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20921-400 Rio de Janeiro - RJ, Brazil
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33institutetext: LESIA, Observatoire de Paris, Université PSL, Sorbonne
Université, Université de Paris, CNRS, 92190 Meudon, France 44institutetext:
Universidade Federal do Rio de Janeiro - Observatório do Valongo, Ladeira
Pedro Antônio 43, CEP 20.080-090, Rio de Janeiro - RJ, Brazil 55institutetext:
Federal University of Technology - Paraná (UTFPR-Curitiba), Rua Sete de
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Space Institute, UCF, 12354 Research Parkway, Partnership 1 building, Room
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University of Namur, Rue de Bruxelles 61, 5000 Namur, Belgium
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CNRS, Laboratoire Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice Cedex
4, France 1111institutetext: Space Telescope Science Institute, Baltimore,
Maryland, USA 1212institutetext: Aix Marseille Université, CNRS, CNES, LAM
(Laboratoire d’Astrophysique de Marseille) UMR 7326, 13388, Marseille, France
1313institutetext: Canada-France-Hawaii Telescope, 65-1238 Mamalahoa Highway,
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IMCCE, Observatoire de Paris, PSL Research University, CNRS, Sorbonne
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Technologies, 07058 Antalya, Turkey 1717institutetext: TÜBİTAK National
Observatory, Akdeniz University Campus, 07058 Antalya, Turkey
1818institutetext: Institute of Physics, Federal University of Uberlândia,
Uberlândia-MG, Brazil 1919institutetext: UNESP - São Paulo State University,
Grupo de Dinâmica Orbital e Planetologia, CEP 12516-410, Guaratinguetá, SP,
Brazil 2020institutetext: Universidade Estadual de Ponta Grossa, O.A. - DEGEO,
Ponta Grossa (PR), Brazil 2121institutetext: Internationale Amateursternwarte
(IAS) e. V., Mittelstr. 6, 15749 Mittenwalde, Germany 2222institutetext:
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Physique and Observatoire du Mont-Mégantic, Université de Montréal, C.P. 6128,
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Institut Trottier de Recherche sur les exoplanètes, Université de Montréal
2525institutetext: Harvard-Smithsonian Center for Astrophysics, 60 Garden
Street, Cambridge, MA 02138, USA 2626institutetext: Gemini Observatory/NSF’s
NOIRLab, Hilo, Hawaii, USA 2727institutetext: Cabrillo College Astronomy, USA
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Atmospheric Research Center, Graduate School of Science, Tohoku University,
6-3 Aramaki -aza-aoba, Aoba-ku, Sendai 980-8578, Japan 3939institutetext:
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allée Marcel Leclerc, bâtiment B, 13008 Marseille, France 4141institutetext:
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Tokyo, Japan 4949institutetext: School of Medicine Department of Basic
Sciences University of Occupational and Environmental Health, Japan 1-1
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Observatory

# The two rings of (50000) Quaoar

C. L. Pereira 1122<EMAIL_ADDRESS>B. Sicardy 33 B. E. Morgado 441122
F. Braga-Ribas 551122 E. Fernández-Valenzuela 6677 D. Souami Fulbright
Visiting Scholar (2022 - 2023) at University of California, Berkeley3388991010
B. J. Holler 1111 R. C. Boufleur 1122 G. Margoti 55 M. Assafin 4422 J. L.
Ortiz 77 P. Santos-Sanz 77 B. Epinat 12121313 P. Kervella 33 J. Desmars
14141515 R. Vieira-Martins 1122 Y. Kilic 16161717 A. R. Gomes-Júnior
1818191922 J. I. B. Camargo 1122 M. Emilio 20201155 M. Vara-Lubiano 77 M.
Kretlow 7721212222 L. Albert 23232424 C. Alcock 2525 J. G. Ball 2626 K. Bender
2727 M. W. Buie 2828 K. Butterfield 2929 M. Camarca 3030 J. H. Castro-Chacón
3131 R. Dunford 2929 R. S. Fisher 3232 D. Gamble 29293333 J. C. Geary 2525 C.
L. Gnilka 3434 K. D. Green 3535 Z. D. Hartman 2626 C-K. Huang 3636 H.
Januszewski 1313 J. Johnston 3737 M. Kagitani 3838 R. Kamin 2929 J. J.
Kavelaars 3939 J. M. Keller 3737 K. R. de Kleer 3030 M. J. Lehner 3636 A.
Luken 3232 F. Marchis 40404141 T. Marlin 3030 K. McGregor 4242 V. Nikitin
29294343 R. Nolthenius 2727 C. Patrick 2929 S. Redfield 4242 A. W. Rengstorf
4444 M. Reyes-Ruiz 3131 T. Seccull 2626 M. F. Skrutskie 4545 A. B. Smith 2626
M. Sproul 4646 A. W. Stephens 2626 A. Szentgyorgyi 2525 S. Sánchez-Sanjuán
3131 E. Tatsumi 47474848 A. Verbiscer 4545 S-Y. Wang 3636 F. Yoshida 49495050
R. Young 5151 Z-W. Zhang 3636

( Received 10 March 2023 / Accepted 17 April 2023 )

###### Abstract

Context.  Quaoar is a classical trans-Neptunian object (TNO) with an area-
equivalent diameter of 1100 km and an orbital semi-major axis of 43.3
astronomical units. Based on stellar occultations observed between 2018 and
2021, an inhomogeneous ring (Q1R, i.e., Quaoar’s first ring) has been detected
around this body.

Aims.  A new stellar occultation by Quaoar was observed on August 9, 2022,
with the aim of improving Quaoar’s shape models and the physical parameters of
Q1R, while searching for additional material around the body.

Methods.  The occultation provided nine effective chords across Quaoar,
pinning down its size, shape, and astrometric position. Large facilities, such
as Gemini North and the Canada-France-Hawaii Telescope (CFHT), were used to
obtain high acquisition rates and signal-to-noise ratios. The light curves
were also used to characterize the Q1R ring (radial profiles and orbital
elements).

Results.  Quaoar’s elliptical fit to the occultation chords yields the limb
with an apparent semi-major axis of $579.5\pm 4.0$ km, apparent oblateness of
$0.12\pm 0.01$, and area-equivalent radius of $543\pm 2$ km. Quaoar’s limb
orientation is consistent with Q1R and Weywot orbiting in Quaoar’s equatorial
plane. The orbital radius of Q1R is refined to a value of $4,057\pm 6$ km. The
radial opacity profile of the more opaque ring profile follows a Lorentzian
shape that extends over 60 km, with a full width at half maximum (FWHM) of
$\sim$5 km and a peak normal optical depth of 0.4. Besides the secondary
events related to the already reported rings, new secondary events detected
during the August 2022 occultation in three different data sets are consistent
with another ring around Quaoar with a radius of $2,520\pm 20$ km, assuming
the ring is circular and co-planar with Q1R. This new ring has a typical width
of 10 km and a normal optical depth of $\sim$0.004. Just as Q1R, it also lies
outside Quaoar’s classical Roche limit.

###### Key Words.:

methods: data analysis – methods: observational – techniques: photometric –
Kuiper belt objects: individual: Quaoar – Planets and satellites: rings

##  1 Introduction

In the last decade, three ring systems have been discovered around minor
bodies in the outer Solar System: the Centaur Chariklo (Braga-Ribas et al.
2014), the dwarf planet Haumea (Ortiz et al. 2017), and the Trans-Neptunian
Object (TNO) (50000) Quaoar (Morgado et al. 2023). Dense material has also
been detected around the Centaur Chiron (Ruprecht et al. 2015; Ortiz et al.
2015; Sickafoose et al. 2020). However, the nature of this material, namely,
whether it is a permanent or transient ring or a dust shell, is still a matter
of debate.

Quaoar’s ring, referred to as Q1R hereafter, was detected during several
stellar occultations observed between 2018 and 2021 (Morgado et al. 2023). Q1R
has a radius of about 4,100 km with significant azimuthal variations in the
optical depth, ranging between 0.004 and 0.1-0.7, and in width, ranging from 5
km to 300 km. Like Chariklo’s and Haumea’s rings, Quaoar’s Q1R ring orbits
close to the 1/3 spin-orbit resonance (SOR) with the central body, suggesting
a link between this resonance and the ring (Salo et al. 2021; Sicardy et al.
2021; Morgado et al. 2023). Meanwhile, a unique property of Q1R is its
location, which is far outside Quaoar’s classical Roche limit. This limit is
estimated to be at 1,780 km from the body center, assuming particles with a
bulk density of $\rho=0.4\,\rm{g}\,\rm{cm}^{-3}$. Outside the Roche limit,
rings should accrete into satellites over timescales of less than 100 years
(Kokubo et al. 2000; Takeda & Ida 2001). However, collisions more elastic than
previously considered for Saturn’s ring may maintain a ring unaccreted at
distances greater than the Roche limit (Morgado et al. 2023). The 6:1 mean-
motion resonance (MMR) of Quaoar and its satellite, Weywot, may contribute to
the confinement of the ring since the satellite’s eccentricity creates an
equilibrium region capable of concentrating ring material in an arc over a
longitude interval (Morgado et al. 2023). The confinement of the rings may
also be due to the presence of putative ”shepherd” satellites.

This letter presents observations of a stellar occultation by Quaoar that
occurred on August 9, 2022. This event was observed in Hawaii, the continental
US, and Mexico. The high image acquisition rate and high signal-to-noise-
ratios (S/Ns) obtained at Mauna Kea with the ‘Alopeke camera at the Gemini
North 8.1-m telescope and Wide-field InfraRed Camera (WIRCam) at the Canada-
France-Hawaii 3.6-m Telescope (CFHT) allowed the dense part of Q1R to be
radially resolved and its optical depth to be probed in the r’, z’ and Ks
bands. Furthermore, this event revealed a new ring around Quaoar, hereafter
called the Q2R (Quaoar’s second ring).

##  2 Prediction and observation

The occultation of August 9, 2022 was predicted within the framework of the
European Research Council (ERC) Lucky Star
project111https://lesia.obspm.fr/lucky-star. The campaign was managed by the
Occultation Portal website222https://occultation.tug.tubitak.gov.tr described
in Kilic et al. (2022). The occultation shadow crossed the continental United
States, Mexico, and the Hawaiian archipelago with a geocentric shadow velocity
of 17.6 km s-1. The shadow path and the sites involved in the observation
campaign are shown in Appendix 6. More details on the occulted star and Quaoar
are provided in Table 3.

Table 1:  More details about the occulted star and Quaoar.

Occulted star
---
Epoch | 2022-08-09 06:34:04 UTC
Source ID |  Gaia DR3 4098214367441486592
Star position |  $\alpha_{\star}$= 18h21m42s.86965 $\pm$ 0.2404 mas
at epoch11$1$The star position was taken from the Gaia Data Release 3 (GDR3) star catalog (Gaia Collaboration et al. 2022) and is propagated to the event epoch using the formalism of Butkevich & Lindegren (2014) applied with SORA (Gomes-Júnior et al. 2022). |  $\delta_{\star}$= -15∘12′45′′.9639 $\pm$ 0.2191 mas
Magnitudes22$2$J, H, and K from the NOMAD catalog (Zacharias et al. 2004). | G = 15.3; RP = 14.1; BP = 16.8;
| J = 12.0; H = 11.0; K = 10.7
Apparent diameter33$3$Limb-darkened angular diameter estimated using a fit of the spectral energy distribution of the star with (Castelli & Kurucz 2003) atmosphere models, with the reddening definition of (Fitzpatrick 1999). | 0.44 mas / 1.33 km
RUWE44$4$Renormalized unit weight error (Lindegren et al. 2021). | 0.959
(50000) Quaoar
Ephemeris version | NIMAv1655$5$Obtained from the Numerical Integration of the Motion of an Asteroid (NIMA, Desmars et al. 2015), based on astrometric positions derived from previous stellar occultations and the Minor Planet Center (MPC) database, available at https://lesia.obspm.fr/lucky-star/obj.php?p=958
Geocentric Distance | 41.983157 au
Apparent Magnitude | V = 18.8
Mass66$6$Obtained from the method of Vachier et al. (2012). |  $(1.2\pm 0.05)\times 10^{21}$ kg
Rotation period77$7$Assuming the double-peaked light curve from Ortiz et al. (2003). |  $17.6788\pm 0.0004$ h
Weywot’s orbital pole66$6$Obtained from the method of Vachier et al. (2012).
RA | 17h44${}^{m}\leavevmode\nobreak\ \pm$ 40m
DEC | +50${}^{\circ}\leavevmode\nobreak\ \pm$ 6∘
333

Table 6 summarizes the observational circumstances. Among the stations, CFHT
was equipped with the WIRCam (Puget et al. 2004) at 2.15 $\mathrm{\mu}$m (Ks-
band) with an 8.8 Hz acquisition rate. The nearby Gemini North telescope used
the Alopeke instrument (Scott et al. 2021) to simultaneously record the event
with the z’ (947 nm) and r’ (620 nm) bands at a 10 Hz cadence. The images at
the Tohoku University Haleakala Observatory (TUHO) in Hawaii and at the
Transneptunian Automated Occultation Survey (TAOS II) in Baja California
(Mexico) were taken with no filter to maximize the S/N. Q2R was not detected
in these observations, likely due to insufficient S/N values. At the
University of California, Santa Cruz (UCSC - California, US), the Dunrhomin
Observatory (Colorado, US), Sommers-Bausch Observatory (Colorado, US),
Nederland (Colorado, US), and a mobile station in Bonny Doon Eco Reserve
(California, US), only the main body was recorded due to a low S/N.

##  3 Data analysis

The data sets were analyzed using the standard photometry procedures of the
Platform for Reduction of Astronomical Images Automatically (PRAIA, Assafin et
al. 2011). The flux of the occulted star was corrected by the flux of nearby
reference stars to account for sky transparency variations and was normalized
to the total star$+$Quaoar flux outside the occultation.

The Stellar Occultation Reduction and Analysis package (SORA, Gomes-Júnior et
al. 2022) was used to derive the ingress and egress instants of the
occultation by the main body, assuming an opaque body without atmosphere. More
details are presented in Appendix A. The extremities of the resulting chords
were finally used to fit an elliptical model to Quaoar’s limb at the moment of
the occultation.

In the case of rings, the events were fitted using the square box model
described in Appendix E and pipelines based on the SORA package. This provides
the radial width of the ring $W_{\rm r}$ (assuming a circular ring), its
normal opacity, $p_{\rm N}$, and normal optical depth, $\tau_{\rm N}$, from
which the equivalent width, $E_{\rm p}=W_{\rm r}p_{\rm N}$, and equivalent
depth, $A_{\tau}=W_{\rm r}\tau_{\rm N}$, are derived.

In the case of the CFHT and Gemini data, the acquisition rate and S/N were
high enough to provide resolved profiles of the dense part of Q1R. Then,
$E_{\rm p}$ and $A_{\tau}$ can be obtained directly from the numerical
calculations of the integrals $A_{\tau}=\int_{W_{\rm r}}{(v_{\rm r}\tau_{\rm
N})dt}$ and $E_{\rm p}=\int_{W_{\rm r}}{(v_{\rm r}p_{\rm N})dt}$ (i.e.,
without using the square model described above), $v_{\rm r}$ is the star
velocity perpendicular to the ring, measured in the ring plane.

The dense part of Q1R has a radial profile reminiscent of the Saturnian F-ring
core (Bosh et al. 2002). More precisely, it is consistent with a Lorentzian
shape used here to provide the full width at half maximum (FWHM) of the
profile and the event’s timing, from which its location relative to Quaoar is
obtained.

##  4 Results

Figure 1: Detection of Q1R, Q2R, and the main body in the Gemini (z’) light
curve. The normalized flux is plotted in black vs. time (UTC), while the best-
fitting models are over-plotted in red. The shallow events caused by Q1R
(ingress) and Q2R (ingress & egress) are fitted by a square model. A
Lorentzian function fits the dense part of Q1R at egress; see Sect. 3. The
spikes observed during the disappearance and the re-appearance of the star
behind Quaoar stem from the diffraction by the sharp opaque limb of the body.
Note: there is a variation in the flux close to -110 and 290 seconds, caused
by sudden variations in the seeing, and they can be neglected. More detailed
views of Q1R and Q2R obtained at other telescopes are displayed in Figs. 2, 7,
8, and 9.

###  4.1 Main body

The event on August 9, 2022 provided ten chords across Quaoar’s main body.
However, Gemini used a dual camera at two wavelengths, resulting in nine
effective chords across the body, that is, $N$=18 chord extremities with the
timings listed in Table 5.

We used the SORA package to fit an ellipse to those extremities projected in
the sky plane. This ellipse is defined by $M=5$ parameters and the fit has
$N-M=13$ degrees of freedom. The fitted parameters are the ephemeris offsets
$f_{\rm c}$ and $g_{\rm c}$ to apply to Quaoar’s center in right ascension and
declination, respectively, the apparent semi-major axis $a^{\prime}$ of the
limb, its apparent oblateness
$\epsilon^{\prime}=(a^{\prime}-b^{\prime})/a^{\prime}$, where $b^{\prime}$ is
the apparent semi-minor axis of the limb, and $P$ is the position angle of
$b^{\prime}$. The elliptical fit also considers a $\sigma_{\rm model}$, which
is the uncertainty of the ellipse associated with the existence of putative
topographic features on the surface of Quaoar. From the methodology presented
by (Johnson & McGetchin 1973), we estimated that Quaoar might support
topographic features of about 5 km, given Quaoar’s density
$\rho=1.99\,\rm{g}\,\rm{cm}^{-3}$ and strength $\rm{S}=0.0303\times
10^{9}\,dynes\,cm^{-2}$, consistent with an ice-rich composition. The standard
deviation of the observed radial residuals is $\sim$8 km, providing an upper
limit for topographic features consistent with this prediction. Considering
the equivalent radius in area as 542 km, this corresponds to 1.5%. The fit
results are listed in Table 4. The chords and the best-fitting elliptical limb
are plotted in Fig. 3, with more details given in Fig. 5.

Table 2: Retrieved parameters of Quaoar’s body and its rings11$1$Quaoar’s
limb, Quaoar’s geocentric position, and Q1R’s geometry are given at epoch
August 9th, 2022, 06:34:03.560 UTC. The fits use the star position and the
NIMAv16 ephemeris for Quaoar (Table 3). .

Quaoar’s limb fitting
---
Apparent semi-major axis |  $a^{\prime}=579.5\pm 4.0$ km
Apparent oblateness | $\epsilon^{\prime}=0.12\pm 0.01$
Position angle |  $P=345.2\pm 1.2$ degrees
Ephemeris offset |  $f_{\rm c}=-24.7\pm 2.0$ km
|  $g_{\rm c}=-13.9\pm 2.1$ km
$R_{\rm equiv}$ |  $543\pm 2$ km
$\chi^{2}$ per degree of freedom | 0.986
Quaoar’s geocentric |  $\alpha$= 18h21m42s.8677703 $\pm$ 0.249 mas
position (ICRS) |  $\delta$= -15∘12′ 45′′.829691 $\pm$ 0.230 mas
Q1R’s preferred pole |  $\alpha_{\rm P}$= 17h19m16s.8 $\pm$ 55s.2
position (ICRS)22$2$The preferred solution is co-planar with Quaoar’s equatorial plane and Weywot’s orbit within $5\pm 7$ deg. The complementary position $\alpha_{\rm P}$= 10h00${}^{m}\pm 05^{m}$ and $\delta_{\rm P}$= +79∘21${}^{\prime}\pm 08^{\prime}$, is not considered here as it provides a less satisfactory fit to the Q1R detections and yields an inclination of 45 degrees with respect to Weywot’s orbit, see Sec. 4.2 |  $\delta_{\rm P}$= +53∘27’$\pm$ 18’
Q1R’s geometry
Radius |  $4,057\pm 6$ km
Opening angle |  $B=-20.0\pm 0.3$ degrees
Position angle |  $P=350.2\pm 0.2$ degrees
$\chi^{2}$ per degree of freedom | 1.633
Q2R’s geometry33$3$Assuming a circular ring co-planar with Q1R. The error bar
for the Q2R ring radius may be underestimated since we only have two effective
detections, and several parameters were assumed.
Radius |  $2,520\pm 20$ km
444

###  4.2 Ring Q1R

The Q1R ring was detected on both sides of the main body in the Gemini and
CFHT light curves, and their physical properties were determined as described
in Sect. 3. Although the detections of the dense part of the Q1R in the Gemini
and CFHT light curves indicate the presence of diffuse material around the
ring, the quality of the TAOS II and TUHO light curves only allows for the
narrow and dense part of the ring to be detected. Therefore, the parameters
$E_{\rm p}$ and $A_{\tau}$ were determined by the square box model for these
data.

Due to the low optical depth of the Q1R segment intercepted before the closest
approach, only the light curves obtained at Gemini and CFHT have sufficient
S/N for detection (see an example in Fig. 1 and more details in Figs. 2, 7, 8,
and 9). The Gemini detection in the r’ bandpass seems more sharply defined
than its counterpart in z’, but this effect remains marginal considering the
S/N. The CFHT light curve shows a drop simultaneously, but it may be affected
by a seeing deterioration that compromises an accurate determination of the
ring boundaries. Even so, the central times of the Gemini and CFHT detections
are all consistent at the 1-$\sigma$ level. The parameters of Q1R for all the
detections are presented in Table 7.

We have combined the data from this work and the previous observations to
improve the Q1R orbital parameters, assuming a fixed ring pole orientation
between 2018 and 2022, as per Morgado et al. (2023). We tested a range of pole
orientations and ring radii using a $\chi^{2}$ statistic, resulting in two
complementary solutions presented in Table 4. The preferred solution has a
$\chi^{2}$ per degree of freedom $\chi^{2}_{\rm pdf}=1.6$ and shows a better
agreement with observations than the mirror solution with $\chi^{2}_{\rm
pdf}=4.1$. Moreover, the preferred solution is co-planar with Weywot’s orbit
to within $5\pm 7$ deg, as expected from a primordial disk surrounding Quaoar
that evolved into a ring and formed its satellite, while the mirror solution
is inclined by $45\pm 7$ degrees with respect to Weywot’s orbit.

The position angles of Quaoar’s projected pole ($345.2\pm 1.2$ deg) and Q1R
($350.2\pm 0.2$ deg) are misaligned by $\sim$5 degrees (Table 4). Thus, our
results are consistent with Q1R orbiting close to Quaoar’s equatorial plane,
assuming that the body is an oblate spheroid. Part of this misalignment could
stem from the fact that Quaoar is a triaxial ellipsoid (or a body with a more
complex shape). Hence, the position angle of Quaoar’s limb does not
necessarily coincide with the position angle of Quaoar’s pole.

###  4.3 The discovery of a new ring around Quaoar

Figure 2:  Detection of the Q2R ring in the Gemini and CFHT light curves. The
data points are plotted in black, and the red lines are the square box model
fits derived from the quantities listed in Table 7. An arbitrary offset was
applied in the vertical direction for clarity. The time axis is relative to
August 9, 2022 at 06:34:49.26 UTC, the time of the closest approach of Mauna
Kea to Quaoar’s shadow center.

The unique photometric quality of the Gemini and CFHT data allowed for the
detection of additional material around Quaoar (Fig. 1). These data sets
reveal additional secondary events symmetrically located with respect to
Quaoar (Fig. 2). Two such events are simultaneously detected in the Gemini
z’-band and CFHT Ks-band light curves before the closest approach, with
detections reaching around 5.5$\sigma$ and 5.2$\sigma$, respectively.
Conversely, the light curves display simultaneous events after the closest
approach with significant detections, standing at 5.7$\sigma$, 3.7$\sigma$,
and 4.7$\sigma$ for the Gemini z’, Gemini r’, and CFHT data sets,
respectively. Assuming that the light curves have a normal distribution, the
probability that individual points of equivalent width
E${}_{\rm{p}}(3\sigma)>12$ km occur randomly in each light curve is
$\rm{p}\approx 1.4\times 10^{-3}$, with p approaching zero for values larger
than Ep. While the two light curves of the Gemini instrument may be correlated
as they were taken at the same telescope, the Gemini and CFHT data are
independent in terms of fast-seeing fluctuations. Using Poisson statistics, we
estimate that the probability that the simultaneous events in the Gemini and
CFHT data occur randomly due to the seeing fluctuations is very low, with
$\rm{p}\approx 10^{-6}$.

Some dips in flux were observed in the vicinity of Q2R detection in the region
after the closest approach (Fig. 2). These detections are at the detection
limit of our data, so we cannot say whether these are due to seeing variations
or additional structures. Furthermore, we do not observe counterparts of these
dips in the region before the closest approach.

All these detections are consistent with a new circular ring (Q2R) co-planar
and concentric with Q1R, orbiting at $2,520\pm 20$ km from Quaoar (Table 4).
The radial width, optical depth, equivalent width, and equivalent depth of Q2R
are listed in Table 7. They were obtained as described in Appendix E.

##  5 Discussion and conclusions

Figure 3:  Representation of our results on Quaoar’s shape (center) and the
detection of the two rings Q1R (outer ring) and Q2R (inner ring). The red
segments correspond to the 1-$\sigma$ error bars on the particular events. The
orbit of Q1R is determined from a simultaneous fit using the present work and
previous detections of 2018, 2019, 2020, and 2021 reported by Morgado et al.
(2023) (see Section 4.2). The solution for the orbit of the new Q2R ring
assumes that this ring is co-planar and concentric with Q1R. The central part
of the plot (occultation by the solid body) is enlarged in Fig. 5. In yellow,
we show the 1/3 SOR resonance with Quaoar, and in teal is the 5/7 SOR
resonance. The purple ellipse represents 6/1 MMR with Weywot (considering the
double-peaked rotation period), and the green ellipse presents the expected
Roche limit, considering particles with a bulk density of
$\rho=0.4\,\rm{g}\,\rm{cm}^{-3}$. The arrow shows the star’s motion relative
to Quaoar. Note: the orbital radius of Weywot is about three times larger than
that of Q1R, and thus it is not shown in this representation.

The stellar occultation of August 9, 2022 provided nine effective occultation
chords obtained in Hawaii, Mexico, and the continental United States. They
constrained Quaoar’s shape, providing an apparent semi-major axis of
$a^{\prime}=579.5\pm 4.0$ km, an apparent oblateness
$\epsilon^{\prime}=0.12\pm 0.01$, and an area-equivalent radius of $R_{\rm
equiv}=a^{\prime}\sqrt{1-\epsilon^{\prime}}=543\pm 2$ km. Using an absolute
magnitude of $H_{\rm V}=2.73\pm 0.06$ (Fornasier et al. 2013), this yields a
geometric albedo of $p_{\rm V}=0.124\pm 0.006$. Our value of $R_{\rm equiv}$
differs from that of Braga-Ribas et al. (2013), $555.0\pm 2.5$ km, by about 12
km, that is, at the 4-$\sigma$ level. This difference could be evidence that
Quaoar is not an oblate spheroid, but rather a triaxial ellipsoid or a body
with a more complex shape; alternatively, it may be caused by a change in the
aspect-angle since 2011, as observed from Earth.

The continuous region and the dense part of Q1R were detected during this
event. The dense part was radially resolved and showed a Lorentzian profile
reminiscent of Saturn’s F ring (Bosh et al. 2002) or Neptune’s ring arcs
(Nicholson et al. 1990; Sicardy et al. 1991). This contrasts with the sharp
edges observed for Chariklo’s main ring C1R (Bérard et al. 2017; Morgado et
al. 2021). The dense part of Q1R is detected over a radial width of $\sim$60
km with a peak optical depth of $\tau_{\rm N}\approx 0.4$, an FWHM Lorentzian
width of 5 km and an equivalent width $E_{\rm p}$ of around 2 km (see Table
7). The values presented here are more precise than (but consistent with)
those published by Morgado et al. (2023).

The detections of the dense part of Q1R in 2021 imply a minimum arc length of
365 km, corresponding to an azimuthal extension of about 5.1 degrees (Morgado
et al. 2023). The detections from CFHT, Gemini, TUHO, and TAOS II in 2022
(Figs. 8, 9) suggest a minimum arc-length of 226 km or $\sim$3.2 degrees.
Since 2011, we have obtained 19 cuts of the Q1R with a sufficient S/N to
detect the densest region. The limited azimuthal extent of the two detections
of the dense regions means that they could both be parts of the same arc-like
structure. If this structure was detected two times among the 19 cuts, then
its extent can be estimated using Poisson statistics. This analysis yields an
arc length with a 70% chance of falling between 18 and 72 degrees. In this
case, all of the detections in either 2021 or 2022 would each be correlated
samples of one part of this arc.

The more tenuous component of Q1R is radially resolved and shows no marked
structures, being consistent with a square model within our S/N limits. The
best light curves (Gemini and CFHT) provide a typical width of 80-100 km and a
typical normal optical depth of 0.003 (and, thus, an equivalent width of
$\sim$0.3 km) for that component at the longitude where it was detected (Table
7).

Our detections, combined with those reported by Morgado et al. (2023),
improved the orbital elements of Q1R. They are consistent with a circular ring
of radius $4,057.2\pm 5.8$ km (Table 4) corresponding to about 7.5$\times$
Quaoar’s area-equivalent radius $R_{\rm equiv}$. This value coincides with the
6/1 MMR with Weywot ($4,020\pm 60$ km) and within 3-$\sigma$ of the 1/3 SOR
with Quaoar ($4,200\pm 60$ km), considering the double-peaked rotation period
of $17.6788\pm 0.0004$ h (Ortiz et al. 2003).

Our preferred solution for Q1R’s orbital pole is consistent with this ring
being co-planar with Weywot’s orbit. Moreover, the apparent semi-major axes of
Quaoar’s limb and Q1R’s orbit are aligned at the 4-$\sigma$ level, suggesting
that Q1R lies in Quaoar’s equatorial plane, as expected from a colliding ring
system (Kokubo et al. 2000).

Our data reveal a new ring (Q2R) around Quaoar. The detections are consistent
with a circular ring of radius $2,520\pm 20$ km co-planar with Q1R (Table 4,
Figs. 2 and 3). It has a typical width of 10 km, an optical depth of about
0.004, and an equivalent depth of about 0.04 km (see accurate values in Table
7). Although closer to Quaoar than Q1R, Q2R is at 4.6$\times$ Quaoar’s radius,
also outside Quaoar’s Roche limit, which is estimated to be around 1,780 km,
assuming the ring particle density as $\rho=0.4\,\rm{g}\,\rm{cm}^{-3}$
(Morgado et al. 2023).

Using previously determined values for Quaoar’s mass and rotation period
(Table 3), we derived a 5/7 SOR radius of $2,525\pm 35$ km. This coincides
with Q2R’s radius, $2,520\pm 20$ km, to within the 1-$\sigma$ error bars. Like
the 1/3 SOR, the 5/7 SOR is a second-order resonance and, as such, may play an
essential role in the confinement of Q2R. However, more solid determinations
of Q2R’s orbit and Quaoar’s shape are required.

Table 7 shows that differences in the equivalent widths of Q1R and Q2R are
observed between the z’, r’, and Ks filters. The significance and
interpretation of these differences require more analysis and will be
discussed in a forthcoming publication. Moreover, the high S/N obtained at
Gemini and CFHT will be used to detect or put a stringent upper limit on a
putative atmosphere. Finally, the comparison of past (and possibly future)
results derived from multi-chord occultations by Quaoar’s main body will be
used to constrain its shape better. This will be important for better
understanding the dynamics of Quaoar’s ring system, particularly under the
effect of spin-orbit resonances with the body.

###### Acknowledgements.

C.L.P is thankful for the support of the CAPES and FAPERJ/DSC-10
(E26/204.141/2022). This work was carried out within the “Lucky Star” umbrella
that agglomerates the efforts of the Paris, Granada, and Rio teams, funded by
the European Research Council under the European Community’s H2020 (ERC Grant
Agreement No. 669416). This study was financed in part by the National
Institute of Science and Technology of the e-Universe project (INCT do
e-Universo, CNPq grant 465376/2014-2). This study was financed in part by
CAPES - Finance Code 001. The following authors acknowledge the respective
CNPq grants: B.E.M. 150612/2020-6; F.B.R. 314772/2020-0; R.V.M. 307368/2021-1;
M.A. 427700/2018-3, 310683/2017-3, 473002/2013-2; J.I.B.C. 308150/2016-3,
305917/2019-6. R.C.B acknowledge the FAPERJ grant E26/202.125/2020. E.F.-V.
acknowledges financial support from the Florida Space Institute and the Space
Research Initiative. J.L.O., P.S-S., M.V-L, and M. K. acknowledge financial
support from the grant CEX2021-001131-S funded by MCIN/AEI/
10.13039/501100011033, they also acknowledge the financial support by the
Spanish grants PID2020-112789GB-I00 from AEI and Proyecto de Excelencia de la
Junta de Andalucía PY20-01309. Funding for RECON was provided by grants from
USA: NSF AST-1413287, AST-1413072, AST-1848621, and AST-1212159. We thank
RECON observers Doug Thompson, Ken Conway, Dorey Conway, Terry Miller, David
Schulz, Michael von Schalscha, and Matt Christensen for their efforts in
collecting data. Based on observations obtained with WIRCam, a joint project
of CFHT, Taiwan, Korea, Canada, France, at the Canada-France-Hawaii Telescope
(CFHT) which is operated from the summit of Maunakea by the National Research
Council of Canada, the Institut National des Sciences de l’Univers of the
Centre National de la Recherche Scientifique of France, and the University of
Hawaii. The observations at the Canada-France-Hawaii Telescope were performed
with care and respect from the summit of Maunakea which is a significant
cultural and historic site. We thank Marc Baril and Tom Vermeulen for their
time dedicated to the observation performed at Canada-France-Hawaii Telescope
(CFHT). Based on observations obtained at the international Gemini
Observatory, a program of NSF’s NOIRLab, which is managed by the Association
of Universities for Research in Astronomy (AURA) under a cooperative agreement
with the National Science Foundation on behalf of the Gemini Observatory
partnership: the National Science Foundation (United States), National
Research Council (Canada), Agencia Nacional de Investigación y Desarrollo
(Chile), Ministerio de Ciencia, Tecnología e Innovación (Argentina),
Ministério da Ciência, Tecnologia, Inovações e Comunicações (Brazil), and
Korea Astronomy and Space Science Institute (Republic of Korea). This work
made use of data from GN-2022B-DD-101 observing program and were obtained with
the High-Resolution Imaging instrument(s) ‘Alopeke (and/or Zorro). ‘Alopeke
(and/or Zorro) was funded by the NASA Exoplanet Exploration Program and built
at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P.
Horch, and Emmett Quigley. ‘Alopeke (and/or Zorro) was mounted on the Gemini
North (and/or South) telescope of the international Gemini Observatory, a
program of NSF’s NOIRLab, which is managed by the Association of Universities
for Research in Astronomy (AURA) under a cooperative agreement with the
National Science Foundation. This work was enabled by observations made from
the Gemini North telescope, located within the Maunakea Science Reserve and
adjacent to the summit of Maunakea. We are grateful for the privilege of
observing the Universe from a place that is unique in both its astronomical
quality and its cultural significance. This research used sora, a python
package for stellar occultations reduction and analysis, developed with the
support of ERC Lucky Star and LIneA/Brazil, within the collaboration of Rio-
Paris-Granada teams. This work profited from unpublished occultations by
Quaoar made at SOAR (SO2019A-003) and the Pico dos Dias Observatory
(OP2019A-004) to improve the accuracy of the ephemeris NIMAv16.

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##  Appendix A Main body occultation and elliptical fit

The light curves obtained with the Gemini and CFHT telescopes present
prominent diffraction signatures in the main body occultation mainly due to
the high acquisition rate, the large wavelength, and high S/N of the light
curves. During the occultation, Quaoar was at a geocentric distance of
41.983157 au, implying a Fresnel scale, $F=\sqrt{\lambda D/2}$, from 1.5 to
2.6 km, for visible to near-infrared (CFHT Ks-band) wavelengths, respectively.
Considering the apparent velocity of the event as 17.57 km s-1 and the
acquisition rate of the Gemini and CFHT instruments, we obtained a spatial
resolution of these light curves of $\sim$1.8 km. These high-S/N light curves
have a Fresnel diffraction effect on the same order of magnitude as the
instrumental response (e.g., the integration time).

Figure 4: Gemini z’ light curve (black dots) and the modeled light curve (red)
considering the local normal star velocity, the wavelength and apparent star
size. The geometric model is in blue. The normalized flux is plotted as a
function of the seconds before and after the local closest approach
(2022-08-09 06:34:49.26 UT). Table 3:  Ingress and egress times for Quaoar’s
main body.

Site | Ingress | Egress
---|---|---
| (hh:mm:ss.s) | (hh:mm:ss.s)
CFHT | 06:34:23.34 (0.02) | 06:35:11.42 (0.01)
Gemini (z’) | 06:34:23.29 (0.01) | 06:35:11.29 (0.01)
Gemini (r’) | 06:34:23.27 (0.03) | 06:35:11.28 (0.02)
TUHO | 06:34:23.5 (0.2) | 06:35:18.1 (0.2)
TAOS II | 06:30:38.3 (0.2) | 06:31:35.1 (0.1)
Dunrhomin | 06:30:03.3 (0.4) | 06:31:02.8 (0.9)
Sommers-Bausch | 06:30:02.6 (0.2) | 06:31:03.8 (0.5)
UCSC | 06:31:20.3 (0.1) | 06:32:00.9 (0.1)
Nederland | 06:30:03.4 (0.5) | 06:31:04.4 (0.8)
Bonny Doon | 06:31:20.7 (0.4) | 06:32:01.3 (0.4)
555Times are in UT and the error bars in parentheses are in seconds and given
at the 1-$\sigma$ level. The significant differences in the CFHT and Gemini
instants are due to the length of the chords, with the CFHT being $\sim 1$ km
longer than the Gemini, as expected from the limb.

With the limb-darkened angular diameter of the star estimated and considering
the observation band pass and its FWHM, we also fit the local normal velocity
($v_{\perp}$). By minimizing the $\chi^{2}$ statistic, we found the velocity
and the ingress (egress) time that best explained the observed light curve.
This was applied to the Gemini z’-band data since this was our best light
curve. The radial velocities obtained are $v_{\perp}$ = 10.2 km s-1 and
$v_{\perp}$ = 16.6 km s-1 for ingress and egress instants, respectively. The
separation between the chords obtained from the Gemini and CFHT when projected
onto the sky plane was about 160 meters, a value smaller than the star’s
apparent diameter. Therefore, the velocities were considered the same at both
sites. This procedure resulted in an optimal fit for both diffraction spikes
and the baseline occultation by the main body. Figure 4 presents an example of
this fit with the Gemini z’ light curve plotted with the modeled curve.

Ingress and egress times for the main body occultation are presented in Table
5. The instant uncertainties of the Gemini and CFHT light curves are limited
by occultation modeling, not data quality. The elliptical fit parameters to
the chord extremities are presented in Fig. 5.

Figure 5: Close-up view of Fig. 3 shows the best elliptical fit (black
ellipse) and ellipses within 1-$\sigma$ region (grey) to the ten chords
derived from the timings in Table 5. The grey 1-$\sigma$ fits differ only
slightly from the black best-fit. The red segments represent the 1-$\sigma$
error bars in the ingress and egress times. At this scale, the Gemini and CFHT
chords are superimposed, and their error bars would be too small to be seen
(i.e., they are here represented by red squares, much bigger than their actual
sizes).

##  Appendix B Observational circumstances

The observational circumstances for this occultation campaign are presented in
Table 6. We present in this work the analysis of ten data sets with positive
detections for the main body, of which four are detections of the Q1R ring and
three are detections of both the Q1R and Q2R rings. In addition, we have more
eighteen sites that participated in the observation campaign, but reported
cloudy skies. The target star was too faint for the setup in Woodside (CA, US)
and Summerland BC, CA, even with 4 and 3 seconds of exposure, respectively; in
addition, the Summerland data was acquired while clouds were drifting through
the field.

Table 4: Observational circumstances. | Longitude [∘ ’ ”] |  | Aperture (mm) | Exp. Time | Status
---|---|---|---|---|---
Site | Latitude [∘ ’ ”] | Observers | Detector | Cycle | Light curve
| Altitude [m] |  | Filter | (s) | Dispersion
CFHT | -155 28 07.93 | H. Januszewski | 3,580 | 0.113 | Positive
MaunaKea | 19 49 31.01 | M. Baril, B. Epinat | WIRCam | 0.113 | 0.013
HI, US | 4,206.1 | T. Vermeulen | Ks |  |
Gemini-North | -155 28 08.6 | T. Seccull | 8,100 | 0.100 | Positive
MaunaKea | 19 49 26 | A. Stephens | ’Alopeke | 0.101 | z’: 0.012
HI, US | 4,213 |  | r’ & z’ |  | r’: 0.022
TUHO | -156 15 27 | E. Tatsumi | 600 | 1.500 | Positive
Haleakala | 20 42 25.14 | F. Yoshida | ASI178MM | 1.634 | 0.098
HI, US | 3,051.8 | M. Kagitani | None |  |
TAOS II | -115 27 47.88 | J. H. Castro-Chacón | 1,300 | 0.200 | Positive
San Pedro Martir | 31 02 36.26 | S. Sánchez-Sanjuán | Andor Ixon 888 | 0.205 | 0.274
BC, MX | 2,824.8 |  | None |  |
Dunrhomin Obs. | -105 09 46.57 |  | 280 | 2 | Positive
Longmont | 40 15 08.46 | M. Buie | QHY174M-GPS | 2 | 0.229
CO, US | 1,592.8 |  | None |  |
Sommers-Bausch Obs. | -105 15 47.09 | J. Keller | 508 | 0.9 | Positive
Boulder | 40 00 13.32 | J. Johnston | QHY174M-GPS | 0.9 | 0.259
CO, US | 1,645 |  | None |  |
| -105 26 44.05 | M. Skrutskie | 200 | 2 | Positive
Nederland | 39 59 13.95 | A. Verbiscer | QHY174M-GPS | 2 | 0.313
CO, US | 2,492.6 |  | 570 nm |  |
Bonny Doon Eco Reserve | -122 08 18.62 |  | 203 | 0.260 | Positive
Santa Cruz | 37 03 03.12 | K. Bender | Watec 910HX/RC | 0.260 | 0.538
CA, US | 488.4 |  | None |  |
UCSC | -122 04 46.20 |  | 203 | 0.500 | Positive
Santa Cruz | 37 01 03.47 | R. Nolthenius | Watec 910HX/RC | 0.534 | 0.556
CA, US | 344.0 |  | None |  |
| -119 40 12 |  | 457 | 3 |
Summerland | 49 36 00 | D. Gamble | QHY174M-GPS | 3 | Negative
BC, CA | 485 |  | None |  |
| -122 15 30.0 |  | 114 | 4 |
Woodside | 37 23 36.2 | F. Marchis | eVscope/IMX347 | 4 | Inconclusive
CA, US | 367 |  | None |  |
Naylor Obs. | -76 53 43.48 | R. Kamin | 355.6 | 0.5 |
Harrisburg | 40 08 55.60 | R. Young | QHY174M-GPS | 0.5 | Overcast
PA, US | 181.6 |  |  |  |
Nikitin Home | -105 11 00.96 |  | 279.4 | 6 |
Gunbarrel | 40 04 08.61 | V. Nikitin | QHY174-GPS | 6 | Overcast
CO, US | 1,594 |  | None |  |
Van Vleck Obs. | -72 39 33.12 | K. McGregor | 610 |  |
Middletown | 41 33 18.62 | S. Redfield | Apogee E2V | No data | Overcast
CT, US | 68.9 |  | Luminance |  |
Skychariot | -74 58 49.28 |  | 406.4 |  |
Shohola | 41 21 39.94 | M. Sproul | ASI1600MC | No data | Overcast
PA, US | 414.6 |  | None |  |
NIRo - CAC | -87 22 31.08 |  | 508 |  |
Lowell | 41 16 15.03 | A. W. Rengstorf | FLI PL09000 | No data | Overcast
IN, US | 195.7 |  | I |  |
Table 1: [continued.] Observational circumstances.

| Longitude [∘ ’ ”] |  | Aperture (mm) | Exp. Time | Status
---|---|---|---|---|---
Site | Latitude [∘ ’ ”] | Observers | Detector | Cycle | Light curve
| Altitude [m] |  | Filter | (s) | Dispersion
Jimginny Obs. | -88 07 0.00 |  | 356 |  |
Naperville | 41 45 32.40 | R. Dunford | QHY174M-GPS | No data | Overcast
IL, US | 230 |  | None |  |
Pine Mountain Obs. | -120 56 28.8 | S. Fisher | 610 |  |
Bend | 43 47 30.3 | A. Luken | Andor Xyla | No data | Overcast
OR, US | 1,920 |  | None |  |
Palomar Obs. | -116 51 53.5 | E. Fernandez-Valenzuela | 5,100 |  |
San Diego | 33 21 21.5 | K.de Kleer | CHIMERA | No data | Overcast
CA, US | 1,712 | T. Marlin, M. Camarca |  |  |
Westport Astronomical Obs. | -73 19 39.0 |  | 356 |  |
Westport | 41 10 16.0 | K. D. Green | QHY174M-GPS | No data | Overcast
CT, US | 87 |  | None |  |
| -119 09 39 |  | 280 |  |
Yerington | 38 59 28 | M. Christensen | QHY174M-GPS | No data | Overcast
NV, US | 1,340 |  | None |  |
| -121 18 55 |  | 280 |  |
Bend | 44 03 29 | A-M. Eklund | QHY174M-GPS | No data | Overcast
OR, US | 1,105 |  | None |  |
| -114 37 40 | D. Thompson | 280 |  |
Yuma | 32 41 34 | K. Conway | QHY174M-GPS | No data | Overcast
CA, US | 62 | D. Conway | None |  |
| -119 49 06 |  | 280 |  |
Reno | 39 32 42 | T. Stoffel | QHY174M-GPS | No data | Overcast
NV, US | 1,387 |  | None |  |
| -120 10 24 | T. Miller | 280 |  |
Cedarville | 41 31 45 | D. Schulz | QHY174M-GPS | No data | Overcast
CA, US | 1,420 |  | None |  |
| -113 51 46 | C. Patrick | 305 |  |
Kingman | 35 24 43 | K. Butterfield | QHY174M-GPS | No data | Overcast
AZ, US | 966 |  | 0.5 focal reducer |  |
| -116 45 29 |  | 280 |  |
Beatty | 36 54 04 | J. Heller | QHY174M-GPS | No data | Overcast
CA, US | 999 |  | None |  |
| -121 23 56 |  | 280 |  |
Burney | 41 02 45 | M. von Schalscha | QHY174M-GPS | No data | Overcast
CA, US | 1,012 |  | None |  |
| -119 44 59 |  | 304 |  |
Gardnerville | 38 56 29 | J. Bardecker | Watec 910-HX | No data | Overcast
NV, US | 1,449 |  | None |  |

666TAOS II: The Transneptunian Automated Occultation Survey. CFHT: Canada-
France-Hawaii Telescope. TUHO: Tohoku University Haleakala Observatory. UCSC:
University of California, Santa Cruz. NIRo: Northwest Indiana Robotic -
Calumet Astronomy Center.

##  Appendix C Occultation map

Figure 6 presents the stellar occultation map with all the sites involved in
this campaign. The green dots mark the positive detections and orange dots are
stations that reported cloudy weather. The white dot represents the Woodside
observation with the eVscope where the event was not detected due to the low
S/N of the data set. The continuous line represents the limit of the Quaoar’s
shadow projected over the Earth. The dashed-lines represent the Q1R and Q2R
projections, as annotated in the image.

Figure 6: Occultation map with positive detections (black dots) and sites that
reported cloudy weather (orange dots). The white dot represents no data due to
the instrumental limits. In red we display a site with data but no occultation
detected. The black solid lines mark Quaoar’s shadow limits and the black
dashed lines mark the Q1R and Q2R projections. The black arrow in the bottom
right indicates the direction of motion of the shadow.

##  Appendix D Light curves

Figure 7 displays all the normalized light curves obtained during the stellar
occultation by (50000) Quaoar of August 9, 2022. The stellar flux has been
corrected for sky transparency fluctuations using reference stars. The time
scale is relative to the closest approach time at each site. Besides the
occultation by Quaoar and Q1R, the Gemini and CFHT light curves revealed the
new Q2R ring. The Tohoku (TUHO) and TAOS II data allowed for the detection of
the densest region of the Q1R ring. Due to the S/N limitations, the Nederland,
Dunrhomin, Sommers-Bausch, UCSC, and Bonny Doon stations detected only the
occultation by the main body. Some data sets presented dropped frames during
the photometric process, such as the Durhomin and Nederland. This occurred due
to the degradation of the sky quality shortly after the occultation of the
main body, in addition to clouds crossing the field of view of the telescope.

We noticed that the Gemini and CFHT light curves were misaligned in time,
probably caused by an offset in the Gemini data. As we have a reliable time
source for the CFHT data, we aligned the centers of these chords by applying
an offset of $+\,0.27$ seconds on the Gemini chord, obtaining values of
$\chi^{2}_{\rm{pdf}}$ closer to 1. Although very close, this result is
slightly better than applying an offset to the CFHT chord.

Figure 7: All the positive light curves obtained during the August 9, 2022
stellar occultation by Quaoar. The black dots represent the data points. These
light curves are plotted as a function of the time in seconds relative to the
closest approach for each site. The green and gray vertical lines stand for
Q2R and Q1R, respectively. When detected, these lines stand for calculated
detection times. For light curves where these secondary structures were not
detected, the green and gray lines mark theoretical times expected from our
best-fit solution for a circular ring. The horizontal gray dashed lines
represent the baseline and minima of the stellar fluxes.

##  Appendix E Modeled ring detections

The ring events were fit using the models described in Elliot et al. (1984)
and Bérard et al. (2017), that is, square boxes with uniform opacity and sharp
edges. As with the main body occultation modeling, the modeled curve was
convolved with observation bandpass, apparent star diameter, and instrumental
response, but now fitting an opacity for the square box. The outputs of a
given fit are the width, $W$, of the event, its apparent opacity, $p^{\prime}$
(corresponding to the depth of the observed stellar flux drop), and its
apparent optical depth, $\tau^{\prime}=-\ln(1-p^{\prime})$. The geometry of
the ring (assumed here to be circular) is defined by its opening angle, $B$,
and the position angle, $P$, of its apparent semi-minor axis, counted
positively from local celestial north to the celestial east direction. This
provides the radial width, $W_{\rm r}$, the normal opacity, $p_{\rm N}=|\sin
B|\leavevmode\nobreak\ (1-\sqrt{1-p^{\prime}})$, and the normal optical depth,
$\tau_{\rm N}=\tau^{\prime}\leavevmode\nobreak\ |\sin B|/2$, of the ring from
each event. The factor of 2 in the formula for $\tau_{\rm N}$ is due to the
fact that Airy diffraction by individual ring particles results in a loss of
light, resulting in an observed optical depth to be twice as large as would be
measured at the ring level; see more details in Cuzzi (1985); Roques et al.
(1987). We note that the formulae for $p_{\rm N}$ and $\tau_{\rm N}$ are valid
only for mono-layer and poly-layer rings, respectively. The equivalent width
(respectively, depth) $E_{\rm p}=W_{\rm r}p_{\rm N}$ (respectively,
$A_{\tau}=W_{\rm r}\tau_{\rm N}$) measures the amount of ring material that
blocked the stellar rays in the mono-layer (resp. poly-layer) case.

A Lorentzian function is fitted to the dense part of the Q1R in the Gemini and
CFHT light curves, converted from flux to normal optical depth as a function
of radial distance in the ring plane. The area under the curve equals the
equivalent width, Aτ, of the ring, and the full-width half minimum (FWHM)
gives us the approximate width of the ring’s core. The position of the
function’s valley gives us the average radial distance between the ring and
Quaoar’s center. The Ep value was obtained from the integral of the ring
profile in the curve of the normal opacity, $p_{\rm{N}}$, as a function of the
radial distance in the sky plane. This modeling considers the stellar apparent
diameter of 1.33 km, which has negligible influence on the ring profile.

Figs. 8 and 9 present all light curves in which the Q1R and Q2R rings were
detected. The TAOS II light curve, used for detecting of the densest region of
the Q1R ring, was obtained by applying aperture photometry to stacked images
in order to increase the S/N. Stacking was performed using Python routines
built from the astropy library as described in Morgado et al. (2019), where
each new image is the average of six original images, resulting in a temporal
resolution of 1.2 seconds for each stacked image.

Table 2: Physical parameters of rings Q1R and Q2R.

Ring | Detection | Mid-time | $r$ | $W_{\rm r}$ | $p_{\rm N}$ | $\tau_{\rm N}$ | $\rm{E}_{\rm p}$ | $\rm{A}_{\tau}$
---|---|---|---|---|---|---|---|---
August 9th, 2022 | (km) | (km) | (km) | (km)
Q1R- ing | CFHT | 06:31:15.8 (0.2) | 4,009.9 (1.8) | 105.5 (5.6) | 0.0030 (0.0003) | 0.0030 (0.0003) | 0.31 (0.02) | 0.31 (0.02)
Gemini (z’) | 06:31:15.72 (0.04) | 3,995.4 (0.3) | 76.4 (0.8) | 0.0026 (0.0001) | 0.0026 (0.0001) | 0.20 (0.01) | 0.20 (0.01)
Gemini (r’) | 06:31:15.7 (0.1) | 3,995.9 (0.8) | 85.8 (1.1) | 0.0032 (0.003) | 0.0032 (0.0003) | 0.28 (0.02) | 0.28 (0.02)
Q1R- egr |  CFHTa | 06:38:32.581 (0.003) | 4,123.69 (0.05) | 6.1 (0.1) | 0.330 (0.007) | 0.395 (0.009) | 2.01 (0.03) | 2.41 (0.04)
Gemini (z’)a | 06:38:32.444 (0.003) | 4,123.11 (0.02) | 5.3 (0.1) | 0.353 (0.008) | 0.417 (0.007) | 1.87 (0.02) | 2.21 (0.02)
Gemini (r’)a | 06:38:32.429 (0.005) | 4,122.85 (0.04) | 5.1 (0.1) | 0.341 (0.009) | 0.398 (0.008) | 1.74 (0.03) | 2.03 (0.03)
TUHO | 06:38:37.4 (0.7) | 4,107.2 (7.6) | 71 (17) | 0.03 (0.01) | 0.03 (0.01) | 2.3 (0.9) | 2.4 (1.1))
TAOS II | 06:34:55.8 (0.5) | 4,125.7 (4.4) | 34 (16) | 0.05 (0.02) | 0.05 (0.03) | 1.6 (1.1) | 1.7 (1.3)
Q2R- ing | CFHT | 06:32:43.9 (0.2) | 2,490.4 (1.6) | 16.5 (3.0) | 0.0022 (0.0004) | 0.0022 (0.0004) | 0.036 (0.006) | 0.036 (0.006)
Gemini (z’) | 06:32:43.61 (0.06) | 2,493.8 (0.6) | 11.3 (0.8) | 0.0034 (0.0005) | 0.0034 (0.0005) | 0.038 (0.006) | 0.039 (0.006)
Gemini (r’) |  n.d.b |  n.d.b |  n.d.b |  n.d.b |  n.d.b |  n.d.b |  n.d.b
Q2R- egr | CFHT | 06:37:00.20 (0.01) | 2,540.7 (0.1) | 10.8 (0.2) | 0.0049 (0.0003) | 0.0049 (0.0003) | 0.052 (0.001) | 0.053 (0.001)
Gemini (z’) | 06:37:00.09 (0.02) | 2,540.45 (0.1) | 6.3 (0.3) | 0.0047 (0.0005) | 0.0048 (0.0006) | 0.030 (0.002) | 0.030 (0.002)
Gemini (r’) | 06:36:59.93 (0.02) | 2,528.3 (1.1) | 9.1 (2.5) | 0.003 (0.001) | 0.003 (0.001) | 0.030 (0.008) | 0.030 (0.008)
777The normal opacity, $p_{\rm N}$, and normal optical depth, $\tau_{\rm N}$,
were calculated from the ring opening angle, $B$, and position angle, $P$, on
August 9, 2022, derived from the orientation of the body obtained from the
2018-2022 data. The other parameters are: Mid-time in UT and the error bars in
parentheses given in seconds; $r$, the radial distance from Quaoar’s center in
kilometers; $\rm{W}_{\rm r}$, the radial width in kilometers; $\rm{E}_{\rm p}$
and $\rm{A}_{\tau}$, the equivalent width and equivalent depth in kilometers
respectively. The terms “ing” and “egr” stand for ingress and egress,
respectively, and refer to the fact that the detection occurred before and
after the occultation by Quaoar’s main body, respectively. The error bars in
parentheses are at the 1-$\sigma$ level. aFrom Lorentzian fit. The width
$W_{\rm r}$ is defined as the FWHM of the $\tau_{\rm{N}}$ profile. The Ep and
Aτ values where obtained from respective integrals in the ring profile (see
Sec. 3); bNot detected.

Figure 8: Fits to the Q1R and Q2R light curve data taken with Gemini (z’),
Gemini (r’), and CFHT (Ks) (the corresponding filters are indicated in
parentheses). These light curves are plotted as a function of the time in
seconds relative to the local closest approach (C/A). The blue dots represent
residuals with an arbitrary vertical offset for clarity. Note: the y-axis
scale is different in the rightmost panel in each row (dense part of Q1R).

Figure 9: Fits to the Q1R dense ring in light curve data taken with TUHO and
TAOS II telescopes, from top to bottom. TAOS II data are stacked every six
images. These light curves are plotted as a function of the time in seconds
relative to the local closest approach (C/A), with a different y-axis scale in
the rightmost panel in each row (dense part of Q1R). The green vertical dashed
lines represent the theoretical times for the Q1R and Q2R rings. Note:
although marginal, the two data points that detect the Q1R ring in the TAOS II
light curve are statistically significant at the 4.2-$\sigma$ level.


# On the Maximum of Dependent Gaussian Random Variables: A Sharp Bound for the
Lower Tail

Miles E. Lopes  Department of Statistics, University of California, Davis;
Davis, California 95616<EMAIL_ADDRESS>
###### Abstract.

Although there is an extensive literature on the maxima of Gaussian processes,
there are relatively few non-asymptotic bounds on their lower-tail
probabilities. In the context of a finite index set, this paper offers such a
bound, while also allowing for many types of dependence. Specifically, let
$(X_{1},\dots,X_{n})$ be a centered Gaussian vector, with standardized
entries, whose correlation matrix $R$ satisfies $\max_{i\neq
j}R_{ij}\leq\rho_{0}$ for some constant $\rho_{0}\in(0,1)$. Then, for any
$\epsilon_{0}\in(0,\sqrt{1-\rho_{0}})$, we establish an upper bound on the
probability $\mathbb{P}(\max_{1\leq i\leq
n}X_{i}\leq\epsilon_{0}\sqrt{2\log(n)})$ that is a function of
$\rho_{0},\,\epsilon_{0},$ and $n$. Furthermore, we show the bound is sharp,
in the sense that it is attained up to a constant, for each $\rho_{0}$ and
$\epsilon_{0}$.

###### Key words and phrases:

Gaussian processes, comparison inequalities, small deviations

###### 2000 Mathematics Subject Classification:

Primary 60G15; Secondary 60E15.

The author was supported in part by NSF Grant DMS 1613218.

##  1\. Introduction

The maxima of Gaussian processes play an essential role in many aspects of
probability and statistics, and the literature describing them is highly
developed [LLR83, Adl90, Lif95, LT13, Tal14]. Within this area, a variety of
questions are related to showing that the maximum of a process is unlikely to
deviate far above, or below, its mean. However, in comparison to the set of
tools for handling the upper tail of a maximum, there are relatively few
approaches for the lower tail. (Additional commentary related to this
distinction may be found in [Tal14, p.viii] [LS01, Sec.4.2] [Lif95, Sec.18].)

In this paper, we consider the setting of a finite index set, where
$X=(X_{1},\dots,X_{n})$ is a Gaussian vector with $\mathbb{E}[X_{i}]=0$ and
$\operatorname{var}(X_{i})=1$ for all $1\leq i\leq n$. The correlation matrix
of $X$ is denoted by $R\in\mathbb{R}^{n\times n}$, and throughout the paper,
we make the basic assumption that there is some fixed constant
$\rho_{0}\in(0,1)$ such that

(1.1) $\max_{i\neq j}R_{ij}\leq\rho_{0}.$

In particular, the matrix $R$ is not required to be invertible. Letting

(1.2) $M_{n}(X)=\max_{1\leq i\leq n}X_{i},$

our aim is to derive a non-asymptotic upper bound on $\mathbb{P}(M_{n}(X)\leq
t)$, where $t$ is a suitable point in the lower tail.

###  1.1. Background.

To provide background on lower-tail bounds for $M_{n}(X)$, we now briefly
review some leading results. Under the preceding conditions, the well-known
concentration inequality for Lipschitz functions of Gaussian vectors implies
that for any $s>0$,

(1.3) $\mathbb{P}\Big{(}M_{n}(X)\ \leq\ \text{med}[M_{n}(X)]-s\Big{)}\ \leq\
e^{-s^{2}/2},$

where $\text{med}[\cdot]$ is any median [ST74, Bor75, LT13]. Although this
bound is broadly applicable, it can fail to describe lower-tail probabilities
smaller than $\mathcal{O}(n^{-1})$. To see this, consider using (1.3) to bound
the probability $\mathbb{P}(M_{n}(X)\leq\delta_{0}\text{med}[M_{n}(X)])$ for
some $\delta_{0}\in(0,1)$. If the entries of $X$ are independent, then
$\text{med}[M_{n}(X)]=\sqrt{2\log(n)}(1+o(1))$, and it follows that (1.3)
cannot give a bound better than $\mathcal{O}(n^{-1})$ in this case.
Furthermore, such a bound is far too large. In fact, when the entries of $X$
are independent, the actual size of this probability is smaller than
$\mathcal{O}(n^{-\kappa_{0}})$ for _any_ $\kappa_{0}>1$, no matter which
$\delta_{0}$ is chosen. This limitation is also of more general interest,
because it is a prototypical instance of the “superconcentration phenomenon”,
which has been a topic of much ongoing research activity [Cha14].

Very recently, a breakthrough result of Paouris and Valettas [PV18, eqn. 1.5]
improved upon (1.3) by showing that the inequality

(1.4) $\mathbb{P}\Big{(}M_{n}(X)\leq\text{med}[M_{n}(X)]-s\Big{)}\ \leq\
\textstyle\frac{1}{2}\exp\Big{\\{}-\textstyle\frac{\pi}{1024}\frac{s^{2}}{\operatorname{var}[M_{n}(X)]}\Big{\\}},$

holds for any $s>0$. The improvement over (1.3) is most readily apparent when
the entries of $X$ are independent, because the bound
$\operatorname{var}(M_{n}(X))=\mathcal{O}(1/\log(n))$ is known to hold in that
case [Cha14, Prop. 9.5]. Notably, the result underlying (1.4) is also quite
general, since it allows $M_{n}(X)$ to be replaced with any convex function of
$X$. Another interesting feature of (1.4) is that it specifically describes
the lower tail, and does not have an analogue for the upper tail (whereas the
bound (1.3) does).

Despite the progress achieved by the bound (1.4), it is natural to suspect
that the constant $\frac{\pi}{1024}$ can be sharpened, and our main result
(Theorem 2.2) leads to such an improvement in the following way. First note
that if $\delta_{0}\in(0,1)$ is fixed, then (1.4) becomes

(1.5) $\mathbb{P}\Big{(}M_{n}(X)\ \leq\
\delta_{0}\,\text{med}[M_{n}(X)]\Big{)}\ \leq\
\textstyle\frac{1}{2}\exp\Big{\\{}-\textstyle\frac{\pi(1-\delta_{0})^{2}}{1024}\frac{\text{med}[M_{n}(X)]^{2}}{\operatorname{var}[M_{n}(X)]}\Big{\\}}.$

In the particular case that $R_{ij}=\rho_{0}$ for all $i\neq j$, it is
possible to show that

(1.6) $\lim_{n\to\infty}\
\textstyle\frac{\text{med}[M_{n}(X)]^{2}}{\operatorname{var}[M_{n}(X)]}\frac{1}{2\log(n)}\
=\ \frac{1-\rho_{0}}{\rho_{0}},$

which is handled later in Proposition 4.3. Hence, in a worst-case sense with
respect to the dependence condition (1.1), the exponent in (1.5) cannot be
asymptotically better than

(1.7)
$-\textstyle\frac{2\pi}{1024}\frac{1-\rho_{0}}{\rho_{0}}(1-\delta_{0})^{2}\,\log(n).$

As an improvement, our main result provides an inequality similar to (1.5),
which holds uniformly with respect to (1.1), and has an exponent equal to

(1.8) $\ \ \ \ \ \ \ \ \ \,\ \ \ \
-\textstyle\frac{1-\rho_{0}}{\rho_{0}}(1-\delta_{0})^{2}\,\log(n)+\mathcal{O}(\log\log(n)).$

Thus, up to a $\log\log(n)$ term, the $\frac{2\pi}{1024}$ in (1.7) can be
replaced with 1. (From a purely heuristic standpoint, this substitution makes
sense, because it corresponds to replacing $\frac{\pi}{1024}$ with
$\frac{1}{2}$ in (1.4), which gives the usual form of a Gaussian tail bound.)

One more result to mention is a lower-tail bound due to Hartigan [Har14,
Theorem 3.4]. This bound is of a different form than those mentioned
previously, and is based upon the “residual variables” defined by
$V_{i}=X_{i}-\mathbb{E}[X_{i}|X_{1},\dots,X_{i-1}]$. If we put
$\kappa(n,\varepsilon)=2\log(n/\sqrt{2\pi})-2\log\log(1/\varepsilon)$ for some
$\varepsilon\in(0,1)$, and define $\sigma_{n}^{2}=\min_{1\leq i\leq
n}\operatorname{var}(V_{i})$, then as long as $\kappa(n,\varepsilon)\geq 6$,
the bound may be written as

(1.9) $\mathbb{P}\bigg{(}M_{n}(X)\ \leq\
\sigma_{n}\sqrt{\kappa(n,\varepsilon)-\log(\kappa(n,\varepsilon))}-\sqrt{1-\sigma_{n}^{2}}|\Phi^{-1}(\varepsilon)|\bigg{)}\
\leq\ 2\varepsilon,$

where $\Phi$ is the standard normal distribution function.111When referencing
the main result in the paper [Har14], note that a square root is omitted, and
that the quantity $\sigma\tau\Phi^{-1}(\alpha)$ should be replaced with
$\tau\Phi^{-1}(\alpha)$. In addition, the $\lambda_{n}\sqrt{2\log(n)}$ in the
abstract should be replaced with $\sqrt{2\lambda_{n}\log(n)}.$ For a suitable
choice of $\varepsilon$, this bound has some qualitative similarities with the
Paouris-Valettas bound (1.4) and our main result in Theorem 2.2. However, due
to the intricate dependence on the parameters
$(\varepsilon,n,\sigma_{n},\rho_{0})$, it is not obvious if the bound (1.9) is
sharp under (1.1). By contrast, the simple form of the bound given in Theorem
2.2 makes it possible to see immediately that the bound is sharp. Also, it is
notable that all three arguments underlying (1.4), (1.9), and Theorem 2.2 are
entirely different.

###  1.2. Remarks on proof and outline

At a high level, our proof is based on the classical idea of comparing $X$
with another Gaussian vector whose entries are equicorrelated. This idea seems
to have its roots in the early work of Berman [Ber62, Ber64] on the maxima of
stationary Gaussian processes, and hinges on Slepian’s lemma [Sle62]: If
$X^{\prime}\in\mathbb{R}^{n}$ is a centered Gaussian vector with standardized
entries, and if its correlation matrix $R^{\prime}$ satisfies

$R_{ij}\ \leq\ R_{ij}^{\prime}$

for all $(i,j)$, then the inequality

(1.10) $\mathbb{P}\big{(}M_{n}(X)\leq t\big{)}\ \leq\
\mathbb{P}\big{(}M_{n}(X^{\prime})\leq t\big{)}$

holds for all $t\in\mathbb{R}$. For the purpose of deriving an upper bound on
$\mathbb{P}(M_{n}(X)\leq t)$, the equicorrelated case where
$R_{ij}^{\prime}=\rho_{0}$ for all $i\neq j$ is of special importance, because
it is “maximal” with respect to (1.10). More precisely, for any fixed
$\rho_{0}$, the inequality (1.10) holds over the entire set of matrices $R$
with $\max_{i\neq j}R_{ij}\leq\rho_{0}$.

Although this general strategy has been used in a variety of works to analyze
$M_{n}(X)$, the associated results have been predominantly asymptotic, and
focused on deviations of order $o(1)$. (We refer to the papers [Ber64, PI67,
PI69, MY75], as well as Chapter 6 of the book [LLR83] for an overview.)
Alternatively, we study deviations that are proportional to $\sqrt{\log(n)}$,
which extend much deeper into the lower tail.

The remainder of the paper is organized as follows. After presenting our main
result in Section 2, we discuss some consequences in Section 3, showing how
the result can be improved when $R$ has additional structure, and how it can
be applied to small-ball probabilities. The proof of the main result is given
in Section 4. With regard to notation, the symbols $c$ and $C$ always denote
constants that do not depend on $n$, but their dependence on other parameters
may change from line to line. The abbreviation
$\ell_{2}(n):=\log\log(\max\\{n,3\\})$ is also used.

##  2\. Main result

To clarify the statement of our main result, we first state a basic
proposition describing the sizes of $\mathbb{E}[M_{n}(X)]$ and
$\text{med}[M_{n}(X)]$ under (1.1). More specifically, the proposition shows
that the value $\sqrt{2(1-\rho_{0})\log(n)}$ is a natural reference level for
a lower-tail bound. Although this fact could be considered well known to
specialists, it is not easily referenced in the form given below, and so we
provide a short proof at the end of Section 4.

######  Proposition 2.1.

Let $\mu_{n}$ stand for either $\mathbb{E}[M_{n}(X)]$ or
$\textup{med}[M_{n}(X)]$. Fix a constant $\rho_{0}\in(0,1)$, and suppose the
condition (1.1) holds. Then, there is a universal constant $c_{0}>0$, such
that

(2.1) $\begin{split}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mu_{n}&\ \geq\
\sqrt{2(1-\rho_{0})\log(n)}-c_{0}\sqrt{\ell_{2}(n)}.\end{split}$

Furthermore, if $R_{ij}=\rho_{0}$ for all $i\neq j$, then

(2.2) $\mu_{n}\ \leq\ \sqrt{2(1-\rho_{0})\log(n)}.$

The following theorem is the main result of the paper.

######  Theorem 2.2.

Fix two constants $\delta_{0},\rho_{0}\in(0,1)$, and suppose the condition
(1.1) holds. In addition, define the constants

(2.3) $\alpha_{0}=\textstyle\frac{(1-\rho_{0})(1-\delta_{0})^{2}}{\rho_{0}}\ \
\ \ \text{ and }\ \ \ \
\beta_{0}=\textstyle\frac{(1-\rho_{0})(1-\delta_{0})}{\rho_{0}}.$

Then, there is a constant $C>0$ depending only on $(\delta_{0},\rho_{0})$,
such that

(2.4) $\
\mathbb{P}\bigg{(}M_{n}(X)\,\leq\,\delta_{0}\sqrt{2(1-\rho_{0})\log(n)}\bigg{)}\
\leq\ C\,n^{-\alpha_{0}}\log(n)^{\frac{\beta_{0}-1}{2}}.$

Furthermore, the bound (2.4) is sharp in the sense that if $R_{ij}=\rho_{0}$
for all $i\neq j$, then there is a constant $c>0$ depending only on
$(\delta_{0},\rho_{0})$, such that

(2.5)
$\mathbb{P}\bigg{(}M_{n}(X)\,\leq\,\delta_{0}\sqrt{2(1-\rho_{0})\log(n)}\bigg{)}\
\geq\ c\,n^{-\alpha_{0}}\log(n)^{\frac{\beta_{0}-1}{2}}.$

To comment on some basic features of the theorem, first note that the dominant
exponent satisfies $\alpha_{0}\to\infty$ as $\rho_{0}\to 0$. Hence, the bound
respects the fact that the lower-tail probability decays faster than any power
of $n^{-1}$ when the entries of $X$ are independent (as discussed in Section
1.1). Second, the theorem conforms with the reference level motivated by
Proposition 2.1, since (2.4) implies that $\text{med}[M_{n}(X)]$ cannot be
much less than $\sqrt{2(1-\rho_{0})\log(n)}$.

##  3\. Some consequences

This brief section presents two direct corollaries of Theorem 2.2.

###  3.1. Taking advantage of correlation structure

If the matrix $R$ satisfies $\max_{i\neq j}R_{ij}=\rho_{0}$, but most off-
diagonal entries are substantially less than $\rho_{0}$, then the upper bound
in Theorem 2.2 can be conservative. Fortunately, the following simple
corollary shows this is not necessarily a problem. Roughly speaking, if there
is a number $\tilde{\rho}_{0}\in(0,\rho_{0})$ such that most off-diagonal
entries satisfy $R_{ij}\leq\tilde{\rho}_{0}$, then Theorem 2.2 can be
essentially re-used with $\rho_{0}$ being replaced by $\tilde{\rho}_{0}$.
Furthermore, this can be done even when $\rho_{0}=1$.

######  Corollary 3.1.

Fix two constants $\delta_{0},\tilde{\rho}_{0}\in(0,1)$. Suppose there is an
index set $J\subset\\{1,\dots,n\\}$ with cardinality $\tilde{n}\geq 2$, such
that $R_{ij}\leq\tilde{\rho}_{0}$ for all distinct $i,j\in J$. Lastly, define
the constants

$\tilde{\alpha}_{0}\ =\
\textstyle\frac{(1-\tilde{\rho}_{0})(1-\delta_{0})^{2}}{\tilde{\rho}_{0}}\text{
\ \ \ \ and \ \ \ \ }\tilde{\beta}_{0}\ =\
\textstyle\frac{(1-\tilde{\rho}_{0})(1-\delta_{0})}{\tilde{\rho}_{0}}.$

Then, there is a constant $C>0$ depending only on
$(\delta_{0},\tilde{\rho}_{0})$ such that

(3.1)
$\mathbb{P}\bigg{(}M_{n}(X)\,\leq\,\delta_{0}\sqrt{2(1-\tilde{\rho}_{0})\log(\tilde{n})}\bigg{)}\
\leq\
C\,\tilde{n}^{-\tilde{\alpha}_{0}}\,\log(\tilde{n})^{\frac{\tilde{\beta}_{0}-1}{2}}.$

If detailed knowledge of $R$ is available, the corollary can be further
refined by carefully choosing $\tilde{\rho}_{0}$ and $J$. Indeed, there is a
tradeoff involved, because a smaller value of $\tilde{\rho}_{0}$ improves the
exponent $\tilde{\alpha}_{0}$, but reduces $\tilde{n}$.

###  3.2. Upper bounds for small-ball probabilities

Small-ball probabilities are known to be an important ingredient in the study
of Gaussian processes [LS01], and they have the form
$\mathbb{P}(\|X\|_{\infty}\leq t)$, where $\|X\|_{\infty}=\max_{1\leq i\leq
n}|X_{i}|$, and $t$ is suitably small. Such probabilities are also a special
case of the previous lower-tail probabilities, since they can be written as
$\mathbb{P}(M_{2n}(Y)\leq t)$ with $Y=(-X,X)\in\mathbb{R}^{2n}$.

One of the well-known challenges involved in analyzing small-ball
probabilities is that relatively few general results are available. In this
regard, two noteworthy bounds are due to Latała-Oleszkiewicz [LO05, Theorem
4], and Paouris-Valettas [PV18, Theorem 3.3]. When $X\in\mathbb{R}^{n}$ is a
centered Gaussian vector with standardized entries, these bounds respectively
state that if $\delta_{0}\in(0,1/2)$, then

(3.2) $\ \ \ \mathbb{P}\Big{(}\|X\|_{\infty}\leq\delta_{0}m_{n}\Big{)}\ \leq\
\textstyle\frac{1}{2}\exp\Big{\\{}-\frac{1}{4}m_{n}^{2}\log(\frac{1}{2\delta_{0}})\Big{\\}},$

and

(3.3) $\mathbb{P}\Big{(}\|X\|_{\infty}\leq\delta_{0}m_{n}\Big{)}\ \leq\
\textstyle\frac{1}{2}\exp\Big{\\{}-c\,\textstyle\frac{m_{n}^{2}}{\varsigma_{n}^{2}}\log(\frac{1}{\delta_{0}})\Big{\\}},$

where $m_{n}=\text{med}[\|X\|_{\infty}]$,
$\varsigma_{n}^{2}=\operatorname{var}[\|X\|_{\infty}]$, and $c>0$ is a
universal constant. Due to the fact that the constant $c>0$ is not easily
specified, the first bound (3.2) has an advantage insofar as its exponent is
explicit. On the other hand, the second bound (3.3) may be preferred if
$\varsigma_{n}^{2}\to 0$ as $n\to\infty$ (as in the case of independent
variables, cf. [Cha14, Prop. 9.5]).

In cases where $\delta_{0}$ is not too small, the following corollary of
Theorem 2.2 can enhance some aspects of the bounds (3.2) and (3.3).

######  Corollary 3.2.

Fix two numbers $\delta_{0},\rho_{0}\in(0,1)$, and let
$(\alpha_{0},\beta_{0})$ be as in (2.3). Also, let $C>0$ be as in (2.4), and
suppose $\max_{i\neq j}|R_{ij}|\leq\rho_{0}$. Then,

(3.4) $\
\mathbb{P}\Big{(}\|X\|_{\infty}\leq\delta_{0}\sqrt{2(1-\rho_{0})\log(2n)}\Big{)}\
\leq\ C\,(2n)^{-\alpha_{0}}\log(2n)^{\frac{\beta_{0}-1}{2}}.$

In addition, if $R_{ij}\in[0,\rho_{0}]$ for all $i\neq j$, then

(3.5)
$\mathbb{P}\Big{(}\|X\|_{\infty}\leq\delta_{0}\sqrt{2(1-\rho_{0})\log(n)}\Big{)}\
\leq\ C^{2}\,n^{-2\alpha_{0}}\log(n)^{\beta_{0}-1}.$

Although it is difficult to make an exact comparison with the bounds (3.2) and
(3.3), it is possible to point out some basic differences. To do this, recall
that $\alpha_{0}$ can be made arbitrarily large when $\rho_{0}$ is small
enough, and that $m_{n}\leq\sqrt{2\log(2n)}$ (cf. [Kwa94]). Based on these
observations, the right side of (3.4) will be of smaller order than the right
side of (3.2) when $\rho_{0}$ is sufficiently small (for a given
$\delta_{0}$). With regard to the bound (3.3), it is known from [Cha14, Prop.
9.5] that there is a universal constant $c_{1}>0$ for which

$\frac{1}{\varsigma_{n}^{2}}\ \geq\
\frac{c_{1}}{\rho_{0}+\textstyle\frac{1}{\log(n)}}.$

Hence, when $\rho_{0}$ is small, the bounds (3.4) and (3.3) are roughly
similar, except that the exponent in (3.4) is explicit. On this point, it is
also worth mentioning that even if the $c$ in (3.3) can be specified, it is
still difficult to obtain good constants when bounding $\varsigma_{n}^{2}$.

_Proof of Corollary 3.2._ The first bound (3.4) follows immediately from
Theorem 2.2 by considering the vector $Y=(X,-X)\in\mathbb{R}^{2n}$. To prove
the second bound (3.5), let $R^{\prime}\in\mathbb{R}^{n\times n}$ denote the
correlation matrix whose off-diagonal entries are equal to $\rho_{0}$, and let
$X^{\prime}\sim N(0,R^{\prime})$. In addition, let
$Y^{\prime}\in\mathbb{R}^{2n}$ be the random vector obtained by concatenating
two independent copies of $X^{\prime}$. Due to the assumption that
$R_{ij}\in[0,\rho_{0}]$ for all $i\neq j$, the correlation matrices of $Y$ and
$Y^{\prime}$ are related by the following entrywise inequality,

(3.6) $\text{cor}(Y)\ =\ \Big{(}\begin{array}[]{cc}R&\\!\\!-R\\\
-R&R\end{array}\Big{)}\ \leq\ \Big{(}\begin{array}[]{cc}R^{\prime}&\\!\\!0\\\
0&R^{\prime}\end{array}\Big{)}\ =\ \text{cor}(Y^{\prime}).$

Therefore, the vectors $Y$ and $Y^{\prime}$ satisfy the conditions of
Slepian’s lemma (which allows for singular correlation matrices). In turn, if
we define the number

(3.7) $t_{n}=\delta_{0}\sqrt{2(1-\rho_{0})\log(n)},$

then we obtain

(3.8) $\begin{split}\mathbb{P}\big{(}\|X\|_{\infty}\leq t_{n}\big{)}&\ =\
\mathbb{P}\big{(}M_{2n}(Y)\leq t_{n}\big{)}\\\\[8.5359pt] &\ \leq\
\mathbb{P}\big{(}M_{2n}(Y^{\prime})\leq t_{n}\big{)}\\\\[8.5359pt] &\ =\
\Big{\\{}\mathbb{P}\big{(}M_{n}(X^{\prime})\leq
t_{n}\big{)}\Big{\\}}^{2}\\\\[8.5359pt] &\ \leq\
C^{2}\,n^{-2\alpha_{0}}\log(n)^{\beta_{0}-1},\end{split}$

as needed.∎

##  4\. Proofs for Section 2

Remark. Throughout the proofs, we always assume that $n$ is sufficiently large
for any given expression to make sense — since this can be accommodated by an
adjustment of constants in the results.

Proof of the upper bound (2.4). Let $X^{\prime}\sim N(0,R^{\prime})$ be an
equicorrelated Gaussian vector, as in the proof of Corollary 3.2. It follows
that if $Z_{0},Z_{1},\dots,Z_{n}$ denote independent $N(0,1)$ random
variables, then the coordinates of $X^{\prime}$ may be jointly represented as

(4.1) $X_{i}^{\prime}=\sqrt{\rho_{0}}Z_{0}+\sqrt{1-\rho_{0}}Z_{i}.$

This yields a valuable representation of the maximum,

(4.2) $M_{n}(X^{\prime})=\sqrt{\rho_{0}}Z_{0}+\sqrt{1-\rho_{0}}\max_{1\leq
i\leq n}Z_{i}.$

In other words, $M_{n}(X^{\prime})$ is the convolution of the variables on the
right side. Recalling the definition of $t_{n}$ in (3.7), a simple calculation
then gives an exact formula,

(4.3) $\mathbb{P}\Big{(}M_{n}(X^{\prime})\leq
t_{n}\Big{)}=\int_{-\infty}^{\infty}\psi_{n}(s)ds,$

where the integrand is defined by

(4.4) $\psi_{n}(s)\ =\
\phi(s)\Phi^{n}\Big{(}\textstyle\frac{t_{n}-\sqrt{\rho_{0}}s}{\sqrt{1-\rho_{0}}}\Big{)},$

with $\phi$ being the standard normal density.

The remainder of the proof consists in bounding integral
$\int_{-\infty}^{\infty}\psi_{n}(s)ds$ over the intervals $(-\infty,-c_{n}]$,
$[-c_{n},0]$, and $[0,\infty)$, where we put

(4.5)
$c_{n}=\sqrt{\textstyle\frac{2(1-\rho_{0})\log(n)}{\rho_{0}}}\Big{\\{}1-\delta_{0}-\textstyle\frac{\frac{1}{4}\ell_{2}(n)}{\log(n)}\Big{\\}}.$

The choice of this value as a cut-off point for decomposing the integral (4.3)
is an important element of the proof. In particular, the $1/4$ coefficient on
the $\ell_{2}(n)/\log(n)$ term is essential.

Handling the integral over $[-c_{n},0]$ is the most involved portion of the
proof, and is postponed to Lemma 4.1. Once the proof of that lemma is
complete, it will be straightforward to control the integral over
$[0,\infty)$, which is done in Lemma 4.2. For the moment, we only handle the
interval $(-\infty,-c_{n}]$, since it requires no further preparation. Indeed,
we have

(4.6) $\begin{split}\int_{-\infty}^{-c_{n}}\psi_{n}(s)&\ \leq\
\Phi(-c_{n})\\\\[5.69046pt] &\ \leq\
\frac{e^{-c_{n}^{2}/2}}{\sqrt{2\pi}c_{n}}\ \ \ \ \text{(Mill's
inequality)}\\\\[11.38092pt] &\ \leq\
\textstyle\frac{C}{\sqrt{\log(n)}}\,n^{-\alpha_{0}}\,\exp\big{\\{}\frac{\beta_{0}}{2}\ell_{2}(n)\big{\\}}\\\\[8.5359pt]
&\ =\ Cn^{-\alpha_{0}}\log(n)^{\frac{\beta_{0}-1}{2}}\end{split}$

where $C$ only depends on $\delta_{0}$ and $\rho_{0}$. This completes the
proof of the upper bound (2.4).

Proof of the lower bound (2.5). It suffices to derive a lower bound on the
integral $\int_{-\infty}^{-c_{n}}\psi_{n}(s)ds$. Using the fact that the
function
$\Phi^{n}\Big{(}\textstyle\frac{t_{n}-\sqrt{\rho_{0}}s}{\sqrt{1-\rho_{0}}}\Big{)}$
is decreasing in $s$, we have

(4.7) $\int_{-\infty}^{-c_{n}}\psi_{n}(s)ds\ \geq\
\Phi^{n}\Big{(}\textstyle\frac{t_{n}+\sqrt{\rho_{0}}c_{n}}{\sqrt{1-\rho_{0}}}\Big{)}\Phi(-c_{n}).$

If we define the number

(4.8) $\begin{split}r_{n}&\ =\
\textstyle\frac{t_{n}+\sqrt{\rho_{0}}c_{n}}{\sqrt{1-\rho_{0}}}\\\\[5.69046pt]
&\ =\
\sqrt{2\log(n)}\Big{\\{}1-\textstyle\frac{\frac{1}{4}\ell_{2}(n)}{\log(n)}\Big{\\}},\end{split}$

then Mill’s inequality gives

(4.9) $\begin{split}\Phi(r_{n})&\ \geq\
1-\textstyle\frac{1}{\sqrt{2\pi}r_{n}}e^{-r_{n}^{2}/2}\\\\[5.69046pt] &\ \geq\
1-\textstyle\frac{C}{\sqrt{\log(n)}}e^{-\log(n)}e^{\frac{1}{2}\ell_{2}(n)}\\\\[5.69046pt]
&\ =\ 1-\textstyle\frac{C}{n}.\end{split}$

Hence, the limit $(1-C/n)^{n}\to\exp(-C)$ as $n\to\infty$ shows that
$\Phi^{n}(r_{n})$ is lower bounded by a positive constant for all large $n$.
Finally, regarding the factor $\Phi(-c_{n})$ in (4.7), the lower-bound version
Mill’s inequality [Dur05, Theorem 1.4] gives

(4.10) $\begin{split}\Phi(-c_{n})&\ \geq\
\Big{\\{}\textstyle\frac{1}{c_{n}}-\textstyle\frac{1}{c_{n}^{3}}\Big{\\}}\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}c_{n}^{2}},\\\\[8.5359pt]
&\ \geq\ cn^{-\alpha_{0}}\log(n)^{\frac{1}{2}(\beta_{0}-1)},\end{split}$

where the second step follows from a calculation very similar to (4.6). This
completes the proof of the lower bound (2.5) and the theorem. ∎

######  Lemma 4.1.

There is a constant $C>0$ depending only on $(\rho_{0},\delta_{0})$ such that

(4.11) $\int_{-c_{n}}^{0}\psi_{n}(s)ds\ \leq\
C\,n^{-\alpha_{0}}\log(n)^{\frac{1}{2}(\beta_{0}-1)}.$

###### Proof.

To simplify the analysis of $\psi_{n}(s)$, we will parameterize $s$ as
function of a number $\delta$ lying in an interval $\mathcal{I}_{n}$ defined
by

$\mathcal{I}_{n}=\Big{[}0\,,\,1-\delta_{0}-\textstyle\frac{\frac{1}{4}\ell_{2}(n)}{\log(n)}\Big{]}.$

Specifically, we write

$s=s_{n}(\delta)\ =-a_{n}\delta,$

where

$a_{n}=\sqrt{\textstyle\frac{2(1-\rho_{0})\log(n)}{\rho_{0}}}.$

In particular, note that $s_{n}(\cdot)$ maps $\mathcal{I}_{n}$ to the interval
$[-c_{n},0]$. As a means of simplifying the factor
$\Phi^{n}\big{(}\textstyle\textstyle\frac{t_{n}-\sqrt{\rho_{0}}s}{\sqrt{1-\rho_{0}}}\big{)}$
in the definition of $\psi_{n}(s)$, let

(4.12)
$\begin{split}u_{n}(\delta)&=\textstyle\frac{t_{n}-\sqrt{\rho_{0}}s_{n}(\delta)}{\sqrt{1-\rho_{0}}}\\\\[8.5359pt]
&=\sqrt{2\log(n)}(\delta_{0}+\delta).\end{split}$

It follows that for all $\delta\in\mathcal{I}_{n}$,

(4.13)
$\log\psi_{n}(s_{n}(\delta))=-\textstyle\frac{(1-\rho_{0})\cdot\log(n)\cdot\delta^{2}}{\rho_{0}}+n\log\Phi(u_{n}(\delta))-\log(\sqrt{2\pi}).$

Due to the lower-bound form of Mill’s inequality, we have

(4.14) $\begin{split}\Phi(u_{n}(\delta))&\ \leq\
1-\textstyle\frac{1}{\sqrt{2\pi}}\Big{\\{}\textstyle\frac{1}{u_{n}(\delta)}-\textstyle\frac{1}{u_{n}^{3}(\delta)}\Big{\\}}\displaystyle\exp({-\textstyle\frac{1}{2}u_{n}(\delta)^{2}}),\\\\[8.5359pt]
&\ =\ 1-b_{n}(\delta)\end{split}$

where $b_{n}(\delta)$ is defined by the last line. When $n$ is sufficiently
large, it is simple to check that the condition $0<b_{n}(\delta)<1$ holds for
all $\delta\in\mathcal{I}_{n}$, which gives

$\log(1-b_{n}(\delta))\ \leq\ -b_{n}(\delta).$

Combining the last few steps, the following bound holds for all
$\delta\in\mathcal{I}_{n}$,

(4.15) $\begin{split}n\log\Phi(u_{n}(\delta))\ &\
\leq-n\,b_{n}(\delta)\\\\[5.69046pt] &\ \leq\
\textstyle\frac{-cn}{\sqrt{\log(n)}}\exp\big{\\{}-\log(n)(\delta_{0}+\delta)^{2}\big{\\}},\end{split}$

where $c>0$ depends only on $\delta_{0}$ and $\rho_{0}$.

The work up to this point provides us with a useful majorant for $\psi_{n}$.
By looking at the equation (4.13) and the bound (4.15), it is clear that if we
define the function

(4.16)
$\begin{split}f_{n}(\delta)&=\textstyle\frac{(1-\rho_{0})\log(n)\delta^{2}}{\rho_{0}}+\textstyle\frac{cn}{\sqrt{\log(n)}}\displaystyle\exp\big{\\{}{-\log(n)(\delta_{0}+\delta)^{2}}\big{\\}},\end{split}$

then the bound

(4.17) $\psi_{n}(s_{n}(\delta))\ \leq\ \exp\\{-f_{n}(\delta)\\}$

holds for all $\delta\in\mathcal{I}_{n}$. Integrating this bound gives

(4.18) $\int_{-c_{n}}^{0}\psi_{n}(s)ds\ \leq\
a_{n}\\!\int_{\mathcal{I}_{n}}\exp\\{-f_{n}(\delta)\\}d\delta.$

We now parameterize $\delta$ as a function of a positive number $\eta$ using

$\delta=\delta_{n}(\eta)=1-\delta_{0}-\textstyle\frac{\eta\ell_{2}(n)}{\log(n)}.$

Specifically, if we define the interval

$\mathcal{J}_{n}=\Big{[}\textstyle\frac{1}{4}\,,\,\frac{(1-\delta_{0})\log(n)}{\ell_{2}(n)}\Big{]},$

then $\delta_{n}(\cdot)$ maps $\mathcal{J}_{n}$ to $\mathcal{I}_{n}$, and the
integral bound (4.18) becomes

(4.19) $\begin{split}\int_{-c_{n}}^{0}\psi_{n}(s)ds&\
\leq\textstyle\frac{a_{n}\ell_{2}(n)}{\log(n)}\displaystyle\int_{\mathcal{J}_{n}}\exp\\{-f_{n}(\delta_{n}(\eta))\\}d\eta.\end{split}$

The remainder of the proof will be divided into two parts, in which the
integral over $\mathcal{J}_{n}$ is decomposed with the subintervals

$\mathcal{J}_{n}^{\prime}=\Big{[}\textstyle\frac{1}{4}\,,\,\textstyle\frac{\log(n)^{1/4}}{\ell_{2}(n)}\Big{]}\
\ \ \ \ \text{ and }\ \ \ \ \
\mathcal{J}_{n}^{\prime\prime}=\Big{[}\textstyle\frac{\log(n)^{1/4}}{\ell_{2}(n)},\textstyle\frac{(1-\delta_{0})\log(n)}{\ell_{2}(n)}\Big{]}.$

In handling these subintervals, it will be convenient to label the summands of
$f_{n}$ in line (4.16) according to

$f_{n}(\delta)=g_{n}(\delta)+h_{n}(\delta).$

_The integral over $\mathcal{J}_{n}^{\prime}$._ By expanding out the square
$\delta_{n}(\eta)^{2}$, and dropping the smallest positive term, the following
bound holds for any $\eta$,

$\exp\big{\\{}\\!-\\!g_{n}(\delta_{n}(\eta))\big{\\}}\ \leq\
n^{-\alpha_{0}}\exp\Big{\\{}2\beta_{0}\ell_{2}(n)\eta\Big{\\}}.$

In addition, if we expand the square $(\delta_{0}+\delta_{n}(\eta))^{2}$, and
use the fact that every $\eta\in\mathcal{J}_{n}^{\prime}$ is bounded above by
$\log(n)^{1/4}/\ell_{2}(n)$, then a short calculation gives

$\begin{split}-h_{n}(\delta_{n}(\eta))&\ \leq\
-ce^{\ell_{2}(n)(2\eta-\frac{1}{2})},\end{split}$

for small enough $c>0$. Directly combining the last two steps gives

(4.20) $\exp\big{\\{}\\!-\\!f_{n}(\delta_{n}(\eta))\big{\\}}\ \leq\
n^{-\alpha_{0}}\exp\Big{\\{}2\beta_{0}\eta\ell_{2}(n)-ce^{\ell_{2}(n)(2\eta-\frac{1}{2})}\Big{\\}}.$

To simplify the previous bound, define

$x(\eta)=2\eta-\textstyle\frac{1}{2}.$

Since $x(\eta)$ is non-negative for all $\eta\in\mathcal{J}_{n}^{\prime}$, we
may approximate $\exp\\{\ell_{2}(n)x(\eta)\\}$ from below using a second-order
Taylor expansion

$e^{\ell_{2}(n)x(\eta)}\ \geq\
1+\ell_{2}(n)x(\eta)+\textstyle\frac{1}{2}\ell_{2}(n)^{2}x(\eta)^{2}.$

After some arithmetic, the bound (4.20) becomes

(4.21) $\begin{split}\exp\big{\\{}\\!-\\!f_{n}(\delta_{n}(\eta))\big{\\}}&\
\leq\ e^{\frac{\beta_{0}}{2}\ell_{2}(n)}\cdot e^{-c}\cdot
n^{-\alpha_{0}}\cdot\varphi_{n}(x(\eta)),\end{split}$

where we define the function

$\varphi_{n}(x)=\exp\Big{\\{}-\big{[}\textstyle\frac{c}{2}\ell_{2}(n)^{2}\big{]}x^{2}+\big{[}(\beta_{0}-c)\ell_{2}(n)\big{]}x\Big{\\}}.$

Integrating the bound (4.21) over $\mathcal{J}_{n}^{\prime}$, we obtain

(4.22)
$\small\begin{split}\textstyle\frac{a_{n}\ell_{2}(n)}{\log(n)}\displaystyle\int_{\mathcal{J}_{n}^{\prime}}\exp\\{-f_{n}(\delta_{n}(\eta))\\}d\eta&\
\ \leq\ \ C\cdot\ell_{2}(n)\cdot\log(n)^{\frac{1}{2}(\beta_{0}-1)}\cdot
n^{-\alpha_{0}}\displaystyle\int_{0}^{\infty}\textstyle\varphi_{n}(x)dx.\end{split}$

To handle the last integral, note that the function $\varphi_{n}(x)$ can be
written in the form $\varphi_{n}(x)=\exp(-ax^{2}+bx)$, and that the following
elementary bound holds for any $a>0$ and $b\in\mathbb{R}$,

$\int_{0}^{\infty}e^{-ax^{2}+bx}dx\ \leq\
\textstyle\frac{\sqrt{\pi}}{\sqrt{a}}\exp\Big{(}\textstyle\frac{b^{2}}{4a}\Big{)}.$

Therefore,

$\displaystyle\int_{0}^{\infty}\varphi_{n}(x)dx\ \leq\
\textstyle\frac{C}{\ell_{2}(n)}.$

Combining this with the bound (4.22) completes the work on
$\mathcal{J}_{n}^{\prime}$.

_The integral over $\mathcal{J}_{n}^{\prime\prime}$._ First note that
$\exp\\{-f_{n}(\delta)\\}\leq\exp\\{-h_{n}(\delta)\\}$. Also, the function
$\exp\\{-h_{n}(\delta_{n}(\eta))\\}$ is decreasing in $\eta$, and so if we
denote the left endpoint of $\mathcal{J}_{n}^{\prime\prime}$ as
$\eta_{n}:=\log(n)^{1/4}/\ell_{2}(n)$, then we have the following height-width
integral bound

$\begin{split}\textstyle\frac{a_{n}\ell_{2}(n)}{\log(n)}\displaystyle\int_{\mathcal{J}_{n}^{\prime\prime}}\exp\big{\\{}\\!-\\!f_{n}(\delta_{n}(\eta))\big{\\}}d\eta&\
\leq\
\textstyle\frac{a_{n}\ell_{2}(n)}{\log(n)}\cdot|\mathcal{J}_{n}^{\prime\prime}|\cdot\exp\\{-h_{n}(\delta_{n}(\eta_{n}))\\}\\\\[8.5359pt]
&\ \leq\
C\cdot\sqrt{\log(n)}\cdot\exp\Big{\\{}\textstyle\frac{-c}{\sqrt{\log(n)}}e^{2\log(n)^{1/4}}\Big{\\}}.\end{split}$

It is simple to check that this bound is of smaller order than
$n^{-\alpha_{0}}\log(n)^{\frac{1}{2}(\beta_{0}-1)}$. This completes the proof
of Lemma 4.1∎

######  Lemma 4.2.

There is a constant $C>0$ depending only on $(\delta_{0},\rho_{0})$ such that

(4.23) $\int_{0}^{\infty}\psi_{n}(s)ds\ \leq\
C\,n^{-\alpha_{0}}\log(n)^{\frac{1}{2}(\beta_{0}-1)}.$

###### Proof.

The function
$\Phi^{n}\Big{(}\textstyle\frac{t_{n}-\sqrt{\rho_{0}}s}{\sqrt{1-\rho_{0}}}\Big{)}$
is decreasing in $s$, and so

$\int_{0}^{\infty}\psi_{n}(s)ds\leq\
\Phi^{n}\Big{(}\textstyle\frac{t_{n}}{\sqrt{1-\rho_{0}}}\Big{)}.$

Then, the inequality (4.15) gives

(4.24)
$\begin{split}\Phi^{n}\Big{(}\textstyle\frac{t_{n}}{\sqrt{1-\rho_{0}}}\Big{)}&\
\leq\ \exp(-h_{n}(0))\\\\[8.5359pt] &\ =\
\exp\Big{\\{}-\textstyle\frac{cn}{\sqrt{\log(n)}}e^{-\log(n)\delta_{0}^{2}}\Big{\\}}\\\\[8.5359pt]
&\ =\
\exp\Big{\\{}\textstyle\frac{-cn^{1-\delta_{0}^{2}}}{\sqrt{\log(n)}}\Big{\\}},\end{split}$

which is clearly of smaller order than the stated bound. ∎

### Proof of Proposition 2.1

We first show (2.1) in the case where $\mu_{n}=\mathbb{E}[M_{n}(X)]$. Let
$X^{\prime}\in\mathbb{R}^{n}$ be as in the proof of Corollary 3.2. By
integrating the inequality (1.10), Slepian’s lemma implies

(4.25) $\mathbb{E}\big{[}M_{n}(X)\big{]}\ \geq\
\mathbb{E}\big{[}M_{n}(X^{\prime})\big{]}.$

In turn, the representation (4.2) gives

(4.26) $\begin{split}\mathbb{E}\big{[}M_{n}(X^{\prime})\big{]}&\ =\
\sqrt{1-\rho_{0}}\,\mathbb{E}\Big{[}\max_{1\leq i\leq
n}Z_{i}\Big{]}\\\\[8.5359pt] &\ \geq\
\sqrt{2(1-\rho_{0})\log(n)}-c_{0}\sqrt{\ell_{2}(n)}\end{split}$

for some universal constant $c_{0}>0$, where the second step can be obtained
from [Mas07, p.66]. To handle $\mu_{n}=\text{med}[M_{n}(X)]$, note that

$\Big{|}\mathbb{E}[M_{n}(X)]-\text{med}[M_{n}(X)]\Big{|}\ \leq\
\sqrt{\operatorname{var}[M_{n}(X)]}\ \leq 1,$

where the second step follows from the Gaussian Poincaré inequality [BLM13,
Theorem 3.20].

To show the upper bound (2.2), note that we may take $X=X^{\prime}$. When
$\mu_{n}=\mathbb{E}[M_{n}(X^{\prime})]$, the result follows from the first
line of (4.26) and the standard inequality $\mathbb{E}[\max_{1\leq i\leq
n}Z_{i}]\leq\sqrt{2\log(n)}$. When $\mu_{n}=\text{med}[M_{n}(X^{\prime})]$,
the result follows from the fact that
$\text{med}[M_{n}(X^{\prime})]\leq\mathbb{E}[M_{n}(X^{\prime})]$, since
$M_{n}(\cdot)$ is a continuous convex function [Kwa94]. ∎

######  Proposition 4.3.

If $R_{ij}=\rho_{0}$ for all $i\neq j$, then

(4.27) $\lim_{n\to\infty}\
\textstyle\frac{\textup{med}[M_{n}(X)]^{2}}{\operatorname{var}[M_{n}(X)]}\frac{1}{2\log(n)}\
=\ \frac{1-\rho_{0}}{\rho_{0}}.$

###### Proof.

Due to Proposition 2.1,

$\lim_{n\to\infty}\textstyle\frac{\text{med}[M_{n}(X)]^{2}}{2\log(n)}=1-\rho_{0}.$

To show that $\operatorname{var}[M_{n}(X)]\to\rho_{0}$, the representation
(4.2) gives

(4.28)
$\begin{split}\operatorname{var}[M_{n}(X)]&=\rho_{0}+(1-\rho_{0})\operatorname{var}\Big{[}\max_{1\leq
i\leq n}Z_{i}\Big{]}\\\\[8.5359pt]
&=\rho_{0}+\mathcal{O}\big{(}\textstyle\frac{1}{\log(n)}\big{)},\end{split}$

where the second line follows from [Cha14, Prop. 9.5].∎

### Acknowledgement

The author thanks Junwen Yao for carefully reading an initial draft of this
work.

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