\section{Introduction}

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\section{Introduction}
\begin{figure}[t]
 \centering
  \begin{subfigure}{0.33\linewidth}
  \begin{minipage}[t]{0.33\linewidth}
	\includegraphics[width=1.0in]{1/11.jpg}\vspace{0.1cm}
	\includegraphics[width=1.0in]{1/14.png}
 	\end{minipage}
 \caption*{Input}
\end{subfigure}\hspace{-1mm}
\begin{subfigure}{0.33\linewidth}
\begin{minipage}[t]{0.33\linewidth}
\includegraphics[width=1.0in]{1/12.png}\vspace{0.1cm}
\includegraphics[width=1.0in]{1/15.png}
\end{minipage}
\caption*{MIRNet}
\end{subfigure}\hspace{-1mm}
\begin{subfigure}{0.33\linewidth}
\begin{minipage}[t]{0.33\linewidth}
\includegraphics[width=1.0in]{1/13.png}\vspace{0.1cm}
\includegraphics[width=1.0in]{1/16.png}
\end{minipage}
\caption*{Ours}
\end{subfigure}
\caption{Visual comparision with supervised low light image enhancement method MIRNet~\cite{han2020mirnet}.The proposed method can well imporve the contrast and reduce noises and at same time reduce the color bias.}
\vspace{-0.5cm}
\end{figure}
Low-light Image Enhancement is a fundamental task in low-level vision and an important pre-processing step in many other vision tasks. images captured under the unsuitable lighting are either too dark or too bright. The art of recovering the original clean image from its corrupted measurements is studied under the image restoration task. It is an ill-posed inverse problem,due to the existence of many possible solutions. While we enhancing the brightness, we also need to tackle the color distortion, the amplified noise, the loss of detail and texture information and the blurred edges. Traditional methods usually address low-light image enhancement by histogram equalization(HE)-based approaches~\cite{pizer1987adaptive,pizer1990contrast,2007Brightness}, Retinex Theory~\cite{1978The,jobson1997properties,Jobson1997A}, Gamma Correction~\cite{rahman2016adaptive}, etc. However, these methods need to manually set parameters,have poor generalization ability and often result in visible noise for real low-light images. With the rapid development of deep learning technology, numerous advanced approaches~\cite{2017LLNet,2018LLCNN} have been developed and achieve impressive success. By involving the strategy of a noise-robust autoencoder-based way and ReLU activate function into deep architectures, LLNet~\cite{2017LLNet} is proposed to enlighten images with minimum pixel-level saturation and achieves a much higher peak signal (PSNR) to noise ratio than conventional state-of-the-art approaches~\cite{guo2016lime}. For the pursuit of highly accurate restoring results,some follow-up works have been proposed to decomposes low-light input into reflectance and illumination maps~\cite{wei2018deep,zhang2019kindling}. To involve multi-scale information,some most advanced end-to-end methods~\cite{2020Attention,han2020mirnet} apply U-net~\cite{ronneberger2015u} as their basic structure and add some dense residual block in each level. Although these methods obtain competitive performance on benchmark datasets, their heavy inference time hinder their application.

To seek a better trade-off between image denosing performance and the consumption of computational resources, Densely self-guided Wavelet network (DSWN)~\cite{2020Densely} is proposed for image denoising task by a top-down guidance strategy. DSWN generates multi-resolution inputs with the discrete wavelet  transformation (DWT) and inverse discrete wavelet transform (IDWT) before any convolutional operation. Large-scale contextual information extracted at low resolution is gradually propagated into the higher resolution sub-networks to guide the feature extraction processes at these scales. Using such a structure,DSWN is able to achieve a better denoising performance than U-net with less runtime and GPU memory.

Inspired by DSWN, we proposed a  Attention based Broadly Self-guided Network (ABSGN) (as is shown in Figure 2) which is able to improve performance of MIRNet (as is shown in Figure 1) and require less runtime than the state-of-the-art densely networks based on U-net structure. In ABSGN, we design Global Spatial Attention (GSA) Block (as shown in Figure 3) to better get the global information at the lowest resolution level. Then, we embed dilation convolution into Densely conneted Block to enlarge receptive Field using the self-guided strategy~\cite{gu2019self}, which we called Multi-level Guided Dense Block (MGDB), To achieve a better performance, we adopt more MGDB blocks with dense connections at the full resolution level. We combine discrete wavelet transformation (DWT) and a convolution layer with 3x3 convolution kernel to achieve the upsampling process, Corresponding to which we use inverse discrete wavelet transform (IDWT) and a convolution layer with 3x3 convolution kernel to achieve the downsampling process. Such a design can better extract the information of multi-scale feature maps. In addition,wavelet has been applied to denoising task in traditional methods\cite{sardy2001robust}. Utilizing wavelet transform to incorporate multi-scale information makes it possible for the network to have time-frequency analysis capabilities. Such a structure is able to help our network to deal with the low-light image enhancement task at different exposures.Our main contributions are concluded as follows:
\begin{itemize}
    \item As we konw,we firstly introduce Self-guided Network and Multi-level wavelet transform into low-light image enhancement,which can accelerate the inference speed and avoid down-sampling information loss.
    \item We embed dilation convolution into Dense Block to enlarge the receptive Field using the self-guided strategy,which achieve a higher PSNR and preserve more details.
    \item We propose a new global Attention module called GSA,which has better global feature extraction capability and further improve image restoration performance.
    \item We design a broadly self-guided wavelet network which outperforms conventional methods and is more efﬁcient than the state-of-the-art low-light image enhancement networks with dense blocks.
\end{itemize}

\section{Related works}
In this section, we briefly introduce some works related to our research. First, we review some deep learning based low-light image Enhancement, then we discuss some previous works including Attention mechanism and Feature extraction.
\subsection{Deep Neural Networks for low-light image enhancement}
In recent years, researches have shown that deep learning technologies outperform traditional methods on low-light image Enhancement by extracting more suitable image features~\cite{2018GLADNet}. including supervised learning,unsupervised learning and zero-shot learning. Most of the early works based on supervised learning train the networks. As we know,K.G.Lore et.al~\cite{2017LLNet} firstly designed a deep neural networks with stacked sparse denosing autoencoder perform low-light image enhancement with minimum pixel-level saturation.Using feedforward CNN with different Gaussian convolution kernels, MSR-Net~\cite{shen2017msr} is able to simulate the pipeline of Multi-scale Retinex (MSR) for directly learning end-to-end mapping between dark and bright images. GLADNet~\cite{2018GLADNet} is able to calculates global illumination and has global illumination-aware, which better handle a wide range of light levels.  By introducing Retinex theory to explicitly decompose the image into reflectance and illumination, RetinexNet~\cite{wei2018deep} is able to enhance the lightnesss over illumination. The similar work KinD~\cite{zhang2019kindling} additionally introduce degradation removal in the reflectance to improve the quality of the restored image. KinD++ is the improved version of KinD,which Effectively improve the quality of reflection map by using Multi-scale illumination attention(MISA) module. Considering both Retinex model and GAN, RDGAN~\cite{2019RDGAN} further improved the restored image quality by embedding a GAN network.

Besides, by involving the strategy of generative adversarial network (GAN) into deep architectures, EnlightenGan~\cite{2021EnlightenGAN} is proposed to handle the limitation of low/normal-light pairs, which learns a one-to-many relation from low-light to normal-light image space without paired datasets. To overcome the lack of paired training data, Zhang et.al ~\cite{zhang2021self} presented a self-supervised low-light Image Enhancemnet, which fully make use of Retinex theroy to decompose the low-light image and enhance the reflectance map to view as the final restored image. In addition, Zero-DCE~\cite{2020Zero} estimates the pixel-wise and the high-order curves for dynamic range adjustment of a low-light image in an unsupervised way, which use meticulously designed non-reference loss functions for training. Recently, Syed Waqas Zamir et al~\cite{han2020mirnet} proposed a Multiscale Residual Block (MRB) that uses both residual learning and Attention Unit as its basic structure, maximizing feature reuse and achieving a significant improvement in the performance of low light image enhancement.
\subsection{Attention mechanism}
The attention mechanism has been studied in a wide variety of computer vision problems, which try to mimic the human visual system in making substantial use of contextual information in understanding RGB images. This involves discarding unwanted regions in the image, while focusing on more important parts containing rich features of our vision task. Based on such an idea,SENet~\cite{hu2018squeeze} is to learn feature weights according to loss through the network,so that the effective feature map weight is large, and the invalid or small effect is small. Using such a strategy to train the model to achieve better results. Further, Woo,S.et.al propose Channel Attention Module (CBA) and Spatial Attention module (SPA) to pay attention to different components. In this paper,we propose a novel Global attention CNN model. In which,We use the spatial Attention to improve the attention to different area in feature maps. Since the feature map of each channel has different contributions to the following network. Therefore, we also embed CBA into our MGDB to extract more useful information for low-light image enhancement. The structure of channel attention is illustrated in Figure 2.
\subsection{Feature Extraction}
PixelShuffle and wavelet transform have been proposed to replace pooling and interpolation to avoid information loss. Multi-level wavelet transform is considered by DSWN~\cite{2020Densely} to achieve better receptive field size and avoid down-sampling information loss by embedding wavelet transform into CNN architecture. DSWN owns more power to model both spatial context and inter-subband dependency by embedding DWT and IDWT to CNN. In this paper,our proposed network adopts the same method as DSWN to extract multi-scale information with a totally different architecture from DSWN.

As we know, stacking multiple convolutional layers can help us effectively extract high-dimensional information in the feature map. However, this will lead to a substantial increase in network parameters, requiring the massive amount of training data to prevent overfitting. In addition, the size of the convolution kernel used in the convolution layer will bring different receptive fields to feature extraction. The larger the size of the convolution kernel used, the larger the receptive field. But it will bring heavy computation and slower running time. With Densely Connected Residual Block, DRNet~\cite{2018Image} mitigates the problems of overfitting, vanishing gradient, and training instability during training very deep and wide networks. Moreover, it can improve the propagation and reuse of features by creating direct connections from the previous layers to the subsequent layers. Recently, Injecting holes in the standard convolution map, The dilated convolutions support exponential expansion of the receptive field without loss of resolution or coverage. The number of parameters associated with each layer is identical. The receptive field grows exponentially. Utilizing the dilation convolutions and Self-guided leanring strategy, MLGRB~\cite{WANG2019206} can extract the spatial contextual conformation to enhance further the feature representation ability. Inspired by MLGRB, we adopts the similar architecture as MLGRB to further acquire multi-scale information in different resolution space.
\begin{figure*}[t]
  \centering

   \includegraphics[width=1.0\linewidth]{test11.png}
   \caption{An illustration of our proposed network}
\end{figure*}
\section{Attention based Broadly Self-guided  Network}
In this section, we firstly introduce the overall network structure and then introduce the details of ABSGN.
\subsection{Overall Structure of ABSGN}
Our proposed network use a top-down self-guidance architecture to better exploit image multi-scale information. Information extracted at low resolution is gradually propagated into the higher resolution sub-networks to guide the feature extraction processes. Firstly, we pass the input image through a Conv+PReLU layers to obtain a main feature map with 32 channels. Then instead of PixelShuffle and PixelUnShuffle,DWT and IDWT are used to generate multi-scale inputs. ABSGN uses wavelet transform to transform the main feature map to three smaller scales. At the lowest resolution layer, we offer a Global Spatial Attention (GSA) Block including spatial attention module~\cite{2018CBAM} to be aware of the global context/color
information. Apart from the lowest solution level, we add Multi-level Guided Densely Block (MGDB) one or two blocks  as shown in Figure 2. At the full resolution layer, we adopt Dense connetions consists of two MGDB to imporove the reuse of the main feature map. In the rest of this paper, we simply refer to the advantages of GSA and MGDB. we also make a comparative experiment to prove our conclusion. In addition, we ﬁnd batch normalization is harmful for the denoising performance and only use one normalization layer in this network.
\subsection{Detail Structure of ABSGN}
different form DSWN, we fistly acquire the main feature map with 32 channels from the input image by a Conv+PReLU layer. the aim of which is to increase the feature map by increasing the number of channel. we use DWT and a Conv+PReLU layer to finish the downsampling process. Using a Conv+PReLU layer to decrease the number of channels,which not only can decrease the calculation but also fine-tune the down-sampled feature map, Performing the above downsampling operation three times in sequence to obtain feature maps of different sizes.

The top level of ABSGN works on the smallest spatial resolution to extract large scale information.As is shown in Figure 3, the top sub-network use the GSA block to get the global information. which contains two Conv+PReLU layers and AdaptiveAvgPool2d, AdaptiveMaxPool2d, interpolation (the Resize block in Figure 3) and Spatial Attention module (SPA). Particulary, given an input feature map, i.e.X, with a size of $H\times W\times C$, AdaptiveAvgPool2d and AdaptiveMaxPool2d is employed to extract the representative information. Average of them is used for the global information producing a feature map with a size of $1\times 1\times C$. Then, an interpolation function is utilized to upscale the feature map with global information which is processed by a Conv+PReLU to shrink the number of channels, yielding a global feature map with a size of $H\times W\times C_1$.
\begin{figure}[t]
   \centering
   \includegraphics[width=1.0\linewidth]{module.png}
   \caption{Diagram of Global Spatial Attention (GSA) block}
\end{figure}
Then we apply SPA block to increase attention to different areas of the global feature map. The SPA block simultaneously applies max-pooling and average-pool Along channel dimention, then cancatenate the two feature maps to generate feature descriptors, the purpose of which is to highlight the information area. the feature descriptors then generates a spatial attention map through a convolution layer, Finally the map normalized by Activation function (Sigmoid) multiplies the input to obtain the output as optimized global feature map. Finally a concat function and a Conv+PReLU function are employed to combine the input feature map (encoding local information) and the optimized global feature map (encoding global information) to produce an output feature map with $H\times W\times C$.

Corresponds to the above downsampling process, we use IDWT and a Conv+PReLU layer to finish the upsampling process. At the middle two levels, 1x1 convolutional kernel layers are used to merge information extracted from different resolution. The network structure of the middle sub-networks take MGDB to fully extract the feature inforamation, which embed dilation convolution with different dilation factor into Dense Block. As is shown in Figure 2, we apply self-guided strategy to use the dilation covolution with greater dilation factor to guide the dilation covolution with the smaller one. before the final output, we add Channel Attention module~\cite{2018CBAM} to enhance the attention to different channels.

As for the full resolution level, we add more MGDB to reuse the main feature map, which enhance the feature extraction capability of ABSGN, after merging information from all the scales, we use two Conv+PReLU layers to acqiure the final output as the restored image. By adding gradient loss, our network is able to achieve better retention of details without reducing PSNR. Inspired by a new joint loss function~\cite{2015Loss}, our network uses L1 loss plus SSIM loss for training. The total loss is as follows:
\begin{equation}
\mit L_{ABSGN} = \gamma\mit L_{SSIM}(I,\hat{I})+(1-\gamma)\mit L_{l1}(I,\hat{I})
\end{equation}
where $\gamma \in [0,1]$ is the weight to balance the two terms. Here we choose the value of the $\gamma$ is 0.16.
\section{Experiment}
In this section, we ﬁrst introduce the training details and provide experimental results on different datasets. Then we compare ABSGN with several state-of-the-art low-light image enhancement approaches.
\begin{figure*}[t]
 \centering
 \begin{subfigure}{0.2\linewidth}
 \centering
 \includegraphics[width=1.3in]{4/41.png}
 \caption*{(1) Input}
 \end{subfigure}\hspace{-2mm}
 \begin{subfigure}{0.2\linewidth}
  \centering
 \includegraphics[width=1.3in]{4/42.png}
 \caption*{(2) LIME}
 \end{subfigure}\hspace{-2mm}
  \begin{subfigure}{0.2\linewidth}
   \centering
 \includegraphics[width=1.3in]{4/43.png}
 \caption*{(3) GLADNet}
 \end{subfigure}\hspace{-2mm}
  \begin{subfigure}{0.2\linewidth}
   \centering
 \includegraphics[width=1.3in]{4/44.png}
 \caption*{(4) EnlightenGAN}
 \end{subfigure}\hspace{-2mm}
  \begin{subfigure}{0.2\linewidth}
  \centering
 \includegraphics[width=1.3in]{4/45.png}
 \caption*{(5) KinD++}
 \end{subfigure}

 \begin{subfigure}{0.2\linewidth}
 \centering
 \includegraphics[width=1.3in]{4/46.png}
 \caption*{(6) Zero-DCE}
 \end{subfigure}\hspace{-2mm}
 \begin{subfigure}{0.2\linewidth}
  \centering
 \includegraphics[width=1.3in]{4/47.png}
 \caption*{(7) RUAS}
 \end{subfigure}\hspace{-2mm}
  \begin{subfigure}{0.2\linewidth}
   \centering
 \includegraphics[width=1.3in]{4/48.png}
 \caption*{(8) MIRNet}
 \end{subfigure}\hspace{-2mm}
  \begin{subfigure}{0.2\linewidth}
   \centering
 \includegraphics[width=1.3in]{4/49.png}
 \caption*{(9) Ours}
 \end{subfigure}\hspace{-2mm}
  \begin{subfigure}{0.2\linewidth}
  \centering
 \includegraphics[width=1.3in]{4/410.png}
 \caption*{(10) Reference}
 \end{subfigure}
\caption{Restoring results of the conventional methods and the proposed method on LOL dataset(lighter)}
\end{figure*}

\begin{figure*}[t]
 \centering
 \begin{subfigure}{0.2\linewidth}
 \centering
 \includegraphics[width=1.3in]{5/51.png}
 \caption*{(1) Input}
 \end{subfigure}\hspace{-2mm}
 \begin{subfigure}{0.2\linewidth}
  \centering
 \includegraphics[width=1.3in]{5/52.png}
 \caption*{(2) LIME}
 \end{subfigure}\hspace{-2mm}
  \begin{subfigure}{0.2\linewidth}
   \centering
 \includegraphics[width=1.3in]{5/53.png}
 \caption*{(3) GLADNet}
 \end{subfigure}\hspace{-2mm}
  \begin{subfigure}{0.2\linewidth}
   \centering
 \includegraphics[width=1.3in]{5/54.png}
 \caption*{(4) EnlightenGAN}
 \end{subfigure}\hspace{-2mm}
  \begin{subfigure}{0.2\linewidth}
  \centering
 \includegraphics[width=1.3in]{5/55.png}
 \caption*{(5) KinD++}
 \end{subfigure}

 \begin{subfigure}{0.2\linewidth}
 \centering
 \includegraphics[width=1.3in]{5/56.png}
 \caption*{(6) Zero-DCE}
 \end{subfigure}\hspace{-2mm}
 \begin{subfigure}{0.2\linewidth}
  \centering
 \includegraphics[width=1.3in]{5/57.png}
 \caption*{(7) RUAS}
 \end{subfigure}\hspace{-2mm}
  \begin{subfigure}{0.2\linewidth}
   \centering
 \includegraphics[width=1.3in]{5/58.png}
 \caption*{(8) MIRNet}
 \end{subfigure}\hspace{-2mm}
  \begin{subfigure}{0.2\linewidth}
   \centering
 \includegraphics[width=1.3in]{5/59.png}
 \caption*{(9) Ours}
 \end{subfigure}\hspace{-2mm}
  \begin{subfigure}{0.2\linewidth}
  \centering
 \includegraphics[width=1.3in]{5/510.png}
 \caption*{(10) Reference}
 \end{subfigure}
\caption{Restoring results of the conventional methods and the proposed method on LOL dataset(Darker)}
\end{figure*}
\subsection{Experimental Setting}
Low-Light (LOL) dataset~\cite{wei2018deep} is a publicly available dark-light paired images dataset in the real sense. The low-light images are collected by changing exposure time and ISO. It contains 500 images in total, The resolution of each of these images is 400 × 600. we use 485 images of them for training, and the rest for evaluation as suggested by Most state-of-the-art low-light image enhancement solutions select the LOL as their training dataset~\cite{2018GLADNet,zhang2019kindling}. To train our ABSGN model, we also use LOL dataset as training and validation dataset for low-light image enhancement task. The LOL dataset consists of two parts: the real-world dataset and the synthetic dataset. Because the LOL synthetic dataset cannot simulate the degradation of real-world images very well, we deprecate the synthetic dataset where low-light images are synthesized from normal-light images and only use the real-world dataset. Our datasets for comparision also includes LIME~\cite{guo2016lime}, MEF~\cite{2015Perceptual}, DICM~\cite{2013Contrast} which are used by some recent low-light image enhancment networks~\cite{zhang2019kindling}. For these datasets lack of reference images, we use non-reference metric including No-reference Image Quality and Uncertainly Evaluator (UNIQUE)~\cite{zhang2021uncertainty} and Blind/Referenceless Image Spatial Quality Evaluator (BRISQUE)~\cite{2012No}. In addition, we also choose another commonly used data set called MIT-Adobe FiveK dataset for training and evaluation. MIT-Adobe FiveK~\cite{2011Learning} contains 5000 images of various indoor and outdoor scenes captured with DSLR cameras in different lighting conditions. The tonal attributes of all images are manually adjusted by five different trained photographers (labelled as experts A to E). we choose the enhanced images of expert C as the ground-truth. Moreover, the first 4500 images are used for training and the last 500 for testing. When training our model, we randomly crop 256 × 256 patches from the training images. The input patches of the proposed network are randomly flipped and rotated for data augmentation. The parameters of network are Kaiming initialized~\cite{2015Delving}. We train the whole network for 300 epochs overall. The learning rate is initialized as $1\times10^{-4}$ at the ﬁrst 200 epochs and reduce to $5\times10^{-5}$ in the next 50 epochs. We ﬁnetune our model at the last 50 epochs with a $1\times10^{-5}$ learning rate. For optimization, we use Adam optimizer ~\cite{2014Adam} with $\beta{1} = 0.5$,$\beta2 = 0.999$ and batch size equals to 5. We use L1 loss and SSIM loss for the total loss function, L1 loss is a PSNR-oriented optimization in the training process ~\cite{2015Loss}. SSIM loss can keep the overall structure. We adopt PSNR,SSIM~\cite{2004Image}, LPIPS ~\cite{zhang2018unreasonable} and FSIM ~\cite{zhang2011fsim} as the quantative metrics to measure the performance of our method. We use the Pytorch framework to build our model on the platform with Nvidia TITAN XP GPU and an Intel(R) Xeon(R) E5-2620 v4 2.10GHz CPU.

\subsection{Evaluation on LOL Dataset}
We compare our proposed network with several state-of-the-art low-light image enhancement solutions: MSRCR~\cite{Jobson1997A}, LIME, NPE, JED~\cite{2018Joint}, MBLLEN~\cite{lv2018mbllen}, RetinexNet, GLADNet, RDGAN, KinD++, Zero-DCE, EnlightenGAN and MIRNet. To compare the performance, we determined to take above measures as the objective and subjective measurements.
\begin{table}[t]\small
\centering
\caption{quantitative comparision of serveral metric between our method and state-of-the-art methods on LOL dataset.}
\begin{tabular}{c|cccc}
\hline
  \textbf{Method} & \textbf{SSIM}$\uparrow$ & \textbf{PSNR}$\uparrow$ & \textbf{LPIPS}$\downarrow$  &\textbf{FSIM}$\uparrow$   \\ \hline
    Input                     & 0.1914             & 7.77             & 0.4173            &0.7190      \\\hline
    MSRCR                     & 0.4615             & 13.17            & 0.4404            &0.8450         \\\hline
    LIME                      & 0.4449             & 16.76            & 0.4183            &0.8549       \\\hline
    NPE                       & 0.4839             & 16.97            & 0.4156            &0.8964    \\\hline
    JED                       & 0.6509             & 13.69            & 0.3549            &0.8812       \\\hline
    MBLLEN                    & 0.7247             & 17.86            & 0.3672            &0.9262           \\\hline
    RetinexNet                & 0.4249             & 16.77            & 0.4670            &0.8642           \\\hline
    GLADNet                      & 0.6820             & 19.72            & 0.3994            &0.9329           \\\hline
    RDGAN                     & 0.6357             & 15.94            & 0.3985            &0.9276           \\\hline
    KinD++                    & 0.8203             & 21.30            & 0.1614            &0.9424           \\\hline
    Zero-DCE                  & 0.5623             & 14.87            & 0.3852            &0.9276           \\\hline
    EnlightenGAN              & 0.6515             & 17.48            & 0.3903            &0.9226          \\\hline
    MIRNet                    & 0.8321             & 24.14            & 0.0846            &0.9547           \\\hline
 \textbf{ours} & { \textbf{0.8680}} & { \textbf{24.59}} & {\textbf{0.0772}} & {\textbf{0.9659}} \\ \hline
\end{tabular}
\end{table}
\begin{table}[ht]\scriptsiz
\caption{$UNIQUE\uparrow/BRISQUE\uparrow$ Comparison on NPE, LIME, MEF, DICM}
\centering
\renewcommand{\arraystretch}{2.0}
\begin{tabular}{c|ccc|c}
\hline
\textbf{Method}      &       \textbf{LIME}   &     \textbf{MEF}   &   \textbf{DICM}  &\textbf{Average}\\ [3pt]\hline
    Dark             & 0.826/21.81       & 0.738/23.56       &0.795/21.57    &0.786/22.31  \\[3pt]
    PIE              & 0.791/22.72       & 0.752/11.02       &0.791/21.72    &0.778/18.49   \\[3pt]
    LIME            & 0.774/20.44       & 0.722/15.25       &0.758/23.48    &0.751/19.72  \\[3pt]
    MBLLEN          & 0.768/30.26       & 0.717/37.44       &0.787/32.44    &0.757/33.38     \\[3pt]
    RetinexNet       & 0.794/31.47       & 0.755/20.08       &0.770/29.53    &0.773/27.03   \\[3pt]
    KinD            & 0.766/39.29       & 0.747/31.36       &0.776/32.71    &0.763/34.45   \\[3pt]
    Zero-DCE         & 0.811/21.40       & 0.762/16.84       &0.777/27.35    &0.783/21.86    \\[3pt]
    MIRNet           & 0.814/33.73       & 0.768/21.45       &0.812/33.71    &0.798/29.63   \\[3pt]\hline
 \textbf{ours}  & { \textbf{0.827 / 32.23}} & { \textbf{0.784 / 38.52}} & { \textbf{0.804 / 33.23}}  & { \textbf{0.806 / 34.66}}\\[3pt] \hline
\end{tabular}
\vspace{-2mm}
\end{table}
As shown in table 1, our method achieves the best performance in PSNR, SSIM, LPIPS and FSIM, MIRNet is the second best method, Although MIRNet adpots more complicated structure and has more than ten times the number of parameters, ABSGN is still able to achieve better PSNR and SSIM than MIRNet on average. Figure 4 and Figure 5 show some example results in LOL dataset with different light levels. we can see that our ABSGN shows the best PSNR and SSIM in different light levels. There also show some detail results of several low-light image enhancement, where KinD++ is a classical decomposing network and MIRNet has excellent feature extraction capabilities. We can see all the three low-light image enhancment networks are able to achieve a obvious improvement compared with low light images. ABSGN is better than KinD++ and MIRNet in some details such as texture details of the book in Figure 4 and the color of the embroidery in Figure 5. Our proposed method is able to handle different light levels and reserve more details at the same time.
\begin{figure*}[t]
\centering
 \begin{subfigure}{0.24\linewidth}
  \begin{minipage}[t]{0.24\linewidth}
	\includegraphics[width=4cm,height=3cm]{11/71.JPG}\vspace{0.1cm}
	\includegraphics[width=4cm,height=3cm]{7/75.png}\vspace{0.1cm}
	\includegraphics[width=4cm,height=3cm]{7/79.png}
 	\end{minipage}
 	\caption*{Input}
\end{subfigure}\hspace{-0.1cm}
\begin{subfigure}{0.24\linewidth}
\begin{minipage}[t]{0.24\linewidth}
\includegraphics[width=4cm,height=3cm]{11/72.png}\vspace{0.1cm}
\includegraphics[width=4cm,height=3cm]{7/76.png}\vspace{0.1cm}
\includegraphics[width=4cm,height=3cm]{7/710.png}
\end{minipage}
\caption*{KinD++}
\end{subfigure}\hspace{-0.1cm}
\begin{subfigure}{0.24\linewidth}
\begin{minipage}[t]{0.24\linewidth}
\includegraphics[width=4cm,height=3cm]{11/73.png}\vspace{0.1cm}
\includegraphics[width=4cm,height=3cm]{7/77.png}\vspace{0.1cm}
\includegraphics[width=4cm,height=3cm]{7/711.png}
\end{minipage}
\caption*{MIRNet}
\end{subfigure}\hspace{-0.1cm}
\begin{subfigure}{0.24\linewidth}
\begin{minipage}[t]{0.24\linewidth}
\includegraphics[width=4cm,height=3cm]{11/74.png}\vspace{0.1cm}
\includegraphics[width=4cm,height=3cm]{7/78.png}\vspace{0.1cm}
\includegraphics[width=4cm,height=3cm]{7/712.png}
\end{minipage}
\caption*{Ours}
\end{subfigure}
\caption{Visual Comparision on DICM,LIME,MEF from top to bottom. from left to right:Dark,KInD++,MIRNet,ours }
\vspace{-0.2cm}
\end{figure*}
\begin{figure*}[t]
 \centering
 \begin{subfigure}{0.33\linewidth}
 \centering
 \includegraphics[width=1.5in]{6/61.png}
 \caption*{(1) Input}
 \end{subfigure}\hspace{-2mm}
 \begin{subfigure}{0.33\linewidth}
  \centering
 \includegraphics[width=1.5in]{6/62.png}
 \caption*{(2) HDRNet~\cite{gharbi2017deep}}
 \end{subfigure}\hspace{-2mm}
  \begin{subfigure}{0.33\linewidth}
   \centering
 \includegraphics[width=1.5in]{6/63.png}
 \caption*{(3) DeepUPE~\cite{2019Underexposed}}
 \end{subfigure}

 \begin{subfigure}{0.33\linewidth}
 \centering
 \includegraphics[width=1.5in]{6/64.png}
 \caption*{(4) MIRNet}
 \end{subfigure}\hspace{-2mm}
 \begin{subfigure}{0.33\linewidth}
  \centering
 \includegraphics[width=1.5in]{6/65.png}
 \caption*{(5) ABSGN(ours)}
 \end{subfigure}\hspace{-2mm}
  \begin{subfigure}{0.33\linewidth}
   \centering
 \includegraphics[width=1.5in]{6/66.png}
 \caption*{(6) References}
 \end{subfigure}
\caption{Restoring results of the conventional methods and the proposed method on MIT-Adobe FiveK dataset}
\end{figure*}
Table 2 shows the comparision on LIME, MEF and DICM datasets.Our model has the best average UNIQUE and the best average BRISQUE. From figure 6, we can also conclude a similar conclusion to the LIME, MEF and DICM datasets. At a darker region, Restoring networks tend to smooth the noise too much, because the network is difficult to distinguish true details from the dark region. ABSGN can better preserve details at dark regions,such as soils and floors.

\subsection{Evaluation on MIT-Adobe FiveK dataset}
We report PSNR/SSIM values of our method and several other techniques in Table 3 for the MIT-Adobe FiveK datasets,respectively. It can be seen that our Network achieves significant improvements over previous approaches.Notably,when compared to the recent best methods, ABSGN obtains 1.52 dB performance gain over over MIRNet on the Adobe-Fivek dataset.
\begin{table}[t]\footnotesize
\caption{Image Enhancement between our method and state-of-the-art methods on MIT-Adobe FiveK dataset.}
\centering
\renewcommand{\arraystretch}{2.0}
\begin{tabular}{cccccc}
\hline
  \textbf{Method}      & \textbf{HDRNet}   &   \textbf{DPE} & \textbf{DeepUPE}  &\textbf{MIRNet} &\textbf{ours}\\ \hline
    PSNR               & 21.96           &22.15    &23.04  &23.73  &25.25  \\
    SSIM               & 0.866           &0.850    &0.893  &0.925  &0.931    \\\hline
\end{tabular}
\vspace{-3mm}
\end{table}
We also show visual results in Figure 7. Compared to other techniques, our method generates enhanced images that are natural and vivid in appearance and have better global and local contrast.
\subsection{Ablation study}
In this part, to prove the effectiveness of the proposed module and the neccessity of introducing the method, we have made comparative Experiment. besides, we did another comparative experiment to emphasize the advantage and practical value of ABSGN.
\subsubsection{Contribution of our proposed module}
This ablation study is to answer the question that why not just adpot Dense Connetly Residual (DCR) Block to get the local information and using Global Attention Aware (GIA) module like GIANet~\cite{meng2020gia} to get the global information like many end-to-end Network, if the network adopt the conventional Global Attention module,Whether there will be better effect. As is mentioned in Sec.3.1, the MGDB adpot the similar self-guided strategy to introduce the dilation convolution,which has larger receptive ﬁeld and better local feature extraction capacity. the CNN based method GSA module is introduced in our proposed method to avoid the problems that too long convolution layers cause the loss of global information.

In Table 4, we show the ablation study of these two modules.we use the DenseRes Block to replace our MGDB to test the effect. In Addition, we try to apply the custom convolution to take place of all the dilation convolutions, which will tell our the necessity of introducing Dilation Convolution (DC). The results suggest that MGDB can get more local information and better performance in SSIM/PSNR. Similarity, we respectively use SPA module and GIA module substitute for our GSA module to prove the advantage of our proposed module. As is shown in Table 4, Compared to SPA module and GIA module,our module has irreplaceable advantage. these above module can’t replace our GSA module to extract the global information.
\begin{table}[t]\smal
\caption{Comparision experiment using different modules on LOL dataset.}
\centering
\setlength{\tabcolsep}{2mm}
\begin{tabular}{c|ccccc}
\hline
  \textbf{Method} & \textbf{SPA} &  \textbf{GIA} & \textbf{DCR}  &  \textbf{w/o DC}  & \textbf{ours} \\ [4pt]\hline
    PSNR(dB)        & 22.30           & 23.42        & 22.67           & 23.16              &24.59      \\
    SSIM            & 0.8456          & 0.8515       & 0.8483          & 0.8546             &0.8675       \\\hline
\end{tabular}}
\end{table}
\subsubsection{Adavantage of ABSGN}
We compare BSWN with other state-of-art deep learning low-light image enhancement methods in terms of the number of parameter,the inference time and UQI~\cite{wang2002universal} in LOL Dataset. As is shown in tabe 5, our model has the best UQI, meaning our results has closest information with the reference images.

\begin{table}[t]
\caption{Runtime cost and performance comparison of our mothod and other state-of-art deep learning methods on the LOL dataset.}
\centering
\begin{tabular}{c|cc|c}
\toprule
Deep learning Method & Params & Time cost & UQI$\uparrow$\\
\midrule
    MBLLEN      & 1.95M    & 80ms    & 0.8261 \\
    RetinexNet  & 9.2M     & 20ms    & 0.9110 \\
    GLAD        & 11M      & 25ms    & 0.9204 \\
    RDGAN       & 4.2M     & 30ms    & 0.8296 \\
    Zero-DCE    & 0.97M    & 2ms     & 0.7205 \\
    ElightenGAN & 33M      & 20ms    & 0.8499 \\
    KinD++      & 35.7M    & 280ms   & 0.9482 \\
    MIRNet      & 365M     & 340ms   & 0.9556 \\
    Our model   & 33M      & 14ms    & 0.9589 \\
\bottomrule
\end{tabular}
\label{tab:example}
\end{table}
\begin{figure}[ht]
  \centering
   \includegraphics[width=1.0\linewidth]{scatter_best1.jpg}
   \caption{Runtime cost and SSIM comparison of our method and other state-of-the-art deep learning methods on the LOLdataset.}
\end{figure}
Although the parameters of our model is a little heavy, MIRNet is ten times bigger than ours. what’s more, inference speed of our model is very fast. As is shown in Figure 8, we tested all of the deep learing models on the same platform. The closer the method is to the upper left corner in the figure, the faster the model is and the higher the SSIM is. As is shown in the last colume of table in table 5. our model only take 14 ms to process a real-world low-light image with resolution of 600x400 from the LOL dataset. The speed of our model ranks second among all of the compared methods. Although Zero-DCE has very fast inference speed,which is only to deal with a little darker image. the biggest problem of Zero-DCE is difficult to restore the dark image especially in the night, which leads to its lack of strong applicability and application prospects. In order to reduce the depth of the network, make full use of the system's ability to parallelize, reduce the depth of the network by strengthening the width of the network, our model apply self-guidance strategy to parallel deal with the feature map in different resolutions. This is why our model can run at such a high speed keeping excellent restoring capacity.

\section{Conclution}
In this paper, we proposed a Attention based Broadly Self-guided Network (ABSGN) for low-light image enhancement. ABSGN adopts a top-down manner to restore the low-light images. We use wavelet transform and a conv+PReLU layer to generate input variations with different spatial resolutions. At the lowest solution, we design GSA module to fully collect the global information. Then, we embed a MGDB into the middle two low spatial resolution levels to fully get local information. respectively, At the full resolution level, we employ more MGDB and a Dense connections to further reuse the information of the main feature map from the input image. The proposed ABSGN was validated on image restoration and real world low-light image enhancement benchmark and ABSGN is able to generate higher quality results than the compared state-of-the-art methods. Further, our ABSGN has excellent inference speed, which has good practical value and application prospects.
{\small
\bibliographystyle{ieee_fullname}


\section{Introduction}
\subsection{Canonical partition functions for log-Coulomb gases}\label{1_1}
Let $X$ be a topological space endowed with a metric $d$ and a finite positive Borel measure $\mu$ satisfying $\mu^N\{(x_1,\dots,x_N)\in X^N:x_i=x_j\text{ for some $i\neq j$}\}=0$ for every $N\geq 1$. A \emph{log-Coulomb gas} in $X$ is a statistical model described as follows: Consider $N\geq 1$ particles with fixed charge values $\mathfrak{q}_1,\dots,\mathfrak{q}_N\in\R$ and corresponding variable locations $x_1,\dots,x_N\in X$. Whether the charge values are distinct or not, we assume particles are distinguished by the labels $1,2,\dots,N$, so that unique configurations of the system correspond to unique tuples $(x_1,\dots,x_N)\in X^N$. Each tuple is called a \emph{microstate} and has an \emph{energy} defined by $$E(x_1,\dots,x_N):=\begin{cases}-\sum_{1\leq i<j\leq N}\mathfrak{q}_i\mathfrak{q_j}\log d(x_i,x_j)&\text{if $x_i\neq x_j$ for all $i<j$},\\\infty&\text{otherwise}.\end{cases}$$ Note that $E^{-1}(\infty)$ has measure 0 in $X^N$ by our choice of $\mu$, and that $E$ is identically zero if $N=1$. We assume the system is in thermal equilibrium with a heat reservoir at some \emph{inverse temperature} $\beta>0$, so that its microstates form a canonical ensemble distributed according to the density $e^{-\beta E(x_1,\dots,x_N)}$. The \emph{canonical partition function} $\beta\mapsto Z_N(X,\beta)$ is defined as the total mass of this density, namely
\begin{equation}\label{canonical}
Z_N(X,\beta):=\int_{X^N}e^{-\beta E(x_1,\dots,x_N)}\,d\mu^N=\int_{X^N}\prod_{1\leq i<j\leq N}d(x_i,x_j)^{\mathfrak{q}_i\mathfrak{q}_j\beta}\,d\mu^N~.
\end{equation}
Given $(X,d,\mu)$ and $\mathfrak{q}_1,\dots,\mathfrak{q}_N$, the explicit formula for $Z_N(X,\beta)$ is of primary interest, as it yields fundamental relationships between the observable parameters of the system and its temperature. For instance, the system's dimensionless free energy, mean energy, and energy fluctuation (variance) are respectively given by $-\log Z_N(X,\beta)$, $-\partial/\partial\beta\log Z_N(X,\beta)$, and $\partial^2/\partial\beta^2\log Z_N(X,\beta)$, all of which are functions of $\beta$ (and hence of temperature). Below are two examples in which the formula for $Z_N(X,\beta)$ is known.

\begin{example}\label{arch}
If $X=\R$ with the standard metric $d$, the standard Gaussian measure $\mu$, and charge values $\mathfrak{q}_1=\dots=\mathfrak{q}_N=1$, then \emph{Mehta's integral formula} \cite{FW} states that $Z_N(\R,\beta)$ converges absolutely for all complex $\beta$ with $\re(\beta)>-2/N$, and in this case $$Z_N(\R,\beta)=\prod_{j=1}^N\frac{\Gamma(1+j\beta/2)}{\Gamma(1+\beta/2)}~.$$ At the special values $\beta\in\{1,2,4\}$, the probability density $\frac{1}{Z_N(\R,\beta)}e^{-\beta E(x_1,\dots,x_N)}$ coincides with the joint density of the $N$ eigenvalues $x_1,\dots,x_N$ (counted with multiplicity) of the $N\times N$ Gaussian orthogonal $(\beta=1)$, unitary $(\beta=2)$, and symplectic $(\beta=4)$ matrix ensembles.
\end{example}

\begin{example}\label{Z_3}
If $X=\Z_p$ with $d(x,y)=|x-y|_p$, the Haar probability measure $\mu$, and $N=3$ charges with $\mathfrak{q}_1=1$, $\mathfrak{q}_2=2$, and $\mathfrak{q}_3=3$, then a general theorem in \cite{Web20} implies that $Z_3(\Z_p,\beta)$ converges absolutely for all complex $\beta$ with $\re(\beta)>-1/6$, and in this case $$Z_3(\Z_p,\beta)=\frac{p^{11\beta}}{p^{2+11\beta}-1}\cdot\left((p-1)(p-2)+(p-1)^2\left[\frac{1}{p^{1+2\beta}-1}+\frac{1}{p^{1+3\beta}-1}+\frac{1}{p^{1+6\beta}-1}\right]\right)~.$$
\end{example}

Note that $\beta\mapsto Z_N(X,\beta)$ always extends to a complex domain containing the line $\re(\beta)=0$. To simultaneously treat all possible choices of $\mathfrak{q}_i$, we extend further to a subset of $\C^{N(N-1)/2}$ as follows:

\begin{definition}\label{gen_Z}
For any $N\geq 0$ and $(X,d,\mu)$ as above, we write $\bm{s}$ for a complex tuple $(s_{ij})_{1\leq i<j\leq N}$ (the empty tuple if $N=0$ or $N=1$) and define $\mathcal{Z}_0(X,\bm{s}):=1$ and $$\mathcal{Z}_N(X,\bm{s}):=\int_{X^N}\prod_{1\leq i<j\leq N}d(x_i,x_j)^{s_{ij}}\,d\mu^N\qquad\text{for $N\geq 1$.}$$
\end{definition}

Once the formula and domain for $\bm{s}\mapsto\mathcal{Z}_N(X,\bm{s})$ are known, then for any choice of $\mathfrak{q}_1,\dots,\mathfrak{q}_N\in\R$, the formula and domain for $\beta\mapsto Z_N(X,\beta)$ follow by specializing $\mathcal{Z}_N(X,\bm{s})$ to ${s_{ij}=\mathfrak{q}_i\mathfrak{q}_j\beta}$. Thus, given $(X,d,\mu)$, the main problem is to determine the formula and domain of $\bm{s}\mapsto\mathcal{Z}_N(X,\bm{s})$.

\subsection{$p$-fields, projective lines, and splitting chains}\label{1_2}
Of our two main goals in this paper, the first is to determine the explicit formula and domain of $\bm{s}\mapsto\mathcal{Z}_N(\P^1(K),\bm{s})$, where $K$ is a $p$-field and its projective line $\P^1(K)$ is endowed with a natural metric and measure. To make this precise, we briefly recall well-known properties of $p$-fields (see \cite{Weil}, for instance) and establish some notation. Fix a $p$-field $K$, write $|\cdot|$ for its canonical absolute value, write $d$ for the associated metric (i.e., $d(x,y):=|x-y|$), and define $$R:=\{x\in K:|x|\leq 1\}\qquad\text{and}\qquad P:=\{x\in K:|x|<1\}~.$$ The closed unit ball $R$ is the maximal compact subring of $K$, the open unit ball $P$ is the unique maximal ideal in $R$, and the group of units is $R^\times=R\setminus P=\{x\in K:|x|=1\}$. The residue field $R/P$ is isomorphic to $\F_q$ for some prime power $q$, and there is a canonical isomorphism of the cyclic group $(R/P)^\times$ onto the group of $(q-1)$th roots of unity in $K$. Fixing a primitive such root $\xi\in K$ and sending $P\mapsto 0$ extends the isomorphism $(R/P)^\times\to\{1,\xi,\dots,\xi^{q-2}\}$ to a bijection $R/P\to\{0,1,\xi,\dots,\xi^{q-2}\}$ with inverse $y\mapsto y+P$. Therefore $\{0,1,\xi,\dots,\xi^{q-2}\}$ is a complete set of representatives for the cosets of $P\subset R$, i.e.,
\begin{equation}\label{R_decomp}
R=P\sqcup\underbrace{(1+P)\sqcup(\xi+P)\sqcup\dots\sqcup(\xi^{q-2}+P)}_{R^\times}~.
\end{equation}
Fix a uniformizer $\pi\in K$ (any element satisfying $P=\pi R$) and let $\mu$ be the unique additive Haar measure on $K$ satisfying $\mu(R)=1$. The open balls in $K$ are precisely the sets of the form $y+\pi^vR$ with $y\in K$ and $v\in\Z$, and every such ball is compact with measure equal to its radius, i.e., $\mu(y+\pi^vR)=|\pi^v|=q^{-v}$. In particular, each of the $q$ cosets of $P$ in \eqref{R_decomp} is a compact open ball with measure (and radius) $q^{-1}$, and two elements $x,y\in R$ satisfy $|x-y|=1$ if and only if they belong to different cosets. Henceforth, we reserve the symbols $K$, $|\cdot|$, $d$, $R$, $P$, $q$, $\xi$, $\pi$, and $\mu$ for the items above, and we distinguish $|\cdot|$ from the standard absolute value on $\C$ by writing $|\cdot|_\C$ for the latter.\\

We now recall some useful facts from \cite{FiliPetsche} in our present notation. The projective line of $K$ is the quotient space $\P^1(K)=(K^2\setminus\{(0,0)\})/\sim$, where $(x_0,x_1)\sim(y_0,y_1)$ if and only if $y_0=\lambda x_0$ and $y_1=\lambda x_1$ for some $\lambda\in K^\times$. Thus we may understand $\P^1(K)$ concretely as the set of symbols $[x_0:x_1]$ with $(x_0,x_1)\in K^2\setminus\{(0,0)\}$, subject to the relation $[\lambda x_0:\lambda x_1]=[x_0:x_1]$ for all $\lambda\in K^\times$ and endowed with the topology induced by the quotient $(x_0,x_1)\mapsto[x_0:x_1]$. The projective line is compact and metrizable by the \emph{spherical metric} $\delta:\P^1(K)\times\P^1(K)\to\{0\}\cup\{q^{-v}:v\in\Z_{\geq 0}\}$, which is defined via
\begin{equation}\label{sph_met_def}
\delta([x_0:x_1],[y_0:y_1]):=\frac{|x_0y_1-x_1y_0|}{\max\{|x_0|,|x_1|\}\cdot\max\{|y_0|,|y_1|\}}~.
\end{equation}
In particular, every open set in $\P^1(K)$ is a union of balls of the form
\begin{equation}\label{ball_def}
B_v[x_0:x_1]:=\{[y_0:y_1]\in\P^1(K):\delta([x_0:x_1],[y_0:y_1])\leq q^{-v}\}
\end{equation}
with $[x_0:x_1]\in\P^1(K)$ and $v\in\Z_{\geq0}$, and every such ball is open and compact. The projective linear group $PGL_2(R)$ is the quotient of $GL_2(R)=\{A\in M_2(R):\det(A)\in R^\times\}$ by its center, namely $Z=\{\left(\begin{smallmatrix}\lambda&0\\0&\lambda\end{smallmatrix}\right):\lambda\in R^\times\}\cong R^\times$. It is straightforward to verify that the rule $$\phi[x_0:x_1]:=[ax_0+bx_1:cx_0+dx_1]~,$$ where $\phi\in PGL_2(R)$ and $\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\in GL_2(R)$ is any representative of $\phi$, gives a well-defined transitive action of $PGL_2(R)$ on $\P^1(K)$.

\begin{lemma}[$PGL_2(R)$-invariance \cite{FiliPetsche}]\label{invariance} The spherical metric satisfies $$\delta(\phi[x_0:x_1],\phi[y_0:y_1])=\delta([x_0:x_1],[y_0:y_1])$$ for all $\phi\in PGL_2(R)$ and all $[x_0:x_1],[y_0:y_1]\in\P^1(K)$. There is also a unique Borel probability measure $\nu$ on $\P^1(K)$ satisfying $$\nu(\phi(M))=\nu(M)$$ for all $\phi\in PGL_2(R)$ and all Borel subsets $M\subset\P^1(K)$. In particular, for each $v\in\Z_{\geq0}$ the relation $\phi(B_v[x_0:x_1])=B_v(\phi[x_0:x_1])$ defines a transitive $PGL_2(R)$ action on the set of balls of radius $q^{-v}$, and thus $\nu(B_v[x_0:x_1])$ depends only on $v$.
\end{lemma}

It is routine to verify that $\mu^N\{(x_1,\dots,x_N)\in(y+\pi^vR)^N:x_i=x_j\text{ for some $i\neq j$}\}=0$ and $$\nu^N\{([x_{1,0}:x_{1,1}],\dots,[x_{N,0}:x_{N,1}])\in(\P^1(K))^N:[x_{i,0}:x_{i,1}]=[x_{j,0}:x_{j,1}]\text{ for some $i\neq j$}\}=0$$ for all $N\geq 1$, so we have suitable metrics and measures to define log-Coulomb gases in $X=y+\pi^vR$ and $X=\P^1(K)$. \Cref{gen_Z} specializes to these $X$ as follows:

\begin{definition}\label{main_Z_def}
With $N\geq 0$ and $\bm{s}$ as before, we have $\mathcal{Z}_0(y+\pi^vR,\bm{s})=\mathcal{Z}_0(\P^1(K),\bm{s})=1$, and
\begin{align*}
\mathcal{Z}_N(y+\pi^vR,\bm{s})&=\int_{(y+\pi^vR)^N}\prod_{1\leq i<j\leq N}|x_i-x_j|^{s_{ij}}\,d\mu^N\qquad\text{and}\\
\mathcal{Z}_N(\P^1(K),\bm{s})&=\int_{(\P^1(K))^N}\prod_{1\leq i<j\leq N}\delta([x_{i,0}:x_{i,1}],[x_{j,0}:x_{j,1}])^{s_{ij}}\,d\nu^N
\end{align*}
for $N\geq 1$. The first integral is independent of $y$, and equal to $\mathcal{Z}_N(R,\bm{s})$ if $v=0$ or $\mathcal{Z}_N(P,\bm{s})$ if $v=1$.
\end{definition}
The formulas and domains of absolute convergence for the integrals above can be stated neatly in terms of the following items from \cite{Web20}:

\begin{definition}[Splitting chains]\label{splchdef}
A \emph{splitting chain} of order $N\geq 2$ and length $L\geq 1$ is a tuple $\spl=(\ptn_0,\dots,\ptn_L)$ of partitions of $[N]=\{1,\dots,N\}$ satisfying $$\{[N]\}=\ptn_0>\ptn_1>\ptn_2>\dots>\ptn_L=\{\{1\},\dots,\{N\}\}~.$$
\begin{itemize}
\item[(a)] Each non-singleton part $\lambda\in\ptn_0\cup\ptn_1\cup\dots\cup\ptn_L$ is called a \emph{branch} of $\spl$. We write $\mathcal{B}(\spl)$ for the set of all branches of $\spl$, i.e., $$\mathcal{B}(\spl):=(\ptn_0\cup\dots\cup\ptn_{L-1})\setminus\ptn_L~.$$
\item[(b)] Each $\lambda\in\mathcal{B}(\spl)$ must appear in a final partition $\ptn_\ell$ before refining into two or more parts in $\ptn_{\ell+1}$, so we define its \emph{depth} $\ell_\spl(\lambda)\in\{0,1,\dots,L-1\}$ and \emph{degree} $\deg_\spl(\lambda)\in\{2,3,\dots,N\}$ by $$\ell_\spl(\lambda):=\max\{\ell:\lambda\in\ptn_\ell\}\qquad\text{and}\qquad\deg_\spl(\lambda):=\#\{\lambda'\in\ptn_{\ell_\spl(\lambda)+1}:\lambda'\subset\lambda\}~.$$
\item[(c)] We say that $\spl$ is \emph{reduced} if each $\lambda\in\mathcal{B}(\spl)$ satisfies $\lambda\in\ptn_\ell\iff\ell=\ell_\spl(\lambda)$.
\end{itemize}
Write $\mathcal{R}_N$ for the set of reduced splitting chains of order $N$ and define $$\Omega_N:=\bigcap_{\substack{\lambda\subset[N]\\\#\lambda>1}}\left\{\bm{s}:\re(e_\lambda(\bm{s}))>0\right\}\qquad\text{where}\qquad e_\lambda(\bm{s}):=\#\lambda-1+\sum_{\substack{i<j\\i,j\in\lambda}}s_{ij}~.$$
\end{definition}

Proposition 3.15 and Theorem 2.6(c) in \cite{Web20} imply the following proposition, which shall be generalized slightly in order to prove the main results of this paper:

\begin{proposition}\label{R_prop}
For $N\geq 2$, the integral $\mathcal{Z}_N(R,\bm{s})$ converges absolutely if and only if $\bm{s}\in\Omega_N$, and in this case it can be written as the finite sum $$\mathcal{Z}_N(R,\bm{s})=q^{\sum_{i<j}s_{ij}}\sum_{\spl\in\mathcal{R}_N}\prod_{\lambda\in\mathcal{B}(\spl)}\frac{(q-1)_{\deg_{\spl}(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}~.$$ Here $(q-1)_n$ stands for the degree $n$ falling factorial $(z)_n=z(z-1)\dots(z-n+1)\in\Z[z]$ evaluated at the integer $z=q-1$.
\end{proposition}

\section{Statement of results}\label{2}
\subsection{The projective analogue}\label{2_1}

Our first main result is the following analogue of \Cref{R_prop}:

\begin{theorem}\label{main1}
For $N\geq 2$, the integral $\mathcal{Z}_N(\P^1(K),\bm{s})$ converges absolutely if and only if $\bm{s}\in\Omega_N$, and in this case it can be written as the finite sum $$\mathcal{Z}_N(\P^1(K),\bm{s})=\frac{1}{(q+1)^{N-1}}\sum_{\spl\in\mathcal{R}_N}\frac{q^{N+\sum_{i<j}s_{ij}}+1-\deg_{\spl}([N])}{q+1-\deg_{\spl}([N])}\prod_{\lambda\in\mathcal{B}(\spl)}\frac{(q-1)_{\deg_{\spl}(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}~.$$ The summand for each $\spl\in\mathcal{R}_N$ is defined for all prime powers $q$, as the denominator $q+1-\deg_\spl([N])$ is cancelled by the factor $(q-1)_{\deg_\spl([N])-1}$ inside the product over $\lambda\in\mathcal{B}(\spl)$.
\end{theorem}

The evident similarities between \Cref{R_prop} and \Cref{main1} follow from explicit relationship between the metrics and measures on $K$ and those on $\P^1(K)$. These relationships also play a role in the upcoming results, so they are worth recalling now. Note that $[x_0:x_1]\neq[1:0]$ if and only if $x_1\neq 0$, in which case $x=x_0/x_1$ is the unique element of $K$ satisfying $[x:1]=[x_0:x_1]$. Therefore the rule $\iota(x):=[x:1]$ defines a bijection $\iota:K\to\P^1(K)\setminus\{[1:0]\}$ and relates the metric structures of $K$ and $\P^1(K)$ in a simple way: Given $x,y\in K$, \eqref{sph_met_def} implies $\delta(\iota(x),[1:0])=(\max\{1,|x|\})^{-1}$ and
\begin{equation}\label{metric_rule}
\delta(\iota(x),\iota(y))=\begin{cases}|x-y|&\text{if $x,y\in R$},\\1&\text{if $x\in R$ and $y\notin R$},\\|1/x-1/y|&\text{if }x,y\notin R.\end{cases}
\end{equation}
Using the definitions \eqref{ball_def} and $v_K(x):=-\log_q|x|$ for $x\in K^\times$, along with \eqref{metric_rule} and the \emph{strong triangle equality} (i.e., $|x+y|=\max\{|x|,|y|\}$ whenever $|x|\neq|y|$), one easily verifies that
\begin{equation}\label{ball_rule}
\iota(y+\pi^vR)=\begin{cases}B_v(\iota(y))&\text{if }y\in R,\\B_{v-2v_K(y)}(\iota(y))&\text{if }y\notin R,\end{cases}
\end{equation}
whenever $y\in K$ and $v\in\Z_{>0}$. That is, $\iota$ sends the open ball of radius $r\in(0,1)$ centered at $y\in K$ onto the open ball of radius $r/\max\{1,|y|^2\}$ centered at $\iota(y)\in\P^1(K)\setminus\{[1:0]\}$, so $\iota:K\to\P^1(K)\setminus\{[1:0]\}$ is a homeomorphism that restricts to an isometry on $R$ and a contraction on $K\setminus R$.

The map $\iota$ also relates the measures on $K$ and $\P^1(K)$ in a simple way: Given $v>0$ and a complete set of representatives $y_1,\dots,y_{q^v}\in R$ for the cosets of $\pi^vR\subset R$, applying \eqref{ball_rule} to the partition $R=(y_1+\pi^vR)\sqcup\dots\sqcup(y_{q^v}+\pi^vR)$ yields $$\iota(R)=B_v[y_1:1]\sqcup\cdots\sqcup B_v[y_{q^v}:1]~.$$ Therefore $PGL_2(R)$-invariance of $\nu$ (\Cref{invariance}) implies $\nu(\iota(R))=q^v\cdot\nu(B_v[0:1])$. On the other hand, $$\iota(K\setminus R)=\iota(\{x:|x|\geq q\})=\{\iota(x):\delta(\iota(x),[1:0])\leq q^{-1}\}=B_1[1:0]\setminus\{[1:0]\}$$ implies $\iota(R)\sqcup B_1[1:0]=\iota(R)\sqcup\iota(K\setminus R)\sqcup\{[1:0]\}=\P^1(K)$, which has measure 1. But $\nu(\iota(R))=q\cdot\nu(B_1[1:0])$, so the measure of $\iota(R)$ must be $q/(q+1)$, and therefore every ball $B_v[x_0:x_1]\subset\P^1(K)$ with $v>0$ has measure $q^{-v}\cdot q/(q+1)$. Combining this with \eqref{ball_rule}, one concludes that the measure $\nu$ on $\P^1(K)\setminus\{[1:0]\}$ pulls back along $\iota$ to an explicit measure on $K$:
\begin{equation}\label{measure_rule}
\nu(\iota(M))=\frac{q}{q+1}\int_M\left(\max\{1,|x|^2\}\right)^{-1}\,d\mu\qquad\text{for any Borel subset }M\subset K~.
\end{equation}

Finally, \eqref{R_decomp} and \eqref{ball_rule} give a nice refinement of $\P^1(K)=\iota(R)\sqcup B_1[1:0]$ in terms of the $(q-1)$th roots of unity in $K$, which should be understood as the projective analogue of \eqref{R_decomp}:
\begin{equation}\label{P(K)_decomp}
\P^1(K)=B_1[0:1]\sqcup\underbrace{B_1[1:1]\sqcup B_1[\xi:1]\sqcup\dots\sqcup B_1[\xi^{q-2}:1]}_{\iota(R^\times)}\sqcup B_1[1:0]~.
\end{equation}
Indeed, all $q+1$ of the parts in the partition are balls with measure $1/(q+1)$ and radius $q^{-1}$, and two points $[x_0:x_1],[y_0:y_1]\in\P^1(K)$ satisfy $\delta([y_0:y_1],[y_0:y_1])=1$ if and only if $[x_0:x_1]$ and $[y_0:y_1]$ belong to different parts. Note that $\iota$ sends $R^\times$ onto the ``equator" $\iota(R^\times)$, i.e., the set of points in $\P^1(K)$ with $\delta$-distance 1 from both the ``south pole" $[0:1]$ and the ``north pole" $[1:0]$.

\subsection{Relationships between grand canonical partition functions}\label{2_2}
So far we have only considered log-Coulomb gases with $N$ labeled (and hence distinguishable) particles. Our second main result concerns the situation in which all particles are identical with charge $\mathfrak{q}_i=1$ for all $i$, in which case a microstate $(x_1,\dots,x_N)\in X^N$ is regarded as unique only up to permutations of its entries. Since the energy $E(x_1,\dots,x_N)$ and measure on $X^N$ are invariant under such permutations, each unlabeled microstate makes the contribution $e^{-\beta E(x_1,\dots,x_N)}d\mu^N$ to the integral $Z_N(X,\beta)$ in \eqref{canonical} precisely $N!$ times. Therefore the canonical partition function for the unlabeled microstates is given by $Z_N(X,\beta)/N!$. We further assume that the system exchanges particles with the heat reservoir with chemical potential $\eta$ and define the \emph{fugacity parameter} $f=e^{\eta\beta}$. In this situation the particle number $N\geq 0$ is treated as a random variable and the canonical partition function is replaced by the \emph{grand canonical partition function}
\begin{equation}\label{granddef}
Z(f,X,\beta):=\sum_{N=0}^\infty Z_N(X,\beta)\frac{f^N}{N!}
\end{equation}
with the familiar convention $Z_0(X,\beta)=1$. Many properties of the system can be deduced from the grand canonical partition function. For instance, if $\beta>0$ is fixed and $Z_N(X,\beta)$ is sub-exponential in $N$, then $Z(f,X,\beta)$ is analytic in $f$ and the expected number of particles in the system is given by $f\frac{\partial}{\partial f}\log(Z(f,X,\beta))$. The canonical partition function for each $N\geq 0$ can also be recovered by evaluating the $N$th derivative of $Z(f,X,\beta)$ with respect to $f$ at $f=0$.

We are interested in the examples $Z(f,R,\beta)$, $Z(f,P,\beta)$, and $Z(f,\P^1(K),\beta)$, which turn out to share several common properties and simple relationships. By setting $s_{ij}=\beta$ in \Cref{main_Z_def}, one sees that $|Z_N(R,\beta)|_\C$, $|Z_N(P,\beta)|_\C$, and $|Z_N(\P^1(K),\beta)|_\C$ are bounded above by 1 for all $N\geq 0$ and all $\beta>0$, and hence $Z(f,R,\beta)$, $Z(f,P,\beta)$, and $Z(f,\P^1(K),\beta)$ are analytic in $f$ when $\beta>0$. Sinclair recently found an elegant relationship between the first two, which is closely related to the partition of $R$ in \eqref{R_decomp}:

\begin{proposition}[The $q$th Power Law \cite{ChrisA}]\label{Chris}
For $\beta>0$ we have $$Z(f,R,\beta)=(Z(f,P,\beta))^q~.$$
\end{proposition}

Roughly speaking, the $q$th Power Law states that a log-Coulomb gas in $R$ exchanging energy and particles with a heat reservoir ``factors" into $q$ identical sub-gases (one in each coset of $P$) that exchange energy and particles with the reservoir. For $\beta>0$, note that the series equation $Z(f,R,\beta)=(Z(f,P,\beta))^q$ is equivalent to the coefficient identities
\begin{equation}\label{Rcoeff}
\frac{Z_N(R,\beta)}{N!}=\sum_{\substack{N_0+\dots+N_{q-1}=N\\N_0,\dots,N_{q-1}\geq 0}}\prod_{k=0}^{q-1}\frac{Z_{N_k}(P,\beta)}{N_k!}\qquad\text{for all $N\geq 0$.}
\end{equation}
The $\beta=1$ case of \eqref{Rcoeff} is given in \cite{BGMR}, in which the positive number $Z_N(R,1)/N!$ is recognized as the probability that a random monic polynomial in $R[x]$ splits completely in $R$. The more general $\beta>0$ case given in \cite{ChrisA} makes explicit use of the partition of $R$ into cosets of $P$ (as in \eqref{R_decomp}). In \Cref{3}, we will use the analogous partition of $\P^1(K)$ into $q+1$ balls (recall \eqref{P(K)_decomp}) to show that
\begin{equation}\label{P(K)coeff}
\frac{Z_N(\P^1(K),\beta)}{N!}=\sum_{\substack{N_0+\dots+N_q=N\\N_0,\dots,N_q\geq 0}}\prod_{k=0}^q\left(\frac{q}{q+1}\right)^{N_k}\frac{Z_{N_k}(P,\beta)}{N_k!}\qquad\text{for all $\beta>0$ and $N\geq 0$,}
\end{equation}
which immediately implies our second main result:
\begin{theorem}[The $(q+1)$th Power Law]\label{main2}
For all $\beta>0$ we have $$Z(f,\P^1(K),\beta)=(Z(\tfrac{qf}{q+1},P,\beta))^{q+1}~.$$
\end{theorem}

Like the $q$th Power Law, the $(q+1)$th Power Law roughly states that a log-Coulomb gas in $\P^1(K)$ exchanging energy and particles with a heat reservoir ``factors" into $q+1$ identical sub-gases in the balls $B_1[0:1]$, $B[1:1]$, $B[\xi:1]$, $\dots$, $B[\xi^{q-2}:1]$, $B_1[1:0]$ (each isometrically homeomorphic to $P$), with fugacity $\frac{qf}{q+1}$. The $q$th Power Law allows the $(q+1)$th Power Law to be written more crudely as
\begin{equation}\label{P(K)=RP}
Z(f,\P^1(K),\beta)=Z(\tfrac{qf}{q+1},R,\beta)\cdot Z(\tfrac{qf}{q+1},P,\beta)~,
\end{equation}
which is to say that the gas in $\P^1(K)$ ``factors" into two sub-gases: one in $\iota(R)$ and one in $B[1:0]$ (which are respectively isometrically homeomorphic to $R$ and $P$), both with fugacity $\frac{qf}{q+1}$.

\subsection{Functional equations and a quadratic recurrence}\label{2_3}
Although \Cref{R_prop} and \Cref{main1} provide explicit formulas for $Z_N(R,\beta)$ and $Z_N(\P^1(K),\beta)$, they are not efficient for computation because they require a complete list of reduced splitting chains of order $N$. For a practical alternative, we take advantage of both Power Laws and the following ideas from \cite{BGMR} and \cite{ChrisA}: Apply $Z(f,P,\beta)\cdot\frac{\partial}{\partial f}$ to the equation $Z(f,R,\beta)=(Z(f,P,\beta))^q$ to get $$Z(f,P,\beta)\cdot\frac{\partial}{\partial f}Z(f,R,\beta)=q\cdot Z(f,R,\beta)\cdot\frac{\partial}{\partial f}Z(f,P,\beta)~,$$ then expand both sides as power series in $f$ to obtain the coefficient equations
\begin{equation}\label{coeff}
\sum_{k=1}^N\frac{Z_{N-k}(P,\beta)}{(N-k)!}\frac{Z_k(R,\beta)}{(k-1)!}=q\cdot\sum_{k=1}^N\frac{Z_{N-k}(R,\beta)}{(N-k)!}\frac{Z_k(P,\beta)}{(k-1)!}\qquad\text{for all }N\geq 1.
\end{equation}
The identities $Z_j(P,\beta)=q^{-j-\binom{j}{2}\beta}Z_j(R,\beta)$ follow easily from \Cref{main_Z_def} and eliminate all instances of $Z_j(P,\beta)$ in \eqref{coeff} while introducing powers of the form $q^{-j-\binom{j}{2}\beta}$. For $N\geq 2$, a careful rearrangement of these powers, the factorials, and the terms in \eqref{coeff} yields the explicit recurrence $$\frac{Z_N(R,\beta)}{N!q^{\frac{1}{2}\binom{N}{2}\beta}}=\sum_{k=1}^{N-1}\frac{k}{N}\cdot\frac{\sinh\left(\frac{\log(q)}{2}\left[\left(N+\binom{N}{2}\beta\right)\left(1-\frac{2k}{N}\right)+1\right]\right)}{\sinh\left(\frac{\log(q)}{2}\left[\left(N+\binom{N}{2}\beta\right)-1\right]\right)}\cdot\frac{Z_{N-k}(R,\beta)}{(N-k)!q^{\frac{1}{2}\binom{N-k}{2}\beta}}\cdot\frac{Z_k(R,\beta)}{k!q^{\frac{1}{2}\binom{k}{2}\beta}}~.$$ The expression at left is identically 1 if $N=0$ or $N=1$, so induction confirms that it is polynomial in ratios of hyperbolic sines for all $N\geq 0$. In particular, its dependence on $q$ is carried only by the factor $\log(q)$ appearing inside the hyperbolic sines, which motivates the following lemma:

\begin{lemma}[The Quadratic Recurrence]\label{quad_rec}
Set $F_0(t,\beta)=F_1(t,\beta)=1$ for all $\beta\in\C$ and all $t\in\R$. For $N\geq 2$, $\re(\beta)>-2/N$, and $t\in\R$, define $F_N(t,\beta)$ by the recurrence $$F_N(t,\beta):=\begin{cases}\displaystyle{\sum_{k=1}^{N-1}\frac{k}{N}\cdot\frac{\sinh\left(\frac{t}{2}\left[\left(N+\binom{N}{2}\beta\right)\left(1-\frac{2k}{N}\right)+1\right]\right)}{\sinh\left(\frac{t}{2}\left[\left(N+\binom{N}{2}\beta\right)-1\right]\right)}\cdot F_{N-k}(t,\beta)\cdot F_k(t,\beta)}&\text{if $t\neq 0$,}\\ \\\displaystyle{\sum_{k=1}^{N-1}\frac{k}{N}\cdot\frac{\left(N+\binom{N}{2}\beta\right)\left(1-\frac{2k}{N}\right)+1}{\left(N+\binom{N}{2}\beta\right)-1}\cdot F_{N-k}(0,\beta)\cdot F_k(0,\beta)}&\text{if $t=0$.}\end{cases}$$
\begin{enumerate}
\item[(a)] For fixed $N\geq 2$ and fixed $t$, the function $\beta\mapsto F_N(t,\beta)$ is holomorphic for $\re(\beta)>-2/N$.
\item[(b)] For fixed $N\geq 2$ and fixed $\beta$, the function $t\mapsto F_N(t,\beta)$ is defined, smooth, and even on $\R$.
\end{enumerate}
\end{lemma}
Both parts of the Quadratic Recurrence are straightforward to verify by induction. An interesting and immediate consequence of The Quadratic Recurrence and the preceding discussion is the formula $$Z_N(R,\beta)=N!q^{\frac{1}{2}\binom{N}{2}\beta}F_N(\log(q),\beta)~.$$ It offers a computationally efficient alternative to the ``univariate case" of  \Cref{R_prop} (when $s_{ij}=\beta$ for all $i<j$) and extends $Z_N(R,\beta)$ to a smooth function of $q\in(0,\infty)$. Moreover, it transforms nicely under the involution $q\mapsto q^{-1}$: $$Z_N(R,\beta)\big|_{q\mapsto q^{-1}}=N!q^{-\frac{1}{2}\binom{N}{2}\beta}F_N\left(\log(q^{-1}),\beta\right)=N!q^{-\frac{1}{2}\binom{N}{2}\beta}F_N\left(\log(q),\beta\right)=q^{-\binom{N}{2}\beta}Z_N(R,\beta)~.$$

The Quadratic Recurrence serves the projective analogue as well. Expanding \eqref{P(K)=RP} into powers of $f$ yields the coefficient equations
\begin{equation}\label{Z_N/N!}
\frac{Z_N(\P^1(K),\beta)}{N!}=\sum_{k=0}^N\left(\frac{q}{q+1}\right)^N\frac{Z_{N-k}(R,\beta)}{(N-k)!}\frac{Z_k(P,\beta)}{k!}\qquad\text{for all }N\geq 0,
\end{equation}
and the identities $Z_j(P,\beta)=q^{-j-\binom{j}{2}\beta}Z_j(R,\beta)$ and $Z_j(R,\beta)=j!q^{\frac{1}{2}\binom{j}{2}\beta}F_j(\log(q),\beta)$ allow the $k$th summand to be rewritten as $$\left(\frac{q}{q+1}\right)^N\frac{Z_{N-k}(R,\beta)}{(N-k)!}\frac{Z_k(P,\beta)}{k!}=\frac{q^{\frac{1}{2}\left(N+\binom{N}{2}\beta\right)\left(1-\frac{2k}{N}\right)}}{\left(2\cosh\left(\frac{\log(q)}{2}\right)\right)^N}\cdot F_{N-k}(\log(q),\beta)\cdot F_k(\log(q),\beta)\qquad\text{for $N\geq 1$}.$$ Thus, adding two copies of the sum in \eqref{Z_N/N!} together, pairing the $k$th term of the first copy with the $(N-k)$th term of the second copy, and dividing by $2$ gives $$\frac{Z_N(\P^1(K),\beta)}{N!}=\sum_{k=0}^N\frac{\cosh\left(\frac{\log(q)}{2}\left(N+\binom{N}{2}\beta\right)\left(1-\frac{2k}{N}\right)\right)}{\left(2\cosh\left(\frac{\log(q)}{2}\right)\right)^N}\cdot F_{N-k}(\log(q),\beta)\cdot F_k(\log(q),\beta)\qquad\text{for $N\geq 1$},$$ which is valid for $\re(\beta)>-2/N$. Through this formula, $Z_N(\P^1(K),\beta)$ clearly extends to a smooth function of $q\in(0,\infty)$ and is invariant under the involution $q\mapsto q^{-1}$. We conclude this section by summarizing these observations:

\begin{theorem}[Efficient Formulas and Functional Equations]
Suppose $N\geq 2$ and $\re(\beta)>-2/N$, and define $(F_k(t,\beta))_{k=0}^N$ as in \Cref{quad_rec}. The $N$th canonical partition functions are given by the formulas
\begin{align*}
Z_N(R,\beta)&=N!q^{\frac{1}{2}\binom{N}{2}\beta}F_N(\log(q),\beta)\qquad\text{and}\\ \\
Z_N(\P(K),\beta)&=N!\sum_{k=0}^N\frac{\cosh\left(\frac{\log(q)}{2}\left(N+\binom{N}{2}\beta\right)\left(1-\frac{2k}{N}\right)\right)}{\left(2\cosh\left(\frac{\log(q)}{2}\right)\right)^N}\cdot F_{N-k}(\log(q),\beta)\cdot F_k(\log(q),\beta)~,
\end{align*}
which extend $Z_N(R,\beta)$ and $Z_N(\P^1(K),\beta)$ to smooth functions of $q\in(0,\infty)$ satisfying $$Z_N(R,\beta)\big|_{q\mapsto q^{-1}}=q^{-\binom{N}{2}\beta}Z_N(R,\beta)\qquad\text{and}\qquad Z_N(\P^1(K),\beta)\big|_{q\mapsto q^{-1}}=Z_N(\P^1(K),\beta)~.$$
\end{theorem}

It should be noted here that the $q\mapsto q^{-1}$ functional equation at left is a special case of the one proved in \cite{DenMeus}, and that both functional equations closely resemble the ones in \cite{Voll10}.

\newpage
\section{Proofs of the main results}\label{3}
This section will establish the proofs of \Cref{main1,main2}. The common step in both is a decomposition of $(\P^1(K))^N$ into $(q+1)^N$ cells that are isometrically isomorphic to $P^N$, which combines with the metric and measure properties in \Cref{1_2} to create the key relationship between the canonical partition functions for $X=\P^1(K)$ and $X=P$. We will prove this relationship first, then conclude the proofs of \Cref{main1,main2} in their own subsections.

\subsection{Decomposing the integral over $(\P^1(K))^N$}
We begin with an integer $N\geq 2$ that shall remain fixed for the rest of this section, reserve the symbol $\bm{s}$ for a complex tuple $(s_{ij})_{1\leq i<j\leq N}$, and fix the following notation to better organize the forthcoming arguments:
\begin{notation}
Let $I$ be a subset of $[N]=\{1,\dots,N\}$.
\begin{itemize}
\item For any set $X$ we write $X^I$ for the product $\prod_{i\in I}X=\{x_I=(x_i)_{i\in I}:x_i\in X\}$ and assume $X^I$ has the product topology if $X$ is a topological space.
\item We write $\mu^I$ for the product Haar measure on $K^I$ satisfying $\mu^I(R^I)=1$, and we make this consistent for $I=\varnothing$ by giving the singleton space $K^\varnothing=R^\varnothing=\{0\}$ measure 1. We also write $\nu^I$ for the product measure on $(\P^1(K))^I$, with the same convention for $I=\varnothing$.
\item For a measurable subset $X\subset K$ we set $\mathcal{Z}_\varnothing(X,\bm{s}):=1$ and $$\mathcal{Z}_I(X,\bm{s}):=\int_{X^I}\prod_{\substack{i<j\\i,j\in I}}|x_i-x_j|^{s_{ij}}\,d\mu^I\qquad\text{if}\quad I\neq\varnothing~.$$ Note that $\mathcal{Z}_I(X,\bm{s})$ is constant with respect to the entry $s_{ij}$ if $i\in[N]\setminus I$ or $j\in[N]\setminus I$, and it is equal to $\mathcal{Z}_N(X,\bm{s})$ if $I=[N]$.
\item Using the constant $q=\#(R/P)$, we write $(I_0,\dots,I_q)\vdash[N]$ for an \emph{ordered} partition of $[N]$ into at most $q+1$ parts. That is, $(I_0,\dots,I_q)\vdash[N]$ means $I_0,\dots,I_q$ are $q+1$ disjoint ordered subsets of $[N]$ with union equal to $[N]$, where some $I_k$ may be empty.
\end{itemize}
\end{notation}

In addition to the above, it will be useful to consider $I$-analogues of splitting chains:

\begin{definition}\label{Isplchdef}
Suppose $I\subset[N]$. An \emph{$I$-splitting chain} of length $L\geq 0$ is a tuple $\spl=(\ptn_0,\dots,\ptn_L)$ of partitions of $I$ satisfying $$\{I\}=\ptn_0>\ptn_1>\ptn_2>\dots>\ptn_L=\{\{i\}:i\in I\}~.$$ If $\#I\geq 2$, we define $\mathcal{B}(\spl)$, $\ell_\spl(\lambda)$, and $\deg_\spl(\lambda)\in\{2,3,\dots,\#I\}$ just as in \Cref{splchdef}. Otherwise $\mathcal{B}(\spl)$ will be treated as the empty set and there is no need to define $\ell_\spl$ or $\deg_\spl$. Finally, we call an $I$-splitting chain $\spl$ \emph{reduced} if each $\lambda\in\mathcal{B}(\spl)$ satisfies $\lambda\in\ptn_\ell\iff\ell=\ell_\spl(\lambda)$, write $\mathcal{R}_I$ for the set of reduced $I$-splitting chains, and define $$\Omega_I:=\bigcap_{\substack{\lambda\subset I\\\#\lambda>1}}\left\{\bm{s}:\re(e_\lambda(\bm{s}))>0\right\}\qquad\text{where}\qquad e_\lambda(\bm{s}):=\#\lambda-1+\sum_{\substack{i<j\\i,j\in\lambda}}s_{ij}~.$$
\end{definition}

Note that $\mathcal{R}_\varnothing=\varnothing$ because $I=\varnothing$ has no partitions, $\Omega_\varnothing=\C^{N(N-1)/2}$ because $\Omega_\varnothing$ is an intersection of subsets of $\C^{N(N-1)/2}$ taken over an empty index set, and $e_\varnothing(\bm{s})=-1$ for a similar reason. For each singleton $\{i\}$, the set $\mathcal{R}_{\{i\}}$ is comprised of a single splitting chain of length zero, $\Omega_{\{i\}}=\C^{N(N-1)/2}$ for the same reason as the $I=\varnothing$ case, and similarly $e_{\{i\}}(\bm{s})=0$. At the other extreme, taking $I=[N]$ in \Cref{Isplchdef} recovers \Cref{splchdef} and gives $\Omega_I=\Omega_N$.

\begin{proposition}\label{Iold}
For any $v\in\Z$ and any nonempty subset $I\subset[N]$, the integral $\mathcal{Z}_I(\pi^vR,\bm{s})$ converges absolutely if and only if $\bm{s}\in\Omega_I$, and in this case
$$\mathcal{Z}_I(\pi^vR,\bm{s})=\frac{1}{q^{(v-1)(e_I(\bm{s})+1)+\#I}}\sum_{\spl\in\mathcal{R}_I}\prod_{\lambda\in\mathcal{B}(\spl)}\frac{(q-1)_{\deg_{\spl}(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}~.$$ In particular, we recover \Cref{R_prop} by taking $I=[N]$ and $v=0$.
\end{proposition}

\begin{proof}
First suppose $I$ is a singleton, so that the product inside the integral $Z_I(\pi^vR,\bm{s})$ is empty and hence $$Z_I(\pi^vR,\bm{s})=\int_{(\pi^vR)^I}\,d\mu^I=\int_{\pi^vR}\,d\mu=q^{-v}~.$$ This integral is constant, and hence absolutely convergent, for all $\bm{s}\in\C^{N(N-1)/2}=\Omega_I$. On the other hand, $\mathcal{R}_I$ consists of a single $I$-splitting chain, namely the one-tuple $\spl=(\{I\})$. Then $\mathcal{B}(\spl)=\varnothing$ and $e_I(\bm{s})=0$ imply $$\frac{1}{q^{(v-1)(e_I(\bm{s})+1)+\#I}}\sum_{\spl\in\mathcal{R}_I}\prod_{\lambda\in\mathcal{B}(\spl)}\frac{(q-1)_{\deg_{\spl}(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}=\frac{1}{q^{(v-1)\cdot1+1}}\prod_{\lambda\in\varnothing}\frac{(q-1)_{\deg_{\spl}(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}=q^{-v}$$ as well, so the claim holds for any singleton subset $I\subset[N]$. Now suppose $I$ is not a singleton. By relabeling $I$ we may assume $I=[n]$ where $2\leq n\leq N$. By Proposition 3.15 and Lemma 3.16(c) in \cite{Web20}, the integral $$\mathcal{Z}_I(R,\bm{s})=\mathcal{Z}_n(R,\bm{s})=\int_{R^n}\prod_{1\leq i<j\leq n}|x_i-x_j|^{s_{ij}}\,d\mu^N$$ converges absolutely if and only if $\bm{s}$ belongs to the intersection
\begin{equation}\label{polytope}
\bigcap_{\spl\in\mathcal{R}_n}\bigcap_{\lambda\in\mathcal{B}(\spl)}\left\{\bm{s}:\re(e_{\lambda}(\bm{s}))>0\right\}~,
\end{equation}
and for such $\bm{s}$ we have $$\mathcal{Z}_n(R,\bm{s})=q^{e_{[n]}(\bm{s})+1-n}\sum_{\spl\in\mathcal{R}_n}\prod_{\lambda\in\mathcal{B}(\spl)}\frac{(q-1)_{\deg_{\spl}(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}~.$$ Changing variables in the integral $\mathcal{Z}_n(R,\bm{s})$ by the homothety $R^n\to(\pi^vR)^n$ gives $$\mathcal{Z}_N(\pi^vR,\bm{s})=q^{-v(e_{[N]}(\bm{s})+1)}\cdot\mathcal{Z}_n(R,\bm{s})=\frac{1}{q^{(v-1)(e_{[n]}(\bm{s})+1)+n}}\sum_{\spl\in\mathcal{R}_n}\prod_{\lambda\in\mathcal{B}(\spl)}\frac{(q-1)_{\deg_{\spl}(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}~,$$ and the first equality implies that the domain of absolute convergence for $\mathcal{Z}_n(\pi^vR,\bm{s})$ is also the intersection appearing in \eqref{polytope}. But every subset $\lambda\subset[n]$ with $\#\lambda>1$ appears as a branch in at least one reduced splitting chain of order $n$, so the intersection in \eqref{polytope} is precisely $\Omega_{[n]}$. Therefore the claim holds for $I=[n]$, and hence for any non-singleton subset $I\subset[N]$.\\
\end{proof}

The $v=1$ case of \Cref{Iold} has an important relationship with the main result of this section, which is the following theorem.

\begin{theorem}\label{main}
For each $N\geq 2$, the integral $\mathcal{Z}_N(\P^1(K),\bm{s})$ converges absolutely if and only if $\bm{s}\in\Omega_N$, and in this case $$\mathcal{Z}_N(\P^1(K),\bm{s})=\left(\frac{q}{q+1}\right)^N\sum_{(I_0,\dots,I_q)\vdash[N]}\prod_{k=0}^q\mathcal{Z}_{I_k}(P,\bm{s})~.$$
\end{theorem}

\begin{proof}
The partition of $\P^1(K)$ in \eqref{P(K)_decomp} can be rewritten in the form $$\P^1(K)=\bigsqcup_{k=0}^q\phi_k(B_1[0:1])~,$$ where $\phi_k\in PGL_2(R)$ is the element represented by $\left(\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right)$ if $k=0$, $\left(\begin{smallmatrix}1&\xi^{k-1}\\0&1\end{smallmatrix}\right)$ if $0<k<q$, or $\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$ if $k=q$. This leads to a partition of the $N$-fold product, $$(\P^1(K))^N=\bigsqcup_{(I_0,\dots,I_q)\vdash[N]}C(I_0,\dots,I_q)~,$$ where each part is a ``cell" of the form
\begin{align*}
C(I_0,\dots,I_q)&:=\{([x_{1,0}:x_{1,1}],\dots,[x_{N,0}:x_{N,1}])\in(\P^1(K))^N:[x_{i,0}:x_{i,1}]\in\phi_k(B_1[0:1])\iff i\in I_k\}\\
&=\prod_{k=0}^q(\phi_k(B_1[0:1]))^{I_k}~.
\end{align*}
Accordingly, the integral $\mathcal{Z}_N(\P^1(K),\bm{s})$ breaks into a sum of integrals of the form
\begin{equation}\label{cube_integral}
\int_{C(I_0,\dots,I_q)}\prod_{1\leq i<j\leq N}\delta([x_{i,0}:x_{i,1}],[x_{j,0}:x_{j,1}])^{s_{ij}}\,d\nu^N~,
\end{equation}
summed over all $(I_0,\dots,I_q)\vdash[N]$. Since every cell $C(I_0,\dots,I_q)$ has positive measure, the integral $\mathcal{Z}_N(\P^1(K),\bm{s})$ converges absolutely if and only if the integral in \eqref{cube_integral} converges absolutely for all $(I_0,\dots,I_q)\vdash[N]$. Fix one $(I_0,\dots,I_q)$ for the moment. By \eqref{P(K)_decomp} and the definition of the $\phi_k$'s above, note that the entries of each tuple $([x_{1,0}:x_{1,1}],\dots,[x_{N,0}:x_{N,1}])\in C(I_0,\dots,I_q)$ satisfy $\delta([x_{i,0}:x_{i,1}],[x_{j,0}:x_{j,1}])^{s_{ij}}=1$ if and only if $i$ and $j$ belong to different parts of $(I_0,\dots,I_q)$. Therefore the integrand in \eqref{cube_integral} factors as
\begin{align*}
\prod_{1\leq i<j\leq N}\delta([x_{i,0}:x_{i,1}],[x_{j,0}:x_{j,1}])^{s_{ij}}&=\prod_{k=0}^q\prod_{\substack{i<j\\i,j\in I_k}}\delta([x_{i,0}:x_{i,1}],[x_{j,0}:x_{j,1}])^{s_{ij}}~,
\end{align*}
and the measure on $C(I_0,\dots,I_q)$ factors in a similar way, namely $\prod_{k=0}^qd\nu^{I_k}$ where $\nu^{I_k}$ is the product measure on $(\P^1(K))^{I_k}$. Now Fubini's Theorem for positive functions and $PGL_2(R)$-invariance give
\begin{align*}
\int_{C(I_0,\dots,I_q)}&\Bigg|\prod_{1\leq i<j\leq N}\delta([x_{i,0}:x_{i,1}],[x_{j,0}:x_{j,1}])^{s_{ij}}\Bigg|_\C\,d\nu^N\\
&=\prod_{k=0}^q\int_{(\phi_k(B_1[0:1]))^{I_k}}\Bigg|\prod_{\substack{i<j\\i,j\in I_k}}\delta([x_{i,0}:x_{i,1}],[x_{j,0}:x_{j,1}])^{s_{ij}}\Bigg|_\C\,d\nu^{I_k}\\
&=\prod_{k=0}^q\int_{(B_1[0:1])^{I_k}}\Bigg|\prod_{\substack{i<j\\i,j\in I_k}}\delta([x_{i,0}:x_{i,1}],[x_{j,0}:x_{j,1}])^{s_{ij}}\Bigg|_\C\,d\nu^{I_k}~,
\end{align*}
so the integral in \eqref{cube_integral} converges absolutely if and only if all $q+1$ of the integrals of the form
\begin{equation}\label{I_k_integral}
\int_{(B_1[0:1])^{I_k}}\prod_{\substack{i<j\\i,j\in I_k}}\delta([x_{i,0}:x_{i,1}],[x_{j,0}:x_{j,1}])^{s_{ij}}\,d\nu^{I_k}
\end{equation}
converge absolutely. The change of variables $P^{I_k}\to(B_1[0:1])^{I_k}$ given by $\iota:P\to B_1[0:1]$ in each coordinate, along with \eqref{metric_rule}, \eqref{ball_rule}, and \eqref{measure_rule}, allows the integral in \eqref{I_k_integral} to be rewritten as $(\frac{q}{q+1})^{\#I_k}\mathcal{Z}_{I_k}(P,\bm{s})$, and thus \Cref{Iold} implies that it converges absolutely if and only if $\bm{s}\in\Omega_{I_k}$. Therefore the integral over $C(I_0,\dots,I_q)$ in \eqref{cube_integral} converges absolutely if and only if $\bm{s}\in\Omega_{I_0}\cap\dots\cap\Omega_{I_q}$, and in this case Fubini's Theorem for absolutely integrable functions, $PGL_2(R)$-invariance, and the change of variables above allow it to be rewritten as
\begin{align*}
\int_{C(I_0,\dots,I_q)}&\prod_{1\leq i<j\leq N}\delta([x_{i,0}:x_{i,1}],[x_{j,0}:x_{j,1}])^{s_{ij}}\,d\nu^N\\
&=\prod_{k=0}^q\int_{(B_1[0:1])^{I_k}}\prod_{\substack{i<j\\i,j\in I_k}}\delta([x_{i,0}:x_{i,1}],[x_{j,0}:x_{j,1}])^{s_{ij}}\,d\nu^{I_k}\\
&=\prod_{k=0}^q\left(\frac{q}{q+1}\right)^{\#I_k}\mathcal{Z}_{I_k}(P,\bm{s})\\
&=\left(\frac{q}{q+1}\right)^N\prod_{k=0}^q\mathcal{Z}_{I_k}(P,\bm{s})~.
\end{align*}

Finally, since $\mathcal{Z}_N(\P^1(K),\bm{s})$ is the sum of these integrals over all $(I_0,\dots,I_q)\vdash[N]$, it converges absolutely if and only if $$\bm{s}\in\bigcap_{(I_1,\dots,I_q)\vdash[N]}\left(\Omega_{I_1}\cap\dots\cap\Omega_{I_q}\right)=\bigcap_{\substack{I\subset[N]\\\#I>1}}\Omega_I~.$$ The last equality of intersections holds because each subset $I\subset[N]$ with $\#I>1$ appears as a part in at least one of the ordered partitions $(I_1,\dots,I_q)\vdash[N]$, and because none of the parts with $\#I_k\leq 1$ affect the intersection (because $\Omega_{I_k}=\C^{N(N-1)/2}$ for such $I_k$). The intersection of $\Omega_I$ over all $I\subset[N]$ with $\#I>1$ is clearly equal to $\Omega_{[N]}=\Omega_N$ by \Cref{Isplchdef}, so the proof is complete.\\
\end{proof}

\subsection{Finishing the proof of Theorem 2.1}
\Cref{main} established that the integral $\mathcal{Z}_N(\P^1(K),\bm{s})$ converges absolutely if and only if $\bm{s}\in\Omega_N$, and for such $\bm{s}$ it gave
\begin{equation}\label{sum1}
\mathcal{Z}_N(\P^1(K),\bm{s})=\left(\frac{q}{q+1}\right)^N\sum_{(I_0,\dots,I_q)\vdash[N]}\prod_{k=0}^q\mathcal{Z}_{I_k}(P,\bm{s})~.
\end{equation}
It remains to show that the righthand sum can be converted into the sum over $\spl\in\mathcal{R}_N$ proposed in \Cref{main1}.

\begin{proof}[Proof of \Cref{main1}] We begin by breaking the terms of the sum in \eqref{sum1} into two main groups. The simpler group is indexed by those $(I_0,\dots,I_q)$ with $I_j=[N]$ for some $j$ and $I_k=\varnothing$ for all $k\neq j$, in which case $\mathcal{Z}_{I_j}(P,\bm{s})=\mathcal{Z}_N(P,\bm{s})$ and $\mathcal{Z}_{I_k}(P,\bm{s})=1$ for all $k\neq j$. Therefore each of the group's $q+1$ terms (one for each $j\in\{0,\dots,q\}$) contributes the quantity $\prod_{k=0}^q\mathcal{Z}_{I_k}(P,\bm{s})=\mathcal{Z}_N(P,\bm{s})$ to the sum in \eqref{sum1} for a total contribution of
\begin{equation}\label{term1}
(q+1)\mathcal{Z}_N(P,\bm{s})=\frac{q+1}{q^N}\sum_{\spl\in\mathcal{R}_N}\prod_{\lambda\in\mathcal{B}(\spl)}\frac{(q-1)_{\deg_\spl(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}
\end{equation}
by the $v=1$ and $I=[N]$ case of \Cref{Iold}. The other group of terms is indexed by the ordered partitions $(I_0,\dots,I_q)\vdash[N]$ satisfying $I_0,\dots,I_q\subsetneq[N]$. To deal with them carefully, we fix one such $(I_0,\dots,I_q)$ for the moment, and note that the number $d$ of nonempty parts $I_k$ must be at least 2. Thus we have indices $k_1,\dots,k_d\in\{0,\dots,q\}$ with $I_{k_j}\neq\varnothing$, and for every $k\in\{0,\dots,q\}\setminus\{k_1,\dots,k_d\}$ we have $I_k=\varnothing$ and hence $\mathcal{Z}_{I_k}(P,\bm{s})=1$. For the nonempty sets $I_{k_j}$, \Cref{Iold} expands $\mathcal{Z}_{I_{k_j}}(P,\bm{s})$ as a sum over $\mathcal{R}_{I_{k_j}}$ (whose elements shall be denoted $\spl_j$ instead of $\spl$) and hence
\begin{align*}
\prod_{k=0}^q\mathcal{Z}_{I_k}(P,\bm{s})&=\prod_{j=1}^d\frac{1}{q^{\#I_{k_j}}}\sum_{\spl_j\in\mathcal{R}_{I_{k_j}}}\prod_{\lambda\in\mathcal{B}(\spl_j)}\frac{(q-1)_{\deg_{\spl_j}(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}\\
&=\frac{1}{q^N}\sum_{(\spl_1,\dots,\spl_d)\in\mathcal{R}_{I_{k_1}}\times\cdots\times\mathcal{R}_{I_{k_d}}}\prod_{j=1}^d\prod_{\lambda\in\mathcal{B}(\spl_j)}\frac{(q-1)_{\deg_{\spl_j}(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}\\
&=\frac{1}{q^N}\sum_{(\spl_1,\dots,\spl_d)\in\mathcal{R}_{I_{k_1}}\times\cdots\times\mathcal{R}_{I_{k_d}}}\prod_{\lambda\in\mathcal{B}(\spl_1)\sqcup\cdots\sqcup\mathcal{B}(\spl_d)}\frac{(q-1)_{\deg_{\spl_j}(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}~.
\end{align*}
We now make use of a simple correspondence between the tuples $(\spl_1,\dots,\spl_d)\in\mathcal{R}_{I_{k_1}}\times\cdots\times\mathcal{R}_{I_{k_d}}$ and the reduced splitting chains $\spl=(\ptn_0,\ptn_1,\dots,\ptn_L)\in\mathcal{R}_N$ satisfying $\ptn_1=\{I_{k_1},\dots,I_{k_d}\}$. To establish it, note that each $\spl\in\mathcal{R}_N$ corresponds uniquely to its branch set $\mathcal{B}(\spl)$ (Lemma 2.5(b) of \cite{Web20}), which generalizes in an obvious way to reduced $I$-splitting chains (for any nonempty $I\subset[N]$). Now if $\spl=(\ptn_0,\ptn_1,\dots,\ptn_L)\in\mathcal{R}_N$ satisfies $\ptn_1=\{I_{k_1},\dots,I_{k_d}\}$, the corresponding branch set $\mathcal{B}(\spl)$ decomposes as $$\mathcal{B}(\spl)=\{[N]\}\sqcup\bigsqcup_{j=1}^d\{\lambda\in\mathcal{B}(\spl):\lambda\subset I_{k_j}\}~.$$ Each of the sets $\{\lambda\in\mathcal{B}(\spl):\lambda\subset I_{k_j}\}$ is the branch set $\mathcal{B}(\spl_j)$ for a unique $\spl_j\in\mathcal{R}_{I_{k_j}}$, so in this sense $\spl$ ``breaks" into a unique tuple $(\spl_1,\dots,\spl_d)\in\mathcal{R}_{I_{k_1}}\times\cdots\times\mathcal{R}_{I_{k_d}}$. On the other hand, any tuple $(\spl_1,\dots,\spl_d)\in\mathcal{R}_{I_{k_1}}\times\cdots\times\mathcal{R}_{I_{k_d}}$ can be ``assembled" as follows. Since $\{I_{k_1},\dots,I_{k_d}\}$ is a partition of $[N]$, taking the union of the $d$ branch sets $\mathcal{B}(\spl_1),\dots,\mathcal{B}(\spl_d)$ and the singleton $\{[N]\}$ forms the branch set $\mathcal{B}(\spl)$ for a unique $\spl\in\mathcal{R}_N$. It is clear that ``breaking" and ``assembling" are inverses, giving a correspondence $\mathcal{R}_N\longleftrightarrow\mathcal{R}_{I_{k_1}}\times\cdots\times\mathcal{R}_{I_{k_d}}$ under which each identification $\spl\longleftrightarrow(\spl_1,\dots,\spl_d)$ amounts to a branch set equation, i.e., $$\mathcal{B}(\spl)\setminus\{[N]\}=\mathcal{B}(\spl_1)\sqcup\cdots\sqcup\mathcal{B}(\spl_d)~.$$ In particular, each $\lambda\in\mathcal{B}(\spl)\setminus\{[N]\}$ is contained in exactly one $\mathcal{B}(\spl_j)$, and $\deg_\spl(\lambda)=\deg_{\spl_j}(\lambda)$ by \Cref{splchdef} in this case. These facts allow the sum over $\mathcal{R}_{I_1}\times\cdots\times\mathcal{R}_{I_d}$ above to be rewritten as a sum over all $\spl\in\mathcal{R}_N$ with $\ptn_1=\{I_{k_1},\dots,I_{k_q}\}$, and each product over $\lambda\in\mathcal{B}(\spl_{k_1})\sqcup\cdots\sqcup\mathcal{B}(\spl_{k_d})$ inside it is simply a product over $\lambda\in\mathcal{B}(\spl)\setminus\{[N]\}$. We conclude that an ordered partition $(I_0,\dots,I_q)\vdash[N]$ with $I_0,\dots,I_q\subsetneq[N]$ contributes the quantity
\begin{equation}\label{term2}
\prod_{k=0}^q\mathcal{Z}_{I_k}(P,\bm{s})=\frac{1}{q^N}\sum_{\substack{\spl\in\mathcal{R}_N\\\ptn_1=\{I_{k_1},\dots,I_{k_d}\}}}\prod_{\lambda\in\mathcal{B}(\spl)\setminus\{[N]\}}\frac{(q-1)_{\deg_\spl(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}
\end{equation}
to the sum in \eqref{sum1}, where $\{I_{k_1},\dots,I_{k_d}\}$ is the (unordered) subset of nonempty parts in that particular ordered partition. We must now total the contribution in \eqref{term2} over all possible $(I_0,\dots,I_q)\vdash[N]$ with $I_0,\dots,I_q\subsetneq[N]$. Given a partition $\{\lambda_1,\dots,\lambda_d\}\vdash[N]$ with $d\geq 2$, note that there are precisely $(q+1)_d=(q+1)\cdot(q)_{d-1}$ ordered partitions $(I_0,\dots,I_q)\vdash[N]$ such that $\{I_{k_1},\dots,I_{k_d}\}=\{\lambda_1,\dots,\lambda_d\}$. Therefore summing \eqref{term2} over all $(I_0,\dots,I_q)\vdash[N]$ with $I_0,\dots,I_q\subsetneq[N]$ gives
\begin{align*}
\sum_{\substack{(I_0,\dots,I_q)\vdash[N]\\I_0,\dots,I_q\subsetneq[N]}}\prod_{k=0}^q\mathcal{Z}_{I_k}(P,\bm{s})&=\frac{1}{q^N}\sum_{\substack{(I_0,\dots,I_q)\vdash[N]\\I_0,\dots,I_q\subsetneq[N]}}\sum_{\substack{\spl\in\mathcal{R}_N\\\ptn_1=\{I_{k_1},\dots,I_{k_d}\}}}\prod_{\lambda\in\mathcal{B}(\spl)\setminus\{[N]\}}\frac{(q-1)_{\deg_\spl(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}\\
&=\frac{q+1}{q^N}\sum_{\substack{\{\lambda_1,\dots,\lambda_d\}\vdash[N]\\d\geq 2}}(q)_{d-1}\sum_{\substack{\spl\in\mathcal{R}_N\\\ptn_1=\{\lambda_1,\dots,\lambda_d\}}}\prod_{\lambda\in\mathcal{B}(\spl)\setminus\{[N]\}}\frac{(q-1)_{\deg_\spl(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}~.
\end{align*}
Given a partition $\{\lambda_1,\dots,\lambda_d\}\vdash[N]$, those splitting chains $\spl\in\mathcal{R}_N$ with $\ptn_1=\{\lambda_1,\dots,\lambda_d\}$ all have $\deg_\spl([N])=\#\ptn_1=d$ by \Cref{splchdef}. Moreover, no $\spl\in\mathcal{R}_N$ is missed or repeated in the sum of sums above, so it can be rewritten as
\begin{align*}
\sum_{\substack{(I_0,\dots,I_q)\vdash[N]\\I_0,\dots,I_q\subsetneq[N]}}\prod_{k=0}^q\mathcal{Z}_{I_k}(P,\bm{s})&=\frac{q+1}{q^N}\sum_{\spl\in\mathcal{R}_N}(q)_{\deg_\spl([N])-1}\prod_{\lambda\in\mathcal{B}(\spl)\setminus\{[N]\}}\frac{(q-1)_{\deg_\spl(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}\\
&=\frac{q+1}{q^N}\sum_{\spl\in\mathcal{R}_N}\frac{(q)_{\deg_\spl([N])-1}}{(q-1)_{\deg_\spl([N])-1}}\cdot(q^{e_{[N]}(\bm{s})}-1)\prod_{\lambda\in\mathcal{B}(\spl)}\frac{(q-1)_{\deg_\spl(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}\\
&=\frac{q+1}{q^N}\sum_{\spl\in\mathcal{R}_N}\frac{q^{N+\sum_{i<j}s_{ij}}-q}{q+1-\deg_\spl([N])}\prod_{\lambda\in\mathcal{B}(\spl)}\frac{(q-1)_{\deg_\spl(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}~.
\end{align*}
Note that the summand for each $\spl\in\mathcal{R}_N$ is still defined for any prime power $q$ since the denominators $(q-1)_{\deg_\spl([N])-1}$ and $q+1-\deg_\spl([N])$ (which vanish when $q=\deg_\spl([N])-1$) are cancelled by the numerator $(q-1)_{\deg_\spl([N])-1}$ appearing in the product over $\lambda\in\mathcal{B}(\spl)$. Finally, we evaluate the righthand side of \eqref{sum1} by combining the sum directly above with that in \eqref{term1} and multiplying through by $(\frac{q}{q+1})^N$. This yields the desired formula for $\mathcal{Z}_N(\P^1(K),\bm{s})$:
\begin{align*}
\mathcal{Z}_N(\P^1(K),\bm{s})&=\frac{1}{(q+1)^{N-1}}\sum_{\spl\in\mathcal{R}_N}\left(1+\frac{q^{N+\sum_{i<j}s_{ij}}-q}{q+1-\deg_\spl([N])}\right)\prod_{\lambda\in\mathcal{B}(\spl)}\frac{(q-1)_{\deg_\spl(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}\\
&=\frac{1}{(q+1)^{N-1}}\sum_{\spl\in\mathcal{R}_N}\frac{q^{N+\sum_{i<j}s_{ij}}+1-\deg_\spl([N])}{q+1-\deg_\spl([N])}\prod_{\lambda\in\mathcal{B}(\spl)}\frac{(q-1)_{\deg_\spl(\lambda)-1}}{q^{e_\lambda(\bm{s})}-1}
\end{align*}
\end{proof}

\subsection{Finishing the proof of Theorem 2.3}

Our final task is to prove the $(q+1)$th Power Law, which we noted in \Cref{2_2} is equivalent to the equations in \eqref{P(K)coeff}. That is, it remains to prove $$\frac{Z_N(\P^1(K),\beta)}{N!}=\sum_{\substack{N_0+\dots+N_q=N\\N_0,\dots,N_q\geq 0}}\prod_{k=0}^q\left(\frac{q}{q+1}\right)^{N_k}\frac{Z_{N_k}(P,\beta)}{N_k!}\qquad\text{for all $\beta>0$ and $N\geq 0$}.$$

\begin{proof}
Fix $N\geq 0$ and $\beta>0$, and fix $\bm{s}$ via $s_{ij}=\beta$ for all $i<j$, so that $\mathcal{Z}_N(\P^1(K),\bm{s})=Z_N(\P^1(K),\beta)$ and $\mathcal{Z}_I(P,\bm{s})=Z_{\#I}(P,\beta)$ for any subset $I\subset[N]$. The formula in \Cref{main} relates these functions of $\beta$ via
\begin{align*}
Z_N(\P^1(K),\beta)&=\mathcal{Z}_N(\P^1(K),\bm{s})\\
&=\left(\frac{q}{q+1}\right)^N\sum_{(I_0,\dots,I_q)\vdash[N]}\prod_{k=0}^q\mathcal{Z}_{\#I_k}(P,\bm{s})\\
&=\sum_{(I_0,\dots,I_q)\vdash[N]}\prod_{k=0}^q\left(\frac{q}{q+1}\right)^{\#I_k}Z_{\#I_k}(P,\beta)~.
\end{align*}
For each choice of $q+1$ ordered integers $N_0,\dots,N_q\geq 0$ satisfying $N_0+\dots+N_q=N$, there are precisely $$\binom{N}{N_0,\dots,N_q}=\frac{N!}{N_0!\cdots N_q!}$$ ordered partitions $(I_0,\dots,I_q)\vdash[N]$ satisfying $\#I_0=N_0,\dots,\#I_q=N_q$. Finally, grouping ordered partitions according to all possible ordered integer choices establishes the desired equation:
\begin{align*}
\frac{Z_N(\P^1(K),\bm{s})}{N!}&=\frac{1}{N!}\cdot\sum_{(I_0,\dots,I_q)\vdash[N]}\prod_{k=0}^q\left(\frac{q}{q+1}\right)^{\#I_k}Z_{\#I_k}(P,\beta)\\
&=\frac{1}{N!}\cdot\sum_{\substack{N_0+\dots+N_q=N\\N_0,\dots,N_q\geq 0}}\binom{N}{N_0,\dots,N_q}\prod_{k=0}^q\left(\frac{q}{q+1}\right)^{N_k}Z_{N_k}(P,\beta)\\
&=\sum_{\substack{N_0+\dots+N_q=N\\N_0,\dots,N_q\geq 0}}\prod_{k=0}^q\left(\frac{q}{q+1}\right)^{N_k}\frac{Z_{N_k}(P,\beta)}{N_k!}~.
\end{align*}
\end{proof}

\noindent\textbf{Acknowledgements}: I would like to thank Clay Petsche for confirming several details about the measure and metric on $\P^1(K)$, and I would like to thank Chris Sinclair for many useful suggestions regarding the Power Laws and the Quadratic Recurrence.