\\}\cap\\{0\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}\partial
B_{\varepsilon}(0)\\}\big{)}.\end{split}$

The last conditional probability is similar to that appearing in the first
line of equation (2.23) of [1], it is the probability of a similar
connectivity event and the conditioning is of the same type. This implies that
the arguments used to show the existence of the limit in the first line of
(2.23) of [1] can be used to show the existence of the limit of the
conditional probability in the last line of (2.16), so we can define

(2.17)
$\displaystyle\begin{split}&\mathbb{P}\big{(}z\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}0|\\{z\longleftrightarrow\partial
B_{\varepsilon}(z)\\}\cap\\{0\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}\partial
B_{\varepsilon}(0)\\}\big{)}\\\ &\quad:=\lim_{a\to
0}\mathbb{P}^{a}\big{(}z^{a}\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}0|\\{z^{a}\longleftrightarrow\partial
B_{\varepsilon}(z^{a})\\}\cap\\{0\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}\partial
B_{\varepsilon}(0)\\}\big{)}\end{split}$

and

(2.18)
$\displaystyle\begin{split}&\rho_{\mathbb{H}}(z):=\varepsilon^{-7/16}\mathbb{P}\big{(}z\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}0|\\{z\longleftrightarrow\partial
B_{\varepsilon}(z)\\}\cap\\{0\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}\partial
B_{\varepsilon}(0)\\}\big{)},\end{split}$

valid for all $\varepsilon>0$ sufficiently small.

Now, keeping in mind Fig. 2, consider the event $\mathcal{A}^{a}$ that $z^{a}$
is connected to $0$ by an open path in the discrete upper half-plane (i.e.
$z^{a}\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}0$) _and_
$\hat{z}^{a}$ is connected to $0^{\prime}$ by an open path in the discrete
_lower_ half-plane, where $\hat{z}^{a}$ is the image of $z^{a}$ under a
reflection and a translation that map the discrete upper half-plane to the
discrete lower half-plane, mapping $0$ to $0^{\prime}$. $\mathcal{A}^{a}$ can
be written as the intersection
$\mathcal{A}^{a}=\mathcal{A}^{a}_{u}\cap\mathcal{A}^{a}_{l}$ of two
_independent_ events,
$\mathcal{A}^{a}_{u}=z^{a}\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}0$
and a corresponding event $\mathcal{A}^{a}_{l}$ in the discrete lower half-
plane, so that, by reflection and translation invariance,

(2.19)
$\displaystyle\mathbb{P}^{a}(\mathcal{A}^{a})=\Big{(}\mathbb{P}^{a}\big{(}z^{a}\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}0\big{)}\Big{)}^{2}.$

If we write

(2.20) $\displaystyle f(0,z,\bar{z}):=\lim_{a\to
0}\pi_{a}^{-2}\overline{\pi}_{a}^{-2}\mathbb{P}^{a}(\mathcal{A}^{a}),$

since $\mathcal{A}^{a}$ is a connectivity event of the same type as those
considered in Theorem 1.1 of [1], the arguments in the proof of Theorem 1.1 of
[1] can be used to show that the limit in (2.20) exists in $(0,\infty)$.

We can define $f(z_{0},z_{1},z_{2})$ for any triple of points
$z_{0},z_{1},z_{2}$ that form an isosceles triangle, with $z_{0}$ denoting the
vertex opposite the base of the triangle, using (2.20) in the obvious way,
i.e., considering the two half-planes bounded by the line through $z_{0}$
perpendicular to the base of the triangle and placing the origin at $z_{0}$.
It is clear, then, that $f$ is invariant under translations and rotations.

We now show that $f$ scales covariantly under dilations. Firstly, note that
(2.19) implies

(2.21) $\displaystyle f(0,z,\bar{z})=\big{(}\rho_{\mathbb{H}}(z)\big{)}^{2}.$

Next, consider $s>0$ and take $\varepsilon>0$ such that both $z$ and $sz$ are
at distance greater than $2\varepsilon$ from the real line. Then we have

(2.22) $\displaystyle\begin{split}&\rho_{\mathbb{H}}(sz)=\lim_{a\to
0}\pi_{a}^{-1}\overline{\pi}_{a}^{-1}\mathbb{P}^{a}\big{(}sz^{a}\longleftrightarrow
0\big{)}\\\
&\quad=\varepsilon^{-7/16}\mathbb{P}\big{(}sz\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}0|\\{sz\longleftrightarrow\partial
B_{\varepsilon}(sz)\\}\cap\\{0\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}\partial
B_{\varepsilon}(0)\\}\big{)}\\\
&\quad=\varepsilon^{-7/16}\mathbb{P}\big{(}z\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}0|\\{z\longleftrightarrow\partial
B_{\varepsilon/s}(z)\\}\cap\\{0\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}\partial
B_{\varepsilon/s}(0)\\}\big{)},\end{split}$

where the last equality follows from the scale invariance of the scaling limit
of critical percolation [3, 5]. This implies that

(2.23)
$\displaystyle\begin{split}&\rho_{\mathbb{H}}(sz)=s^{-7/16}(\varepsilon/s)^{-7/16}\mathbb{P}\big{(}z\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}0|\\{z\longleftrightarrow\partial
B_{\varepsilon/s}(z)\\}\cap\\{0\stackrel{{\scriptstyle\mathbb{H}}}{{\longleftrightarrow}}\partial
B_{\varepsilon/s}(0)\\}\big{)}\\\
&\qquad\quad=s^{-7/16}\rho_{\mathbb{H}}(z),\end{split}$

which proves the covariance of $\rho_{\mathbb{H}}$, and therefore of $f$ (by
(2.21)), under scale transformations and is consistent with (2.14).

Given the covariance properties of $f$, standard considerations (see, e.g.,
Section 4.3.1 of [10] or the proof of Theorem 4.5 of [2]) imply that

(2.24) $\displaystyle
f(0,z,\bar{z})=\frac{C}{|z|^{a}|\bar{z}|^{b}|z-\bar{z}|^{c}},$

for some constant $C\in(0,\infty)$ and with $a=b=5/48+2/3-5/48=2/3$ and
$c=5/48+5/48-2/3=-11/24$. Therefore, combining (2.24) and (2.21), we have that

(2.25)
$\displaystyle\rho_{\mathbb{H}}(z)=\sqrt{f(0,z,\bar{z})}=\sqrt{C}|z-\bar{z}|^{11/48}|z|^{-1/3}|\bar{z}|^{-1/3}.$

Writing $z=x+iy=re^{i\theta}$, this gives

(2.26)
$\displaystyle\rho_{\mathbb{H}}(z)=C_{\mathbb{H}}\,y^{11/48}\,|z|^{-2/3}=C_{\mathbb{H}}\,r^{-7/16}\,(\sin\theta)^{11/48},$

where $C_{\mathbb{H}}=2^{11/48}\sqrt{C}$. ∎

We conclude this paper with an observation. The first part of the proof of
Theorem 2.2 (up to equation (LABEL:eq:scale-invariance)) applies to more
general “bulk-boundary” connection probabilities, involving multiple points in
the upper half-plane and on the real line. In general, there is no known
explicit formula, but one can still prove the existence of the limit and its
scale covariance.

######  Theorem 2.3.

Consider critical site percolation on the triangular lattice. Given
$z_{1},\ldots,z_{k}$ in the upper half-plane and
$x_{1},\ldots,x_{n}\in\mathbb{R}$, for any sequences of vertices
$z_{1}^{a},\ldots,z_{k}^{a}$ and $x_{1}^{a}\ldots,x^{a}_{n}$ of $a\mathbb{T}$
in the discrete upper half-plane with $z^{a}_{1}\to z_{1},\ldots,z^{a}_{k}\to
z_{k}$ and $x^{a}_{1}\to x_{1},\ldots,x^{a}_{n}\to x_{n}$ as $a\to 0$, the
limit

(2.27) $\displaystyle\lim_{a\to
0}\pi_{a}^{-k}\overline{\pi}_{a}^{-n}\mathbb{P}^{a}\big{(}z^{a}_{i}\longleftrightarrow
x^{a}_{j}\;\forall
i,j\big{)}=:\rho_{\mathbb{H}}(z_{1},\ldots,z_{k};x_{1},\ldots,x_{n})$

exists in $(0,\infty)$. Moreover, for any $s>0$,

(2.28)
$\displaystyle\rho_{\mathbb{H}}(sz_{1},\ldots,sz_{k};sx_{1},\ldots,sx_{n})=s^{-5k/48-n/3}\rho_{\mathbb{H}}(z_{1},\ldots,z_{k};x_{1},\ldots,x_{n}).$

###### Idea of the proof..

The existence of the limit follows from the same arguments used in the proof
of Theorem 1.1 of [1] and of the limit in (2.16), with the difference that now
one needs to condition on $k+n$ one-arm events in the upper half-plane, at
$z^{a}_{1},\dots,z^{a}_{k}$ and $x^{a}_{1},\ldots,x^{a}_{n}$. The scale
covariance is proved in the same way as (LABEL:eq:scale-invariance), with the
exponent coming from the exponents in (2.1) and (2.2). ∎

Acknowledgments: The author thanks Peter Kleban and Robert Ziff for an
interesting correspondence and for comments on a draft of the paper.

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