Youri Peskine
email: youri.peskine@eurecom.fr

Damir Korenčić

Ivan Grubišić
email: grubisic@irb.hr

Paolo Papotti
email: paolo.papotti@eurecom.fr

Raphael Troncy
email: raphael.troncy@eurecom.fr

Paolo Rosso
email: upvprosso@dsic.upv.es

Definitions Matter: Guiding GPT for Multi-label Classification

Large language models have recently risen in popularity due to their ability to perform many natural language tasks without requiring any fine-tuning. In this work, we focus on two novel ideas: (1) generating definitions from examples and using them for zero-shot classification, and (2) investigating how an LLM makes use of the definitions. We thoroughly analyze the performance of GPT-3 model for fine-grained multi-label conspiracy theory classification of tweets using zero-shot labeling. In doing so, we asses how to improve the labeling by providing minimal but meaningful context in the form of the definitions of the labels. We compare descriptive noun phrases, humancrafted definitions, introduce a new method to help the model generate definitions from examples, and propose a method to evaluate GPT-3's understanding of the definitions. We demonstrate that improving definitions of class labels has a direct consequence on the downstream classification results.

Introduction

Recent success of Large Language Models (LLMs) is due to their superior performance on various tasks [START_REF] Qin | Is ChatGPT a General-purpose Natural Language Processing Task Solver[END_REF][START_REF] Liu | Summary of ChatGPT-Related Research and Perspective Towards the Future of Large Language Models[END_REF][START_REF] Yang | Harnessing the power of llms in practice: A survey on chatgpt and beyond[END_REF], such as text generation [START_REF] Dai | AugGPT: Leveraging ChatGPT for Text Data Augmentation[END_REF][START_REF] Khalil | Will ChatGPT Get You Caught? Rethinking of Plagiarism Detection[END_REF], summarization [START_REF] Wang | Zero-Shot Cross-Lingual Summarization via Large Language Models[END_REF], question answering [START_REF] Peinl | Evaluation of medium-large language models at zero-shot closed book generative question answering[END_REF][START_REF] Tan | Can chatgpt replace traditional kbqa models? an in-depth analysis of the question answering performance of the gpt llm family[END_REF], information retrieval [START_REF] Omar | ChatGPT versus Traditional Question Answering for Knowledge Graphs: Current Status and Future Directions Towards Knowledge Graph Chatbots[END_REF], machine translation [START_REF] Hendy | How Good Are GPT Models at Machine Translation? A Comprehensive Evaluation[END_REF][START_REF] Jiao | Is ChatGPT A Good Translator? Yes With GPT-4 As The Engine[END_REF], and inductive reasoning [START_REF] Wei | Chain-of-Thought Prompting Elicits Reasoning in Large Language Models[END_REF][START_REF] Kojima | Large Language Models are Zero-Shot Reasoners[END_REF][START_REF] Zhong | Can ChatGPT Understand Too? A Comparative Study on ChatGPT and Fine-tuned BERT[END_REF]. However, LLMs had much less success with solving specific tasks [START_REF] Qin | Is ChatGPT a General-purpose Natural Language Processing Task Solver[END_REF], such as text classification, where they still lag behind fine-tuned transformer models [START_REF] Yang | Harnessing the power of llms in practice: A survey on chatgpt and beyond[END_REF]. Class definitions are commonly communicated to LLMs in the form of zero-shot prompts and few-shot exam- * Equal contribution ples, but both approaches are inferior to "defining" classes via a train set of annotated examples.

Misinformation has been a major research topic in the last few years, due to its significant impact on society. Social media have been a large vector of fake-news, especially during the COVID-19 pandemic. We revisit the problem of LLM zeroshot classification on the use-case of fine-grained multi-label conspiracy theory detection in tweets [START_REF] Langguth | COCO: an Annotated Twitter Dataset of COVID-19 Conspiracy Theories[END_REF], using GPT-3 model [START_REF] Brown | Language Models are Few-Shot Learners[END_REF]. We show that the challenging task of distinguishing conspiracy theories related to COVID-19 provides valuable insights into the capabilities and limitations of GPT-3. In particular, we aim at exploring the role played by the definition of the labels in zero-shot classification and explore the ability of GPT-3 in generating such definitions from a small set of labelled examples. Additionally, we propose a method to test the ability of GPT-3 to understand the definitions and apply them to classification1 .

Related Work

Zero-shot learning [START_REF] Larochelle | Zero-Data Learning of New Tasks[END_REF] is a machine learning problem where a model predicts classes without access to training data. It is based on the knowledge transfer from seen to unseen classes [START_REF] Ye | Zero-shot Text Classification via Reinforced Self-training[END_REF], using auxiliary information about the classes that can be provided to the model, such as text descriptions [START_REF] Larochelle | Zero-Data Learning of New Tasks[END_REF][START_REF] Ba | Predicting Deep Zero-Shot Convolutional Neural Networks Using Textual Descriptions[END_REF], semantic attributes [START_REF] Yongqin Xian | Zero-Shot Learning -A Comprehensive Evaluation of the Good, the Bad and the Ugly[END_REF][START_REF] Christoph H Lampert | Attribute-Based Classification for Zero-Shot Visual Object Categorization[END_REF] or concepts defined in ontologies [START_REF] Wang | Zero-shot recognition via semantic embeddings and knowledge graphs[END_REF]Zhang et al., 2019a). Within the domain of natural language processing (NLP), especially text classification, the zero-shot learning approach has been used in a variety of tasks, such as argument [START_REF] Zhang | Zero-shot label-aware event trigger and argument classification[END_REF], keywords [START_REF] Nam | All-in text: Learning document, label, and word representations jointly[END_REF], intent [START_REF] Liu | Reconstructing Capsule Networks for Zero-shot Intent Classification[END_REF] and utterance [START_REF] Yann N Dauphin | Zero-Shot Learning for Semantic Utterance Classification[END_REF] classification.

Unlike the traditional approach for zero-shot text classification, LLMs are capable of solving zero-shot classification with no auxiliary information [START_REF] Zhu | Can ChatGPT Reproduce Human-Generated Labels? A Study of Social Computing Tasks[END_REF][START_REF] Törnberg | ChatGPT-4 Outperforms Experts and Crowd Workers in Annotating Political Twitter Messages with Zero-Shot Learning[END_REF][START_REF] Kuzman | Chatgpt: Beginning of an end of manual annotation? use case of automatic genre identification[END_REF]. However, there is evidence that LLMs can outperform crowd workers in zero-shot text annotation tasks [START_REF] Gilardi | ChatGPT Outperforms Crowd-Workers for Text-Annotation Tasks[END_REF]. A different approach is to fine-tune the LLM with examples made of manually crafted lists of dictionary/encyclopedia entries related to the target label [START_REF] Gao | The Benefits of Label-Description Training for Zero-Shot Text Classification[END_REF] or to fine-tune smaller pre-trained language models with examples retrieved [START_REF] Yu | Re-Gen: Zero-Shot Text Classification via Training Data Generation with Progressive Dense Retrieval[END_REF] or generated by LLMs [START_REF] Zhang | Generation-driven Contrastive Self-training for Zero-shot Text Classification with Instructiontuned GPT[END_REF][START_REF] Ye | ZeroGen: Efficient Zero-shot Learning via Dataset Generation[END_REF][START_REF] Meng | Generating Training Data with Language Models: Towards Zero-Shot Language Understanding[END_REF].

Approaches that use class definitions for constructing zero-shot classifiers and prompts have been proposed, both for CNNs and transformers (Zhang et al., 2019b;[START_REF] Yin | Benchmarking zero-shot text classification: Datasets, evaluation and entailment approach[END_REF], and for LLMs [START_REF] Brown | Language Models are Few-Shot Learners[END_REF][START_REF] Gilardi | ChatGPT Outperforms Crowd-Workers for Text-Annotation Tasks[END_REF]. However, they do not experiment with different variants of class definitions, but use one definition type for classification and report the results. Additionally, to the best of our knowledge, no previous research has used LLMs to generate a class-specific definition directly from examples and used that information for zero-shot text classification as we investigate in this work.

Semantic abilities of LLMs are commonly evaluated on NL understanding and reasoning tasks [START_REF] Yang | Harnessing the power of llms in practice: A survey on chatgpt and beyond[END_REF], but work on targeted evaluation of fine-grained semantic properties is scarce. [START_REF] Sahu | Unpacking Large Language Models with Conceptual Consistency[END_REF] propose to evaluate the LLM's comprehension of query-related concepts by using a knowledge graph. To the best of our knowledge, there is no previous work focused on the ability LLMs to understand and apply definitions.

Methodology

In this section, we first describe the dataset used for conspiracy theory classification. Then, we present the approaches to definition-based zero-shot classification using GPT-3, and describe the method for evaluation of GPT-3's understanding of definitions. In this work, we use the term 'definition' to denote any additional explanation of the textual labels used to describe the class.

Conspiracy Theory Classification Dataset

The COCO dataset contains Twitter posts annotated w.r.t. 12 named COVID-19 related conspiracy theories [START_REF] Langguth | COCO: an Annotated Twitter Dataset of COVID-19 Conspiracy Theories[END_REF]. Part of this dataset was used in MediaEval FakeNews challenge [START_REF] Pogorelov | Combining Tweets and Connections Graph for FakeNews Detection at MediaEval[END_REF], where participants had to detect mentions of 9 conspiracy theories2 given the tweet texts. In this work, we use the Me-diaEval dataset, on which the best approaches used fine-tuned CT-BERT models [START_REF] Curran Associates | COVID-Twitter-BERT: A Natural Language Processing Model to Analyse COVID-19 Content on Twitter[END_REF][START_REF] Peskine | Detecting COVID-19-related conspiracy theories in tweets[END_REF][START_REF] Korenčić | Tackling Covid-19 Conspiracies on Twitter using BERT Ensembles, GPT-3 Augmentation, and Graph NNs[END_REF]. The test set has not been shared publicly before early 2023, we can safely use it in our experiments as it is not part of the training data of GPT-3 that contains data up to September 20213 .

Zero-shot Classification & Definitions

We leverage GPT-3 to perform multi-label zeroshot conspiracy theory detection on the test set of the MediaEval data. In particular, we perform binary classification for each conspiracy category, labeling each tweet as either mentioning the conspiracy or not.

Our baseline method relies on zero-shot (ZS) conspiracy theory classification from the textual label of the classes only (e.g. 'Anti-vaccination', 'Harmful Radiation', 'Satanism', etc). This assesses if the knowledge encoded in GPT-3 is able to differentiate between similar conspiracy theories.

Our next two approaches aim at improving the model's understanding of the label by providing more context in the prompt, specifically with a short definition of the label. We compare two types of definitions: Human-Written (HW) and Example-Generated (EG).

The HW definitions are given in the dataset overview paper [START_REF] Langguth | COCO: an Annotated Twitter Dataset of COVID-19 Conspiracy Theories[END_REF], and are part of the guidelines that were given to the human annotators of the data. Despite the definitions being well-written, annotators had to regularly discuss their understanding of the categories, suggesting the difficulty of the task at hand4 .

The EG definitions are generated with GPT-3 from the training set, by providing GPT-3 with 25 examples of tweets mentioning a given conspiracy theory and 25 examples of tweets not related to the conspiracy theory. We use 5 different random seeds to randomly select the example tweets5 , resulting in 45 definitions generated in total. We then ask the model to come up with a short textual description that could separate the sets of tweets. In this setting, we do not provide the textual label of the conspiracy theory, but we only give example tweets to the model. This prevents the model to rely on some of its pre-trained knowledge from reading the textual label. Examples of definitions which have been generated are in Appendix A.

For prompting the model we rely on simple prompts, using both OpenAI's 'system' and 'user' roles in our request. The 'system' message contains a description of the task, while the 'user' message contains the tweet's content to be classified. For the classification of the tweets, the definition is appended at the end of the 'system' message. Example prompts used to generate EG definitions and to annotate conspiracy theories are provided in Appendix B.

Definition Understanding

Approach of definition-based zero-shot classification leads to the question whether GPT-3 is able to correctly "interpret" definitions and "apply" them to text classification, which, in our case amounts to detection of conspiracy categories in texts. We propose two tests aimed at assessing if GPT-3 indeed "understands" the definitions given in the prompts.

The general approach is to use semantic similarity to measure how similarity between definitions correlates with the output of the definition-based classifiers, which we view as a result of GPT-3's "interpretation" and "application" of a definition. For example, one expectation is that similar definitions should lead to similar outputs. We perform the tests using the 45 example-generated definitions, which represent a challenging test case of mutually close definitions -randomly varied and derived from related categories. We define the semantic similarity of two definitions as cosine similarity of their embeddings, using state-of-art6 sentence embedding model [START_REF] Reimers | Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks[END_REF].

The first test of GPT-3's "understanding" of the definitions measures whether EG definitions more similar to HW ones guide the model to produce better classification results. This is achieved by correlating the similarity between the EG definitions and the corresponding HW ones, and the performance of the classifiers based on the generated definitions.

The second test measures whether mutually similar EG definitions guide the model to produce similar predictions. This is achieved by correlating the similarity between two EG definitions on one side, and the similarity of the corresponding classifiers' predictions on the other side. Similarity between two sets of predicted binary labels is calculated using Cohen's κ, a chance-corrected measure of annotator agreement.

Results

Conspiracy Theory Classification

In this section, we discuss the results of the different approaches on the classification of the full test set, totalling 823 tweets. Average results are in Table 1, and per-category results are in Figure 1. We use Matthews correlation coefficient, Precision, Recall and F1 score to compute the classification performance.

Results show that both EG and HW definitions outperform the ZS baseline. It supports the claim that GPT-3 is capable of leveraging the knowledge provided via the definitions to perform classification and, therefore, that definitions of the labels can be used to guide the model to better perform NLP tasks. While EG definitions do not reach the same performance as HW ones, they can still be used to significantly improve classification accuracy, especially in cases where the HW definition is not available. Our method shows that we can infer a textual description from examples and that GPT-3 can use it to better annotate future samples. Indeed, the usage of EG definitions leads to an average relative gain of around +10% in MCC, Precision and F1 scores compared to the ZS baseline. HW definitions see an even greater improvement of around +30% in average, showing the importance of a welldefined definition. However, these results are still far from the state-of-the-art CT-BERT fine-tuning methods.

Figure 1 reports the performances for all approaches per conspiracy theory. We observe a general trend with definitions having a positive impact on the performance for most conspiracy theories. However, some concepts are seemingly harder for GPT-3 to produce useful definitions, such as Satanism, where the EG definitions lead to worse results than the ZS baseline. Also, some conspiracies are more robust to the EG definitions, as the variance is low and changing the samples lead to similar results, such as Intentional Pandemic, or Fake Virus. Lastly, some EG definitions lead to better results than the HW ones, suggesting that with a better sampling of the examples, this method could generate higher-performing definitions.

Definition Understanding Tests

The Spearman's rank correlation coefficients between semantic similarity of the definitions and the results of the definition-based zero-shot classifiers are shared in Table 2. The strength of the correlations is fair, which supports the claim that GPT-3 is able to correctly interpret the definitions and apply them to conspiracy detection. Namely, higher similarity between EG and HW definitions leads to more accurate classifications, which suggest that the model can translate better definitions into better predictions. Additionally, higher similarity between two EG definitions correlates with higher agreement between their corresponding predictions, which suggest that the model translates similar definitions into similar predictions. An interesting question that stems from the variation of the definitions is whether the performance increase is a result of the quality or the quantity of information in the definitions. To address this question we correlated the length of the 45 EG definitions measured by the number of tokens with their classification performance measured by MCC. We found a lack of correlation -a very small ρ of 0.062. We take this as evidence supporting the claim that the performance depends on the quality, and not on the quantity, of information in a definition.

Discussion & Future Work

While classification of conspiracy theories using EG definitions is done in a zero-shot fashion, the generation of the definitions still relies on annotated examples. This is different than standard in-context few-shot classification as these examples do not need to be part of the classification prompt. Indeed, it can be seen as a way to compress the information from few-shot examples into a shorter descriptive context that can be appended in the zero-shot prompt. Further experiments could explore this approach and compare it to standard in-context few-shot classification.

The correlation tests of definition understanding in Section 4.2 support the claim that GPT-3 can indeed interpret and apply the definitions correctly. This is complemented by the results in Section 4.1 which show that better definitions lead to better results. However, further testing should be done on more LLMs and with other corpora. Such experiments are an interesting direction for future work with the potential to shed light on the semantic capabilities of LLMs.

The results of the definition-based zero-shot classifiers imply several practical recommendations and potential applications, all of which represent topics for future work. They include the use of the (high recall) classifiers to create more balanced samples for labeling, application of the classifiers to detect annotation errors, recommendations for mitigation of the low precision, and use of EG definitions for few-shot learning. More details on these topics can be found in Appendix C.

Conclusion

In this work, we analyze the impact of label definitions on the performance of GPT-3 zero-shot classification, on a challenging task of fine-grained conspiracy theory detection. We show that the use of better definitions leads to a significant gain in most metrics (MCC, Precision, F1). We also demonstrate an approach of generating definitions from examples. Human-Written definitions still provide better results, while example-generated definitions show promising performance. Additionally, we successfully tested GPT-3's ability to understand and apply these definitions for classification.

Limitations

We conduct our experiments using only the GPT-3.5 model, which is not open sourced, and is accessible only as a cloud service which might incur high usage costs, and forces the users to rely on a third-party service.

The experiments are conducted on conspiracy theory classification, and thus the performance im-provements might not be directly applicable to other multi-label classification tasks.

We mainly focus on providing better definitions for labels, but other factors, such as the quality of the input data, the prompt, and the model architecture can also play a role in improving classification performance.

The definition understanding analysis is based on several complex artifacts: LLMs, the model of semantic similarity, and the human-crafted definitions and annotations. Therefore, there might be other viable explanations of the obtained positive correlations, including unexpected interactions.

Ethics Statement

Our study aims to improve the classification of conspiracy theories in social media posts, which might assist in detecting and mitigating misinformation, thus contributing to a more reliable and trustworthy online environment.

Biases We are aware of the biases of LLMs in classification tasks [START_REF] Bender | On the Dangers of Stochastic Parrots: Can Language Models Be Too Big?[END_REF]. Using language models to generate definitions might lead to unintended biases in the generated definitions, which could impact the performance of the classifier. However, our goal is to study the impact of definitions on the performance of GPT-3, which can bring insights on how to reduce bias with the right prompts and more diverse training data.

Environmental Impact The use of large-scale Transformers requires a lot of computations and GPUs/TPUs for training, which contributes to global warming [START_REF] Strubell | Energy and policy considerations for modern deep learning research[END_REF]. This is a smaller issue in our case, as we do not train such models from scratch; rather, we fine-tune them on relatively small datasets or use models for inference in zero-shot settings. The definition of the concept is a conspiracy theory that suggests the existence of a deep state that is orchestrating the COVID-19 pandemic and blocking the release of Hydroxychloroquine, a cure for the virus. This theory also involves the belief that the pandemic is being used to push liberal agendas, create economic recession, help China's economy, and stop Trump rallies. It is often associated with the QAnon movement and involves the idea that Dr. Fauci is a Deep State Killer.

A Examples of Definitions

A.1.2 Example Generated (Seed 1)

The definition of the concept is the use of hydroxychloroquine (HCQ) as a possible treatment for COVID-19.

A.1.3 Human Written

Suppressed cures: Narratives which propose that effective medications for COVID-19 were available, but whose existence or effectiveness has been denied by authorities, either for financial gain by the vaccine producers or some other harmful intent.

A.2 Definitions of Harmful Radiation Conspiracy Category

A.2.1 Example Generated (Seed 4)

The definition of the concept is a conspiracy theory that claims that the COVID-19 pandemic is not caused by a virus, but rather by the implementation of 5G technology and that the pandemic is being used as a distraction to implement a digital currency and implantable microchips for population control. This theory has been widely discredited by scientific evidence and health organizations.

A.2.2 Example Generated (Seed 1)

The definition of the concept is a conspiracy theory that claims that the COVID-19 pandemic is caused by the deployment of 5G wireless networks. This theory suggests that the radiation from 5G towers weakens the immune system, making people more susceptible to the virus, or that the virus is somehow being transmitted through the 5G network. Despite being debunked by scientists and health experts, this theory has gained traction among some individuals and has led to the destruction of 5G towers in some areas.

A.2.3 Human Written

Harmful radiation: Narratives that connect COVID-19 to wireless transmissions, especially from 5 G equipment, claiming for example that 5 G is deadly and that COVID-19 is a coverup, or that 5 G allows mind control via microchips injected in the bloodstream.

A.3 Definitions of Depopulation Conspiracy Category

A.3.1 Example Generated (Seed 2)
The definition of the concept is depopulation, which refers to the deliberate reduction of the world's population through various means such as disease, vaccines, and other methods. The tweets in the first set suggest that there is a group of people, including Bill Gates, who are actively working towards depopulation as part of their agenda. The tweets in the second set, on the other hand, do not mention depopulation and focus on other topics related to COVID-19 such as conspiracy theories, mental health, and political issues.

A.3.2 Example Generated (Seed 3)

The definition of the concept is depopulation, which refers to the deliberate reduction of the human population through various means such as dis-ease, sterilization, or other forms of population control. The tweets in the first set suggest that some individuals believe that the COVID-19 pandemic and the development of vaccines are part of a larger depopulation agenda orchestrated by individuals such as Bill Gates. These beliefs are not present in the second set of tweets, which focus on other topics related to COVID-19 such as its impact on mental health, the effectiveness of vaccines, and political responses to the pandemic.

A.3.3 Human Written

Depopulation: Conspiracy theories on population reduction or population growth control suggest that either COVID-19 or the vaccines are being used to reduce population size, either by killing people or by rendering them infertile. In some cases, this is directed against specific ethnic groups.

A.4 Definitions of Satanism Conspiracy Category

A.4.1 Example Generated (Seed 0)

The definition of the concept is the belief in a conspiracy theory that the COVID-19 pandemic is part of a larger plan orchestrated by a group of powerful individuals or organizations, such as the Illuminati, the New World Order, or the Deep State, to exert more control over the population and ultimately lead to the implementation of the "mark of the beast," which could be a microchip or a vaccine that would be mandatory and used to monitor and control people's actions and movements. This belief often involves the idea that the pandemic is a hoax or exaggerated, and that the real agenda behind it is to push for a global government and depopulation.

A.4.2 Example Generated (Seed 3)

The definition of the concept is the belief that the COVID-19 vaccine or any other form of mandatory vaccination is the "Mark of the Beast" as described in the Book of Revelation. This belief is often associated with conspiracy theories involving the government, deep state, and Luciferian Freemasons who are seen as trying to control and enslave the population through the use of tracking chips and microchipped vaccines. The concept is rooted in religious and apocalyptic beliefs and is often used to justify opposition to vaccination and other public health measures.

A.4.3 Human Written

Satanism: Narratives in which the perpetrators are alleged to be some kind of satanists, perform objectionable rituals, or make use of occult ideas or symbols. May involve harm or sexual abuse of children, such as the idea that global elites harvest adrenochrome from children.

B Prompt Description

C Recommendations For Practical Use

In this section, we elaborate on some recommendations for applications of definition-based zero-shot classifiers. These recommendations are mainly motivated by the classification results from Section 4.1.

Fixing the class imbalance for labeling Recall of the definition-based zero-shot classifiers is high and comparable to the recall of the fine-tuned model. Therefore, a possible application of such classifiers is the selection of text data for labeling, with the goal of fixing the class imbalance, i.e., increasing the expected proportion of positive examples. This approach could help mitigate the rarity of positive examples in many text classification use-cases, such as various misinformation detection scenarios.

Correcting annotation errors Another potential application of the definition-based zero-shot classifiers is detecting and correcting annotation errors. The approach we propose is to perform error analysis of the classifiers based on human definitions, which are commonly used for text annotation. As suggested by low precision scores (see Table 1), the number of false positives is high -on average 145.89 texts per category for the test set of 830 texts. However, the number of false negatives is lower and more tractable (on average 27.11 texts per category). Additionally, high recall implies that the texts tend to be correctly detected as nonconspiracies, so the false negatives also seem more likely to identify examples wrongly annotated as conspiracies.

Our preliminary analysis indicates that this is indeed the case. We randomly selected 5 false negative texts per category and checked the annotations using the category definitions from [START_REF] Langguth | COCO: an Annotated Twitter Dataset of COVID-19 Conspiracy Theories[END_REF]. We found, on average, 3.8 labeling errors per category (76% of inspected texts).

Mitigating the low precision

The classification results in Table 1 show that the definition-based zero-shot classifiers suffer from low precision. This means that there is a high occurrence of false positives -texts belonging to other related categories being recognized as adhering to the definition of the category being classified. A possible remedy for this could be to upgrade the category definitions with text explicitly excluding similar categories.

Example-generated definitions use cases for fewshot learning An interesting use-case of EG definitions is the fact that they serve as a way to encode a lot of information into a shorter paragraph. Indeed, the LLMs can provide a descriptive definition of the task from a set of examples. This way, rather than providing all the examples each time we want to annotate a sample, we can provide a much shorter context, allowing to reduce the prompt size, and thus the cost, significantly.

Also, the quality of the definition matters, meaning we can actually use a more powerful model (such as GPT-4) to generate the definition, but still use a cheaper model to run the annotation (such as . This allows to annotate large amount of data with a higher-quality definition without increasing the cost by much.

Figure 1 :

 1 Figure 1: MCC score on the test set. Error bars show the minimum and maximum values (5 random seeds)

  Results of the two definition understanding tests based on semantic similarity and classification results. Top row contains Spearman's correlations of similarity between EG and HW definitions, and performance of EG zero-shot classifiers. Bottom row contains correlations of similarity between pairs of EG definitions, and cohen's kappa of their classification.

Table 1 :

 1 Performance of the LLM and transformer models using macro-averaging.

	Approach	MCC Precision Recall F1
	Zero-shot w/ Example-generated definitions 0.442 0.371 0.398 0.331	0.852 0.440 0.831 0.485
	w/ Human-written definitions	0.516 0.464	0.823 0.555
	CT-BERT ensembling	0.780 0.779	0.849 0.810

The code is available at: https://github.com/ dkorenci/gpt-def-zeroshot

Suppressed Cures (SUP), Behaviour Control (BHC), Anti-Vaccination (AVX), Fake Virus (FAK), Intentional Pandemic (INT), Harmful Radiation (HAR), Depopulation (DEP), New World Order (NWO), Satanism (SAT)

3 According to https://platform.openai.com/docs/ models/gpt-3-

54 The authors report a 92% inter-annotator agreement and more than half tweets had at least one disagreement.

Tweets in both sets can also support other conspiracy theories (multi-label classification problem)

We use top-ranked all-mpnet-base-v2 model: https: //www.sbert.net/docs/pretrained_models.html

Acknowledgements

This work has been partially supported by CHIST-ERA within the CIMPLE project (CHIST-ERA-19-XAI-003) and by ANR within the ECLADATTA project (ANR-22-CE23-0020). The work at the Universitat Politècnica de València is carried out in the framework of the XAI-DisInfodemics research project on eXplainable AI for disinformation and conspiracy detection during infodemics (Grant PLEC2021-007681), funded by MCIN/AEI/ 10.13039/501100011033 and by European Union NextGeneration EU/PRTR.


Jonas Lampart
email: lampart@math.cnrs.fr

David Mitrouskas
email: david.mitrouskas@ist.ac.at

Krzysztof Myśliwy
email: krzysztof.mysliwy@ist.ac.at

On the global minimum of the energymomentum relation for the polaron

For the Fröhlich model of the large polaron, we prove that the ground state energy as a function of the total momentum has a unique global minimum at momentum zero. This implies the non-existence of a ground state of the translation invariant Fröhlich Hamiltonian and thus excludes the possibility of a localization transition at finite coupling.

Introduction

The polaron describes an electron interacting with the quantized optical modes of a polarizable lattice. It poses a classic problem in solid state physics that was initiated by Landau's one-page paper [START_REF] Landau | Über die Bewegung der Elektronen in Kristallgitter[END_REF] about the possibility of self-trapping of an electron by way of deformation of the lattice. The existence of self-trapping for the polaron was debated in the physics literature over many decades, see e.g. [START_REF] Fisher | Ground-state symmetry of a generalized polaron[END_REF][START_REF] Gerlach | Absence of phonon-induced localization for the free optical polaron and the corresponding Wannier exciton-phonon system[END_REF][START_REF] Gerlach | Analytical properties of polaron systems or: Do polaronic phase transitions exist or not?[END_REF][START_REF] Luttinger | Generalized path-integral formalism of the polaron problem and its second-order semi-invariant correction to the ground-state energy[END_REF][START_REF] Matz | Dynamical Theory of the Large Polaron: Fock Approximation[END_REF][START_REF] Peeters | On the existence of a phase transition for the Fröhlich polaron[END_REF][START_REF] Spohn | Roughening and pinning transitions for the polaron[END_REF][START_REF] Tjablikow | Adiabatische Form der Störungstheorie im Problem der Wechselwirkung eines Teilchens mit einem gequantelten Feld[END_REF][START_REF] Toyozawa | Self-Trapping of an electron by the acoustical mode of lattice vibration[END_REF]. Mathematically it is related to the existence of a normalizable ground state for the translation invariant Hamiltonian. While for weak coupling, the non-existence of ground states is almost immediately evident, the conclusion is less obvious for strong coupling. In fact, for the semi-classical theory, which is known to be accurate in the limit of strong coupling [START_REF] Donsker | Asymptotics for the polaron[END_REF][START_REF] Lieb | Exact ground state energy of the strong-coupling polaron[END_REF], the translational symmetry is broken on the level of the ground state [START_REF] Lieb | Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation[END_REF][START_REF] Pekar | Untersuchung über die Elektronentheorie der Kristalle[END_REF]. This supported the idea about the existence of a localization transition at finite coupling.

In this work, we give a mathematical proof that such a transition does not occur for the Fröhlich Hamiltonian. This Hamiltonian models the large polaron, where the lattice is described by a continuous quantum field with constant dispersion relation. We prove that the translation invariant Hamiltonian does not have a ground state, irrespective of the value of the coupling. To achieve this, we study the energy-momentum relation, that is, the ground state energy as a function of the conserved total momentum, and show that it attains its global unique minimum at momentum zero. The non-existence of ground states is then a simple consequence of this property. Quoting [6, p. 1575], this solves a physically obvious but mathematically long-standing conjecture.

Following H. Fröhlich [START_REF] Fröhlich | Theory of electrical breakdown in ionic crystals[END_REF][START_REF] Fröhlich | Electrons in lattice fields[END_REF] the large polaron is defined on the Hilbert space

H = L 2 (R 3 , dx) ⊗ F (1.1)
with F = ∞ n=0 n sym L 2 (R 3 ) the bosonic Fock space, and governed by the Hamiltonian

H = -∆ x + N + ϕ(v x ), (1.2)
where -∆ x = (-i∇ x ) 2 , x and -i∇ x describing the position and momentum of the electron and N = dΓ(1) denotes the number operator on Fock space (modeling the energy of the phonon field, whose dispersion relation is constant). The interaction is described by the linear field operator

ϕ(v x ) = a(v x ) + a * (v x ) with v x (y) = v(y -x), v(y) = √ α 1 |y| 2 .
(1.

3)

The bosonic creation and annihilation operators, a * , a, satisfy the usual canonical commutation relations. After setting ℏ = 1 and the mass of the electron equal to 1/2, the polaron model depends on a single dimensionless parameter α > 0. In the present work, the choice of α is not relevant, as our statements shall hold equally for all values of the coupling constant. An important property of the Fröhlich Hamiltonian H is that it defines a translationinvariant system, in the sense that it commutes with the total momentum operator [H, -i∇ x + P f ] = 0 (1.4) where P f = dΓ(-i∇) describes the momentum of the phonons. As a consequence, it is possible to simultaneously diagonalize the total momentum and the energy. This simultaneous diagonalization is implemented best by the Lee-Low-Pines transformation [START_REF] Lee | The motion of slow electrons in a polar crystal[END_REF] S : H → ⊕ R 3 F dP defined by F •e iP f x where F indicates the Fourier transformation w.r.t. x. As is easily revealed by a direct computation, S(-i∇ x + P f )S * = ⊕ R 3 P dP and SHS * = ⊕ R 3 H(P ) dP with

H(P ) = (P -P f ) 2 + N + ϕ(v).
(1.5)

The fiber operator H(P ) acts on F and describes the system at total momentum P ∈ R 3 . Irrespectively of whether we take (1.2) or (1.5) as our starting point, the definition of the model is somewhat formal, since v / ∈ L 2 . A common way to define the Fröhlich model is to start from the quadratic form associated with H(P ) (or H) and show that it is an infinitesimal perturbation of the form associated with the non-interacting operator (α = 0). By the KLMN Theorem this implies the existence of a unique self-adjoint operator associated with the form of H(P ) (or H), which we call the fibered Fröhlich Hamiltonian. For convenience of the reader we give the details in the appendix.

The main topic of this paper is the energy-momentum relation of the polaron, defined as the lowest possible energy as a function of the total momentum E α (P ) := inf σ(H(P )).

(1.6)

By rotation invariance of the Fröhlich Hamiltonian, E α (P ) = E α (RP ) for all rotations R ∈ O(3) and hence E α (P ) = E rad,α (|P |). It was shown by L. Gross [START_REF] Gross | Existence and uniqueness of physical ground states[END_REF] that E α (0) ≤ E α (P ) for all P ∈ R 3 . As recently pointed out in a paper by Dybalski and Spohn [START_REF] Dybalski | Effective mass of the polaron -revisited[END_REF], it is further conjectured that P = 0 is the unique global minimum E α (0) < E α (P ) for all P ̸ = 0.

(1.7)

While this inequality is not surprising from the physics point of view, its mathematical proof turns out to be less obvious. In fact already Gross' proof for showing that P = 0 is a global minimum requires a considerable amount of work. In a small neighborhood around zero the strict inequality can be inferred from the finiteness of the effective mass [START_REF] Dybalski | Effective mass of the polaron -revisited[END_REF] (the effective mass is defined by (M eff,α ) -1 = E ′′ rad,α (P )| P =0 ). The goal of this paper is to prove (1.7) for all non-zero P ∈ R 3 . A proof was proposed in 1991 by Gerlach and Löwen [START_REF] Gerlach | Analytical properties of polaron systems or: Do polaronic phase transitions exist or not?[END_REF]. However, their argument is not completely rigorous and its validity has been recently debated [START_REF] Dybalski | Effective mass of the polaron -revisited[END_REF]. Our aim is to follow the idea suggested by Gerlach and Löwen and turn it into a complete mathematical proof. To this end we also benefit from results on the Fröhlich Hamiltonian that became available more recently.

Approaching the problem from a different angle, Dybalski and Spohn provide a proof of (1.7) under the assumption that a central limit theorem for the polaron path measure with two-sided pinning holds, see [START_REF] Dybalski | Effective mass of the polaron -revisited[END_REF]Sec. 5], in particular Theorem 5.3 and Conjecture 5.2 therein. To our knowledge, this particular version of the central limit theorem, however, has not been rigorously established so far.

At the time of finalizing our paper, we were informed by Polzer about his new work [START_REF] Polzer | Renewal approach for the energy-momentum relation of the Fröhlich polaron[END_REF] that presents a proof of (1.7) based on a strategy, also in the probabilistic framework, that does not rely on the validity of the central limit theorem with two-sided pinning. In this new approach the polaron path measure is represented as a mixture of Gaussian measures and analyzed with techniques from renewal theory (see also [START_REF] Betz | A functional central limit theorem for polaron path measures[END_REF][START_REF] Betz | Effective mass of the Polaron: a lower bound[END_REF][START_REF] Mukherjee | Identification of the polaron measure I: Fixed coupling regime and the central limit theorem for large times[END_REF]). Among other interesting properties, [START_REF] Polzer | Renewal approach for the energy-momentum relation of the Fröhlich polaron[END_REF] shows that E rad,α (|P |) is a monotone increasing function with strict monotonicity below the essential spectrum, i.e., in particular in a neighborhood around P = 0.

In our proof we follow an operator-theoretic approach that is inspired by [START_REF] Faris | Invariant cones and uniqueness of the ground state for fermion systems[END_REF][START_REF] Fröhlich | Existence of dressed one-electron states in a class of persistent models[END_REF][START_REF] Gerlach | Analytical properties of polaron systems or: Do polaronic phase transitions exist or not?[END_REF][START_REF] Lampart | The resolvent of the Nelson Hamiltonian improves positivity[END_REF][START_REF] Lampart | On Nelson-type Hamiltonians and abstract boundary conditions[END_REF][START_REF] Møller | The translation invariant massive nelson model: I. The bottom of the spectrum[END_REF][START_REF] Miyao | Nondegeneracy of ground states in nonrelativistic quantum field theory[END_REF][START_REF] Spohn | The polaron at large total momentum[END_REF]. Except for the following lemma, whose proof is based on results from [START_REF] Møller | The polaron revisited[END_REF] (see Section 2), our presentation is fully self-contained. 1Lemma 1.1. Let α > 0 and set E α (P ) = inf σ(H(P )).

(i) σ ess (H(P )) = [E α (0) + 1, ∞) for all P ∈ R 3 (ii) The set of global minima of P → E α (P ) is non-empty.
Before we continue with our main results, let us give some more remarks on the context of the problem. For the non-interacting model, i.e. for α = 0, one easily sees that E 0 (P ) = min{P 2 , 1} with P 2 describing the region where the full momentum is on the electron whereas in the constant region it is favorable that the total momentum is carried by a single phonon with the electron at rest. The expectation from physics is that the form of E α (P ) remains qualitatively the same also for non-zero α. Lemma 1.1(i), also known as the HVZ Theorem, confirms this for the essential spectrum, whose threshold corresponds to an electron at rest and a free phonon. One expects that the shape of E α (P ) remains essentially a parabola in a certain region around zero (though with a negative shift E α (0) and with different curvature at P = 0) and that it approaches E α (0) + 1 as |P | → ∞. That the expected parabola gives in fact an exact upper bound, E α (P ) ≤ E α (0) + P 2 /(2M eff,α ) for all P and α, was shown in [START_REF] Polzer | Renewal approach for the energy-momentum relation of the Fröhlich polaron[END_REF]. The mathematical verification of the described picture is particularly interesting in the strong coupling limit α → ∞, where one can analyze the asymptotic expansion of E α (P ) and M eff,α . For results about the strong-coupling limit see [2-5, 21, 22, 27, 38].

As already alluded to at the beginning of the paper, the form of the energy-momentum relation is also related to the existence of a ground state of the translation invariant Hamiltonian H. For weak coupling the non-existence is physically evident since for α = 0 no eigenstates exist. For strong coupling, on the other hand, there is a leading-order contribution of the classical polarization field and hence the interaction cannot be viewed as a perturbation of a translation-invariant Hamiltonian anymore. In fact, in the limit of strong coupling, the ground state energy approaches the corresponding semiclassical energy [START_REF] Donsker | Asymptotics for the polaron[END_REF][START_REF] Lieb | Exact ground state energy of the strong-coupling polaron[END_REF][START_REF] Pekar | Untersuchung über die Elektronentheorie der Kristalle[END_REF] which is known to possess a manifold of localized ground states [START_REF] Lieb | Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation[END_REF]. This supported the idea that also for the full quantum model, the translational symmetry may be broken on the level of the ground state for sufficiently large coupling, thus resulting in a transition from a delocalized regime (weak coupling) to a localized one (strong coupling). The existence of a ground state for H, however, would require that the set of global minima of E α (P ) is a set of non-zero Lebesgue measure. As our result implies that the global minimum is unique, this possibility is ruled out and thus we obtain a rigorous proof (see Corollary 1.3) of the non-existence of such a localization transition at any finite value of the coupling parameter.

Main results

The next statement is our main result.

Theorem 1.2. The energy-momentum relation E α (P ) = inf σ(H(P )) has its unique global minimum at P = 0, i.e. for every α > 0 and every P ̸ = 0, we have the strict inequality

E α (0) < E α (P ).
(

As a corollary we obtain that the Fröhlich Hamiltonian H does not possess a ground state, irrespective of the value of the coupling constant.

Corollary 1.3. There exists no α ∈ (0, ∞) such that inf σ(H) is an eigenvalue of H.

Proof. Suppose on the contrary that there exists an α ∈ (0, ∞) such that H has a ground state Ψ α ∈ H . By the Lee-Low-Pines transformation (see introduction),

inf σ(H) = R 3 dP ⟨Ψ α (P ), H(P )Ψ α (P )⟩ R 3 dP ∥Ψ α (P )∥ 2 ≥ R 3 dP E α (P )∥Ψ α (P )∥ 2 R 3 dP ∥Ψ α (P )∥ 2 (1.9) with Ψ α (P ) = F [e iP f x Ψ α ](P ) ∈ F where F is the Fourier transform w.r.t x. Note that ∥Ψ α (P )∥ 2 ∈ L 1 (R 3 ) by Parseval and e iP f x Ψ α ∈ L 2 (R 3 , F). Now by Theorem 1.2 the set {P ∈ R 3 | E α (P ) = E α (0)
} has Lebesgue measure zero, and hence the right side of (1.9) is strictly larger than E α (0). On the other hand, (1.8) together with the fiber decomposition of H imply that inf σ(H) = E α (0), which yields a contradiction.

Remark 1.1. Our results do not depend on the precise form of v. The choice of the sign of v in (1.3) does not matter, as the operator with form factor ṽ(y) = e iθ v(y) is unitarily equivalent to the one with v via Γ(e iθ ). Moreover, our proof can easily be generalized to all positive (up to a global phase) v satisfying v(k)(k 2 + 1) -s ∈ L 2 (R 3 ) for some 0 ≤ s < 1/2. For the sake of conciseness we focus on (1.3)

Proof of Theorem 1.2

To prove Theorem 1.2 we introduce a suitable (non-physical) auxiliary Hamiltonian and analyze the degeneracy of its ground state. On the one hand, it shall be easily seen that the multiplicity is strictly larger than one if we assume that P → E α (P ) attains its global minimum for some q ̸ = 0. On the other hand, this cannot be the case as it will follow by an abstract result that the ground state of this auxiliary operator is non-degenerate. As mentioned in the introduction, this idea goes back to Gerlach and Löwen [START_REF] Gerlach | Analytical properties of polaron systems or: Do polaronic phase transitions exist or not?[END_REF].

By Lemma 1.1 (ii) there exists a q ∈ R 3 such that E α (q) ≤ E α (P ) for all P ∈ R 3 and by rotational symmetry, we can assume that q = (|q|, 0, 0). For said q we choose ℓ = ℓ(q) > 0 such that q ∈ (2π/ℓ)Z 3 and consider the Hilbert space

H ℓ = L 2 (Λ ℓ ) ⊗ F, (2.1)
where Λ ℓ = [0, ℓ] 3 is the three-dimensional cube of side length ℓ with periodic boundary conditions, and F denotes again the bosonic Fock space over L 2 (R 3 ). On this space we define (in the sense of quadratic forms)

h ℓ = (-i∇ x -P f ) 2 + N + ϕ(v) (2.2)
where -i∇ x now denotes the gradient on L 2 (Λ ℓ ). The reason for restricting the variable x to the torus is that h ℓ now has a ground state. Indeed, since x only appears in the gradient, and since the corresponding momenta are restricted to (2π/ℓ)Z 3 , one can readily diagonalize

h ℓ = P ∈ 2π ℓ Z 3 π ℓ,P ⊗ H(P ) (2.3)
with π ℓ,P = |φ ℓ,P ⟩⟨φ ℓ,P | the projection onto the normalized plane wave φ ℓ,P = ℓ -3/2 e -iP x ∈ L 2 (Λ ℓ ) with momentum P . Here it is important to note that the Fock space component is described by the fiber Hamiltonian H(P ) as defined in (1.5). Also note that since H(P ), P ∈ R 3 , is self-adjoint and bounded from below, so is h ℓ . By Lemma 1.1 we know that E α (q) < inf σ ess (H(q)), hence there is an eigenfunction ξ(q) ∈ F such that H(q)ξ(q) = E α (q)ξ(q). Since E α (q) ≤ E α (P ) for all P ∈ R 3 , the wave function φ ℓ,q ⊗ ξ(q) ∈ H ℓ is a ground state of h ℓ with eigenvalue E α (q). The next lemma, whose proof is postponed to the next section, shows that this ground state is actually unique. Proposition 2.1. For all α > 0 and any ℓ > 0 for which inf σ(h ℓ ) is an eigenvalue (in particular for the value ℓ(q) chosen above), this eigenvalue has multiplicity one.

With this at hand, we can prove Theorem 1.2 by way of contradiction. To this end assume that q ̸ = 0. By rotational symmetry E α (q) = E α (-q) and, since E α (-q) < inf σ ess (H(-q)) by Lemma 1.1, the fiber Hamiltonian H(-q) has an eigenfunction ξ(-q) ∈ F with eigenvalue E α (-q). This implies that there is a second ground state of h ℓ , given by φ ℓ,-q ⊗ ξ(-q) ∈ H ℓ , that is orthogonal to φ ℓ,q ⊗ ξ(q). Thus we arrive at a contradiction to the statement of Proposition 2.1, and therefore we can rule out that q is non-zero. This shows that P → E α (P ) can attain its global minimum only at q = 0, hence the proof of Theorem 1.2 is complete.

In the remainder of this subsection we prove Lemma 1.1. For that purpose we first state and prove the following general result.

Lemma 2.2. Let A n , n ∈ N, be a sequence of self-adjoint operators satisfying σ ess (A n ) = [a n , b n ] with -∞ ≤ a n ≤ b n ≤ ∞. If A n n→∞ ---→ A in norm-resolvent sense for some self-adjoint operator A, then a n n→∞ ---→ a, b n n→∞ ---→ b for some -∞ ≤ a ≤ b ≤ ∞ and σ ess (A) = [a, b]. Proof of Lemma 2.2. First assume that a n , b n n→∞ ---→ a, b for some -∞ ≤ a ≤ b ≤ ∞. Since σ ess (A n ) ̸ = ∅, the essential spectrum of A cannot be empty [36, Thm. XIII.23]. Now, for ε > 0 we choose N (ε) such that [a n , b n ] ⊂ [a -ε, b + ε] for all n ≥ N (ε). It follows that σ(A n ) is purely discrete in (-∞, a -ε) ∪ (b + ε, ∞) for all n ≥ N (ε)
. By [START_REF] Oliveira | Intermediate spectral theory and quantum dynamics[END_REF]Prop. 11.4.31] this implies that σ(A) is purely discrete on the same interval and thus, using that ε was arbitrary,

σ ess (A) ⊂ [a, b]. Since σ ess (A) ̸ = ∅, we get σ ess (A) = {a} if a = b.
For a < b, let t ∈ (a, b)∩ρ(A). By norm-resolvent convergence we have that if t ∈ ρ(A), then t ∈ ρ(A n ) for all n sufficiently large [32, Prop. 10. To remove the initial assumption on a n , b n , assume there is no a ∈ [-∞, ∞] such that a n n→∞ ---→ a (the same argument applies to b n ). Then there are at least two subsequences of a n with limits a, ã ∈ [-∞, ∞], a ̸ = ã. Applying the argument from above to the two subsequences (and the associated subsequences of A n ) separately leads to the contradiction that

[a, b] = σ ess (A) = [ã, b].
Proof of Lemma 1.1. The proof is based on the results from [START_REF] Møller | The polaron revisited[END_REF]. Since the latter considers more regular polaron models with v ∈ L 2 (R 3 ), we need to introduce the fiber Hamiltonians with UV cutoff. We define H Λ (P ) as in (1.5) with form factor v Λ defined by vΛ = χ Λ v where

χ Λ is the characteristic function k → χ [0,Λ) (|k|). Since v Λ ∈ L 2 (R 3
), it is easy to verify that H Λ (P ) is self-adjoint and bounded from below. Denoting E α,Λ (P ) = inf σ(H Λ (P )), we can quote two important results from [START_REF] Møller | The polaron revisited[END_REF]: For every Λ > 0

(i') σ ess (H Λ (P )) = [E α,Λ (0) + 1, ∞), (ii') lim |P |→∞ |E α,Λ (P ) -inf σ ess (H Λ (P ))| = 0.
These statements are given in [START_REF] Møller | The polaron revisited[END_REF]Thm. 2.1. and Thm. 2.4] for the choice ω ≡ 1.

To show (i), note that by Proposition A.2, we also have H Λ (P ) Λ→∞ ---→ H(P ) in normresolvent sense, which implies E α,Λ (0) Λ→∞ ---→ E α (0) = inf σ(H(0)). In fact, by [START_REF] Oliveira | Intermediate spectral theory and quantum dynamics[END_REF]Prop. 10.2.4a] there exists a sequence of elements t Λ ∈ σ(H Λ (0)) converging to E α (0) as Λ → ∞, in particular t Λ ≥ E α,Λ (0) for all Λ, which shows that every accumulation point of the sequence E α,Λ (0) is no larger than E α (0). On the other hand, by [START_REF] Oliveira | Intermediate spectral theory and quantum dynamics[END_REF]Prop. 10.2.4b] E α,Λ (0) has no accumulation point below E α (0), and thus E α,Λ (0) Λ→∞ ---→ E α (0), and hence (i) is a consequence of (i') and Lemma 2.2.

To show (ii), we follow the argument from [START_REF] Dybalski | Effective mass of the polaron -revisited[END_REF]Sec. 4]. First note that by the Lee-Low-Pines transformation E := inf P ∈R 3 E α (P ) coincides with inf σ(H) and thus E > -∞. (ii) states that M = {P ∈ R 3 | E α (P ) = E} is a non-empty set. Assuming the opposite, M = ∅, implies the existence of a sequence (P j ) j∈N ⊂ R 3 , |P j | → ∞, satisfying lim j→∞ E α (P j ) = E. W.l.o.g. we can choose P j such that |P j | is monotone increasing and

E α (P j ) ∈ (E, E + 1 4 ) for all j ≥ 1. Further consider f ∈ C ∞ c (R) with supp(f ) ⊆ (E -1 2 , E + 1 2
) and f (s) = 1 for all s ∈ (E -

sup P ∈R 3 ||f (H Λ (P )) -f (H(P ))|| ≥ lim Λ→∞ sup j≥j Λ ||f (H Λ (P j )) -f (H(P j ))|| = lim Λ→∞ sup j≥j Λ ||f (H(P j ))|| (2.4)
where the last step follows from (i') and (ii') if we choose j Λ large enough. In more detail, for every Λ, we can choose j Λ such that σ(H

Λ (P j )) ∩ (-∞, E α,Λ (0) + 3 4 ) = ∅ for all j ≥ j Λ . Since E ≤ E α (0) ≤ E α,Λ (0) + 1 4 (here we use again E α,Λ (0)
Λ→∞ ---→ E α (0)), this implies σ(H Λ (P j )) ∩ (-∞, E + 1 2 ] = ∅ for all j ≥ j Λ , and thus f (H Λ (P j )) = 0. By assumption, however, inf σ(H(P j )) ∈ (E, E + 1 4 ) for all j ≥ 1 and thus the right side of (2.4) is one, which is a contradiction.

Proof of Proposition 2.1

In this section we show that the resolvent of h ℓ is a positivity improving operator w.r.t. a suitable Hilbert cone using a strategy that was applied to the renormalized Nelson model in [START_REF] Lampart | The resolvent of the Nelson Hamiltonian improves positivity[END_REF]. (Compared to the latter we face the additional difficulty that h ℓ is an operator on the tensor product H ℓ and not only on F). The statement of Proposition 2.1 will then follow from a Perron-Frobenius type argument due to Faris [START_REF] Faris | Invariant cones and uniqueness of the ground state for fermion systems[END_REF]. Let

C := Ψ ∈ H ℓ ∀n ∈ N 0 : (-1) n Ψ (n) (x, y 1 , . . . , y n ) ≥ 0 . (2.5)
This defines a (Hilbert) cone in H ℓ in the sense of [START_REF] Faris | Invariant cones and uniqueness of the ground state for fermion systems[END_REF]. A bounded operator K on H ℓ is called

• positivity preserving (with respect to C) if Ku ∈ C for any u ∈ C, and

• positivity improving (with respect to C) if ⟨Ku, v⟩ > 0 for any u, v ∈ C \ {0}.

We recall that v(y) = √ α|y| -2 > 0 and thus

v(k) = √ α(4π|k|) -1 . (2.6)
Taking into account the alternating sign in the definition of C, the interaction energy is negative on C, i.e., ⟨Ψ, a(v)Ψ⟩ ≤ 0 for Ψ ∈ C. Moreover, ϕ(-v) is positivity preserving, which is the key property to be used in the proof of the positivity-improving property in Proposition 2.5 below. Now consider the self-adjoint positive operator

h ℓ,0 = (-i∇ x -P f ) 2 + N + 1 (2.7)
acting on the Hilbert space H ℓ = L 2 (Λ ℓ ) ⊗ F, which is essentially h ℓ without the interaction term ϕ(v).

Lemma 2.3. The resolvent h -1 ℓ,0 is positivity preserving with respect to the cone C. Its restriction to the zero-phonon space L 2 (Λ ℓ ) is positivity improving with respect to the cone of positive functions in L 2 (Λ ℓ ).

Proof. The restriction of h -1

ℓ,0 to L 2 (Λ ℓ ) is given by the periodization of the resolvent of -∆ on R 3 , i.e. it has the integral kernel

k∈Z 3 e -|x-x ′ -kℓ| 4π|x -x ′ -kℓ| , (2.8)
which converges due to the exponential decay of the numerator and is strictly positive. Similarly, the integral kernel of h -1 ℓ,0 on the n-phonon space F (n) is (as one readily checks using the Fourier transform)

k∈Z 3 e - √ n+1|x-x ′ -kℓ| 4π|x -x ′ -kℓ| n j=1 δ(y j -x -y ′ j + x ′ + kℓ), (2.9)
which is a non-negative distribution.

Lemma 2.4. The operator a(v)h -1 ℓ,0 is bounded, and 1 + a(v)h -1 ℓ,0 is invertible, with bounded inverse given by

(1 + a(v)h -1 ℓ,0 ) -1 = ∞ j=0 a(-v)h -1 ℓ,0 j ,
where the sum converges in the operator norm.

Proof. For n ∈ N we have using the Fourier transform and Cauchy-Schwarz

∥a(v)h -1/2 ℓ,0 Ψ (n) ∥ 2 L 2 (Λ ℓ )⊗F (n-1) = n p∈ 2π ℓ Z 3 R 3(n-1) dk 1 • • • dk n-1 dk n v(k n )Ψ (n) (p, k 1 , . . . , k n ) ((p + k 1 + • • • k n ) 2 + n + 1) 1/2 2 ≤ n∥Ψ (n) ∥ 2 sup k∈R 3 R 3 dξ |v(ξ)| 2 (k + ξ) 2 + n + 1 ≤ (n + 1) 1/2 ∥Ψ (n) ∥ 2 α (4π) 2 R 3 dη η 2 (η 2 + 1)
.

(2.10)

where we used (2.6) and performed the change of variables η = ξ/ √ n + 1 in the last step. We thus have the inequality

∥a(v)h -1/2 ℓ,0 Ψ∥ ≤ C∥(N + 1) 1/4 Ψ∥, (2.11)
which implies that

∥a(v)h -1 ℓ,0 Ψ∥ ≤ C∥(N + 1) -1/4 Ψ∥ (2.12)
for some constant C that depends on α. For j ∈ N this implies ∥(a(v)h -1 ℓ,0 ) j Ψ∥ ≤ C∥(a(v)h -1 ℓ,0 ) j-1 (N + j) -1/4 Ψ∥ ≤ C j (j!) -1/4 ∥Ψ∥.

(2.13) Since C j (j!) -1/4 is summable, the Neumann series converges and one easily checks that the limit is the inverse of 1 + a(v)h -1 ℓ,0 .

With the operator h ℓ,0 we can write the auxiliary operator (2.2) as

h ℓ + 1 = (1 + a(v)h -1 ℓ,0 )h ℓ,0 (1 + h -1 ℓ,0 a * (v)) =:K -a(v)h -1 ℓ,0 a * (v) =:T = K + T.
Proposition 2.5. Let α > 0. For all λ > -inf σ(h ℓ ) the resolvent (h ℓ + λ) -1 is positivity improving with respect to C.

Proof. It is sufficient to prove the statement for one λ > -inf σ(h ℓ ), by analyticity of the resolvent (see also [START_REF] Møller | The polaron revisited[END_REF]Lem. A.4]). For sufficiently large λ we shall first prove

(h ℓ + 1 + λ) -1 = (K + T + λ) -1 = (K + λ) -1 1 + T (K + λ) -1 -1 = (K + λ) -1 ∞ j=0 -T (K + λ) -1 j . (2.14)
To this end, we show that T is infinitesimally K-bounded. In fact, by (2.11),

∥T Ψ∥ ≤ C∥(N + 1) 1/2 Ψ∥ ≤ ε∥N Ψ∥ + C ε ∥Ψ∥, (2.15)
for arbitrary ε > 0, and moreover,

∥N Ψ∥ ≤ ∥N (1 + h -1 ℓ,0 a * (v))Ψ∥ + ∥N h -1 ℓ,0 a * (v)Ψ∥ ≤ ∥h ℓ,0 (1 + h -1 ℓ,0 a * (v))Ψ∥ + C∥(N + 1) 3/4 Ψ∥ ≤ ∥(1 + a(v)h -1 ℓ,0 ) -1 ∥∥KΨ∥ + 3 4 ∥N Ψ∥ + ( 1 4 C 4 + 3 4 )∥Ψ∥. (2.16)
Proceeding with (2.14), we use that -T is positivity preserving, since it is the product of the three positivity preserving operators h -1 ℓ,0 (see Lemma 2.3) and a(-v), a * (-v) (note the alternating sign in the definition of C). The claim will thus follow if we can prove that (K + λ) -1 improves positivity. As K ≥ 1, this holds for all λ > -1 if it holds for λ = 0.

Let 0 ̸ = Ψ, Φ ∈ C and let n, m be so that Ψ

(n) ̸ = 0, Φ (m) ̸ = 0. Set ϕ := (a(-v)h -1 ℓ,0 ) m Φ (m) and ψ := (a(-v)h -1 ℓ,0 ) n Ψ (n)
. Since a(-v) and h -1 ℓ,0 are positivity preserving, ϕ, ψ are nonnegative elements of L 2 (Λ ℓ ). Since v > 0, they do not vanish identically. We thus have

⟨Φ, K -1 Ψ⟩ = ∞ j=0 a(-v)h -1 ℓ,0 j Φ, h -1 ℓ,0 ∞ k=0 a(-v)h -1 ℓ,0 k Ψ ≥ ⟨ϕ, h -1 ℓ,0 ψ⟩ > 0, (2.17)
since h ℓ,0 improves positivity on L 2 (Λ ℓ ) by Lemma 2.3. This proves the claim.

Proof of Proposition 2.1. The proposition is proved using the positivity-improving property of h ℓ following arguments of [START_REF] Faris | Invariant cones and uniqueness of the ground state for fermion systems[END_REF]. Let H R ℓ be the subspace of real-valued functions in H ℓ . Since h ℓ is invariant under complex conjugation, it can be restricted to an operator on H R ℓ . Moreover, any eigenvalue e of h ℓ is also an eigenvalue of this restriction and their multiplicity is the same, since the R-linear map of multiplication by the imaginary unit is an isomorphism between the real and imaginary subspaces of ker(h ℓ -e).

It is thus sufficient to prove that inf σ(h ℓ ) is a simple eigenvalue of h ℓ | H R ℓ . This is equivalent to proving that, for λ > -inf σ(h ℓ ), the space of real eigenfunctions of (h ℓ +λ) -1 for its largest eigenvalue e := (inf σ(h ℓ ) + λ) -1 > 0 has dimension one.

Let Ψ ∈ H R ℓ be a normalized eigenfunction of (h ℓ + λ) -1 with eigenvalue e. We can write

Ψ = Ψ + -Ψ -, (2.18)
where

Ψ (n) ± = (-1) n max{±Ψ (n) , 0}, (2.19)
so Ψ ± ∈ C and ⟨Ψ + , Ψ -⟩ = 0. We then have

e = ⟨Ψ, (h ℓ + λ) -1 Ψ⟩ = ⟨Ψ + , (h ℓ + λ) -1 Ψ + ⟩ + ⟨Ψ -, (h ℓ + λ) -1 Ψ -⟩ -2 ⟨Ψ -, (h ℓ + λ) -1 Ψ + ⟩ ≥0 ≤ ⟨(Ψ + + Ψ -), (h ℓ + λ) -1 (Ψ + + Ψ -)⟩ ≤ e, (2.20)
since e is the largest eigenvalue and ∥Ψ + + Ψ -∥ = ∥Ψ∥ = 1. We must thus have equality in (2.20), so

⟨Ψ -, (h ℓ + λ) -1 Ψ + ⟩ = 0. ( 2

.21)

Since (h ℓ + λ) -1 improves positivity this implies that either Ψ + or Ψ -are equal to zero, i.e. Ψ ∈ C or -Ψ ∈ C. Now assume there exist two orthogonal real eigenfunctions Φ, Ψ ∈ ker((h ℓ + λ) -1 -e). By changing signs if necessary, we may assume that Φ, Ψ ∈ C \ {0}. Then ⟨Φ, Ψ⟩ = e -1 ⟨Φ, (h ℓ + λ) -1 Ψ⟩ > 0, (2.22) a contradiction, so e is a simple eigenvalue.

A Definition of the Fröhlich Hamiltonian

In this section we define the Fröhlich Hamiltonian as a self-adjoint operator following the ideas of [START_REF] Lampart | On Nelson-type Hamiltonians and abstract boundary conditions[END_REF]. For a different proof, based on a commutator trick introduced in [START_REF] Lieb | Exact ground state energy of the strong-coupling polaron[END_REF][START_REF] Lieb | Ground-state energy and effective mass of the polaron[END_REF], see [START_REF] Griesemer | Self-adjointness and domain of the Fröhlich Hamiltonian[END_REF]. We will give the construction for the fiber operators H(P ), from which H can be obtained by reversing the Lee-Low-Pines transformation. Consider the quadratic form Q P (Ψ) := ⟨Ψ, (H(P ) + 1)Ψ⟩ = Ψ, ((P -P f ) 2 + N + 1)Ψ + 2 Re Ψ, a(v)Ψ .

(A.1)

Introducing the positive operator H 0 := (P -P f ) 2 + N + 1, one easily sees that Q P is well defined on D(H 1/2 0 ). Moreover, by inequality (A.4) below, Q P is an infinitesimal perturbation of the form of H 0 . Consequently, by the KLMN Theorem there exists a unique self-adjoint so that H Λ (P ) + 1 = K Λ (P ) + T Λ (P ), similarly as in (A.3). Then with z = 1 ± i (H Λ (P ) + z) -1 -(H(P ) + z) -1 = (K Λ (P ) + T Λ (P ) ± i) -1 -(K(P ) + T (P ) ± i) -1 = (H Λ (P ) + z) -1 (a(v -v Λ )H -1 0 H 0 (1 + H -1 0 a * (v))(H(P ) + z) -1 (A. with lim Λ→∞ C Λ = 0. Then, since N is H(P )-bounded by construction of H(P ), the expression (A.9) also tends to zero in norm as Λ → ∞, which proves the claim.

  2.4]. Since ρ(A n ) ⊂ [a n , b n ] c n→∞ ---→ [a, b] c , this contradicts the assumption t ∈ (a, b). Hence (a, b) ⊂ σ(A), and since the spectrum is closed, [a, b] ⊂ σ(A). This proves σ ess (A) = [a, b] also for a < b.

7 )+

 7 (H Λ (P ) + z) -1 (1 + a(v Λ )H -1 0 )H 0 (1 + H -1 0 a * (v -v Λ )))(H(P ) + z) -1 (A.8)+ (H Λ (P ) + z) -1 (T (P ) -T Λ (P ))(H(P ) + z) -1 . (A.9)By an analogous bound to (2.10), we have∥a(v -v Λ )H ∥a(v -v Λ )H -1 0 ∥ tends to zero as Λ → ∞. Since H 0 (1 + H -1 0 a * (v))(H(P ) + z) -1is a bounded operator by construction of H(P ), this shows that (A.7) tends to zero in norm.Since|v Λ (k)| ≤ |v(k)| we have ∥a(v Λ )H -1/2 0 Ψ∥ ≤ C∥(N + 1) 1/4 Ψ∥ (A.11)uniformly in Λ. With this, the bounds from the proof of Proposition A.1 hold for H Λ (P ) uniformly in Λ and (A.8) tends to zero for the same reason as (A.7).To conclude, note that by (A.10) and (A.11) we have ∥(T Λ (P ) -T (P ))Ψ∥ ≤ C Λ ∥(N + 1) 3/4 Ψ∥ (A.12)

In this regard, let us note that we shall not make use of Gross' inequality E α (0) ≤ E α (P )[START_REF] Gross | Existence and uniqueness of physical ground states[END_REF], which would evidently imply property (ii).

Acknowledgements D.M. and K.M. thank Robert Seiringer for helpful discussions.

Funding Financial support from the Agence Nationale de la Recherche (ANR) through the projects ANR-17-CE40-0016, ANR-17-CE40-0007-01, ANR-17-EURE-0002 (J.L.) and from the European Union's Horizon 2020 research and innovation programme under the Maria Sk lodowska-Curie grant agreement No. 665386 (K.M.) is gratefully acknowledged.

Data Availability Statement Data sharing not applicable to this article as no datasets were generated or analysed.

operator whose form is Q P . To make this more explicit, rewrite Q P as

From this, we see that Q P is associated with the operator

= K(P ) + T (P ). (A.3)

Proposition A.1. For α > 0 and P ∈ R 3 the quadratic form Q P is the form of the self-adjoint operator H(P ) + 1 given by (A.3) with domain

Proof. By the same proof as in (2.11) we have for

Hence, by the proof of Lemma 2.4, the operator 1 + a(v)H -1 0 and its adjoint are boundedly invertible. This implies that D(H(P )) is dense, since D(H 0 ) is. Moreover, K(P ) is a symmetric, invertible operator and thus self-adjoint. The proof is completed by showing that T (P ) is infinitesimally K(P )-bounded as in (2.15), (2.16).

In the proof of Lemma 1.1 we used that the operators with cutoff H Λ (P ) converge to H(P ) in norm-resolvent sense. We give a proof of this fact in the following proposition (see also [START_REF] Griesemer | Self-adjointness and domain of the Fröhlich Hamiltonian[END_REF]).

Proof. Recall that H 0 = (P -P f ) 2 + N + 1. The bounds on the difference of the resolvents will manifestly be independent of P , so we will not emphasize this at every step. Let Conflict of interest There are no conflicts of interest to declare.


Christophe Profeta

A remark on positive sojourn times of symmetric processes

Keywords: Occupation times, Stable processes, Brownian motion. 2010 Mathematics Subject Classification: 60J55; 60G52, 60J65

We show that under some slight assumptions the positive sojourn time of a product of symmetric processes converges towards one half as the number of processes increases. Monotony properties are then exhibited in the case of symmetric stable processes, and used, via a recurrence relation, to get upper and lower bounds on the moments of the occupation time (in the first and third quadrants) for 2-dimensional Brownian motion. Explicit values are also given for the second and third moments in the n-dimensional Brownian case.

Introduction

In this note, we are interested in the study of the random variables

A n = 1 0 1 { n i=1 X (i) u ≥0} du
where X (1) , . . . , X (n) are independent and identically distributed symmetric processes. The random variable A n may be interpreted as the time spent by an n-dimensional process (with independent components) in some symmetric orthants.

When n = 1, the random variable A 1 has been widely studied for several families of processes. The most celebrated example is the case of symmetric Lévy processes X (1) = L (1) such that P(L (1) 1 = 0) = 0, for which it is known that A 1 follows the classic arcsine law (see Getoor and Sharpe [START_REF] Getoor | On the arcsine laws for Lévy processes[END_REF]) :

P(A 1 ∈ dz) = 1 π z(1 -z)
dz, z ∈ (0, 1).

When n = 2, the random variable A 2 corresponds to the time spent by a planar symmetric process X (1) + iX (2) in the first and third quadrant of the complex plane. In the special case of the planar Brownian motion B (1) + iB (2) , a first attempt to find the law of A 2 has been undertaken by Ernst and Shepp [START_REF] Ernst | On occupation times of the first and third quadrants for planar Brownian motion[END_REF] in which the authors try to compute the double Laplace transform of A 2 . More generally, the study of the sojourn times of planar Brownian motion in a cone has already attracted much attention. In particular it was proven by Mountford [START_REF] Mountford | Limiting behaviour of the occupation of wedges by complex Brownian motion[END_REF] that if C is a closed convex cone of magnitude θ with vertex at 0, then there exist two constants κ 1 and κ 2 such that

κ 1 t 1/ξ ≤ P 1 0 1 {(B (1) u ,B (2)
u )∈C} du ≤ t ≤ κ 2 t 1/ξ , t ∈ [0, 1], (1)
with ξ = 2 π (2π -θ). The first moments of this random variable were computed by Desbois [START_REF] Desbois | Occupation times for planar and higher dimensional Brownian motion[END_REF], see also Bingham and Doney [START_REF] Bingham | On higher-dimensional analogues of the arcsine law[END_REF] in the special case θ = π 2 . Analogues of (1) for n-dimensional Brownian motion were obtained by Meyre and Werner [START_REF] Meyre | On the occupation times of cones by Brownian motion[END_REF] and Nakayama [START_REF] Nakayama | On the asymptotic behavior of the occupation time in cones of d-dimensional Brownian motion[END_REF], in which the exponent ξ is related to the first eigenvalue of the Laplacian operator -∆/2. In addition to these bounds, the strong arcsine law (see [START_REF] Bingham | The strong arcsine law in higher dimensions[END_REF]) gives the asymptotics of the sojourn time of a n-dimensional Brownian motion (B

(i) , 1 ≤ i ≤ n) in the positive orthant : 1 ln(t) t 1 n i=1 1 {B (i) u ≥0} du u a.s. ----→ t→+∞ 1 2 n .
Observe that the summation here is logarithmic : we refer to [START_REF] Bingham | Summability methods and almost-sure convergence[END_REF] for a general discussion between summability methods and limits of occupation times. In this note, we shall first study the limit of the variables A n as the dimension n goes to ∞. Theorem 1. Let (X (i) , i ≥ 1) be independent and identically distributed symmetric processes.

1. The strong law of large numbers holds for the sequence (A n , n ≥ 1) :

1 k k n=1 A n a.s. -----→ k→+∞ 1 2 .
2. Assume that for a.e. u ∈ (0, 1), the random variables X

u have no atoms at 0 and that for a.e. 0 < u < s < 1, 0 < P X (1) u ≥ 0, X

s ≥ 0 < 1 2 . ( (1)
) 2
Then, for any p > 0,

A n L p -----→ n→+∞ 1 2 .

Assume furthermore that

1 0 1 0 1 P X (1) u ≤ 0, X (1)
s ≥ 0 du ds < +∞. (3)
Then,

A n a.s. -----→ n→+∞ 1 2 .
When thinking of symmetric Lévy processes, an interpretation of this result is as follows : the usual arcsine law essentially explains that, although L (1) is centered, there is a high probability that it spends more time on one side of the axis than on the other one. As the number of Lévy processes increases, so do the changes of sign of the product, hence the resulting process spends a more balanced time on each side of the abscissa axis.

Remark 2. Note that an assumption such as (2) is necessary to obtain the L p -convergence. Indeed, let for instance (X i , i ≥ 1) be a family of i.i.d. symmetric random variables admitting a density. Define the processes :

X (i) t = tX i (t ≥ 0)
which do not satisfy the assumption P(X

u ≥ 0, X

≥ 0) < 1 2 . (1) s
In this case, the random variables A n all have the same law :

A n

(law) = 1 2 (δ 0 + δ 1 )
and the L p -convergence of Theorem 1 cannot hold.

Example 3. Assumption (3) is for instance satisfied by symmetric α-stable Lévy processes L with α > 1. Indeed, for 0 < u < s, using the symmetry, independent increments and scaling properties, we first deduce that :

P(L s ≤ 0, L u ≥ 0) = P s u -1 1/α L 1 ≥ Z 1 , Z 1 ≥ 0
where Z 1 is a symmetric α-stable r.v. independent from L 1 . Next, for ν > 0 small enough, applying Fubini's theorem :

+∞ 0 t -ν-1 P(t 1/α L 1 ≥ Z 1 , Z 1 ≥ 0)dt = 1 ν E L να 1 1 {L1≥0} E Z -να 1 1 {Z1≥0} = 1 4ν E [|L 1 | να ] E |Z 1 | -να = 1 4α 2 ν Γ(ν) Γ(να) cos(ναπ/2) Γ(-ν) Γ(-να) cos(ναπ/2) = 1 2αν
sin(ναπ/2) sin(νπ) cos(ναπ/2) .

Therefore, using the inverse mapping for the Mellin transform (see for instance Janson [START_REF] Janson | Moments of Gamma type and the Brownian supremum process area[END_REF]), we obtain the asymptotics

F (t) := P t 1/α L 1 ≥ Z 1 , Z 1 ≥ 0 ∼ t→0 + 1 πα sin(π/α) t 1/α
hence, by a change of variable

1 0 1 0 1 P(L u ≤ 0, L s ≥ 0) du ds = 2 1 0 s 0 1 F (s/u -1) du ds = +∞ 0 1 F (t)(t + 1) 2 dt < +∞ which is Assumption (3).
The remainder of the paper is organized as follows : we prove Theorem 1 in Section 2, then study some monotony properties of A n when dealing with stable processes in Section 3, and finally compute the first moments of A n and give some bounds on A 2 for Brownian motion in Section 4.

Proof of Theorem 1

Proof. We start with the law of large numbers of Point 1. Let us first define the centered random variables

A * n = A n - 1 2
and observe that these random variables are uncorrelated. Indeed, decomposing A k+n and using the tower property of conditional expectations for k ≥ 1

E[A * n A * n+k ] = E[A n A n+k ] - 1 4 = E A n 1 0 1 { n i=1 X (i) u ≥0} 1 { n+k i=n+1 X (i) u ≥0} + 1 { n i=1 X (i) u ≤0} 1 { n+k i=n+1 X (i) u ≤0} du - 1 4 = E A n 1 0 1 { n i=1 X (i) u ≥0} 1 2 + 1 { n i=1 X (i) u ≤0} 1 2 du - 1 4 = 1 2 E[A n ] - 1 4 = 0.
Now, since the random variables (A * n , n ≥ 1) are uniformly bounded by 1, the result will follow from Theorem 1 in [START_REF] Lyons | Strong laws of large numbers for weakly correlated random variables[END_REF] after having checked that

k≥1 1 k E   1 k k n=1 A * n 2   < ∞.
But this is immediate since developing the square and applying Fubini's theorem :

k≥1 1 k E   1 k k n=1 A * n 2   = k≥1 1 k 3 k n=1 E (A * n ) 2 ≤ k≥1 1 k 2 < +∞, hence we conclude that 1 k k n=1 A * n a.s. -----→ k→+∞ 0,
which is Point 1 of Theorem 1.

To prove the L p -convergence, let us consider, for n ≥ 1, the function

F n : [0, 1] 2 → [0, 1] defined by F n (u, s) = P n i=1 X (i) u ≥ 0, n i=1 X (i) s ≥ 0 .
By symmetry and since there are no atoms at 0, we may decompose F n+1 as

F n+1 (u, s) = 2 P n i=1 X (i) u ≥ 0, n i=1 X (i) s ≥ 0 P X (n+1) u ≥ 0, X (n+1) s ≥ 0 + 2 P n i=1 X (i) u ≤ 0, n i=1 X (i) s ≥ 0 P X (n+1) u ≤ 0, X (n+1) s ≥ 0 (4)
and rewrite this under the form

F n+1 (u, s) = 2F n (u, s)F 1 (u, s) + 2 1 2 -F n (u, s) 1 2 -F 1 (u, s) = 4 F n (u, s) - 1 4 F 1 (u, s) - 1 4 + 1 4 .
In particular, we deduce by iteration that

F n+1 (u, s) - 1 4 = F n (u, s) - 1 4 (4F 1 (u, s) -1) = 1 4 (4F 1 (u, s) -1) n+1 . (5)
Now, for a.e. u = s, we have by assumption -1 < 4F 1 (u, s) -1 < 1, so we may let n → +∞ to obtain

F n (u, s) -----→ n→+∞ 1 4 .
Finally, applying the dominated convergence theorem

E A n - 1 2 2 = E A 2 n - 1 4 = 1 0 1 0 F n (u, s) duds - 1 4 -----→ n→+∞ 0
which proves the L 2 -convergence of Theorem 1, hence the L p -convergence for any 0 < p ≤ 2 by Hölder's inequality. But, since for any n ∈ N, |A n -1 2 | ≤ 1, we further obtain that for p ≥ 2

E A n - 1 2 p ≤ E A n - 1 2 2 -----→ n→+∞ 0 which ends the proof of Point 2.
Finally, to get the a.s. convergence of Point 3, we apply Fubini's theorem to obtain the bound, thanks to (5) :

+∞ n=1 E A n - 1 2 2 = +∞ n=1 1 0 1 0 F n (u, s) - 1 4 duds = 1 4 1 0 1 0 4F 1 (u, s) -1 2 -4F 1 (u, s) du ds ≤ 1 16 1 0 1 0 1 P(X (1)
u ≤ 0, X

≥ 0) du ds < +∞. (1) s
The a.s. convergence then follows from the usual application of the Bienaymé-Tchebychev inequality and the Borel-Cantelli lemma.

Monotonicity for stable processes

We now assume that (X (i) = L (i) ) i≥1 are independent symmetric α-stable Lévy processes with α ∈ (0, 2] defined on a probability space (Ω, F ∞ , P). From Point 2 of Theorem 1, we deduce that for any p > 0 :

E[A p n ] -----→ n→+∞ 1 2 p .
When dealing with stable processes, it turns out that the sequence (E[A p n ], n ≥ 1) is monotone, according to the value of p (i.e. to the convexity of the function x → x p ). Proposition 4. Let p > 0 be fixed. The sequence

(E[A p n ], n ≥ 1) is decreasing if p ∈ N * increasing if 0 < p < 1.
As a consequence, for any λ ∈ R and n ≥ 1 :

E e λAn+1 ≤ E e λAn .
For symmetric stable Lévy processes, the r.v.'s (A n , n ≥ 1) are thus ordered via moment-generating functions or Laplace transforms.

Proof. We start the proof with a simple lemma.

Lemma 5. Let n ≥ 1 and (X i ) i≤n be i.i.d. symmetric random variables with common density f and let (A i ) i≤n be random variables, independent from the (X i ) i≤n and such that P (

n i=1 A i > 0) = 1.
Then, the function

t → P n i=1 (X i + tA i ) ≥ 0 is increasing from 1/2 to 1.
Proof. Observe first that by conditioning on the distribution of the sequence (A i ) i≤n

P n i=1 (X i + tA i ) ≥ 0 = R . . . R P n i=1 (X i + ta i ) ≥ 0 P(A 1 ∈ da 1 , . . . , A n ∈ da n )
we only need to prove that the function

Ψ n (t) = P n i=1 (X i + ta i ) ≥ 0
is increasing, under the assumption n i=1 a i > 0. Next, for any n ≥ 1, we have Ψ n (0) = 1/2 and lim t→+∞ Ψ n (t) = 1. We shall prove that Ψ n is increasing by induction on n. For n = 1, the result is clear since in this case a 1 > 0. Assume now that n ≥ 2 and that Ψ n-1 is increasing from 1/2 to 1. Since the (X i ) are independent, we may decompose

Ψ n (t) = P(X n + ta n ≥ 0)P n-1 i=1 (X i + ta i ) ≥ 0 + P(X n + ta n ≤ 0)P n-1 i=1 (X i + ta i ) ≤ 0 .
We now separate two cases.

1. Assume first that a n > 0. Then n-1 i=1 a i > 0 and differentiating

Ψ ′ n (t) = a n f (-ta n )Ψ n-1 (t) + P(X n + ta n ≥ 0)Ψ ′ n-1 (t) -a n f (-ta n )(1 -Ψ n-1 (t)) -P(X n + ta n ≤ 0)Ψ ′ n-1 (t) = a n f (-ta n ) (2Ψ n-1 (t) -1) + Ψ ′ n-1 (t) (P(X n + ta n ≥ 0) -P(X n + ta n ≤ 0)) .
Since Ψ n-1 (t) ≥ 1/2 and P(X n + ta n ≥ 0) > P(X n + ta n ≤ 0), we deduce from the recursion hypothesis that Ψ ′ n (t) > 0.

2. Assume now that a n < 0. Then n-1 i=1 a i < 0 and we deduce from the symmetry of X 1 and X n that2

Ψ n (t) = P(X n + t(-a n ) ≤ 0)P (X 1 + t(-a 1 )) n-1 i=2 (X i + ta i ) ≤ 0 + P(X n + t(-a n ) ≥ 0)P (X 1 + t(-a 1 )) n-1 i=2 (X i + ta i ) ≥ 0 .
The result then follows from the first case, since -a n > 0 and -n-1 i=1 a i > 0.

We now come back to the proof of Proposition 4. To simplify the notation, we set :

P (n) u = n i=1 L (i) u .
Let us consider the function F : R + → [0, 1] defined by :

F (x) = E 1 0 1 {(x+Zu)P (n) u ≥0} du p
where Z is another α-stable Lévy process independent from the (L (i) ). We shall prove that F is increasing on [0, +∞). By the change of variable u = x α s and scaling, we have

F (x) = x αp E 1/x α 0 1 {(1+Zs)P (n) s ≥0} ds p .
Differentiating with respect to x and going back to the original variable, we obtain

F ′ (x) = αp x E 1 0 1 {(x+Zu)P (n) u ≥0} du p - 1 0 1 {(x+Zu)P (n) u ≥0} du p-1 1 {(x+Z1)P (n) 1 ≥0}
.

Applying Fubini's theorem, we need to prove that 1 0 . . .

1 0 P p i=1 (x + Z ui )P (n) ui ≥ 0 du 1 . . . du p ≥ 1 0 . . . 1 0 P p-1 i=1 (x + Z ui )P (n) ui ≥ 0 ∩ (x + Z 1 )P (n) 1 ≥ 0 du 1 . . . du p .
We shall in fact simply prove that the inequality holds on the integrands :

P p-1 i=1 (x + Z ui )P (n) ui ≥ 0 ∩ (x + Z up )P (n) up ≥ 0 ≥ P p-1 i=1 (x + Z ui )P (n) ui ≥ 0 ∩ (x + Z 1 )P (n) 1 ≥ 0 . (6)
where we may assume, up to renaming the variables, that 0 ≤ u 1 ≤ u 2 ≤ . . . ≤ u p ≤ 1. To simplify the notation, let us introduce the measure Q defined, for Λ ∈ F ∞ , by :

Q(Λ) = P Λ p-1 i=1 (x + Z ui )P (n) ui ≥ 0 .
Dividing both sides of (6) by P p-1 i=1 (x + Z ui )P

(n) ui ≥ 0 , we are thus led to prove that the function

t → Q (x + Z t+up-1 )P (n) t+up-1 ≥ 0 is decreasing on [0, 1 -u p-1 ].
Applying the Markov property, we may decompose

Q (x + Z t+up-1 )P (n) t+up-1 ≥ 0 = Q x + Z up-1 + t 1/α Z 1 n i=1 L (i) up-1 + t 1/α L (i) 1 ≥ 0
where Z 1 and ( L (i) 1 ) are independent symmetric α-stable random variables, independent from Z and the (L (i) ). Observe furthermore that by definition of Q :

Q (x + Z up-1 ) n i=1 L (i) up-1 > 0 = Q (x + Z up-1 )P (n) up-1 > 0 = 1.
Therefore, applying Lemma 5 with the sequences

(X i , 1 ≤ i ≤ n) = ( L (i) 1 , 1 ≤ i ≤ n), X n+1 = Z 1 and (A i , 1 ≤ i ≤ n) = (L (i) up-1 , 1 ≤ i ≤ n), A n+1 = x + Z up-1 ,
we deduce by composition that the function t → Q (x + Z t+up-1 )P

(n)

t+up-1 ≥ 0 is decreasing on [0, 1u p-1 ] (in fact on [0, +∞)), hence the function F is increasing on [0, +∞). It remains then to apply the dominated convergence theorem, upon noticing that

E[A p n+1 ] = F (0) ≤ lim x→+∞ F (x) = E[A p n ]
which yields the proof for integer values.

Summing the different moments, we deduce that for λ ≥ 0 :

E e λAn+1 ≤ E e λAn
Next, by symmetry

E e λ(1-An+1) ≤ E e λ(1-An) ⇐⇒ E e -λAn+1 ≤ E e -λAn
hence, for 0 < p < 1 :

+∞ 0 λ -p-1 1 -E e -λAn dλ ≤ +∞ 0 λ -p-1 1 -E e -λAn+1 dλ which is exactly E[A p n ] ≤ E[A p n+1 ] (0 < p < 1).
Remark 6. We give below an example of a process satisfying Assumption ( 2), but for which the sequence (E[A 2 n ], n ≥ 1) is not decreasing. Take for instance

X (1) t = B t 1 {t≤1/2} -B t-1/2 1 {t>1/2} and X (2) t = W t 1 {t≤1/2} -W t-1/2 1 {t>1/2}
where B and W are two independent Brownian motions started from 0. Then :

A 1 = 1 2 and A 2 = 2 1/2 0 1 {BtWt≥0} dt (law) = 1 0 1 {BuWu≥0} du hence (see next Section for the value of E[A 2 2 ]) E A 2 1 = 1 4 < E A 2 2 = 3 8 - 1 2π 2 .
4 A study of moments in the Brownian case

Second and third moments

The first three moments are easy to compute in the Brownian case. Indeed, from Formula (5), the second moment of A n equals:

E[A 2 n ] = 1 4 + 1 2 1 0 s 0 (4F 1 (u, s) -1) n du ds
where, from Bingham and Doney [START_REF] Bingham | On higher-dimensional analogues of the arcsine law[END_REF], the quadrant probability is given for 0 ≤ u ≤ s by :

F 1 (u, s) = 1 4 + 1 2π arcsin u s .
After some changes of variables and successive integrations by parts, we deduce that

E[A 2 n ] = 1 4 + 1 8π n π 0 t n sin(t)dt = 1 4 + (-1) ⌊n/2⌋+1 n!(n -2⌊n/2⌋ -1) 8π n + 1 8 ⌊n/2⌋ k=0 (-1) k n! (n -2k)! π -2k . By symmetry, since E[(1 -A n ) 3 ] = E[A 3 n ],
we further obtain :

E A 3 n = 3 2 E A 2 n - 1 4 .
We give below the first values of the second and third moments, in which the decreasing property may be observed.

n E[A 2 n ] E[A 3 n ] 1 3 8 ≃ 0, 375 5

Higher moments for two Brownian motions

Getting the explicit values of higher moments seems a hard task as outlined in several papers [START_REF] Bingham | On higher-dimensional analogues of the arcsine law[END_REF][START_REF] Desbois | Occupation times for planar and higher dimensional Brownian motion[END_REF][START_REF] Ernst | On occupation times of the first and third quadrants for planar Brownian motion[END_REF].

We propose here a method to get lower and upper bounds on these moments. Recall the moments of the arcsine distribution :

E[A p 1 ] = 1 2 2p p 2p = (2p)! 2 2p (p!) 2 = Γ(p + 1/2) √ πΓ(p + 1) ∼ p→+∞ 1 √ πp Proposition 7.
For any p ≥ 1, we have and

E[A p 2 ] ≤ 1 2p + 1 8 π 2 3 F 2 1/2 1/2 1 p + 3/2 3/2 ; 1 + 1 π 2 p-1 k=0 2 (p -k) 2 E[A k 1 ]
E[A p 2 ] ≥ 1 2p + 1 8 π 2 3 F 2 1/2 1/2 1 p + 3/2 3/2 ; 1 + 1 π 2 p-1 k=0 2 (p -k) 2 E[A k 2 ]
where 3 F 2 denotes the usual generalized hypergeometric function, see [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF]Section 9.1]. Note that both bounds are the same when p equals 1 and 2. Asymptotically, we further obtain that

6 π 2 p ≤ E[A p 2 ] ≤ 1 3 √ πp (p → +∞).
In particular, this implies that the r.v. A 2 cannot follow a Beta distribution (hence neither a generalized arcsine distribution). Indeed, otherwise the Beta distribution would be β

1 2 + 4 π 2 -4 , 1 2 + 4 π 2 -4 , since then E[β] = 1 2 , E[β 2 ] = 3 8 - 1 2π 2 and E[β 3 ] = 5 16 - 3 4π 2 .
But, as p → +∞, we would have

E[β p ] = O 1 p 1 2 + 4 π 2 -4
which would contradict the lower bound since 1 2 + 4 π 2 -4 > 1. Numerical computations are shown in Figure 1, in which it is seen that the lower bound is clearly the better one.

Proof. Let B and W be two independent Brownian motions and define

M p (x) = +∞ 0 e -t/2 E t 0 1 {(x+Bu)Wu>0} du p dt so that E[A p 2 ] = M p (0) 2 p+1 p! .
Applying first the Markov property at the stopping time T x = inf{u ≥ 0, x + B u = 0} and Fubini's theorem, we deduce that

E t 0 1 {(x+Bu)Wu>0} du p = E t 0 1 {Wu>0} du p 1 {Tx>t} + E Tx 0 1 {Wu>0} du + t Tx 1 {(x+Bu)Wu>0} du p 1 {Tx≤t} = t p E [A p 1 ] P(T x > t) + p k=0 p k E   Tx 0 1 {Wu>0} du p-k t-Tx 0 1 { Bs(WT x + Ws)>0} du k 1 {Tx≤t}  
where B and W are two independent Brownian motions, independent from B and W . We now take the Laplace transform of both sides. Applying the Fubini-Tonelli theorem and a change of variable, we obtain

M p (x) = E [A p 1 ] +∞ 0 e -t/2 t p P(T x > t)dt + p k=0 p k E   e -Tx/2 Tx 0 1 {Wu>0} du p-k M k (W Tx )   = R p-1 (x) + E e -Tx/2 M p (W Tx )
where, by scaling, R p-1 is defined by

R p-1 (x) = E[A p 1 ] +∞ 0 e -t/2 t p P(T x > t)dt+ p-1 k=0 p k E e -Tx/2 T p-k x 1 0 1 {Wu>0} du p-k M k ( T x W 1 ) .
We thus obtain the relation, since E e -Tx/2 = e -|x| ,

M p (x) -M p (0) = R p-1 (x) -(1 -e -|x| )M p (0) + E e -Tx/2 (M p (W Tx ) -M p (0)) . (7)
The expectation on the right-hand side may be computed to give

E e -Tx/2 (M p (W Tx ) -M n (0)) = 2 +∞ 0 x √ 2πt 3 e -x 2 2t -t/2 dt +∞ 0 1 √ 2πt e -z 2 2t (M p (z) -M p (0))dz = 2 +∞ 0 x π K 1 ( √ x 2 + z 2 ) √ x 2 + z 2 (M p (z) -M p (0))dz
where K ν denotes the modified Bessel function of the second kind of order ν. Integrating (7) with respect to K 0 (x) dx x , we deduce from the formulae [5, p.377 n 0 (33)] and [7, p.695 n 0 (9)]

+∞ 0 K 1 ( √ x 2 + z 2 ) √ x 2 + z 2 K 0 (x)dx = π 2z K 0 (z) and +∞ 0 (1 -e -x )K 0 (x) dx x = π 2 8 that +∞ 0 (M p (x)-M p (0))K 0 (x) dx x = +∞ 0 (R p-1 (x)-(1-e -|x| )M p (0))K 0 (x) dx x + +∞ 0 K 0 (z)(M p (z)-M p (0)) dz z hence M p (0) = 8 π 2 +∞ 0 R p-1 (x)K 0 (x) dx x . (8)
Remark 8. Note that we might have used the Kontorovitch-Lebedev transform (see for instance [START_REF] Mckean | A winding problem for a resonator driven by a white noise[END_REF]) to get a recurrence relation between M p (x) and R p-1 (x) :

M p (x) = 2 π 2 +∞ 0 cosh(πγ/2) cosh(πγ/2) -1 K iγ (x) +∞ 0 K iγ (z)R p-1 (z) dz z γ sinh(πγ)dγ
but this leads to quite complicated calculations, even for p = 3.

We shall rather find bounds on R p-1 . Plugging the expression of R p-1 in (8), we first need to compute : Next from Section 3 and the proof of monotony of the moments, we deduce that

M k (0) ≤ M k ( T x W 1 ) ≤ M k (+∞)
hence, going back to the expression of R p-1 , it remains to compute

p k E[A p-k 1 ]2 k+1 k!E[A k 2 ]
Then, using the explicit value of the moments of the arcsine distribution, we finally obtain after some simplifications

E[A p 2 ] ≥ 1 2p + 1 8 π 2 3 F 2 1/2 1/2 1 p + 3/2 3/2 ; 1 + 1 π 2 p-1 k=0 2 (p -k) 2 E[A k 2 ]
which is the announced result. The computations for the upper bound are similar.

Figure 1 :

 1 Figure 1: Monte Carlo simulation of E[A p 2 ] for 1 ≤ p ≤ 500.

√ 2 Γ

 2 π2 -k (Γ(p -k)) (p -k + 1/2)

LaMME, Université d' Évry -Val d'Essonne, Bâtiment I.B.G.B.I., 3ème étage,

Bd. de France, 91037 EVRY CEDEX. E-mail: christophe.profeta@univ-evry.fr

with the usual convention that empty products are 1

Acknowledgments. We thank the referees for pointing out the existence of the strong arcsine law and suggesting various improvements in the presentation of the proofs of the paper. This research benefited from the support of the "Chaire Marchés en Mutation", Fédération Bancaire Française.