(polarized) electromagnetic waves.

###  VII.2 Mesoscopic optics in fractal and long-range correlated media

Light propagation in positively-correlated media is characterized by a non-
exponential decay of the coherent intensity. As discussed in Sec. IV.2,
materials exhibiting fractal heterogeneities in the form of non-scattering
regions of varying sizes can lead in certain conditions to a superdiffusive
behavior Burioni  _et al._ (2014); Savo  _et al._ (2014). Anomalous transport
processes Klages  _et al._ (2008) and dynamics on fractal networks Nakayama
_et al._ (1994) have a long history, but optical studies on fractal media have
so far been mostly concerned with structure factor measurements in the single
scattering regime Lin  _et al._ (1989) (note that the optical properties of
semicontinuous metal films near percolation, on which there exists a vast
litterature Shalaev (2007), strongly rely on near-field plasmonic effects and
not on light transport). Coherent optical phenomena in “Lévy-like” media have
been sparsely addressed until now Burresi  _et al._ (2012). Multiple
scattering formalisms have been extended to media described by fractal
dimensions Akkermans  _et al._ (1988); Wang and Lu (1994) or exhibiting
superdiffusion Bertolotti  _et al._ (2010b), disregarding however several
difficulties related to the definition of the self-energy and the effective
index Tarasov (2015), and perhaps more importantly to the quenched nature of
disorder Burioni  _et al._ (2014); Barthelemy  _et al._ (2010). All in all,
the development of a rigorous ab-initio theory for multiple light scattering
in strongly-heterogeneous materials would be a formidable achievement.

Numerical and experimental studies on 1D and quasi-1D Lévy-like systems have
revealed anomalous conductance fluctuations and scaling Fernández-Marín  _et
al._ (2014); Ardakani and Nezhadhaghighi (2015); Lima  _et al._ (2019).
Higher-dimensional systems are likely to exhibit a similarly rich physics,
which fully remains to be explored. One example is the critical dimension of 2
above which the Anderson transition exists Abrahams  _et al._ (1979) that may
be lowered, depending on the fractality or lacunarity of the system. Optical
experiments and numerical simulations could be performed in this regard on
high-index planar photonic structures similarly to Riboli  _et al._ (2014),
giving access to LDOS statistics, or to Yamilov  _et al._ (2014) for
transmittance and internal light intensity measurements.

An alternative route for the experimental study of mesoscopic phenomena in
long-range correlated systems could rely on disordered photonic network Gaio
_et al._ (2019) – an optical implementation of random graphs Janson  _et al._
(2011) –, wherein light propagates through 1D waveguides and is scattered at
the waveguide vertices. The waveguide lengths and vertices connectivity thus
take the role of structural correlations. The network is a low-dimensional
medium embedded in three-dimensional space, and allow to design light
transport and the optical modes. Complex networks with finely-controlled
parameters can be fabricated by self-assembly Gaio  _et al._ (2019), by
standard lithography techniques or implemented on macroscopic systems Lepri
_et al._ (2017).

###  VII.3 Towards novel applications

The sensitivity of the LDOS to the local environment discussed in Sec. V.4
makes quantum emitters interesting optical probes of nanostructured materials
Pelton (2015). Many studies have reported the dramatic impact of the local
morphology of a complex medium on the spontaneous emission statistics from
neighboring fluorescent molecules or quantum dots Krachmalnicoff  _et al._
(2010); Sapienza  _et al._ (2011); Birowosuto  _et al._ (2010); de Sousa  _et
al._ (2016); Riboli  _et al._ (2017). A key question to address will be
whether optical measurements mediated by near-field probes could reveal
statistical information on an unknown morphology, which could be extremely
interesting for the remote monitoring of structural phases deep inside a 3D
volume de Sousa  _et al._ (2016). This would require a deep understanding on
the relation between subwavelength-scale correlations and near-field
phenomena. In addition to LDOS measurements, measuring the cross spectral
density of states (CDOS) Cazé  _et al._ (2013), which describes mode
connectivity in structured media and could be obtained from coherence
measurements on the light emitted from two classical or quantum dipole sources
Canaguier-Durand  _et al._ (2019), may bring precious additional information.
First CDOS measurements have recently been realized in the microwave regime
Rustomji  _et al._ (2021).

The coherent control of light waves in disordered media is an important branch
of research in multiple light scattering that has been powered in recent years
by wavefront shaping techniques Rotter and Gigan (2017). We have seen in Sec.
V that structural correlations can result into strong spectral variations of
scattering, transport and localization, suggesting that correlated disorder
could yield higher degrees of spectral and spatial control, with applications
in optical imaging. Disordered media are also being exploited as an
unconventional platform for quantum walks and quantum state engineering
Defienne  _et al._ (2016); Leedumrongwatthanakun  _et al._ (2020). These are
very delicate experiments that require lossless materials, so far attempted in
multimode fibers, but which could benefit in the future from correlated
disordered media for multiplexing and spectral resolution.

The design of visual appearance is another aspect of research on correlated
disorder that has considerably grown in importance in recent years. As
beautifully illustrated by many diverse examples in the living world, the
interplay or order and disorder has a direct impact on appearance at
macroscopic scales, yielding increased transparency or whiteness, iridescent
or non-iridescent colors [Sec. VI.3]. The numerical modelling of realistic
correlated materials, considering for instance local imperfections Chung  _et
al._ (2012) and large-scale random variations Chan  _et al._ (2019) in certain
ordered systems, will be an essential development of the field in the near
future. Our perception of objects indeed relies on many attributes of visual
appearance Hunter and Harold (1987) – not only color, but also gloss, haze,
translucency, texture, etc. –, which are affected by multiple scattering and
are rarely considered in full. Understanding how optical properties created by
structural correlations at the microscopic scale translate into visual effects
at the macroscopic scale is a great and exciting challenge for the coming
years. Beyond appearance, correlated disordered media will play an important
role in thermal management, for example for radiative cooling Wang and Zhao
(2020), where broadband light control is needed from inexpensive self-
assembled media. Correlated disordered materials could be used to realize
multifunctional materials, where optical (visual) properties could be combined
with desired thermal, electrical, mechanical or tribological functionalities.
Last but not least, efforts should be amplified to develop and promote low-
carbon footprint, ecologically-responsible material synthesis, which can be
best achieved in disordered assemblies as those discussed in this review.

## Acknowledgments

We are very grateful to Aristide Dogariu (CREOL, University of Central
Florida, USA) and Akhlesh Lakhtakia (Penn State University, USA) for fruitful
discussions. KV acknowledges funding from the french National Agency for
Research (ANR) via the projects “NanoMiX” (ANR-16-CE30-0008) and “Nano-
Appearance” (ANR-19-CE09-0014-01). RP and RC acknowledge funding from LABEX
WIFI (Laboratory of Excellence within the French Program ”Investments for the
Future”) under references ANR-10-LABX-24 and ANR-10-IDEX-0001-02 PSL*. This
research was supported by the Swiss National Science Foundation through
project numbers 169074, 188494 (FS) and 197146 (LSFP). RS and SV acknowledge
funding by the Engineering and Physical Sciences Research Council (EPSRC). SV
acknowledges the European Research Council (ERC-2014-STG H2020 639088).

##  Appendix A Green functions in Fourier space

###  A.1 Dyadic Green tensor

We consider a statistically homogeneous and translationally-invariant medium.
The dyadic Green tensor in a uniform background medium with auxiliary
permittivity $\epsilon_{\text{b}}$ is given by Eq. (14) and related to its
Fourier transform as

$\displaystyle{\bf G}_{\text{b}}({\bf r}-{\bf
r}^{\prime})=\frac{1}{(2\pi)^{3}}\int{\bf G}_{\text{b}}({\bf k})e^{{\rm i}{\bf
k}\cdot({\bf r}-{\bf r}^{\prime})}d{\bf k}.$ (139)

The Green tensor in Fourier space is given by

$\displaystyle{\bf G}_{\text{b}}({\bf k})$ $\displaystyle=$
$\displaystyle\left[k^{2}\bm{\mathbf{1}}-{\bf k}\otimes{\bf
k}-k_{\text{b}}^{2}\bm{\mathbf{1}}\right]^{-1}$ (140) $\displaystyle=$
$\displaystyle\left[-k_{\text{b}}^{2}\frac{{\bf k}\otimes{\bf
k}}{k^{2}}+\left(k^{2}-k_{\text{b}}^{2}\right)\left(\bm{\mathbf{1}}-\frac{{\bf
k}\otimes{\bf k}}{k^{2}}\right)\right]^{-1}$ $\displaystyle=$
$\displaystyle\frac{1}{k_{\text{b}}^{2}}\left[-\frac{{\bf k}\otimes{\bf
k}}{k^{2}}+\frac{k_{\text{b}}^{2}}{k^{2}-(k_{\text{b}}+{\rm
i}0)^{2}}\left(\bm{\mathbf{1}}-\frac{{\bf k}\otimes{\bf
k}}{k^{2}}\right)\right].$

The small imaginary part ${\rm i}0$ introduced here is relevant in integrals
involving ${\bf G}_{\text{b}}({\bf k})$. Using Eq. (31), the Green tensor can
finally be rewritten as

$\displaystyle{\bf G}_{\text{b}}({\bf k})$ $\displaystyle=$
$\displaystyle\frac{1}{k_{\text{b}}^{2}}\left[-\frac{{\bf k}\otimes{\bf
k}}{k^{2}}+\text{PV}\left\\{\frac{k_{\text{b}}^{2}}{k^{2}-k_{\text{b}}^{2}}\right\\}\left(\bm{\mathbf{1}}-\frac{{\bf
k}\otimes{\bf k}}{k^{2}}\right)\right]$ (143) $\displaystyle+{\rm
i}\pi\delta(k^{2}-k_{\text{b}}^{2})\left(\bm{\mathbf{1}}-\frac{{\bf
k}\otimes{\bf k}}{k^{2}}\right).$

###  A.2 Dressed Green tensor

Following Eq. (55), the Green tensor can be decomposed into local and non-
local terms, the latter being also known as the Lorentz propagator. In Fourier
space, we thus have

$\displaystyle\bm{\mathbf{g}}_{\text{b}}({\bf k})$ $\displaystyle=$
$\displaystyle-\frac{1}{3k_{\text{b}}^{2}}+\int_{r<a}{\bf G}_{\text{b}}({\bf
r})e^{-{\rm i}{\bf k}\cdot{\bf r}}d{\bf r},$ (144) $\displaystyle\tilde{{\bf
G}}_{\text{b}}({\bf k})$ $\displaystyle\equiv$ $\displaystyle{\bf
G}_{\text{b}}({\bf k})-\bm{\mathbf{g}}_{\text{b}}({\bf k}).$ (145)

Specific expressions for arbitrary values of the radius $a$ are provided by
Bedeaux and Mazur (1973). For $k_{\text{b}}a\ll 1$, we have

$\displaystyle\bm{\mathbf{g}}_{\text{b}}({\bf
r})=-\frac{1}{3k_{\text{b}}^{2}}\delta({\bf r}-{\bf
r}^{\prime})\bm{\mathbf{1}},$ (146)

which simply leads to

$\displaystyle\tilde{{\bf G}}_{\text{b}}({\bf k})$ $\displaystyle\equiv$
$\displaystyle{\bf G}_{\text{b}}({\bf
k})+\frac{\bm{\mathbf{1}}}{3k_{\text{b}}^{2}}.$ (147)

Let us now consider a medium composed of small volume elements within the
Maxwell-Garnett approximation. Setting the auxiliary permittivity to that of
the host medium, $\epsilon_{\text{b}}=\epsilon_{\text{h}}$, the average
transition operator describing scattering by a single volume element is then
given by

$\displaystyle\left\langle\tilde{{\bf T}}({\bf
k})\right\rangle=k_{\text{h}}^{2}\rho\alpha_{0}\bm{\mathbf{1}},$ (148)

with $\alpha_{0}$ the polarizability. The Fourier transform of the propagator
$\hat{\bm{\mathbf{\mathcal{G}}}}_{\text{b}}$ defined in Eq. (67) then reads

$\displaystyle\hat{{\bf G}}_{\text{h}}({\bf k})$ $\displaystyle=$
$\displaystyle\left[\bm{\mathbf{1}}-k_{\text{h}}^{2}\rho\alpha_{0}\tilde{{\bf
G}}_{\text{h}}({\bf k})\right]^{-1}\tilde{{\bf G}}_{\text{h}}({\bf k})$
$\displaystyle=$
$\displaystyle\left[\bm{\mathbf{1}}-\frac{\rho\alpha_{0}}{3}\bm{\mathbf{1}}-\rho\alpha_{0}k_{\text{h}}^{2}{\bf
G}_{\text{h}}({\bf k})\right]^{-1}\left({\bf G}_{\text{h}}({\bf
k})+\frac{\bm{\mathbf{1}}}{3k_{\text{h}}^{2}}\right).$

Taking into account the definition of the Maxwell-Garnett permittivity
$\epsilon_{\text{MG}}$ in Eq. (75), the dressed propagator can eventually be
written as

$\displaystyle\hat{{\bf G}}_{\text{h}}({\bf k})$ $\displaystyle=$
$\displaystyle\left(\frac{\epsilon_{\text{MG}}+2\epsilon_{\text{h}}}{3\epsilon_{\text{h}}}\right)^{2}\left({\bf
G}_{\text{MG}}({\bf
k})+\frac{\epsilon_{\text{MG}}}{\epsilon_{\text{MG}}+2\epsilon_{\text{h}}}\frac{\bm{\mathbf{1}}}{k_{\text{MG}}^{2}}\right),$

where ${\bf G}_{\text{MG}}({\bf k})$ is the Green tensor with $k_{\text{h}}$
replaced by the Maxwell-Garnett wave number
$k_{\text{MG}}=k_{0}\sqrt{\epsilon_{\text{MG}}}$.

In absence of absorption, the imaginary part of $\hat{{\bf G}}_{\text{h}}$ is
given by

$\displaystyle\text{Im}\hat{{\bf G}}_{\text{h}}({\bf k})$ $\displaystyle=$
$\displaystyle\pi\left(\frac{\epsilon_{\text{MG}}+2\epsilon_{\text{h}}}{3\epsilon_{\text{h}}}\right)^{2}\delta(k^{2}-k_{\text{MG}}^{2})\left(\bm{\mathbf{1}}-\frac{{\bf
k}\otimes{\bf k}}{k^{2}}\right),$

leading to Eq. (76).

##  Appendix B Derivation of Eq. (26)

In the Fourier domain and by making use of the average Green function

$\left\langle\bm{\mathbf{G}}(\bm{\mathbf{k}},\omega)\right\rangle=\left[k^{2}\bm{\mathbf{P}}(\hat{\bm{\mathbf{k}}})-k_{b}^{2}\bm{\mathbf{1}}-\bm{\mathbf{\Sigma}}(\bm{\mathbf{k}})\right]^{-1},$
(153)

we obtain

$\left\\{\bm{\mathbf{1}}\otimes\left[k^{\prime
2}\bm{\mathbf{P}}(\hat{\bm{\mathbf{k^{\prime}}}})-\bm{\mathbf{\Sigma}}^{*}(\bm{\mathbf{k}}^{\prime})\right]-\left[k^{2}\bm{\mathbf{P}}(\hat{\bm{\mathbf{k}}})-\bm{\mathbf{\Sigma}}(\bm{\mathbf{k}})\right]\otimes\bm{\mathbf{1}}\right\\}\\\
\cdot\bm{\mathbf{C}}(\bm{\mathbf{k}},\bm{\mathbf{k}}^{\prime})=\left[\left\langle\bm{\mathbf{G}}(\bm{\mathbf{k}})\right\rangle\otimes\bm{\mathbf{1}}-\bm{\mathbf{1}}\otimes\left\langle\bm{\mathbf{G}}^{*}(\bm{\mathbf{k}}^{\prime})\right\rangle\right]\\\
\cdot\int\bm{\mathbf{\Gamma}}(\bm{\mathbf{k}},\bm{\mathbf{\upkappa}},\bm{\mathbf{k}}^{\prime},\bm{\mathbf{\upkappa}}^{\prime})\cdot\bm{\mathbf{C}}(\bm{\mathbf{\upkappa}},\bm{\mathbf{\upkappa}}^{\prime})\frac{\mathrm{d}\bm{\mathbf{\upkappa}}}{8\pi^{3}}\frac{\mathrm{d}\bm{\mathbf{\upkappa}}^{\prime}}{8\pi^{3}}$
(154)

where we have neglected the source term
$\left\langle\bm{\mathbf{E}}\right\rangle\otimes\left\langle\bm{\mathbf{E}}^{*}\right\rangle$
and used the tensorial relation

$(\bm{\mathbf{1}}\otimes\bm{\mathbf{B}}-\bm{\mathbf{A}}\otimes\bm{\mathbf{1}})^{-1}\cdot(\bm{\mathbf{A}}^{-1}\otimes\bm{\mathbf{1}}-\bm{\mathbf{1}}\otimes\bm{\mathbf{B}}^{-1})=\bm{\mathbf{A}}^{-1}\otimes\bm{\mathbf{B}}^{-1}.$
(155)

$\bm{\mathbf{1}}$ is the identity tensor. Since we are dealing with dilute
media and since the longitudinal part of the Green tensor is irrelevent
regarding light transport, we now consider the transverse approximation which
consists in taking the transverse part of all operators involved in Eq. (154)
Barabanenkov  _et al._ (1995); Cherroret  _et al._ (2016). Using the
definitions

$\displaystyle\bm{\mathbf{\Gamma}}_{\perp}(\bm{\mathbf{k}},\bm{\mathbf{\upkappa}},\bm{\mathbf{k}}^{\prime},\bm{\mathbf{\upkappa}}^{\prime})$
$\displaystyle=\bm{\mathbf{P}}(\bm{\mathbf{u}})\otimes\bm{\mathbf{P}}(\bm{\mathbf{u}}^{\prime})\cdot\bm{\mathbf{\Gamma}}(\bm{\mathbf{k}},\bm{\mathbf{\upkappa}},\bm{\mathbf{k}}^{\prime},\bm{\mathbf{\upkappa}}^{\prime}),$
$\displaystyle\bm{\mathbf{C}}_{\perp}(\bm{\mathbf{k}},\bm{\mathbf{k}}^{\prime})$
$\displaystyle=\bm{\mathbf{P}}(\bm{\mathbf{u}})\otimes\bm{\mathbf{P}}(\bm{\mathbf{u}}^{\prime})\cdot\bm{\mathbf{C}}(\bm{\mathbf{k}},\bm{\mathbf{k}}^{\prime}),$

we obtain

$\left[k^{\prime
2}-k^{2}-\Sigma_{\perp}^{*}(\bm{\mathbf{k}}^{\prime})+\Sigma_{\perp}(\bm{\mathbf{k}})\right]\bm{\mathbf{C}}_{\perp}(\bm{\mathbf{k}},\bm{\mathbf{k}}^{\prime})\\\
=\left[\left\langle G_{\perp}(\bm{\mathbf{k}})\right\rangle-\left\langle
G_{\perp}^{*}(\bm{\mathbf{k}}^{\prime})\right\rangle\right]\\\
\times\int\bm{\mathbf{\Gamma}}_{\perp}(\bm{\mathbf{k}},\bm{\mathbf{\upkappa}},\bm{\mathbf{k}}^{\prime},\bm{\mathbf{\upkappa}}^{\prime})\cdot\bm{\mathbf{C}}_{\perp}(\bm{\mathbf{\upkappa}},\bm{\mathbf{\upkappa}}^{\prime})\frac{\mathrm{d}\bm{\mathbf{\upkappa}}}{8\pi^{3}}\frac{\mathrm{d}\bm{\mathbf{\upkappa}}^{\prime}}{8\pi^{3}}.$
(156)

Still considering that we have statistical homogeneity for the disordered
medium, we have
$\bm{\mathbf{\Gamma}}(\bm{\mathbf{r}}^{\prime},\bm{\mathbf{r}}^{\prime\prime},\bm{\mathbf{\uprho}}^{\prime},\bm{\mathbf{\uprho}}^{\prime\prime})=\bm{\mathbf{\Gamma}}(\bm{\mathbf{r}}^{\prime}+\Delta\bm{\mathbf{r}},\bm{\mathbf{r}}^{\prime\prime}+\Delta\bm{\mathbf{r}},\bm{\mathbf{\uprho}}^{\prime}+\Delta\bm{\mathbf{r}},\bm{\mathbf{\uprho}}^{\prime\prime}+\Delta\bm{\mathbf{r}})$
which leads to

$\bm{\mathbf{\Gamma}}_{\perp}(\bm{\mathbf{k}},\bm{\mathbf{\upkappa}},\bm{\mathbf{k}}^{\prime},\bm{\mathbf{\upkappa}}^{\prime})=8\pi^{3}\delta(\bm{\mathbf{k}}-\bm{\mathbf{k}}^{\prime}-\bm{\mathbf{\upkappa}}+\bm{\mathbf{\upkappa}}^{\prime})\\\
\times\bar{\bm{\mathbf{\Gamma}}}_{\perp}(\bm{\mathbf{k}},\bm{\mathbf{\upkappa}},\bm{\mathbf{k}}^{\prime},\bm{\mathbf{\upkappa}}^{\prime})$
(157)

in the Fourier space. By a change of variable, we also now define the
correlation

$\bm{\mathbf{L}}_{\perp}(\bm{\mathbf{q}},\bm{\mathbf{k}})\equiv\bm{\mathbf{C}}_{\perp}\left(\bm{\mathbf{k}}+\frac{\bm{\mathbf{q}}}{2},\bm{\mathbf{k}}-\frac{\bm{\mathbf{q}}}{2}\right)$
(158)

which leads to this new form of the Bethe-Salpeter equation in the Fourier
domain

$\left[\left(\bm{\mathbf{k}}-\frac{\bm{\mathbf{q}}}{2}\right)^{2}-\left(\bm{\mathbf{k}}+\frac{\bm{\mathbf{q}}}{2}\right)^{2}-\Sigma_{\perp}^{*}\left(\bm{\mathbf{k}}-\frac{\bm{\mathbf{q}}}{2}\right)+\Sigma_{\perp}\left(\bm{\mathbf{k}}+\frac{\bm{\mathbf{q}}}{2}\right)\right]\\\
\times\bm{\mathbf{L}}_{\perp}(\bm{\mathbf{q}},\bm{\mathbf{k}})=\left[\left\langle
G_{\perp}\left(\bm{\mathbf{k}}+\frac{\bm{\mathbf{q}}}{2}\right)\right\rangle-\left\langle
G_{\perp}^{*}\left(\bm{\mathbf{k}}-\frac{\bm{\mathbf{q}}}{2}\right)\right\rangle\right]\\\
\times\int\bar{\bm{\mathbf{\Gamma}}}_{\perp}\left(\bm{\mathbf{k}}+\frac{\bm{\mathbf{q}}}{2},\bm{\mathbf{k}}^{\prime}+\frac{\bm{\mathbf{q}}}{2},\bm{\mathbf{k}}-\frac{\bm{\mathbf{q}}}{2},\bm{\mathbf{k}}^{\prime}-\frac{\bm{\mathbf{q}}}{2}\right)\cdot\bm{\mathbf{L}}_{\perp}\left(\bm{\mathbf{k}}^{\prime},\bm{\mathbf{q}}\right)\frac{\mathrm{d}\bm{\mathbf{k}}^{\prime}}{(2\pi)^{3}}$
(159)

which is Eq. (26).

##  Appendix C Configurational average for statistically homogeneous systems

###  C.1 Particle correlation functions

We consider a random ensemble of $N$ particles circumscribed in a volume $V$
and centered at positions ${\bf R}=\left[{\bf R}_{1},{\bf R}_{2},\dots\bf
R_{N}\right]$. A specific configuration is described by a normalized
probability distribution $P^{(N)}$ such that

$\displaystyle P^{(N)}({\bf R}_{1},\cdots,{\bf R}_{N})\mathrm{d}{\bf
R}_{1}\cdots\mathrm{d}{\bf R}_{N}$ (160)

is the probability of finding a configuration in which particle $j$ is
centered between ${\bf R}_{j}$ and ${\bf R}_{j}+\mathrm{d}{\bf R}_{j}$.

Assuming that the particles are spherically symmetric (otherwise, the
distribution whould also include orientational variables, $\Omega_{j}$) and
identical, the distribution is symmetric in the labels $1,\dots,N$ and we can
define the $M$-particle density $\rho^{(M)}({\bf R}_{1},\cdots,{\bf R}_{M})$
as the probability of finding a configuration of $M$ particles whatever the
configuration of the remaining $N-M$ particles

$\displaystyle\rho^{(M)}({\bf R}_{1},\cdots,{\bf R}_{M})=\frac{N!}{(N-M)!}$
$\displaystyle\times$ $\displaystyle\int P^{(N)}({\bf R}_{1},\cdots,{\bf
R}_{N})\mathrm{d}{\bf R}_{M+1}\cdots\mathrm{d}{\bf R}_{N}$

The instantaneous particle number density $\rho({\bf r})$ for a given
configuration of particles, ${\bf R}=\left[{\bf R}_{1},\cdots,{\bf
R}_{N}\right]$, is defined as

$\displaystyle\rho({\bf r})$ $\displaystyle\equiv$
$\displaystyle\sum_{j=1}^{N}\delta({\bf r}-{\bf R}_{j}),$ (162)

and its configurational average $\langle\rho({\bf r})\rangle$ is given by

$\displaystyle\langle\rho({\bf r})\rangle$ $\displaystyle=$
$\displaystyle\left\langle\sum_{j}\delta({\bf r}-{\bf R}_{j})\right\rangle$
$\displaystyle=$ $\displaystyle N\int\cdots\int P^{(N)}({\bf r},{\bf
R}_{2},\cdots,{\bf R}_{N})\mathrm{d}{\bf R}_{2}\cdots\mathrm{d}{\bf R}_{N}.$

In the limit of infinite system size and assuming a statistically homogeneous
and isotropic medium (i.e. all properties are statistically invariant by
translation and rotation), both the number of particles and the volume of the
material tend to infinity (i.e. $\\{N,V\\}\to\infty$), and one can define an
average particle number density, $\rho\equiv\left\langle\rho({\bf
r})\right\rangle=N/V$, that is constant.

One can then define the $n$-particle probably density functions $\rho_{n}({\bf
r}_{1},{\bf r}_{2},...,{\bf r}_{n})$ as

$\displaystyle\rho_{n}({\bf r}_{1},\cdots,{\bf
r}_{n})\equiv\left\langle\sum_{\begin{subarray}{c}j_{1},\cdots j_{s}=1\\\
j_{1}\neq j_{2}\cdots\neq j_{s}\end{subarray}}^{N}\delta({\bf r}_{1}-{\bf
R}_{j_{1}})\cdots\delta({\bf r}_{s}-{\bf R}_{j_{s}})\right\rangle,$ (165)

as well as the $n$-particle correlation function

$\displaystyle g_{n}({\bf r}_{1},{\bf r}_{2},...,{\bf
r}_{n})\equiv\frac{\rho_{n}({\bf r}_{1},{\bf r}_{2},...,{\bf
r}_{n})}{\rho^{n}}.$ (166)

The important quantity in most systems is the pair correlation function
$g_{2}({\bf r}_{1},{\bf r}_{2})$, that describes the conditional probability
of finding a particle at ${\bf r}_{2}$ given at particle fixed at ${\bf
r}_{1}$. For isotropic media, $g_{2}$ only depends on the radial distance
$r_{12}=|{\bf r}_{1}-{\bf r}_{2}|$. The total correlation function $h_{2}({\bf
r})$ defined as

$\displaystyle h_{2}({\bf r})\equiv g_{2}({\bf r})-1,$ (167)

has the benefit of converging to zero at separation distances larger than a
correlation length $\ell_{\text{c}}$.

###  C.2 Fluctuations of the number of particles in a volume

The fluctuations of the number of particles $N$ in a given volume $v$ can be
defined as

$\displaystyle\delta_{N}\equiv\frac{1}{\rho v}\left(\left\langle
N^{2}\right\rangle-\left\langle
N\right\rangle^{2}\right)=\frac{1}{v}\int_{v}\frac{\left\langle\Delta\rho({\bf
r})\Delta\rho({\bf r}^{\prime})\right\rangle}{\rho}d{\bf r}d{\bf r}^{\prime},$
(168)

with

$\frac{\left\langle\Delta\rho({\bf r}_{1})\Delta\rho({\bf
r}_{2})\right\rangle}{\rho}=\frac{\left\langle\rho({\bf r}_{1})\rho({\bf
r}_{2})\right\rangle-\rho^{2}}{\rho}\\\
=\frac{1}{\rho}\Bigg{[}\left\langle\sum_{a=1}^{N}\delta({\bf r}_{1}-{\bf
R}_{a})\delta({\bf r}_{2}-{\bf R}_{a})\right\rangle\\\
+\left\langle\sum_{a}\sum_{b\neq a}\delta({\bf r}_{1}-{\bf R}_{a})\delta({\bf
r}_{2}-{\bf R}_{b})\right\rangle-\rho^{2}\Bigg{]}\\\ =\delta({\bf r}_{1}-{\bf
r}_{2})+\rho\left(\frac{\rho_{2}({\bf r}_{1}-{\bf
r}_{2})}{\rho^{2}}-1\right)\\\ \equiv\delta({\bf r}_{1}-{\bf r}_{2})+\rho
h_{2}({\bf r}_{1}-{\bf r}_{2}).$ (169)

If $v$ is a sphere of radius $R_{s}$, one gets Van Kranendonk and Sipe (1977);
Torquato and Stillinger (2003)

$\displaystyle\delta_{N}$ $\displaystyle=$ $\displaystyle 1$ (170)
$\displaystyle+$ $\displaystyle\rho\int
h_{2}(r_{12})\left[1-\frac{3}{4}\frac{r}{R_{s}}+\frac{1}{16}\left(\frac{r}{R_{s}}\right)^{3}\right]4\pi
r^{2}dr_{12}$ $\displaystyle\sim$ $\displaystyle 1+\rho\int h_{2}(r_{12})4\pi
r^{2}dr_{12},$

where, in the last step, we assumed that $R_{s}$ is larger than the
correlation length $\ell_{\text{c}}$ of $h_{2}(r)$. Expressions exist also for
two-dimensional systems and non-spherical excluded volumes Torquato and
Stillinger (2003).

The static structure factor $S({\bf k})$ is related to the Fourier transform
of $h_{2}({\bf r})$ via the expression

$\displaystyle S({\bf k})$ $\displaystyle\equiv$ $\displaystyle 1+\rho
h_{2}({\bf k}),$ (171)

with $h_{2}({\bf k})=\int h_{2}({\bf r})e^{-i{\bf k}\cdot{\bf r}}d{\bf r}$.

##  Appendix D Local density of states and quasinormal modes

The local density of states (LDOS) is defined from the projected LDOS, Eq.
(126), as

$\rho_{e}(\mathbf{r},\omega)=\frac{2\omega}{\pi
c^{2}}\text{Im}\left[\text{Tr}\mathbf{G}(\mathbf{r},\mathbf{r},\omega)\right].$
(172)

We will now express this quantity in terms of the eigenmodes of the system.

The eigenmodes of non-conservative (non-Hermitian) systems, also known as
quasinormal modes (QNMs), are described by complex frequencies
$\tilde{\omega}_{m}=\omega_{m}-i\gamma_{m}/2$ and normalized fields
$\tilde{\mathbf{E}}_{m}(\mathbf{r})$, the non-zero imaginary part steming from
leakage. QNMs have a long history Baum (1976); Ching  _et al._ (1998) and are
receiving considerable attention from the photonics community since a few
years Lalanne  _et al._ (2018). Following a recent QNM formalism Sauvan  _et
al._ (2013, 2014); Yan  _et al._ (2018), we can write the field $\mathbf{E}$
generated by a dipole emitter in terms of QNMs as

$\mathbf{E}(\mathbf{r},\omega)=\sum_{m}\alpha_{m}(\omega)\tilde{\mathbf{E}}_{m}(\mathbf{r}),$
(173)

with $\alpha_{m}$ the excitation coefficients, defined as

$\alpha_{m}(\omega)=-\frac{\omega}{2\epsilon_{0}\left(\omega-\tilde{\omega}_{m}\right)}\mathbf{p}\cdot\tilde{\mathbf{E}}_{m}(\mathbf{r}).$
(174)

Quite expectedly, the efficiency of excitation of a mode depends on the
amplitude of the QNM field at the dipole position and the spectral distance
with the resonance frequency. Having further that
$\mathbf{E}(\mathbf{r},\omega)=\mu_{0}\omega^{2}\mathbf{G}(\mathbf{r},\mathbf{r}^{\prime},\omega)\mathbf{p}$,
one arrives to a modal decomposition of the dyadic Green function

$\mathbf{G}(\mathbf{r},\mathbf{r}^{\prime},\omega)=-\frac{c^{2}}{2\omega}\sum_{m}\frac{\tilde{\mathbf{E}}_{m}(\mathbf{r})\otimes\tilde{\mathbf{E}}_{m}(\mathbf{r}^{\prime})}{\omega-\tilde{\omega}_{m}}.$
(175)

Note that Eq. (175) can also be obtained from the Mittag-Leffler theorem which
introduces the residues of the Green tensor at the QNM complex frequencies
Muljarov and Langbein (2016). Inserting Eq. (175) in Eq. (172) finally leads
to

$\rho_{e}(\mathbf{r},\omega)=-\frac{1}{\pi}\text{Im}\left[\sum_{m}\frac{\text{Tr}\left[\tilde{\mathbf{E}}_{m}(\mathbf{r})\otimes\tilde{\mathbf{E}}_{m}(\mathbf{r})\right]}{\omega-\tilde{\omega}_{m}}\right].$
(176)

The LDOS is now explicitly expressed as a sum over resonant modes. To further
convince ourselves, we can take the limit of vanishing leakage, in which case
both $\mathbf{E}_{m}$ and $\tilde{\omega}_{m}$ tend to become real. Using
$\lim_{\eta\rightarrow
0^{+}}\frac{1}{x+i\eta}=\text{PV}\left[\frac{1}{x}\right]-i\pi\delta(x)$, we
arrive to the well-known expression of the LDOS for conservative systems
Novotny and Hecht (2012); Carminati  _et al._ (2015)

$\rho_{e}(\mathbf{r},\omega)=\sum_{m}|\tilde{\mathbf{E}}_{m}(\mathbf{r})|^{2}\delta(\omega-\omega_{m}).$
(177)

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