# Theory of mobility edge and non-ergodic extended phase in coupled random
matrices

Xiaoshui Lin  CAS Key Laboratory of Quantum Information, University of Science
and Technology of China, Hefei, 230026, China  Guang-Can Guo  CAS Key
Laboratory of Quantum Information, University of Science and Technology of
China, Hefei, 230026, China  Synergetic Innovation Center of Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, China  Hefei National Laboratory, University of
Science and Technology of China, Hefei 230088, China  Ming Gong
<EMAIL_ADDRESS>CAS Key Laboratory of Quantum Information, University of
Science and Technology of China, Hefei, 230026, China  Synergetic Innovation
Center of Quantum Information and Quantum Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China  Hefei National Laboratory,
University of Science and Technology of China, Hefei 230088, China

###### Abstract

The mobility edge, as a central concept in disordered models for localization-
delocalization transitions, has rarely been discussed in the context of random
matrix theory (RMT). Here we report a new class of random matrix model by
direct coupling between two random matrices, showing that their overlapped
spectra and un-overlapped spectra exhibit totally different scaling behaviors,
which can be used to construct tunable mobility edges. This model is a direct
generalization of the Rosenzweig-Porter model, which hosts ergodic, localized,
and non-ergodic extended (NEE) phases. A generic theory for these phase
transitions is presented, which applies equally well to dense, sparse, and
even corrected random matrices in different ensembles. We show that the phase
diagram is fully characterized by two scaling exponents, and they are mapped
out in various conditions. Our model provides a general framework to realize
the mobility edges and non-ergodic phases in a controllable way in RMT, which
pave avenue for many intriguing applications both from the pure mathematics of
RMT and the possible implementations of ME in many-body models, chiral
symmetry breaking in QCD and the stability of the large ecosystems.

††preprint: APS/123-QED

Random matrix theory (RMT) dealing with statistics of eigenvalues and
observations (associated with the eigenvectors) in large random matrices
(Forrester, 2010; Brody  _et al._ , 1981; Mehta, 2004; Livan  _et al._ , 2018)
is one of the most interdisciplinary fields in physics and probability theory.
Initially developed by Wigner and Dyson in the analysis of nucleus spectra
(Wigner, 1955; Dyson, 1962a), the RMT has since found numerous applications in
different fields, including quantum chaos (Bohigas  _et al._ , 1984; Berry
_et al._ , 1977; Kos  _et al._ , 2018), Anderson localization (Mirlin, 2000;
Goda  _et al._ , 2006), many-body localization (MBL) (Pal and Huse, 2010;
Nandkishore and Huse, 2015; Abanin  _et al._ , 2019; Alet and Laflorencie,
2018; Imbrie, 2016), quantum chromodynamics (QCD) Stephanov (1996);
Verbaarschot (1994); Kieburg  _et al._ (2013); Verbaarschot and Wettig (2000)
and ecological stability May (1972); Serván  _et al._ (2018); Grilli  _et al._
(2017), etc.. However, the central concept of mobility edge (ME) for energy-
resolved localization-delocalization transition, which has been widely
explored in the single particle disordered models Mott (1987); Semeghini  _et
al._ (2015); Pixley  _et al._ (2015); Biddle  _et al._ (2009); Biddle and Das
Sarma (2010); Ganeshan  _et al._ (2015); Wang  _et al._ (2020, 2022); An  _et
al._ (2021), and many-body disordered models (Wei  _et al._ , 2019; Luitz  _et
al._ , 2015; Brighi  _et al._ , 2020; Deng  _et al._ , 2017; Chanda  _et al._
, 2020; Nag and Garg, 2017; Modak and Mukerjee, ; De Roeck  _et al._ , 2016;
De Roeck and Huveneers, 2017; Zhang  _et al._ , 2022; Lazarides  _et al._ ,
2015; Lüschen  _et al._ , 2018; Kohlert  _et al._ , 2019), has rarely been
discussed in the context of RMT. To the best of our knowledge, it has only
been discussed recently in the Lévy matrices Cizeau and Bouchaud (1994);
Tarquini  _et al._ (2016); Biroli and Tarzia (2021); Aggarwal  _et al._
(2019); Sarkar  _et al._ (2023). However, this model, with a heavy-tail
distribution, is much more challenge to be realized in the current
experiments, in which the uniform and Gaussian random variables are much more
appealing. Therefore, as an open question, the general construction of random
matrices for tunable ME without a heavy-tail distribution and the underlying
mechanism for ME is still unclear, which also limits our understanding of
many-body ME in realistic physical models De Roeck  _et al._ (2016); De Roeck
and Huveneers (2017); Zhang  _et al._ (2022); Lazarides  _et al._ (2015);
Lüschen  _et al._ (2018); Kohlert  _et al._ (2019).

Figure 1:  (a) Overlapped and un-overlapped spectra between $H_{0}$ and
$H_{1}$. The states in the energy window $[E-\Gamma/2,E+\Gamma/2]$ are
hybridized by the off-diagonal block coupling. (b)-(c) The scaling of $\Gamma$
versus system size $N$ in the overlapped (b) and un-overlapped spectra (c).
(d)-(e) The phase diagram of the model (1) for overlapped spectra (d); and un-
overlapped spectra (e), with $\gamma_{\text{ergo}}$ ($\gamma_{\text{AL}}$)
being the boundary between the NEE phase and ergodic (localized) phase.

Here, we propose a coupled random matrix model for the realization of ME and
the non-ergodic extended (NEE) phase. This model with a block structure is
constructed from two independent random matrices with their coupling strength
scaled as $N^{-\gamma/2}$ and coupling connectance scaled as $N^{\nu}$, where
$\gamma$ and $\nu$ are two major parameters to characterize all phases. The
motivation behind this construction is three-fold: (I) With only direct
coupling between different random matrices, the overlapped and un-overlapped
spectra would exhibit different scaling behaviors (see Fig. 1 (a)), giving
rise to ME. This is different from the RP ensemble with only one single block
(see S1 and S2 in Ref. SI ). (II) The coupling will become irrelevant when
$\gamma$ is large and relevant when $\gamma$ is small (see Fig. 1 (b) - (c)),
giving rise to ergodic, localized, and NEE phases by varying of $\gamma$ and
$\nu$, similar to the Rosenzweig-Porter ensemble Rosenzweig and Porter (1960);
Kravtsov  _et al._ (2015); Bogomolny and Sieber (2018); Facoetti  _et al._
(2016); Khaymovich  _et al._ (2020); Biroli and Tarzia (2021); De Tomasi and
Khaymovich (2022); Buijsman and Lev (2022) and $\beta$-ensemble Das and Ghosh
(2022); Das  _et al._ (2023). (III) The block structure of our model is
closely related to the Hamiltonian of many-body disordered models, in which
the block structure is due to the symmetry Giraud  _et al._ (2022). This
construction can give some new insights into the appearance of many-body ME
and the quantum avalanche phenomenon in MBL Thiery  _et al._ (2018); Rubio-
Abadal  _et al._ (2019); See Toh  _et al._ (2022).

Physical model and methods: The coupled random matrix model can be written as

$\mathcal{H}=\begin{pmatrix}H_{0}&0\\\
0&H_{1}\end{pmatrix}+\frac{g}{N^{\gamma/2}}\begin{pmatrix}0&V\\\
V^{\dagger}&0\end{pmatrix},$ (1)

where $H_{0}$ and $H_{1}$ are $N\times N$ diagonal random matrix belonging to
Poisson Ensemble (PE) with $(H_{\sigma})_{ii}$ uniformly distributed in
$(M_{\sigma}-U_{\sigma}/2,M_{\sigma}+U_{\sigma}/2)$, for $\sigma=0$, 1. The
entries of $V$ are chosen to be independently distributed as
$P(V_{ij})=(1-c)\delta(V_{ij})+h(V_{ij})c$ Not , where $c=N^{\nu-1}$
($0\leq\nu\leq 1$) controls the degree of connectance of the matrix, and
$h(x)=\exp(-x^{2}/2)/\sqrt{2\pi}$. The constant $g$ is not essential when
$\nu\neq 0$, thus hereafter we set $g=1$ (Kravtsov  _et al._ , 2015). We
expect this model can host the ergodic, localized, and NEE phases, and tunable
MEs. The general theory for the NEE phase and ME, and verification of their
universality are the main tasks of this work.

We will verify the predictions from the statistics of eigenvalues and
eigenvectors. The level-spacing ratio is defined as
$r_{n}=\min(s_{n},s_{n+1})/\max(s_{n},s_{n+1})$, where $s_{n}=E_{n+1}-E_{n}$
is the level-spacing between two adjacent energy levels. Its mean value is
taken to be $\langle r\rangle\approx 0.5307$ for the ergodic and NEE phases
while $\langle r\rangle\approx 0.386$ for the localized phase Kravtsov  _et
al._ (2015); Khaymovich  _et al._ (2020). Next, the fractal dimension of wave
functions is defined as $D_{q}(E_{n},N)=-\ln(\sum_{m}|\langle
m|\psi_{n}\rangle|^{2q})/[\ln(2N)(1-q)]$ with $|\psi_{n}\rangle$ being the
eigenvectors. In the large $N$ limit,
$D_{q}(E_{n})=\lim_{N\rightarrow\infty}D_{q}(E_{n},N)$, distinguishing the
localized ($D_{q}=0$), ergodic ($D_{q}=1$), and NEE ($0<D_{q}<1$) phases
Kravtsov  _et al._ (2015).

Theory of phase transition and phase diagram: We first present our theory of
phase transition with only PEs in the diagonal block of Eq. 1. When $\gamma\gg
1$ (or $g=0$), the influence of $V$ can be neglected, yielding the
eigenvectors $H_{\sigma}|i\sigma\rangle=b_{i\sigma}|i\sigma\rangle$, where
$|i\sigma\rangle=|\psi_{i,\sigma}\rangle$, with $b_{i,\sigma}$ being the
corresponding eigenvalue. When $\gamma$ around unity, coupling between these
two matrices is relevant, which can be understood from Fermi’s Golden Rule as
Sakurai and Commins (1995); Micklitz  _et al._ (2022); Venturelli  _et al._
(2023)

$\Gamma(b_{i\sigma})=2\pi\sum_{j,\sigma^{\prime}}|\langle
i\sigma|T(b_{i\sigma})|j\sigma^{\prime}\rangle|^{2}\delta(b_{i\sigma}-b_{j\sigma^{\prime}}),$
(2)

with $T(E)$ the transition operator. The quantity $\Gamma$ can be interpreted
as the bandwidth of the state $|i\sigma\rangle$ perturbed by $V$, which
implies that the eigenstates within the energy window of
$[b_{i\sigma}-\Gamma/2,b_{i\sigma}+\Gamma/2]$ are strongly hybridized
Venturelli  _et al._ (2023). Thus, the two transition points
$\gamma_{\text{ergo}}$ and $\gamma_{\text{AL}}$, which denote the ergodic-NEE
transition and NEE-localized transition points (see Fig. 1 (d)), can be
characterized by the following $\beta$ functions

$\displaystyle\beta_{\text{ergo}}=\frac{d\ln(\Gamma/U)}{d\ln(N)},\quad\beta_{\text{AL}}=\frac{d\ln(\Gamma/\delta_{\text{eff}})}{d\ln(N)},$
(3)

with $U\propto N^{0}$ the width of the spectra for global coupling and
$\delta_{\text{eff}}$ the effective averaged level-spacing for local coupling.
This yields Fig. 1 (b) - (c). When $\nu=1$, we have $\delta_{\text{eff}}\sim
UN^{-1}$, in agreement with the averaged level-spacing of $H_{0}$ and $H_{1}$.
When $0<\nu<1$, we have $\delta_{\text{eff}}\sim UN^{-\nu}$ because only
$N^{\nu}$ states are coupled by the coupling $V$ on average. Physically, in
the language of renormalization group theory, $\beta<0$ ($\beta>0$) means that
$\Gamma/U$ or $\Gamma/\delta_{\text{eff}}$ are irrelevant (relevant) and flows
to zero (infinite) with the increasing of $N$. Therefore, the condition
$\beta=0$ is marginal, which determines the value of $\gamma_{\text{ergo}}$
and $\gamma_{\text{AL}}$.

Figure 2:  (a) The level-spacing ratio $\langle r\rangle_{E}$ against index
$\gamma$ and energy $E$ with $N=2^{12}$ and $\nu=1$. (b) The fractal dimension
$D_{2}(E)$ against $\gamma$ and $E$. The horizontal lines in (a) and (b)
denote $E=0.6$ and $E=1.5$. (c) A detail plot of $\langle r\rangle_{E}$ at
different sizes. The cross points are at $\gamma=3/2$ and $\gamma=2$ (see the
two grey vertical lines) for $\beta_{\text{AL}}$ in the overlapped and un-
overlapped regimes, respectively. (d) The fractal dimension $D_{2}(E)$ for
$|E-0.6|<\delta$ and $|E-1.5|<\delta$, denoted as $E=0.6$ and 1.5, with
$\delta=0.05$ for $\beta_{\text{ergo}}=1$.

One of the central observations is that the overlapped and un-overlapped
spectra will exhibit totally different behaviors, as pointed out in (I). In
the overlapped spectra (see Fig. 1 (a)), we have $\langle
i\sigma|V|j\sigma^{\prime}\rangle=V_{ij}/N^{\gamma/2}\neq 0$ for
$b_{i\sigma}\sim b_{j\sigma^{\prime}}$ from direct couplings. Taking ensemble
average and energy conservation, we obtain the decay rate to the leading order
(see S3 in Ref. SI )

$\Gamma_{\text{ov}}=2\pi\sum_{j}|V_{ij}|^{2}N^{-\gamma}\sim
2\pi\rho(b_{i\sigma})N^{\nu-\gamma}.$ (4)

In contrast, in the un-overlapped spectra, the first-order process is
forbidden and the second-order term is dominated, with $T(b_{i\sigma})\approx
V(E-H_{1-\sigma})^{-1}V$, yielding

$\Gamma_{\text{un}}=\frac{2\pi}{N^{2\gamma}}\sum_{j\neq i}|\langle
i\sigma|T(b_{i\sigma})|j\sigma\rangle|^{2}\propto\frac{2\pi\rho(b_{i\sigma})}{N^{2(\gamma-\nu)}}.$
(5)

Thus the two regimes have totally different scaling behaviors with respect to
$N$. It should be noticed that we have neglected the possible higher-order
terms (see S3 in Ref. SI ), which will not affect the existence of the NEE
phase and MEs in our model. Combining Eq. 3 \- Eq. 5 lead to $\beta$ function
for the overlapped spectra as

$\beta_{\text{ergo}}=\nu-\gamma,\quad\beta_{\text{AL}}=2\nu-\gamma,$ (6)

and that for the un-overlapped spectra as

$\beta_{\text{ergo}}=2(\nu-\gamma),\quad\beta_{\text{AL}}=3\nu-2\gamma.$ (7)

$\beta=0$ determines the phase boundaries, thus we have
$\gamma_{\text{ergo}}=\nu$, $\gamma_{\text{AL}}=2\nu$ for the overlapped
spectra and $\gamma_{\text{ergo}}=\nu$, $\gamma_{\text{AL}}=3\nu/2$ for the
un-overlapped spectra, yielding the phase diagram in Fig. 1 (d) and (e). So
the MEs in the context of correlation of energy levels exist when
$3\nu/2<\gamma<2\nu$ and the MEs in the context of wave function exist when
$\nu<\gamma<2\nu$. It is found that $\gamma_{\text{ergo}}=\nu$ for both
overlapped and un-overlapped spectra, and when $\gamma<\nu$ (for
$\beta_{\text{ergo}}>0$), all states are strongly coupled. from strong global
coupling.

The above analysis is a generalization of the approach presented in RP model
(Khaymovich  _et al._ , 2020; Biroli and Tarzia, 2021) based on the divergence
of $\sum_{j}|H_{ij}|$ and $\sum_{j}|H_{ij}|^{2}$, which also yield Eq. 6 RPn .
Our approach has a somewhat much clearer physical meaning, and meanwhile can
be generalized to block coupled models. We find that in the regime
$\gamma\in(\nu,3\nu/2)$ (un-overlapped regime) and $\gamma\in(\nu,2\nu)$
(overlapped regime), the states can be coupled only through consequential
coupling between neighboring energy states and form mini-bands in the local
spectrum Khaymovich  _et al._ (2020); Venturelli  _et al._ (2023). The above
results are significant because, in the sparse matrix limit of $\nu=0$, we
have $\gamma_{\text{ergo}}=\gamma_{\text{AL}}=0$, which corresponds to the
many-body disordered spin models and Hubbard models widely studied in the
literature Chanda  _et al._ (2020); Deng  _et al._ (2017); Wei  _et al._
(2019) and sparse matrix researched by Mirlin et. al. Mirlin and Fyodorov
(1991); Fyodorov and Mirlin (1991). In this condition, $g$ is marginal, and a
finite $g_{c}$ is required for the localization-delocalization transition.

Figure 3: The same as that in Fig. 2 but with $\nu=0.6$. All the major
features of phase transition in Fig. 2 are maintained, except that in (c) the
two cross points happen at $\gamma_{\text{AL}}=1.39$ in the overlapped spectra
and $\gamma_{\text{AL}}=1.16$ in the un-overlapped spectra; while from the
theoretical prediction it should be $1.2$ and $0.9$, respectively. This may
come from the finite size effect (see S4 of Ref. SI , from which we find the
apparent cross points $\gamma_{c}$ shift towards the theoretic expectation
with the increase of $N$) or the neglect of the higher-order terms in
perturbation theory. In (d) from the fractal dimension $D_{2}(E)$ for $E=0.6$
and $E=1.5$, we still have $\beta_{\text{ergo}}=\nu=0.6$; see Fig. 2.

It is extremely intriguing to discuss the properties of the NEE phase, whose
fractal dimension is not an integer. The support set of the eigenstates is
simply given by the bandwidth $\Gamma$ divided by the averaged level spacing
$\delta_{\text{eff}}\sim N^{-\nu}$, which is a result of the local coupling.
Thus the non-zero component of the new eigenstate scales as
$\mathcal{N}\sim\Gamma/N^{-\nu}$ and it can be written as
$|\psi_{i,\sigma}^{\prime}\rangle=\sum_{j,\sigma^{\prime},|b_{i\sigma}-b_{j\sigma^{\prime}}|<\Gamma/2}c_{j,\sigma^{\prime}}|j\sigma^{\prime}\rangle$,
with $c_{j,\sigma^{\prime}}\sim(\Gamma N^{\nu})^{-1/2}$ for normalization
Venturelli  _et al._ (2023). The fractal dimension of these wave functions is
given by $D_{q}=\lim_{N\rightarrow\infty}\ln((\Gamma
N^{\nu})^{1-q})/(\ln(2N)(1-q))$. Using the expression of $\Gamma$ in Eq. 2
will yield

$D_{q}^{\text{ov}}=2\nu-\gamma,\quad D_{q}^{\text{un}}=3\nu-2\gamma,$ (8)

for $\gamma_{\text{ergo}}<\gamma<\gamma_{\text{AL}}$. Thus, the NEE phase,
without $q$-dependence, is fractal but not multifractal, setting it apart from
from that in the RP ensemble with a heavy-tail distribution (Khaymovich  _et
al._ , 2020; Biroli and Tarzia, 2021). With heavy-tail distributed elements,
the existence of ME has also been reported Tarquini  _et al._ (2016); Aggarwal
_et al._ (2019). However, the underlying mechanism is different.

With this theoretical analysis, it is essential to verify the above results
numerically. We use $M_{0}=0$, $M_{1}=1$ and $U_{0}=U_{1}=2$ in Eq. 1. Thus,
the spectra of $H_{0}$ and $H_{1}$ overlap within energy regime $[0,1]$. The
results with $\nu=1$ are shown in Fig. 2, which agree excellently with the
theoretical prediction in Eq. 6 and Eq. 7. The results with $\nu=0.6$ are
presented in Fig. 3. We find that the numerical results indicate
$\gamma_{\text{ergo}}=1.16$ and $\gamma_{\text{AL}}=1.39$, being slightly
different from the prediction based on perturbation theory. This should come
from the finite size effect and we show in S4 SI  that these two limits can be
approached in the large $N$ limit.

Figure 4:  (a) The averaged level-spacing ratio $\langle r\rangle_{E}$ against
index $\gamma$ and energy $E$. The horizontal lines denote $E=0.6$ and
$E=1.5$. (b) A detailed plot of $\langle r\rangle_{E}$ at different sizes. The
cross points are at $\gamma=3/2$ and $\gamma=2$, corresponding to the grey
vertical lines. (c) The fractal dimension $D_{2}(E)$ for $E=0.6$ and $E=1.5$.
(d)-(f) The distribution function for finite-size $D_{2}(E,N)$ with $0<E<1$ at
sizes $N=2^{6}$ (blue), $N=2^{8}$ (green), $N=2^{9}$ (orange), $N=2^{10}$
(red). We use $\gamma=0.5$ in (d), $\gamma=1.5$ in (e), and $\gamma=2.5$ in
(f). The insets show the averaged $D_{2}(E,N)$ of states over $E\in[0,1]$
against $1/\ln(2N)$. The solid lines are fitted according to $\langle
D_{2}(E,N)\rangle_{E}=A/\ln(2N)+D_{2}(E)$.

Universality of this mechanism: Since the block structure, instead of the
symmetry, is most essential for our model, we expect our results to be also
applicable when $H_{0}$, $H_{1}$, and $V$ are replaced by other random
matrices. However, the transition point and the associated $\beta$ functions
may be affected by correlations in $V$. To this end, we set $H_{0}$ as PE
whose diagonal elements are distributed in the interval $[0,2]$ and $H_{1}$ as
Gaussian orthogonal ensemble (GOE) whose elements are independent Gaussian
distributed with mean zero and variance $1/\sqrt{2N}$ (see other choices of
$H_{0}$ and $H_{1}$ in S5 of Ref. SI ). By choosing these two ensembles, their
spectra are overlapped in the interval $[0,1]$. Then we set $\nu=1$ for the
off-diagonal block and expect $\gamma_{\text{ergo}}=1$ and
$\gamma_{\text{AL}}=2$. When $\gamma>\gamma_{\text{AL}}$, the coupling is
irrelevant, yielding a coexistence phase of the ergodic phase and localized
phase in the overlapped spectra with some different statisticsGiraud  _et al._
(2022); Berry and Robnik (1984). In the un-overlapped spectra, the states are
either localized or ergodic depending on their energy. When
$\gamma<\gamma_{\text{ergo}}$, the off-diagonal coupling becomes relevent with
the increasing of $N$, yielding the ergodic phase for the whole spectra. The
most intriguing physics happens in the intermediate region of $\gamma$, the
coupling between $H_{0}$ and $H_{1}$ induces competition of ergodic and
localized states, leaving their fate undetermined. We perform a calculation
based on the exact diagonalization method to examine the phase diagram, which
is presented in Fig. 4. The level-spacing ratio $\langle r\rangle_{E}$
exhibits two different cross points for overlapped and un-overlapped spectra,
which is the same as the results in Fig. 2. However, $\langle r\rangle_{E}\neq
0.386$ and $0<D_{2}(E)<1$ in the overlapped spectra when $\gamma>2$,
corresponds to a coexistence phase of ergodic and localized states (Berry and
Robnik, 1984; Giraud  _et al._ , 2022). This is further verified by the
distribution of finite-size $D_{2}(E,N)$ as shown in Fig. 4 (d), (e), and (f).
These results indicate that there is only one peak of $P(D_{2}(E,N))$ for the
ergodic and NEE phases while the distribution of $P(D_{2}(E,N))$ of the
coexistence phase exhibits two peaks as shown in Fig. 4 (f).

Another important extension of this theory is replacing the Gaussian
distributed $V$ with a circular orthogonal ensemble (COE), with $V^{T}V=1$
Dyson (1962b); Forrester (2010). This is expected if we set $H_{0}$ and
$H_{1}$ as random matrices, which can be diagonalized using two unitary
matrices $U_{1}$ and $U_{2}$. Then the off-diagonal matrix should be
$V=U_{1}^{\dagger}TU_{2}$, which can be a COE when $T=\mathbb{I}$ is unity.
This matrix, unlike the Gaussian counterpart, is correlated, and it is
expected that $\langle(V_{ij})^{2}\rangle\sim N^{-1}$, since
$\text{Tr}(V^{\dagger}V)=N$. As compared with the Gaussian distributed $V$, we
have to define the effective parameters as $\gamma_{\text{eff}}=1+\gamma$,
following the above procedure. It yields $\gamma_{\text{ergo}}=0$,
$\gamma_{\text{AL}}=1$ for the overlapped spectra and
$\gamma_{\text{ergo}}=0$, $\gamma_{\text{AL}}=1/2$ for the un-overlapped
spectra. The verification of the $\gamma_{\text{AL}}$ is presented in Fig. 5,
setting $V$ to be a COE, showing $\gamma_{\text{AL}}=1/2$ and $1$ in these two
different energy regimes. Here, we also chose the energy regime $[0,1]$ as the
overlapped spectra.

Figure 5:  (a) The averaged level-spacing ratio $\langle r\rangle_{E}$ against
index $\gamma$ within energy windows $|E-1.5|\leq 0.01$ and $|E-0.6|\leq 0.01$
with $V$ being COE. The cross points are at $\gamma=1/2$ and $\gamma=1$,
corresponding to the grey vertical lines. (b) The fractal dimension $D_{2}(E)$
around $E=0.6$ (overlapped) and $E=1.5$ (un-overlapped), showing
$\gamma_{\text{AL}}=1$ (overlapped) and $\gamma_{\text{AL}}=1/2$ (un-
overlapped).

Conclusion and Remarks: We present a new class of random matrices based on two
coupled matrices and show that the overlapped and un-overlapped spectra have
totally different scaling behaviors, which can be used to construct tunable
MEs. This theory applies to dense, sparse, and even corrected random matrices.
Our study opens several intriguing research directions. Firstly, it provides a
universal framework for the realization of MEs in random matrices and even
realistic models, which offer exceptional opportunities for the experimental
detection of many-body MEs. Secondly, our approach could be applied to the
possible MEs in non-hermitian random models with overlapped spectra in the
complex plane (De Tomasi and Khaymovich, 2022; Hamazaki  _et al._ , 2019; Gong
_et al._ , 2018; Ashida  _et al._ , 2021). Finally, it can be generalized to
multi-block random matrices, which are used for MBL in realistic models.
Applications of our model may include ME and localization-delocalization
transitions in many-body models (Thiery  _et al._ , 2018; Rubio-Abadal  _et
al._ , 2019; See Toh  _et al._ , 2022), chiral symmetry breaking of the Dirac
operator in QCD Stephanov (1996); Verbaarschot (1994); Verbaarschot and Wettig
(2000) and stability of large ecological communities May (1972); Serván  _et
al._ (2018); Grilli  _et al._ (2017); Eco . This model itself is also of
general interest in pure mathematics in context of RMT Forrester (2010).

Acknowledgments: This work is supported by the National Natural Science
Foundation of China (NSFC) with No. 11774328, and Innovation Program for
Quantum Science and Technology (No. 2021ZD0301200 and No. 2021ZD0301500).

## References

  * Forrester (2010) Peter J. Forrester,  _Log-Gases and Random Matrices (LMS-34)_ (Princeton University Press, Princeton, 2010).
  * Brody  _et al._ (1981) T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, and S. S. M. Wong, “Random-matrix physics: spectrum and strength fluctuations,” Rev. Mod. Phys. 53, 385–479 (1981).
  * Mehta (2004) Madan Lal Mehta,  _Random matrices_ (Elsevier, 2004).
  * Livan  _et al._ (2018) Giacomo Livan, Marcel Novaes, and Pierpaolo Vivo, “Introduction to random matrices theory and practice,” Monograph Award 63 (2018).
  * Wigner (1955) Eugene P. Wigner, “Characteristic Vectors of Bordered Matrices With Infinite Dimensions,” Annals of Mathematics 62, 548 (1955).
  * Dyson (1962a) Freeman J. Dyson, “A Brownian‐Motion Model for the Eigenvalues of a Random Matrix,” J. Math. Phys. 3, 1191 (1962a).
  * Bohigas  _et al._ (1984) O. Bohigas, M. J. Giannoni, and C. Schmit, “Characterization of chaotic quantum spectra and universality of level fluctuation laws,” Phys. Rev. Lett. 52, 1 (1984).
  * Berry  _et al._ (1977) Michael Victor Berry, M. Tabor, and John Michael Ziman, “Level clustering in the regular spectrum,” Proc. R. Soc. Lond. A 356, 375 (1977).
  * Kos  _et al._ (2018) Pavel Kos, Marko Ljubotina, and Tomaž Prosen, “Many-Body Quantum Chaos: Analytic Connection to Random Matrix Theory,” Phys. Rev. X 8, 021062 (2018).
  * Mirlin (2000) Alexander D. Mirlin, “Statistics of energy levels and eigenfunctions in disordered systems,” Physics Reports 326, 259–382 (2000).
  * Goda  _et al._ (2006) Masaki Goda, Shinya Nishino, and Hiroki Matsuda, “Inverse anderson transition caused by flatbands,” Phys. Rev. Lett. 96, 126401 (2006).
  * Pal and Huse (2010) Arijeet Pal and David A. Huse, “Many-body localization phase transition,” Phys. Rev. B 82, 174411 (2010).
  * Nandkishore and Huse (2015) Rahul Nandkishore and David A. Huse, “Many-Body Localization and Thermalization in Quantum Statistical Mechanics,” Annual Review of Condensed Matter Physics 6, 15 (2015).
  * Abanin  _et al._ (2019) Dmitry A. Abanin, Ehud Altman, Immanuel Bloch, and Maksym Serbyn, “Colloquium: Many-body localization, thermalization, and entanglement,” Rev. Mod. Phys. 91, 021001 (2019).
  * Alet and Laflorencie (2018) Fabien Alet and Nicolas Laflorencie, “Many-body localization: an introduction and selected topics,” Comptes Rendus Physique 19, 498 (2018).
  * Imbrie (2016) John Z. Imbrie, “Diagonalization and Many-Body Localization for a Disordered Quantum Spin Chain,” Phys. Rev. Lett. 117, 027201 (2016).
  * Stephanov (1996) M. A. Stephanov, “Random matrix model of qcd at finite density and the nature of the quenched limit,” Phys. Rev. Lett. 76, 4472–4475 (1996).
  * Verbaarschot (1994) Jacobus Verbaarschot, “Spectrum of the qcd dirac operator and chiral random matrix theory,” Phys. Rev. Lett. 72, 2531–2533 (1994).
  * Kieburg  _et al._ (2013) Mario Kieburg, Jacobus J. M. Verbaarschot, and Savvas Zafeiropoulos, “Spectral properties of the wilson-dirac operator and random matrix theory,” Phys. Rev. D 88, 094502 (2013).
  * Verbaarschot and Wettig (2000) J. J. M. Verbaarschot and T. Wettig, “Random matrix theory and chiral symmetry in qcd,” Annual Review of Nuclear and Particle Science 50, 343 – 410 (2000).
  * May (1972) Robert M. May, “Will a large complex system be stable?” Nature 238, 413–414 (1972).
  * Serván  _et al._ (2018) Carlos A. Serván, José A. Capitán, Jacopo Grilli, Kent E. Morrison, and Stefano Allesina, “Coexistence of many species in random ecosystems,” Nature Ecology & Evolution 2, 1237–1242 (2018).
  * Grilli  _et al._ (2017) Jacopo Grilli, Matteo Adorisio, Samir Suweis, György Barabás, Jayanth R. Banavar, Stefano Allesina, and Amos Maritan, “Feasibility and coexistence of large ecological communities,” Nature Communications 8, 14389 (2017).
  * Mott (1987) N Mott, “The mobility edge since 1967,” Journal of Physics C: Solid State Physics 20, 3075 (1987).
  * Semeghini  _et al._ (2015) G. Semeghini, M. Landini, P. Castilho, S. Roy, G. Spagnolli, A. Trenkwalder, M. Fattori, M. Inguscio, and G. Modugno, “Measurement of the mobility edge for 3D Anderson localization,” Nat. Phys 11, 554 (2015).
  * Pixley  _et al._ (2015) J. H. Pixley, Pallab Goswami, and S. Das Sarma, “Anderson localization and the quantum phase diagram of three dimensional disordered dirac semimetals,” Phys. Rev. Lett. 115, 076601 (2015).
  * Biddle  _et al._ (2009) J. Biddle, B. Wang, D. J. Priour, and S. Das Sarma, “Localization in one-dimensional incommensurate lattices beyond the Aubry-Andr\’e model,” Phys. Rev. A 80, 021603 (2009).
  * Biddle and Das Sarma (2010) J. Biddle and S. Das Sarma, “Predicted Mobility Edges in One-Dimensional Incommensurate Optical Lattices: An Exactly Solvable Model of Anderson Localization,” Phys. Rev. Lett. 104, 070601 (2010).
  * Ganeshan  _et al._ (2015) Sriram Ganeshan, J. H. Pixley, and S. Das Sarma, “Nearest Neighbor Tight Binding Models with an Exact Mobility Edge in One Dimension,” Phys. Rev. Lett. 114, 146601 (2015).
  * Wang  _et al._ (2020) Yucheng Wang, Xu Xia, Long Zhang, Hepeng Yao, Shu Chen, Jiangong You, Qi Zhou, and Xiong-Jun Liu, “One-Dimensional Quasiperiodic Mosaic Lattice with Exact Mobility Edges,” Phys. Rev. Lett. 125, 196604 (2020).
  * Wang  _et al._ (2022) Yunfei Wang, Jia-Hui Zhang, Yuqing Li, Jizhou Wu, Wenliang Liu, Feng Mei, Ying Hu, Liantuan Xiao, Jie Ma, Cheng Chin, and Suotang Jia, “Observation of interaction-induced mobility edge in an atomic aubry-andré wire,” Phys. Rev. Lett. 129, 103401 (2022).
  * An  _et al._ (2021) Fangzhao Alex An, Karmela Padavić, Eric J. Meier, Suraj Hegde, Sriram Ganeshan, J. H. Pixley, Smitha Vishveshwara, and Bryce Gadway, “Interactions and mobility edges: Observing the generalized aubry-andré model,” Phys. Rev. Lett. 126, 040603 (2021).
  * Wei  _et al._ (2019) Xingbo Wei, Chen Cheng, Gao Xianlong, and Rubem Mondaini, “Investigating many-body mobility edges in isolated quantum systems,” Phys. Rev. B 99, 165137 (2019).
  * Luitz  _et al._ (2015) David J. Luitz, Nicolas Laflorencie, and Fabien Alet, “Many-body localization edge in the random-field Heisenberg chain,” Phys. Rev. B 91, 081103 (2015).
  * Brighi  _et al._ (2020) Pietro Brighi, Dmitry A. Abanin, and Maksym Serbyn, “Stability of mobility edges in disordered interacting systems,” Phys. Rev. B 102, 060202 (2020).
  * Deng  _et al._ (2017) Dong-Ling Deng, Sriram Ganeshan, Xiaopeng Li, Ranjan Modak, Subroto Mukerjee, and J. H. Pixley, “Many-body localization in incommensurate models with a mobility edge,” Annalen der Physik 529, 1600399 (2017).
  * Chanda  _et al._ (2020) Titas Chanda, Piotr Sierant, and Jakub Zakrzewski, “Many-body localization transition in large quantum spin chains: The mobility edge,” Phys. Rev. Research 2, 032045 (2020).
  * Nag and Garg (2017) Sabyasachi Nag and Arti Garg, “Many-body mobility edges in a one-dimensional system of interacting fermions,” Phys. Rev. B 96, 060203 (2017).
  * (39) Ranjan Modak and Subroto Mukerjee, “Many-Body Localization in the Presence of a Single-Particle Mobility Edge,” Phys. Rev. Lett. 115, 230401.
  * De Roeck  _et al._ (2016) Wojciech De Roeck, Francois Huveneers, Markus Müller, and Mauro Schiulaz, “Absence of many-body mobility edges,” Phys. Rev. B 93, 014203 (2016).
  * De Roeck and Huveneers (2017) Wojciech De Roeck and François Huveneers, “Stability and instability towards delocalization in MBL systems,” Phys. Rev. B 95, 155129 (2017).
  * Zhang  _et al._ (2022) Yu Zhang, Bozhen Zhou, Haiping Hu, and Shu Chen, “Localization, multifractality, and many-body localization in periodically kicked quasiperiodic lattices,” Phys. Rev. B 106, 054312 (2022).
  * Lazarides  _et al._ (2015) Achilleas Lazarides, Arnab Das, and Roderich Moessner, “Fate of Many-Body Localization Under Periodic Driving,” Phys. Rev. Lett. 115, 030402 (2015).
  * Lüschen  _et al._ (2018) Henrik P. Lüschen, Sebastian Scherg, Thomas Kohlert, Michael Schreiber, Pranjal Bordia, Xiao Li, S. Das Sarma, and Immanuel Bloch, “Single-Particle Mobility Edge in a One-Dimensional Quasiperiodic Optical Lattice,” Phys. Rev. Lett. 120, 160404 (2018).
  * Kohlert  _et al._ (2019) Thomas Kohlert, Sebastian Scherg, Xiao Li, Henrik P. Lüschen, Sankar Das Sarma, Immanuel Bloch, and Monika Aidelsburger, “Observation of Many-Body Localization in a One-Dimensional System with a Single-Particle Mobility Edge,” Phys. Rev. Lett. 122, 170403 (2019).
  * Cizeau and Bouchaud (1994) P. Cizeau and J. P. Bouchaud, “Theory of Lévy matrices,” Phys. Rev. E 50, 1810 (1994).
  * Tarquini  _et al._ (2016) E. Tarquini, G. Biroli, and M. Tarzia, “Level Statistics and Localization Transitions of L\’evy Matrices,” Phys. Rev. Lett. 116, 010601 (2016).
  * Biroli and Tarzia (2021) G. Biroli and M. Tarzia, “Lévy-Rosenzweig-Porter random matrix ensemble,” Phys. Rev. B 103, 104205 (2021).
  * Aggarwal  _et al._ (2019) Amol Aggarwal, Patrick Lopatto, and Horng-Tzer Yau, “GOE Statistics for Levy Matrices,”  (2019), arXiv:1806.07363 [math-ph].
  * Sarkar  _et al._ (2023) Madhumita Sarkar, Roopayan Ghosh, and Ivan M. Khaymovich, “Tuning the phase diagram of a rosenzweig-porter model with fractal disorder,” Phys. Rev. B 108, L060203 (2023).
  * (51) See Supplemental Material for (I) Characterization of the Rosenzweig-Porter (RP) ensemble (II) Phase diagram of the sparse Rosenzweig-Porter ensemble. (III) The derivation of Eq. 4 and Eq. 5 in the main text. (IV) Finite-size analysis of the phase boundary. (V) Coupled random matrice with Gaussian unitary ensembles.
  * Rosenzweig and Porter (1960) Norbert Rosenzweig and Charles E. Porter, “”repulsion of energy levels” in complex atomic spectra,” Phys. Rev. 120, 1698–1714 (1960).
  * Kravtsov  _et al._ (2015) V E Kravtsov, I M Khaymovich, E Cuevas, and M Amini, “A random matrix model with localization and ergodic transitions,” New J. Phys. 17, 122002 (2015).
  * Bogomolny and Sieber (2018) E. Bogomolny and M. Sieber, “Eigenfunction distribution for the Rosenzweig-Porter model,” Phys. Rev. E 98, 032139 (2018).
  * Facoetti  _et al._ (2016) Davide Facoetti, Pierpaolo Vivo, and Giulio Biroli, “From non-ergodic eigenvectors to local resolvent statistics and back: A random matrix perspective,” EPL 115, 47003 (2016).
  * Khaymovich  _et al._ (2020) I. M. Khaymovich, V. E. Kravtsov, B. L. Altshuler, and L. B. Ioffe, “Fragile extended phases in the log-normal Rosenzweig-Porter model,” Phys. Rev. Res. 2, 043346 (2020).
  * De Tomasi and Khaymovich (2022) Giuseppe De Tomasi and Ivan M. Khaymovich, “Non-Hermitian Rosenzweig-Porter random-matrix ensemble: Obstruction to the fractal phase,” Phys. Rev. B 106, 094204 (2022).
  * Buijsman and Lev (2022) Wouter Buijsman and Yevgeny Bar Lev, “Circular Rosenzweig-Porter random matrix ensemble,” SciPost Phys. 12, 082 (2022).
  * Das and Ghosh (2022) Adway Kumar Das and Anandamohan Ghosh, “Nonergodic extended states in the $\beta$ ensemble,” Phys. Rev. E 105, 054121 (2022).
  * Das  _et al._ (2023) Adway Kumar Das, Anandamohan Ghosh, and Ivan M. Khaymovich, “Absence of mobility edge in short-range uncorrelated disordered model: Coexistence of localized and extended states,” (2023), arXiv:2305.02351 [cond-mat.dis-nn] .
  * Giraud  _et al._ (2022) Olivier Giraud, Nicolas Macé, Éric Vernier, and Fabien Alet, “Probing symmetries of quantum many-body systems through gap ratio statistics,” Phys. Rev. X 12, 011006 (2022).
  * Thiery  _et al._ (2018) Thimothée Thiery, François Huveneers, Markus Müller, and Wojciech De Roeck, “Many-body delocalization as a quantum avalanche,” Phys. Rev. Lett. 121, 140601 (2018).
  * Rubio-Abadal  _et al._ (2019) Antonio Rubio-Abadal, Jae-yoon Choi, Johannes Zeiher, Simon Hollerith, Jun Rui, Immanuel Bloch, and Christian Gross, “Many-body delocalization in the presence of a quantum bath,” Phys. Rev. X 9, 041014 (2019).
  * See Toh  _et al._ (2022) Jun Hui See Toh, Katherine C McCormick, Xinxin Tang, Ying Su, Xi-Wang Luo, Chuanwei Zhang, and Subhadeep Gupta, “Many-body dynamical delocalization in a kicked one-dimensional ultracold gas,” Nat. Phys. 18, 1297–1301 (2022).
  * (65) The other choices of random entries with finite variance are equivalent. This is due to the central limit theorem, where the sum of all identical distributed random entries converges to the Gaussian function. When the variance of the random entries does not exist, the model is equivalent to the Lévy random matrix.
  * Sakurai and Commins (1995) Jun John Sakurai and Eugene D Commins, “Modern quantum mechanics, revised edition,” (1995).
  * Micklitz  _et al._ (2022) Tobias Micklitz, Alan Morningstar, Alexander Altland, and David A. Huse, “Emergence of fermi’s golden rule,” Phys. Rev. Lett. 129, 140402 (2022).
  * Venturelli  _et al._ (2023) Davide Venturelli, Leticia F. Cugliandolo, Grégory Schehr, and Marco Tarzia, “Replica approach to the generalized Rosenzweig-Porter model,” SciPost Phys. 14, 110 (2023).
  * (69) For a single random matrix, the criterion of $\langle\sum_{j}|H_{ij}|\rangle$ will yields $N^{\nu-\gamma/2}$. Furthermore, we have $(\sum_{j}|H_{ij}|)^{2}\sim N^{\nu}\sum_{j}|H_{ij}|^{2}$. With these two results, we can obtain Eq. 6 in the main text.
  * Mirlin and Fyodorov (1991) A. D. Mirlin and Y. V. Fyodorov, “Universality of level correlation function of sparse random matrices,” J. Phys. A: Math. Gen. 24, 2273 (1991).
  * Fyodorov and Mirlin (1991) Yan V. Fyodorov and Alexander D. Mirlin, “Localization in ensemble of sparse random matrices,” Phys. Rev. Lett. 67, 2049 (1991).
  * Berry and Robnik (1984) M. V. Berry and M. Robnik, “Semiclassical level spacings when regular and chaotic orbits coexist,” J. Phys. A: Math. Gen. 17, 2413 (1984).
  * Dyson (1962b) Freeman J Dyson, “A Brownian-motion model for the eigenvalues of a random matrix,” J. Math. Phys. 3, 1191 (1962b).
  * Hamazaki  _et al._ (2019) Ryusuke Hamazaki, Kohei Kawabata, and Masahito Ueda, “Non-hermitian many-body localization,” Phys. Rev. Lett. 123, 090603 (2019).
  * Gong  _et al._ (2018) Zongping Gong, Yuto Ashida, Kohei Kawabata, Kazuaki Takasan, Sho Higashikawa, and Masahito Ueda, “Topological phases of non-hermitian systems,” Phys. Rev. X 8, 031079 (2018).
  * Ashida  _et al._ (2021) Yuto Ashida, Zongping Gong, and Masahito Ueda, “Non-hermitian physics,” Advances in Physics 69, 249 – 435 (2021).
  * (77) For a large system described by $dX/dt=AX$ May (1972), where $X$ is a $N$ column vector and $A=-\mathbb{I}+B$ is the matrix at the fixed point, with $I$ being a unity matrix and $B$ being a random matrix. The variation of the eigenvalues by Eq. 4 in the main text yields $\sigma^{2}=\langle b_{ij}^{2}\rangle N^{\nu}$, thus phase transition happens at $\langle b_{ij}^{2}\rangle N^{\nu}=1$. Following our theory one may consider the stability of coupled large ecosystems.


# Geometric Approach For Majorizing Measures

Shih Yu Chang Shih Yu Chang is with the Department of Applied Data Science,
San Jose State University, San Jose, CA, U. S. A. (e-mail:
shihyu.chang@sjsu.edu).

###### Abstract

Gaussian processes can be considered as subsets of a standard Hilbert space,
but the geometric understanding that would relate the size of a set with the
size of its convex hull is still lacking. In this work, we adopt a geometric
approach to the majorizing measure problem by identifying the covering number
relationships between a given space $T$ and its convex hull, represented by
$T_{h}$. If the space $T$ is a closed bounded polyhedra in $\mathbb{R}^{n}$,
we can evaluate the volume ratio between the space $T$ and its convex hull
obtained by the Quickhull algorithm. If the space $T$ is a general compact
object in $\mathbb{R}^{n}$ with non-empty interior, we first establish a more
general reverse Brunn-Minkowski inequality for nonconvex spaces which will
assist us to bound the volume of $T_{h}$ in terms of the volume of $T$ if
$T_{h}$ can be acquired by the finite average of the space $T$ with respect to
the Minkowski sum. If the volume ratio between the space $T$ and the space
$T_{h}$ is obtained, the covering number ratio between the space $T$ and the
space $T_{h}$ can also be obtained which will be used to build majorizing
measure inequality. For infinite dimensional space, we show that the constant
$L$ at majorizing measure inequality may not always exist and the existence
condition will depend on geometric properties of $T$.

Index terms— Generic chaining, majorizing measure, convex hull, reverse Brunn-
Minkowski inequality, Minkowski sum.

##  1 Introduction

In order to estimate the supremum of a stochastic process, generic chaining
method is a powerful tool for this aspect. This method stemmed from the
classical chaining method and the later majorizing measures method, which were
developed by several great scholars, Kolmogorov, Dudley, Fernique and
Talagrand, to understand the supremum properties of stochastic processes.

Let us recall the following definition.

###### Definition 1

Given a metric space $(T,d(\cdot,\cdot))$, we define
$\gamma_{\alpha}(T,d(\cdot,\cdot))$ as

$\displaystyle\gamma_{\alpha}(T,d(\cdot,\cdot))=\inf\sup\limits_{t\in
T}\sum\limits_{m\geq 0}2^{m/\alpha}\Delta(\mathrm{A}_{n}(t)),$ (1)

where the infimum is taken over all admissible sequences and
$\Delta(\mathrm{A}_{n}(t))$ denotes the diameter of $\mathrm{A}_{n}(t)$ with
respect to the metric $d(\cdot,\cdot)$.

In the Gaussian case, that is when $X_{t}=\sum\limits_{t_{k}\in T}t_{k}g_{k}$,
where $g_{k}$ are i.i.d. standard Gaussians, we have the celebrated
Fernique–Talagrand majorizing measure theorem as:

$\displaystyle\frac{1}{L}\gamma_{2}(T,d(\cdot,\cdot))\leq\mathbb{E}\sup\limits_{t\in
T}X_{t}\leq L\gamma_{2}(T,d(\cdot,\cdot)).$ (2)

The majorizing measure theorem is a main method used to prove Theorem 2.11.1
in [1]. Gaussian processes can be seen as subsets of a standard Hilbert space,
but the geometric understanding that would relate the size of a set with the
size of its convex hull is still insufficient [1]. The purpose of this work is
to establish Theorem 2.11.1 in [1] based on a geometrical approach, i.e., we
wish to show

$\displaystyle\gamma_{2}(T_{h})\leq L\gamma_{2}(T),$ (3)

where $T_{h}$ is the convex hull for the original space $T$.

We consider both finite dimensional and infinite dimensional situations. For a
finite dimensional situation, we consider two subcases: polyhedra and general
object in $\mathbb{R}^{n}$. If the space $T$ is a closed bounded polyhedra in
$\mathbb{R}^{n}$, we determine the volume ratio between the space $T$ and its
convex hull obtained by the Quickhull algorithm, denoted as $T_{h}$. If the
space $T$ is a general compact object in $\mathbb{R}^{n}$ with non-empty
interior, we first prepare a more general reverse Brunn-Minkowski inequality
for nonconvex spaces which will help us to bound the volume of $T_{h}$ in
terms of the volume of $T$ if $T_{h}$ can be obtained by the finite average of
the space $T$ with respect to the Minkowski sum. If the volume ratio between
the space $T$ and the space $T_{h}$ is obtained, the covering number ratio
between the space $T$ and the space $T_{h}$ can also be obtained which will be
used to establish Theorem 2.11.1. For infinite dimensional space, since the
important part in the geometric approach to prove Theorem 2.11.1 in [1] is to
know the the covering number (metric entropy) relations between the space $T$
and its convex hull space $T_{h}$, the authors in [2] have results about the
estimation of the covering number of $T_{h}$ given the covering number of $T$.
We demonstrate that the constant $L$ at Theorem 2.11.1 in [1] may not always
exist and the existence will depend on geometric properties of $T$.

The rest of this paper is organized as follows. The volume ratio for the
polyhedra and the general object in $\mathbb{R}^{n}$ is discussed in Section
2. In Section 3, we bound the covering number of $T_{h}$ in terms of the
covering number of $T$ for polyhedra and general situations. Majorizing
measuring theorem is proved from a geometric perspective in Section 4. For the
space with infinite dimension, we provide the condition for the existence of
the constant $L$ in Eq. (3) at Section 5.

##  2 Volume Ratio Between $T$ and $T_{h}$

In this section, we will consider the volume ratio between the original
geometric space and its convex hull. All objects are assumed to be compact
with non-empty interior. In Section 2.1, we will consider $T$ as a polyhedra
object and the general geometric objects in Eucleduean space is discussed in
Section 2.2. If the space $T$ is a convex space, its convex hull space,
denoted as $T_{h}$, is idential to $T$. Therefore, we will focus the volume
difference for the situation that $T$ is not a convex space.

###  2.1 Polyhedra Case

We assume that the space $T$ is a closed bounded polyhedra in $\mathbb{R}^{n}$
having an orientable boundary, denoted as $\partial T$, which can be
triangulated into a set of $(n-1)$-dimensional simplices. From Eq. (2.1) in
[3], we have the volume of $T$ expressed as

$\displaystyle\mbox{Vol}_{n}\left(T\right)$ $\displaystyle=$
$\displaystyle\sum\limits_{\sigma\in\partial
T}\frac{1}{n!}\mbox{det}\left(v_{1}(\sigma),\cdots,v_{n}(\sigma)\right),$ (4)

where $v_{i}(\sigma)$ are the vertices of the $(n-1)$-simplex $\sigma$ ordered
according to the orientation of $\sigma$. The other volume formula provided by
Eq. (2.2) in [3] can be expressed as

$\displaystyle\mbox{Vol}_{n}\left(T\right)$ $\displaystyle=$
$\displaystyle(-1)^{(n-1}\sum\limits_{\sigma\in\partial
T}\left(\frac{1}{n}\sum\limits_{i=1}^{n}v_{i}[n]\right)\frac{1}{(n-1)!}\mbox{det}\left(\begin{array}[]{ccc}1&\cdots&1\\\
v_{1}[\neg n]&\cdots&v_{n}[\neg n]\\\ \end{array}\right),$ (7)

where $v_{i}[n]$ is the $n$-th coordinate of the point $v_{i}$, and
$v_{i}[\neg n]$ is obtained by deleting the $n$-th coordinate from $v_{i}$.

Following Lemma 1 is provided to give the volume ratio between $T_{h}$ and
$T$.

###### Lemma 1

Given a compact and connected polyhedra $T$ in $\mathbb{R}^{n}$, we have
volume ratio between $T_{h}$ and $T$, denoted as $R_{\mbox{Poly},n}$, which
can be expressed as

$\displaystyle
R_{\mbox{Poly},n}=\frac{\mbox{Vol}_{n}\left(T_{h}\right)}{\mbox{Vol}_{n}\left(T\right)},$
(8)

where $\mbox{Vol}_{n}\left(T_{h}\right)$ and $\mbox{Vol}_{n}\left(T\right)$
are volumes of $T_{h}$ and $T$.

Proof: We can apply the Quickhull algorithm to convert $T$ into $T_{h}$, see
[4]. From Theorem 2.3 in [4], they show that the Quickhull algorithm can
produce the convex hull of a set of points (polyhedra) in $\mathbb{R}^{n}$.
Then, we can apply Eq. (4) or Eq. (7) to evaluate
$\mbox{Vol}_{n}\left(T_{h}\right)$. Then, this Lemma is proved. $\hfill\Box$

Note that $R_{\mbox{Poly},n}\geq 1$ and the equality holds when $T$ is a
convex set.

###  2.2 General Case

In this section, we will extend polyhedra to general objects in
$\mathbb{R}^{n}$. Our main tool is to reverse Brunn–Minkowski theorem. Let us
recall the traditional reverse Brunn–Minkowski inequality first. There is a
constant $C_{1}$, independent of $n$, such that for any two centrally
symmetric convex objects $A$ and $B$ in $\mathbb{R}^{n}$, there are volume-
preserving linear maps $\rho$ and $\varrho$ from $\mathbb{R}^{n}$ to
$\mathbb{R}^{n}$ such that for any real numbers $s,t>0$ and any natural number
$m$, we have

$\displaystyle[\mbox{Vol}_{n}\left(s\rho(A)\oplus
t\varrho(B)\right)]^{1/m}\leq
C_{1}\left(s[\mbox{Vol}_{n}(\rho(A))]^{1/m}+t[\mbox{Vol}_{n}(\varrho(B))]^{1/m}\right),$
(9)

where $\oplus$ is the Minkowski sum operator between two geometric objects.
However, traditional reverse Brunn–Minkowski theorem requires that the objects
$A$ and $B$ are centrally symmetric convex bodies. We will prove a more
general Theorem 1 that relaxs the objects $A$ and $B$ required to be centrally
symmetric convex objects.

Given a compact object $D$ in $\mathbb{R}^{n}$, we have its circumscribed
ball, denoted as $S_{D}$, and the ratio between $\mbox{Vol}(S_{D})$ and
$\mbox{Vol}(D)$ is represented by $\beta_{D}$, i.e.,

$\displaystyle\beta_{D}=\frac{\mbox{Vol}(S_{D})}{\mbox{Vol}(D)}.$ (10)

Then, we have the following theorem about more general class of objects in
$\mathbb{R}^{n}$.

###### Theorem 1

Given two compact objects with non-empty interior in $\mathbb{R}^{n}$, denoted
as $A$ and $B$, two volume-preserving linear maps $\rho,\varrho$, any real
numbers $s,t>0$ and any natural number $m$, we have

$\displaystyle[\mbox{Vol}_{n}\left(s\rho(A)\oplus
t\varrho(B)\right)]^{1/m}\leq
C_{1}\left(s[\beta_{\rho(A)}\mbox{Vol}_{n}(\rho(A))]^{1/m}+t[\beta_{\varrho(B)}\mbox{Vol}_{n}(\varrho(B))]^{1/m}\right),$
(11)

where $C_{2}$ is a constant.

Proof: Since we have

$\displaystyle[\mbox{Vol}_{n}\left(s\rho(A)\oplus t\varrho(B)\right)]^{1/m}$
$\displaystyle\leq_{1}$ $\displaystyle[\mbox{Vol}_{n}\left(sS_{\rho(A)}\oplus
tS_{\varrho(B)}\right)]^{1/m}$ (12) $\displaystyle\leq_{2}$ $\displaystyle
C_{1}\left(s[\mbox{Vol}_{n}(S_{\rho(A)})]^{1/m}+t[\mbox{Vol}_{n}(S_{\varrho(B)})]^{1/m}\right)$
$\displaystyle=$ $\displaystyle
C_{1}\left(s[\beta_{\rho(A)}\mbox{Vol}_{n}(\rho(A))]^{1/m}+t[\beta_{\varrho(B)}\mbox{Vol}_{n}(\varrho(B))]^{1/m}\right),$

where the inequality $\leq_{1}$ 111We can consider to apply other center
symmetric shapes, like $p$-ball, to provide better circumscribed fit of the
original shapes $\rho(A)$ or $\varrho(B)$ in $\leq_{1}$ of Eq. (12), see [5,
6]. This means that we can provide tighter bound in $\leq_{1}$ of Eq. (12)
according to the geometric properties of $\rho(A)$ or $\varrho(B)$. comes from
the fact that the volume of an object is less or equal than its circumscribed
ball; the inequality $\leq_{2}$ comes from Eq. (9); the lasy equality comes
from the definition $\beta$ provided by Eq. (10). $\hfill\Box$

Given an object $A$ in $\mathbb{R}^{n}$, we define $A(k)$ as

$\displaystyle A(k)=\frac{1}{k}\oplus_{i=1}^{k}A_{i},$ (13)

where $A_{i}=A$. Then, we have the following Lemma about the volume $A(k)$ and
the volume $A(k-1)$.

###### Lemma 2

Let $A$ be a compact objects with non-empty interior in $\mathbb{R}^{n}$, then
we have

$\displaystyle\mbox{Vol}_{n}(A(k))\leq\left(\frac{2C^{(k-1)}_{2}}{k}+\frac{C_{2}(C^{(k-2)}_{2}-1)}{k(C_{2}-1)}\right)\mbox{Vol}_{n}(A),$
(14)

where $C_{2}=C_{1}C^{\prime}_{2}$ and $C^{\prime}_{2}$ is an upper bound for
$\beta_{A(k)}$.

Proof: From [7], we know that $A(k)\rightarrow A_{h}$ when
$k\rightarrow\infty$, where $A_{h}$ is the convex hull of $A$. Since $A_{h}$
is also a volume bounded object, we can upper bound the ratio parameter
$\beta_{A(k)}$ by $C^{\prime}_{2}$ for all $k$, where $C^{\prime}_{2}$ is a
positive number.

Because we have

$\displaystyle\mbox{Vol}_{n}(A(k))$ $\displaystyle=$
$\displaystyle\mbox{Vol}_{n}\left(\frac{k-1}{k}A(k-1)+\frac{1}{k}A\right)$
(15) $\displaystyle\leq_{1}$ $\displaystyle
C_{1}\left(\frac{k-1}{k}[\beta_{A(k-1)}\mbox{Vol}_{n}(A(k-1))]+\frac{1}{k}[\beta_{A}\mbox{Vol}_{n}(A)]\right)$
$\displaystyle\leq_{2}$ $\displaystyle
C_{1}\left(\frac{k-1}{k}[C^{\prime}_{2}\mbox{Vol}_{n}(A(k-1))]+\frac{1}{k}[C^{\prime}_{2}\mbox{Vol}_{n}(A)]\right)$

where the inequality $\leq_{1}$ is obtained from Theorem 1 by setting $m=1$;
the inequality $\leq_{2}$ is obtained by using the upper bound
$C^{\prime}_{2}$ for $\beta_{A(k-1)}$ and $\beta_{A}=\beta_{A(1)}$.

By applying Eq. (15) iterativelys, we can uppder the volume of $A(k)$ by the
volume of $A$ as

$\displaystyle\mbox{Vol}_{n}(A(k))\leq\left(\frac{2C^{(k-1)}_{2}}{k}+\frac{C_{2}(C^{(k-2)}_{2}-1)}{k(C_{2}-1)}\right)\mbox{Vol}_{n}(A),$
(16)

where $C_{2}=C_{1}C^{\prime}_{2}$. This Lemma is proved. $\hfill\Box$

Following Lemma 3 is provided to give the volume ratio between $T_{h}$ and $T$
when $T$ is a general compact object with non-empty interior in
$\mathbb{R}^{n}$.

###### Lemma 3

Given $T$ as a compact object with non-empty interior in $\mathbb{R}^{n}$ and
we assume that $A(k_{h})=A_{h}$ for some natural number $k_{h}$. This means
that $A(k_{h}+n)$ will be equal to $A_{h}$ for any natural number $n$ since
$A(k_{h})$ is a convex set already. We have volume ratio between $T_{h}$ and
$T$, denoted as $R_{\mbox{Gen},n}$, which can be expressed as

$\displaystyle R_{\mbox{Gen},n}$ $\displaystyle=$
$\displaystyle\frac{\mbox{Vol}_{n}\left(T_{h}\right)}{\mbox{Vol}_{n}\left(T\right)}$
(17) $\displaystyle\leq$
$\displaystyle\left(\frac{2C^{(k_{h}-1)}_{2}}{k_{h}}+\frac{C_{2}(C^{(k_{h}-2)}_{2}-1)}{k_{h}(C_{2}-1)}\right).$

Proof: This Lemma can be obtained directly from Lemma 2. $\hfill\Box$

##  3 Covering Number Bounds by Volume

In this section, we will try to upper bound the covering number of $T_{h}$ by
the covering number of $T$ via covering number bounds by volume. We begin with
the covering number definition.

###### Definition 2

We use $N(A,d(\cdot,\cdot),\epsilon)$ to represent the covering number of the
space $A$ with $\epsilon$-ball with distance metric function $d(\cdot,\cdot)$.
Then, $N(A,d(\cdot,\cdot),\epsilon)$ can be defined as

$\displaystyle N(T,d(\cdot,\cdot),\epsilon)\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\min\\{n:\mbox{the number of $\epsilon$-ball to cover $T$}\\}.$
(18)

From Theorem 14.2 in [8], we have

$\displaystyle\left(\frac{1}{\epsilon}\right)^{n}\frac{\mbox{Vol}(A)}{\mbox{Vol}(\mathbf{B})}\leq
N(A,d(\cdot,\cdot),\epsilon)\leq\frac{\mbox{Vol}(A\oplus\frac{\epsilon}{2}\mathbf{B})}{\mbox{Vol}(\frac{\epsilon}{2}\mathbf{B})}\leq_{1}\left(\frac{3}{\epsilon}\right)^{n}\frac{\mbox{Vol}(A)}{\mbox{Vol}(\mathbf{B})}$
(19)

where $\mathbf{B}$ is the unit norm ball, and the inequality $\leq_{1}$ is
valid if $A$ is a convex set and $\epsilon\mathbf{B}\subset A$.

We will have the following two Lemmas to upper bound the covering number of
$T_{h}$ by the covering number of $T$ with respect to different types of $T$.

###### Lemma 4

Given a compact and connected polyhedra $T$ in $\mathbb{R}^{n}$, we have the
following relation between the covering number of $T_{h}$ and the covering
number of $T$:

$\displaystyle N(T_{h},d(\cdot,\cdot),\epsilon)$ $\displaystyle\leq$
$\displaystyle R_{\mbox{Poly},n}3^{n}N(T,d(\cdot,\cdot),\epsilon),$ (20)

where $R_{\mbox{Poly},n}$ is the volume ratio given by Lemma 1.

Proof:

Because $T_{h}$ is a convex set, we have

$\displaystyle N(T_{h},d(\cdot,\cdot),\epsilon)$ $\displaystyle\leq$
$\displaystyle\left(\frac{3}{\epsilon}\right)^{n}\frac{\mbox{Vol}(T_{h})}{\mbox{Vol}(\mathbf{B})}$
(21) $\displaystyle=_{1}$
$\displaystyle\left(\frac{3}{\epsilon}\right)^{n}\frac{R_{\mbox{Poly},n}\mbox{Vol}(T)}{\mbox{Vol}(\mathbf{B})}$
$\displaystyle=$ $\displaystyle
R_{\mbox{Poly},n}3^{n}\left(\frac{1}{\epsilon}\right)^{n}\frac{\mbox{Vol}(T)}{\mbox{Vol}(\mathbf{B})}$
$\displaystyle\leq$ $\displaystyle
R_{\mbox{Poly},n}3^{n}N(T,d(\cdot,\cdot),\epsilon),$

where the equality $=_{1}$ comes from Lemma 1, and the first and the last
inequalities are obtained from Eq. (19). $\hfill\Box$

###### Lemma 5

Given a compact object $T$ with non-empty interior in $\mathbb{R}^{n}$ and
assume that $T_{h}=\frac{1}{k}\oplus_{i=1}^{k}T$ with finite $k$, we have the
following relation between the covering number of $T_{h}$ and the covering
number of $T$:

$\displaystyle N(T_{h},d(\cdot,\cdot),\epsilon)$ $\displaystyle\leq$
$\displaystyle R_{\mbox{Gen},n}3^{n}N(T,d(\cdot,\cdot),\epsilon),$ (22)

where $R_{\mbox{Gen},n}$ is the volume ratio given by Lemma 3.

Proof:

Because $T_{h}$ is a convex set, we have

$\displaystyle N(T_{h},d(\cdot,\cdot),\epsilon)$ $\displaystyle\leq$
$\displaystyle\left(\frac{3}{\epsilon}\right)^{n}\frac{\mbox{Vol}(T_{h})}{\mbox{Vol}(\mathbf{B})}$
(23) $\displaystyle=_{1}$
$\displaystyle\left(\frac{3}{\epsilon}\right)^{n}\frac{R_{\mbox{Gen},n}\mbox{Vol}(T)}{\mbox{Vol}(\mathbf{B})}$
$\displaystyle=$ $\displaystyle
R_{\mbox{Gen},n}3^{n}\left(\frac{1}{\epsilon}\right)^{n}\frac{\mbox{Vol}(T)}{\mbox{Vol}(\mathbf{B})}$
$\displaystyle\leq$ $\displaystyle
R_{\mbox{Gen},n}3^{n}N(T,d(\cdot,\cdot),\epsilon),$

where the equality $=_{1}$ comes from Lemma 3, and the first and the last
inequalities are obtained from Eq. (19). $\hfill\Box$

##  4 Majorizing Measures By Geometry Approach

From Lemma 4 and Lemma 5, we are ready to prove Theorem 2.11.1 in [1]
geometrically at Eucledean space with dimension $n$.

###### Theorem 2

Given the space $T$ as a closed bounded polyhedra in $\mathbb{R}^{n}$ having
an orientable boundary, we have

$\displaystyle\gamma_{\alpha}(T_{h},d(\cdot,\cdot))\leq
L_{\mbox{Poly}}\gamma_{\alpha}(T,d(\cdot,\cdot)),$ (24)

where $L_{\mbox{Poly}}$ is the constant depending on the underlying geometry
of the space $T$ and $\alpha$.

Proof:

Because we have

$\displaystyle\gamma_{\alpha}(T_{h},d(\cdot,\cdot))$
$\displaystyle\leqslant_{\alpha}$ $\displaystyle\int_{0}^{\infty}\left(\log
N(T_{h},d(\cdot,\cdot),\epsilon)\right)^{1/\alpha}d\epsilon$ (25)
$\displaystyle\leq_{1}$ $\displaystyle\int_{0}^{\infty}\left(\log
R_{\mbox{Poly},n}3^{n}N(T,d(\cdot,\cdot),\epsilon)\right)^{1/\alpha}d\epsilon$
$\displaystyle\leq$
$\displaystyle\left(\frac{\log(R_{\mbox{Poly},n}3^{n})}{\log
2}+1\right)^{1/\alpha}\int_{0}^{\infty}\left(\log
N(T,d(\cdot,\cdot),\epsilon)\right)^{1/\alpha}d\epsilon$
$\displaystyle\leqslant_{\alpha}$
$\displaystyle\left(\frac{\log(R_{\mbox{Poly},n}3^{n})}{\log
2}+1\right)^{1/\alpha}\gamma_{\alpha}(T,d(\cdot,\cdot)),$

where the inequality $\leq_{1}$ comes from Lemma 4, and the first and last
inequalities come from Theorem 1.2 in [9]. This Lemma is proved. $\hfill\Box$

###### Theorem 3

Given a compact object $T$ in $\mathbb{R}^{n}$ such that
$T_{h}=\frac{1}{k}\oplus_{i=1}^{k}T$ with finite $k$, we have

$\displaystyle\gamma_{\alpha}(T_{h},d(\cdot,\cdot))\leq
L_{\mbox{Gen}}\gamma_{\alpha}(T,d(\cdot,\cdot)),$ (26)

where $L_{\mbox{Gen}}$ is the constant depending on the underlying geometry of
the space $T$ and $\alpha$.

Proof:

Since we have

$\displaystyle\gamma_{\alpha}(T_{h},d(\cdot,\cdot))$
$\displaystyle\leqslant_{\alpha}$ $\displaystyle\int_{0}^{\infty}\left(\log
N(T_{h},d(\cdot,\cdot),\epsilon)\right)^{1/\alpha}d\epsilon$ (27)
$\displaystyle\leq_{1}$ $\displaystyle\int_{0}^{\infty}\left(\log
R_{\mbox{Gen},n}3^{n}N(T,d(\cdot,\cdot),\epsilon)\right)^{1/\alpha}d\epsilon$
$\displaystyle\leq$
$\displaystyle\left(\frac{\log(R_{\mbox{Gen},n}3^{n})}{\log
2}+1\right)^{1/\alpha}\int_{0}^{\infty}\left(\log
N(T,d(\cdot,\cdot),\epsilon)\right)^{1/\alpha}d\epsilon$
$\displaystyle\leqslant_{\alpha}$
$\displaystyle\left(\frac{\log(R_{\mbox{Gen},n}3^{n})}{\log
2}+1\right)^{1/\alpha}\gamma_{\alpha}(T,d(\cdot,\cdot)),$

where the inequality $\leq_{1}$ comes from Lemma 5, and the first and last
inequalities come from Theorem 1.2 in [9]. This Lemma is proved. $\hfill\Box$

##  5 Infinite Dimensional $T$ Discussion

From Section 4, the key part in the geometric approach to prove Theorem 2.11.1
in [1] is to represent the covering number $N(T_{h},d(\cdot,\cdot),\epsilon)$
in terms of the covering number $N(T,d(\cdot,\cdot),\epsilon)$. We will show
that, when $T$ is a subset of an infinite dimensional space, e.g., Banach
space or Hilbert space, the constant $L$ at Eq. (3) may not always exist. The
existence will depend on geometric properties of $T$.

According to Section 2 in [2], the authors list some specific estimates of
$\log N(T_{h},d(\cdot,\cdot),\epsilon)$ under various common assumptions on
$\log N(T,d(\cdot,\cdot),\epsilon)$. Here, we assume $p=2$ and $q=1$ (convex
hull) in those parameters used in Section 2 in [2].

If $\log
N(T,d(\cdot,\cdot),\epsilon)=O(\epsilon^{-\chi}\left|\log\epsilon\right|^{\psi})$
for $\chi>2$, and $\psi\in\mathbb{R}$, we have $\log
N(T_{h},d(\cdot,\cdot),\epsilon)=O(\epsilon^{-\chi}\left|\log\epsilon\right|^{\psi})$.
In this situation, we can find a constant, say $C_{3}$, to upper bound the
following ratio between the covering number $N(T_{h},d(\cdot,\cdot),\epsilon)$
and the covering number $N(T,d(\cdot,\cdot),\epsilon)$:

$\displaystyle\frac{\log N(T_{h},d(\cdot,\cdot),\epsilon)}{\log
N(T,d(\cdot,\cdot),\epsilon)}\leq C_{3}.$ (28)

According to the proof in Theorem 2 (or Theorem 3), the ratio between
$\frac{\gamma_{\alpha}(T_{h},d(\cdot,\cdot))}{\gamma_{\alpha}(T,d(\cdot,\cdot))}$
can be upper bound by the integration with respect to $\epsilon$ from 0 to
$\Delta(T)$ for contant integrand $C_{3}$.

If $\log
N(T,d(\cdot,\cdot),\epsilon)=O(\epsilon^{-2}\left|\log\epsilon\right|^{\psi})$
for $\psi>-2$, we have $\log
N(T_{h},d(\cdot,\cdot),\epsilon)=O(\epsilon^{-2}\left|\log\epsilon\right|^{2+\psi})$.
In this situation, we can find a function of $\epsilon$ to upper bound the
following ratio between the covering number $N(T_{h},d(\cdot,\cdot),\epsilon)$
and the covering number $N(T,d(\cdot,\cdot),\epsilon)$:

$\displaystyle\frac{\log N(T_{h},d(\cdot,\cdot),\epsilon)}{\log
N(T,d(\cdot,\cdot),\epsilon)}\leq C_{4}\left|\log\epsilon\right|^{2}.$ (29)

According to the proof in Theorem 2 (or Theorem 3), the ratio between
$\frac{\gamma_{\alpha}(T_{h},d(\cdot,\cdot))}{\gamma_{\alpha}(T,d(\cdot,\cdot))}$
can be upper bound by the integration with respect to $\epsilon$ from 0 to
$\Delta(T)$ for the integrand as $C_{4}\left|\log\epsilon\right|^{2}$ since
such integration exists.

If $\log
N(T,d(\cdot,\cdot),\epsilon)=O(\epsilon^{-2}\left|\log\epsilon\right|^{\psi})$
for $\psi=-3$, we have $\log N(T_{h},d(\cdot,\cdot),\epsilon)=$
$O(\epsilon^{-2}\left(\left|\log\left|\log\epsilon\right|\right|\right)^{2+\psi})$.
In this situation, we can find a function of $\epsilon$ to upper bound the
following ratio between the covering number $N(T_{h},d(\cdot,\cdot),\epsilon)$
and the covering number $N(T,d(\cdot,\cdot),\epsilon)$:

$\displaystyle\frac{\log N(T_{h},d(\cdot,\cdot),\epsilon)}{\log
N(T,d(\cdot,\cdot),\epsilon)}\leq
C_{5}\frac{\left|\log\epsilon\right|^{3}}{\left|\log\left|\log\epsilon\right|\right|}.$
(30)

According to the proof in Theorem 2 (or Theorem 3), the ratio between
$\frac{\gamma_{\alpha}(T_{h},d(\cdot,\cdot))}{\gamma_{\alpha}(T,d(\cdot,\cdot))}$
can be upper bound by the integration with respect to $\epsilon$ from 0 to
$\Delta(T)$ for the integrand as
$C_{5}\frac{\left|\log\epsilon\right|^{3}}{\log\left|\log\epsilon\right|}$.
However, if the diameter $\Delta(T)$ equals one, this integral does not exist.

Therefore, the existence of the constant $L$ at Theorem 2.11.1 in [1] can be
summarized by the following Theorem 4

###### Theorem 4

Suppose we have a function $f(\epsilon)$ which satisfies the following:

$\displaystyle\frac{\log N(T_{h},d(\cdot,\cdot),\epsilon)}{\log
N(T,d(\cdot,\cdot),\epsilon)}\leq f(\epsilon),$ (31)

given $0<\epsilon\leq\Delta(T)$. If the following integral exists:

$\displaystyle\int_{0}^{\Delta(T)}f(\epsilon),$ (32)

we can find the constant $L$ at Eq. (3).

Proof:

The proof is the same as the argument used in the proof for Theorem 2 or
Theorem 3. $\hfill\Box$

## References

  * [1] M. Talagrand, “Upper and lower bounds for stochastic processes. decomposition theorems,” _Ergebnisse der Mathematik und ihrer Grenzgebiete_ , vol. 60, 2021\.
  * [2] J. Cockreham, F. Gao, and Y. Yang, “Metric entropy of $q$-hulls in Banach spaces of type-$p$,” _Proceedings of the American Mathematical Society_ , vol. 145, no. 12, pp. 5205–5214, 2017.
  * [3] E. L. Allgower and P. H. Schmidt, “Computing volumes of polyhedra,” _Mathematics of Computation_ , vol. 46, no. 173, pp. 171–174, 1986.
  * [4] C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” _ACM Transactions on Mathematical Software (TOMS)_ , vol. 22, no. 4, pp. 469–483, 1996.
  * [5] J. Bastero, J. Bernués, and A. Pena, “An extension of milman’s reverse brunn-minkowski inequality,” _arXiv preprint math/9501210_ , 1995.
  * [6] A. Litvak, “On the constant in the reverse brunn-minkowski inequality for p-convex balls, convex geometric analysis,(berkeley, ca, 1996), 129–137,” _Math. Sci. Res. Inst. Publ_ , vol. 34.
  * [7] M. Fradelizi, M. Madiman, A. Marsiglietti, and A. Zvavitch, “The convexification effect of minkowski summation,” _EMS Surveys in Mathematical Sciences_ , vol. 5, no. 1, pp. 1–64, 2018.
  * [8] “14.1 covering and packing,” http://www.stat.yale.edu/~yw562/teaching/598/lec14.pdf, accessed: 2016-05-30.
  * [9] M. Talagrand, “Majorizing measures without measures,” _Annals of probability_ , pp. 411–417, 2001.