Search

This work elaborates on a detailed analysis of the novel characteristics of gravitational waves (GWs) generated by extreme mass ratio inspirals (EMRIs) within the framework of modified gravity (MOG). Our study begins by exploring the geometrical and dynamical properties of the Kerr-MOG spacetime. We employ the numerical kludge (NK) method for waveform simulations and reveal that the parameter $\alpha$, representing deviations from general relativity (GR), significantly impacts the frequencies of geodesic orbits and, consequently, the EMRI waveforms. However, the waveform confusion problem remains mainly unresolved, posing a challenge in distinguishing between the underlying gravitational theories based on the observed EMRI waveforms. Notably, by incorporating the effects of radiation reaction, we observe a substantial reduction in the waveform overlap over time. This reduction could enhance our ability to discern between different waveforms over an extended period., Comment: 25 pages, 14 figures

In this work, we explore the effect of deformation of the nuclei on collective flow in relativistic heavy-ion collisions. The parameter associated with the geometrical deformation in the Glauber model is tuned to reproduce the empirical multiplicity probability distributions correctly. Subsequently, the particle spectra and collective flows for Au+Au and U+U collisions are evaluated using a hybrid hydrodynamic code CHESS. We analyze the effects of the degrees of freedom associated with the IC on the final-state flow harmonics by exploring the parameter space of the former. The connection between the deformation parameters, specifically $\beta_2$ and $\beta_4$, and the flow anisotropies is scrutinized. In particular, deviations in elliptic flow at $p_{\mathrm{T}}\sim 2$ GeV are observed at smaller values of $\beta_2$ in Au+Au collisions. On the other hand, for U+U collisions, the averaged overall flow harmonics are found to be less sensitive to the geometrical parameters. Despite the difference in the model's specifications, our findings largely confirm those obtained in the literature employing different approaches, which indicate that flow harmonics can be used as a sensible probe for the initial geometry fluctuations and to discriminate between different theoretical models., Comment: 17 pages, 10 figures

In the black hole perturbation theory framework, two different physical pictures for echoes in compact objects have been proposed. The first mechanism interprets echoes as repeated reflections of gravitational waves within a potential well, where the echo period is defined by twice the distance related to the spatial displacement operator that separates two local maxima of the effective potential. The second mechanism associates echoes with a discontinuity in the effective potential, potentially associated with specific accretion processes, without necessarily introducing a second local maximum in the effective potential. This discontinuity leads to echo signals that are typically attenuated over time more quickly, with their period dictated by the characteristics of the transfer amplitudes. In both scenarios, the echoes correspond to a new category of quasinormal modes with minor real parts, with their period connected to the spacing between successive modes in the frequency domain. This work elaborates on a unified framework in compact stars that encompasses both echo mechanisms. It suggests that these two types of echoes derive from different physical origins and can be independently triggered. The occurrence and interplay between these two types of echoes are demonstrated through numerical simulations. %The observational relevance of this study is also addressed., Comment: 17 pages and 6 figures

We use the SLE$_\kappa$ loop measure to construct a natural representation of the Virasoro algebra of central charge $c = c(\kappa) \le 1$. In particular, we introduce a non-degenerate bilinear Hermitian form (not positive-definite) using the SLE loop measure and show that the representation is (indefinite) unitary. Our proof relies on the infinitesimal conformal restriction property of the SLE loop measure., Comment: 24 pages, added more references

The P\"ochl-Teller effective potential mimics an asymptotically de Sitter black hole bounded by an event horizon and a cosmological one. Owing to the benefit of being analytically soluble, the asymptotic quasinormal modes in the modified P\"oschl-Teller potential have been extensively explored in the literature by various authors, and the results bear distinct features. Specifically, for small discontinuities placed at the potential's peak, Skakala and Visser showed that the resulting modes lie primarily along the imaginary frequency axis, in line with the numerical results encountered for most black hole metrics. However, it was also suggested that under ultraviolet perturbations, asymptotic modes are expected to lie parallel to the real axis, closely intervening with recent developments on spectral instability. In this work, by numerical and semi-analytical approaches, we aim to resolve the above apparent ambiguity. The numerical scheme is based on an improved version of the matrix method, which is implemented in compactified hyperboloidal coordinates on the Chebyshev grid. It is demonstrated that both asymptotic behaviors indeed agree with the numerical findings, which is somewhat to one's surprise. Specifically, we report the emergence of a novel branch of purely imaginary modes originating from a bifurcation in the asymptotic quasinormal mode spectrum. Moreover, we demonstrate how the bifurcation and asymptotic modes evolve as the discontinuity moves away from the potential's peak, furnishing a dynamic picture as the spectral instability unfolds. It is further argued that they can be partly attributed to the observed parity-dependent deviations occurring for the low-lying perturbed modes of the original P\"oschl-Teller effective potential., Comment: 41 pages, 7 figures, and 2 tables

This paper briefly introduces the solutions developed by our team, HFUT-VUT, for Track 1 of self-supervised heart rate measurement in the 3rd Vision-based Remote Physiological Signal Sensing (RePSS) Challenge hosted at IJCAI 2024. The goal is to develop a self-supervised learning algorithm for heart rate (HR) estimation using unlabeled facial videos. To tackle this task, we present two self-supervised HR estimation solutions that integrate spatial-temporal modeling and contrastive learning, respectively. Specifically, we first propose a non-end-to-end self-supervised HR measurement framework based on spatial-temporal modeling, which can effectively capture subtle rPPG clues and leverage the inherent bandwidth and periodicity characteristics of rPPG to constrain the model. Meanwhile, we employ an excellent end-to-end solution based on contrastive learning, aiming to generalize across different scenarios from complementary perspectives. Finally, we combine the strengths of the above solutions through an ensemble strategy to generate the final predictions, leading to a more accurate HR estimation. As a result, our solutions achieved a remarkable RMSE score of 8.85277 on the test dataset, securing \textbf{2nd place} in Track 1 of the challenge.

In education data mining (EDM) communities, machine learning has achieved remarkable success in discovering patterns and structures to tackle educational challenges. Notably, fairness and algorithmic bias have gained attention in learning analytics of EDM. With the increasing demand for the right to be forgotten, there is a growing need for machine learning models to forget sensitive data and its impact, particularly within the realm of EDM. The paradigm of selective forgetting, also known as machine unlearning, has been extensively studied to address this need by eliminating the influence of specific data from a pre-trained model without complete retraining. However, existing research assumes that interactive data removal operations are conducted in secure and reliable environments, neglecting potential malicious unlearning requests to undermine the fairness of machine learning systems. In this paper, we introduce a novel class of selective forgetting attacks designed to compromise the fairness of learning models while maintaining their predictive accuracy, thereby preventing the model owner from detecting the degradation in model performance. Additionally, we propose an innovative optimization framework for selective forgetting attacks, capable of generating malicious unlearning requests across various attack scenarios. We validate the effectiveness of our proposed selective forgetting attacks on fairness through extensive experiments using diverse EDM datasets.

Pre-trained vision-language models learn massive data to model unified representations of images and natural languages, which can be widely applied to downstream machine learning tasks. In addition to zero-shot inference, in order to better adapt pre-trained models to the requirements of downstream tasks, people usually use methods such as few-shot or parameter-efficient fine-tuning and knowledge distillation. However, annotating samples is laborious, while a large number of unlabeled samples can be easily obtained. In this paper, we investigate a novel "pre-trained annotating - weakly-supervised learning" paradigm for pre-trained model application and experiment on image classification tasks. Specifically, based on CLIP, we annotate image samples with multiple prompt templates to obtain multiple candidate labels to form the noisy partial label dataset, and design a collaborative consistency regularization algorithm to solve this problem. Our method simultaneously trains two neural networks, which collaboratively purify training labels for each other and obtain pseudo-labels for self-training, while adopting prototypical similarity alignment and noisy supervised contrastive learning to optimize model representation. In experiments, our method achieves performances far beyond zero-shot inference without introducing additional label information, and outperforms other weakly supervised learning and few-shot fine-tuning methods, and obtains smaller deployed models. Our code is available at: \url{https://anonymous.4open.science/r/Co-Reg-8CF9}.

Higher-degree polynomial interpolations carried out on uniformly distributed nodes are often plagued by {\it overfitting}, known as Runge's phenomenon. This work investigates Runge's phenomenon and its suppression in various versions of the matrix method for black hole quasinormal modes. It is shown that an appropriate choice of boundary conditions gives rise to desirable suppression of oscillations associated with the increasing Lebesgue constant. For the case of discontinuous effective potentials, where the application of the above boundary condition is not feasible, the recently proposed scheme with delimited expansion domain also leads to satisfactory results. The onset of Runge's phenomenon and its effective suppression are demonstrated by evaluating the relevant waveforms. Furthermore, we argue that both scenarios are either closely related to or practical imitations of the Chebyshev grid. The implications of the present study are also addressed., Comment: 17 pages, 4 figures and 4 tables

In the framework of black hole perturbation theory, this work investigates the standing wave solutions in Reissner-Nordtsr\"om (RN) anti-de Sitter (AdS) spacetimes with a naked singularity. These solutions can be viewed as a specific class of quasinormal modes exhibiting distinct characteristics. The imaginary parts of their frequencies are numerically vanishing, allowing them to persist over an extended period. Besides, these modes are predominantly stationary in terms of the evolution of spacetime waveforms. The numerical calculations are carried out employing the finite difference method, and the quasinormal frequencies extracted by the Prony method are shown to be consistent with those obtained using the matrix method. The obtained waveforms and quasinormal frequencies are shown to be drastically different from those of an extreme RN-AdS black hole. As the quasinormal modes are primarily dissipative, the non-dissipative standing waves are attributed to the nature that the singularity can neither be a sink nor a source of the gravitational system., Comment: 15 pages and 8 figures

The critical two-dimensional Brownian loop-soup is an infinite collection of non-interacting Brownian loops in a planar domain that possesses some combinatorial features related to the notion of indistinguishability of bosons. The properly renormalized occupation time field of this collection of loops is known to be distributed like the properly defined square of a Gaussian free field. In the present paper, we investigate aspects of the question about how much information these fields provide about the loop-soup. Among other things, we show that the exact set of points that are actually visited by some loops in the loop-soup is not determined by these fields. We further prove that given the fields, a dense family of special points will each have a conditional probability 1/2 of being part of the loop-soup. We also exhibit another instance where the possible decompositions (given the field) into individual loops and excursions can be grouped into two clearly different groups, each having a conditional probability 1/2 of occurring., Comment: 29 pages

We show that the occupation measure of planar Brownian motion exhibits a constant height gap of $5/\pi$ across its outer boundary. This property bears similarities with the celebrated results of Schramm--Sheffield [18] and Miller--Sheffield [12] concerning the height gap of the Gaussian free field across SLE$_4$/CLE$_4$ curves. Heuristically, our result can also be thought of as the $\theta \to 0^+$ limit of the height gap property of a field built out of a Brownian loop soup with subcritical intensity $\theta>0$, proved in our recent paper [3]. To obtain the explicit value of the height gap, we rely on the computation by Garban and Trujillo Ferreras [1] of the expected area of the domain delimited by the outer boundary of a Brownian bridge., Comment: 23 pages, 5 figures

Damour-Solodukhin wormholes are intriguing theoretical constructs, closely mimicking many properties of black holes. This study delves into two distinct characteristics of the waveforms emitted from such wormholes, namely, the late-time tails and echoes, which can substantially be used to distinguish its identity. Notably, both features appear in the latter stages of quasinormal oscillations and stem from the singularities of the Green's function. The late-time tail, on the one hand, arises due to the branch cuts in the relevant Green's function. Within the Damour-Solodukhin wormhole paradigm, singularities are present in both ingoing and outgoing waveforms, which entails a generalization of the existing recipe for black hole metrics. On the other hand, the echoes are attributed to a new set of quasinormal poles, supplementing those of the respective black holes, reminiscent of the scenario where the spacetime metric possesses a discontinuity. It is inferred that both features are observationally relevant in distinguishing a wormhole from its black hole counterpart. Moreover, we suggest a potential interplay concerning the late-time evolution between the two mechanisms in question., Comment: 20 pages, 5 figures

Holographic superconductor phase transition and spontaneous scalarization are triggered by the instability of the underlying vacuum black hole spacetime. Although both hairy black hole solutions are closely associated with the tachyonic instability of the scalar degree of freedom, they are understood to be driven by distinct causes. It is, therefore, interesting to explore the interplay between the two phenomena in the context of a scenario where both mechanisms are present. To this end, we investigate the Einstein-scalar-Gauss-Bonnet theory in asymptotically anti-de Sitter spacetime with a Maxwell field. On the one hand, the presence of the charged scalar and Maxwell fields in anti-de Sitter spacetime furnishes the celebrated framework for a holographic superconductor. On the other hand, the non-minimal Gauss-Bonnet coupling between the scalar field and the gravitational sector triggers spontaneous scalarization. However, near the transition curve, the two phases are found to be largely indistinguishable regarding both the radial profile and effective potential. This raises the question of whether the hairy black holes triggered by different mechanisms are smoothly joined by a phase transition or whether these are actually identical solutions. To assess the transition more closely, we evaluate the phase diagram in terms of temperature and chemical potential and discover a smooth but first-order transition between the two hairy solutions by explicitly evaluating Gibbs free energy and its derivatives. In particular, one can elaborate a thermodynamic process through which a superconducting black hole transits into a scalarized one by raising or decreasing the temperature. Exhausting the underlying phase space, we analyze the properties and the interplay between the two hairy solutions., Comment: 24 pages and 10 figures

Despite the recent progress in deep neural networks (DNNs), it remains challenging to explain the predictions made by DNNs. Existing explanation methods for DNNs mainly focus on post-hoc explanations where another explanatory model is employed to provide explanations. The fact that post-hoc methods can fail to reveal the actual original reasoning process of DNNs raises the need to build DNNs with built-in interpretability. Motivated by this, many self-explaining neural networks have been proposed to generate not only accurate predictions but also clear and intuitive insights into why a particular decision was made. However, existing self-explaining networks are limited in providing distribution-free uncertainty quantification for the two simultaneously generated prediction outcomes (i.e., a sample's final prediction and its corresponding explanations for interpreting that prediction). Importantly, they also fail to establish a connection between the confidence values assigned to the generated explanations in the interpretation layer and those allocated to the final predictions in the ultimate prediction layer. To tackle the aforementioned challenges, in this paper, we design a novel uncertainty modeling framework for self-explaining networks, which not only demonstrates strong distribution-free uncertainty modeling performance for the generated explanations in the interpretation layer but also excels in producing efficient and effective prediction sets for the final predictions based on the informative high-level basis explanations. We perform the theoretical analysis for the proposed framework. Extensive experimental evaluation demonstrates the effectiveness of the proposed uncertainty framework.

Multi-camera perception tasks have gained significant attention in the field of autonomous driving. However, existing frameworks based on Lift-Splat-Shoot (LSS) in the multi-camera setting cannot produce suitable dense 3D features due to the projection nature and uncontrollable densification process. To resolve this problem, we propose to regulate intermediate dense 3D features with the help of volume rendering. Specifically, we employ volume rendering to process the dense 3D features to obtain corresponding 2D features (e.g., depth maps, semantic maps), which are supervised by associated labels in the training. This manner regulates the generation of dense 3D features on the feature level, providing appropriate dense and unified features for multiple perception tasks. Therefore, our approach is termed Vampire, stands for "Volume rendering As Multi-camera Perception Intermediate feature REgulator". Experimental results on the Occ3D and nuScenes datasets demonstrate that Vampire facilitates fine-grained and appropriate extraction of dense 3D features, and is competitive with existing SOTA methods across diverse downstream perception tasks like 3D occupancy prediction, LiDAR segmentation and 3D objection detection, while utilizing moderate GPU resources. We provide a video demonstration in the supplementary materials and Codes are available at github.com/cskkxjk/Vampire., Comment: Accepted by AAAI 2024

Background: Ovarian cancer is among the three most frequent gynecologic cancers globally. High-grade serous ovarian cancer (HGSOC) is the most common and aggressive histological type. Guided treatment for HGSOC typically involves platinum-based combination chemotherapy, necessitating an assessment of whether the patient is platinum-resistant. The purpose of this study is to propose a deep learning-based method to determine whether a patient is platinum-resistant using multimodal positron emission tomography/computed tomography (PET/CT) images. Methods: 289 patients with HGSOC were included in this study. An end-to-end SE-SPP-DenseNet model was built by adding Squeeze-Excitation Block (SE Block) and Spatial Pyramid Pooling Layer (SPPLayer) to Dense Convolutional Network (DenseNet). Multimodal data from PET/CT images of the regions of interest (ROI) were used to predict platinum resistance in patients. Results: Through five-fold cross-validation, SE-SPP-DenseNet achieved a high accuracy rate and an area under the curve (AUC) in predicting platinum resistance in patients, which were 92.6% and 0.93, respectively. The importance of incorporating SE Block and SPPLayer into the deep learning model, and considering multimodal data was substantiated by carrying out ablation studies and experiments with single modality data. Conclusions: The obtained classification results indicate that our proposed deep learning framework performs better in predicting platinum resistance in patients, which can help gynecologists make better treatment decisions. Keywords: PET/CT, CNN, SE Block, SPP Layer, Platinum resistance, Ovarian cancer

Deep neural networks have exhibited remarkable performance across a wide range of real-world tasks. However, comprehending the underlying reasons for their effectiveness remains a challenging problem. Interpreting deep neural networks through examining neurons offers distinct advantages when it comes to exploring the inner workings of neural networks. Previous research has indicated that specific neurons within deep vision networks possess semantic meaning and play pivotal roles in model performance. Nonetheless, the current methods for generating neuron semantics heavily rely on human intervention, which hampers their scalability and applicability. To address this limitation, this paper proposes a novel post-hoc framework for generating semantic explanations of neurons with large foundation models, without requiring human intervention or prior knowledge. Our framework is designed to be compatible with various model architectures and datasets, facilitating automated and scalable neuron interpretation. Experiments are conducted with both qualitative and quantitative analysis to verify the effectiveness of our proposed approach.

Although partition temperature derived using the Darwin-Fowler method is exact for simple scenarios, the derivation for complex systems might reside on specific approximations whose viability is not ensured if the thermodynamic limit is not attained. This work elaborates on a related problem relevant to relativistic high-energy collisions. On the one hand, it is simple enough that closed form expression can be obtained precisely for the one-particle distribution function. On the other hand, the resulting expression is not an exponential form, and therefore it is not straightforward that the notion of partition function could be implied. Specifically, we derive the one-particle distribution function for massless particles where the phase space integration is performed exactly for the underlying canonical ensemble consisting of a given number of particles. We discuss the viability of the partition temperature in this case. Possible implications of the obtained results regarding the observed Tsallis distribution in transverse momentum spectra in high-energy collisions are also addressed., Comment: 12 pages, 2 figures, and 1 table

Hypergraph Neural networks (HyperGNNs) and hypergraph signal denoising (HyperGSD) are two fundamental topics in higher-order network modeling. Understanding the connection between these two domains is particularly useful for designing novel HyperGNNs from a HyperGSD perspective, and vice versa. In particular, the tensor-hypergraph convolutional network (T-HGCN) has emerged as a powerful architecture for preserving higher-order interactions on hypergraphs, and this work shows an equivalence relation between a HyperGSD problem and the T-HGCN. Inspired by this intriguing result, we further design a tensor-hypergraph iterative network (T-HGIN) based on the HyperGSD problem, which takes advantage of a multi-step updating scheme in every single layer. Numerical experiments are conducted to show the promising applications of the proposed T-HGIN approach., Comment: 5 pages, accepted by EUSIPCO 2023

We derive an exact expression for the celebrated backbone exponent for Bernoulli percolation in dimension two at criticality. It turns out to be a root of an elementary function. Contrary to previously known arm exponents for this model, which are all rational, it has a transcendental value. Our derivation relies on the connection to the SLE$_\kappa$ bubble measure, the coupling between SLE and Liouville quantum gravity, and the integrability of Liouville conformal field theory. Along the way, we derive a formula not only for $\kappa=6$ (corresponding to percolation), but for all $\kappa \in (4,8)$., Comment: 60 pages, 15 figures; simplified arguments in Appendix C; minor improvements of presentations

Multiplicity fluctuations play a crucial role in relativistic heavy-ion collisions. In this work, we explore how the multiplicity fluctuations can be effectively suppressed in the measurement of particle correlations. In particular, through proper normalization, particle correlations can be evaluated in a manner irrelevant to multiplicity. When the multiplicity fluctuations are adequately extracted, Monte Carlo simulations show that the remaining correlations possess distinct features buried in the otherwise overwhelming fluctuations. Moreover, we argue that such a normalization scheme naturally agrees with the multi-particle correlator, which can be evaluated using the Q-vectors. The implications of the present study in the data analysis are also addressed., Comment: 15 pages, 3 figures

In this work, we explore the effects of surrounding dark matter featuring different equations of state on the axial gravitational quasinormal modes of supermassive black holes situated at the center of galaxies. Our attention primarily rests on dark matter exhibiting a spike structure, originating from relativistic Bondi accretion through an adiabatic process, which diminishes at a certain distance from the black hole. We analyze how varying the equation of state of the dark matter influences the properties of the spacetime in the black hole's vicinity. Our findings reveal that different states of dark matter spikes correspondingly affect the black hole's quasinormal modes. In particular, we identify deviations in both the ringing frequency and damping time, reaching magnitudes of up to $10^{-3}$ for certain parameter values. These variations can potentially be detected by upcoming space-borne detectors. Our findings thus indicate the feasibility of discerning and limiting the essential properties of dark matter surrounding supermassive black holes using future gravitational wave detections, particularly in the case of extreme mass ratio inspiral systems., Comment: 23 pages, 6 figures, 1 table

In this paper, we present our solution to the MuSe-Personalisation sub-challenge in the MuSe 2023 Multimodal Sentiment Analysis Challenge. The task of MuSe-Personalisation aims to predict the continuous arousal and valence values of a participant based on their audio-visual, language, and physiological signal modalities data. Considering different people have personal characteristics, the main challenge of this task is how to build robustness feature presentation for sentiment prediction. To address this issue, we propose exploiting diverse features. Specifically, we proposed a series of feature extraction methods to build a robust representation and model ensemble. We empirically evaluate the performance of the utilized method on the officially provided dataset. \textbf{As a result, we achieved 3rd place in the MuSe-Personalisation sub-challenge.} Specifically, we achieve the results of 0.8492 and 0.8439 for MuSe-Personalisation in terms of arousal and valence CCC.

In the field of monocular 3D detection, it is common practice to utilize scene geometric clues to enhance the detector's performance. However, many existing works adopt these clues explicitly such as estimating a depth map and back-projecting it into 3D space. This explicit methodology induces sparsity in 3D representations due to the increased dimensionality from 2D to 3D, and leads to substantial information loss, especially for distant and occluded objects. To alleviate this issue, we propose MonoNeRD, a novel detection framework that can infer dense 3D geometry and occupancy. Specifically, we model scenes with Signed Distance Functions (SDF), facilitating the production of dense 3D representations. We treat these representations as Neural Radiance Fields (NeRF) and then employ volume rendering to recover RGB images and depth maps. To the best of our knowledge, this work is the first to introduce volume rendering for M3D, and demonstrates the potential of implicit reconstruction for image-based 3D perception. Extensive experiments conducted on the KITTI-3D benchmark and Waymo Open Dataset demonstrate the effectiveness of MonoNeRD. Codes are available at https://github.com/cskkxjk/MonoNeRD., Comment: Accepted by ICCV 2023

Remote photoplethysmography (rPPG) based physiological measurement is an emerging yet crucial vision task, whose challenge lies in exploring accurate rPPG prediction from facial videos accompanied by noises of illumination variations, facial occlusions, head movements, \etc, in a non-contact manner. Existing mainstream CNN-based models make efforts to detect physiological signals by capturing subtle color changes in facial regions of interest (ROI) caused by heartbeats. However, such models are constrained by the limited local spatial or temporal receptive fields in the neural units. Unlike them, a native Transformer-based framework called Dual-path TokenLearner (Dual-TL) is proposed in this paper, which utilizes the concept of learnable tokens to integrate both spatial and temporal informative contexts from the global perspective of the video. Specifically, the proposed Dual-TL uses a Spatial TokenLearner (S-TL) to explore associations in different facial ROIs, which promises the rPPG prediction far away from noisy ROI disturbances. Complementarily, a Temporal TokenLearner (T-TL) is designed to infer the quasi-periodic pattern of heartbeats, which eliminates temporal disturbances such as head movements. The two TokenLearners, S-TL and T-TL, are executed in a dual-path mode. This enables the model to reduce noise disturbances for final rPPG signal prediction. Extensive experiments on four physiological measurement benchmark datasets are conducted. The Dual-TL achieves state-of-the-art performances in both intra- and cross-dataset testings, demonstrating its immense potential as a basic backbone for rPPG measurement. The source code is available at \href{https://github.com/VUT-HFUT/Dual-TL}{https://github.com/VUT-HFUT/Dual-TL}

Consider a Brownian loop soup $\mathcal{L}_D^\theta$ with subcritical intensity $\theta \in (0,1/2]$ in some 2D bounded simply connected domain. We define and study the properties of a conformally invariant field $h_\theta$ naturally associated to $\mathcal{L}_D^\theta$. Informally, this field is a signed version of the local time of $\mathcal{L}_D^\theta$ to the power $1-\theta$. When $\theta=1/2$, $h_\theta$ is a Gaussian free field (GFF) in $D$. Our construction of $h_\theta$ relies on the multiplicative chaos $\mathcal{M}_\gamma$ associated with $\mathcal{L}_D^\theta$, as introduced in [ABJL23]. Assigning independent symmetric signs to each cluster, we restrict $\mathcal{M}_\gamma$ to positive clusters. We prove that, when $\theta=1/2$, the resulting measure $\mathcal{M}_\gamma^+$ corresponds to the exponential of $\gamma$ times a GFF. At this intensity, the GFF can be recovered by differentiating at $\gamma=0$ the measure $\mathcal{M}_\gamma^+$. When $\theta<1/2$, we show that $\mathcal{M}_\gamma^+$ has a nondegenerate fractional derivative at $\gamma=0$ defining a random generalised function $h_\theta$. We establish a result which is analoguous to the recent work [ALS23] in the GFF case ($\theta=1/2$), but for $h_\theta$ with $\theta \in (0,1/2]$. Relying on the companion article [JLQ23], we prove that each cluster of $\mathcal{L}_D^\theta$ possesses a nondegenerate Minkowski content in some non-explicit gauge function $r \mapsto r^2 |\log r|^{1-\theta+o(1)}$. We then prove that $h_\theta$ agrees a.s. with the sum of the Minkowski content of each cluster multiplied by its sign. We further extend the couplings between CLE$_4$, SLE$_4$ and the GFF to $h_\theta$ for $\theta\in(0,1/2]$. We show that the (non-nested) CLE$_\kappa$ loops form level lines for $h_\theta$ and that there exists a constant height gap between the values of the field on either side of the CLE loops., Comment: 80 pages, 4 figures. Improved presentation compared to first version