# Observation of gamma rays up to 320 TeV from the middle-aged TeV pulsar wind
nebula HESS J1849-000

M. Amenomori  Department of Physics, Hirosaki University, Hirosaki 036-8561,
Japan  S. Asano  Department of Physics, Shinshu University, Matsumoto
390-8621, Japan  Y. W. Bao  School of Astronomy and Space Science, Nanjing
University, Nanjing 210093, China  X. J. Bi  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>D. Chen
National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China  T. L. Chen  Department of Mathematics and Physics, Tibet
University, Lhasa 850000, China  W. Y. Chen  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>Xu Chen  Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100012, China  Y. Chen  School of Astronomy and Space
Science, Nanjing University, Nanjing 210093, China  Cirennima  Department of
Mathematics and Physics, Tibet University, Lhasa 850000, China  S. W. Cui
Department of Physics, Hebei Normal University, Shijiazhuang 050016, China
Danzengluobu  Department of Mathematics and Physics, Tibet University, Lhasa
850000, China  L. K. Ding  Key Laboratory of Particle Astrophysics, Institute
of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China;
<EMAIL_ADDRESS><EMAIL_ADDRESS>J. H. Fang  Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>University of Chinese Academy of Sciences, Beijing 100049, China  K. Fang  Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>C. F. Feng  Institute of Frontier and Interdisciplinary
Science and Key Laboratory of Particle Physics and Particle Irradiation (MOE),
Shandong University, Qingdao 266237, China  Zhaoyang Feng  Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>Z.
Y. Feng  Institute of Modern Physics, SouthWest Jiaotong University, Chengdu
610031, China  Qi Gao  Department of Mathematics and Physics, Tibet
University, Lhasa 850000, China  A. Gomi  Faculty of Engineering, Yokohama
National University, Yokohama 240-8501, Japan  Q. B. Gou  Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>Y.
Q. Guo  Key Laboratory of Particle Astrophysics, Institute of High Energy
Physics, Chinese Academy of Sciences, Beijing 100049, China;
<EMAIL_ADDRESS><EMAIL_ADDRESS>Y. Y. Guo  Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>Y.
Hayashi  Department of Physics, Shinshu University, Matsumoto 390-8621, Japan
H. H. He  Key Laboratory of Particle Astrophysics, Institute of High Energy
Physics, Chinese Academy of Sciences, Beijing 100049, China;
<EMAIL_ADDRESS><EMAIL_ADDRESS>Z. T. He  Department of Physics,
Hebei Normal University, Shijiazhuang 050016, China  K. Hibino  Faculty of
Engineering, Kanagawa University, Yokohama 221-8686, Japan  N. Hotta  Faculty
of Education, Utsunomiya University, Utsunomiya 321-8505, Japan  Haibing Hu
Department of Mathematics and Physics, Tibet University, Lhasa 850000, China
H. B. Hu  Key Laboratory of Particle Astrophysics, Institute of High Energy
Physics, Chinese Academy of Sciences, Beijing 100049, China;
<EMAIL_ADDRESS><EMAIL_ADDRESS>K. Y. Hu  Corresponding author  Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>University of Chinese Academy of Sciences, Beijing
100049, China  J. Huang  Corresponding author  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>H. Y. Jia
Institute of Modern Physics, SouthWest Jiaotong University, Chengdu 610031,
China  L. Jiang  Key Laboratory of Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Sciences, Beijing 100049, China;
<EMAIL_ADDRESS><EMAIL_ADDRESS>P. Jiang  National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, China  H. B. Jin
National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China  K. Kasahara  Faculty of Systems Engineering, Shibaura Institute
of Technology, Omiya 330-8570, Japan  Y. Katayose  Faculty of Engineering,
Yokohama National University, Yokohama 240-8501, Japan  C. Kato  Department of
Physics, Shinshu University, Matsumoto 390-8621, Japan  S. Kato  Corresponding
author  Institute for Cosmic Ray Research, University of Tokyo, Kashiwa
277-8582, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>I.
Kawahara  Faculty of Engineering, Yokohama National University, Yokohama
240-8501, Japan  T. Kawashima  Institute for Cosmic Ray Research, University
of Tokyo, Kashiwa 277-8582, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>K. Kawata  Institute for Cosmic Ray Research,
University of Tokyo, Kashiwa 277-8582, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>M. Kozai  Polar Environment Data Science Center,
Joint Support-Center for Data Science Research, Research Organization of
Information and Systems, Tachikawa 190-0014, Japan  D. Kurashige  Faculty of
Engineering, Yokohama National University, Yokohama 240-8501, Japan  Labaciren
Department of Mathematics and Physics, Tibet University, Lhasa 850000, China
G. M. Le  National Center for Space Weather, China Meteorological
Administration, Beijing 100081, China  A. F. Li  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>Institute of
Frontier and Interdisciplinary Science and Key Laboratory of Particle Physics
and Particle Irradiation (MOE), Shandong University, Qingdao 266237, China
School of Information Science and Engineering, Shandong Agriculture
University, Taian 271018, China  H. J. Li  Department of Mathematics and
Physics, Tibet University, Lhasa 850000, China  W. J. Li  Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>Institute of Modern Physics, SouthWest Jiaotong University, Chengdu 610031,
China  Y. Li  National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100012, China  Y. H. Lin  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>University
of Chinese Academy of Sciences, Beijing 100049, China  B. Liu  Department of
Astronomy, School of Physical Sciences, University of Science and Technology
of China, Hefei 230026, China  C. Liu  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>J. S. Liu
Key Laboratory of Particle Astrophysics, Institute of High Energy Physics,
Chinese Academy of Sciences, Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>L. Y. Liu  National Astronomical Observatories, Chinese
Academy of Sciences, Beijing 100012, China  M. Y. Liu  Department of
Mathematics and Physics, Tibet University, Lhasa 850000, China  W. Liu  Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>H. Lu  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China<EMAIL_ADDRESS><EMAIL_ADDRESS>X. R. Meng  Department of
Mathematics and Physics, Tibet University, Lhasa 850000, China  Y. Meng  Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>University of Chinese Academy of Sciences, Beijing
100049, China  K. Munakata  Department of Physics, Shinshu University,
Matsumoto 390-8621, Japan  K. Nagaya  Faculty of Engineering, Yokohama
National University, Yokohama 240-8501, Japan  Y. Nakamura  Institute for
Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan;
<EMAIL_ADDRESS><EMAIL_ADDRESS>Y. Nakazawa  College of
Industrial Technology, Nihon University, Narashino 275-8575, Japan  H. Nanjo
Department of Physics, Hirosaki University, Hirosaki 036-8561, Japan  C. C.
Ning  Department of Mathematics and Physics, Tibet University, Lhasa 850000,
China  M. Nishizawa  National Institute of Informatics, Tokyo 101-8430, Japan
R. Noguchi  Faculty of Engineering, Yokohama National University, Yokohama
240-8501, Japan  M. Ohnishi  Institute for Cosmic Ray Research, University of
Tokyo, Kashiwa 277-8582, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>S. Okukawa  Faculty of Engineering, Yokohama
National University, Yokohama 240-8501, Japan  S. Ozawa  National Institute of
Information and Communications Technology, Tokyo 184-8795, Japan  X. Qian
National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China  X. L. Qian  Department of Mechanical and Electrical
Engineering, Shangdong Management University, Jinan 250357, China  X. B. Qu
College of Science, China University of Petroleum, Qingdao 266555, China  T.
Saito  Tokyo Metropolitan College of Industrial Technology, Tokyo 116-8523,
Japan  Y. Sakakibara  Faculty of Engineering, Yokohama National University,
Yokohama 240-8501, Japan  M. Sakata  Department of Physics, Konan University,
Kobe 658-8501, Japan  T. Sako  Institute for Cosmic Ray Research, University
of Tokyo, Kashiwa 277-8582, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>T. K. Sako  Institute for Cosmic Ray Research,
University of Tokyo, Kashiwa 277-8582, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>T. Sasaki  Faculty of Engineering, Kanagawa
University, Yokohama 221-8686, Japan  J. Shao  Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>Institute of
Frontier and Interdisciplinary Science and Key Laboratory of Particle Physics
and Particle Irradiation (MOE), Shandong University, Qingdao 266237, China  M.
Shibata  Faculty of Engineering, Yokohama National University, Yokohama
240-8501, Japan  A. Shiomi  College of Industrial Technology, Nihon
University, Narashino 275-8575, Japan  H. Sugimoto  Shonan Institute of
Technology, Fujisawa 251-8511, Japan  W. Takano  Faculty of Engineering,
Kanagawa University, Yokohama 221-8686, Japan  M. Takita  Corresponding author
Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582,
Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>Y. H. Tan  Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>N. Tateyama  Faculty of Engineering, Kanagawa
University, Yokohama 221-8686, Japan  S. Torii  Research Institute for Science
and Engineering, Waseda University, Tokyo 162-0044, Japan  H. Tsuchiya  Japan
Atomic Energy Agency, Tokai-mura 319-1195, Japan  S. Udo  Faculty of
Engineering, Kanagawa University, Yokohama 221-8686, Japan  H. Wang  Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>S. F. Wang  Department of Mathematics and Physics, Tibet
University, Lhasa 850000, China  Y. P. Wang  Department of Mathematics and
Physics, Tibet University, Lhasa 850000, China  Wangdui  Department of
Mathematics and Physics, Tibet University, Lhasa 850000, China  H. R. Wu  Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>Q. Wu  Department of Mathematics and Physics, Tibet
University, Lhasa 850000, China  J. L. Xu  National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, China  L. Xue
Institute of Frontier and Interdisciplinary Science and Key Laboratory of
Particle Physics and Particle Irradiation (MOE), Shandong University, Qingdao
266237, China  Z. Yang  Key Laboratory of Particle Astrophysics, Institute of
High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China;
<EMAIL_ADDRESS><EMAIL_ADDRESS>Y. Q. Yao  National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, China  J. Yin
National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China  Y. Yokoe  Institute for Cosmic Ray Research, University of
Tokyo, Kashiwa 277-8582, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>Y. L. Yu  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China<EMAIL_ADDRESS><EMAIL_ADDRESS>University of Chinese
Academy of Sciences, Beijing 100049, China  A. F. Yuan  Department of
Mathematics and Physics, Tibet University, Lhasa 850000, China  L. M. Zhai
National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China  H. M. Zhang  Key Laboratory of Particle Astrophysics, Institute
of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China;
<EMAIL_ADDRESS><EMAIL_ADDRESS>J. L. Zhang  Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>X.
Zhang  School of Astronomy and Space Science, Nanjing University, Nanjing
210093, China  X. Y. Zhang  Institute of Frontier and Interdisciplinary
Science and Key Laboratory of Particle Physics and Particle Irradiation (MOE),
Shandong University, Qingdao 266237, China  Y. Zhang  Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>Yi
Zhang  Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain
Observatory, Chinese Academy of Sciences, Nanjing 210034, China  Ying Zhang
Key Laboratory of Particle Astrophysics, Institute of High Energy Physics,
Chinese Academy of Sciences, Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>S. P. Zhao  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China<EMAIL_ADDRESS><EMAIL_ADDRESS>Zhaxisangzhu  Department of
Mathematics and Physics, Tibet University, Lhasa 850000, China  X. X. Zhou
Institute of Modern Physics, SouthWest Jiaotong University, Chengdu 610031,
China  Y. H. Zou  Key Laboratory of Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Sciences, Beijing 100049, China;
<EMAIL_ADDRESS><EMAIL_ADDRESS>University of Chinese Academy of
Sciences, Beijing 100049, China

###### Abstract

Gamma rays from HESS J1849$-$000, a middle-aged TeV pulsar wind nebula (PWN),
are observed by the Tibet air shower array and the muon detector array. The
detection significance of gamma rays reaches $4.0\,\sigma$ and $4.4\,\sigma$
levels above $25\,{\rm TeV}$ and $100\,{\rm TeV}$, respectively, in units of
Gaussian standard deviation $\sigma$. The energy spectrum measured between
$40\,{\rm TeV}<E<320\,{\rm TeV}$ for the first time is described with a simple
power-law function of ${\rm d}N/{\rm d}E=(2.86\pm 1.44)\times
10^{-16}(E/40\,{\rm TeV})^{-2.24\pm 0.41}\,{\rm TeV}^{-1}\,{\rm cm}^{-2}\,{\rm
s}^{-1}$. The gamma-ray energy spectrum from the sub-TeV ($E<1\,{\rm TeV}$) to
sub-PeV ($100\,{\rm TeV}<E<1\,{\rm PeV}$) ranges including the results of
previous studies can be modeled with the leptonic scenario, inverse Compton
scattering by high-energy electrons accelerated by the PWN of PSR
J1849$-$0001\. On the other hand, the gamma-ray energy spectrum can also be
modeled with the hadronic scenario in which gamma rays are generated from the
decay of neutral pions produced by collisions between accelerated cosmic-ray
protons and the ambient molecular cloud found in the gamma-ray emitting
region. The cutoff energy of cosmic-ray protons $E_{\rm p,\,cut}$ is estimated
at ${\rm log}_{10}(E_{\rm p,\,cut}/{\rm TeV})=3.73^{+2.98}_{-0.66}$,
suggesting that protons are accelerated up to the PeV energy range. Our study
thus proposes that HESS J1849$-$000 should be further investigated as a new
candidate for a Galactic PeV cosmic-ray accelerator, PeVatron.

Galactic cosmic rays (567) — Pulsar wind nebulae (2215) — Gamma-ray sources
(633) — Gamma-ray astronomy (628) — Gamma-ray observatories (632)

##  1 Introduction

The origin of the knee at around $4\,{\rm PeV}$ in the cosmic-ray (CR) energy
spectrum has been an open question since its discovery (Kulikov & Kristiansen,
1958). One of the major approaches to the problem is observing sub-PeV
$(100\,{\rm TeV}<E<1\,{\rm PeV})$ $\pi^{0}$-decay gamma rays generated by
collisions of PeV CRs accelerated by a source with the ambient interstellar
medium. This strategy helps us discover several candidates of Galactic PeV CR
accelerators, “PeVatrons”, including supernova remnants (SNR), star-forming
regions, and unidentified sources (Amenomori et al., 2021; Cao et al., 2021a;
Albert et al., 2021a; Abeysekara et al., 2021; Fang et al., 2022). On the
other hand, many of the TeV and sub-PeV gamma-ray sources are pulsar wind
nebulae (PWNe), representing the accelerators of high-energy electrons and
positrons (H.E.S.S. collaboration, 2018a, b; Amenomori et al., 2019; Albert et
al., 2021b). In particular, middle-aged PWNe harboring pulsars with
characteristic ages of $\gtrsim 10\,{\rm kyr}$ occupy a large fraction of PWNe
(for example, see H.E.S.S. Collaboration et al. (2019a, b)). Some theories
also discuss that many unidentified sources bright in gamma rays could be
middle-aged or old PWNe (Tibolla et al., 2013; Vorster et al., 2013; Kaufmann
& Tibolla, 2018). Moreover, in terms of CR acceleration, a PWN inside its
precursor SNR could accelerate PeV CRs (Ohira et al., 2018). Therefore,
detailed observations of individual middle-aged PWNe are crucial not only to
study the majority of Galactic gamma-ray sources but also to elucidate the
origin of CRs around the knee energy range.

HESS J1849$-$000 is a TeV gamma-ray source detected by H.E.S.S. (Terrier et
al., 2008; H.E.S.S. collaboration, 2018a). The TeV emission is nearly centered
at an X-ray pulsar PSR J1849$-$0001, which is surrounded by a diffuse
synchrotron X-ray PWN found by previous studies (Terrier et al., 2008;
Gotthelf et al., 2011; Kuiper & Hermsen, 2015; Vleeschower Calas et al.,
2018). The spin period, spin-down luminosity, and characteristic age of PSR
J1849$-$0001 are $P=38.5\,{\rm ms}$, $\dot{E}=9.8\times 10^{36}\,{\rm
erg}\,{\rm s}^{-1}$, and $\tau_{\rm c}=42.9\,{\rm kyr}$, respectively
(Gotthelf et al., 2011). Due to the positional coincidence of the TeV gamma-
ray emission with the pulsar and the energetic nature of the pulsar, HESS
J1849$-$000 is interpreted as the PWN which is powered by PSR J1849$-$0001 and
is in transition from a young, synchrotron-dominated phase to an evolved,
inverse-Compton dominated phase from the comparison of the energy fluxes of
the diffuse keV X-ray and TeV gamma-ray emissions H.E.S.S. collaboration
(2018a). The differential spectrum of the TeV emission has a relatively hard
index of $\simeq 2$ (H.E.S.S. collaboration, 2018a, b), which may allow the
detection of nearby gamma-ray sources in the ultra-high energy range by HAWC
(eHWC J1850$+$001 in $E>56\,{\rm TeV}$) and LHAASO (LHAASO J1849$-$0001 at
$E=100\,{\rm TeV}$) (Abeysekara et al., 2020; Cao et al., 2021a). In
particular, the HAWC collaboration also implies the association of the
observed gamma-ray emission with J1849$-$0001 and points out that such a high-
energy gamma-ray emission may be ubiquitous around powerful pulsars with
$\dot{E}>10^{36}\,{\rm erg}\,{\rm s}^{-1}$ (Albert et al., 2021b). However,
the current situation of the ultra-high-energy gamma-ray observations still
lacks studies of the particle acceleration taking place in this middle-aged
PWN through detailed analysis.

This paper presents the observation of gamma rays from HESS J1849$-$000 up to
the sub-PeV energy range by the Tibet air shower array and the muon detector
array and proposes some possible mechanisms of the gamma-ray emission through
detailed data analysis and discussion. First, Section 2 briefly introduces the
experiment. After presenting the results of the data analysis and the
discussion in Sections 3 and 4, respectively, the conclusion is made in
Section 5.

##  2 Experiment

Tibet air shower (AS) array has been observing CRs above the TeV energy range
since 1991 (Amenomori et al., 1992, 1999, 2009, 2019) at Yangbajing
($90{\fdg}522\,{\rm E}$, $30{\fdg}102{\rm N}$, $4,300\,{\rm m}$ a.s.l.) in
Tibet, China. The surface array consists of 597 plastic scintillation
detectors covering a total geometrical area of $65,700\,{\rm m}^{2}$. These
detectors record the number of secondary particles in ASs and the timings of
detection of these particles to reconstruct the energy and arrival direction
of primary CRs. The surface array is equipped with an underground water-
Cherenkov-type muon detector (MD) array with a geometrical area of
$3,400\,{\rm m}^{2}$ to improve the sensitivity to celestial gamma rays (Sako
et al., 2009; Amenomori et al., 2019). This work utilizes data taken by the
Tibet AS array and the MD array from 2014 February to 2017 May (719 live
days). The data analysis and the Monte Carlo simulation performed in this work
are described in Appendices A and B, respectively.

##  3 Results

###  3.1 Detection of gamma rays from HESS J1849$-$000

Events are taken by opening a circular ON-source window with an angular radius
of $0.6^{\circ}$ centered at HESS J1849$-$000,
$({\alpha},\,{\delta})=(282{\fdg}24,-0{\fdg}04)$ in the J2000 coordinates
(H.E.S.S. collaboration, 2018a). On the other hand, the background is
estimated by counting events in 20 OFF windows, each of which has the same
size as the ON-source window, opened with the equi-zenith angle method
(Amenomori et al., 2003). The events in the ON-source and OFF windows are then
binned by energy into five logarithmically equal bins per decade. After the
event selection described in Appendix A, the detection significance of gamma
rays from HESS J1849$-$000 reaches $4.0\,\sigma$ and $4.4\,\sigma$ levels
above $25\,{\rm TeV}$ and $100\,{\rm TeV}$, respectively, in units of Gaussian
standard deviation $\sigma$ (Li & Ma, 1983).

Figure 1 shows the significance maps of the HESS J1849$-$000 region above
$25\,{\rm TeV}$ (top) and $100\,{\rm TeV}$ (bottom) pixelized by
$0.05^{\circ}\times 0.05^{\circ}$ pixels. The map is smoothed with the point-
spread function (PSF) of the experiment. In the map above $100\,{\rm TeV}$,
the pixel with the maximum significance (let us call it “the brightest pixel”)
is found at $({\alpha},\,{\delta})=(282{\fdg}33,0{\fdg}08)$, deviating by
$0{\fdg}15$ from HESS J1849$-$000\. A toy Monte Carlo simulation shows that
the $68\%$ statistical uncertainty in the brightest pixel’s orientation is
$0.18^{\circ}$. The pointing systematics of the experiment along the right
ascension and the declination are also estimated at $0{\fdg}058$ and
$0{\fdg}055$, respectively, from the data analysis of the gamma rays coming
from the Crab Nebula; see Appendix C. From the above statistical and
systematic uncertainties, the total uncertainty in the center position of the
observed gamma-ray emission with the $68\%$ confidence level is estimated at
$0{\fdg}22$ following the methodology used in the previous studies (H.E.S.S.
collaboration, 2018a; Amenomori et al., 2022). Therefore, the center position
of the sub-PeV gamma-ray emission observed in this study is consistent with
that of HESS J1849$-$000\. On the other hand, HESS J1852$-$000 (H.E.S.S.
collaboration, 2018a) deviates by $0.74^{\circ}$ in angular distance from the
brightest pixel, and the deviation corresponds to $3.0\,\sigma$ significance
taking into account both the uncertainty in the center position of the
observed sub-PeV gamma-ray emission ($0{\fdg}22$) and that in the position of
HESS J1852$-$000\. The result thus disfavors HESS J1852$-$000 as the origin of
the sub-PeV gamma rays.

Figure 1: Top: significance map of the HESS J1849$-$000 region above $25\,{\rm
TeV}$, pixelized by $0.05^{\circ}\times 0.05^{\circ}$ pixels and smoothed with
the PSF. The white dashed line indicates the Galactic plane. Positions of
nearby celestial objects in the sky region, including galactic SNRs (and the
candidates), PSR J1849$-$0001, and gamma-ray sources, are plotted with symbols
as presented in the right legend, and their extensions (if they have) are
indicated with circles with the same colors as the symbols. Data on these
nearby objects are cited from H.E.S.S. collaboration (2018a) (H.E.S.S.);
Abeysekara et al. (2017), Albert et al. (2020), and Abeysekara et al. (2020)
(HAWC); Cao et al. (2021a) (LHAASO); Abdollahi et al. (2020) (4FGL sources,
Fermi-LAT); Anderson et al. (2017) (SNRs); Manchester et al. (2005) (PSR
J1849$-$0001). The green contours show the ${}^{12}{\rm CO}$ ($J=1-0$) line
emission which is found near HESS J1849$-$000 in the analysis of the archive
data published by the FUGIN survey (Umemoto et al., 2017). The emission is
integrated with the velocity range of $93-100\,{\rm km\,s^{-1}}$ and the
contour levels are 20, 30, 40, and 50 ${\rm K\,km\,s^{-1}}$. The lower-left
inset shows the PSF. Bottom: significance map above $100\,{\rm TeV}$ smoothed
with the PSF.

###  3.2 Energy spectrum

The energy spectrum of gamma rays from HESS J1849$-$000 is measured for the
first time in an energy range between $40\,{\rm TeV}<E<320\,{\rm TeV}$ as
shown with the red points in Figure 2. The differential energy flux in each
energy bin is calculated only if the detection significance of gamma rays
exceeds $2\,\sigma$, otherwise, the $99\%$ upper limit on the flux is
calculated. Table 1 summarizes the result of the calculation. A simple power-
law (PL) function can be fitted to the measured spectrum and the best-fit
result is ${\rm d}N/{\rm d}E=(2.86\pm 1.44)\times 10^{-16}(E/40\,{\rm
TeV})^{-2.24\pm 0.41}\,{\rm TeV}^{-1}\,{\rm cm}^{-2}\,{\rm s}^{-1}$ with
$\chi^{2}/{\rm d.o.f.}=0.47/3$. The experiment’s absolute energy scale
uncertainty of $12\%$ (Amenomori et al., 2009) dominates the systematic
uncertainty in the flux normalization of $\simeq 27\%$. The fraction of
contamination to the number of events above $100\,{\rm TeV}$ from the lower
energy range due to the finite energy resolution is estimated at $\simeq
20\%$. Furthermore, there may be a spillover of gamma rays from the nearby
gamma-ray source HESS J1852$-$000 into the ON-source window (see also Figure
1). Assuming a two-dimensional Gaussian with a $0.28^{\circ}$ extension for
the morphology of HESSJ1852$-$000 (H.E.S.S. collaboration, 2018a), the 95%
upper limit on its integral gamma-ray flux above $100\,{\rm TeV}$ is
calculated as $2.08\times 10^{-15}\,{\rm cm}^{-2}\,{\rm s}^{-1}$ if the
spectrum extends beyond the sub-PeV range. The spillover into the ON-source
window is thus estimated at $<3.65\times 10^{-16}\,{\rm cm}^{-2}\,{\rm
s}^{-1}$, which is less than $20\%$ of the integral flux of HESS J1849$-$000
above $100\,{\rm TeV}$, $1.93^{+0.85}_{-0.65}\times 10^{-15}\,{\rm
cm}^{-2}\,{\rm s}^{-1}$. According to Vernetto & Lipari (2016), the
attenuation of gamma rays due to the interactions with the interstellar
radiations on the way to Earth is $\simeq 12\%$ at $150\,{\rm TeV}$ if the
distance to HESS J1849$-$000 is assumed as $7\,{\rm kpc}$ (H.E.S.S.
collaboration, 2018b).

Table 1: Differential energy flux or the 99% upper limit on the flux
calculated for each energy bin.

$E$ (${\rm TeV}$) | Flux (${\rm TeV}^{-1}\,{\rm cm}^{-2}\,{\rm s}^{-1}$) | Significance ($\sigma$)
---|---|---
$29.4$ | $<1.96\times 10^{-15}$ | $1.7$
$48.2$ | $2.10^{+1.31}_{-1.11}\times 10^{-16}$ | 2.1
$76.4$ | $4.97^{+3.65}_{-2.71}\times 10^{-17}$ | 2.3
$121$ | $2.82^{+1.57}_{-1.13}\times 10^{-17}$ | 3.8
$194$ | $6.83^{+6.26}_{-3.80}\times 10^{-18}$ | 2.8
$316$ | $2.84^{+3.09}_{-1.73}\times 10^{-18}$ | 2.7

Note. — The third column presents the significance of the event excess in the
ON-source window over the background calculated with Equation (17) of Li & Ma
(1983), which corresponds to the detection significance of gamma rays from
HESS J1849$-$000

Figure 2: Top: gamma-ray energy spectrum of HESS J1849$-$000 modeled with the
leptonic scenario using Naima (Zabalza, 2015). The red data points are the
result obtained in this work and the other points are cited from Acero et al.
(2013) (Fermi-LAT, cyan); H.E.S.S. collaboration (2018a) (H.E.S.S., blue); Cao
et al. (2021a) (LHAASO, green). The error bars of the data points show
$1\,\sigma$ statistical uncertainty and the downward arrows present the $99\%$
upper limits. The black curve shows the best-fit result to the whole spectrum,
and the thick and thin gray bands show the $1\,\sigma$ and $2\,\sigma$
confidence intervals of the fit, respectively. Bottom: gamma-ray energy
spectrum modeled with the hadronic scenario. For detailed descriptions, see
Sections 4.1 and 4.2.

##  4 Discussion

###  4.1 Leptonic scenario

The gamma-ray energy spectrum from the sub-TeV ($E<1\,{\rm TeV}$) to the sub-
PeV ranges observed by Fermi-LAT (Acero et al., 2013), H.E.S.S. (H.E.S.S.
collaboration, 2018a), LHAASO (Cao et al., 2021a), and this work is modeled
with the leptonic scenario using Naima (Zabalza, 2015). In this scenario,
gamma rays are generated from inverse Compton scattering (ICS) off the
interstellar radiations by high-energy electrons (Khangulyan et al., 2014).
The target radiation field is assumed to consist of the cosmic microwave
background and near- (temperature of $20\,{\rm K}$ and energy density of
$0.75\,{\rm eV}\,{\rm cm}^{-3}$) and far-infrared blackbody components
($3,000\,{\rm K}$ and $1.26\,{\rm eV}\,{\rm cm}^{-3}$) whose energy densities
are determined using GALPROP (Porter et al., 2017). The distance to HESS
J1849$-$000 is assumed as $7\,{\rm kpc}$, the same as that assumed for PSR
J1849$-$0001 in previous studies (Gotthelf et al., 2011; H.E.S.S.
collaboration, 2018b). The energy spectrum of parent electrons is assumed to
follow a simple PL function

$\frac{{\rm d}N_{\rm e}}{{\rm d}E}=A_{\rm e}\bigg{(}\frac{E}{10\,{\rm
TeV}}\bigg{)}^{-\alpha_{\rm e}}\,\,{\rm eV}^{-1}$ (1)

where the normalization constant $A_{\rm e}$ and the spectral index
$\alpha_{\rm e}$ are free parameters. Under the above model assumptions, Naima
computes the posterior distributions of the parameters and gives the median
value and its uncertainty corresponding to the central $68\%$ of each
distribution. The best-fit result of the modeling is shown with the black
curve in the top panel of Figure 2 and the spectral parameters are estimated
at ${\rm log}_{10}A_{\rm e}=31.98^{+0.06}_{-0.07}$ and $\alpha_{\rm
e}=2.46^{+0.08}_{-0.07}$. Assuming a PL function with an exponential cutoff
(ECPL) for the electron spectrum does not improve the goodness of fit, and the
$95\%$ lower limit on the cutoff energy is estimated at $740\,{\rm TeV}$. The
obtained limit is extremely high but is not implausible because the
acceleration of PeV electrons takes place in the Crab Nebula (Cao et al.,
2021b). A broken PL function is not significantly preferred as the electron
spectrum either and the current gamma-ray observations can be well explained
with ICS by electrons following a simple PL spectrum.

In the leptonic scenario considered above, the gamma rays observed can be
interpreted as ICS radiations from high-energy electrons accelerated by the
PWN of PSR J1849$-$0001\. On the other hand, the total energy of electrons
above $100\,{\rm GeV}$ is calculated as $2.8^{+1.0}_{-0.7}\times 10^{47}\,{\rm
erg}$, which occupies only $\simeq 2\%$ of the total spin-down energy of PSR
J1849$-$0001 ($\simeq 1.3\times 10^{49}\,{\rm erg}$). If the rest of the spin-
down energy is assigned to the magnetic field inside the diffuse X-ray nebula
around the pulsar ($\simeq 75^{{}^{\prime\prime}}$ in radius or $\simeq
2.5\,{\rm pc}$ at the distance of $7\,{\rm kpc}$ according to Gotthelf et al.
(2011)), then the magnetic field strength $B$ in the nebula would be $\sim
400\,\mu{\rm G}$. Such a strong field, however, leads to the synchrotron X-ray
nebula far outshining that observed with the energy flux of $\sim
10^{-12}\,{\rm erg\,cm^{-2}\,s^{-1}}$ in $2-10\,{\rm keV}$ (Gotthelf et al.,
2011; Kuiper & Hermsen, 2015; Vleeschower Calas et al., 2018), which can be
reproduced with $B\sim 2\,\mu{\rm G}$ under a simple one-zone model in the
spectral fit assuming the aforementioned PL electron population. Furthermore,
the one-zone model would not be adequate to interpret the observations since
the extensions of the X-ray nebula and TeV gamma-ray emission ($\simeq
0.09^{\circ}$, H.E.S.S. collaboration (2018b)) are much different, and more
detailed theoretical modeling is thus required when one considers the leptonic
scenario.

###  4.2 Hadronic scenario

On the other hand, the gamma-ray energy spectrum can also be explained in the
context of the hadronic scenario. This scenario supposes that CR protons
generate $\pi^{0}$-decay gamma rays through interactions with the ambient
interstellar medium (Kafexhiu et al., 2014). To investigate the environment
around the source, archive data of the ${}^{12}{\rm CO}\,(J=1-0)$ line
emission published by the FUGIN survey (Umemoto et al., 2017) is analyzed. The
analysis finds a $\sim 20\,{\rm pc}$ size molecular cloud in the velocity
range of $93-100\,{\rm km}\,{\rm s}^{-1}$ corresponding to the distance around
$7\,{\rm kpc}$, which resides at the west side of HESS J1849$-$000 as shown by
green contours in Figure 1. The cloud has the ${}^{12}{\rm CO}$ line intensity
of $\sim 20\,{\rm K}\,{\rm km\,s^{-1}}$ and can provide a gas density of
$\gtrsim 10$ protons ${\rm cm}^{-3}$ assuming the canonical CO-to-${\rm
H}_{2}$ factor of $2\times 10^{20}\,{\rm cm}^{-2}\,({\rm K\,km\,s^{-1}})^{-1}$
(Bolatto et al., 2013). A wealthy amount of target material for CRs is also
implied by the high hydrogen column density of $\simeq 4.5\times 10^{22}\,{\rm
cm}^{-2}$ toward the source direction (Gotthelf et al., 2011). The region
bright in gamma rays above $100\,{\rm TeV}$ shown in Figure 1 is positionally
consistent with the cloud within the uncertainty discussed in Section 3.1.

The modeling of the gamma-ray energy spectrum from the sub-TeV to the sub-PeV
ranges is performed under the hadronic scenario assuming an ECPL function for
the spectrum of CR protons:

$\frac{{\rm d}N_{\rm p}}{{\rm d}E}=A_{\rm p}\bigg{(}\frac{E}{10\,{\rm
TeV}}\bigg{)}^{-\alpha_{\rm p}}\,{\rm exp}\bigg{(}-\frac{E}{E_{\rm
p,\,cut}}\bigg{)}\,\,{\rm eV}^{-1}$ (2)

where the normalization constant $A_{\rm p}$, the spectral index $\alpha_{\rm
p}$, and the cutoff energy $E_{\rm p,\,cut}$ are free parameters. The density
of the interstellar medium is assumed as $10$ protons ${\rm cm}^{-3}$ based on
the aforementioned radio data analysis. The result of the modeling is shown in
the bottom panel of Figure 2, and the spectral parameters are determined as
${\rm log}_{10}A_{\rm p}=33.93^{+0.09}_{-0.11}$, $\alpha_{\rm
p}=2.01^{+0.12}_{-0.21}$, and ${\rm log}_{10}(E_{\rm p,\,cut}/{\rm
TeV})=3.73^{+2.98}_{-0.66}$, respectively; it is suggested that the cutoff
energy reaches the PeV range. The total energy of the CR protons between
$1\,{\rm TeV}<E<10\,{\rm PeV}$ is calculated as $(1.1\pm 0.2)\times
10^{49}\,{\rm erg}$.

The aforementioned spectral index ($\simeq 2$) implies CR protons are
accelerated by shock waves of the PWN or the precursor SNR. However, given the
high cutoff energy beyond $1\,{\rm PeV}$, the acceleration of CR protons in a
PWN-SNR composite system may provide an adequate explanation. Ohira et al.
(2018) pointed out that the PWN adiabatically compressed by the reverse shock
of the precursor SNR can produce PeV CR protons. The total energy of $\sim
10^{49}\,{\rm erg}$ given to particles by the PWN-SNR system (Gelfand et al.,
2009) is also consistent with our result. Moreover, such a composite system
can also accelerate and leptons and would produce an X-ray nebula as observed
in the adiabatically compressed PWN with the amplified magnetic field (Gelfand
et al., 2009; Ohira et al., 2018). On the other hand, there are still no
theories that systematically study the distribution of high-energy particles
generated in a PWN-SNR system simulating detailed particle acceleration
processes and the resultant spectrum of their non-thermal radiations.
Neutrinos inevitably generated in the hadronic interaction are not currently
detected and the contribution to gamma-ray flux from hadrons is not well
constrained (Huang & Li, 2022). Future deep observation of neutrinos by
IceCube-Gen2 (Aartsen et al., 2021) and of sub-PeV gamma rays by ALPACA (Kato
et al., 2021), CTA (The Cherenkov Telescope Array Consortium, 2017), and SWGO
(Hinton & the SWGO Collaboration, 2021) have great importance to find clear
evidence for the acceleration of PeV CRs in this energetic source.

##  5 Conclusion

This paper presents the results of the observation of gamma rays from HESS
J1849$-$000 by the Tibet AS array and the MD array and discusses the origin of
the observed gamma-ray emission. The detection significance of gamma rays
above $25\,{\rm TeV}$ and $100\,{\rm TeV}$ reaches $4.0\,\sigma$ and
$4.4\,\sigma$ levels, respectively. The energy spectrum first measured between
$40\,{\rm TeV}<E<320\,{\rm TeV}$ is described with a simple power-law function
of ${\rm d}N/{\rm d}E=(2.86\pm 1.44)\times 10^{-16}(E/40\,{\rm TeV})^{-2.24\pm
0.41}\,{\rm TeV}^{-1}\,{\rm cm}^{-2}\,{\rm s}^{-1}$. The gamma-ray energy
spectrum from the sub-TeV to sub-PeV range including the results of previous
studies can be modeled with the leptonic scenario in which high-energy
electrons accelerated by the PWN of PSR J1849$-$0001 radiate gamma rays
through ICS. However, the scenario requires detailed theoretical modeling
other than the one-zone model to explain the compact X-ray nebula around PSR
J1849$-$0001 and the relatively small energy given to the electrons above
$100\,{\rm GeV}$ ($\simeq 2\%$ of the spin-down power of PSR J1849$-$0001). On
the other hand, the hadronic scenario can also model the spectrum, suggested
by the detection of a molecular cloud found in the gamma-ray emitting region.
The cutoff energy of CR protons is estimated at ${\rm log}_{10}(E_{\rm
p,\,cut}/{\rm TeV})=3.73^{+2.98}_{-0.66}$, suggesting that CR protons are
accelerated up to the PeV energy range in HESS J1849$-$000\. As a possible way
to explain the observed gamma rays, it is proposed that HESS J1849$-$000 may
be a PWN-SNR composite system. The PWN-SNR system can provide the site to
accelerate both PeV CR protons and high-energy leptons and realize the
observed complex of non-thermal radiations from the X-ray to sub-PeV gamma-ray
range. Note that, however, there is still much room to theoretically study the
particle distribution and the resultant non-thermal radiation spectra in such
a composite system. In addition, future observations of neutrinos and gamma
rays in the higher energy range than explored in this work are important to
determine the dominant contribution from hadrons or leptons to the observed
gamma-ray radiations and to obtain evidence for the acceleration of PeV CRs.
It is worth emphasizing that HESS J1849$-$000 needs to be further investigated
as one of the PeVatron candidates from the aspects of both theories and
observations.

The collaborative experiment of the Tibet Air Shower Arrays has been conducted
under the auspices of the Ministry of Science and Technology of China and the
Ministry of Foreign Affairs of Japan. This work was supported in part by a
Grant-in-Aid for Scientific Research on Priority Areas from the Ministry of
Education, Culture, Sports, Science and Technology, and by Grants-in-Aid for
Science Research from the Japan Society for the Promotion of Science in Japan.
This work is supported by the National Natural Science Foundation of China
under Grants No. 12227804, No. 12275282, No. 12103056 and No. 12073050, and
the Key Laboratory of Particle Astrophysics, Institute of High Energy Physics,
CAS. This work is also supported by the joint research program of the
Institute for Cosmic Ray Research (ICRR), the University of Tokyo. S. Kato is
supported by JST SPRING, Grant Number JPMJSP2108. This publication makes use
of data from FUGIN, FOREST Unbiased Galactic plane Imaging survey with the
Nobeyama 45-m telescope, a legacy project in the Nobeyama 45-m radio
telescope. Nobeyama Radio Observatory is a branch of the National Astronomical
Observatory of Japan, National Institutes of Natural Sciences.

##  Appendix A Data analysis

Selection criteria for AS events are the same as those introduced by Amenomori
et al. (2019), except for the maximum of the analyzed zenith-angle range, the
threshold energy, and the cut using the total number of muons recorded by the
MD array. The first condition is changed from $40^{\circ}$ to $50^{\circ}$ to
improve statistics because HESS J1849$-$000 culminates only at the meridian
zenith of $30{\fdg}1$ at the Tibet site. Our analysis using the Monte Carlo
(MC) simulation described in Appendix B finds that this extension increases
the photon statistics by $\simeq 30\%$ above $\simeq 30\,{\rm TeV}$, and
accordingly the threshold energy of the analysis is fixed at $25\,{\rm TeV}$.
Second, the MD cut condition is optimized for this work with the method
presented by Amenomori et al. (2022) using gamma-ray events generated with the
MC simulation and the background events from the experimental data. The
resultant cut requires events to satisfy $\Sigma N_{\mu}<2.8\times
10^{-4}(\Sigma\rho/{\rm m}^{-2})^{1.4}$ or $\Sigma N_{\mu}<0.4$ where $\Sigma
N_{\mu}$ denotes the total number of muons recorded with the MD array and
$\Sigma\rho$ is the sum of particle number densities recorded with the
scintillation detectors of the AS array. The MD cut can reject more than
$99.9\%$ of background CRs in the gamma-ray equivalent energy range above
$100\,{\rm TeV}$ while keeping $\simeq 80\%$ of gamma rays from HESS
J1849$-$000 in the same energy range. The energy and angular resolutions of
the AS array for $100\,{\rm TeV}$ gamma rays after the event selection are
evaluated as $\simeq 30\%$ and $0.28^{\circ}$ (50% containment), respectively.
After the event selection, $1.3\times 10^{7}$ events are left for later
analysis.

##  Appendix B Monte Carlo simulation

Using the air-shower simulation software Corsika v7.4000 (Heck et al., 1998),
a total number of $1.1\times 10^{8}$ gamma rays following a power-law spectrum
with an index of 2 are generated between $300\,{\rm GeV}<E<100\,{\rm PeV}$ and
injected into the atmosphere from a hypothetical point source at
$\delta=0^{\circ}$. The air-shower development in the atmosphere is simulated
until they reach the altitude of the Tibet AS array. The source extension,
spectral index, and flux normalization of HESS J1849$-$000 are appropriately
taken into account in the MC data analysis.

The air-shower events are randomly thrown over the circle with a $300\,{\rm
m}$ radius centered at the AS array, and the detector responses for these
events are simulated with Geant 4 v10.0 (Agostinelli et al., 2003). The energy
loss of shower particles in the scintillation detectors and the soil layer is
calculated. For the particles that reach MDs, their Cherenkov photon emission
in the water layer of MDs is simulated. The photons are reflected on the walls
and floor of each MD and collected by a 20-inch photomultiplier suspended
downward at the ceiling, where the number of photoelectrons is counted. The
simulation data are processed through our analysis pipeline in the same way as
the experimental data.

##  Appendix C Pointing systematics of the experiment

To estimate the pointing systematics of the experiment, events from the
direction of the Crab Nebula are analyzed in the zenith-angle range larger
than the meridian zenith of HESS J1849$-$000 ($30{\fdg}1$). The center of the
gamma-ray emission above $25\,{\rm TeV}$ from the direction of the nebula is
determined with the two-dimensional unbinned maximum likelihood analysis by
fitting an axisymmetric Gaussian to the distribution of events. The resultant
center is $(\alpha,\,\delta)=(83{\fdg}644\pm 0{\fdg}062,\,22{\fdg}030\pm
0{\fdg}053)$, deviated by $0{\fdg}010\pm 0{\fdg}057$ and $0{\fdg}015\pm
0{\fdg}053$ in real angle along the right ascension and the declination,
respectively, from the position of the Crab pulsar of
$(\alpha,\,\delta)=(83{\fdg}633,\,22{\fdg}015)$111http://simbad.u-strasbg.fr/simbad/sim-
id?Ident=Crab+Pulsar. The pointing systematics is estimated by summing the
apparent deviation and its statistical error in quadrature, which results in
$0{\fdg}058$ and $0{\fdg}055$ along the right ascension and the declination,
respectively.

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c\sqrt{n}\right]}{t}+\mathrm{Pr}\left[w_{max}\geq c\sqrt{n}\right].$ (12)

To bound $\mathbb{E}\left[K_{4}\mid w_{max}\leq c\sqrt{n}\right]$, we note
that a random weight $w_{i}$ fulfills

$\displaystyle\mathrm{Pr}\left[w_{i}\leq x\mid w_{max}\leq c\sqrt{n}\right]$
$\displaystyle=\frac{\mathrm{Pr}\left[\bigcap_{j}w_{j}\leq c\sqrt{n}\cap
w_{i}\leq x\right]}{\mathrm{Pr}\left[\bigcap_{j}w_{j}\leq c\sqrt{n}\right]}$
$\displaystyle=\frac{\mathrm{Pr}\left[w_{i}\leq
x\right]}{\mathrm{Pr}\left[w_{i}\leq c\sqrt{n}\right]}$

and since $\mathrm{Pr}\left[w_{i}\leq c\sqrt{n}\right]=1-o(1)$, we get that
the density of $w_{i}$ is $\rho_{w_{i}}(x)=c^{\prime}x^{-\beta}$ for some
$c^{\prime}\geq 0$ independent of $x$. Using that, we use Theorem 1.2 to bound

$\displaystyle\begin{split}&\mathrm{Pr}\left[U_{k}\text{ is clique }\mid
w_{k}\leq c\sqrt{n}\right]\leq\\\ &\hskip
28.45274pt\Theta(1)\int_{w_{0}}^{c\sqrt{n}}\cdots\int_{w_{0}}^{c\sqrt{n}}\prod_{1\leq
i<j\leq
k}\left(\frac{\kappa_{ij}}{n}\right)^{1-\varepsilon}w_{1}^{-\beta}\cdots
w_{k}^{-\beta}\mathrm{d}w_{1}\ldots\mathrm{d}w_{k}\end{split}$
$\displaystyle=\Theta(1)n^{-\binom{k}{2}(1-\varepsilon)}\left(\int_{w_{0}}^{c\sqrt{n}}w^{(k-1)(1-\varepsilon)-\beta}\mathrm{d}w\right)^{k}$

for some function $\varepsilon(n)=o(1)$. Because $k\geq 4$ and $\beta=3$, we
observe that the exponent $(k-1)(1-\varepsilon)-\beta$ is greater than $-1$
for sufficiently large $n$, and hence, the above integral evaluates to

$\displaystyle\mathrm{Pr}\left[U_{k}\text{ is clique }\mid w_{k}\leq
c\sqrt{n}\right]=c^{k((k-1)(1-\varepsilon)+1-\beta)}\Theta(1)n^{\frac{k}{2}(1-\beta)}.$

For $k=4$, we obtain

$\displaystyle\mathrm{Pr}\left[U_{k}\text{ is clique }\mid w_{k}\leq
c\sqrt{n}\right]\leq\Theta(1)c^{4}n^{-4}$

and accordingly

$\displaystyle\mathbb{E}\left[K_{4}\mid w_{max}\leq
c\sqrt{n}\right]=\binom{n}{4}\mathrm{Pr}\left[U_{k}\text{ is clique }\mid
w_{k}\leq c\sqrt{n}\right]\leq\Theta(1)c^{4}.$

By (4.23) and (11), this implies

$\displaystyle\mathrm{Pr}\left[K_{4}\geq
t\right]=\Theta(1)c^{4}t^{-1}+\mathcal{O}(1)c^{-2}.$

Setting $c=t^{1/5}$ yields $\mathrm{Pr}\left[K_{4}\geq
t\right]=\mathcal{O}(1)(t^{-1/5}+t^{-2/5})$ and shows that the probability
that $K_{4}\geq t$ can be made arbitrarily small by increasing $t$. To bound
the clique number, we note that the existence of a clique of size $k$ implies
the existence of $\binom{k}{4}$ cliques of size $4$ and so,

$\displaystyle\mathrm{Pr}\left[\omega(G)\geq
k\right]\leq\mathrm{Pr}\left[K_{4}\geq\binom{k}{4}\right],$

which can be made arbitrarily small by choosing $k$ large enough. Hence the
probability that the clique number grows as any superconstant function
$f(n)=\omega(1)$ is in $o(1)$, which shows that the clique number is in
$\mathcal{O}(1)$ a.a.s.

Finally, we deal with the case where $\beta>3$, where we show that, in this
case, there are no cliques of size $4$ or larger a.a.s.

######  Theorem 4.24.

Let $G=\operatorname{GIRG}(n,\beta,w_{0},d)$ be a standard GIRG with $\beta>3$
and $d=\omega(\log(n))$. Then, $\mathbb{E}[K_{k}]=o(1)$ for $k\geq 4$ and,
thus, $\omega(G)\leq 3$ a.a.s.

######  Proof 4.25.

We use a similar strategy as in the above paragraph. In analogy to Equation
10, we now have

$\displaystyle\mathrm{Pr}\left[w_{max}\geq
n^{\alpha}\right]\leq\Theta(1)n^{1-(1-\beta)\alpha},$

which is $o(1)$ if $\alpha>\frac{1}{\beta-1}$. For some $\alpha$ in the range
$1/2>\alpha>\frac{1}{\beta-1}$, we get

$\displaystyle\begin{split}&\mathrm{Pr}\left[U_{k}\text{ is clique }\mid
w_{k}\leq n^{\alpha}\right]\leq\\\ &\hskip
28.45274pt\Theta(1)\int_{w_{0}}^{n^{\alpha}}\cdots\int_{w_{0}}^{n^{\alpha}}\prod_{1\leq
i<j\leq
k}\left(\frac{\kappa_{ij}}{n}\right)^{1-\varepsilon}w_{1}^{-\beta}\cdots
w_{k}^{-\beta}\mathrm{d}w_{1}\ldots\mathrm{d}w_{k}\end{split}$
$\displaystyle=\Theta(1)n^{-\binom{k}{2}(1-\varepsilon)}\left(\int_{w_{0}}^{n^{\alpha}}w^{(k-1)(1-\varepsilon)-\beta}\mathrm{d}w\right)^{k}$
$\displaystyle=\Theta(1)n^{\left(\alpha-\frac{1}{2}\right)k(k-1)(1-\varepsilon)+\alpha
k(1-\beta)}=o\left(n^{\alpha k(1-\beta)}\right).$

Accordingly,

$\displaystyle\mathbb{E}\left[K_{4}\mid w_{max}\leq
c\sqrt{n}\right]=\binom{n}{4}\mathrm{Pr}\left[U_{k}\text{ is clique }\mid
w_{k}\leq c\sqrt{n}\right]=\Theta(1)n^{4+4\alpha(1-\beta)}=o(1).$

as $\alpha(1-\beta)<-1$. By Markov’s inequality this implies

$\displaystyle\mathrm{Pr}\left[K_{4}\geq 1\right]\leq\mathbb{E}\left[K_{4}\mid
w_{max}\leq n^{\alpha}\right]+\mathrm{Pr}\left[w_{max}\geq
n^{\alpha}\right]=o(1)$

as desired.

##  5 Asymptotic Equivalence with IRGs

We continue by investigating the infinite-dimensional limit of our model,
i.e., the case where $n$ is fixed and $d$ goes to infinity. We prove that in
this situation, the GIRG model becomes in a strong sense equivalent to the
non-geometric IRG model. That is, we prove that the total variation distance
of the distribution over all possible graphs with $n$ vertices goes to $0$ as
$d\rightarrow\infty$. We prove the following

See 1.1

We split the proof of this theorem by considering the case of an $L_{p}$-norm
for $1\leq p<\infty$ and the case of $L_{\infty}$-norm separately. Our
investigations in Section 1.4 further show why RGGs on the torus become
equivalent to non-geometric models as $d$ tends to infinity and why this is
not the case if we use the hypercube instead as previously observed in [18,
13].

###  5.1 Equivalence for $L_{p}$-norms with $1\leq p<\infty$

Our argument builds upon a multivariate central-limit-theorem similar to the
one used by Devroye et al. who establish a similar statement for SRGGs [17] .

Before starting the proof, we introduce some necessary auxiliary statements.
Our argumentation builds upon the following Berry-Esseen theorem introduced in
[36].

######  Theorem 5.1 (Theorem 1.1 in [36]).

Let $Z_{1},\ldots,Z_{d}$ be independent zero-mean random variables taking
values in $\mathbb{R}^{m}$. Let further $Z\coloneqq\sum_{i=1}^{d}Z_{i}$ and
assume that the covariance matrix of $Z$ is the identity matrix. Let
$X\in\mathbb{R}^{m}$ be a random variable following the standard $m$-variate
normal distribution $\mathcal{N}(0,I)$. Then for any convex set
$A\subseteq\mathbb{R}^{m}$, we have

$\left|\mathrm{Pr}\left[Z\in A\right]-\mathrm{Pr}\left[X\in
A\right]\right|\leq(42d^{1/4}+16)\sum_{i=1}^{d}\mathbb{E}\left[\left\|Z_{i}\right\|^{3}\right]$

where $\left\|Z_{i}\right\|$ is the $L_{2}$-norm of $Z_{i}$.

This illustrates that for $d\rightarrow\infty$, the (random) distance between
two vertices behaves like a Gaussian random variable. Throughout this section,
we use the following notation. For any $u,v\in V$, we define
$\Delta_{(u,v)}\in\mathbb{R}^{d}$ as the component-wise distance of $u$ and
$v$, i.e.,
$\Delta_{(u,v),i}\coloneqq\left|\mathbf{x}_{ui}-\mathbf{x}_{vi}\right|_{C}=\min\\{\left|\mathbf{x}_{ui}-\mathbf{x}_{vi}\right|,1-\left|\mathbf{x}_{ui}-\mathbf{x}_{vi}\right|\\}$.
We may express

$\displaystyle\mathrm{Pr}\left[u\sim
v\right]=\mathrm{Pr}\left[\left\|\Delta_{(u,v)}\right\|_{p}\leq
t_{uv}\right]=\mathrm{Pr}\left[\sum_{i=1}^{d}\Delta_{(u,v),i}^{p}\leq
t_{uv}^{p}\right]$

and we further note that $\Delta_{(u,v),i}$ and $\Delta_{(u,v),j}$ are i.i.d.
random variables. Define

$\displaystyle\mu\coloneqq\mathbb{E}\left[\Delta_{(u,v),i}^{p}\right]\hskip
14.22636pt\text{and}\hskip
14.22636pt\sigma^{2}\coloneqq\text{Var}\left[\Delta_{(u,v),i}^{p}\right].$

and let the random variable $Z_{(u,v),i}$ be defined as

$\displaystyle
Z_{(u,v),i}\coloneqq\frac{\Delta_{(u,v),i}^{p}-\mu}{\sqrt{d}\sigma}.$

Now define $Z_{(u,v)}\coloneqq\sum_{i=1}^{d}Z_{(u,v),i}$ and observe that this
allows us to express

$\displaystyle\mathrm{Pr}\left[u\sim
v\right]=\mathrm{Pr}\left[\sum_{i=1}^{d}\Delta_{(u,v),i}^{p}\leq
t_{uv}^{p}\right]=\mathrm{Pr}\left[Z_{(u,v)}\leq\frac{t_{uv}^{p}-d\mu}{\sqrt{d}\sigma}\right].$

Working with $Z_{(u,v)}$ instead of $\Delta_{(u,v)}$ has the advantage that we
have $\mathbb{E}\left[Z_{(u,v)}\right]=0$ and

$\displaystyle\text{Var}\left[Z_{(u,v)}\right]=\sum_{i=1}^{d}\text{Var}\left[Z_{(u,v),i}\right]=\frac{1}{d\sigma^{2}}\sum_{i=1}^{d}\text{Var}\left[\Delta_{(u,v),i}^{p}\right]=1.$

These properties are useful when applying Theorem 5.1. Now recall that
$t_{uv}$ is defined such that the marginal connection probability
$\mathrm{Pr}\left[u\sim v\right]$ is equal to $\min\left\\{1,\frac{\lambda
w_{u}w_{v}}{n}\right\\}$, which is required in order to ensure that
$\mathbb{E}\left[\deg(v)\right]\propto w_{v}$ for all $v$. We use this to
establish the following lemma describing the asymptotic behavior of the
threshold $t_{uv}$.

######  Lemma 5.2.

Let $G=\operatorname{GIRG}(n,\beta,w_{0},d)=(V,E)$ be a standard GIRG with
$L_{p}$-norm for $p\in[1,\infty)$. Denote by $\Phi$ the cumulative density
function of the standard Gaussian distribution, i.e.,
$\Phi(x)=\sqrt{2\pi}\int_{-\infty}^{x}e^{-t^{2}/2}dt$. Then for any $u,v\in V$
with $u\neq v$ and $\frac{\lambda w_{u}w_{v}}{n}<1$, we have

$\lim_{d\rightarrow\infty}\frac{t_{uv}^{p}-d\mu}{\sqrt{d}\sigma}=\Phi^{-1}\left(\frac{\lambda
w_{u}w_{v}}{n}\right).$

######  Proof 5.3.

Let $u,v$ be fixed. In the remainder of this proof, we abbreviate $Z_{(u,v)}$
with $Z$, and $Z_{(u,v),i}$ with $Z_{i}$. For every $c\in\mathbb{R}$, define
the set $A_{c}=\left\\{x\in\mathbb{R}\mid x\leq c\right\\}$. Let further
$X\sim\mathcal{N}(0,1)$ be a standard Gaussian random variable. We get from
Theorem 5.1 that

$\displaystyle\left|\mathrm{Pr}\left[Z\leq c\right]-\mathrm{Pr}\left[X\leq
c\right]\right|$
$\displaystyle\leq(42d^{1/4}+16)\sum_{i=1}^{d}\mathbb{E}\left[\left|Z_{i}\right|^{3}\right]$
$\displaystyle=\frac{42d^{1/4}+16}{d^{3/2}\sigma^{3}}\sum_{i=1}^{d}\mathbb{E}\left[\left|\Delta_{(u,v),i}^{p}-\mu\right|^{3}\right]$
$\displaystyle\leq\frac{42d^{1/4}+16}{d^{1/2}\sigma^{3}}$ (13)
$\displaystyle=o_{d}(1)$ (14)

because $\Delta_{(u,v),i}^{p}-\mu\in[-1,1]$, which shows that $Z$ converges to
a standard Gaussian random variable as $d\rightarrow\infty$. In particular,
this statement is true for $c=\Phi^{-1}\left(\frac{\lambda
w_{u}w_{v}}{n}\right)$. At the same time,

$\displaystyle\mathrm{Pr}\left[X\leq\Phi^{-1}\left(\frac{\lambda
w_{u}w_{v}}{n}\right)\right]=\frac{\lambda
w_{u}w_{v}}{n}=\mathrm{Pr}\left[Z\leq\frac{t_{uv}^{p}-d\mu}{\sqrt{d}\sigma}\right],$

where the second step follows by the definition of $t_{uv}$ and $Z$. Hence, by
(5.3),

$\displaystyle\lim_{d\rightarrow\infty}\left|\mathrm{Pr}\left[Z\leq\Phi^{-1}\left(\frac{\lambda
w_{u}w_{v}}{n}\right)\right]-\mathrm{Pr}\left[Z\leq\frac{t_{uv}^{p}-d\mu}{\sqrt{d}\sigma}\right]\right|=0.$

Since the function $f(c)=\mathrm{Pr}\left[Z\leq c\right]$ converges to the
cumulative density function $\Phi$ of the standard Gaussian distribution and
since this function is continuous and strictly monotonically increasing, we
infer that

$\displaystyle\lim_{d\rightarrow\infty}\frac{t_{uv}^{p}-d\mu}{\sqrt{d}\sigma}=\Phi^{-1}\left(\frac{\lambda
w_{u}w_{v}}{n}\right).$

With this, we prove the main theorem of this section.

######  Proof 5.4 (Proof of Theorem 1.1 for $p\in[1,\infty)$).

As $n$ is fixed, the set $\mathcal{G}(n)$ is finite and so it suffices to show
that for all $H\in\mathcal{G}(n)$, we have

$\lim_{d\rightarrow\infty}\mathrm{Pr}\left[G_{\text{GIRG}}=H\right]=\mathrm{Pr}\left[G_{\text{IRG}}=H\right].$
(15)

First of all, we note that for any $u,v\in V$ with $\frac{\lambda
w_{u}w_{v}}{n}\geq 1$, $u$ and $v$ are guaranteed to be connected in both
$G_{\text{GIRG}}$ and $G_{\text{IRG}}$. Hence, for every $H\in\mathcal{G}(n)$
in which $u$ and $v$ are not connected, we get
$\mathrm{Pr}\left[G_{\text{GIRG}}=H\right]=\mathrm{Pr}\left[G_{\text{IRG}}=H\right]=0$.
For this reason, it suffices to show Equation 15 for all $H\in\mathcal{G}(n)$
in which all $u,v\in V$ with $u\neq v$ and $\frac{\lambda w_{u}w_{v}}{n}\geq
1$ are connected.

Let $H$ be an arbitrary but fixed such graph. We define the set
$\mathcal{Q}=\\{(u,v)\mid 1\leq u<v\leq n,\frac{\lambda w_{u}w_{v}}{n}<1\\}$,
which contains all pairs of vertices that are not connected with probability
$1$. For any event $\mathbf{E}$, we denote by $\mathds{1}(\mathbf{E})$ the
random variable that is $1$ if $\mathbf{E}$ occurs and $0$ otherwise.
Similarly, we denote by $H_{(u,v)}$ an indicator variable that is $1$ if the
edge $\\{u,v\\}$ is present in $H$ and $0$ otherwise. Furthermore, for every
$(u,v)\in\mathcal{Q}$, we define $\mathcal{N}_{(u,v)}$ to be an independent
standard Gaussian random variable. Then,

$\mathrm{Pr}\left[G_{\text{IRG}}=H\right]=\mathrm{Pr}\left[\bigcap_{(u,v)\in\mathcal{Q}}\left(\mathds{1}\left(\mathcal{N}_{(u,v)}\leq\Phi^{-1}\left(\frac{\lambda
w_{u}w_{v}}{n}\right)\right)=H_{(u,v)}\right)\right].$

Furthermore, recall the definition of $Z_{(u,v)}$ and observe

$\mathrm{Pr}\left[G_{\text{GIRG}}=H\right]=\mathrm{Pr}\left[\bigcap_{(u,v)\in\mathcal{Q}}\left(\mathds{1}\left(Z_{(u,v)}\leq\frac{t_{uv}^{p}-d\mu}{\sqrt{d}\sigma}\right)=H_{(u,v)}\right)\right].$

In addition, we define the random graph $\tilde{G}$ in which all $u\neq v\in
V$ with $\frac{\lambda w_{u}w_{v}}{n}\geq 1$ are guaranteed to be connected,
and in which for every $(u,v)\in\mathcal{Q}$, the edge ${u,v}$ present if and
only if $Z_{(u,v)}\leq\Phi^{-1}\left(\frac{\lambda w_{u}w_{v}}{n}\right)$.
Accordingly,

$\mathrm{Pr}\left[\tilde{G}=H\right]=\mathrm{Pr}\left[\bigcap_{(u,v)\in\mathcal{Q}}\left(\mathds{1}\left(Z_{(u,v)}\leq\Phi^{-1}\left(\frac{\lambda
w_{u}w_{v}}{n}\right)\right)=H_{(u,v)}\right)\right].$

From Lemma 5.2, we get that

$\displaystyle\lim_{d\rightarrow\infty}\frac{t_{uv}^{p}-d\mu}{\sqrt{d}\sigma}=\Phi^{-1}\left(\frac{\lambda
w_{u}w_{v}}{N}\right)$

and so,

$\lim_{d\rightarrow\infty}\left|\mathrm{Pr}\left[G_{\text{GIRG}}=H\right]-\mathrm{Pr}\left[\tilde{G}=H\right]\right|=0.$

It therefore only remains to show
$\lim_{d\rightarrow\infty}\mathrm{Pr}\left[\tilde{G}=H\right]=\mathrm{Pr}\left[G_{\text{IRG}}=H\right]$.

For this, we let $m=|\mathcal{Q}|$ and we define the random vector
$Z_{i}\in\mathbb{R}^{m}$ that has the random variables $Z_{(u,v),i}$ as its
components for all $(u,v)\in\mathcal{Q}$. We further define
$Z\coloneqq\sum_{i=1}^{d}Z_{i}$. We use Theorem 5.1 and define the set
$A\subseteq\mathbb{R}^{m}$ such that

$\displaystyle x\in
A\Leftrightarrow\forall(u,v)\in\mathcal{Q}:\begin{cases}x_{(u,v)}\leq\Phi^{-1}\left(\frac{\lambda
w_{u}w_{v}}{n}\right)&\text{if }H_{(u,v)}=1\\\
x_{(u,v)}>\Phi^{-1}\left(\frac{\lambda
w_{u}w_{v}}{n}\right)&\text{otherwise.}\end{cases}$

It is easy to observe that $A$ is convex. We get
$\mathrm{Pr}\left[\tilde{G}=H\right]=\mathrm{Pr}\left[Z\in A\right]$ and
$\mathrm{Pr}\left[G_{\text{IRG}}=H\right]=\mathrm{Pr}\left[X\in A\right]$
where $X$ is a random variable from the standard $m$-variate normal
distribution.

We further note that for all $(u,v)\in\mathcal{Q}$, the random variables
$\Delta_{(u,v),1},\ldots,\Delta_{(u,v),d}$ are independent, which implies that
$Z_{1},\ldots,Z_{d}$ are independent as well. Furthermore, they have
expectation $0$ and for all $1\leq i,j\leq d,(u,v),(s,t)\in\mathcal{Q}$ with
$(u,v)\neq(s,t)$, the random variables $\Delta_{(u,v),i}$ and $\Delta_{st,j}$
are independent, even if $i=j$ and $\\{u,v\\}\cap\\{s,t\\}\neq\emptyset$
(because the torus is a homogeneous space). This implies that also
$Z_{(u,v),i}$ and $Z_{(s,t),j}$ as well as $Z_{(u,v)}$ and $Z_{(s,t)}$ are
independent. Hence, $\text{Cov}\left[Z_{(u,v)},Z_{(s,t)}\right]=0$.
Accordingly, the covariance matrix of $Z$ is the identity matrix. Thus,
Theorem 5.1 implies

$\displaystyle\left|\mathrm{Pr}\left[Z\in A\right]-\mathrm{Pr}\left[X\in
A\right]\right|$
$\displaystyle\leq(42d^{1/4}+16)\sum_{i=1}^{d}\mathbb{E}\left[\left\|Z_{i}\right\|^{3}\right]$
$\displaystyle=(42d^{1/4}+16)\sum_{i=1}^{d}\mathbb{E}\left[\left(\sum_{(u,v)\in\mathcal{Q}}Z_{(u,v),i}^{2}\right)^{3/2}\right]$
$\displaystyle=(42d^{1/4}+16)\sum_{i=1}^{d}\frac{1}{d^{3/2}\sigma^{3}}\mathbb{E}\left[\left(\sum_{(u,v)\in\mathcal{Q}}\left(\Delta_{(u,v),i}^{p}-\mu\right)^{2}\right)^{3/2}\right]$
$\displaystyle\leq(42d^{1/4}+16)\sum_{i=1}^{d}\frac{m^{3/2}}{d^{3/2}\sigma^{3}}$
$\displaystyle=(42d^{1/4}+16)\frac{dm^{3/2}}{d^{3/2}\sigma^{3}}$
$\displaystyle=\frac{m^{3/2}}{\sigma^{3}}\left(\frac{42}{d^{1/4}}+\frac{16}{d^{1/2}}\right)=o_{d}(1),$

as $\Delta_{(u,v),i}^{p}-\mu\in[-1,1]$ and $m\leq\binom{n}{2}$. This shows

$\displaystyle\lim_{d\rightarrow\infty}\mathrm{Pr}\left[\tilde{G}=H\right]=\mathrm{Pr}\left[G_{\text{IRG}}=H\right],$

as desired.

As mentioned before, the above result helps in getting an intuition for how
the choice of the underlying ground space of geometric random graphs affects
the impact of an increasing dimensionality. Recall that RGGs on the hypercube
do _not_ converge to Erdős–Rényi graphs as $n$ is fixed and
$d\rightarrow\infty$ [13, 18]. However, our results imply that they do when
choosing the torus as ground space. These apparent disagreements are despite
the fact that we apply the central limit theorem similarly.

As discussed before, the above proof relies on the fact that, for independent
zero-mean variables $Z_{1},\dots,Z_{d}$, the covariance matrix of the random
vector $Z=\sum_{i=1}^{d}Z_{i}$ is the identity matrix. This is due to the fact
that the torus is a homogeneous space, implying that the probability measure
of a ball of radius $r$ is the same, regardless of where this ball is
centered. This makes the random variables $Z_{(u,v)}$ and $Z_{(u,s)}$
independent. As a result their covariance is $0$ although both “depend” on the
position of $u$.

For the hypercube, this is not the case. Although the distance of two vertices
can analogously be defined as a sum of independent, zero-mean random vectors
over all dimensions just like we do above, (the only difference being that
$\Delta_{(u,v),i}$ is now the distance between $u,v$ in dimension $i$ in the
hypercube, leading to different values of $\mu$ and $\sigma^{2}$) the random
variables $Z_{(u,v)}$ and $Z_{(u,s)}$ do _not_ have a covariance of $0$.

In fact, one can verify that for every $1\leq i\leq d$, there is a slightly
positive covariance between $\Delta_{(u,v),i}$ and $\Delta_{(u,s),i}$ (equal
to $1/180$). This transfers to a covariance between $Z_{(u,v)}$ and
$Z_{(u,s)}$, which stays constant as $d$ grows, since

$\displaystyle\text{Cov}[Z_{(u,v)},Z_{(u,s)}]$
$\displaystyle=\mathbb{E}\left[Z_{(u,v)}\cdot Z_{(u,s)}\right]$
$\displaystyle=\sum_{i=1}^{d}\sum_{j=1}^{d}\mathbb{E}\left[Z_{(u,v),i}\cdot
Z_{(u,s),j}\right]$
$\displaystyle=\frac{1}{d\sigma^{2}}\sum_{i=1}^{d}\text{Cov}[\Delta_{(u,v),i}^{p}\Delta_{(u,s),i}^{p}]$
$\displaystyle=\frac{\text{Cov}[\Delta_{(u,v),1}^{p}\Delta_{(u,s),1}^{p}]}{\sigma^{2}},$

where we used that $\mathbb{E}\left[Z_{(u,v),i}\cdot Z_{(u,s),j}\right]=0$ if
$i\neq j$. Accordingly, the covariance matrix $\Sigma$ of $Z$ is not the
identity matrix. Nevertheless, our proof from the previous section still
applies if we replace $Z$ by $Y=\sum_{i=1}^{d}\Sigma^{-1}Z_{i}$. Now $Y$ is
the sum of independent random vectors and has the identity matrix as its
covariance matrix, so Theorem 5.1 remains applicable. Furthermore,
$\mathbb{E}\left[||\Sigma^{-1}Z_{i}||^{3}\right]$ is still proportional to
$d^{-3/2}$ and thus remains bounded such that $Y$ converges to a standard
$m$-dimensional normal vector. This implies that $Z$ converges to a random
vector from $\mathcal{N}(0,\Sigma)$ showing that RGGs on the hypercube
converge to a non-geometric model where the probability that any fixed graph
is sampled can be described – like above – as the probability that
$Z\sim\mathcal{N}(0,\Sigma)$ falls into the convex set $A$. In this model,
however, the edges are not drawn independently, as $\Sigma$ is not the
identity matrix. In fact, for any three vertices $s,u,v$, the components
$Z_{(u,v)}$ and $Z_{(u,s)}$ are slightly positively correlated, so there is a
slightly higher probability that $s,u,v$ form a triangle than in a
corresponding Erdős–Rényi graph. This leads to a higher tendency to form
cliques, which is in accordance with the observations from Erba et al. [18].

###  5.2 Asymptotic Equivalence for $L_{\infty}$-norm

In this section, we prove that our model also loses its geometry if
$L_{\infty}$-norm is used. We use a different technique to prove this theorem,
as the $L_{\infty}$-distance between two vertices is not a sum of independent
random variables anymore and central limit theorems do not apply anymore.
Instead our argument builds upon the bounds we establish in Section 4.1.

######  Proof 5.5 (Proof of Theorem 1.1 for $L_{\infty}$-norm).

We show that for all $H\in\mathcal{G}(n)$,

$\displaystyle\lim_{d\rightarrow\infty}\mathrm{Pr}\left[G_{\text{GIRG}}=H\right]=\mathrm{Pr}\left[G_{\text{IRG}}=g\right].$
(16)

We start by establishing a way to compute $\mathrm{Pr}\left[G=H\right]$ for
any random variable $G$ representing a distribution over all graphs in
$\mathcal{G}(n)$. For this, we denote by $E(H)$ the set of edges of a graph
$H=(V,E)\in\mathcal{G}(n)$. We further let $\binom{V}{2}$ be the set of all
possible edges on the vertex set $V$. Now, for any $H\in\mathcal{G}$, we have

$\displaystyle\mathrm{Pr}\left[G=H\right]=\mathrm{Pr}\left[E(G)\supseteq
E(H)\right]-\sum_{E(H)\subset\mathcal{A}\subseteq\binom{V}{2}}\mathrm{Pr}\left[E(G)=\mathcal{A}\right].$

That is, we may express the probability that $G$ is sampled as the probability
that a supergraph of $G$ is sampled minus the probability that a any proper
supergraph of $G$ is sampled. Now, for any
$E(H)\subset\mathcal{A}\subseteq\binom{V}{2}$, the probability
$\mathrm{Pr}\left[E(G)=\mathcal{A}\right]$ is the probability that $G$ is a
specific graph with at least $|E(H)|+1$ edges. Now, we may repeatedly
substitute terms of the form $\mathrm{Pr}\left[E(G)=\mathcal{A}\right]$ in the
same way until we have an (alternating) sum consisting only of terms that have
the form $\mathrm{Pr}\left[E(G)\supseteq\mathcal{A}\right]$ for some
$E(H)\subset\mathcal{A}\subseteq\binom{V}{2}$. That is, we may calculate the
probability $\mathrm{Pr}\left[G=H\right]$ even if we only know
$\mathrm{Pr}\left[E(G)\supseteq\mathcal{A}\right]$ for any
$\mathcal{A}\subseteq\binom{V}{2}$.

As $n$ is fixed, in order to prove our statement in Equation 16, it suffices
to prove that, for each $\mathcal{A}\subseteq\binom{V}{2}$, we have

$\displaystyle\lim_{d\rightarrow\infty}\mathrm{Pr}\left[E(G_{\text{GIRG}})\supseteq\mathcal{A}\right]=\mathrm{Pr}\left[E(G_{\text{IRG}})\supseteq\mathcal{A}\right]=\prod_{\\{i,j\\}\in\mathcal{A}}\frac{\kappa_{ij}}{n}.$

Using Theorem 1.2, we get that

$\displaystyle\mathrm{Pr}\left[E(G_{\text{GIRG}})\supseteq\mathcal{A}\right]=\prod_{\\{i,j\\}\in\mathcal{A}}\left(\frac{\kappa_{ij}}{n}\right)^{1\mp\mathcal{O}_{d}\left(\frac{\ln(n)}{d}\right)}.$

For $d\rightarrow\infty$, this clearly converges to
$\mathrm{Pr}\left[E(G_{\text{IRG}})\supseteq\mathcal{A}\right]$ and the proof
is finished.

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