# Is ChatGPT Involved in Texts? Measure the Polish Ratio to Detect ChatGPT-
Generated Text

Lingyi Yang1, Feng Jiang 1 2 3, Haizhou Li 1 2
1 School of Data Science, The Chinese University of Hong Kong, Shenzhen, China
2 Shenzhen Research Institute of Big Data, China
3 School of Information Science and Technology, University of Science and
Technology of China, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Corresponding Author

###### Abstract

The remarkable capabilities of large-scale language models, such as ChatGPT,
in text generation have impressed readers and spurred researchers to devise
detectors to mitigate potential risks, including misinformation, phishing, and
academic dishonesty. Despite this, most previous studies have been
predominantly geared towards creating detectors that differentiate between
purely ChatGPT-generated texts and human-authored texts. This approach,
however, fails to work on discerning texts generated through human-machine
collaboration, such as ChatGPT-polished texts. Addressing this gap, we
introduce a novel dataset termed HPPT (ChatGPT-polished academic abstracts),
facilitating the construction of more robust detectors. It diverges from
extant corpora by comprising pairs of human-written and ChatGPT-polished
abstracts instead of purely ChatGPT-generated texts. Additionally, we propose
the "Polish Ratio" method, an innovative measure of the degree of modification
made by ChatGPT compared to the original human-written text. It provides a
mechanism to measure the degree of ChatGPT influence in the resulting text.
Our experimental results show our proposed model has better robustness on the
HPPT dataset and two existing datasets (HC3 and CDB). Furthermore, the "Polish
Ratio" we proposed offers a more comprehensive explanation by quantifying the
degree of ChatGPT involvement.

##  1 Introduction

From GPT-2 [1] to GPT-3 [2], the emergence of Large Language Models (LLM) has
made it possible that machines can generate a variety of high-quality texts
that are quite similar to human language, making it hard to distinguish
between human-generated and AI-generated texts. The release of ChatGPT [3]
contributed to the widespread use of AI-generated text tools while leading to
concerns that the abuse of ChatGPT may bring misleading information, phishing,
and academic dishonesty [4]. Therefore, some studies [5] attempt to build
detectors based on the pre-trained language model for ChatGPT-generated texts
to prevent the abuse of the powerful ChatGPT.

However, detectors that perform well in intra-domain scenarios have poor
robustness due to being easily attacked by carefully designed strategies such
as paraphrasing and polishing [6], which refers to the process of revising and
improving the original text. Although retrieval [7] can help mitigate this
problem, it still poses significant challenges, such as high costs and lack of
explanation. Moreover, as pointed out by the Association for Computational
Linguistics (ACL), it is necessary to establish a grading system for the
degree to which large models participate in the paper generation process
111https://2023.aclweb.org/blog/ACL-2023-policy.

In order to identify ChatGPT-polished texts and provide users with more
intuitive explanations, we create a novel dataset called HPPT (ChatGPT-
polished academic abstracts instead of fully generated ones) for training a
detector and also propose the Polish Ratio method which measures the degree of
modification made by ChatGPT compared to the original human-written text.
Through experiments, we have demonstrated the effectiveness of our model in
accurately identifying polished texts trained on our novel dataset. Moreover,
our explanation method, the Polish Ratio, has shown promising results on both
our own dataset and other datasets that have not been seen before: there are
significant distinct distributions in the predicted Polish Ratio of human-
written, ChatGPT-polished, and ChatGPT-generated texts. Our contributions are:
a) we build a Human-ChatGPT Polished Paired abstracT (HPPT) dataset,
containing polished and original paired abstracts with similarity measurements
as the degree of polishing. b) We propose the Polish Ratio to help explain the
detection model indicating the modification degree of the text by ChatGPT. c)
The experimental results show that our model performs better than other
baselines on three datasets. The code and dataset are available:
https://github.com/Clement1290/ChatGPT-Detection-PR-HPPT.

##  2 Related Work

Along with the appearance of large language models such as ChatGPT, some
detection algorithms are proposed to prevent the abuse of such powerful AI-
generated text models. Recent detection methods can be roughly grouped into
two categories [8]: white-box and black-box detectors. The white-box detector
needs to access the distributed probability or vocabulary of the target
language model, while the black-box detector only checks the output text of
the target model. Table 1 summarizes recent algorithms that detect ChatGPT-
generated texts.

Detector | Open Source | Black Box | Detection basis | Explanation
---|---|---|---|---
| DetectGPT
---
[9]
✓ | ✗ | Probability Score | /
| Watermarks
---
[10]
✓ | ✗ | Red & Green Tokens | /
| Roberta-based-detector
---
[5]
✓ | ✓ | Output String | /
| OpenAI
---
[11]
✗ | ✓ | Output String | /
| GPTZero
---
✗ | ✓ | Output String | /
| DistilBERT-based-detector
---
[12]
✗ | ✓ | Output String | SHAP
| Ours
---
✓ | ✓ | Output String | Polish Ratio

Table 1: Summary of recent algorithms detecting ChatGPT-generated texts.

[13] is a popular white-box detector, which takes advantage of the probability
and entropy of the linguistic patterns, while DetectGPT [9] utilizes the
observation that ChatGPT texts tend to lie in areas where the log probability
function has negative curvature to conduct zero-shot detection. Watermark-
based detection methods [10, 14] need to divide the vocabulary into a
randomized set of words that is softly prompted to use during sampling
("green" tokens) and the remaining words ("red" tokens), and modify the logits
embedding when the model generates the results.

Training another classifier as the detector with labeled data [11, 12, 5] is a
mainstream black-box method. [5] finetunes the Roberta on the HC3 (Human
ChatGPT Comparison Corpus) dataset to obtain an effective detector. However,
the simple ChatGPT-generated texts in the HC3 dataset make the model trained
on it vulnerable to being attacked using the polishing strategy, and the
robustness is not ensured. Moreover, some effective detectors are not open-
source and are directly used for commercial operations, such as GPTZero
222https://gptzero.me/ and OriginalityAI 333https://originality.ai/.

On the other hand, the existing black-box detectors rarely provide
explanations for the prediction. [15] proposed Local Interpretable Model-
agnostic Explanations (LIME) to explain the predictions of any classifier in
an interpretable and faithful manner by learning an interpretable model
locally around the prediction. [16] proposed the SHapley Additive exPlanations
(SHAP) method to assign each feature an importance value for a particular
prediction. Although they perform effectively on basic NLP tasks such as
sentiment analysis [17], they do not provide a convincing explanation to users
in ChatGPT detection task [12], compelling us to step forward and address this
gap.

##  3 Method

To facilitate detecting ChatGPT-polished texts and offer more intuitive
explanations to assist final judgment, we first collect human-written
abstracts and polish all of them using ChatGPT forming Human-ChatGPT Polished
Paired abstracT (HPPT) dataset. Additionally, for each abstract pair in the
dataset, we furnish three different similarity metrics (Jaccard Distance,
Levenshtein Distance, and BERT semantic similarity) between the human-written
abstract and the corresponding abstract polished by ChatGPT. Based on the
data, we train the Roberta model as the detector to conduct the detection
task. Meanwhile, we also train the Polish Ratio model to explain the detection
result to reveal the degree of ChatGPT involvement. The overall detection
process is shown in Figure 1.

Figure 1: The study design of our detection method.

###  3.1 HPPT Dataset: Human-ChatGPT Polished Paired abstractT

Since the existing datasets are specifically designed for purely ChatGPT-
generated detection, such as HC3 [5] and ignore the ChatGPT-polished texts, we
first construct the Human-ChatGPT Polished Paired abstracT (HPPT) dataset.
Specifically, we collect human-written abstracts of accepted papers from
several popular NLP academic conferences and polish all of them using ChatGPT
444The prompt is ”please polish the following sentences:¡abstracts¿”. We also
tested the prompt ”please rewrite the following sentences:¡abstracts¿” and
found that there is no big difference using ”polish” or ”rewrite”. The
differences of Levenshtein Distance or Jaccard Distance between using “polish”
and “rewrite” for most sample pairs are within the range of $0.1$.. The texts
in our dataset are paired, making it easy to observe the difference between
human-written and ChatGPT-polished texts. Overall, we collect 6050 pairs of
abstracts and corresponding polished versions from the ACL anthology (ACL,
EMNLP, COLING, and NAACL) in the past five years (2018-2022): 2525 are from
ACL, 914 are from EMNLP, 1572 are from COLING, and 1039 are from NAACL.

To uncover the distinctions between human-written and ChatGPT-polished texts,
we compute their similarities using three metrics: BERT semantic
similarity555BERT semantic similarity refers to the cosine similarity between
two sentences’ embeddings using the BERT model. Here we use the SciBERT model
[18]., Jaccard Distance, and Levenshtein Distance (details are presented in
section 3.3). As shown in Figure 2, the paired texts are semantically similar
in BERT semantic similarity, which shows it is difficult to distinguish human-
written and ChatGPT-Polished abstracts from BERT-based semantics. But Jaccard
Distance and Levenshtein Distance provide a better way to distinguish them as
they align closely with the Gaussian distribution, making them suitable for
measuring the degree of ChatGPT involvement.

Jaccard Distance

Levenshtein Distance

BERT semantic similarity

Figure 2: Similarity distribution of polished abstracts in our HPPT dataset.

###  3.2 Detection: Roberta-based black box model

Following the previous work [5], we treat the detection as a binary
classification task and build a black-box detector utilizing the Roberta model
[19] because the model of ChatGPT is not accessible, as discussed in the
Section 2. Unlike them, we regard the original abstract without polishing as
human-written, and its corresponding ChatGPT polished abstract is regarded as
ChatGPT involved.

###  3.3 Explanation

Due to the black box nature of many existing detectors, they are unable to
provide explanations that are crucial for differentiating ChatGPT-Polished
text. Therefore, we employ two independent explanation methods GLTR (Giant
Language model Test Room) and Polish Ratio (PR). This approach is favored over
posterior explanation methods such as SHAP, ensuring more comprehensive
justifications for our final judgments.

  * $\bullet$

GLTR: Giant Language model Test Room [13] is a statistical method used for AI-
generated text detection (mainly for GPT-2). It assumes that human-written
text uses a wider subset of the distribution under a model. Although it was
initially designed for GPT-2-generated text detection, we formulate a
hypothesis that the distribution of GPT-2 and ChatGPT-generated texts is
similar in some way since both are AI-generated texts. It uses three criteria
to help detection: (1) The probability of the word, as shown in Equation 1;
(2) The absolute rank of a word that is the rank of the Equation 1; (3) The
entropy of the predicted distribution, as shown in Equation 2.

$Prob_{i}=p_{det}(X_{i}|X_{1:i-1})$ (1)

$Entropy_{i}=-\sum_{w}p_{det}(X_{i}=w|X_{1:i-1})log(\\\
p_{det}(X_{i}=w|X_{1:i-1}))$ (2)

Specifically, if a text is written by a human, a word should have a low
probability, which leads to a higher top rank and the entropy also should be
large.

We consider the GLTR as our baseline for the explanation method as we have
discovered that the method is effective in explaining the distinction between
human-written and entirely ChatGPT-generated texts. However, it fails to
provide adequate explanations when ChatGPT-polished texts are present (refer
to Section 4.3 for further details). To address this issue, we adopt a new
method called the Polish Ratio.

  * $\bullet$

PR: Polish Ratio is a new metric we propose to measure the degree of ChatGPT
involvement for a text. It measures the degree of text modification before and
after revision using ChatGPT by calculating the similarity score between
original human-written texts and the corresponding polished texts. In our
dataset HPPT, we take two metrics Jaccard Distance and Levenshtein Distance
(Levenshtein Distance is normalized by the maximum length of the two
sequences) as the Polish Ratio. Since ChatGPT generates the text using tokens,
we evaluate the distances based on white-space tokenized words. As shown in
Equation 3, Jaccard distance measures the dissimilarity between sets by
comparing the size of their intersection and union. Levenshtein distance
measures the difference between two sequences by counting the minimum number
of edits (deletion, insertion, and substitution), as shown in Equation 4.

$J(A,B)=1-\frac{{|A\cap B|}}{{|A\cup B|}}$ (3)

Where:

    * –

$A$, $B$ are the set of words in the document text strings.

    * –

$|\cdot|$ represents the cardinality (number of elements) of the set.

$\text{lev}(a,b)=\begin{cases}\max(|a|,|b|)\ \ \text{if }\min(|a|,|b|)=0\\\
\min\left\\{\begin{aligned} &\text{lev}(a[2:],b[2:])+1,\\\
&\text{lev}(a[2:],b[1:])+1,\\\
&\text{lev}(a[1:],b[2:])+1\end{aligned}\right\\}\text{otherwise}\end{cases}$
(4)

Where:

    * –

$a$, $b$ represent a list of white-space tokenized words in the document.

    * –

$|\cdot|$ represents the length of the word list.

    * –

$\text{lev}(a[i:],b[j:])$ represents the Levenshtein distance between the
sublist of $a$ starting at index $i$ and $b$ starting at index $j$.

Therefore, we regard the PR model as the regression model where either the
Jaccard distance or normalized Levenshtein distance of the polished texts is
the target value of the Polish Ratio. In ideal conditions, the predicted PR
value of an abstract should approach 0 for a human-written one and should be
close to 1 when ChatGPT revises a majority of words in the abstract. We use
the Roberta-based model to extract feature $\mathbf{x_{i}}\in\mathbb{R}^{768}$
of each sentence $i$ and use an MLP to conduct the final regression task. The
sigmoid function is chosen as the activation function to ensure the range of
the regression result is in $[0,1]$. Compared to other explanation indices
like confidence level, our PR method takes advantage of the paired abstracts
before and after polishing to measure how much the ChatGPT involvement is,
which can give a more independent and convincing explanation.

##  4 Experiment and analysis

###  4.1 Experiment Setup

####  4.1.1 Dataset

We conduct experiments on the following three datasets to demonstrate the
effectiveness of our model.

  * •

HPPT

It is our built ChatGPT-polished dataset, which consists of 6050 pairs of
recent abstracts and corresponding polished versions. Meanwhile, to measure
the degree of ChatGPT involvement in the text, we also provide the Levenshtein
distance and Jaccard distance of the polished abstracts compared with their
corresponding human-written ones as the labeled PR value and label 0 as the PR
value of those human-written abstracts. We randomly partition the HPPT into
the train, test, and validation sets by $6:3:1$ to train and test our model
(Roberta-HPPT).

  * •

HC3 [5]

The HC3 (Human ChatGPT Comparison Corpus)
dataset666https://huggingface.co/datasets/Hello-SimpleAI/HC3/tree/main is one
of the most popular ChatGPT detection datasets containing question-answering
QA pairs. The human answers are collected from publicly available question-
answering datasets and wiki text, while the answers provided by ChatGPT are
obtained from its preview website through manual input of questions for each
interaction. We choose its English version corpus, which consists of 85,449 QA
pairs (24,322 questions, 58,546 human answers, and 26,903 ChatGPT answers). We
randomly partition the HC3 into the train, test, and validation sets by
$6:3:1$ and regard the answer text as the input of our detection model to
ensure the detector’s versatility. In addition, it is also used to train our
baseline model (Roberta-HC3).

  * •

CDB [20]

The ChatGPT-Detector-Bias
Dataset777https://huggingface.co/datasets/WxWx/ChatGPT-Detector-Bias (CDB) is
a mixed dataset consisting of 749 text samples derived from human, ChatGPT and
GPT-4. The human data is from TOEFL essays in a Chinese educational forum, US
College Admission essays, and scientific abstracts from Stanford’s course
CS224n. The GPT data consists of samples polished by both ChatGPT and GPT-4,
as well as essays fully generated by ChatGPT using different prompts. We also
take it as the harder dataset to test the detectors’ generalization ability
because it not only contains the GPT-4-generated or GPT-4-polished text but
also contains well-designed prompt engineering ChatGPT-generated text and the
human writing samples from both native and non-native English writers.

####  4.1.2 Reproduction details

During training our detection model, we use the batch size $16$, learning rate
$2e-5$ and the maximum epoch is set to 10. The model is chosen as the best one
on the validation set. While training our PR model, we use batch size $4$,
learning rate $1e-5$, and the maximum epoch is set to 15.

###  4.2 Detection Result

Table 2 shows several baselines and our model performance on three datasets.
Our model performs well on both in-domain dataset HPPT and out-of-domain
datasets (HC3 and CDB), suggesting that our model trained on the polished HPPT
dataset is more robust than other models. Although the Roberta-HPPT model is
only trained on HPPT, it achieves comparable performance compared to the SOTA
model in HC3, with only a 3% difference and better than DetectGPT. On the more
difficult dataset (CBD), our model significantly outperforms other baselines
with 88.15% accuracy. Specifically, our model only drops 6% on the out-of-
domain dataset while the Roberta-HC3 and the DetectGPT drop by nearly 40%,
demonstrating the strong robustness of our model. The reason is that our model
is trained on the ChatGPT-polished text instead of ChatGPT-generated text,
which can tackle more difficult samples such as GPT-4-generated,
GPT-4-polished, and well-designed prompt engineering ChatGPT-generated texts.

Test Dataset | ACC | AUROC
---|---|---
HPPT | HC3 | CDB | HPPT | HC3 | CDB
GPTZero | - | - | 0.4406 | - | - | 0.6818
| OpenAI-GPT-2
---
- | - | 0.4633 | - | - | 0.5604
OriginalityAI | - | - | 0.4967 | - | - | 0.5721
DetectGPT | 0.5129 | 0.8309 | 0.4593 | 0.6876 | 0.9058 | 0.7308
| Roberta-HC3
---
0.5285 | 0.9991 | 0.5848 | 0.7946 | 1 | 0.7526
| Roberta
---
-HPPT (ours)
0.9465 | 0.9671 | 0.8825 | 0.9947 | 0.9931 | 0.9518
Table 2: The detection performance of some popular detection models. The
evaluation metrics are Area under ROC curve (AUROC) and accuracy (ACC). The
results of GPTZero, OpenAl-GPT-2 [21], and OriginalityAI detectors are derived
from the data presented in [20].

###  4.3 Explanation Analysis

Although our model achieves high accuracy in detecting ChatGPT-polished texts,
it still needs an explanation of the degree of ChatGPT involvement in the
text. Motivated by this, we adopt two explanation methods (GLTR and Polish
Ratio) to measure them.

####  4.3.1 GLTR

(a) Sample 1: human-written

(b) Sample 2: ChatGPT-generated

(c) Sample 3: human-written

(d) Sample 4: ChatGPT-polished

Figure 3: The visualization result of some sample texts with the help of the
GLTR demo: http://gltr.io./dist/index.html. A word that ranks within the top
10 probability is highlighted in green, top 100 in yellow, top 1,000 in red,
and the rest in purple. Sample 1 and Sample 2 are chosen from the HC3 test
set, while Sample 3 and Sample 4 are chosen from the HPPT test set.

Figure 3 shows the visualization of the probability, absolute rank, and the
distribution’s entropy of two pairs of texts from HC3 and HPPT. A higher
concentration of green tokens indicates that the words are selected from the
top-ranked probability generated by the LLM system. Comparing samples 1 and 2,
we find that the difference between human-written and ChatGPT-generated texts
is noticeable. Texts entirely generated by ChatGPT prefer to use those top-
rank probability words, which validates our hypothesis in Section 3.3.
However, it is relatively hard to distinguish human-written and ChatGPT-
polished texts, as illustrated in samples 3 and 4. The reason is that the
fraction of top-rank probability words employed in polished texts decreases
significantly, approaching that of human-written texts. As a result, the GLTR
explanation mechanism proves challenging in this scenario.

####  4.3.2 Polish Ratio Regression

Therefore, we propose Polish Ratio to explain the detection result of both
entirely ChatGPT-generated and ChatGPT-polished texts. As shown in Table 3, we
try Jaccard distance and Levenshtein distance as the true label of the Polish
Ratio respectively. Considering the influence of outliers and the unbalanced
samples because of 0 PR ones (human-written abstracts), we try different loss
functions to test their effectiveness. Suppose the target value for the Polish
Ratio of each text sample $i$ is $y_{i}$ and the predicted value is
$\hat{y}_{i}$, the different losses $\ell_{i}$ for sample $i$ are defined as:

$\mathcal{L}_{1}(y_{i},\hat{y_{i}})=|y_{i}-\hat{y}_{i}|$ (5)
$Smooth\mathcal{L}_{1}(y_{i},\hat{y_{i}})=\begin{cases}0.5\cdot(y_{i}-\hat{y_{i}})^{2}&\text{if
}|y_{i}-\hat{y_{i}}|<\beta\\\
|y_{i}-\hat{y_{i}}|-0.5\cdot\beta&\text{otherwise}\end{cases}$ (6)
$\mathcal{L}_{MSE}(y_{i},\hat{y_{i}})=(y_{i}-\hat{y}_{i})^{2}$ (7)

where $\mathcal{L}_{1}$ is L1-loss, $Smooth\mathcal{L}_{1}$ is smooth L1-loss
and $\mathcal{L}_{MSE}$ is the mean square error while the loss for all $N$
samples is calculated as $\ell(y,\hat{y})=\frac{1}{N}\sum_{i=1}^{N}\ell_{i}$.
Through the experiment, we find that MSE is the best choice.

| Similarity metrics
---|---
Loss function | Jaccard distance | Levenshtein distance
L1-loss | 0.0837 | 0.0832
Smooth L1-loss ($\beta=0.1$) | 0.0779 | 0.0866
Mean Square Error | 0.0728 | 0.0813
Table 3: Comparison of mean absolute error (MAE) values for PR regression
using different loss functions and similarity metrics on the HPPT test set.
Lower values indicate better performance.

(a) Jaccard distance

(b) Levenshtein distance

Figure 4: Differences between predicted PR for human-written texts (HW), ChatGPT-polished texts (CP) and ChatGPT-generated texts (CG): HW and CP are directly from the HPPT testset where CG are from the HC3 testset. Text |  Source |  PR
---|---|---
1a) Theoretical work in morphological typology offers the possibility of measuring morphological diversity on a continuous scale. However, literature in Natural Language Processing (NLP) typically labels a whole language with a strict type of morphology, e.g. fusional or agglutinative. In this work, we propose to reduce the rigidity of such claims, by quantifying morphological typology at the word and segment level. We consider Payne (2017)’s approach to classify morphology using two indices: synthesis (e.g. analytic to polysynthetic) and fusion (agglutinative to fusional). For computing synthesis, we test unsupervised and supervised morphological segmentation methods for English, German and Turkish, whereas for fusion, we propose a semi-automatic method using Spanish as a case study. Then, we analyse the relationship between machine translation quality and the degree of synthesis and fusion at word (nouns and verbs for English-Turkish, and verbs in English-Spanish) and segment level (previous language pairs plus English-German in both directions). We complement the word-level analysis with human evaluation, and overall, we observe a consistent impact of both indexes on machine translation quality. |  Human |  0.14%
1b) Theoretical work in morphological typology provides a means of quantifying morphological diversity on a continuous scale. However, NLP literature often labels a language with a strict morphological type, such as fusional or agglutinative. This work aims to reduce the rigidity of these claims by quantifying morphological typology at both the word and segment level. We adopt Payne (2017)’s approach , which classifies morphology using two indices: synthesis (e.g. analytic to polysynthetic) and fusion (agglutinative to fusional). To compute synthesis, we evaluate unsupervised and supervised morphological segmentation methods for English, German, and Turkish. For fusion, we propose a semi-automatic method using Spanish as a case study. We examine the relationship between machine translation quality and synthesis and fusion at both the word level (nouns and verbs for English-Turkish and verbs in English-Spanish) and segment level (English-German in both directions). We supplement the word-level analysis with human evaluation, and we observe a consistent impact of both indices on machine translation quality. |  ChatGPT-polished |  34.48%
2a) Framing is a communication strategy to bias discussion by selecting and emphasizing. Frame detection aims to automatically analyze framing strategy. Previous works on frame detection mainly focus on a single scenario or issue, ignoring the special characteristics of frame detection that new events emerge continuously and policy agenda changes dynamically. To better deal with various context and frame typologies across different issues, we propose a two-stage adaptation framework. In the framing domain adaptation from pre-training stage, we design two tasks based on pivots and prompts to learn a transferable encoder, verbalizer, and prompts.In the downstream scenario generalization stage, the transferable components are applied to new issues and label sets. Experiment results demonstrate the effectiveness of our framework in different scenarios. Also, it shows superiority both in full-resource and low-resource conditions. |  Human |  0.14%
2b) The communication strategy of framing involves selecting and emphasizing certain aspects in order to bias discussion. To analyze this strategy, frame detection is used. However, previous works in this field have mainly focused on a single scenario or issue, ignoring the fact that new events and policy agendas are constantly emerging. To address this issue, we propose a two-stage adaptation framework. The first stage involves adapting the framing domain through pre-training, using two tasks based on pivots and prompts to learn a transferable encoder, verbalizer, and prompts. In the second stage, the transferable components are applied to new issues and label sets. Our framework has been shown to be effective in different scenarios, and to perform better than other methods, both in conditions of full resources and low resources. |  ChatGPT-polished |  66.61%
Table 4: Sample cases of the Polish Ratio based on the Levenshtein distance
where the parts edited by ChatGPT are highlighted in red. From a subjective
standpoint, we can discern the rationale behind our PR regression task: the PR
score for human-written abstracts tends to be close to 0, indicating no
modification. Also, larger PR means more parts introduced by ChatGPT as
represented by more red parts.

The different distributions of PR values given by our model on test data in
Figure 4 show that our PR explanation model demonstrates strong generalization
capabilities. It not only effectively distinguishes between human-written and
polished texts but also successfully discriminates ChatGPT-generated texts
that are not encountered during the training stage. While both Jaccard
distance and Levenshtein distance serve as suitable PR metrics to
differentiate the CP and CG groups, the use of Levenshtein distance proves to
be more effective, where the median Polish Ratio of ChatGPT-polished texts is
around 0.5, while that of ChatGPT-generated texts is around 0.65. According to
the violin plot shown in Figure 4(b), we suggest a Polish Ratio greater than
0.2 indicates ChatGPT involvement, and a value greater than 0.6 indicates that
ChatGPT generates most words.

###  4.4 Case study for Polish Ratio

Our Polish Ratio is also accurate in the case study. Table 4 is the case study
of the Polish Ratio regression on the index Levenshtein Distance. Indeed, it
can effectively differentiate the extent of modification introduced by
ChatGPT. For instance, the degree of text modification by ChatGPT varies
significantly between text (1b) and text (2b), as highlighted in red, leading
to a notable difference in their PR values, which proves the effectiveness of
our PR method.

| Precision | Recall | Support | Model
---|---|---|---|---
Human | 51.47% | 99.46% | 738 | Roberta-HC3
ChatGPT | 92.00% | 6.23% | 738
Human | 99.4% | 89.84% | 738 | Roberta-HPPT
ChatGPT | 90.70% | 99.46% | 738
Table 5: Detailed recall and precision for the Roberta-HC3 and Robarta-HPPT in
HPPT test Corpus.

###  4.5 Error Analysis

We conduct an error analysis to gain insights into our model’s learning
process and assess the efficacy of our explanation methods. We focus on
understanding the reasons behind the mistakes the baseline model and our model
made in the HPPT dataset, shown in Table 5.

The result shows that the baseline model only trained on the purely ChatGPT-
generated text exhibited confusion when encountering a polishing attack, as
evidenced by the classification of nearly all polished samples as human-
written. In contrast, our model (Roberta-HPPT) tends to make mistakes for
originally human-written texts, as shown in Table 5. By exploring the Polish
Ratio of these misclassified samples, we find misclassified samples have a
relatively high PR which originally should be close to 0 PR for human-written
samples, as shown in Figure 5. This discovery indicates that our PR method can
explain the reasons why the detection model makes mistakes: the writing style
of original human-written texts may be similar to that of ChatGPT.

Figure 5: Differences between human-written samples and misclassified samples
whose ground truth is human-written in the test set: the mean value of PR for
misclassified samples is around 0.3, which makes our detection model confused.

###  4.6 Generalization to the Chinese language and different LLMs

The framework of our ChatGPT involvement detection method also works
effectively on Chinese texts. We collect 1,232 abstracts from the CSL: a
large-scale Chinese Scientific Literature dataset [22] that contains 396,209
Chinese papers’ meta-information including title, abstract, keywords, academic
category and discipline. Then we use ChatGPT to polish 888The prompt is ”
请润色以下文本： ¡abstracts¿” (Translation: ”Please polish the following
sentences:¡abstracts¿”). the selected 1,232 Chinese abstracts and calculate
the Jaccard Distance and Levenshtein Distance (normalized) between the
polished and the corresponding human-written abstract based on Chinese
characters. The 1,232 pairs of abstracts are randomly partitioned into the
train, test, and validation sets by $6:3:1$. Following the framework mentioned
in Section 3, we train the detection and PR model 999We use the Chinese pre-
trained Roberta model: https://huggingface.co/hfl/chinese-roberta-wwm-ext to
investigate whether our proposed method is generalizable to the Chinese
language. Through experiments, we discover that the detection accuracy
($98.24\%$) significantly outperforms DetectGPT ($53.25\%$) on our collected
Chinese test sets, and our PR model can also distinguish human-written,
ChatGPT-polished, and ChatGPT-generated texts 101010Human-written and ChatGPT-
polished texts are from our collected test sets. ChatGPT-generated texts are
100 scientific abstracts additionally generated purely by ChatGPT through the
prompt: ” 请写一篇200字左右科学文献中文摘要 ” (Translation: ”Please write a Chinese abstract
of scientific literature of about 200 words”).  effectively as shown in Figure
6.

(a) Jaccard distance

(b) Levenshtein distance

Figure 6: Differences between predicted PR for human-written texts (HW),
ChatGPT-polished texts (CP) and ChatGPT-generated texts (CG).

Our proposed "Polish Ratio" method can also be generalizable to other LLMs
like Llama2 [23]. Following the framework shown in Figure 1, we replaced the
ChatGPT with the Llama2-7b model and polished the same abstracts in HPPT.
Under the same experiment settings as reported in section 4.1.2, our method
can effectively distinguish human-written (HW), Llama-polished (LP), and
Llama-generated (LG) texts, as shown in Figure 7. Both results suggest that
our proposed method can be generalized to different languages and work on
different kinds of powerful LLMs.

(a) Jaccard distance

(b) Levenshtein distance

Figure 7: Differences between predicted PR for human-written texts (HW),
Llama-polished texts (LP) and Llama-generated texts (LG).

##  5 Conclusion

In this paper, we focus on building a more robust ChatGPT detector with
independent explanations by detecting the ChatGPT-polished text. We first
develop a dataset called Human-ChatGPT Polished Paired abstracTs (HPPT). Then
we train a Roberta-based model that can distinguish human-written texts and
ChatGPT-polished texts. We also propose a Polish Ratio method to indicate the
degree of ChatGPT involvement in the text. Experiments demonstrate that our
model achieves better robustness than the baselines. Equipped with the help of
the robust detector and Polish Ratio explanation we proposed, users can make
an accurate and convincing judgment of the suspected texts. In the future, we
will focus on investigating the specific patterns ChatGPT prefers to use from
the perspective of some dynamic features of the text.

## Acknowledgement

This research is supported by the internal project of Shenzhen Science and
Technology Research Fund (Fundamental Research Key Project Grant No.
JCYJ20220818103001002), and the Internal Project Fund from Shenzhen Research
Institute of Big Data under Grant No. T00120-220002.

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# Convergence of the logarithm of the characteristic polynomial of unitary
Brownian motion in Sobolev space

Johannes<EMAIL_ADDRESS>Mathematical Institute,
University of Oxford, Isao<EMAIL_ADDRESS>Department
of Statistics, University of Warwick

###### Abstract

We prove that the convergence of the real and imaginary parts of the logarithm
of the characteristic polynomial of unitary Brownian motion toward Gaussian
free fields on the cylinder, as the matrix dimension goes to infinity, holds
in certain suitable Sobolev spaces, which we believe to be optimal. This is
the natural dynamical analogue of the result for a fixed time by Hughes,
Keating and O’Connell [1]. A weak kind of convergence is known since the work
of Spohn [2], which was widely improved recently by Bourgade and Falconet [3].
In the course of this research we also proved a Wick-type identity, which we
include in this paper, as it might be of independent interest.

##  1 Introduction

As unitary Brownian motion preserves the Haar measure on the unitary group
$U(n)$, to many results of Haar distributed unitary matrices there is a
corresponding dynamical result for a unitary Brownian motion $U$ at
equilibrium. This is in particular the case for some properties of the
eigenvalues, whose dynamics have been studied first by Dyson [4], who computed
a stochastic differential equation describing their evolution. In this paper,
we intend to achieve such a transition from static to dynamic for the Hughes-
Keating-O’Connell theorem on the large $n$ limit of the logarithm $\log p_{n}$
of the characteristic polynomial.

Characteristic polynomials of random matrices are fundamental objects in
random matrix theory. They are closely related to the theory of log-correlated
fields and to Gaussian multiplicative chaos [1, 3, 5]. In the case of Haar-
distributed matrices from the classical compact groups, there are also
remarkable similarities between the statistics of the characteristic
polynomial and those of the Riemann zeta function and other number-theoretic
$L$-functions, which led to a number of very precise conjectures for those
$L$-functions [6, 7, 8, 9, 10] \- see [11] for a review.

The real and imaginary part of the logarithm $\log p_{n}$ of the
characteristic polynomial also enters the wide family of _linear statistics_
of the eigenvalues $\lambda_{1},\dots,\lambda_{n}$, that is functions that can
be expressed as $\sum_{i=1}^{n}f(\lambda_{i})$. This family has received much
attention already, both in the static [12, 13, 14, 15] and dynamical [2, 16]
frameworks. Except for [12] in the static case, all these papers assume too
much regularity on $f$ to be applicable directly to $\log p_{n}$, for the type
of convergence they use is too strong. However, it is still possible to use
the result of Spohn [2] to identify the large $n$ limit of $\Re\log p_{n}$ and
$\Im\log p_{n}$ as Gaussian free fields, and prove a weak type of convergence
(see Lemma 3.1 below or the remark (i) below Theorem 1 in [2] ).

In a recent paper [3], Bourgade and Falconet gave the first dynamical
extension of Fisher-Hartwig asymptotics. Those asymptotics allowed them to
give a new proof and improvement of Spohn’s result. They also used those
asymptotics to prove that $|p_{n}|^{\alpha}$, for certain $\alpha$ and when
properly normalized, converge to a Gaussian multiplicative chaos measure
associated to the Gaussian free field $h$ on the cylinder, i.e. informally the
exponential of a multiple of $h$.

The goal of this paper is to specify some Sobolev spaces, which we think to be
optimal, in which we prove the convergence of $\Re\log p_{n}$ and $\Im\log
p_{n}$. 333Several Sobolev spaces are involved because we can improve the
regularity with respect to one of the parameters at the cost of sacrificing
some regularity with respect to the other parameter. This is the natural
dynamical version of the corresponding stationary result for Haar-distributed
unitary matrices by Hughes, Keating and O’Connell [1], who proved that for any
fixed time the logarithm of the characteristic polynomial converges to a
generalized Gaussian field on the unit circle.

In the last section, we state and prove an identity that allows to express the
second moment of the trace of arbitrary products of a GUE matrix $H$ and an
independent CUE matrix $U$ in terms of moments of $U$ only. When the dimension
$n$ is large enough, the Diaconis-Shahshahani theorem on moments of traces of
unitary matrices [13] allows to then compute this new expression explicitly as
a polynomial in $n$.

###  1.1 Context

We let $U_{n}:[0,\infty)\rightarrow U(n)$ be a unitary Brownian motion started
from Haar measure (for a precise definition see Section 2.1), and define its
characteristic polynomial as

$p_{n}(t,\theta):=\text{det}\left(I_{n}-e^{-i\theta}U_{n}(t)\right)=\prod_{k=1}^{n}(1-e^{i(\theta_{k}(t)-\theta)}),\quad(\theta,t)\in[0,2\pi)\times[0,\infty),$

where $0\leq\theta_{1}(t)<...<\theta_{n}(t)<2\pi$ denote the eigenangles of
unitary Brownian motion. We define its logarithm by

$\displaystyle\begin{split}\log
p_{n}(t,\theta):=&\sum_{k=1}^{n}\log(1-e^{i(\theta_{k}(t)-\theta)}),\end{split}$

with the branches on the RHS being the principal branches, such that

$\Im\log(1-e^{i(\theta_{k}(t)-\theta)})\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right],$

with $\Im\log 0:=\pi/2$.

Hughes, Keating and O’Connell proved that for any fixed time $t\geq 0$, $\log
p_{n}(t,\cdot)$ converges to a generalized Gaussian field. Their result,
reformulated to our setting, is as follows:

######  Theorem 1.1 (Hughes, Keating, O’Connell [1]).

For any $\epsilon>0$ and any fixed $t\geq 0$, the sequence of random functions
$\left(\log p_{n}(t,\cdot)\right)_{n\in\mathbb{N}}$ converges in distribution
in $H_{0}^{-\epsilon}(S^{1})$ to the generalized Gaussian field

$\displaystyle X(\theta)=$
$\displaystyle\sum_{k=1}^{\infty}\frac{A_{k}}{\sqrt{k}}e^{ik\theta},$

where $A_{k}$ is a complex Gaussian whose real and imaginary parts are
independent centered Gaussians with variance $1/(2k)$.

It is natural thus to assume that in the dynamic case, i.e. when considering
$\log p_{n}$ also as a function of $t$, that the limit (in an appropriate
function space) would be given by

$\displaystyle
X(t,\theta)=\sum_{k=1}^{\infty}\frac{A_{k}(t)}{\sqrt{k}}e^{ik\theta},$ (1)

where $A_{k}(\cdot)$, $k\in\mathbb{N}$, are independent complex Ornstein-
Uhlenbeck processes started from their stationary distribution, i.e. (up to a
linear time change) solutions to the SDEs

$\displaystyle\text{d}A_{k}(t)=-kA_{k}(t)\text{d}t+\text{d}\left(W_{k}(t)+i\tilde{W}_{k}(t)\right),$
(2)

with $A_{k}(0)$ being a complex Gaussian whose real and imaginary parts are
independent Gaussians with variance $1/(2k)$, and $(W_{k}(t))_{t\geq 0}$,
$(\tilde{W}_{k}(t))_{t\geq 0}$, $k\in\mathbb{N}$, denoting real standard
Brownian motions.

Our main result proves precisely that (for a definition of the Sobolev spaces
$H^{s}([0,T])$ and $H^{-\epsilon}_{0}(S^{1})$ see Section 2.2):

######  Theorem 1.2 (Main Result).

For any $s\in(0,\frac{1}{2})$, $\epsilon>s$ and $T>0$, the sequence of random
fields $\left(\log p_{n}(\cdot,\cdot)\right)_{n\in\mathbb{N}}$ converges in
distribution in the tensor product of Hilbert spaces $H^{s}([0,T])\otimes
H^{-\epsilon}_{0}(S^{1})$ to the generalized Gaussian field $X$ in (1).

A calculation shows that the covariance functions of $\Re X$ and $\Im X$ are
given by

$\displaystyle\mathbb{E}(\Re X(t,\theta),\Re X(t^{\prime},\theta^{\prime}))$
$\displaystyle=\mathbb{E}(\Im X(t,\theta),\Im
X(t^{\prime},\theta^{\prime}))=\frac{1}{2}\log\frac{\max(e^{-t},e^{-t^{\prime}})}{|e^{-t}e^{i\theta}-e^{-t^{\prime}}e^{i\theta^{\prime}}|}.$

The centered Gaussian fields $\Re X$ and $\Im X$ with such a covariance
function have been identified as Gaussian free fields on the infinite cylinder
$\mathbb{R}\times\mathbb{R}/2\pi\mathbb{Z}$ in [3, Section 2.2].

######  Remark 1.3.

Theorem 1.2 implies that there is a trade-off between regularity in $\theta$
and regularity in $t$. We believe that the regularity we obtain is optimal, in
the sense that for $s=1/2$ or $\epsilon=s$, $X$ is almost surely not an
element of the tensor product of $H^{s}([0,T])\otimes
H^{-\epsilon}_{0}(S^{1})$ anymore.

While the limiting field is rotationally invariant from an infinitesimal point
of view, this is not the case for $\log p_{n}$ with finite $n$. In particular,
one can exchange the regularity in the variable $t$ with the regularity in the
variable $\theta$ for the limiting field, but for our proof of convergence to
work, the Sobolev regularity $-\epsilon$ in the variable $\theta$ needs to be
negative which is not the case for the Sobolev regularity $s$ in the variable
$t$.

Just like in the stationary case, the Gaussian field $X$ can’t be defined
pointwise as its variance at each point is infinite, but it can still be
”exponentiated” to build a Gaussian multiplicative chaos (GMC) measure. When
we let $h(t,\theta)$ denote the real part of $X(t,\theta)$, and denote by
$h_{\delta}(t,\theta)$ a mollification of $h$, then for
$\gamma\in(0,2\sqrt{2})$ the random measures

$\displaystyle e^{\gamma
h(t,\theta)}\text{d}\theta\text{d}t:=\lim_{\delta\rightarrow 0}e^{\gamma
h_{\delta}(t,\theta)-\frac{\gamma^{2}}{2}\mathbb{E}(h_{\delta}(t,\theta))}\text{d}\theta\text{d}t$

exist and are non-trivial, where the limit is in probability w.r.t. the
topology of weak convergence of measures on
$\mathbb{R}\times\mathbb{R}/2\pi\mathbb{Z}$, see [17] for a self-contained
proof of this fact. Bourgade and Falconet proved that exponentiating
$\log|p_{n}(t,\theta)|$ in this way, and then taking the large $n$ limit,
gives the same limiting measure as when first taking the large $n$ limit to
obtain the Gaussian free field $h$, and then exponentiating it. Their result
is the dynamical analogue to Webb’s result for fixed $t$ and the measures
being on the unit circle [16], and its precise statement is as follows:

######  Theorem 1.4 (Bourgade, Falconet [3]).

For every $\gamma\in(0,2\sqrt{2})$ it holds that

$\displaystyle\lim_{n\rightarrow\infty}\frac{|p_{n}(t,\theta)|^{\gamma}}{\mathbb{E}\left(|p_{n}(t,\theta)|^{\gamma}\right)}\text{d}\theta\text{d}t=e^{\gamma
h(t,\theta)}\text{d}\theta\text{d}t,$

where the convergence is in distribution in the space of Radon measures on the
infinite cylinder $\mathbb{R}\times\mathbb{R}/2\pi\mathbb{Z}$, equipped with
the topology of weak convergence.

Our main result complements their asymptotics in that it shows in which
Sobolev spaces the convergence of the underlying fields $\log|p_{n}|$ and
$\Im\log p_{n}$ to the Gaussian free field $h$ holds.

Further, Theorem 1.2 is related to the below result by Spohn, which we also
use in our proof. For real-valued functions $f\in
H_{0}^{3/2+\epsilon}(S^{1},\mathbb{R})$, $\epsilon>0$, Spohn considered linear
statistics of the eigenvalues $e^{i\theta_{1}(t)},...,e^{i\theta_{n}(t)}$ of
unitary Brownian motion (in fact he more generally considered interacting
particles on the unit circle with different repulsion strengths):

$\xi_{n}(t,f):=\sum_{j=1}^{n}f(e^{i\theta_{j}(t)}),\qquad(t,f)\in[0,\infty)\times
H^{3/2+\epsilon}_{0}(S^{1},\mathbb{R}).$

Since $H_{0}^{-3/2-\epsilon}(S^{1},\mathbb{R})$ is the dual space of
$H_{0}^{3/2+\epsilon}(S^{1},\mathbb{R})$, one can consider $\xi_{n}$ as a
random continuous map $t\mapsto\xi_{n}(t,\cdot)\in
H_{0}^{-3/2-\epsilon}(S^{1},\mathbb{R})$.

######  Theorem 1.5 (Spohn [2]).

For any $\epsilon>0$, as $n\rightarrow\infty$, $\xi_{n}(t,f)$ converges to a
stationary solution of the SDE

$\;{\rm d}\xi(t,f)=\xi(t,-\sqrt{-\partial_{\theta}^{2}}f)\;{\rm d}t+\;{\rm
d}\mathcal{W}(t,f^{\prime}),$

where $\;{\rm d}\mathcal{W}$ is a white noise given by

$\mathbb{E}[\;{\rm d}\mathcal{W}(t,f)\;{\rm
d}\mathcal{W}(s,g)]=2\delta(t-s)\;{\rm d}s\;{\rm
d}t\frac{1}{2\pi}\int_{0}^{2\pi}f(e^{i\theta})g(e^{i\theta})\;{\rm d}\theta,$

and where the convergence is in distribution in
$\mathcal{C}(\mathbb{R},H^{-3/2-\epsilon}(S^{1},\mathbb{R}))$, endowed with
the topology of locally uniform convergence. The stationary distribution is
given by a Gaussian with covariance

$\displaystyle\mathbb{E}\left(\xi(t,f)\xi(t,g)\right)=\sum_{k\neq
0}|k|f_{k}g_{k}.$

Here, $\sqrt{-\partial_{\theta}^{2}}f$ is simply the function whose
$j^{\text{th}}$ Fourier coefficient is $|j|$ times the $j^{\text{th}}$ Fourier
coefficient of $f$. This result shows in particular that the $k^{\text{th}}$
Fourier coefficient of $\ln p_{n}$ converges toward $\frac{A_{k}}{\sqrt{k}}$
(see Lemma 3.1 below). Further, during the proof of Theorem 1.2, we will need
the following result from Bourgade and Falconet [3, Corollary 3.2]:

######  Corollary 1.6 (Bourgade, Falconet).

Let $(z_{1}(t),...,z_{n}(t))_{t\geq 0}$ denote the eigenvalue process of
unitary Brownian motion, started at Haar measure, and denote
$\operatorname{sgn}(x)=1_{x>0}-1_{x<0}$. For $f,g\in
H_{0}^{1/2}(S^{1},\mathbb{R})$, we have for every $n\in\mathbb{N}$ and $t\geq
0$,

$\displaystyle\mathbb{E}\Big{[}\Big{(}\sum_{j=1}^{n}f(z_{j}(0))\Big{)}\Big{(}\sum_{j=1}^{n}g(z_{j}(t))\Big{)}\Big{]}=\sum_{|k|\leq
n-1}f_{k}g_{-k}\operatorname{sgn}(k)e^{-|k|t}\frac{\sinh(\frac{k^{2}t}{n})}{\sinh(\frac{kt}{n})}+\sum_{|k|\geq
n}f_{k}g_{-k}e^{-\frac{k^{2}t}{n}}\frac{\sinh(kt)}{\sinh(\frac{kt}{n})}.$

##  2 Mathematical Preliminaries

###  2.1 Unitary Brownian motion

Brownian motion $(U_{n}(t))_{t\geq 0}$ on the unitary group $U(n)$ is the
diffusion governed by the stochastic differential equation

$\displaystyle\text{d}U_{n}(t)=\sqrt{2}U_{n}(t)\text{d}B_{n}(t)-U_{n}(t)\text{d}t,$

with $(B_{n}(t))_{t\geq 0}$ denoting a Brownian motion on the space of skew-
Hermitian matrices. That is

$\displaystyle B_{n}(t)=\sum_{k=1}^{n^{2}}X_{k}\tilde{B}^{(k)}(t),$

where $\tilde{B}^{(k)}$, $k=1,...,n^{2}$, are independent one-dimensional
standard Brownian motions, and where the matrices $X_{k}$, $k=1,...,n^{2}$,
are an orthonormal basis of the real vector space of skew-Hermitian matrices
w.r.t. the scalar product $\langle A,B\rangle:=n\operatorname{Tr}(AB^{*})$.
One such basis is given by the matrices
$\frac{1}{\sqrt{2n}}(E_{k,l}-E_{l,k})$,
$\frac{i}{\sqrt{2n}}(E_{k,l}+E_{l,k})$, $1\leq k<l\leq n$, and
$\frac{i}{\sqrt{n}}E_{k,k}$, $1\leq k\leq n$.

######  Remark 2.1.

Unitary Brownian motion is usually defined using a different normalisation,
i.e. satisfying the SDE $\;{\rm d}\tilde{U}_{n}(t)=\tilde{U}_{n}(t)\;{\rm
d}B_{n}(t)-\frac{1}{2}\tilde{U}_{n}(t)$. With this normalisation the generator
is given by one half times the Laplacian on $U(n)$, which is the usual
definition of Brownian motion on a Riemannian manifold. The relation between
the two normalisations is $\tilde{U}_{n}(2t)=U_{n}(t)$.

In this paper we always consider unitary Brownian motion started from Haar
measure on $U(n)$, which is its stationary distribution. Thus $U_{n}(t)$ is
Haar distributed for all $t\geq 0$.

###  2.2 Sobolov spaces and their Tensor Product

Consider the space of square integrable $\mathbb{C}$-valued functions on the
unit circle, with vanishing mean:

$L^{2}_{0}(S^{1})=\left\\{f(\theta)=\sum_{k\in\mathbb{Z}}f_{k}e^{ik\theta}:\sum_{k\in\mathbb{Z}}|f_{k}|^{2}<\infty,f_{0}=0\right\\}.$

For $s\geq 0$, we define $H^{s}_{0}(S^{1})$ as the restriction of
$L^{2}_{0}(S^{1})$ w.r.t. the functions for which the inner product

$\langle f,g\rangle_{s}=\sum_{k\in\mathbb{Z}}|k|^{2s}f_{k}\overline{g_{k}}$

is finite. For $s\leq 0$, we define $H^{s}_{0}(S^{1})$ as the completion of
$L^{2}_{0}(S^{1})$ w.r.t. this scalar product. Note that
$\left(H_{0}^{s}(S^{1}),\langle\cdot,\cdot\rangle_{s}\right)$ is a Hilbert
space for all $s\in\mathbb{R}$. For $s\geq 0$ it is a subspace of
$H^{0}_{0}(S^{1})=L_{0}^{2}(S^{1})$, i.e. the space of square-integrable
functions with zero mean, while for $s<0$, $H_{0}^{s}(S^{1})$ can be
interpreted as the dual space of $H_{0}^{-s}(S^{1})$, i.e. as a space of
generalized functions defined up to additive constant.

For $T>0$, and $s\in(0,1)$, we define the fractional Sobolev space
$H^{s}([0,T])$ as the subspace of $L^{2}([0,T])$, where the Slobodeckij inner
product

$(f,g)_{s}:=\int_{0}^{T}f(t)\overline{g(t)}\text{d}t+\int_{0}^{T}\int_{0}^{T}\frac{(f(t)-f(u))(\overline{g(t)-g(u)})}{|t-u|^{1+2s}}\;{\rm
d}u\;{\rm d}t$

is finite. Note that $(H^{s}([0,T]),(\cdot,\cdot)_{s})$ is a Hilbert space for
all $s>0$.

######  Remark 2.2.

For the fact that the fractional Sobolev spaces defined through Fourier series
or through the Slobodeckij norm agree, the reader can consult e.g. [18].

For $s>0$ and $\epsilon>0$ we let $H^{s}([0,T])\otimes
H^{-\epsilon}_{0}(S^{1})$ denote the tensor product of Hilbert spaces
$H^{s}([0,T])$ and $H^{-\epsilon}_{0}(S^{1})$. Since the inner product on that
space is determined by

$\displaystyle\langle f\otimes g,h\otimes k\rangle_{s,-\epsilon}$
$\displaystyle=(f,h)_{s}\langle g,k\rangle_{-\epsilon}$
$\displaystyle=\int_{0}^{T}f(t)\overline{h(t)}\text{d}t\langle
g,k\rangle_{-\epsilon}+\int_{0}^{T}\int_{0}^{T}\frac{(f(t)-f(u))\overline{(h(t)-h(u))}}{|t-u|^{1+2s}}\;{\rm
d}u\;{\rm d}t\langle g,k\rangle_{-\epsilon},$
$\displaystyle=\int_{0}^{T}\langle
f(t)g,h(t)k\rangle_{-\epsilon}\text{d}t+\int_{0}^{T}\int_{0}^{T}\frac{\langle(f(t)-f(u))g,(h(t)-h(u))k\rangle_{-\epsilon}}{|t-u|^{1+2s}}\;{\rm
d}u\;{\rm d}t,$

we obtain

$\langle F,G\rangle_{s,-\epsilon}=\int_{0}^{T}\langle
F(t,\cdot),G(t,\cdot)\rangle_{-\epsilon}\;{\rm
d}t+\int_{0}^{T}\int_{0}^{T}\frac{\langle
F(t,\cdot)-F(u,\cdot),G(t,\cdot)-G(u,\cdot)\rangle_{-\epsilon}}{|t-u|^{1+2s}}\;{\rm
d}u\;{\rm d}t,$

first when $F$ and $G$ are linear combinations of pure tensor products, and
then for all $F,G\in H^{s}([0,T])\otimes H^{-\epsilon}_{0}(S^{1})$ by density
and continuity.

##  3 Proof of the main result Theorem 1.2

The proof strategy is as in the stationary case in [1]: we treat $\left(\log
p_{n}\right)_{n\in\mathbb{N}}$ as a sequence in $H^{s}([0,T])\otimes
H^{-\epsilon}_{0}(S^{1})$, and show that if any of its subsequences has a
limit then that limit has to be $X$. We do this by showing that the finite-
dimensional distributions of $\left(\log p_{n}\right)_{n\in\mathbb{N}}$, i.e.
the distributions of finite sets of Fourier coefficients at a finite number of
times, converge to those of $X$. We then show that the set $\left(\log
p_{n}\right)_{n\in\mathbb{N}}$ is tight in $H^{s}([0,T])\otimes
H^{-\epsilon}_{0}(S^{1})$. Since $H^{s}([0,T])\otimes
H^{-\epsilon}_{0}(S^{1})$ is complete and separable, Prokhorov’s theorem
implies that the closure of $\left(\log p_{n}\right)_{n\in\mathbb{N}}$ is
sequentially compact w.r.t. the topology of weak convergence. In particular
this means that every subsequence of $\left(\log
p_{n}\right)_{n\in\mathbb{N}}$ has a weak limit $H^{s}([0,T])\otimes
H^{-\epsilon}_{0}(S^{1})$. Since any such limit has to be $X$ it follows that
the whole sequence $\left(\log p_{n}\right)_{n\in\mathbb{N}}$ must converge
weakly to $X$.

We recall that

$\log(1-z)=-\sum_{k=1}^{\infty}\frac{z^{k}}{k}$

for $|z|\leq 1$, where for $z=1$ both sides equal $-\infty$. By using the
identity $\log\det=\operatorname{Tr}\log$ we see that the Fourier expansion of
$\log p_{n}$ w.r.t. the spacial variable $\theta$ is given as follows:

$\displaystyle\begin{split}\log
p_{n}(t,\theta)=&-\sum_{k=1}^{\infty}\frac{\operatorname{Tr}(U_{n}^{k}(t))}{k}e^{-ik\theta}.\end{split}$

######  Lemma 3.1.

Let $((\log p_{n})_{k}(t))_{k\geq 1}$ be the Fourier coefficients of $(\log
p_{n})(t,\cdot)$. The process $(t,k)\mapsto(\log p_{n})_{k}(t)$ converges in
finite-dimensional distributions towards the complex Ornstein-Uhlenbeck
process $(t,k)\mapsto A_{k}(t)$ defined in (2).

###### Proof.

We prove convergence of the finite-dimensional distributions by showing that
for any $l\in\mathbb{N}$ and $0\leq t_{1}<t_{2}<...<t_{l}\leq T$, as
$n\rightarrow\infty$:

$\displaystyle\Big{(}(\log p_{n})_{1}(t_{1}),...,(\log p_{n})_{l}(t_{1}),(\log
p_{n})_{1}(t_{2}),...,(\log p_{n})_{l}(t_{2}),...,(\log
p_{n})_{1}(t_{l}),...,(\log p_{n})_{l}(t_{l})\Big{)}$ $\displaystyle\hskip
113.81102pt\overset{(d)}{\longrightarrow}\Big{(}A_{1}(t_{1}),...,A_{l}(t_{1}),A_{1}(t_{2}),...,A_{l}(t_{2}),...,A_{1}(t_{l}),...,A_{l}(t_{l})\Big{)}.$

Let $e_{k}:\theta\mapsto e^{ik\theta}$. Then, using the notations of Theorem
1.5, $\log(p_{n})_{k}(t)=\xi_{n}\big{(}t,\tfrac{e_{k}}{k}\big{)}$. Thus
Spohn’s theorem, combined with the continuous mapping theorem with the
appropriate continuous map
$\mathcal{C}(\mathbb{R},H^{-3/2-\epsilon}(S^{1},\mathbb{R}))\rightarrow\mathbb{R}^{l^{2}}$,
implies that

$\displaystyle\Big{(}(\log p_{n})_{1}(t_{1}),...,(\log p_{n})_{l}(t_{1}),(\log
p_{n})_{1}(t_{2}),...,(\log p_{n})_{l}(t_{2}),...,(\log
p_{n})_{1}(t_{l}),...,(\log p_{n})_{l}(t_{l})\Big{)}$ $\displaystyle\hskip
56.9055pt\overset{(d)}{\longrightarrow}\Big{(}\xi\big{(}t_{1},\tfrac{e_{1}}{1}\big{)},\dots,\xi\big{(}t_{1},\tfrac{e_{l}}{l}\big{)},\xi\big{(}t_{2},\tfrac{e_{1}}{1}\big{)},\dots,\xi\big{(}t_{2},\tfrac{e_{l}}{l}\big{)},...\xi\big{(}t_{l},\tfrac{e_{1}}{1}\big{)},\dots,\xi\big{(}t_{l},\tfrac{e_{l}}{l}\big{)}\Big{)}.$

Combining the real and imaginary part of $e_{k}$, we obtain that the SDE for
$\xi(\cdot,\frac{e_{k}}{k})$ reduces to

$\;{\rm
d}\xi\left(t,\frac{e_{k}}{k}\right)=-k\xi\left(t,\frac{e_{k}}{k}\right)\;{\rm
d}t+dB_{k}(t),$

where $B_{k}$ is a complex Brownian motion, i.e. a process whose real and
imaginary parts are independent standard Brownian motions. Besides, the
Brownian motions $(B_{k})_{k\geq 0}$ are independent, so that $(A_{k})_{k\geq
1}$ and $\big{(}\xi\big{(}\cdot,\tfrac{e_{k}}{k}\big{)}\big{)}_{k\geq 1}$ are
equal in distribution, which concludes the proof. ∎

We proceed to show tightness of $\left(\log p_{n}\right)_{n\in\mathbb{N}}$ in
$H^{s}([0,T])\otimes H^{-\epsilon}_{0}(S^{1})$, i.e. for every $\delta>0$ we
construct a compact $K_{\delta}\subset H^{s}([0,T])\otimes
H^{-\epsilon}_{0}(S^{1})$ for which

$\sup_{n\in\mathbb{N}}\mathbb{P}\left(\log p_{n}\in
K_{\delta}^{c}\right)<\delta.$

We let $0<s^{\prime}<\epsilon^{\prime}$ such that
$0<s<s^{\prime}<\epsilon^{\prime}<\epsilon$, and choose

$\displaystyle K_{\delta}=\left\\{F\in H^{s}([0,T])\otimes
H_{0}^{-\epsilon}(S^{1}):||F||^{2}_{s^{\prime},-\epsilon^{\prime}}\leq
C_{\delta}\right\\},$

for a $C_{\delta}$ depending on $\delta$. By Lemma 3.2 below we see that
$K_{\delta}$ is compact in $H^{s}([0,T])\otimes H_{0}^{-\epsilon}(S^{1})$, and
by Lemma 3.3 below we see that $\sup_{n\in\mathbb{N}}\mathbb{E}\left(||\log
p_{n}||^{2}_{s^{\prime},-\epsilon^{\prime}}\right)<\infty$. Thus, when
choosing $C_{\delta}$ large enough, we see that

$\displaystyle\sup_{n\in\mathbb{N}}\mathbb{P}\left(\left(\log p_{n}\right)\in
K_{\delta}^{c}\right)=$
$\displaystyle\sup_{n\in\mathbb{N}}\mathbb{P}\left(||\log
p_{n}||_{s,-\epsilon}^{2}>C_{\delta}\right)$ $\displaystyle\leq$
$\displaystyle\frac{\sup_{n\in\mathbb{N}}\mathbb{E}\left(||\log
p_{n}||^{2}_{s^{\prime},-\epsilon^{\prime}}\right)}{C_{\delta}^{2}}$
$\displaystyle<$ $\displaystyle\delta,$

which shows tightness of $\log p_{n}$ and thus together with Lemma 3.1 proves
our Theorm 1.2.

######  Lemma 3.2.

Let $0<s<s^{\prime}<\epsilon^{\prime}<\epsilon$. Then, the inclusion of
$H^{s^{\prime}}([0,T])\otimes H_{0}^{-\epsilon^{\prime}}(S^{1})$ into
$H^{s}([0,T])\otimes H_{0}^{-\epsilon}(S^{1})$ is compact.

###### Proof.

From the Kondrachov embedding theorem, the inclusion $\iota_{1}$ of
$H^{s^{\prime}}([0,T)$ into $H^{s}([0,T])$ is compact, as well as the
inclusion $\iota_{2}$ from $H_{0}^{\epsilon}(S^{1})$ into
$H_{0}^{\epsilon^{\prime}}(S^{1})$. Then the dual operator
$\iota_{2}^{*}:H_{0}^{-\epsilon^{\prime}}(S^{1})\rightarrow
H_{0}^{-\epsilon}(S^{1})$ is also compact. On Hilbert spaces, the tensor
product of two compact operators is also compact (see e.g. [19] 444In [19],
the result is stated for endomorphisms, but this extra assumption is not used
in the proof.), so that $\iota_{1}\otimes\iota_{2}^{*}$ is compact indeed. ∎

######  Lemma 3.3.

For all $s\in\big{(}0,\frac{1}{2}\big{)}$ and all $\epsilon>s$, it holds that
$\sup_{n\in\mathbb{N}}\mathbb{E}\left(||\log
p_{n}||^{2}_{s,-\epsilon}\right)<\infty$.

Proof: We see that

$\displaystyle\mathbb{E}\left(||\log p_{n}||^{2}_{s,-\epsilon}\right)=$
$\displaystyle\mathbb{E}\left(\int_{0}^{T}||\log
p_{n}(\cdot,t)||_{-\epsilon}^{2}\text{d}t\right)+\mathbb{E}\left(\int_{0}^{T}\int_{0}^{T}\frac{||\log
p_{n}(\cdot,t)-\log
p_{n}(\cdot,r)||^{2}_{-\epsilon}}{|t-r|^{2s+1}}\text{d}r\text{d}t\right).$

For the first summand it holds that (with $k\wedge n$ denoting
$\min\\{k,n\\}$)

$\displaystyle\begin{split}\mathbb{E}\left(\int_{0}^{T}||\log
p_{n}(\cdot,t)||^{2}_{-\epsilon}\text{d}t\right)=&\int_{0}^{T}\mathbb{E}\left(\sum_{k=1}^{\infty}k^{-2\epsilon}\frac{\left|\operatorname{Tr}(U_{n}(t)^{k})\right|^{2}}{k^{2}}\right)\text{d}t\\\
=&T\sum_{k=1}^{\infty}k^{-2-2\epsilon}\mathbb{E}\left(\left|\operatorname{Tr}(U_{n}(0)^{k})\right|^{2}\right)\\\
=&T\sum_{k=1}^{\infty}k^{-2-2\epsilon}(k\wedge n)\\\
<&T\sum_{k=1}^{\infty}k^{-1-2\epsilon}<\infty.\end{split}$

For the second summand it holds that:

$\displaystyle\begin{split}&\mathbb{E}\Big{(}\int_{0}^{T}\int_{0}^{T}\frac{||\log
p_{n}(\cdot,t)-\log
p_{n}(\cdot,r)||^{2}_{-\epsilon}}{|t-r|^{2s+1}}\text{d}r\text{d}t\Big{)}\\\
=&\sum_{k=1}^{\infty}k^{-2-2\epsilon}\int_{0}^{T}\int_{0}^{T}\frac{\mathbb{E}\left(|\operatorname{Tr}(U_{n}^{k}(t)-U_{n}^{k}(r))|^{2}\right)}{|t-r|^{2s+1}}\text{d}r\text{d}t\\\
\leq&CT\sum_{k=1}^{\infty}k^{-2-2\epsilon}\int_{0}^{T}\frac{\mathbb{E}\left(|\operatorname{Tr}(U_{n}^{k}(t)-U_{n}^{k}(0))|^{2}\right)}{t^{2s+1}}\text{d}t\\\
\leq&CT\sum_{k=1}^{\infty}k^{-2-2\epsilon}\int_{0}^{k^{-1}}\frac{\mathbb{E}\left(|\operatorname{Tr}(U_{n}^{k}(t)-U_{n}^{k}(0))|^{2}\right)}{t^{2s+1}}\text{d}t\\\
&+CT\sum_{k=1}^{\infty}k^{-2-2\epsilon}\int_{k^{-1}}^{\infty}\frac{4\mathbb{E}\left(|\operatorname{Tr}(U_{n}^{k}(0))|^{2}\right)}{t^{2s+1}}\text{d}t.\end{split}$
(3)

For the second summand in (3) we get

$\int_{k^{-1}}^{\infty}\frac{4\mathbb{E}\left(|\operatorname{Tr}(U_{n}^{k}(0))|^{2}\right)}{t^{2s+1}}\text{d}t=8s(n\wedge
k)k^{2s},$

which is sufficient since $\sum_{k=1}^{\infty}k^{-2-2\epsilon+1+2s}$ is finite
as soon as $s<\epsilon$.

For the first sum in (3) we use Corollary 1.6, which implies that for all
$k\geq 1$

$\displaystyle\begin{split}\mathbb{E}\left(\operatorname{Tr}(U_{n}^{k}(t))\overline{\operatorname{Tr}(U_{n}^{k}(0))}\right)=&1_{k<n}e^{-kt}\frac{\sinh(\frac{k^{2}t}{n})}{\sinh(\frac{kt}{n})}+1_{k\geq
n}e^{-\frac{k^{2}t}{n}}\frac{\sinh(kt)}{\sinh(\frac{kt}{n})}\\\
=&e^{-\frac{k(k\vee n)t}{n}}\frac{\sinh\left(\frac{k(k\wedge
n)t}{n}\right)}{\sinh\left(\frac{kt}{n}\right)},\end{split}$

with $k\vee n:=\max\\{k,n\\}$. Using this, and the fact that $\sinh x\geq x$
and $1/\sinh x\geq 1/x-x/6$ for all $x>0$, we see that for $t<k^{-1}$:

$\displaystyle\mathbb{E}\left(|\operatorname{Tr}(U_{n}^{k}(t)-U_{n}^{k}(0))|^{2}\right)$
$\displaystyle=$
$\displaystyle\mathbb{E}\left(|\operatorname{Tr}(U_{n}^{k}(0))|^{2}\right)+\mathbb{E}\left(|\operatorname{Tr}(U_{n}^{k}(t))|^{2}\right)-2\mathbb{E}\left(\operatorname{Tr}(U_{n}^{k}(t))\overline{\operatorname{Tr}(U_{n}^{k}(0))}\right)$
$\displaystyle=$ $\displaystyle 2(k\wedge n)-2e^{-\frac{k(k\vee
n)t}{n}}\frac{\sinh\left(\frac{k(k\wedge
n)t}{n}\right)}{\sinh\left(\frac{kt}{n}\right)}$ $\displaystyle\leq$
$\displaystyle 2(k\wedge n)-2e^{-\frac{k(k\vee n)t}{n}}\frac{k(k\wedge
n)t}{n}\left(\left(\frac{kt}{n}\right)^{-1}-\frac{kt}{6n}\right)$
$\displaystyle=$ $\displaystyle 2(k\wedge n)-2e^{-\frac{k(k\vee
n)t}{n}}\left(k\wedge n-\frac{k^{2}t^{2}(k\wedge n)}{6n^{2}}\right)$
$\displaystyle=$ $\displaystyle 2e^{-\frac{k(k\vee
n)t}{n}}\frac{k^{2}t^{2}(k\wedge n)}{6n^{2}}+2(k\wedge n)(1-e^{-\frac{k(k\vee
n)t}{n}})$ $\displaystyle\leq$ $\displaystyle 2k^{3}t^{2}+2(k\wedge
n)\frac{k(k\vee n)t}{n}$ $\displaystyle\leq$ $\displaystyle 4k^{2}t.$

Thus we see that when $s<1/2$, the first sum in (3) is bounded by
$T\sum_{k=1}^{\infty}k^{-1-2\epsilon+2s}$, which is finite for $s<\epsilon$.
This finishes the proof. ∎

##  4 A Wick-type identity

In this section we prove LABEL:prop:orbits below, which is about expectations
of the form

$\displaystyle\mathbb{E}\Big{(}\operatorname{Tr}\left(HU^{\sigma_{1}}HU^{\sigma_{2}}\dots
HU^{\sigma_{j}}\right)\overline{\operatorname{Tr}\left(HU^{\sigma_{1}}HU^{\sigma_{2}}\dots
HU^{\sigma_{j}}\right)}\Big{)},$

where $\sigma_{1},...,\sigma_{j}\in\mathbb{Z}$, $U\in U(n)$ is Haar-
distributed and independent from $H$, which is a $GUE(n)$ matrix, i.e.
$H_{ii}\sim\mathcal{N}(0,1)$ for $i=1,...,n$, and $\Re H_{ij}=\Re
H_{ji}\sim\mathcal{N}(0,1/2)$, $\Im H_{ij}=-\Im H_{ji}\sim\mathcal{N}(0,1/2)$
for $1\leq i<j\leq n$, with entries being independent up to the Hermitian
symmetry.

Such expressions appear rather naturally when we consider powers of a unitary
Brownian motion $U$, since for small $s$ it holds that
$U_{t+s}\simeq(1+\sqrt{s}H)U_{t}$.

Let $C_{2j}=\\{\pi\in S_{2j}:\pi^{2}=\text{Id},\forall
l\in\\{1,\dots,2j\\},\pi(l)\neq l\\}$, i.e. $C_{2j}$ is the set of pairings on
$\\{1,...,2j\\}$. Then we see that

$\displaystyle\mathbb{E}\Big{(}\operatorname{Tr}\left(HU^{\sigma_{1}}HU^{\sigma_{2}}\dots
HU^{\sigma_{j}}\right)\overline{\operatorname{Tr}\left(HU^{\sigma_{1}}HU^{\sigma_{2}}\dots
HU^{\sigma_{j}}\right)}\Big{)}$ $\displaystyle=$
$\displaystyle\sum_{i_{1},...,i_{2j},l_{1},...,l_{2j}}\mathbb{E}\left(H_{i_{1}i_{2}}(U^{\sigma_{1}})_{i_{2}i_{3}}\cdots
H_{i_{2j-1}i_{2j}}(U^{\sigma_{j}})_{i_{2j}i_{1}}\overline{H_{l_{1}l_{2}}(U^{\sigma_{1}})_{l_{2}l_{3}}\cdots
H_{l_{2j-1}l_{2j}}(U^{\sigma_{j}})_{l_{2j}l_{1}}}\right)$ $\displaystyle=$
$\displaystyle\sum_{i_{1},...,i_{4j}}\mathbb{E}\left[H_{i_{1}i_{2}}H_{i_{3}i_{4}}\cdots
H_{i_{2j-1}i_{2j}}H_{i_{2j+1}i_{2j+2}}H_{i_{2j+3}i_{2j+4}}\cdots
H_{i_{4j-1}i_{4j}}\right]$ (4) $\displaystyle\hskip
56.9055pt\times\mathbb{E}\left[(U^{\sigma_{1}})_{i_{2}i_{3}}(U^{\sigma_{2}})_{i_{4}i_{5}}\cdots(U^{\sigma_{j}})_{i_{2j}i_{1}}\overline{(U^{\sigma_{1}})_{i_{2j+1}i_{2j+4}}(U^{\sigma_{2}})_{i_{2j+3}i_{2j+6}}\cdots(U^{\sigma_{j}})_{i_{4j-1}i_{2j+2}}}\right]$
$\displaystyle=$ $\displaystyle\sum_{i_{1},...,i_{4j}}\sum_{\pi\in
C_{2j}}\mathbbm{1}_{\forall
l\in\\{1,\dots,2j\\},(i_{2l-1},i_{2l})=(i_{2\pi(l)},i_{2\pi(l)-1})}$
$\displaystyle\hskip
56.9055pt\times\mathbb{E}\left[(U^{\sigma_{1}})_{i_{2}i_{3}}(U^{\sigma_{2}})_{i_{4}i_{5}}\cdots(U^{\sigma_{j}})_{i_{2j}i_{1}}\overline{(U^{\sigma_{1}})_{i_{2j+1}i_{2j+4}}(U^{\sigma_{2}})_{i_{2j+3}i_{2j+6}}\cdots(U^{\sigma_{j}})_{i_{4j-1}i_{2j+2}}}\right].$

The condition $(i_{2l-1},i_{2l})=(i_{2\pi(l)},i_{2\pi(l)-1})\,\,\forall
l\in\\{1,\dots,2j\\}\,\,\forall i_{1},...,i_{4j}\in\\{1,...,n\\}$ allows to
define a map $\pi\mapsto\tilde{\pi}$ from $C_{2j}$ to $C_{4j}$ by the formula

$\displaystyle\tilde{\pi}(2l-1)=$ $\displaystyle
2\pi(l),\quad\tilde{\pi}(2l)=2\pi(l)-1,\quad l=1,...,2j.$

Further we define the pairing $\rho\in C_{4j}$ as

$\displaystyle\rho:=(23)(45)\cdots(2j,1)(2j+1,2j+4)(2j+3,2j+6)\cdots(2j+2l-1,2j+2l+2)\cdots(4j-1,2j+2).$

See Example 4.2 for a list of the pairings $\pi$, $\tilde{\pi}$ and $\rho$,
for $j=2$, and Figure 1 for their depiction.

Note that $\rho$ and all pairings $\tilde{\pi}$ pair even numbers with odd
numbers, thus $\tilde{\pi}\rho$ maps even numbers to even numbers and odd
numbers to odd numbers. Using the pairing $\tilde{\pi}$, the even numbers
$i_{2},i_{4},...i_{4j}$ determine all the odd ones. Thus we see that

$\displaystyle\sum_{i_{1},...,i_{4j}}\sum_{\pi\in C_{2j}}\mathbbm{1}_{\forall
l\in\\{1,\dots,2j\\},(i_{2l-1},i_{2l})=(i_{2\pi(l)},i_{2\pi(l)-1})}$
$\displaystyle\hskip
56.9055pt\times\mathbb{E}\left[(U^{\sigma_{1}})_{i_{2}i_{3}}(U^{\sigma_{2}})_{i_{4}i_{5}}\cdots(U^{\sigma_{j}})_{i_{2j}i_{1}}\overline{(U^{\sigma_{1}})_{i_{2j+1}i_{2j+4}}(U^{\sigma_{2}})_{i_{2j+3}i_{2j+6}}\cdots(U^{\sigma_{j}})_{i_{4j-1}i_{2j+2}}}\right]$
$\displaystyle=$ $\displaystyle\sum_{\pi\in
C_{2j}}\sum_{i_{1},...,i_{4j}}\mathbbm{1}_{\forall
l\in\\{1,\dots,4j\\},i_{l}=i_{\tilde{\pi}(l)}}\mathbb{E}\left[(U^{\sigma_{1}})_{i_{2}i_{\rho(2)}}\cdots(U^{\sigma_{j}})_{i_{2j}i_{\rho(2j)}}\overline{(U^{\sigma_{1}})_{i_{\rho(2j+4)}i_{2j+4}}\cdots(U^{\sigma_{j}})_{i_{\rho(2j+2)}i_{2j+2}}}\right]$
$\displaystyle=$ $\displaystyle\sum_{\pi\in
C_{2j}}\sum_{i_{2},i_{4},...,i_{4j}}\mathbb{E}\left[(U^{\sigma_{1}})_{i_{2}i_{\tilde{\pi}\rho(2)}}\cdots(U^{\sigma_{j}})_{i_{2j}i_{\tilde{\pi}\rho(2j)}}\overline{(U^{\sigma_{1}})_{i_{\tilde{\pi}\rho(2j+4)}i_{2j+4}}\cdots(U^{\sigma_{j}})_{i_{\tilde{\pi}\rho(2j+2)}i_{2j+2}}}\right]$
(5) $\displaystyle=$ $\displaystyle\sum_{\pi\in
C_{2j}}\mathbb{E}\left(\sum_{i_{2},i_{4},...i_{4j}}\prod_{l=1}^{2j}(U^{\hat{\sigma}_{l}})_{i_{2l}i_{\tilde{\pi}\rho(2l)}}\right),$

where

$\displaystyle\hat{\sigma}_{l}=\begin{cases}\sigma_{l},&l=1,2,...,j,\\\
-\sigma_{l-j-1},&l=j+2,...,2j,\\\ -\sigma_{j},&l=j+1.\end{cases}$ (6)

By repeatedly applying $\tilde{\pi}\rho$ to $\\{2,4,...,4j\\}$, we get a
partition of $\\{2,4,...,4j\\}$ into orbits. The set of these orbits we denote
by $\mathcal{O}_{\tilde{\pi}\rho}$. We see that

$\displaystyle\begin{split}&\sum_{\pi\in
C_{2j}}\mathbb{E}\left(\sum_{i_{2},i_{4},...i_{4j}}\prod_{l=1}^{2j}(U^{\hat{\sigma}_{2l}})_{i_{2l}i_{\tilde{\pi}\rho(2l)}}\right)\\\
=&\sum_{\pi\in
C_{2j}}\mathbb{E}\left(\sum_{i_{2},i_{4},...i_{4j}}\prod_{o\in\mathcal{O}_{\tilde{\pi}\rho}}\prod_{w\in
o}(U^{\hat{\sigma}_{w}})_{i_{w}i_{\tilde{\pi}\rho(w)}}\right)\\\
=&\sum_{\pi\in
C_{2j}}\mathbb{E}\left(\prod_{o\in\mathcal{O}_{\tilde{\pi}\rho}}\operatorname{Tr}\left(\prod_{w\in
o}U^{\hat{\sigma}_{w}}\right)\right)\\\ =&\sum_{\pi\in
C_{2j}}\mathbb{E}\left(\prod_{o\in\mathcal{O}_{\tilde{\pi}\rho}}\operatorname{Tr}\left(U^{\sum_{w\in
o}\hat{\sigma}_{w}}\right)\right).\end{split}$ (7)

Putting together (4), (4), (6) and (7), we have proven the following
proposition:

######  Proposition 4.1.

Let $H$ be an $n\times n$ matrix from the $GUE(n)$, and let $U\in U(n)$ be
independent and Haar-distributed. Then for $j\in\mathbb{N}$ and
$\sigma_{1},...,\sigma_{j}\in\mathbb{N}$ it holds that

$\displaystyle\mathbb{E}\Big{(}\operatorname{Tr}\left(HU^{\sigma_{1}}HU^{\sigma_{2}}\dots
HU^{\sigma_{j}}\right)\overline{\operatorname{Tr}\left(HU^{\sigma_{1}}HU^{\sigma_{2}}\dots
HU^{\sigma_{j}}\right)}\Big{)}$ $\displaystyle=$ $\displaystyle\sum_{\pi\in
C_{2j}}\mathbb{E}\left(\prod_{o\in\mathcal{O}_{\tilde{\pi}\rho}}\operatorname{Tr}\left(U^{\sum_{w\in
o}\hat{\sigma}_{w}}\right)\right).$

######  Example 4.2.

For $j=2$ we see that $\rho=(14)(23)(58)(67)$, and (see Figure 1)

$\displaystyle\begin{split}&\pi=(12)(34),\quad\tilde{\pi}=(14)(23)(58)(67),\quad\tilde{\pi}\rho=(2)(4)(6)(8),\\\
&\pi=(13)(24),\quad\tilde{\pi}=(16)(25)(38)(47),\quad\tilde{\pi}\rho=(28)(46),\\\
&\pi=(14)(23),\quad\tilde{\pi}=(18)(27)(36)(45),\quad\tilde{\pi}\rho=(26)(48),\end{split}$

and that $\hat{\sigma}_{2}=\sigma_{1}$, $\hat{\sigma}_{4}=\sigma_{2}$,
$\hat{\sigma}_{6}=-\sigma_{2}$ and $\hat{\sigma}_{8}=-\sigma_{1}$. Thus from
Lemma 4.1 it follows that

$\displaystyle\mathbb{E}\Big{(}\operatorname{Tr}\left(HU^{\sigma_{1}}HU^{\sigma_{2}}\right)\overline{\operatorname{Tr}\left(HU^{\sigma_{1}}HU^{\sigma_{2}}\right)}\Big{)}$
$\displaystyle=$
$\displaystyle\mathbb{E}\left(\operatorname{Tr}U^{\sigma_{1}}\operatorname{Tr}U^{\sigma_{2}}\operatorname{Tr}U^{-\sigma_{2}}\operatorname{Tr}U^{-\sigma_{1}}\right)$
$\displaystyle+\mathbb{E}\left(\operatorname{Tr}U^{\sigma_{1}-\sigma_{1}}\operatorname{Tr}U^{\sigma_{2}-\sigma_{2}}\right)$
$\displaystyle+\mathbb{E}\left(\operatorname{Tr}U^{\sigma_{1}-\sigma_{2}}\operatorname{Tr}U^{\sigma_{2}-\sigma_{1}}\right)$
$\displaystyle=$
$\displaystyle\begin{cases}2\sigma_{1}^{2}+n^{2}+n^{2}&\sigma_{1}=\sigma_{2}\\\
\sigma_{1}\sigma_{2}+n^{2}+|\sigma_{1}-\sigma_{2}|&\sigma_{1}\neq\sigma_{2}\end{cases},$

where the last equality holds for large enough $n$ by Theorem 4.3.

$3$$2$$1$$8$$7$$6$$5$$4$$3$$2$$1$$8$$7$$6$$5$$4$$3$$2$$1$$8$$7$$6$$5$$4$
Figure 1: The pairing $\rho$ is in black, the three pairings $\tilde{\pi}$ in
$\tilde{C}_{8}$ are in red.

######  Theorem 4.3.

(Diaconis, Shahshahani [13]) Let $U$ be a Haar-distributed random matrix in
$U(n)$ and let $Z_{1},...,Z_{k}$ be i.i.d. standard complex Gaussian random
variables. Let $a=(a_{1},...,a_{k})$ and $b=(b_{1},...,b_{k})$ with
$a_{j},b_{j}\in\mathbb{N}$, and let $n\in\mathbb{N}$ be such that

$\displaystyle\max\left\\{\sum_{j=1}^{k}ja_{j},\sum_{j=1}^{k}jb_{j}\right\\}\leq
n.$

then

$\displaystyle\mathbb{E}\left(\prod_{j=1}^{k}((\operatorname{Tr}(U^{j}))^{a_{j}}\overline{(\operatorname{Tr}(U^{j}))^{b_{j}}}\right)=\delta_{ab}\prod_{j=1}^{k}j^{a_{j}}a_{j}!=\mathbb{E}\left(\prod_{j=1}^{k}(\sqrt{j}Z_{j})^{a_{j}}\overline{(\sqrt{j}Z_{j})^{b_{j}}}\right).$

## Acknowledgments

Our work was supported by ERC Advanced Grant 740900 (LogCorRM). Additionally,
Isao Sauzedde was funded by the EPSRC grant EP/W006227/1 during the later
stage of the writing process. We are most grateful to Jon Keating and Hugo
Falconet for very helpful discussions and suggestions, and to Thierry Lévy for
his valuable comments.

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