(eccv) Package eccv Warning: Package ‘hyperref’ is loaded with option
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11institutetext: Institute of Information Engineering, Chinese Academy of
Sciences 22institutetext: School of Cyber Security, University of Chinese
Academy of Sciences
22email: {fuxiaomeng, wangxi1, liqiao, liujin, daijiao<EMAIL_ADDRESS>
# Model Will Tell: Training Membership Inference for Diffusion Models

Xiaomeng Fu 1122 Xi Wang Corresponding Author.11 Qiao Li 11 Jin Liu 1122 Jiao
Dai 11 Jizhong Han 11

###### Abstract

Diffusion models pose risks of privacy breaches and copyright disputes,
primarily stemming from the potential utilization of unauthorized data during
the training phase. The Training Membership Inference (TMI) task aims to
determine whether a specific sample has been used in the training process of a
target model, representing a critical tool for privacy violation verification.
However, the increased stochasticity inherent in diffusion renders traditional
shadow-model-based or metric-based methods ineffective when applied to
diffusion models. Moreover, existing methods only yield binary classification
labels which lack necessary comprehensibility in practical applications. In
this paper, we explore a novel perspective for the TMI task by leveraging the
intrinsic generative priors within the diffusion model. Compared with unseen
samples, training samples exhibit stronger generative priors within the
diffusion model, enabling the successful reconstruction of substantially
degraded training images. Consequently, we propose the Degrade Restore Compare
(DRC) framework. In this framework, an image undergoes sequential degradation
and restoration, and its membership is determined by comparing it with the
restored counterpart. Experimental results verify that our approach not only
significantly outperforms existing methods in terms of accuracy but also
provides comprehensible decision criteria, offering evidence for potential
privacy violations.

###### Keywords:

Membership Inference Diffusion Models Data Privacy

##  1 Introduction

Recently, the emergence of large models, particularly the Diffusion Model [15,
42], has significantly enhanced the authenticity and practicality of
Artificial Intelligence Generated Content (AIGC). However, these data-
intensive models are predominantly trained on extensive unauthorized data [39,
38] scrabbled from the internet, thereby ignoring the proprietary rights and
privacy of the original owners. Consequently, this has amplified the risk of
privacy breaches and copyright disputes for users [23, 10, 2]. For example,
Getty Images has filed a lawsuit against Stability AI recently, accusing that
the company utilized 12 million Getty Images to train its model without
necessary permissions [2]. Furthermore, the improper use of personal identity
to generate facial images also poses a significant concern in the realm of
privacy protection. To address the aforementioned challenges, a pivotal
approach is to ensure that users are informed whether their private data is
being utilized for model training.

(a) The intuition of DRC.

(b) The pipeline of DRC.

Figure 1: (a) The intuition of DRC. A training sample will exhibit a more
pronounced peak compared with an unseen sample. The training sample and the
unseen sample are equally degraded. The degraded training sample can be
restored to its original peak, while the capability is not shared by the
unseen sample. (b) The pipeline of DRC. The original image is first
intentionally degraded, and then restored by the diffusion model. Finally, a
comparison is made between the original image and its restored counterpart to
determine whether the original image is a training member.

We resort to Training Membership Inference (TMI) tasks to empower users with
the ability to discern potential threats to their private data. The TMI task,
also denoted as Membership Inference Attacks (MIAs) [16], is aimed to identify
whether a data sample has been used in the training of a machine learning
model. Historically, TMI methods [40, 17, 24, 27, 37, 44] have been primarily
developed for classification models. Some researchers [5, 12, 14] have
expanded these techniques to encompass generative methods, such as Generative
Adversarial Networks (GANs) [11] and Variational Autoencoders (VAEs) [21].
However, the applicability of existing TMI methods to diffusion models is
limited. Specifically, one prevalent approach in TMI involves training a
shadow model to mimic the target model’s behavior. This technique, however, is
not suitable for diffusion models, as it is computationally intensive to train
a shadow model with comparable parameters. Another category of TMI methods is
metric-based, where a metric (typically the loss value) is selected as a
surrogate for the model’s response towards each sample. The membership of the
sample is then determined based on the numerical value of the metric. These
methods encounter difficulties when dealing with the inherent randomness of
the generation process in diffusion models. Moreover, these methods perceive
TMI as a binary classification task, providing a label without any
explanation. The label-only classification results obtained from these methods
are not trustworthy for end users without some level of technical knowledge,
thereby impeding the practical application in the real world. This highlights
the need for more robust and user-friendly TMI methods for diffusion models.

In this paper, we explore a novel perspective for the TMI task. Large models
inherently possess powerful capabilities for understanding and reasoning.
Taking inspiration from GPT’s self-detection of its generated language [31,
33], we attempt to leverage the inherent generative capability of the
diffusion model to accomplish the TMI task. As illustrated in Figure 1(a), the
training process of the diffusion model is self-referential, with the training
image serving as the objective. Consequently, the sample from the training set
will manifest as a more pronounced peak compared to an unseen sample. By
strategically degrading a training image in a controlled manner, while
ensuring its proximity to the peak, there is a high probability that the
diffusion model can restore it back to its original peak. However, when an
equivalent level of degradation is applied to an unseen image, the diffusion
model may restore it to an alternative peak, obtaining an entirely different
image.

Therefore, we design a novel TMI framework, the Degrade Restore Compare (DRC),
to identify the training membership of a specific image. As depicted in Figure
1(b), our approach involves a three-step process. First, the original image
undergoes intentional degradation. Then, the generative priors of the target
diffusion model are employed to restore the degraded image. Finally, a
comparison is made between the restored image and the original image. The
semantic distance between the original image and its restored counterpart is
utilized to determine whether the original image is a training member. In the
scenario where the original image is a training sample, the degraded image can
be restored to an image almost identical to the original image. Conversely, an
unseen sample may converge to an image entirely distinct from the original.
Our experimental results demonstrate that our method achieves state-of-the-art
performance. Besides, the underlying mechanism of our proposed method is
simple and comprehensible for end users with limited technical knowledge about
computer vision, thereby enhancing the accessibility and applicability of our
research.

In summary, this paper offers the following significant contributions:

  * •

Innovative TMI framework. We introduce the Degrade Restore Compare framework,
a novel approach that capitalizes on the inherent reasoning and generative
capabilities of the diffusion model for TMI tasks.

  * •

Comprehensibility. The fundamental mechanism of our proposed method is
intuitive and comprehensible, enhancing the accessibility and applicability of
our research.

  * •

Accuracy. Experiments show that our proposed method outperforms contemporary
TMI methods by a large margin, demonstrating its effectiveness.

##  2 Related Work and Background

Training Membership Inference. Training Membership Inference, also denoted as
membership inference attack, aims to predict whether a specific data record
has been utilized in the training of a machine learning model. Historically,
TMI methods [40, 17, 24, 27, 37, 44] have been primarily developed for
classification models. The concept is first introduced by Shokri [40], who
employs a series of shadow models to replicate the behavior of the target
model.

Subsequent research [5, 12, 14, 26] has expanded these techniques to include
generative models, such as GAN and VAE. For instance, LOGAN [12] operates
under the assumption that the GAN’s discriminator is more likely to assign a
higher probability to training members. Hilprecht [14] employs records within
close proximity of a target record to approximate the probability of this
record being a member. GAN-Leak [5] posits that members are more closely
aligned with the distribution of generated samples.

While these methods have shown promising results in GAN and VAE [9], they have
demonstrated suboptimal performance when applied to diffusion models [9]. This
is primarily due to the challenges associated with the inherent randomness in
diffusion models. Recently, there have been advancements in Training
Membership Inference methods specifically tailored for diffusion models. A
notable contribution is from Matsumoto [28], who introduced a pioneering TMI
method for diffusion models. This approach utilizes the training loss of the
model and selects a timestep where the divergence of loss between members and
nonmembers is greatest to perform the TMI task. SecMI [9] proposes the concept
of t-error, which represents the posterior estimation error at timestep $t$,
assuming that the t-error of members is smaller than that of nonmembers.
Despite these advancements, these diffusion-specific TMI methods still exhibit
certain limitations. First, their performance has not reached a level that is
robust enough for practical applications. Second, the label-only predictions
generated by these methods lack interpretability, which may lead to a lack of
trust from users.

Diffusion Model. Denoising Diffusion Probabilistic Models (DDPM) [15, 42] have
emerged as a novel branch of generative models. With the impressive ability to
approximate the distribution of image data, diffusion models have made
breakthroughs in many visual content generation tasks [34, 36, 25, 46, 8, 20,
1] and have broken the long-term domination of GANs [7, 29, 30]. DDPMs consist
of a forward and a reverse process. The forward process, also named as the
diffusion process, gradually adds Gaussian noise to the input image $x_{0}$ in
$T$ time steps according to a predefined variance schedule
$\beta_{1},...,\beta_{T}$:

$q(x_{t}|x_{t-1})=\mathcal{N}(x_{t};\sqrt{1-\beta_{t}}x_{t-1},\beta_{t}\mathbf{I})$
(1)

Let $\alpha_{t}=1-\beta_{t}$ and
$\bar{\alpha}_{t}={\textstyle\prod_{s=1}^{t}\alpha_{s}}$, this process can be
simplified to:

$q(x_{t}|x_{0})=\mathcal{N}(x_{t};\sqrt{\bar{\alpha}_{t}}x_{0},(1-\bar{\alpha}_{t})\mathbf{I})$
(2)

When t is large enough, the $\bar{\alpha}$ is approaching 0, making $x_{t}$ an
isotropic Gaussian noise.

The reverse process aims to recover the data distribution from the Gaussian
noise. The reverse process in one step can be represented as:

$p_{\theta}(x_{t-1}|x_{t})=\mathcal{N}(x_{t-1};\mu_{\theta}(x_{t},t),\Sigma_{t})$
(3)

where $\Sigma_{t}$ is a constant depending on the variance schedule
$\beta_{t}$ and $\mu_{\theta}(x_{t},t)$ is determined by a neural network:

$\mu_{\theta}(x_{t},t)=\
\frac{1}{\sqrt{\alpha_{t}}}(x_{t}-\frac{\beta_{t}}{\sqrt{1-\bar{\alpha}_{t}}}\epsilon_{\theta}(x_{t},t))$
(4)

By recursively leveraging the reverse step, Gaussian noise can be recovered to
the original image. To train the DDPM, we first sample an image $x_{0}$, a
timestep $t$ and a random noise $\epsilon\sim\mathcal{N}(0,\mathbf{I})$. We
can obtain a noisy image $x_{t}$ using the forward process (Equation 2). We
then input both the noisy image $x_{t}$ and the timestep $t$ into a U-Net [35]
$\epsilon_{\theta}$ to predict the noise within $x_{t}$. The optimization
objective for the denoising U-Net can be written as:

$\mathcal{L}=\mathbb{E}_{t,x_{0},\epsilon}[||\epsilon-\epsilon_{\theta}(x_{t},t)||_{2}^{2}]$
(5)

To control the randomness of the reverse process, DDIM [41] adjusts the noise
added in each step:

$\displaystyle
x_{t-1}=\sqrt{\bar{\alpha}_{t-1}}\hat{x}_{0}+\sqrt{1-\bar{\alpha}_{t-1}}\hat{\epsilon}$
(6a)
$\displaystyle\hat{x}_{0}=\frac{1}{\sqrt{\bar{\alpha}_{t}}}(x_{t}-\sqrt{1-\bar{\alpha}_{t}}\epsilon_{\theta}(x_{t},t))$
(6b)
$\displaystyle\sqrt{1-\bar{\alpha}_{t-1}}\hat{\epsilon}=\sqrt{1-\bar{\alpha}_{t-1}-\sigma_{t}^{2}}\epsilon_{\theta}(x_{t},t)+\sigma_{t}\epsilon$
(6c)

By adjusting $\sigma_{t}$, we can control the random noise added in each
reverse step. When $\sigma_{t}=0$, the reverse process becomes a deterministic
process.

In the context of our study, the diffusion model serves as the target model.
Considering the practical implications of the TMI task, it is assumed that we
only have black-box access to the victim diffusion model, which also follows
the setting in [9]. This implies that we can only send queries to the victim
model and receive corresponding responses while we have no access to the
internal weights of the target diffusion model.

##  3 Method

###  3.1 Overview

Given the original image $x$, the goal of training membership inference is to
identify whether $x$ is in the training set of the target diffusion model
$\epsilon_{\theta}$. In this work, we propose the Degrade-Restore-Compare
(DRC) framework, which leverages the inherent generative priors in the target
diffusion model to infer the training membership with enhanced accuracy and
comprehensibility of the results. The architecture of DRC is shown in Figure
2. First, we obtain a partially degraded image $x^{D}$ by degrading the face
area of the original image $x$ (Section 3.2). We then exploit the generative
priors of the target model to restore $x^{D}$, thereby obtaining a restored
image $\tilde{x}$ (Section 3.3). Image restoration is an ill-posed problem: a
single degraded image can correspond to multiple potential restored images. If
the restored image $\tilde{x}$ closely resembles the original image, there is
a high probability that the model has encountered $x$ during its training
phase. Thus, we adopt a pretrained network to extract the semantic
representations to compare the original image $x$ with the restored output
$\tilde{x}$, thereby determining whether $x$ is part of the training set
(Section 3.4).

Figure 2: The framework of our proposed DRC. Given the original image record
$x$, we first obtain a partially degraded image $x^{D}$ by degrading the face
area of the original image $x$. We then exploit the generative priors of the
diffusion model $\epsilon_{\theta}$ to restore $x^{D}$, acquiring a restored
image $\tilde{x}$. Finally, we compare the original image $x$ and its restored
image $\tilde{x}$ and compute a membership score to define whether $x$ is in
the training set.

###  3.2 Degrade

As introduced in Figure 1, training samples manifest as more pronounced peaks
compared to unseen samples, which means the degraded training samples can be
restored to the original peaks, even with degradation of a high degree.
However, it remains too challenging to reproduce the original image $x$ from a
fully degraded image. Therefore, we focus on the degradation within the region
of interest (ROI), the inner face region in human face images for example. As
illustrated in Figure 2, we first predict the ROI in the original image,
represented as the ROI mask $M$. We then apply various types of degradation
(e.g., blurring, downsampling) to $x$ to obtain a set of low quality
candidates. $x_{LQ}$ is derived by selecting one candidate from this set. (We
analyze the effects of different degradation types in Section 4.3). Finally,
we utilize $M$ to integrate the original image $x$ and its corresponding fully
degraded $x^{LQ}$. This process can be represented as:

$x_{D}=M~{}\cdot x^{LQ}+(1-M)~{}\cdot x$ (7)

Specifically, we use DINO [4] to predict the ROI mask $M$. DINO is a self-
supervised learning method which can extract meaningful visual features and
producing attention scores for each pixel. We construct the mask $M$ for the
top $p$% of pixels with highest attention scores. The degree of degradation
can be modulated by adjusting the mask ratio. (The impact of varying degrees
of degradation is discussed in Section 4.3).

Algorithm 1 Restore

Input: degraded image $x^{D}$, original image $x$, ROI mask $M$, target
diffusion model $\epsilon_{\theta}$

Output: restored image $\tilde{x}$

$x_{T}\leftarrow x^{D}$

for $t=T,\dots,1$ do

Sample $\epsilon\sim\mathcal{N}(0,\mathbf{I})$

$x^{\prime}\leftarrow\sqrt{\bar{\alpha}_{t}}x+\sqrt{1-\bar{\alpha}_{t}}\epsilon$

$x_{t}\leftarrow M~{}\cdot x_{t}+(1-M)~{}\cdot x^{\prime}$

$\hat{x}_{0}\leftarrow\frac{1}{\sqrt{\bar{\alpha}_{t}}}(x_{t}-\sqrt{1-\bar{\alpha}_{t}}\epsilon_{\theta}(x_{t},t))$

$x_{t-1}\leftarrow\sqrt{\bar{\alpha}_{t-1}}\hat{x}_{0}+\sqrt{1-\bar{\alpha}_{t-1}}\epsilon_{\theta}(x_{t},t)$

end for

$\tilde{x}\leftarrow x_{0}$

return $\tilde{x}$

###  3.3 Restore

We leverage the generative priors of the target diffusion model
$\epsilon_{\theta}$ to restore the degraded image $x^{D}$. More precisely, our
aim is to restore the masked ROI region, conditioned on the surrounding
unmasked pixels. A common approach is to treat the degraded image $x^{D}$ as
$x_{T}$, which serves as the initial point of the reverse process. However,
this approach results in the output irrelevant to the conditional unmasked
pixels. This discrepancy arises because the reverse process comprises $T$
denoising steps and the model progressively forgets the initial conditions
with the step increasing. To overcome this problem, in each denoising step, we
replace the surrounding pixels with $x^{\prime}$ (the prior of the original
image $x$) to force the denoising output aligned with the condition. This
process can be represented as:

$x_{t}\leftarrow M~{}\cdot x_{t}+(1-M)~{}\cdot x\textquoteright$ (8)

Due to the stochastic nature of the diffusion model, we may produce dissimilar
restorations for member data, which will make member data and non-member data
less separable. To eliminate the randomness of the diffusion model, we use
deterministic DDIM sampling when $\sigma_{t}=0$ (Equation 6). Our restoration
process is also summarized in Algorithm 1.

###  3.4 Comparison

By comparing the restored image $\tilde{x}$ with the original image $x$, a
membership score is assigned to $x$. The more similar these two images are,
the larger the membership score is, which demonstrates a higher probability
that $x$ is a member in the training set. The memory mechanism of diffusion
models is predicted on semantic associations rather than pixel-level details
(which will be demonstrated in Section 4.4). Therefore, we compare these two
images from a semantic perspective. Specifically, we first utilize a
pretrained semantic encoder to obtain the semantic representations of these
two images. We then compute the cosine similarity of the semantic
representations as the membership score. This process can be represented as:

$score=\frac{SE(x)\cdot SE(\tilde{x})}{||SE(x)||||SE(\tilde{x})||}$ (9)

where $SE(\cdot)$ is the semantic encoder. In this paper, we adopt a face
recognition network [6] as the semantic encoder for human face images. We
adopt CLIP’s image encoder [32] as the semantic encoder for natural images.

##  4 Experiment

###  4.1 Experimental Setup

Datasets and Diffusion Models. For natural image generation, we evaluate two
commonly used datasets: Cifar10 and Cifar100 [22] datasets. We train the
diffusion model on Cifar10 and Cifar100 following the setting of [9]. For face
generation, we evaluate the CelebA [18] and Flickr-Faces-HQ (FFHQ) [19]
datasets in our experiments. We train the diffusion model on the CelebA [18]
dataset following the architecture of [34]. We also adopt a widely-used open-
sourced diffusion model 111https://github.com/CompVis/latent-diffusion trained
on the FFHQ dataset. Note that this model is officially provided. For Cifar10,
Cifar100 and CelebA datasets, we randomly select half of the training sets to
train the diffusion models, tagged as members. The remaining half is tagged as
non-members. For the open-sourced diffusion model in FFHQ, the member set is
the whole FFHQ dataset 222It should be noted that the diffusion model in FFHQ
may not be trained in the whole FFHQ dataset. Some images are used for
validation only. However, these validation images are not reported. Therefore,
we label the whole FFHQ as the member set. We choose the CelebA dataset as the
non-member set. The experimental results reported in the following are all
based on this label setting.

Evaluation Metrics. To evaluate the performance of our proposed method, we
choose widely used metrics [16] including Accuracy (Acc), Precision, Recall,
and Area-Under-the-ROC-curve (AUC). Acc is the ratio of correctly predicted
samples to the total samples. Precision is the fraction of samples classified
as members that are indeed members of the training set. Recall is the fraction
of the members that are correctly classified as members. For the calculation
of the AUC metric, we refer to [43] which also gives a detailed explanation
about AUC. We also consider the True Positive Rate (TPR) at extremely low
False Positive Rate (FPR) to gauge the confidence level of the prediction, as
suggested by Carlini et al. [3] TPR@1%FPR and<EMAIL_ADDRESS>denote the True
Positive Rate when the False Positive Rate is set at 1% and 0.1% respectively.

Table 1: Performance of our proposed method in CelebA and FFHQ. $\uparrow$ represents that the higher the metric, the better the performance. Bold denotes the best result for each metric. | CelebA | FFHQ
---|---|---
Method | Acc$\uparrow$ | Recall$\uparrow$ | Precision$\uparrow$ | AUC$\uparrow$ | Acc$\uparrow$ | Recall$\uparrow$ | Precision$\uparrow$ | AUC$\uparrow$
$\text{GAN-Leak}_{\textit{ori}}$ [5] | 0.511 | 0.048 | 0.632 | 0.491 | 0.511 | 0.054 | 0.628 | 0.496
$\text{GAN-Leak}_{\textit{inv}}$ [5] | 0.645 | 0.634 | 0.648 | 0.669 | 0.544 | 0.738 | 0.532 | 0.538
NaiveLoss [28] | 0.773 | 0.728 | 0.800 | 0.846 | 0.658 | 0.656 | 0.659 | 0.695
SecMI [9] | 0.779 | 0.804 | 0.766 | 0.848 | 0.629 | 0.630 | 0.629 | 0.656
Ours | 0.962 | 0.962 | 0.962 | 0.989 | 0.753 | 0.622 | 0.843 | 0.811
Table 2: Performance of our proposed method in Cifar10 and Cifar100. $\uparrow$ represents that the higher the metric, the better the performance. Bold denotes the best result for each metric. | Cifar10 | Cifar100
---|---|---
Method | Acc$\uparrow$ | Recall$\uparrow$ | Precision$\uparrow$ | AUC$\uparrow$ | Acc$\uparrow$ | Recall$\uparrow$ | Precision$\uparrow$ | AUC$\uparrow$
$\text{GAN-Leak}_{\textit{ori}}$ [5] | 0.560 | 0.631 | 0.553 | 0.580 | 0.549 | 0.615 | 0.543 | 0.567
$\text{GAN-Leak}_{\textit{inv}}$ [5] | 0.556 | 0.654 | 0.547 | 0.573 | 0.558 | 0.707 | 0.544 | 0.581
NaiveLoss [28] | 0.761 | 0.821 | 0.732 | 0.822 | 0.732 | 0.816 | 0.698 | 0.797
SecMI [9] | 0.825 | 0.877 | 0.794 | 0.886 | 0.800 | 0.866 | 0.765 | 0.869
Ours | 0.874 | 0.920 | 0.843 | 0.931 | 0.852 | 0.916 | 0.812 | 0.919

Implementation Details. Our DRC has two main hyper-parameters: the mask ratio
and the degradation type. The former controls the degraded pixel numbers. The
latter defines the corresponding restoration task (such as denoising,
inpainting, etc.). Due to the varying semantic information contained in
natural images versus facial images, we employ distinct mask ratios. For
CelebA and FFHQ datasets, we mask 20% most important pixels in the original
image (mask ratio=0.2). For Cifar10/100, we mask 50% most important pixels in
the original image (mask ratio=0.5). We choose denoising as the restoration
task. In these settings, our method achieves the best performance. We also
conduct detailed ablations in Section 4.3. Besides, we adopt DDIM sampling
($\sigma_{t}=0$) with an interval of 5 to accelerate the reverse process of
diffusion models. To calculate the Accuracy (Acc), a threshold is needed. We
follow the method in [9] to obtain the threshold.

###  4.2 Comparison to Baselines

We compare our method with the current state-of-the-art MIA method, SecMI [9].
We also compare with other baseline methods including GAN-Leak [5] and
NaiveLoss [28]. As GAN-Leak is designed for GANs and can not be directly used
for diffusion models, we follow the adjustments in [9] to make it available
for diffusion models. (The details can be found in Supplementary Materials.)

Figure 3: The log scaled ROC curves on Cifar10, Cifar100, CelebA and FFHQ datasets. The log scaled ROC provides compelling evidence that our method can substantially perform training membership inference with high prediction confidence.  Table 3: The True Positive Rate (TPR) at very low False Positive Rate (FPR) of our proposed method. TPR@1% and<EMAIL_ADDRESS>stands for TPR when the FPR is set at 1% and 0.1% separately. $\uparrow$ represents that the higher the metric, the better the performance. Bold denotes the best result for each metric. | TPR@1%FPR(%)$\uparrow$ | TPR@0.1%FPR(%)$\uparrow$
---|---|---
Method | Cifar10 | Cifar100 | CelebA | FFHQ | Cifar10 | Cifar100 | CelebA | FFHQ
$\text{GAN-Leak}_{\textit{ori}}$ [5] | 1.28 | 1.97 | 2.22 | 1.80 | 0.12 | 0.18 | 0.51 | 0.55
$\text{GAN-Leak}_{\textit{inv}}$ [5] | 1.31 | 1.89 | 2.50 | 0.63 | 0.09 | 0.21 | 0.22 | 0.01
NaiveLoss [28] | 3.77 | 6.13 | 23.16 | 11.20 | 0.26 | 0.83 | 1.34 | 2.31
SecMI [9] | 6.09 | 9.68 | 4.43 | 9.46 | 0.43 | 1.23 | 0.20 | 2.66
Ours | 12.20 | 16.56 | 80.46 | 30.85 | 0.82 | 2.44 | 54.52 | 18.60

Table 1 and Table 2 presents the performance of our proposed method and other
compared TMI methods. We observe that our method achieves the best performance
on the accuracy, precision, and AUC metrics and outperforms baselines by a
large margin, which shows the superiority of our proposed methods. We also
find that our method is not so effective in the recall metric in FFHQ. This
may be attributed to the varying complexity of different images. Though we
make our best efforts to control the difficulty of image restoration (by
masking areas of the same ratio), the semantic information of masked areas
still varies between different images. More precisely, under the condition of
masking the same number of pixels, some images have simpler semantic
information, while others are too complex for the diffusion model to fully
reproduce. This will make the member samples misclassified to nonmember
samples, resulting in a decrease in recall metric. We also find that in FFHQ,
our method enjoys a high precision of up to 0.843. This is because our method
is based on reproduction. Once a sample is fully reproduced, there is a high
probability that the model has ‘seen’ the sample during its training stage. We
further note that privacy violation detection mandates user trust as a
prerequisite, thus the precision metric assumes a more vital role in this
context. Besides, we will show (in Section 4.3) that by simply aggregating the
restoration results of different degradation types, our method can obtain a
boost in recall and reach a balance between precision and recall. Table 3
presents the True Positive Rate (TPR) at very low False Positive Rate (FPR) of
our method. It can be observed that our method demonstrates significant
success in both of the two evaluations. We also present the log-scaled ROC
curves in Figure 3.

###  4.3 Ablation Study

Restoration Tasks. Different degradation methods will manipulate the original
image to different positions on the learned manifold, which also influences
the initial point of the restoration task. The distance between the degraded
image and the original image determines the difficulty of the restoration task
and thus significantly affects the restoration results. Therefore, we explored
various degradations to construct different restoration tasks (including
deblur, derain, inpaint, super resolution, and denoise), and the results in
CelebA are shown in Table 4.3. It can be seen that the performance gaps among
different restoration tasks are rather small, with a deviation less than 0.01.
The denoising method performs the best, while the inpainting method the worst.
This is because the training objective of the diffusion model is also to
denoise the input image with varying noise levels. The consistency in
objective may recall the model’s memory of the training sample. Compared to
denoising, the inpainting process needs to reproduce the image from an area
without signals, which contradicts the training object most and results in
suboptimal performance.

Table 4: The result of performing different restoration tasks in CelebA. We
have tried five degradation types including deblur, derain, inpaint, super
resolution, and denoise.

| Restoration
---
Task
Acc$\uparrow$ | Recall$\uparrow$ | Precision$\uparrow$ | AUC$\uparrow$
deblur | 0.959 | 0.958 | 0.960 | 0.988
derain | 0.960 | 0.964 | 0.956 | 0.989
inpaint | 0.954 | 0.954 | 0.954 | 0.988
super resolution | 0.959 | 0.960 | 0.958 | 0.989
denoise | 0.962 | 0.962 | 0.962 | 0.989

Table 5: The results of simple aggregation methods FFHQ. We carry the
aggregation by computing the mean and median of the results of different
restoration tasks.

| Aggregation
---
Strategy
Acc$\uparrow$ | Recall$\uparrow$ | Precision$\uparrow$ | AUC$\uparrow$
mean | 0.774 | 0.768 | 0.777 | 0.842
median | 0.759 | 0.738 | 0.770 | 0.823
denoise only | 0.753 | 0.622 | 0.843 | 0.811

Figure 4: We design an aggregation based on the agree numbers of different
restoration tasks. We plot the Precision and Recall in different agree numbers
in FFHQ.

Aggregation of Restoration Methods. We attempt to further aggregate the
results of different restoration methods. We first explore aggregating these
results by simply computing the mean/median of the five restoration methods’
member scores. The results are shown in Table 4.3. We observe that in FFHQ,
the aggregation methods (both mean and median) can balance the recall and
precision metrics. Considering the high precision requirements in practical
privacy violation detection, we design an aggregation method based on voting.
Specifically, we consider a sample as member only if a certain number of the
restoration methods classify this sample as member. We refer to this number as
the ‘agree number’. Figure 4 demonstrates the changes in precision with
different agree numbers. It can be observed that as the agree number
increases, the precision boosts from 0.670 to 0.928, which demonstrates
promising application potential in privacy violation detection. We believe
that if the agree number continues to increase, we will obtain a higher
precision. We also plot changes in recall with different agree numbers (also
in Figure 4). We can observe that the recall metric achieves its peak value
when there is only one restoration method that classifies the sample as
member. This is because with the agree number increasing, the criterion for a
sample to be classified as member becomes more and more strict. For example,
when agree number reaches 5, a sample will be labeled as member only if all
the five restoration methods label it as member. By adjusting the agree
number, we can control the tendency towards precision and recall, allowing for
more flexible usage in real world applications.

Figure 5: The degraded and restored images of our method in different mask
ratios. We show two members and non-members with the mask ratio varying from
0.1 to 0.5. More results can be found in supplementary materials.  Table 6:
Ablations on mask ratio in CelebA. We evaluate the performance with the mask
ratio from 0.1 to 0.5.

mask ratio | Acc$\uparrow$ | Recall$\uparrow$ | Precision$\uparrow$ | AUC$\uparrow$
---|---|---|---|---
0.1 | 0.541 | 0.492 | 0.545 | 0.548
0.2 | 0.962 | 0.962 | 0.962 | 0.989
0.3 | 0.945 | 0.938 | 0.951 | 0.980
0.4 | 0.889 | 0.832 | 0.939 | 0.936
0.5 | 0.816 | 0.704 | 0.907 | 0.869

Degradation Degree. The mask ratio is used to control the image degradation
degree and thereby adjust the difficulty of image restoration. We investigate
the performance under different mask ratios. The qualitative and quantitative
results are shown in Figure 5 and Table 6, respectively. It can be observed
that as the mask ratio increases, the performance of our proposed method first
increases (reaches its optimal level at 0.2) and then decreases. The obtained
experimental results are consistent with the fundamental intuition outlined in
our introduction, which posits that compared to unseen samples, training
samples can be restored to their original state with a higher degree of
degradation. The mask ratio is indicative of the degradation degree. At lower
degradation degrees, both degraded training and unseen samples can be
restored. As the degradation level increases within a certain range, the model
can only restore training samples, resulting in optimal discriminative
performance. However, when the degradation level is beyond the generative
priors, the model is unable to restore either type of degraded sample, leading
to a decline in discriminative capability.

Therefore, we believe there exists a point where the masked semantic area and
unmasked semantic priors reach a balance, achieving the optimal performance of
our proposed method.

Figure 6: The degraded and restored images of our method in different inverted
mask ratios. We show two members and non-members with the mask ratio varying
from 0.1 to 0.5. More results can be found in supplementary materials.  Table
7: Quantitative results with inverted mask ratios varying from 0.2 to 0.5. In
addition to the identity metric (CSIM), we also employ pixel level metrics
($\text{L}_{1}$, MSE) to distinguish between members and non-members.

inverted mask ratio | CSIM | $\text{L}_{1}$ | MSE
---|---|---|---
Acc$\uparrow$ | Recall$\uparrow$ | Precision$\uparrow$ | AUC$\uparrow$ | Acc$\uparrow$ | Recall$\uparrow$ | Precision$\uparrow$ | AUC$\uparrow$ | Acc$\uparrow$ | Recall$\uparrow$ | Precision$\uparrow$ | AUC$\uparrow$
0.2 | 0.585 | 0.844 | 0.556 | 0.608 | 0.655 | 0.778 | 0.624 | 0.699 | 0.662 | 0.648 | 0.667 | 0.702
0.3 | 0.606 | 0.852 | 0.571 | 0.638 | 0.656 | 0.658 | 0.655 | 0.694 | 0.645 | 0.792 | 0.612 | 0.691
0.4 | 0.637 | 0.736 | 0.614 | 0.679 | 0.643 | 0.630 | 0.647 | 0.679 | 0.637 | 0.594 | 0.650 | 0.672
0.5 | 0.540 | 0.746 | 0.626 | 0.698 | 0.629 | 0.522 | 0.664 | 0.657 | 0.619 | 0.582 | 0.629 | 0.649

###  4.4 Further Analysis

Inverted Mask areas. Facial privacy is closely intertwined with our personal
privacy [45, 13]. Consequently, we express particular interest in model
responses to human faces. We further explore what diffusion models tend to
memorize in human face images. In our method, we mask pixels that are most
important in face images. In this experiment, we invert the mask area, i.e. we
mask pixels that are most unimportant in face images. With the inverted mask
ratio increasing, we first mask the background and then remove some of the
hair. It should be noted that in this experiment, the human face of each image
is almost left unmasked. If the background of the restored image is similar to
the original image, it indicates that the model memorizes every pixel in the
image without discrimination, including both the background and the face
geometry. On the contrary, if diffusion models can not restore the masked
background, it indicates that the model focuses more on the face area. This
tendency to memorize face identity reveals an even higher risk of the privacy
leakage. The quantitative and qualitative results are shown in Figure 6 and
Table 7. In our method, we use similarity in identity to distinguish the
members and non-members (termed as CSIM). However, this metric may not work as
the identity information in degraded images is not ruined in this experiment.
Thus, we also use pixel level loss ($\text{L}_{1}$ loss and MSE loss) to
discriminate between members and non-members. From Table 7, we can observe
that pixel level metrics perform slightly better than identity metric (CSIM) .
However, both these metrics show unsatisfactory performance with the best AUC
achieving only 0.702. This indicates that diffusion models are inclined to
ignore information unrelated to identity, such as background, and decorations.
Figure 6 further verifies this indication. In the upper left of Figure 6, the
sunglasses are restored to eyes while the textures in the background are
ignored by the diffusion models, leaving only a blurry area.

##  5 Conclusion

In this study, we introduce DRC, an innovative training membership inference
framework for diffusion models by exploiting the intrinsic reasoning and
generative priors of the diffusion model. Specifically, we first perform
degradation in the high attention area of the original image. Then, we exploit
the generative priors to restore the degraded area. Finally, we compare the
original image with its restored counterpart to determine its membership. Our
experiment results validate the proficiency of our methodology in accurately
distinguishing between members and non-members. Moreover, the underlying
mechanism of our approach is intuitively comprehensible, which enhances the
credibility of our predicted outcomes for users. As a user-friendly technique,
we sincerely anticipate that our approach will make a substantial contribution
to the safeguarding of individual privacy in real-world applications.

## Limitation and Ethical Statement

In this paper, we develop a novel TMI framework for diffusion models. While
our approach demonstrates effectiveness, it is hard to establish sufficient
theoretical evidence. In future work, we will try to provide a rigorous
mathematical proof to validate the theoretical foundation. The goal of this
research is to develop a technique to identify whether a specific sample is
part of the training dataset. The method we propose enjoys numerous
advantageous applications, including privacy violation detection and model
privacy assessment. Despite the potential for malicious entities to exploit
our method for privacy attacks, we emphasize that such attacks can be
mitigated by privacy protection techniques, such as differential privacy.
Moreover, we intend to release our code to facilitate the advancement of
privacy protection technology.

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# The title

L. Kryvonos  E. B. Saff

# On a problem of E. Meckes for the unitary eigenvalue process on an arc

L. Kryvonos  E. B. Saff

###### Abstract

We study the problem [1][1][1]The problem was communicated to the second
author shortly before the untimely death of Professor Meckes. originally
communicated by E. Meckes on the asymptotics for the eigenvalues of the kernel
of the unitary eigenvalue process of a random $n\times n$ matrix. The
eigenvalues $p_{j}$ of the kernel are, in turn, associated with the discrete
prolate spheroidal wave functions. We consider the eigenvalue counting
function $|G(x,n)|:=\\#\\{j:p_{j}>Ce^{-xn}\\}$, ($C>0$ here is a fixed
constant) and establish the asymptotic behavior of its average over the
interval $x\in(\lambda-\varepsilon,\lambda+\varepsilon)$ by relating the
function $|G(x,n)|$ to the solution $J(y)$ of the following energy problem on
the unit circle $S^{1}$, which is of independent interest. Namely, for given
$\theta$, $0<\theta<2\pi$, and given $q$, $0<q<1$, we determine the function
$J(y)=\inf\\{I(\mu):\mu\in\mathcal{P}(S^{1}),\mu(A_{\theta})=y\\}$, where
$I(\mu):=\iint\log\frac{1}{|z-\zeta|}d\mu(z)d\mu(\zeta)$ is the logarithmic
energy of a probability measure $\mu$ supported on the unit circle and
$A_{\theta}$ is the arc from $e^{-i\theta/2}$ to $e^{i\theta/2}$.

MSC: 31A05, 60F10

Keywords: unitary eigenvalue process, discrete prolate spheroidal wave
functions, logarithmic energy problem with constraints

##  1 Introduction

For $n\in\mathbb{N}$, denote by $\mathbb{U}(n)$ the set of $n\times n$ unitary
matrices over $\mathbb{C}$. Since the set $\mathbb{U}(n)$ forms a compact
topological group with respect to matrix multiplication and the usual
topology, there exists a unique probability measure on $\mathbb{U}(n)$ (called
Haar measure) that is invariant under left- and right-translation. In other
words, a distribution of a uniform random element $U_{n}$ of $\mathbb{U}(n)$
is the (Haar) measure $\mu$ on $\mathbb{U}(n)$ such that for any measurable
subset $\mathcal{A}\subset\mathbb{U}(n)$ and any fixed matrix
$M\in\mathbb{U}(n)$,

$\mu(M\mathcal{A})=\mu(\mathcal{A}M)=\mu(\mathcal{A}),$

where $M\mathcal{A}:=\\{MU:\;\;U\in\mathcal{A}\\}$ and
$\mathcal{A}M:=\\{UM:\;\;U\in\mathcal{A}\\}$.

Denote the eigenvalues of a Haar-distributed random unitary $n\times n$ matrix
$U_{n}$, all of which lie on the circle
$\mathbb{S}^{1}:=\\{z\in\mathbb{C}:|z|=1\\}$, by
$\\{e^{i\theta_{1}},\dots,e^{i\theta_{n}}\\}$, $0\leq\theta<2\pi$. Our main
goal is to study the eigenvalue counting function

$\mathcal{N}_{\theta}=\mathcal{N}_{(0,\theta)}:=\\#\\{j:0<\theta_{j}<\theta\\},$
(1)

where $\theta\in(0,2\pi)$ is fixed, by utilizing the determinantal structure
of the eigenvalue process for $\mathcal{N}_{\theta}$ that we describe next.

A point process $\mathcal{X}$ in a locally compact Polish space $X$ is a
random discrete subset of $X$. Let $\mathcal{N}_{A}$ denote the (random)
number of points of $\mathcal{X}$ in $A\subset X$. We say that $\mathcal{X}$
is a determinantal point process if for any finite number of pairwise disjoint
subsets $A_{1},...,A_{k}\subset X$,

$\mathbb{E}\bigg{[}\prod_{j=1}^{k}\mathcal{N}_{A_{j}}\bigg{]}=\int_{A_{1}}...\int_{A_{k}}\det[K(x_{i},x_{j})]_{i,j=1}^{k}d\mu(x_{1})...d\mu(x_{k}),$

for some kernel $K:X\times X\rightarrow[0,1]$ and Borel measure $\mu$ on $X$.
One of the remarkable properties of eigenvalue distributions of matrices from
the compact classical groups is that they are determinantal point processes.
The connection in the case of the unitary group has been known at least since
the work [3] of F. J. Dyson, while for other groups one of the earliest
references known to us is [8] by N. M. Katz and P. Sarnak. We state Theorem
1.1 below for the unitary group only, and refer the reader to [12] for more
general results concerning other groups.

######  Theorem 1.1.

(see [12, Proposition 3.7]) The eigenvalue angles of uniformly distributed
random matrices in $\mathbb{U}(n)$ is a determinantal point process with
respect to uniform measure on $[0,2\pi)$, with kernel

$K_{n}(x,y):=\sum_{j=0}^{n-1}e^{ij(x-y)}.$ (2)

For convenience, we will use an alternative form of the kernel (2):

$K_{n}(x,y):=\begin{cases}\sin(\frac{n(x-y)}{2})/\sin(\frac{(x-y)}{2}),\;\;\;\;\;\;\textnormal{if}\;\;x-y\neq
0,\\\ n,\hskip 122.85876pt\textnormal{if}\;\;x-y=0.\\\ \end{cases}$ (3)

Although the kernels (2) and (3) are different functions, they are unitarily
similar and thus define the same point processes (see [8, Section 5.4]).

Theorem 1.1 allows us to apply the next result.

######  Theorem 1.2.

(J. B. Hough et al., [5], see also [12, Theorem 4.1])

Let $K:X\times X\rightarrow\mathbb{C}$ be a kernel on a locally compact Polish
space $X$ and $\mu$ a Borel measure on $X$ such that the corresponding
integral operator $\mathcal{K}:L^{2}(X,\mu)\rightarrow L^{2}(X,\mu)$ defined
by

$\mathcal{K}(f)(x):=\int K(x,y)f(y)d\mu(y)$

is self-adjoint, nonnegative, and locally trace-class with eigenvalues in
[0,1]. For $D\subset X$ measurable, let
$K_{D}(x,y):=\mathbb{I}_{D}(x)K(x,y)\mathbb{I}_{D}(y)$ be the restriction of
$K$ to $D\times D$. Suppose that $D$ is such that the operator
$\mathcal{K}_{D}$ with kernel $K_{D}$ is trace-class. Denote by
$\\{p_{j}\\}_{j\in\mathcal{J}}$ the eigenvalues of the operator
$\mathcal{K}_{D}$ on $L^{2}(D,\mu)$ (the index set $\mathcal{J}$ may be finite
or countable) and denote by $\mathcal{N}_{D}$ the number of particles of the
determinantal point process with kernel $K$ that lie in $D$. Then

$\mathcal{N}_{D}\overset{d}{=}\sum_{j\in\mathcal{J}}\xi_{j},$

where $``\overset{d}{=}"$ denotes equality in distribution and the $\xi_{j}$
are independent Bernoulli random variables with $\mathbb{P}[\xi_{j}=1]=p_{j}$
and $\mathbb{P}[\xi_{j}=0]=1-p_{j}$.

Let $X=[0,2\pi)$ and $\mu$ be a uniform measure on $X$. Consider an integral
operator $\mathcal{K}_{n}$ with kernel $K_{n}$ from (3). It’s easy to see that
the operator satisfies all conditions of Theorem 1.2. Moreover, since its
kernel is degenerate, $\mathcal{K}_{n}$ has $n$ eigenvalues
$\\{p_{j}\\}_{j=1}^{n}$ (see, for example, [13, Section 3.2]); hence, by
Theorem 1.2, the counting function $\mathcal{N}_{\theta}$, introduced in (1),
is equal in distribution to a sum of $n$ independent Bernoulli random
variables:

$\mathcal{N}_{\theta}\stackrel{{\scriptstyle d}}{{=}}\sum_{j=1}^{n}\xi_{j},$
(4)

where $\mathbb{P}[\xi_{j}=1]=p_{j}$ and $\mathbb{P}[\xi_{j}=0]=1-p_{j}$. We
would like to study the behavior of the $p_{j}$ near zero. Namely, for $C>0$
large and fixed and $x\in\mathbb{R}$, consider

$G(x,n):=\\{j:p_{j}>Ce^{-xn}\\},$ (5)

and let $|G(x,n)|$ be the number of $p_{j}$ in $G(x,n)$. Our main result is
the following theorem that describes the asymptotic behavior of the function
$|G(x,n)|$.

######  Theorem 1.3.

Let $\theta\in(0,2\pi)$. For any fixed $\varepsilon>0$ and
$\lambda\in\mathbb{R}$,

$\frac{1}{2\varepsilon}\int_{\lambda-\varepsilon}^{\lambda+\varepsilon}|G(x,n)|dx=\frac{n}{2\varepsilon}(\Lambda(\lambda+\varepsilon)-\Lambda(\lambda-\varepsilon))-o(n),\;\;n\rightarrow\infty,$
(6)

where $o(n)$ depends on $\lambda$, $\varepsilon$ and the function
$\Lambda(\lambda):=\underset{y\in[0,1]}{\sup}\\{\lambda y-J(y)\\}$ is the
Fenchel-Legendre transform of the rate function
$J(y)=\iint\log\frac{1}{|z-\zeta|}d\nu^{*}_{y}(z)d\nu^{*}_{y}(\zeta)$ with
$\nu^{*}_{y}$ given by (7) of Theorem 1.4 below (with $q$ replaced by $y$).

Furthermore, there exists a constant $c_{0}>0$, such that for all $\lambda\geq
c_{0}$ expression (6) becomes

$\frac{1}{2\varepsilon}\int_{\lambda-\varepsilon}^{\lambda+\varepsilon}|G(x,n)|dx=n-o(n),\;\;n\rightarrow\infty.$

Remark. The above functions $G,\Lambda$ and $J$ depend on the constant
$\theta$, though for simplicity we omit this dependence in our notations.

To prove Theorem 1.3, we use a key observation of T. Liu and E. Meckes [10]
that the random variables $\frac{\mathcal{N}_{\theta}}{n}$ satisfy the large
deviation principle, with the rate function $J(x)$ being a solution to some
equilibrium problem on the unit circle. The function $\widehat{J}(x)$ that
solves an equivalent problem on the interval $[-1,1]$, in turn, is a limit
distribution of zeros of Heine–Stieltjes polynomials $-$ the polynomial
solutions to the generalized Lamé differential equation, and was obtained in
the work [11] by A. Martínez-Finkelshtein and E. B. Saff. In Section 3 we give
an alternative proof of this result as well as establish the following
theorem.

######  Theorem 1.4.

The measure $\nu^{*}=:\nu^{*}_{q}\in\mathcal{P}(\mathbb{S}^{1})$, such that
$I(\nu^{*})=\inf\\{I(\nu):\nu\in\mathcal{P}(\mathbb{S}^{1}),\nu(A_{\theta})=q\\}$,
where $A_{\theta}:=\\{z\in\mathbb{S}^{1}:-\frac{\theta}{2}\leq\arg
z\leq\frac{\theta}{2}\\}$, is unique and

i) if $q\geq\frac{\theta}{2\pi}$, given by

$d\nu^{*}(e^{i\psi})=\frac{\sqrt{|\cos(\psi)-\alpha|}}{2\pi\sqrt{|\cos(\psi)-\cos(\theta/2)|}}d\psi,\;\;\;$
(7)

where $e^{i\psi}\in
A_{\theta}\cup\\{z\in\mathbb{S}^{1}:\arccos(\alpha)\leq\arg\;z\leq
2\pi-\arccos(\alpha)\\},$ with $\alpha$ determined from the equation

$\int_{-1}^{\alpha}\frac{\sqrt{|x-\alpha|}}{\pi\sqrt{|(x+1)(x-\cos(\theta/2))(x-1)|}}dx=1-q;$
(8)

ii) if $q\leq\frac{\theta}{2\pi}$, given by (7), where $e^{i\psi}\in
A^{c}_{\theta}\cup\\{z\in\mathbb{S}^{1}:-\arccos(\alpha)\leq\arg\;z\leq\arccos(\alpha)\\}$
and $\alpha$ is a solution to the equation

$\int_{-1}^{\cos(\theta/2)}\frac{\sqrt{|x-\alpha|}}{\pi\sqrt{|(x+1)(x-\cos(\theta/2))(x-1)|}}dx=1-q.$

Note that, with a suitable change of variables, the operator with kernel (3)
can be written as

$\mathcal{K}_{n}(f)(x)=\int_{-W}^{W}\frac{\sin
n\pi(x-y)}{\sin\pi(x-y)}f(y)dy,$ (9)

with $W=\frac{\theta}{4\pi}$. An alternative way to study the behavior of its
eigenvalues $p_{j}$ is from the operator theory perspective, and below we give
a short summary of known results in this direction.

Operators of the form (9) with $W\in(0,1/2)$, whose eigenfunctions are well-
known discrete prolate spheroidal wave functions, were extensively studied by
D. Slepian in [16]. Along with establishing the asymptotics for the
eigenfunctions of the operator, he also examined the behavior of its
eigenvalues. In particular, he showed in [16] that the first $2nW$ eigenvalues
(arranged in descending order) tend to cluster extremely close to 1, while the
remaining eigenvalues tend to cluster similarly close to 0. These results,
though, do not address precisely how many eigenvalues one can expect to find
between $\varepsilon$ and $1-\varepsilon$. Numerical experiments performed by
S. Karnik, J. Romberg, and M. A. Davenport in [6] suggest that the number of
eigenvalues $p_{j}=p_{j}(W,n)$ in the “transition region” between
$\varepsilon$ and $1-\varepsilon$ behaves like
$\\#\\{j:\varepsilon<p_{j}<1-\varepsilon\\}=O(\log(nW)\log\big{(}\frac{1}{\varepsilon}\big{)})$.
They obtained the nonasymptotic bounds for the number of eigenvalues in
$(\varepsilon,1-\varepsilon)$ that captures the observed logarithmic
dependence of the transition region on $n$ and $\varepsilon$, and compared
them with the previous known nonasymptotic bounds (see [1], [7], [17]).

The paper is organized as follows: in Section 2 we prove Theorem 1.3, in
Section 3 we solve the constrained energy problem on an interval and the unit
circle to find the rate function $J(x)$ from Theorem 1.3.

##  2 Asymptotics for $p_{j}$.

To study the function $|G(x,n)|$, given by (5), we make use of the known large
deviation principle (LDP) due to F. Hiai and D. Petz [4] for the empirical
spectral measure $\mu_{n}$ of $U_{n}$.

######  Definition 2.1.

A sequence of probability Borel measures $\\{P_{n}\\}$ on a topological space
$X$ satisfies an LDP with rate function $I$ and speed $s_{n}$ if for all Borel
sets $\mathcal{B}\subseteq X$,

$-\underset{x\in\mathcal{B}^{0}}{\inf}I(x)\leq\underset{n\rightarrow\infty}{\lim\inf}\frac{1}{s_{n}}\log(P_{n}(\mathcal{B}))\leq\underset{n\rightarrow\infty}{\lim\sup}\frac{1}{s_{n}}\log(P_{n}(\mathcal{B}))\leq-\underset{x\in\overline{\mathcal{B}}}{\inf}I(x)$

######  Theorem 2.2.

(F. Hiai, D. Petz, [4, Section 1])

Let $U_{n}\in\mathbb{U}(n)$ and
$\mu_{n}:=\frac{1}{n}\sum_{j=1}^{n}\delta(e^{i\theta_{j}})$, where
$\\{e^{i\theta_{j}}\\}_{j=1}^{n}$ are the eigenvalues of $U_{n}$ and
$\delta(\zeta)$ denotes the Dirac measure at $\zeta$. Denote by $P_{n}$ the
law of $\mu_{n}$. Then the sequence $\\{P_{n}\\}$ satisfies an LDP on the
space $\mathcal{P}(\mathbb{S}^{1})$ of probability measures on the unit circle
equipped with the topology of weak convergence, with speed $n^{2}$ and
strictly convex rate function

$I(\nu)=-\iint\displaylimits_{\mathbb{S}^{1}\times\mathbb{S}^{1}}\log|z-w|d\nu(z)d\nu(w).$

Let $A_{\theta}$ denote the arc of the circle consisting of those points with
argument in $(0,\theta)$. Then $\nu\mapsto\nu(A_{\theta})$ is a continuous
function of $\nu$ (with respect to the topology of weak convergence), and so
by the contraction principle [2, Section 4.2], the random variables
$\mu_{n}(A_{\theta})=\frac{\mathcal{N}_{\theta}}{n}$ satisfies an $LDP$ on
$[0,1]$ with speed $n^{2}$ and rate function

$J(y)=\inf\\{I(\nu):\nu\in\mathcal{P}(\mathbb{S}^{1}),\nu(A_{\theta})=y\\}.$
(10)

In Section 3 we solve the minimization problem (10) and give an explicit
formula for the function $J(y)$. This allows us to utilize the next result and
investigate the behavior of moment-generating functions of the sequence of
random variables $\\{\frac{\mathcal{N}_{\theta}}{n}\\}$.

######  Theorem 2.3.

(Varadhan’s Integral Lemma, [2, Section 4.3])

Suppose that the family of random variables $\\{Z_{\varepsilon}\\}$ taking
values in a topological space $X$ satisfies the LDP with rate function
$J:X\rightarrow[0,\infty]$. Let $\phi:X\rightarrow\mathbb{R}$ be any
continuous function. Assume further either the tail condition

$\underset{M\rightarrow\infty}{\lim}\underset{\varepsilon\rightarrow
0}{\lim\sup}\;\varepsilon\log\mathbb{E}[e^{\phi(Z_{\varepsilon})/\varepsilon}\mathbb{1}_{\\{\phi(Z_{\varepsilon})>M\\}}]=-\infty,$

or the following moment condition for some $\gamma>1$,

$\underset{\varepsilon\rightarrow
0}{\lim\sup}\;\varepsilon\log\mathbb{E}[e^{\gamma\phi(Z_{\varepsilon})/\varepsilon}]<\infty.$
(11)

Then

$\underset{\varepsilon\rightarrow
0}{\lim}\;\varepsilon\log\mathbb{E}[e^{\phi(Z_{\varepsilon})/\varepsilon}]=\underset{y\in
X}{\sup}\\{\phi(y)-J(y)\\}.$ (12)

Therefore, if we take above $\phi(y)=\lambda y,\;\lambda\in\mathbb{R}$ and
show that condition (11) holds, then Theorem 2.3 implies that the next limit
exist

$\displaystyle\Lambda(\lambda):=\lim_{n\to\infty}\frac{1}{n^{2}}\log\mathbb{E}[e^{\lambda
n\mathcal{N}_{\theta}}].$ (13)

It follows from (12) that the function $\Lambda(\lambda)$ appearing in (6) is
given by the Fenchel-Legendre transform of $J$, i.e.,
$\Lambda(\lambda)=\underset{y\in[0,1]}{\sup}\\{\lambda y-J(y)\\}$. On the
other hand, in the proof of Theorem 1.3 we express the right side of (13) in
terms of the function $|G(x,n)|$ and from there obtain the relationship
between $\Lambda(\lambda)$ and $|G(x,n)|$.

###  2.1 Proof of Theorem 1.3

###### Proof.

As mentioned above, we are going to examine the limit in equation (13). Note
first that if $\lambda<0$, then
$\lim_{n\to\infty}\frac{1}{n^{2}}\log\mathbb{E}[e^{\lambda
n\mathcal{N}_{\theta}}]=0$ trivially, so we restrict our attention to
$\lambda>0$. Since $\mathcal{N}_{\theta}\stackrel{{\scriptstyle
d}}{{=}}\sum_{j=1}^{n}\xi_{j}$ with the $\xi_{j}$ independent Bernoullis with
success probabilities $p_{j}$,

$\displaystyle\frac{1}{n^{2}}\log\mathbb{E}[e^{\lambda
n\mathcal{N}_{\theta}}]=\frac{1}{n^{2}}\sum_{j=1}^{n}\log(p_{j}e^{\lambda
n}+1-p_{j})$ $\displaystyle=\frac{1}{n^{2}}\sum_{j\in
G(\lambda,n)}\log(p_{j}e^{\lambda n}+1-p_{j})+\frac{1}{n^{2}}\sum_{j\notin
G(\lambda,n)}\log(p_{j}e^{\lambda n}+1-p_{j})$ $\displaystyle=:S_{1}+S_{2}$

For $j\notin G(\lambda,n)$ and any $\lambda>0$, $p_{j}\leq p_{j}e^{\lambda
n}\leq C$, and so

$0\leq\sum_{j\notin G(\lambda,n)}\log(p_{j}e^{\lambda n}+1-p_{j})\leq
n\log(C).$

We thus have that $0\leq S_{2}\leq\frac{\log(C)}{n}$.

For $S_{1}$, first observe that

$\log(p_{j}e^{\lambda n}+1-p_{j})=\lambda
n+\log(p_{j})+\log(1+\frac{1-p_{j}}{p_{j}}e^{-\lambda n})$

and that, for $j\in G(\lambda,n)$, $\frac{1-p_{j}}{p_{j}}e^{-\lambda
n}<\frac{1}{C}$, so that

$\displaystyle S_{1}=\frac{\lambda|G(\lambda,n)|}{n}+\frac{1}{n^{2}}\sum_{j\in
G(\lambda,n)}\log(p_{j})+O(\frac{1}{n}),$ (14)

where the implied constant depends only on $C$.

For $j\in G(\lambda,n)$, define $\lambda_{j}$ such that
$p_{j}=Ce^{-\lambda_{j}n}$ and set $\lambda_{0}:=0$ (note that
$\lambda_{j}>0$, all $j$, since $G(\lambda,n)=\emptyset$ for $\lambda\leq 0$).
Then

$\displaystyle\frac{1}{n^{2}}\sum_{j\in
G(\lambda,n)}\log(p_{j})=\frac{\log(C)|G(\lambda,n)|}{n^{2}}-\frac{1}{n}\sum_{j\in
G(\lambda,n)}\lambda_{j}$
$\displaystyle=-\frac{1}{n}\sum_{j=1}^{M}\lambda_{j}+O(\frac{1}{n}),$

where $M=\max\\{j:j\in G(\lambda,n)\\}$.

Define

$g_{n}(\lambda):=\frac{|G(\lambda,n)|}{n}.$

Then, since the $\lambda_{j}$ are in increasing order,
$g_{n}(\lambda_{j+1})-g_{n}(\lambda_{j})=\frac{1}{n}$ for each $j$ and
$g_{n}(\lambda_{1})=0$, by summation by parts,

$\displaystyle-\frac{1}{n}\sum_{j=1}^{M}\lambda_{j}=-\frac{\lambda_{M}}{n}-\sum_{j=1}^{M-1}\lambda_{j}(g_{n}(\lambda_{j+1})-g_{n}(\lambda_{j}))$
$\displaystyle=-\frac{\lambda_{M}}{n}+\lambda_{1}g_{n}(\lambda_{1})-\lambda_{M-1}g_{n}(\lambda_{M})+\sum_{j=2}^{M-1}g_{n}(\lambda_{j})(\lambda_{j}-\lambda_{j-1})$
$\displaystyle=-\frac{\lambda_{M}}{n}-\lambda_{M}g_{n}(\lambda_{M})+\sum_{j=1}^{M}g_{n}(\lambda_{j})(\lambda_{j}-\lambda_{j-1}).$

Since

$\sum_{j=1}^{M}g_{n}(\lambda_{j})(\lambda_{j}-\lambda_{j-1})=\int_{0}^{\lambda_{M}}g_{n}(x)dx,$

we thus have that

$\displaystyle S_{1}=\lambda
g_{n}(\lambda)-\lambda_{M}g_{n}(\lambda_{M})+\int_{0}^{\lambda_{M}}g_{n}(x)dx+O(\frac{1}{n})$
$\displaystyle=\lambda_{M}(g_{n}(\lambda)-g_{n}(\lambda_{M}))+\int_{0}^{\lambda}g_{n}(x)dx+O(\frac{1}{n})$
$\displaystyle=\int_{0}^{\lambda}g_{n}(x)dx+O(\frac{1}{n}).$

As $n\to\infty$, by Theorem 2.3, the limit

$\lim_{n\to\infty}\frac{1}{n^{2}}\log\mathbb{E}[e^{\lambda
n\mathcal{N}_{\theta}}]=\lim_{n\to\infty}\int_{0}^{\lambda}g_{n}(x)dx=\Lambda(\lambda)$
(15)

exists for all $\lambda$. Notice that since we don’t have any additional
information about the behavior of $g_{n}(x)$ for each fixed $x\in[0,\lambda]$,
we can’t interchange the integration with a limit in (15).

For a fixed $\varepsilon>0$, (15) implies

$\int_{\lambda-\varepsilon}^{\lambda+\varepsilon}g_{n}(x)dx\underset{n\rightarrow\infty}{\longrightarrow}\Lambda(\lambda+\varepsilon)-\Lambda(\lambda-\varepsilon),$

therefore we immediately get

$\frac{1}{2\varepsilon}\int_{\lambda-\varepsilon}^{\lambda+\varepsilon}|G(x,n)|dx=\frac{n}{2\varepsilon}(\Lambda(\lambda+\varepsilon)-\Lambda(\lambda-\varepsilon))-o(n).$
(16)

The function $\Lambda(\lambda)$, according to Theorem 2.3, is given by

$\Lambda(\lambda)=\underset{y\in[0,1]}{\sup}\\{\lambda y-J(y)\\}$

with $J(y)=\iint\log\frac{1}{|z-\zeta|}d\nu^{*}_{y}(z)d\nu^{*}_{y}(\zeta)$,
where $\nu^{*}_{y}$ is a measure from Theorem 1.4. It is easy to check that
the left-hand derivative of $J(x)$ at $x=1$ exists, therefore, for all
$\lambda\gg J^{\prime}(1)$ and $\varepsilon$ sufficiently small, (16) becomes

$\frac{1}{2\varepsilon}\int_{\lambda-\varepsilon}^{\lambda+\varepsilon}|G(x,n)|dx=n-o(n)$

that completes the proof. ∎

##  3 Solution to Energy problem

Let $\Sigma\subset\mathbb{C}$ be a compact subset of the complex plane and
$\mathcal{P}(\Sigma)$ the collection of all positive unit Borel measures
supported on $\Sigma$. For a given $0<\theta<2\pi$ we use the notations
$A_{\theta}:=\\{z\in\mathbb{S}^{1}:-\frac{\theta}{2}\leq\arg
z\leq\frac{\theta}{2}\\}$,
$A^{c}_{\theta}:=\\{z\in\mathbb{S}^{1}:\frac{\theta}{2}\leq\arg z\leq
2\pi-\frac{\theta}{2}\\}$ for the corresponding subarcs of the unit circle.

The logarithmic energy $I(\mu)$ and the logarithmic potential $U^{\mu}$ of a
measure $\mu\in\mathcal{P}(\Sigma)$ are defined, respectfully, as

$I(\mu):=\iint\log\frac{1}{|z-\zeta|}d\mu(z)d\mu(\zeta),$ (17)

$U^{\mu}(z):=\int\log\frac{1}{|z-\zeta|}d\mu(\zeta).$ (18)

In this section we study the following two related problems:

Problem I. Given $q$ and $\beta$, with $0<q<1$, $-1<\beta<1$, determine a
measure $\mu\in\mathcal{P}([-1,1])$ that minimizes the energy $I(\mu)$,
subject to the constraint $\mu([\beta,1])=q$.

Problem II. Given $q$ and $\theta$, with $0<q<1$, $0<\theta<2\pi$, determine a
measure $\nu\in\mathcal{P}(\mathbb{S}^{1})$ that minimizes the energy
$I(\nu)$, subject to the constraint $\nu(A_{\theta})=q$.

The limiting version of the Problems I and II with constraints on the mass at
fixed points, $\beta=1$ and $\theta=0$, respectively, were already treated in
a couple of works. The description of the equilibrium charge distribution of
amount $1-q$ on the unit circle when a fixed charge amount $q>0$ is placed at
$t=1$ was done in [9] by M. Lachance, E. B. Saff and R.S. Varga. A similar
problem on an interval was studied in [15] by E. B. Saff, J.L. Ullman and R.S.
Varga, where the the equilibrium charge distribution of amount $1-q_{1}-q_{2}$
was determined when a charge amount $q_{1}>0$ is placed at $t=1$, and a charge
amount $q_{2}>0$ is placed at $t=-1$, $q_{1}+q_{2}<1$. Thus, the solution to
the Problems I and II, given by Theorems 3.1 and 1.4 below, can be considered
as an extension of the results obtained in [9] and [15] (with $q_{2}=0$),
respectively. We show that in the limiting case, when $\beta\rightarrow 1$ in
Problem I, our result recovers the one from [15] .

######  Theorem 3.1.

(A. Martínez-Finkelshtein, E.B. Saff, [11, Section 4])

The measure $\mu^{*}=:\mu^{*}_{q}\in\mathcal{P}([-1,1])$ such that
$I(\mu^{*})=\inf\\{I(\mu):\mu\in\mathcal{P}([-1,1]),\mu([\beta,1])=q\\}$ is
unique and

i) if $q\geq\frac{1}{\pi}\int_{\beta}^{1}\frac{1}{\sqrt{1-x^{2}}}dx$, is given
by

$d\mu^{*}(x)=\frac{\sqrt{|x-\alpha|}}{\pi\sqrt{|(x+1)(x-\beta)(x-1)|}}dx,\;\;\;$
(19)

where $x\in[-1,\alpha]\cup[\beta,1]$, $\alpha\leq\beta$, and with $\alpha$
determined from the equation

$\int_{-1}^{\alpha}\frac{\sqrt{|x-\alpha|}}{\pi\sqrt{|(x+1)(x-\beta)(x-1)|}}dx=1-q;$
(20)

ii) if $q\leq\frac{1}{\pi}\int_{\beta}^{1}\frac{1}{\sqrt{1-x^{2}}}dx$, is
given by (19), where $x\in[-1,\beta]\cup[\alpha,1]$, $\alpha\geq\beta$, and
$\alpha$ is the solution to the equation

$\int_{-1}^{\beta}\frac{\sqrt{|x-\alpha|}}{\pi\sqrt{|(x+1)(x-\beta)(x-1)|}}dx=1-q.$

Note that in the trivial case, when
$q=\frac{1}{\pi}\int_{\beta}^{1}\frac{1}{\sqrt{1-x^{2}}}dx$, we get
$\alpha=\beta$ and so Theorem 3.1 gives, as expected, the arcsine
distribution, $d\mu^{*}(x)=\frac{1}{\pi\sqrt{1-x^{2}}}$.

######  Corollary 3.2.

When $\beta\rightarrow 1$, the density function in (19) becomes

$d\mu^{*}(x)=\frac{\sqrt{|x-\alpha|}}{\pi\sqrt{(x+1)}(1-x)}dx,\;\;\;x\in[-1,\alpha],$

where $\alpha$ is the solution to the equation

$\int_{-1}^{\alpha}\frac{\sqrt{|x-\alpha|}}{\pi\sqrt{(x+1)}(1-x)}dx=1-q.$ (21)

This is precisely the measure obtained in [15] that corresponds to the case
$q_{1}=q$, $q_{2}=0$.

Once Theorem 3.1 is verified, the solution to the Problem II is immediate, as
we show in Section 3.3.

###  3.1 Auxiliary results

We call $\mu\in\mathcal{P}([a,b])$ the unconstrained equilibrium measure on an
interval $[a,b]$ if $\mu$ minimizes the logarithmic energy $I(\mu)$ among all
unit measures in $\mathcal{P}([a,b])$. It is well-known (see e.g. [14, Section
I.3]) that

$d\mu(x)=\frac{1}{\pi}\frac{dx}{\sqrt{(x-a)(b-x)}},\;\;\;x\in[a,b].$

The measure $\mu^{*}$ that minimizes the energy in Problem I can be viewed as
the measure of the form

$\mu^{*}=(1-q)\mu^{*}_{1}+q\mu^{*}_{2},$

where $\mu^{*}_{1}\in\mathcal{P}([-1,\beta])$ and
$\mu^{*}_{2}\in\mathcal{P}([\beta,1])$. In order to characterize the measure
$\mu^{*}$, we refer to [14, Theorem VIII.2.1], which states that a measure
$\mu$ is globally optimal if and only if each component of $\mu$ is optimal
when the other one is kept fixed and regarded as external field. Although the
theorem is originally formulated for the case of closed sets of positive
distance from one another, it is also valid in our setting, when two closed
intervals have disjoint interiors. According to [14, Theorem VIII.2.1], there
exist constants $F_{1}$, $F_{2}$, such that the potential of the measure
$\mu^{*}$ satisfies

$U^{\mu^{*}}(z)\geq F_{1},\;\;\textnormal{q.e.
on}\;[-1,\beta],\;\;\;\;\;U^{\mu^{*}}(z)=F_{1},\;\;\textnormal{q.e.
on}\;\textnormal{supp}\>\mu^{*}_{1},$ (22)

$U^{\mu^{*}}(z)\geq F_{2},\;\;\textnormal{q.e.
on}\;[\beta,1],\;\;\;\;\;U^{\mu^{*}}(z)=F_{2},\;\;\textnormal{q.e.
on}\;\textnormal{supp}\>\mu^{*}_{2}.$ (23)

We remark that a similar reasoning can be applied to characterize the optimal
measure $\nu^{*}\in\mathbb{S}^{1}$ in Problem II.

Our next goal is to show that the measure $\mu^{*}$ is absolutely continuous
with respect to the measure $\sigma_{1}+\sigma_{2}$, where $\sigma_{1}$,
$\sigma_{2}$ are (unconstrained) equilibrium measures on $[-1,\beta]$ and
$[\beta,1]$, respectively, using the following result of de La Vallée Poussin.

######  Theorem 3.3.

(see [14, Theorem IV.4.5.])

Let $\mu$ and $\nu$ be two measures of compact support, and let $\Omega$ be a
domain in which both $U^{\mu}$, $U^{\nu}$ are finite and satisfy with some
constant $c$ the inequality

$U^{\mu}(z)\leq U^{\nu}(z)+c,\;\;\;z\in\Omega.$ (24)

If $A$ is the subset of $\Omega$ in which equality holds in (24), then
$\nu|_{A}\leq\mu|_{A}$; that is, for every Borel set $B\subset A$,
$\nu(B)\leq\mu(B)$.

######  Lemma 3.4.

The measures $\mu^{*}_{1}$ and $\mu^{*}_{2}$ that solve Problem I are
absolutely continuous with respect to the equilibrium measure $\sigma_{1}$ on
$[-1,\beta]$ and, respectively, equilibrium measure $\sigma_{2}$ on
$[\beta,1]$.

###### Proof.

Let $\sigma_{1}\in\mathcal{P}([-1,\beta])$ be the (unconstrained) equilibrium
measure on $[-1,\beta]$. Then $\sigma_{1}$ satisfies

$U^{\sigma_{1}}=C,\;\;\textnormal{ on}\;[-1,\beta],$ (25)

where $C$ is an explicitly known constant. Combining (25) with (22), we obtain

$U^{\sigma_{1}}=U^{\mu^{*}}+C-F_{1},\;\;\;\textnormal{q.e.
on}\;\textnormal{supp}\>\mu^{*}_{1},$ $U^{\sigma_{1}}\leq
U^{\mu^{*}}+C-F_{1},\;\;\;\textnormal{q.e. on}\;[-1,\beta].$ (26)

Since $\|\sigma_{1}\|=\|\mu^{*}\|=1$, the principle of domination for
logarithmic potentials (see, e.g. [14, Theorem II.3.2]) implies that the
inequality (26) holds for every $z\in\mathbb{C}$. By Theorem 3.3, we have
$(1-q)\mu^{*}_{1}\leq\sigma_{1}$. Similarly, we can use the (unconstrained)
equilibrium measure $\sigma_{2}\in\mathcal{P}([\beta,1])$ on $[\beta,1]$ to
show that $q\mu^{*}_{2}\leq\sigma_{2}$. ∎

###  3.2 Proof of Theorem 3.1

###### Proof.

Step 1. Determining the support of $\mu^{*}$.

Let $\sigma\in\mathcal{P}([-1,1])$ denote the (unconstrained) equilibrium
measure on $[-1,1]$ and let $m_{1}:=\sigma([-1,\beta])$,
$m_{2}:=\sigma([\beta,1])$. If $q=m_{2}$, then it’s clear that
$\mu^{*}=\sigma$ and $\textnormal{supp}\;\mu^{*}=[-1,1]$.

We will now assume that $q>m_{2}$ (the proof for the case $q<m_{2}$ is
analogous). We shall show that in this case the support of the minimizer is a
union of two intervals, $[-1,\alpha_{0}]\cup[\beta_{0},1]$,
$\alpha_{0}<\beta_{0}$. For this purpose, we denote as before the restriction
of the measure $\mu^{*}$ to $[-1,\beta]$ and $[\beta,1]$ by $(1-q)\mu^{*}_{1}$
and $q\mu^{*}_{2}$, respectively. First, notice that $U^{\mu^{*}_{2}}(z)$ is a
convex function on $[-1,\beta)$, which follows immediately from

$\frac{d^{2}U^{\mu^{*}_{2}}(z)}{dz^{2}}=\int\frac{d\mu^{*}_{2}(\zeta)}{(z-\zeta)^{2}}\geq
0,\;\;\;\;z\in[-1,\beta).$

Since the measure $\mu^{*}_{1}$ is the solution to the equilibrium problem on
$[-1,\beta]$ with the convex external field $U^{\mu^{*}_{2}}(z)$, then
$\textnormal{supp}\>\mu^{*}_{1}\cap(-1,\beta)$ is an interval (see [14,
Theorem IV.1.10]). By interchanging $\mu^{*}_{1}$ and $\mu^{*}_{2}$ and
repeating the argument, we obtain that
$\textnormal{supp}\>\mu^{*}_{2}\cap(\beta,1)$ is also an interval. It remains
to show that $\alpha_{0}<\beta_{0}$. On the contrary, suppose that
$\alpha_{0}=\beta=\beta_{0}$. Then, by (22) and (23), we have
$U^{\mu^{*}}(z)=F_{1}=F_{2}$ on $[-1,1]$, which implies $\mu^{*}=\sigma$ and
contradicts the assumption $q>m_{2}$.

Step 2. Finding the density function of the minimizing measure.

We are going to apply the standard technique of analyzing the Cauchy transform
of the measure $\mu^{*}$ (see, for example, [14, Section VIII.5]). Namely,
consider the function

$H(z)=\int\frac{d\mu^{*}(\zeta)}{z-\zeta}$ (27)

on the Riemann sphere $\mathbb{\overline{C}}$ cut along
$\textnormal{supp}\>\mu^{*}=[-1,\alpha_{0}]\cup[\beta_{0},1]$. The real part
of $H$ vanishes on the cut since it is the derivative of the equilibrium
potential of $\mu^{*}$ and the latter is constant on
$\textnormal{supp}\>\mu^{*}$. Since $H(\overline{z})=\overline{H(z)}$ and
$H(z)$ is continuous on $(-1,\alpha_{0})\cup(\beta_{0},1)$, the function
$H^{2}(z)$ is analytic on $(-1,\alpha_{0})\cup(\beta_{0},1)$ and therefore
analytic on $\mathbb{\overline{C}}\setminus\\{-1,\alpha_{0},\beta_{0},1\\}$.
Using Lemma 3.4 one can easily verify that $H^{2}(z)$ can have at most simple
poles at the points $\\{-1,\alpha_{0},\beta_{0},1\\}$, and so $H^{2}(z)$ is a
rational function on $\mathbb{\overline{C}}$. Moreover, because
$H^{2}(z)\sim\frac{1}{z^{2}}$ when $z\rightarrow\infty$, we must have

$H^{2}(z)=\frac{(z-A)(z-B)}{(z+1)(z-\alpha_{0})(z-\beta_{0})(z-1)},\;\;\;A,B\in\mathbb{R},$

and thus

$H(z)=\bigg{(}\frac{(z-A)(z-B)}{(z+1)(z-\alpha_{0})(z-\beta_{0})(z-1)}\bigg{)}^{1/2},$

where we take the branch of the square root that is positive for positive $z$.

The function $H(z)$ on the upper part of the cut has the form

$H(z)=\frac{-i\sqrt{|z-A||z-B|}}{\sqrt{|(z+1)(z-\alpha_{0})(z-\beta_{0})(z-1)|}},\;\;\;z\in\textnormal{supp}\>\mu^{*}.$

The Cauchy’s formula gives

$H(z)=\frac{1}{2\pi
i}\oint_{\textnormal{supp}\>\mu^{*}}\frac{H(\zeta)}{\zeta-z}d\zeta=\frac{1}{\pi
i}\int_{\textnormal{supp}\>\mu^{*}}\frac{H(y)}{y-z}dy,$ (28)

where the second integral is taken on the upper part of the cut. Consequently,
(27) together with (28) imply

$d\mu^{*}(y)=\frac{\sqrt{|y-A||y-B|}}{\pi\sqrt{|(y+1)(y-\alpha_{0})(y-\beta_{0})(y-1)}|}dy.$
(29)

Our next goal is to show that in the expression (29) of the density function
$d\mu^{*}(y)$ the constants $A,B$ must be equal to $\alpha_{0}$. Indeed, if we
assume that $A\neq B\neq\alpha_{0}$ and then consider for
$x\in(\alpha_{0},\beta_{0})$

$\frac{dU^{\mu^{*}}(x)}{dx}=-\frac{1}{\pi}\displaystyle\int\limits_{[-1,\alpha_{0}]\cup[\beta_{0},1]}\frac{1}{x-y}\frac{\sqrt{|y-A||y-B|}}{\sqrt{|(y+1)(y-\alpha_{0})(y-\beta_{0})(y-1)}|}dy,$

it’s easy to see that $\frac{dU^{\mu^{*}}(x)}{dx}\rightarrow-\infty$ when
$x\rightarrow\alpha_{0}^{+}$. If $\alpha_{0}<\beta$, then this contradicts
(22), and we conclude that $A=B=\alpha_{0}$. Next, we show that the case
$\alpha_{0}=\beta$ is impossible under the assumption $q>m_{2}$. To show this,
assume the contrary, $\alpha_{0}=\beta$. Then it implies that $A,B$ in (29)
should be equal to $\beta_{0}$, since otherwise we would have
$\frac{dU^{\mu^{*}}(x)}{dx}\rightarrow\infty$ when
$x\rightarrow\beta_{0}^{-}$, that contradicts (23). Thus, the density function
in this case is

$d\mu^{*}(y)=\frac{\sqrt{|y-\beta_{0}|}}{\pi\sqrt{(y+1)(y-\alpha_{0})(y-1)}}dy,$

and, since $\alpha_{0}\leq\beta_{0}$,

$\frac{1}{\pi}\int_{\beta_{0}}^{1}\frac{\sqrt{|y-\beta_{0}|}}{\sqrt{(y+1)(y-\alpha_{0})(y-1)}}dy\leq\frac{1}{\pi}\int_{\beta_{0}}^{1}\frac{dy}{\sqrt{(y+1)(y-1)}}=m_{2},$

that contradicts the assumption $\mu^{*}([\beta,1])=q>m_{2}$ and proves that
we necessarily have $\alpha_{0}<\beta$ as well as $\beta_{0}=\beta$, that
completes the proof.

∎

Figure 1: Graph showing the relationship between the parameters $\alpha$,
$\beta$ and $q$.

###  3.3 Proof of Theorem 1.4

###### Proof.

For a measure $\mu\in\mathcal{P}([-1,1])$ with density $f(x)$, consider its
logarithmic potential

$U^{\mu}(z)=\int_{-1}^{1}\log\frac{1}{|z-x|}f(x)dx$ (30)

Consider the Joukowski map $z=\Psi(\zeta):=\frac{1}{2}(\zeta+\zeta^{-1})$ that
maps the exterior of the unit circle, conformally to
$\mathbb{C}\setminus[-1,1]$. We extend continuously the function $\Psi(\zeta)$
to the unit circle $\mathbb{S}^{1}$ and use the same notation for the
extension. For $z=\Psi(\zeta)\in\mathbb{C}\setminus[-1,1]$ the potential (30)
can be written as

$\begin{split}U^{\mu}(z)&=\int_{-\pi}^{\pi}\log\frac{1}{|\Psi(\zeta)-\Psi(e^{i\psi})|}\widetilde{f}(\psi)d\psi\\\
&=\int_{-\pi}^{\pi}\log\bigg{|}\frac{\zeta-e^{i\psi}}{\Psi(\zeta)-\Psi(e^{i\psi})}\bigg{|}\widetilde{f}(\psi)d\psi+\int_{-\pi}^{\pi}\log\frac{1}{|\zeta-e^{i\psi}|}\widetilde{f}(\psi)d\psi,\end{split}$
(31)

where $x=e^{i\psi}$ and $\widetilde{f}$ is an even function on
$\mathbb{S}^{1}$ with $\int_{\mathbb{S}^{1}}\widetilde{f}d\psi=1$. Let us
denote $d\nu(e^{i\psi}):=\widetilde{f}(\psi)d\psi$. Then we have

$U^{\mu}(z)=h(\zeta)+U^{\nu}(\zeta),\;\;z=\Psi(\zeta)\in\mathbb{C}\setminus[-1,1].$

Since the function

$g(\omega)=\log\bigg{|}\frac{\omega-\zeta}{\Psi(\omega)-\Psi(\zeta)}\bigg{|}$

is harmonic in $\Delta:=\\{z\in\mathbb{C}:|z|>1\\}$ and continuous on the
boundary, we have

$\begin{split}\underset{\zeta\rightarrow
e^{i\varphi}}{\lim}h(\zeta)&=\int_{-\pi}^{\pi}\log\bigg{|}\frac{e^{i\varphi}-e^{i\psi}}{\Psi(e^{i\varphi})-\Psi(e^{i\psi})}\bigg{|}\widetilde{f}(\psi)d\psi\\\
&=\int_{-\pi}^{\pi}\log\frac{1}{|e^{-i\varphi}-e^{i\psi}|}\widetilde{f}(\psi)d\psi+\log
2.\end{split}$ (32)

If $U^{\mu}(z)$ is continuous in $\mathbb{C}$, then in view of (32) we have

$U^{\mu}(\Psi(e^{i\varphi}))=\underset{z\rightarrow\Psi(e^{i\varphi})}{\lim}U^{\mu}(z)=\underset{\zeta\rightarrow
e^{i\varphi}}{\lim}(h(\zeta)+U^{\nu}(\zeta))=2U^{\nu}(e^{i\varphi})+\log 2,$
(33)

where the last equality follows from (32) together with the fact that
$U^{\nu}(\zeta)=U^{\nu}(\overline{\zeta})$, $\zeta\in\overline{\Delta}$.

Now, if we take $\mu$ above to be the measure from (19) with
$\beta=\cos(\frac{\theta}{2})$ and
$q\geq\frac{1}{\pi}\int_{\beta}^{1}\frac{1}{\sqrt{1-x^{2}}}dx=\frac{\theta}{2\pi}$,
then it’s clear that the potential $U^{\mu^{*}}(z)$ is continuous in
$\mathbb{C}$, and

$U^{\mu^{*}}(z)=\frac{1}{2\pi}\int_{S}\log\frac{1}{|z-\Psi(e^{i\psi})|}\frac{\sqrt{|\cos(\psi)-\alpha|}}{\sqrt{|\cos(\psi)-\beta|}}d\psi,$

where the set
$S:=A_{\theta}\cup\\{z\in\mathbb{S}^{1}:\arccos(\alpha)\leq\arg\;z\leq
2\pi-\arccos(\alpha)\\}$, and $\alpha$ is defined by (8). Consequently, (33)
and [14, Theorem VIII.2.1] imply that the measure $\nu^{*}$, where
$d\nu^{*}(e^{i\psi})=\frac{1}{2\pi}\frac{\sqrt{|\cos(\psi)-\alpha|}}{\sqrt{|\cos(\psi)-\beta|}}d\psi$,
is the solution to the Problem II. ∎

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