††thanks: a equally contributed

# Cavitation instabilities in amorphous solids via secondary mechanical
perturbations

Umang A. Dattani1,2,a, Rishabh Sharma3,a, Smarajit Karmakar3,∗ and Pinaki
Chaudhuri1,2<EMAIL_ADDRESS><EMAIL_ADDRESS>1The Institute of
Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
2Homi Bhabha National Institute, Anushaktinagar, Mumbai 400094, India
3Tata Institute of Fundamental Research, 36/P, Gopanpally Village,
Serilingampally Mandal,Ranga Reddy District, Hyderabad, 500046, Telangana,
India

###### Abstract

Amorphous solids are known to fail catastrophically and in some situations,
nano-scaled cavities are believed to play a significant role in the failure.
In a recent work, using numerical simulations, we have shown the
correspondence between cavitation under uniform expansion of amorphous solids
and the yielding under shear. In this study, we probe the stability of
spatially-homogeneous states sampled from expansion trajectories to alternate
modes of driving, viz. macroscopic cyclic shear or local random deformation
via activity. We find that, under cyclic shear and activity, the cavitation
instabilities can occur in expanded states at much higher densities than under
pure uniform-expansion, and the shift in density is determined by the
magnitude of the secondary deformation. We also show that barriers to
cavitation on the energy landscape are much smaller for cyclic-shear and
activity than seen under expansion. Further, we also analyse the spatial
manifestation of cavitation and investigate whether large scale irreversible
plasticity can set in due to the combination of expansion and the secondary
deformation. Overall, our study reveals the interplay between expansion and
other deformation modes leading to cavitation instabilities and the existence
of abundant relaxation pathways for such processes.

††preprint: APS/123-QED

##  I Introduction

The phenomenon of cavitation corresponds to the sudden formation as well as
expansion of a void or bubble in a material that is experiencing a mechanical
tension franc2006fundamentals ; young1999cavitation . Theoretically,
cavitation is perceived to occur when the glass-gas coexistence regime of the
material’s phase diagram is accessed during mechanical loading. In liquids,
cavitation has been extensively studied rayleigh1917viii ; plesset1977bubble ;
maris2000negative ; caupin2006cavitation , especially since collapsing bubbles
often inflicts damages to the adjoining surfaces. On the other hand, in
solids, while cavitation has been observed quite long back gent1959internal ;
gent1970surface , a proper physical understanding of the processes at play
remains missing. In recent times, there is an increased interest in the
phenomenon given its significance in the context of failure of amorphous
solids via crack propagation bouchaud2008fracture ; murali2011atomic ;
singh2016cavitation . Recent experiments have clearly demonstrated that
emergence of isolated cavities and subsequent merger of these micro-cavities
eventually leads to complete failure via fracture CavitationSciadv.abf7293 .
Further, cavitation is often cited as the source of damages in soft disordered
solids like biological tissues raayai2019intimate ; barney2020cavitation .
Also, cavitation rheology is now an emerging protocol to characterize
stability of soft materials to localized perturbations zimberlin2007cavitation
. Therefore, to design better materials which can sustain against such
failure, we need to have a proper physical understanding of the microscopic
processes leading to cavitation in solids.

Given the wide range of applications in which amorphous solids are used,
starting from the nanoscale devices to large appliances, understanding the
failure processes of amorphous solids has been an area of active research over
the last few decades schuh2007mechanical ; Rodney2011 ; bonn2017yield ;
RevModPhys.90.045006 . Typically, hard amorphous solids (e.g. metallic
glasses, silicates, polymer glass etc.) are known to fail catastrophically,
i.e. an usual brittle response which leads to fracture. Till now, much of the
focus has been on shear response of dense glasses, where significant progress
has been made to characterize the detailed elasto-plastic response of the
material including development of mesoscale elasto-plastic models to
understand yielding under shear on a mesoscopic length scale
RevModPhys.90.045006 ; SrikanthPRX ; ContinuumModelling . Not much has been
explored in the context of cavitation in such materials under uniform
expansion or uniaxial deformation process. There are a handful of numerical
studies that recently focused on understanding the microscopic mechanisms of
how cavitation occurs during volumetric strain FalkCavitation ;
PinakiCavitation ; Pablo1 ; KallolCavitation , wherein the emergence of the
cavities has been demonstrated as a yielding process PreviousWork . In fact,
it has been shown PreviousWork  that cavitation instability under uniform
volumetric expansion in a athermal quasistatic limit is a saddle-node
bifurcation in the potential energy landscape where a stable minimum becomes
unstable by merging with the nearest saddle during the expansion process. Such
a process manifests itself via vanishing of an eigenvalue of an appropriate
dynamical matrix (Hessian matrix) of the energy function. This results
suggests an unified picture of failure of amorphous solids on the energy
landscape under various deformation protocols and indicates a clear
possibility of cross coupling between these deformation modes during a natural
deformation process in a material which certainly involves nearly all
deformation paths like shear, volumetric expansion, uniaxial tension etc.
Ramification of these cross coupling can be detrimental to the stability of
the materials under mechanical loading. In fact, recent studies have probed
the interplay of these modes in fracture propagation from a seeded void
LernerBouchbinder . Thus, the fundamental question of how the cavitation
instabilities spontaneously occur in a homogeneous solid via various
combinations of loading needs a systematic study. Further, whether the
location of the instability and thereby the phase diagram gets altered during
these diverse non-equilibrium processes needs to be explored.

In this work, using numerical simulations, we probe cavitation in states
generated during the uniform expansion of a dense amorphous solid, when
subjected to secondary modes of mechanical deformation. We use two different
secondary modes: (i) athermal quasistatic cyclic shear which is a macro-scale
perturbation, and (ii) random local perturbation via self-propelled active
particles, which is a perturbation at micro-scale. In the first scenario it
was done under athermal quasi-static conditions whereas in the second case, we
kept the solid a temperature well below the calorimetric glass transition
temperature ($T_{G}$) defined as a temperature where the relaxation time in
our simulation exceeds $10^{6}$ in the appropriate reduced units. We
demonstrate that irrespective of the form of secondary perturbation,
cavitation happens at densities higher than where it occurs for a pure
expansion protocol. Thus, this indicates that cavitation instabilities couple
to bulk shear and local forcing at these higher densities in conjunction with
the tension generated via expansion. We summarize our findings in the form of
a phase diagram (see Fig.2), wherein the magnitude of the secondary forcing
and the density are the control parameters. Further, we find that cavitation
by itself may not lead to large-scale plasticity and flow. Rather, the
cavitated states that emerge via the secondary forcing can be in an arrested
or aging glassy states, depending on whether they are in an athermal or
thermal conditions, respectively. Only if the magnitude of the forcing is
large enough, an irreversible plasticity occurs via avalanche-like spatial
structures, as observed in the case of athermal cyclic shear. Overall, our
study reveals that the addition of the secondary mechanical perturbations
provide pathways for accessing lurking cavitation instabilities when the
amorphous solid is under tension due to an ongoing expansion process.

The rest of the paper is organized as follows. We first discuss the
deformation protocol we used to study the coupling between different
deformation processes with complete details of the simulation methodology is
described in the Methods section. We then discuss the results and conclude the
paper with implication of these results on understanding microscopic failure
mechanisms of amorphous solids under deformation with the aim to develop
fundamental knowledge for preventing catastrophic failure in these materials
and designing better materials for future applications.

##  II Models and Methods

###  II.1 Model and Simulation Details

For our study, we use the well-characterized two-dimensional model binary
Lennard-Jones mixture (KABLJ) which has $65:35$ composition of the two
species, labelled $A$ and $B$, with interaction parameters $\sigma_{AA}=1.0$,
$\sigma_{BB}=0.88$, $\sigma_{AB}=0.8$, $\epsilon_{AA}=1.0$,
$\epsilon_{BB}=0.5$, $\epsilon_{AB}=1.5$ KobAndersen2D . We smoothen the
Lennard-Jones potential up to first two derivatives. The interactions between
particles $i$ and $j$ take the following form:

$V_{mn}(r_{ij})=4\epsilon_{mn}\left[\left(\frac{\sigma_{mn}}{r_{ij}}\right)^{12}-\left(\frac{\sigma_{mn}}{r_{ij}}\right)^{6}\right]+u(r_{ij})$
(1)

where,

$u(r_{ij})=C_{0}+C_{2}\left(\frac{r_{ij}}{\sigma_{mn}}\right)^{2}+C_{4}\left(\frac{r_{ij}}{\sigma_{mn}}\right)^{4}$
(2)

Here, $m$ and $n$ would correspond to either of the labels $A$ or $B$ which
signify the type of particles. The constants $C_{0}$, $C_{2}$ and $C_{4}$ are
determined by requiring the potential and its first two derivates to be zero
at the cutoff $r_{c}=2.5\sigma_{mn}$. The simulations reported in the paper
are performed on a system size of $N=5000$ with periodic boundary conditions.

Preparation of Initial states:  Cyclic shear: To prepare initial states for
our study, we first equilibriate the system at $T=1.0$ (in LJ units), which is
in the high temperature liquid regime, followed by cooling at a constant rate
of $\dot{T}\approx 10^{-4}$ per MD timestep to $T=0.01$, followed by a quech
to $T=0$. For the active dynamics, the initial states are prepared by cooling
the high temperature liquid at $T=1.0$ to $T=0.01$ at a cooling rate
$\dot{T}=2.5\times 10^{-7}$ per MD timestep. To check the dependence of the
phenomena on preparation-protocol, the results have been cross-checked for
slower cooling rates that generate more stable (lower-energy) amorphous solids
too.

###  II.2 Protocols for mechanical deformations

Here, we outline the various protocols of mechanical deformation that we have
used. Expansion is the primary deformation mode, starting from the high
density amorphous solid. From this expansion trajectory, be it under athermal
conditions or at finite temperatures, initial states are generated at
different densities upon which the secondary deformation is imposed. The
details of the primary and secondary deformations are outlined below.

Athermal quasistatic expansion (AQE):  The solid at $T=0.01$ is quenched to
$T=0$ by conjugate gradient energy minimization. The solid is then subjected
to athemal quasistatic expansion (AQE) Pablo1 ; Pablo2 ; PreviousWork  where,
in each step, a constant volume strain is applied on the system by rescaling
the length of the box by a factor $(1+\epsilon)$ along with affine
transformation of particle coordinates, followed by minimization of the energy
of this strained configuration using the conjugate gradient algorithm. We used
$\epsilon=5\times 10^{-5}$. The configurations from the AQE trajectory, at
different densities, are used as initial states for the athermal quasistatic
cyclic shear procedure detailed below. These simulations are done using the
open source software LAMMPS LAMMPS .

Expansion at finite-temperature: The solid cooled down to $T=0.01$ is
subjected to a uniform volumetric strain by scaling the length of the box as
well as the particle coordinates by a factor of 1 + $\epsilon$ in each time-
step ($dt=0.005$), using $\epsilon=10^{-8}$. Thermostating is achieved by
using Nosé-Hoover chains. The configurations sampled across this finite-
temperature expansion trajectory are used as initial states for imposing the
drive at the microscopic scale via activity as detailed below.

Athermal Quasistatic cyclic shear (AQCS):  In recent times, the response of
amorphous solids to such a shear protocol has been studied in details AQCS ;
ParmarSastryNature . In this protocol, in each step, we apply a shear strain
of magnitude $\delta\gamma\approx 10^{-4}$ by deforming the simulation box and
remapping the particle coordinates, using LAMMPS. The system is then allowed
to relax via energy minimization using conjugate gradient(CG) algorithm. One
cycle of AQCS consists of the following steps: (a) Driving of the system to a
shear strain $\gamma=\gamma_{max}$, using quasistatic steps of magnitude
$\delta\gamma$; (b) Reversal of direction of applied shear strain upto a
strain of $\gamma=-\gamma_{max}$ (c); Another reversal in direction of the
applied strain until the strain reaches $\gamma=0$. Several such cycles are
conducted and the response of the system is studied.

Active dynamics:  To impose driving via activity, we implement run and tumble
dynamics. To a small fraction (2%) of the particles, chosen at random, a
random active force $\vec{f_{0}}=f_{0}(k_{x}\hat{x}+k_{y}\hat{y})$ is imposed.
By restricting the value of k’s to $\pm 1$, the forces added act only along
any of the four diagonal directions. This 4-state clock model is the 2
dimensional analog to the 8-state model used in mandal2016 . The directions of
active forces are randomised after a persistence time $\tau_{p}=1$. In
addition, the conditions $\sum k_{x}=0$ and $\sum k_{y}=0$ are imposed which
in-turn ensure that the net momentum added to the system due to the active
forces was zero at all times.

###  II.3 Measurements

Here, we outline how different observables have been measured, in our
analysis.

Coarse-graining density field: For our analysis, we construct a coarse-grained
density field by using the method described in Ref.TestardBerthierKob  via a
coarse-graining radius of $r=1.5\sigma$. In this coarse-grained density field,
a region of density $\rho<0.5$ is considered as a cavity for marking the
points in the phase diagram.

Overlap function: To check for the glassiness of the spatially inhomogeneous
states that emerge after cavitation has happened, we compute the overlap
correlation function with respect to a state after the cavity is formed. The
correlation function is defined as

$q(t)=\frac{1}{N}\sum_{i=1}^{N}\theta(a-|r_{i}(t)-r_{i}(t_{0})|)$ (3)

Here $\theta(x)$ is the unit step function, and we choose a = 0.3 to allow for
small thermal vibrations about mean. Averaging is done over an ensemble of
trajectories orginating from independent initial states. We show data at
$\rho=1.108$ where we sample initial states from the expansion trajectory and
impose active dynamics, The overlaps were then measured with respect to the
configurations at time $t_{0}$, corresponding to $5\times 10^{6}$ MD steps
after the imposition of active dynamics, by when the cavity has emerged for
all the trajectories within the ensemble.

Mobility maps: To get spatial information about the extent of displacemets at
the local scale, due to the imposed secondary deformation, we construct
mobility maps. For each particle within the system, we compute the squared
distance travelled at time $t$ (or cycle number) relative to the initial state
($t=0$), i.e the time at which secondary deformation is imposed on a state
sampled from the expansion trajectory,

$\Delta^{2}_{i}(t)=\left[\bm{r}_{i}(t)-\bm{r}_{i}(0)\right]^{2}$ (4)

We then colour each particle in the initial configuration ($t=0$) as per the
magnitude of $\Delta^{2}_{i}(t)$.

##  III Perturbation protocol

In Fig.1, we illustrate a schematic of the protocol that we follow, in our
study. When an amorphous solid having attractive interactions is expanded,
either under athermal or thermal conditions, it’s pressure decreases and
becomes negative, and then cavitates via a pressure jump; this is shown via
the pressure ($P$) vs density ($\rho$) line (shown in blue) in the schematic
diagram. The secondary deformation protocol is then imposed on states
generated along that $P$ vs $\rho$ trajectory. In this study, we use two
different types of secondary deformations. In one case, we apply a macroscopic
perturbation in the form of a quasistatic cyclic shear AQCS  having a certain
strain amplitude ($\gamma_{max}$) which becomes a control parameter of our
study. In the second case, we apply random active forcing on a subset of the
particles, with net applied force being zero; this is labeled as active
dynamics; the magnitude of the active forcing ($f_{0}$) is the control
parameter in this case. Earlier studies have demonstrated how the random
active forcing introduces local shear within the system Manning ;
mandal2020extreme . The details for both the methods is discussed in the
Methods section. Note that, recent studies have explored the response of
amorphous solids to athermal quasistic cyclic shear ParmarSastryNature ;
SrikanthPRX  as well as to random active forcing Manning ; mandal2020extreme ,
at high density, and demonstrated that the material yields, i.e. undergoes
largescale irreversible plasticity, when the forcing is large enough. Here, we
are using these protocols as secondary perturbations in conjunction with the
uniform expansion, over a large density range where the pressure experienced
by the material varies from positive to negative, to study the mechanical
response of the amorphous solid, with specific focus on the phenomenon of
cavitation.

Figure 1: A schematic diagram of the protocol followed in our numerical
experiment: states generated during the expansion from a dense spatially
homogeneous amorphous solid are subjected to secondary deformations, either in
the form of a cyclic shear or random active forcing.

##  IV Results

Figure 2: (Left) Phase diagram corresponding to secondary (a) athermal
quasistatic cyclic shear (b) active dynamics at $T=0.01$, imposed on states
obtained via expansion imposed on spatially homogeneous amorphous states. The
pink sector in the phase diagrams corresponds to spatially homogeneous states
and the green sector corresponds to cavitated states. The black vertical line
in (a), (b) demarcates the density at which pressure changes sign. Also shown
in (a), using diamonds and dashes, is the estimated locus for the onset of
irreversible plasticity under applied cyclic shear. (Middle) The blue line
shows pressure vs density for pure expansion using (c) athermal quasistatic
protocol, and (d) at $T=0.01$. The dotted lines originating from this line and
joining circles, squares, diamonds and triangles in both (c) and (d) shows
examples of pressure jumps due to cavitation under secondary cyclic shear and
activity, respectively, starting from spatially homogenous configurations. The
insets in both (c) and (d) show pressure jump, at cavitation, under cyclic
shear and active dynamics respectively, for example trajectories, for
densities of 1.1133 and 1.108, respectively. (Right) For a trajectories marked
in (c) and (d), evolution of energy under secondary deformation: (e) for the
first cycle under cyclic shear using $\gamma_{max}=0.06$, with inset showing
the evolution with cycles, and (f) with time under active dynamics at $T=0.01$
using $f_{0}=3.5$.

Phase diagram:  Our main result, as discussed above, is that cavitation can be
induced earlier, i.e. at higher densities, when the amorphous state is
subjected to a secondary drive, be the cyclic shear or the random active
forcing. This is summarised via the phase diagrams shown in Fig.2(a)-(b),
where we demonstrate the conditions under which cavitation occurs in the space
of density ($\rho$) and the relevant forcing variable, viz. amplitude of
oscillation ($\gamma_{max}$) or magnitude of active forcing ${f_{0}}$. Note
that in the phase diagram, the pink sector corresponds to spatially
homogeneous states and the green sector corresponding to the cavitated states.
It is evident that cavitation occurs at earlier and earlier densities with
increasing magnitude of the secondary forcing, for both the cases studied.
However, it is important to underline that for cavitation to occur, the
ambient pressure has to be negative, i.e. the system has to be in a state of
internal tension, which is generated via the expansion.

Energetics:  Similar to the case of failure via pure expansion, the formation
of a cavity via the secondary forcing is also accompanied by a sudden release
of the internal pressure; see insets of Fig.2(c)-(d) for example trajectories
in the case of cyclic shear and active dynamics, respectively. In the main
panels of Fig.2(c)-(d), we illustrate at a few densities, via dashed lines,
the difference between the initial pressure, prior to the introduction of the
second driving, and the eventual pressure attained after the cavitation. It is
evident that due to the secondary drive, the cavitation process is occurring
at a higher pressure (or lower energy; see Fig.S1 in supplementary
information) compared to the pressures (or energies) at which it would occur
under pure expansion. In Fig.2(e), for an example trajectory, we show how the
energy evolves during the first cycle after the imposition of cyclic shear and
the inset shows how the stroboscopic energy evolves over several cycles; see
supplementary information for the data of the full observation window of 200
cycles. Note that, even at maximum strain during the cycle, where maximal
plasticity is expected to occur, the system’s energy is much less than the
threshold needed for cavitation to occur under pure expansion (marked with
black line in Fig.2(e)). Similarly, in the case of the actively driven system,
we observe in Fig.2(f) that even though energy injection is happening at
microscopic scale via the active forcing, the macroscopic potential energy of
the system is decreasing, i.e. an annealing process is going on after
cavitation and this energy scale is much lower than the situation where
cavities appear under pure expansion. These results suggest that secondary
deformation protocols can provide other lower energy (free energy) barrier
pathways for relaxation eventually leading to cavity formation in these
materials at density which is not accessible purely via one protocol (i.e.
expansion, in this case). In Ref.krishnan2023annealing , it has been shown
that cyclic shear along multiple orthogonal directions can lead to better
annealing than cyclic shear along a single direction. Here, we are able to
show that if various different types of deformation, i.e. volumetric expansion
and shear (both at bulk and local level), can lead to better annealing of
amorphous solids. This also leads to the possibility of achieving stable
(ultra stable) glasses using combination of cyclic volumetric and shear
deformation in these materials, which is a work in progress.

Figure 3: Secondary cyclic shear:  (a) At a fixed density ($\rho=1.1133$),
evolution of rescaled stroboscopic energies, ${|u-u_{0}|/|u_{0}|}$, with
number of cycles, for different $\gamma_{max}$ values as labelled. (b)
Corresponding variation of ensemble averaged asymptotic rescaled energies
(obtained via averaging across the last 20 cycles) with shear amplitude. (c)
Variation of timescales of relaxation to a limit cycle with
$\gamma_{irr}-\gamma_{max}$, for various densities, where $\gamma_{irr}$ is
the estimated yielding threshold for large scale plasticity for each density.
Secondary active dynamics:  (d) At density of $\rho=1.108$, variation of
overlap function, $q(t)$, with time after the formation of cavity for
different values of forcing $f_{0}$ as labelled.

Yielding:  A pertinent question to ask is whether the cavitation induced by
the secondary mechanical perturbation results in large scale plasticity and
subsequent fluidization. We examine this for both the protocols that we have
studied. As already evidenced in Fig.2(e),(f) for example trajectories, the
energy decreases after cavitation, both for cyclic shear and active forcing,
implying that the system is annealing after the cavity has appeared. We need
to check across all accessed state points whether the system reaches an
arrested or aging glassy state, depending upon athermal or thermal conditions,
or whether the system has fluid-like behaviour.

In the case of cyclic shear, we probe this by monitoring, across many cycles
of the applied shear, the stroboscopically measured normalized energy defined
as ${|u-u_{0}|/|u_{0}|}$, where $u_{0}$ is the energy at the start of first
cycle. A typical scenario, at an example density, is shown in Fig.3(a). When
the cyclic shear is imposed, the (normalized) energy of the system starts to
decrease with cycles. For the right combinations of densities and amplitudes
as summarised in the phase diagram, this decrease of energy is accompanied by
formation of a cavity. As the solid undergoes cyclic shear, the energy of
solid decreases in the first few cycles. The decrease could happen due to lack
of initial annealing, as previously reported in studies involving high density
glassy systemsSrikanthAnnealing . Or, it can be due to the formation of a
cavity. We observe that for smaller $\gamma_{max}$, the system eventually
reaches a limit cycle LimitCycleSrikanth ; MaloneyYielding ; DheerajKumar ;
krishnan2023annealing  where either the energy decreases to a constant value
or oscillates between two or more different values. Examples of limit cycles
with a period greater than one are shown in the Supplementary Information. For
larger amplitudes of the oscillatory shear, the system becomes diffusive over
observation timescales. Beyond a certain amplitude, the average energy of the
relaxed solid averaged over last few cycles starts to increase; see Fig.3(b).
The cusp marks the yielding point under cyclic shearSrikanthPRX ;
SrikanthAnnealing . The yield threshold of the lower density solid is higher
than the yield threshold of the homogeneous solid at the beginning of
expansion; the corresponding locus of yield thresholds is marked in the phase
diagram of Fig.2(a) (using diamonds connected with dashes). Further, we track
the time taken for a system to reach a limit cycle $\tau_{rel}$, as a function
of amplitudes is shown in Fig.3(c) at different densities. The power law
observed in the works of Kumar et alDheerajKumar ,
$\tau_{rel}\sim(\gamma_{irr}-\gamma)^{-2.8}$ here too serves as a good guide
to the eye; $\gamma_{irr}$ is the estimated yield threshold under cyclic
shear.

For the case of imposed active dynamics, we do a similar check for large scale
plasticity by computing the overlap function, $q(t)$, relative to a post-
cavitated state. If the system is fluidized, then the overlap function should
completely decorrelate within observation time. However, as observed in
Fig.3(d), this is not the case, implying that the system, even in the presence
of activity, is still in the glassy state, for the range of active forcing
that we have explored. It is possible that higher activity strengths will lead
to fluidization, as has been reported earlier for dense glassy
systemsmandal2016 .

Spatial manifestations:  Finally, we examine the spatial structures emerging
via cavitation induced by the secondary deformation protocol. We construct
maps of the coarse-grained density field TestardBerthierKob . In Fig.4(A), we
show these density maps, after cavitation has occurred, corresponding to the
cyclic shear (top panel) and the active forcing (bottom panel), for some of
the density points marked in Fig.2(c)-(d). We find that, although the initial
states in each case are sampled from the same expansion trajectory, these
cavities occur at different locations in the solid, for both cyclic shear and
active dynamics, if we compare across densities. This suggests abundance of
potential cavitation sites present throughout the solid and also multiple
possible pathways on the energy landscape for the cavitation to occur. In
supplementary material (see Fig.S2 and Fig.S3), we provide illustrative panels
spanning across density and forcing, which show how the density maps reveal
occurrence of cavities, at different locations, depending upon the state
point, once the necessary forcing threshold has been crossed at each density,
providing a pictorial snapshot of the phase diagrams shown in Fig.2.

Further, to characterize the extent of deformation undergone between the
initial state, prior to the application of the secondary drive, and the final
state achieved during the observation time window after the application of the
drive, we construct spatial maps of mobility (See Methods). In Fig.4(B), we
show these maps for the cases in Fig.4(A) where cavitation is observed at
finite secondary drive, viz. subplots (b)-(d) for cyclic shear and (f)-(h) for
active dynamics. We observe large displacements around the location where
cavity is formed for both the secondary deformation protocols. In cyclic
shear, for large $\gamma_{max}$, we see extended spatial structures formed by
the regions undergoing large displacements, as shown in Fig.4(B) (i)-(k),
whereas for smaller $\gamma_{max}$, these structure are more localised around
the cavity; see Supplementary Information. Thus, as $\gamma_{irr}$ is
approached, the spatial scale of plasticity gets more expanded and one expects
these to take avalanche-like structures around yield. In the case of active
dynamics, where we continue to remain glassy and not fluidized within the
scale of forcings explored, the particles with large displacements seem to be
concentrated in localised clusters around the cavity; see Fig.4(B) (l)-(n)
(and Supplementary Information). Perhaps, here too, we can expect extended
structures at larger forcings.

Figure 4: (A) Maps of coarse-grained density field. (a) and (e): cavitated
states due to pure expansion, occurring at $\rho=1.097$ and $1.1028$,
respectively, under athermal quasistatic conditions and at finite temperature
($T=0.01$). Along corresponding trajectories, density maps of cavitated steady
state under secondary drive: (b)-(d) cyclic shear using $\gamma_{max}=0.06$
and shown for $\rho=1.119,1.124,1.130$, (f)-(h) active dynamics using
$f_{0}=3.5$ and shown for $\rho=1.108,1.116,1.132$. (B) Corresponding spatial
maps of squared displacements relative to initial state (i.e. prior to
application of secondary deformation), for (i)-(k) cyclic shear and (l)-(n)
active dynamics.

##  V Discussion

In summary, using numerical simulations, we have provided proof-of-concept
demonstration that in comparison to a case of pure axial expansion, cavitation
can occur at smaller densities when the amorphous solid, expanded from a high
density state, is subjected to secondary mechanical deformations. We have
shown this phenomenon for two examples of secondary deformation, viz. cyclic
shear and active dynamics. Our preliminary analysis (not shown) for uniform
shear also reveals similar behaviour. Thus, it shows how combination and
interplay of different deformation modes can lead to early onset of
cavitation, which evidently would lead to earlier fracture if the expansion
process continues simultaneously. However, note that, for cavitation to
happen, the overall pressure of the system has to be negative, even when
secondary forcing is at play. Thus, the presence of negative tensile stress
that emerged from the strain caused via the expansion is necessary for the
secondary shearing to couple with in order for the relaxation to happen in the
form of cavity formation. Overall, our work has significant implications for
real-life scenarios where typical applications comprise of complex loading
mechanisms which is a mixture of multiple deformation modes, and thus
materials have to be appropriately designed to prevent failure via such cross-
coupling. Further, our study paves the way for more explorations of probing
failure in amorphous solids via complex deformations, across the entire
spectrum of hardness that these materials span.

One of the interesting findings of the work is that the cavitated state that
one obtains via the secondary deformation could be an arrested or glassy
state, depending upon whether the system is athermal or thermal. Or, in other
words, the secondary deformation is providing a pathway to access these local
minima in the energy landscape, at densities where usual axial expansion would
not find them. Further, we demonstrate that multiple local minima, manifested
in the form of different structurally inhomogeneous states, emerge along the
same trajectory, depending upon the density and the strength of the secondary
forcing. This implies that different soft spots get selected for the
cavitation to occur, via the coupling between the different deformation modes
and their relative strengths.

In the context of active dynamics, the emergence of cavities reminds of
motility induced phase separation (MIPS) reported in the active matter systems
cates2015motility , albeit the minority phase being the voids rather than the
more well-studied active droplet. But this would be expected in the high
density limit villarroel2021critical ; PhysRevE.106.L012601 . In most studies
related to MIPS, the active particles interact via repulsive interaction and
the effective attraction emerges out of the active dynamics. In our case, all
the particles have Lennard-Jones interactions, including the small fraction of
particles that are active. The picture that we have in mind is that these
small number of active particles act as random local sources of shear which
collectively organise to help in relaxing a mode that eventually leads to
cavitation. Thus, it is also possible that the MIPS formation in the usual
active systems also has a shearing mechanism at its origin.

In recent times, significant progress has been made towards the development of
simple mesoscale models that use coarse-grained variables like stress and
strain to provide insight into the physical processes at play during the
response exhibited by amorphous solids to mechanical loadings. Not only have
these models been successful in reproducing the phenomenology
nicolas2018deformation  but also have been able to bridge towards microscopic
models liu2021elastoplastic . However, most of the modelling has so far
focussed on the response to applied shear. Our work on the coupling of various
deformation modes and its consequence to failure via cavitation should now
motivate further development of these models towards more complex loading
scenarios.

##  VI Acknowledgements

We thank the HPC facility at IMSc-Chennai and at TIFR-H for providing
computational resources. S.K. would like to acknowledge the support from
Swarna Jayanti Fellowship Grants No. DST/SJF/PSA-01/2018-19 and No.
SB/SFJ/2019-20/05 and Core Research Grant from SERB via grant CRG/2019/005373.
PC acknowledges funding from SERB via grant MTR/2022/001034. We also thank
Peter Sollich for useful discussions.

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# On the mean speed of convergence of empirical and occupation measures in
Wasserstein distance

Emmanuel Boissard  and  Thibaut Le Gouic  Université Paul Sabatier

(Date: August 27, 2024)

###### Abstract.

In this work, we provide non-asymptotic bounds for the average speed of
convergence of the empirical measure in the law of large numbers, in
Wasserstein distance. We also consider occupation measures of ergodic Markov
chains. One motivation is the approximation of a probability measure by
finitely supported measures (the quantization problem). It is found that rates
for empirical or occupation measures match or are close to previously known
optimal quantization rates in several cases. This is notably highlighted in
the example of infinite-dimensional Gaussian measures.

##  1\. Introduction

This paper is concerned with the rate of convergence in Wasserstein distance
for the so-called _empirical law of large numbers_ : let $(E,d,\mu)$ denote a
measured Polish space, and let

(1) $L_{n}=\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}}$

denote the empirical measure associated with the i.i.d. sample $(X_{i})_{1\leq
i\leq n}$ of law $\mu$, then with probability 1, $L_{n}\rightharpoonup\mu$ as
$n\rightarrow+\infty$ (convergence is understood in the sense of the weak
topology of measures). This theorem is also known as Glivenko-Cantelli theorem
and is due in this form to Varadarajan [26].

For $1\leq p<+\infty$, the _$p$ -Wasserstein distance_ is defined on the set
$\mathcal{P}_{p}(E)^{2}$ of couples of measures with a finite $p$-th moment by

$W_{p}^{p}(\mu,\nu)=\inf_{\pi\in\mathcal{P}(\mu,\nu)}\int
d^{p}(x,y)\pi(dx,dy)$

where the infimum is taken on the set $\mathcal{P}(\mu,\nu)$ of probability
measures with first, resp. second, marginal $\mu$, resp. $\nu$. This defines a
metric on $\mathcal{P}_{p}$, and convergence in this metric is equivalent to
weak convergence plus convergence of the moment of order $p$. These metrics,
and more generally the Monge transportation problem from which they originate,
have played a prominent role in several areas of probability, statistics and
the analysis of P.D.E.s : for a rich account, see C. Villani’s St-Flour course
[27].

Our purpose is to give bounds on the mean speed of convergence in $W_{p}$
distance for the Glivenko-Cantelli theorem, i.e. bounds for the convergence
$\mathbb{E}(W_{p}(L_{n},\mu))\rightarrow 0$. Such results are desirable
notably in view of numerical and statistical applications : indeed, the
approximation of a given probability measure by a measure with finite support
in Wasserstein distance is a topic that appears in various guises in the
literature, see for example [15]. The first motivation for this work was to
extend the results obtained by F. Bolley, A. Guillin and C. Villani [5] in the
case of variables with support in $\mathbb{R}^{d}$. As in this paper, we aim
to produce bounds that are non-asymptotic and effective (that is with explicit
constants), in order to achieve practical relevance.

We also extend the investigation to the convergence of occupation measure for
suitably ergodic Markov chains : again, we have practical applications in
mind, as this allows to use Metropolis-Hastings-type algorithms to approximate
an unknown measure (see 1.3 for a discussion of this).

There are many works in statistics devoted to convergence rates in some metric
associated with the weak convergence of measures, see e.g. the book of A. Van
der Vaart and J. Wellner [25]. Of particular interest for us is R.M. Dudley’s
article [11], see Remark Remark.

Other works have been devoted to convergence of empirical measures in
Wasserstein distance, we quote some of them. Horowitz and Karandikar [17] gave
a bound for the rate of convergence of $\mathbb{E}[W_{2}^{2}(L_{n},\mu)]$ to
$0$ for general measures supported in $\mathbb{R}^{d}$ under a moment
condition. M. Ajtai, J. Komlos and G. Tusnady [1] and M.Talagrand [24] studied
the related problem of the average cost of matching two i.i.d. samples from
the uniform law on the unit cube in dimension $d\geq 2$. This line of research
was pushed further, among others, by V. Dobrić and J.E. Yukich [10] or F.
Barthe and C. Bordenave [2] (the reader may refer to this last paper for an
up-to-date account of the Euclidean matching problem). These papers give a
sharp result for measures in $\mathbb{R}^{d}$, with an improvement both over
[17] and [5]. In the case $\mu\in\mathcal{P}(\mathbb{R})$, del Barrio, Giné
and Matran [7] obtain a central limit theorem for $W_{1}(L_{n},\mu)$ under the
condition that $\int_{-\infty}^{+\infty}\sqrt{F(t)(1-F(t))}dt<+\infty$ where
$F$ is the cumulative distribution function (c.d.f.) of $\mu$. In the
companion paper [4], we investigate the case of the $W_{1}$ distance by using
the dual expression of the $W_{1}$ transportation cost by Kantorovich and
Rubinstein, see therein for more references.

Before moving on to our results, we make a remark on the scope of this work.
Generally speaking, the problem of convergence of $W_{p}(L_{n},\mu)$ to $0$
can be divided in two separate questions :

  * •

the first one is to estimate the _mean rate of convergence_ , that is the
convergence rate of $\mathbb{E}[W_{p}(L_{n},\mu)]$,

  * •

while the second one is to study the concentration properties of
$W_{p}(L_{n},\mu)$ around its mean, that is to find bounds on the quantities

$\mathbb{P}(W_{p}(L_{n},\mu)-\mathbb{E}[W_{p}(L_{n},\mu)]\geq t).$

Our main concern here is the first point. The second one can be dealt with by
techniques of measure concentration. We will elaborate on this in the case of
Gaussian measures (see Appendix A), but not in general. However, this is a
well-trodden topic, and some results are gathered in [4].

######  Acknowledgements.

We thank Patrick Cattiaux for his advice and careful reading of preliminary
versions, and Charles Bordenave for introducing us to his work [2] and
connected works.

###  1.1. Main result and first consequences

######  Definition 1.1.

For $X\subset E$, the covering number of order $\delta$ for $X$, denoted by
$N(X,\delta)$, is defined as the minimal $n\in\mathbb{N}$ such that there
exist $x_{1},\ldots,x_{n}$ in $X$ with

$X\subset\bigcup_{j=1}^{n}B(x_{i},\delta).$

Our main statement is summed up in the following proposition.

######  Proposition 1.1.

Choose $t>0$. Let $\mu\in\mathcal{P}(E)$ with support included in $X\subset E$
with finite diameter $d$ such that $N(X,t)<+\infty$. We have the bound :

$\mathbb{E}(W_{p}(L_{n},\mu))\leq
c\left(t+n^{-1/2p}\int_{t}^{d/4}N(X,\delta)^{1/2p}d\delta\right).$

with $c\leq 64/3$.

######  Remark.

Proposition 1.1 is related in spirit and proof to the results of R.M. Dudley
[11] in the case of the bounded Lipschitz metric

$d_{BL}(\mu,\nu)=\inf_{f1-\text{Lip},|f|\leq 1}\int fd(\mu-\nu).$

The analogy is not at all fortuitous : indeed, the bounded Lipschitz metric is
linked to the $1$-Wasserstein distance via the well-known Kantorovich-
Rubinstein dual definition of $W_{1}$ :

$W_{1}(\mu,\nu)=\inf_{f1-\text{Lip}}\int fd(\mu-\nu).$

The analogy stops at $p=1$ since there is no representation of $W_{p}$ as an
empirical process for $p>1$ (there is, however, a general dual expression of
the transport cost). In spite of this, the technique of proof in [11] proves
useful in our case, and the technique of using a sequence of coarser and
coarser partitions is at the heart of many later results, notably in the
literature concerned with the problem of matching two independent samples in
Euclidean space, see e.g. [24] or the recent paper [2].

We now give a first example of application, under an assumption that the
underlying metric space is of finite-dimensional type in some sense. More
precisely, we assume that there exist $k_{E}>0$, $\alpha>0$ such that

(2) $N(E,\delta)\leq k_{E}(\text{Diam }E/\delta)^{\alpha}.$

Here, the parameter $\alpha$ plays the role of a dimension.

######  Corollary 1.2.

Assume that $E$ satisfies (2), and that $\alpha>2p$. With notations as
earlier, the following holds :

$\mathbb{E}[W_{p}(L_{n},\mu)]\leq c\frac{\alpha}{\alpha-2p}\,\text{Diam
}E\,k_{E}^{1/\alpha}n^{-1/\alpha}$

with $c\leq 64/3$.

######  Remark.

In the case of measures supported in $\mathbb{R}^{d}$, this result is neither
new nor fully optimal. For a sharp statement in this case, the reader may
refer to [2] and references therein. However, we recover at least the exponent
of $n^{-1/d}$ which is sharp for $d\geq 3$, see [2] for a discussion. And on
the other hand, Corollary 1.2 extends to more general metric spaces of finite-
dimensional type, for example manifolds.

As opposed to Corollary 1.2, our next result is set in an infinite-dimensional
framework.

###  1.2. An application to Gaussian r.v.s in Banach spaces

We apply the results above to the case where $E$ is a separable Banach space
with norm $\|.\|$, and $\mu$ is a centered Gaussian random variable with
values in $E$, meaning that the image of $\mu$ by every continuous linear
functional $f\in E^{*}$ is a centered Gaussian variable in $\mathbb{R}$. The
couple $(E,\mu)$ is called a (separable) Gaussian Banach space.

Let $X$ be a $E$-valued r.v. with law $\mu$, and define the weak variance of
$\mu$ as

$\sigma=\sup_{f\in E^{*},\,|f|\leq 1}\left(\mathbb{E}f^{2}(X)\right)^{1/2}.$

The small ball function of a Gaussian Banach space $(E,\mu)$ is the function

$\psi(t)=-\log\mu(B(0,t)).$

We can associate to the couple $(E,\mu)$ their Cameron-Martin Hilbert space
$H\subset E$, see e.g. [19] for a reference. It is known that the small ball
function has deep links with the covering numbers of the unit ball of $H$, see
e.g. Kuelbs-Li [18] and Li-Linde [21], as well as with the approximation of
$\mu$ by measures with finite support in Wasserstein distance (the
quantization or optimal quantization problem), see Fehringer’s Ph.D. thesis
[12], Dereich-Fehringer-Matoussi-Scheutzow [8], Graf-Luschgy-Pagès [16].

We make the following assumptions on the small ball function :

  1. (1)

there exists $\kappa>1$ such that $\psi(t)\leq\kappa\psi(2t)$ for $0<t\leq
t_{0}$,

  2. (2)

for all $\varepsilon>0$, $n^{-\varepsilon}=o(\psi^{-1}(\log n))$.

Assumption (2) implies that the Gaussian measure is genuinely infinite
dimensional : indeed, in the case when $\text{dim }K<+\infty$, the measure is
supported in a finite-dimensional Banach space, and in this case the small
ball function behaves as $\log t$.

######  Theorem 1.3.

Let $(E,\mu)$ be a Gaussian Banach space with weak variance $\sigma$ and small
ball function $\psi$. Assume that Assumptions (1) and (2) hold.

Then there exists a universal constant $c$ such that for all

$n\geq(6+\kappa)(\log 2\vee\psi(1)\vee\psi(t_{0}/2)\vee 1/\sigma^{2}),$

the following holds :

(3) $\mathbb{E}(W_{2}(L_{n},\mu))\leq c\left[\psi^{-1}(\frac{1}{6+\kappa}\log
n)+\sigma n^{-1/[4(6+\kappa)]}\right].$

In particular, there is a $C=C(\mu)$ such that

(4) $\mathbb{E}(W_{2}(L_{n},\mu))\leq C\psi^{-1}(\log n).$

Moreover, for $\lambda>0$,

(5) $W_{2}(L_{n},\mu)\leq(C+\lambda)\psi^{-1}(\log n)\text{ with probability
}1-\exp-n\psi^{-1}(\log n)\frac{\lambda^{2}}{2\sigma^{2}}.$

######  Remark.

Note that the choice of $6+\kappa$ is not particularly sharp and may likely be
improved.

In order to underline the interest of the result above, we introduce some
definitions from optimal quantization. For $n\geq 1$ and $1\leq r<+\infty$,
define the optimal quantization error at rate $n$ as

$\delta_{n,r}(\mu)=\inf_{\nu\in\mathcal{P}_{n}}W_{r}(\mu,\nu)$

where the infimum runs on the set $\mathcal{P}_{n}$ of probability measures
with finite support of cardinal bounded by $n$. Under some natural
assumptions, the upper bound of (5) is matched by a lower bound for the
quantization error. Theorem 3.1 in [8] states the following : if for every
$0<\zeta<1$,

$\mu((1-\zeta)\varepsilon B)=o(\mu(\varepsilon B))\text{ as
}\varepsilon\rightarrow 0,$

then

$\delta_{n,r}\gtrsim\psi^{-1}(\log n)$

(where $a_{n}\gtrsim b_{n}$ means $\liminf a_{n}/b_{n}\geq 1$).

In the terminology of quantization, Theorem 1.3 states that the empirical
measure is a rate-optimal quantizer with high probability (under some
assumptions on the small ball function). This is of practical interest, since
obtaining the empirical measure is only as difficult as simulating an instance
of the Gaussian vector, and one avoids dealing with computation of appropriate
weights in the approximating discrete measure.

We leave aside the question of determining the sharp asymptotics for the
average error $\mathbb{E}(W_{2}(L_{n},\mu))$, that is of finding $c$ such that
$\mathbb{E}(W_{2}(L_{n},\mu))\sim c\psi^{-1}(\log n)$. Let us underline that
the corresponding question for quantizers is tackled for example in [22].

###  1.3. The case of Markov chains

We wish to extend the control of the speed of convergence to weakly dependent
sequences, such as rapidly-mixing Markov chains. There is a natural incentive
to consider this question : there are cases when one does not know hom to
sample from a given measure $\pi$, but a Markov chain with stationary measure
$\pi$ is nevertheless available for simulation. This is the basic set-up of
the Markov Chain Monte Carlo framework, and a very frequent situation, even in
finite dimension.

When looking at the proof of Proposition 1.1, it is apparent that the main
ingredient missing in the dependent case is the argument following (18), i.e.
that whenever $A\subset X$ is measurable, $nL_{n}(A)$ follows a binomial law
with parameters $n$ and $\mu(A)$, and this must be remedied in some way. It is
natural to look for some type of quantitative ergodicity property of the
chain, expressing almost-independence of $X_{i}$ and $X_{j}$ in the long range
($|i-j|$ large).

We will consider decay-of-variance inequalities of the following form :

(6) $\text{Var}_{\pi}P^{n}f\leq C\lambda^{n}\text{Var}_{\pi}f.$

In the reversible case, a bound of the type of (6) is ensured by Poincaré or
spectral gap inequalities. We recall one possible definition in the discrete-
time Markov chain setting.

######  Definition 1.2.

Let $P$ be a Markov kernel with _reversible_ measure $\pi\in\mathcal{P}(E)$.
We say that a Poincaré inequality with constant $C_{P}>0$ holds if

(7) $\text{Var}_{\pi}f\leq C_{P}\int f(I-P^{2})fd\pi$

for all $f\in L^{2}(\pi)$.

If (7) holds, we have

$\text{Var}_{\pi}P^{n}f\leq\lambda^{n}\text{Var}_{\pi}f$

with $\lambda=(C_{P}-1)/C_{P}$.

More generally, one may assume that we have a control of the decay of the
variance in the following form :

(8) $\text{Var}_{\pi}P^{n}f\leq C\lambda^{n}\|f-\int fd\pi\|_{L^{p}}.$

As soon as $p>2$, these inequalities are weaker than (6). Our proof would be
easily adaptable to this weaker decay-of-variance setting. We do not provide a
complete statement of this claim.

For a discussion of the links between Poincaré inequality and other notions of
weak dependence (e.g. mixing coefficients), see the recent paper [6].

For the next two theorems, we make the following dimension assumption on $E$ :
there exists $k_{E}>0$ and $\alpha>0$ such that for all $X\subset E$ with
finite diameter,

(9) $N(X,\delta)\leq k_{E}(\text{Diam }X/\delta)^{\alpha}.$

The following theorem is the analogue of Corollary 1.2 under the assumption
that the Markov chain satisfies a decay-of-variance inequality.

######  Theorem 1.4.

Assume that $E$ has finite diameter $d>0$ and (9) holds. Let
$\pi\in\mathcal{P}(E)$, and let $(X_{i})_{i\geq 0}$ be a $E$-valued Markov
chain with initial law $\nu$ such that $\pi$ is its unique invariant
probability. Assume also that (6) holds for some $C>0$ and $\lambda<1$.

Then if $2p>\alpha(1+1/r)$ and $L_{n}$ denotes the occupation measure
$1/n\sum_{i=1}^{n}\delta_{X_{i}}$, the following holds :

$\mathbb{E}_{\nu}\left[W_{p}(L_{n},\pi)\right]\leq
c\frac{\alpha(1+1/r)}{\alpha(1+1/r)-2p}k_{E}^{1/\alpha}d\left(\frac{C\|\frac{d\nu}{d\pi}\|_{r}}{(1-\lambda)n}\right)^{1/[\alpha(1+1/r)]}$

for some universal constant $c\leq 64/3$.

The previous theorem has the drawback of assuming that the state space has
finite diameter. This can be circumvented, for example by truncation
arguments. Our next theorem is an extension to the unbounded case under some
moment conditions on $\pi$. The statement and the proof involve more
technicalities than Theorem 1.4, so we separate the two in spite of the
obvious similarities.

######  Theorem 1.5.

Assume that (9) holds. Let $\pi\in\mathcal{P}(E)$, and let $(X_{i})_{i\geq 0}$
be a $E$-valued Markov chain with initial law $\nu$ such that $\pi$ is its
unique invariant probability. Assume also that (6) holds for some $C>0$ and
$\lambda<1$. Let $x_{0}\in E$ and for all $\theta\geq 1$, denote
$M_{\theta}=\int d(x_{0},x)^{\theta}d\pi$. Fix $r$ and $\zeta>1$ and assume
$2p>\alpha(1+1/r)(1+1/\zeta)$.

There exist two numerical constant $C_{1}(p,r,\zeta)$ and $C_{2}(p,r,\zeta)$
only depending on $p$, $r$ and $\zeta$ such that whenever

$\frac{C\|\frac{d\nu}{d\pi}\|_{r}}{(1-\lambda)n}\leq C_{1}(p,r,\zeta),$

the following holds :

$\mathbb{E}_{\nu}\left[W_{p}(L_{n},\pi)\right]\leq
C_{1}(p,r,\zeta)K(\zeta)\left(\frac{C\|\frac{d\nu}{d\pi}\|_{r}}{(1-\lambda)n}\right)^{1/[\alpha(1+1/r)(1+1/\zeta)]}$

where

$K(\zeta)=\frac{m_{\zeta}}{m_{p}^{\zeta/p}}\vee\frac{m_{\zeta+2p}}{m_{p}^{1+\zeta/p}}\vee
k_{E}^{1/2p(1+1/r)}\frac{2p}{\alpha(1+1/r)}m_{p}^{\alpha/(2p^{2})(1+1/r)}.$

##  2\. Proofs in the independent case

######  Lemma 2.1.

Let $X\subset E$, $s>0$ and $u,v\in\mathbb{N}$ with $u<v$. Suppose that
$N(X,4^{-v}s)<+\infty$. For $u\leq j\leq v$, there exist integers

(10) $m(j)\leq N(X,4^{-j}s)$

and non-empty subsets $X_{j,l}$ of $X$, $u\leq j\leq v$, $1\leq l\leq m(j)$,
such that the sets $X_{j,l}$ $1\leq l\leq m(j)$ satisfy

  1. (1)

for each $j$, $(X_{j,l})_{1\leq l\leq m(j)}$ is a partition of $X$,

  2. (2)

$\text{Diam }X_{j,l}\leq 4^{-j+1}s$,

  3. (3)

for each $j>u$, for each $1\leq l\leq m(j)$ there exists $1\leq l^{\prime}\leq
m(j-1)$ such that $X_{j,l}\subset X_{j-1,l^{\prime}}$.

In other words, the sets $X_{j,l}$ form a sequence of partitions of $X$ that
get coarser as $j$ decreases (tiles at the scale $j-1$ are unions of tiles at
the scale $j$).

###### Proof.

We begin by picking a set of balls $B_{j,l}=B(x_{j,l},4^{-j}s)$ with $u\leq
j\leq v$ and $1\leq l\leq N(X,4^{-j}s)$, such that for all $j$,

$X\subset\bigcup_{l=1}^{N(X,4^{-j}s)}B_{j,l}.$

Define $X_{v,1}=B_{v,1}$, and successively set $X_{v,l}=B_{v,l}\setminus
X_{v,l-1}$. Discard the possible empty sets and relabel the existing sets
accordingly. We have obtained the finest partition, obviously satisfying
conditions (1)-(2).

Assume now that the sets $X_{j,l}$ have been built for $k+1\leq j\leq v$. Set
$X_{k,1}$ to be the reunion of all $X_{k+1,l^{\prime}}$ such that
$X_{k+1,l^{\prime}}\cap B_{k,1}\neq\emptyset$. Likewise, define by induction
on $l$ the set $X_{k,l}$ as the reunion of all $X_{k+1,l^{\prime}}$ such that
$X_{k+1,l^{\prime}}\cap B_{k,l}\neq\emptyset$ and
$X_{k+1,l^{\prime}}\nsubseteq X_{k,p}$ for $1\leq p<l$. Again, discard the
possible empty sets and relabel the remaining tiles. It is readily checked
that the sets obtained satisfy assumptions (1) and (3). We check assumption
(2) : let $x_{k,l}$ denote the center of $B_{k,l}$ and let $y\in
X_{k+1,l^{\prime}}\subset X_{k,l}$. We have

$d(x_{k,l},y)\leq 4^{-k}s+\text{Diam }X_{k+1,l^{\prime}}\leq 2\times 4^{-k}s,$

thus $\text{Diam }X_{k,l}\leq 4^{-k+1}s$ as desired.

∎

Consider as above a subset $X$ of $E$ with finite diameter $d$, and assume
that $N(X,4^{-k}d)<+\infty$. Pick a sequence of partitions $(X_{j,l})_{1\leq
l\leq m(j)}$ for $1\leq j\leq k$, as per Lemma 2.1. For each $(j,l)$ choose a
point $x_{j,l}\in X_{j,l}$. Define the set of points of level $j$ as the set
$L(j)=\\{x_{j,l}\\}_{1\leq l\leq m(j)}$. Say that $x_{j^{\prime},l^{\prime}}$
is an ancestor of $x_{j,l}$ if $X_{j,l}\subset X_{j^{\prime},l^{\prime}}$ : we
will denote this relation by $(j^{\prime},l^{\prime})\rightarrow(j,l)$.

The next two lemmas study the cost of transporting a finite measure $m_{k}$ to
another measure $n_{k}$ when these measures have support in $L(k)$. The
underlying idea is that we consider the finite metric space formed by the
points $x_{j,l}$, $1\leq j\leq k$, as a _metric tree_ , where points are
connected to their ancestor at the previous level, and we consider the problem
of transportation between two masses at the leaves of the tree. The
transportation algorithm we consider consists in allocating as much mass as
possible at each point, then moving the remaining mass up one level in the
tree, and iterating the procedure.

A technical warning : please note that the transportation cost is usually
defined between two probability measures ; however there is no difficulty in
extending its definition to the transportation between two finite measures of
equal total mass, and we will freely use this fact in the sequel.

######  Lemma 2.2.

Let $m_{j}$, $n_{j}$ be measures with support in $L_{j}$. Define the measures
$\tilde{m}_{j-1}$ and $\tilde{n}_{j-1}$ on $L_{j-1}$ by setting

(11) $\displaystyle\tilde{m}_{j-1}(x_{j-1,l^{\prime}})$
$\displaystyle=\sum_{(j-1,l^{\prime})\rightarrow(j,l)}(m_{j}(x_{j,l})-n_{j}(x_{j,l}))\wedge
0,$ (12) $\displaystyle\tilde{n}_{j-1}(x_{j-1,l^{\prime}})$
$\displaystyle=\sum_{(j-1,l^{\prime})\rightarrow(j,l)}(n_{j}(x_{j,l})-m_{j}(x_{j,l}))\wedge
0.$

The measures $\tilde{m_{j-1}}$ and $\tilde{n_{j-1}}$ have same mass, so the
transportation cost between them may be defined. Moreover, the following bound
holds :

(13) $W_{p}(m_{j},n_{j})\leq 2\times
4^{-j+2}d\|m_{j}-n_{j}\|_{TV}^{1/p}+W_{p}(\tilde{m}_{j-1},\tilde{n}_{j-1}).$

###### Proof.

Set $m_{j}\wedge n_{j}(x_{j,l})=m_{j}(x_{j,l})\wedge n_{j}(x_{j,l})$. By the
triangle inequality,

$\displaystyle W_{p}(m,n)\leq$ $\displaystyle W_{p}(m_{j},m_{j}\wedge
n_{j})+\tilde{m}_{j-1}+W_{p}(m_{j}\wedge n_{j}+\tilde{m}_{j-1},m_{j}\wedge
n_{j}+\tilde{n}_{j-1})$ $\displaystyle+W_{p}(m_{j}\wedge
n_{j}+\tilde{n}_{j-1},n_{j}).$

We bound the term on the left. Introduce the transport plan $\pi_{m}$ defined
by

$\displaystyle\pi_{m}(x_{j,l},x_{j,l})$ $\displaystyle=m_{j}\wedge
n_{j}(x_{j,l}),$ $\displaystyle\pi_{m}(x_{j,l},x_{j-1,l^{\prime}})$
$\displaystyle=(m_{j}(x_{j,l})-n_{j}(x_{j,l}))_{+}\text{ when
}(j-1,l^{\prime})\rightarrow(j,l).$

The reader can check that $\pi_{m}\in\mathcal{P}(m_{j},m_{j}\wedge
n_{j}+\tilde{m}_{j-1})$. Moreover,

$\displaystyle W_{p}(m_{j},\tilde{m_{j-1}})$ $\displaystyle\leq\left(\int
d^{p}(x,y)\pi_{m}(dx,dy)\right)^{1/p}$ $\displaystyle\leq
4^{-j+2}d\left(\sum_{l=1}^{m(j)}(m_{j}(x_{j,l})-n_{j}(x_{j,l}))_{+}\right)^{1/p}.$

Likewise,

$W_{p}(n_{j},m_{j}\wedge n_{j}+\tilde{n_{j-1}})\leq
4^{-j+2}d\left(\sum_{l=1}^{m(j)}(n_{j}(x_{j,l})-m_{j}(x_{j,l}))_{+}\right)^{1/p}.$

As for the term in the middle, it is bounded by
$W_{p}(\tilde{m}_{j-1},\tilde{n}_{j-1})$. Putting this together and using the
inequality $x+y\leq 2^{1-1/p}(x^{p}+y^{p})^{1/p}$,, we get

$W_{p}(m_{j},n_{j})\leq
2^{1-1/p}4^{-j+2}d\left(\sum_{l=1}^{m(j)}|m_{j}(x_{j,l})-n_{j}(x_{j,l})|\right)^{1/p}+W_{p}(\tilde{m}_{j-1},\tilde{n}_{j-1}).$

∎

######  Lemma 2.3.

Let $m_{j}$, $n_{j}$ be measures with support in $L_{j}$. Define for $1\leq
j^{\prime}<j$ the measures $m_{j}^{\prime}$, $n_{j}^{\prime}$ with support in
$L_{j}^{\prime}$ by

(14)
$m_{j^{\prime}}(x_{j^{\prime},l^{\prime}})=\sum_{(j^{\prime},l^{\prime})\rightarrow(j,l)}m_{j}(x_{j,l}),\quad
n_{j^{\prime}}(x_{j^{\prime},l^{\prime}})=\sum_{(j^{\prime},l^{\prime})\rightarrow(j,l)}n_{j}(x_{j,l}).$

The following bound holds :

(15) $W_{p}(m_{j},n_{j})\leq\sum_{j^{\prime}=1}^{j}2\times
4^{-j^{\prime}+2}d\|m_{j}^{\prime}-n_{j}^{\prime}\|_{TV}^{1/p}$

###### Proof.

We proceed by induction on $j$. For $j=1$, the result is obtained by using the
simple bound $W_{p}(m_{1},n_{1})\leq d\|m_{1}-n_{1}\|_{\text{TV}}^{1/p}$.

Suppose that (15) holds for measures with support in $L_{j-1}$. By lemma 2.2,
we have

$W_{p}(m_{j},n_{j})\leq 2\times
4^{-j+2}d\|m_{j}-n_{j}\|_{TV}^{1/p}+W_{p}(\tilde{m}_{j-1},\tilde{n}_{j-1})$

where $\tilde{m}_{j-1}$ and $\tilde{n}_{j-1}$ are defined by (11) and (12)
respectively. For $1\leq i<j-1$, define following (14)

$\tilde{m}_{i}(x_{i,l^{\prime}})=\sum_{(i,l^{\prime})\rightarrow(j-1,l)}\tilde{m}_{j-1}(x_{j-1,l}),\quad\tilde{n}_{i}(x_{i,l^{\prime}})=\sum_{(i,l^{\prime})\rightarrow(j-1,l)}\tilde{n}_{j-1}(x_{j-1,l}).$

We have

$W_{p}(m_{j},n_{j})\leq 2\times
4^{-j+2}d\|m_{j}-n_{j}\|_{TV}^{1/p}+\sum_{j^{\prime}=1}^{j-1}2\times
4^{-j^{\prime}+2}d\|\tilde{m}_{i}-\tilde{n}_{i}\|_{TV}^{1/p}.$

To conclude, it suffices to check that for $1\leq i\leq j-1$,
$\|\tilde{m}_{i}-\tilde{n}_{i}\|_{TV}=\|m_{i}-n_{i}\|_{TV}$.

∎

###### Proof of Proposition 1.1..

We pick some positive integer $k$ whose value will be determined at a later
point. Introduce the sequence of partitions $(X_{j,l})_{1\leq l\leq m(j)}$ for
$0\leq j\leq k$ as in the lemmas above, as well as the points $x_{j,l}$.
Define $\mu_{k}$ as the measure with support in $L(k)$ such that
$\mu_{k}(x_{k,l})=\mu(X_{k,l})$ for $1\leq l\leq m(k)$. The diameter of the
sets $X_{k,l}$ is bounded by $4^{-k+1}d$, therefore $W_{p}(\mu,\mu_{k})\leq
4^{-k+1}d$.

Let $L_{n}^{k}$ denote the empirical measure associated to $\mu_{k}$.

For $0\leq j\leq k-1$, define as in Lemma 2.3 the measures $\mu_{j}$ and
$L_{n}^{j}$ with support in $L(j)$ by

(16) $\displaystyle\mu_{j}(x_{j,l^{\prime}})$
$\displaystyle=\sum_{(j,l^{\prime})\rightarrow(k,l)}\mu_{k}(x_{k,l})$ (17)
$\displaystyle L_{n}^{j}(x_{j,l^{\prime}})$
$\displaystyle=\sum_{(j,l^{\prime})\rightarrow(k,l)}L_{n}^{k}(x_{k,l}).$

It is simple to check that $\mu_{j}(x_{j,l})=\mu(X_{j,l})$, and that
$L_{n}^{j}$ is the empirical measure associated with $\mu_{j}$. Applying (15),
we get

(18) $W_{p}(\mu_{k},L_{n}^{k})\leq\sum_{j=1}^{k}2\times
4^{-j+2}d\|\mu_{j}-L_{n}^{j}\|_{TV}^{1/p}.$

Observe that $nL_{n}^{j}(x_{j,l})$ is a binomial law with parameters $n$ and
$\mu(X_{j,l})$. The expectation of $\|\mu_{j}-L_{n}^{j}\|_{TV}$ is bounded as
follows :

$\displaystyle\mathbb{E}(\|\mu_{i}-L_{n}^{i}\|_{TV})$
$\displaystyle=1/2\sum_{l=1}^{m(j)}\mathbb{E}(|(L_{n}^{j}-\mu_{j})(x_{j,l})|)$
$\displaystyle\leq
1/2\sum_{l=1}^{m(j)}\sqrt{\mathbb{E}(|(L_{n}^{j}-\mu_{j})(x_{j,l})|^{2})}$
$\displaystyle=1/2\sum_{l=1}^{m(j)}\sqrt{\frac{\mu(X_{j,l})(1-\mu(X_{j,l}))}{n}}$
$\displaystyle\leq 1/2\sqrt{\frac{m(j)}{n}}.$

In the last inequality, we use Cauchy-Schwarz’s inequality and the fact that
$(X_{j,l})_{1\leq l\leq m(j)}$ is a partition of $X$. Putting this back in
(18), we get

$\displaystyle\mathbb{E}(W_{p}(\mu_{k},L_{n}^{k}))$ $\displaystyle\leq
n^{-1/2p}\sum_{j=1}^{k}2^{1-1/p}4^{(-j+2)}dm(i)^{1/2p}$ $\displaystyle\leq
2^{5-1/p}n^{-1/2p}\sum_{j=1}^{k}4^{-j}dN(X,4^{-j}d)^{1/2p}$ $\displaystyle\leq
2^{6-1/p}/3n^{-1/2p}\int_{4^{-(k+1)}d}^{d/4}N(X,\delta)^{1/2p}d\delta.$

In the last line, we use a standard sum-integral comparison argument.

By the triangle inequality, we have

$W_{p}(\mu,L_{n})\leq
W_{p}(\mu,\mu_{k})+W_{p}(\mu_{k},L_{n}^{k})+W_{p}(L_{n}^{k},L_{n}).$

We claim that $\mathbb{E}(W_{p}(L_{n}^{k},L_{n}))\leq W_{p}(\mu,\mu_{k})$.
Indeed, choose $n$ i.i.d. couples $(X_{i},X_{i}^{k})$ such that
$X_{i}\sim\mu$, $X_{i}^{k}\sim\mu_{k}$, and the joint law of
$(X_{i},X_{i}^{k})$ achieves an optimal coupling, i.e.
$\mathbb{E}|X_{i}-X_{i}^{k}|^{p}=W_{p}^{p}(\mu,\mu^{k})$. We have the
identities in law

$L_{n}\sim\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}},\>L_{n}^{k}\sim\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}^{k}}.$

Choose the transport plan that sends $X_{i}$ to $X_{i}^{k}$ : this gives the
upper bound

$W_{p}^{p}(L_{n},L_{n}^{k})\leq 1/n\sum_{i=1}^{n}|X_{i}-X_{i}^{k}|^{p}$

and passing to expectation proves our claim.

Thus, $\mathbb{E}(W_{p}(\mu,L_{n}))\leq
2W_{p}(\mu,\mu_{k})+\mathbb{E}(W_{p}(\mu_{k},L_{n}^{k}))$. Choose now $k$ as
the largest integer such that $4^{-(k+1)}d\geq t$. This imposes $4^{-k+1}d\leq
16t$, and this finishes the proof.

∎

###### Proof of Corollary 1.2.

It suffices to use Proposition 1.1 along with (2) and to optimize in $t$. ∎

##  3\. Proof of Theorem 1.3.

###### Proof of Theorem 1.3..

We begin by noticing that statement (5) is a simple consequence of statement
(4) and the tensorization of $\mathbf{T}_{2}$ : we have by Corollary A.2

$\mathbb{P}(W_{2}(L_{n},\mu)\geq\mathbb{E}(W_{2}(L_{n},\mu)+t)\leq
e^{-nt^{2}/(2\sigma^{2})},$

and it suffices to choose $t=\lambda\psi^{-1}(\log n)$ to conclude. We now
turn to the other claims.

Denote by $K$ the unit ball of the Cameron-Martin space associated to $E$ and
$\mu$, and by $B$ the unit ball of $E$. According to the Gaussian
isoperimetric inequality (see [19]), for all $\lambda>0$ and $\varepsilon>0$,

$\mu(\lambda K+\varepsilon B)\geq\Phi\left(\lambda+\Phi^{-1}(\mu(\varepsilon
B))\right)$

where $\Phi(t)=\int_{-\infty}^{t}e^{-u^{2}/2}du/\sqrt{2\pi}$ is the Gaussian
c.d.f..

Choose $\lambda>0$ and $\varepsilon>0$, and set $X=\lambda K+\varepsilon B$.
Note

$\mu^{\prime}=\frac{1}{\mu(X)}\mathbf{1}_{X}\mu$

the restriction of $\mu$ to the enlarged ball.

The diameter of $X$ is bounded by $2(\sigma\lambda+\varepsilon)$. The $W_{2}$
distance between $L_{n}$ and $\mu$ is thus bounded as follows :

(19) $W_{2}(L_{n},\mu)\leq
2W_{2}(\mu,\mu^{\prime})+ct+cn^{-1/4}\int_{t}^{(\sigma\lambda+\varepsilon)/2}N(X,\delta)^{1/4}d\delta$

Set

(20) $\displaystyle I_{1}$ $\displaystyle=W_{2}(\mu,\mu^{\prime})$ (21)
$\displaystyle I_{2}$ $\displaystyle=t$ (22) $\displaystyle I_{3}$
$\displaystyle=n^{-1/4}\int_{t}^{(\sigma\lambda+\varepsilon)/2}N(X,\delta)^{1/4}d\delta.$

To begin with, set $\varepsilon=t/2$.

_Controlling $I_{1}$._ We use transportation inequalities and the Gaussian
isoperimetric inequality. By Lemma A.1, $\mu$ satisfies a
$\mathbf{T}_{2}(2\sigma^{2})$ inequality, so that we have

$\displaystyle W_{2}(\mu,\mu^{\prime})$
$\displaystyle\leq\sqrt{2\sigma^{2}H(\mu^{\prime}|\mu)}=\sqrt{-2\sigma^{2}\log\mu(\lambda
K+\varepsilon B)}$
$\displaystyle\leq\sqrt{-2\sigma^{2}\log\Phi(\lambda+\Phi^{-1}(\mu(\varepsilon
B)))}$
$\displaystyle=\sqrt{2}\sigma\sqrt{-\log\Phi(\lambda+\Phi^{-1}(e^{-\psi(t/2)}))}.$

Introduce the tail function of the Gaussian distribution

$\Upsilon(x)=\sqrt{2\pi}^{-1}\int_{x}^{+\infty}e^{-y^{2}/2}dy.$

We will use the fact that $\Phi^{-1}+\Upsilon^{-1}=0$, which comes from
symmetry of the Gaussian distribution. We will also use the bound
$\Upsilon(t)\leq e^{-t^{2}/2}/2$, $t\geq 0$ and its consequence

$\Upsilon^{-1}(u)\leq\sqrt{-2\log u},\quad 0<u\leq 1/2.$

We have

$\Phi^{-1}(e^{-\psi(t/2)})=-\Upsilon^{-1}(e^{-\psi(t/2)})\geq-\sqrt{2\psi(t/2)}$

as soon as $\psi(t/2)\geq\log 2$. The elementary bound $\log\frac{1}{1-x}\leq
2x$ for $x\leq 1/2$ yields

$\displaystyle\sqrt{-2\log\Phi(u)}$
$\displaystyle=\sqrt{2}\left(\log\frac{1}{1-\Upsilon(u)}\right)^{1/2}$
$\displaystyle\leq\sqrt{2}e^{-u^{2}/4}$

whenever $u\geq\Upsilon^{-1}(1/2)=0$. Putting this together, we have

(23) $I_{1}\leq\sqrt{2}\sigma e^{-(\lambda-\sqrt{2\psi(t/2)})^{2}/4}.$

whenever

(24) $\psi(t/2)\geq\log 2\text{ and }\lambda-\sqrt{2\psi(t/2)}\geq 0.$

_Controlling $I_{3}$._ The term $I_{3}$ is bounded by
$1/2n^{-1/4}(\sigma\lambda+t/2)N(X,t)^{1/4}$ (just bound the function inside
by its value at $t$, which is minimal). Denote $k=N(\lambda K,t-\varepsilon)$
the covering number of $\lambda K$ (w.r.t. the norm of $E$). Let
$x_{1},\ldots,x_{k}\in K$ be such that union of the balls
$B(x_{i},t-\varepsilon)$ contains $\lambda K$. From the triangle inequality we
get the inclusion

$\lambda K+\varepsilon B\subset\bigcup_{i=1}^{k}B(x_{i},t).$

Therefore, $N(X,t)\leq N(\lambda K,t-\varepsilon)=N(\lambda K,t/2)$.

We now use the well-known link between $N(\lambda K,t/2)$ and the small ball
function. Lemma 1 in [18] gives the bound

$N(\lambda K,t/2)\leq e^{\lambda^{2}/2+\psi(t/4)}\leq
e^{\lambda^{2}/2+\kappa\psi(t/2)}.$

so that

(25)
$I_{3}\leq\frac{1}{2}(\sigma\lambda+t/2)e^{\frac{\lambda^{2}}{8}+\frac{\kappa}{4}\psi(t/2)-\frac{1}{4}\log
n}.$

Remark that we have used the doubling condition on $\psi$, so that we require

(26) $t/4\leq t_{0}.$

_Final step._ Set now $t=2\psi^{-1}(a\log n)$ and $\lambda=2\sqrt{2a\log n}$,
with $a>0$ yet undetermined. Using (23) and (25), we see that there exists a
universal constant $c$ such that

$\displaystyle\mathbb{E}(W_{2}(L_{n},\mu))\leq$ $\displaystyle
c\left[\psi^{-1}(a\log n)+\sigma e^{-(a/2)\log n}\right.$
$\displaystyle\quad\left.+(\sigma\sqrt{a\log n}+\psi^{-1}(a\log
n))e^{[a(1+\kappa/4)-1/4]\log n}\right].$

Choose $a=1/(6+\kappa)$ and assume $\log n\geq(6+\kappa)(\log
2\vee\psi(1)\vee\psi(t_{0}/2))$. This guarantees that the technical conditions
(24) and (26) are enforced, and that $\psi^{-1}(a\log n)\leq 1$. Summing up,
we get :

$\mathbb{E}(W_{2}(L_{n},\mu))\leq c\left[\psi^{-1}(\frac{1}{6+\kappa}\log
n)+(1+\sigma\sqrt{\frac{1}{6+\kappa}\log n})n^{-1/(12+2\kappa)}\right].$

Impose $\log n\geq(6+\kappa)/\sigma^{2}$ : this ensures
$\sigma\sqrt{\frac{1}{6+\kappa}\log n}\geq 1$. And finally, there exists some
$c>0$ such that for all $x\geq 1$, $\sqrt{\log x}x^{-1/4}\leq c$ : this
implies

$\sqrt{\frac{1}{6+\kappa}\log n}n^{-1/(24+4\kappa)}\leq c.$

This gives

$(1+\sigma\sqrt{\frac{1}{6+\kappa}\log n})n^{-1/(12+2\kappa)}\leq c\sigma
n^{-1/[4(6+\kappa)]}$

and the proof is finished.

∎

##  4\. Proofs in the dependent case

We consider hereafter a Markov chain $(X_{n})_{n\in\mathbb{N}}$ defined by
$X_{0}\sim\nu$ and the transition kernel $P$. Let us denote by

$L_{n}=\sum_{i=1}^{n}\delta_{X_{i}}$

its occupation measure.

######  Proposition 4.1.

Suppose that the Markov chain satisfies (6) for some $C>0$ and $\lambda<1$.
Then the following holds :

(27) $\mathbb{E}_{\nu}(W_{p}(L_{n},\pi))\leq
c\left(t+\left(\frac{C}{(1-\lambda)n}\|\frac{d\nu}{d\pi}\|_{r}\right)^{1/2p}\int_{t}^{d/4}N(X,t)^{1/2p(1+1/r)}dt\right).$

###### Proof.

An application of (15) as in (18) yields

(28) $\mathbb{E}(W_{p}(L_{n},\pi))\leq 2\times 4^{-k+1}d+\sum_{j=1}^{k}2\times
4^{-j+2}d\left(\sum_{l=1}^{m(j)}\mathbb{E}|(L_{n}-\pi)(X_{j,l})|\right)^{1/p}.$

Let $A$ be a measurable subset of $X$, and set
$f_{A}(x)=\mathbf{1}_{A}(x)-\pi(A)$. We have

$\displaystyle\mathbb{E}|(L_{n}-\pi)(A)|$
$\displaystyle=1/n\mathbb{E}_{\nu}|\sum_{i=1}^{n}f_{A}(X_{i})|$
$\displaystyle\leq
1/n\sqrt{\sum_{i=1}^{n}\sum_{j=1}^{n}\mathbb{E}_{\nu}\left[f_{A}(X_{i})f_{A}(X_{j})\right]}.$

Let $\tilde{p},\tilde{q},r\geq 1$ be such that
$1/\tilde{p}+1/\tilde{q}+1/r=1$, and let $s$ be defined by
$1/s=1/\tilde{p}+1/\tilde{q}$. Now, using Hölder’s inequality with $r$ and
$s$,

$\mathbb{E}_{\nu}\left[f_{A}(X_{i})f_{A}(X_{j})\right]\leq\|\frac{d\nu}{d\pi}\|_{r}(\mathbb{E}_{\pi}|f_{A}(X_{i})f_{A}(X_{j})|^{s})^{1/s}.$

Use the Markov property and the fact that $f\mapsto Pf$ is a contraction in
$L^{s}$ to get

$\mathbb{E}_{\nu}\left[f_{A}(X_{i})f_{A}(X_{j})\right]\leq\|\frac{d\nu}{d\pi}\|_{r}\|f_{A}P^{j-i}f_{A}\|_{s}.$

Finally, use Hölder’s inequality with $\tilde{p},\tilde{q}$ : we get

(29)
$\mathbb{E}_{\nu}\left[f_{A}(X_{i})f_{A}(X_{j})\right]\leq\|\frac{d\nu}{d\pi}\|_{r}\|P^{j-i}f_{A}\|_{\tilde{p}}\|f_{A}\|_{\tilde{q}}.$

Set $\tilde{p}=2$ and note that for $1\leq t\leq+\infty$, we have
$\|f_{A}\|_{t}\leq 2\pi(A)^{1/t}$. Use (6) applied to the centered function
$f_{A}$ to get

$\mathbb{E}_{\nu}\left[f_{A}(X_{i})f_{A}(X_{j})\right]\leq
4C\lambda^{j-i}\|\frac{d\nu}{d\pi}\|_{r}\pi(A)^{1-1/r},$

and as a consequence,

(30)
$\mathbb{E}|(L_{n}-\pi)(A)|\leq\frac{1}{\sqrt{n}}\frac{2\sqrt{2C}}{\sqrt{1-\lambda}}\|\frac{d\nu}{d\pi}\|_{r}^{1/2}\pi(A)^{1/2-1/2r}.$

Come back to (28) : we have

$\displaystyle\mathbb{E}(W_{p}(L_{n},\pi))$ $\displaystyle\leq
4^{-k+1}d+32(\frac{2\sqrt{2C}}{\sqrt{1-\lambda}})^{1/p}\|\frac{d\nu}{d\pi}\|_{r}^{1/2p}n^{-1/2p}$
$\displaystyle\quad\times\sum_{j=1}^{k}4^{-j}d\left(\sum_{l=1}^{m(j)}\pi(X_{j,l})^{1/2-1/2r}\right)^{1/p}$
$\displaystyle\leq
4^{-k+1}d+c\left(\frac{C}{(1-\lambda)n}\|\frac{d\nu}{d\pi}\|_{r}\right)^{1/2p}\sum_{j=1}^{k}4^{-j}dm(j)^{1/2p(1+1/r)}$
$\displaystyle\leq
c\left(t+\left(\frac{C}{(1-\lambda)n}\|\frac{d\nu}{d\pi}\|_{r}\right)^{1/2p}\int_{t}^{d/4}N(X,t)^{1/2p(1+1/r)}dt\right).$

∎

###### Proof of Theorem 1.4..

Use (27) and (9) to get

$\mathbb{E}W_{p}(L_{n},\mu)\leq c\left[t+At^{-\alpha/2p(1+1/r)+1}\right]$

where

$A=\frac{2p}{\alpha(1+1/r)}(C/(1-\lambda))^{1/2p}\|\frac{d\nu}{d\pi}\|_{r}^{1/2p}n^{-1/2p}d^{\alpha/2p(1+1/r)}.$

Optimizing in $t$ finishes the proof.

∎

We now move to the proof in the unbounded case.

###### Proof of Theorem 1.5.

We remind the reader that the following assumption stands : for $X\subset E$
with diameter bounded by $d$,

(31) $N(X,\delta)\leq k_{E}(d/\delta)^{\alpha}.$

In the following lines, we will make use of the elementary inequalities

(32) $(x+y)^{p}\leq 2^{p-1}(x^{p}+y^{p})\leq 2^{p-1}(x+y)^{p}.$

_Step 1._

Pick increasing sequence of numbers $d_{i}>0$ to be set later on, and some
point $x_{0}\in E$. Define $C_{1}=B(x_{0},d_{1})$, and $C_{i}=B(x_{0},d_{i})\
B(x_{0},d_{i-1})$ for $i\geq 2$.

The idea is as follows : we decompose the state space $E$ into a union of
rings, and deal separately with $C_{1}$ on the one hand, using the case of
Theorem 1.4 as guideline, and with the union of the $C_{i}$, $i\geq 2$ on the
other hand, where we use more brutal bounds.

We define partial occupation measures

$L_{n}^{i}=1/n\sum_{j=1}^{n}\delta_{X_{j}}\mathbf{1}_{X_{j}\in C_{i}}$

and their masses $m_{i}=L_{n}^{i}(E)$. We have the inequality

(33) $W_{p}^{p}(L_{n},\pi)\leq\sum_{i\geq
1}m_{i}W_{p}^{p}(1/m_{i}L_{n}^{i},\pi).$

On the other hand,

$\displaystyle W_{p}(1/m_{i}L_{n}^{i},\pi)$ $\displaystyle\leq(\int
d(x_{0},x)^{p}d\pi)^{1/p}+(\int d(x_{0},x)^{p}d(1/m_{i}L_{n}^{i}))^{1/p}$
$\displaystyle\leq M_{p}^{1/p}+d_{i},$

so that $W_{p}^{p}(1/m_{i}L_{n}^{i},\pi)\leq
2^{p-1}\left(M_{p}+d_{i}^{p}\right)$ using (32). Also, using (33) and (32)
yields

$W_{p}(L_{n},\pi)\leq
m_{1}^{1/p}W_{p}(1/m_{1}L_{n}^{1},\pi)+2^{1-1/p}\left(\sum_{i\geq
2}m_{i}\left[M_{p}+d_{i}^{p}\right]\right)^{1/p}.$

Pass to expectations to get

(34)
$\mathbb{E}[W_{p}(L_{n},\pi)]\leq\mathbb{E}\left[m_{1}^{1/p}W_{p}(1/m_{1}L_{n}^{1},\pi)\right]+2^{1-1/p}\left(\sum_{i\geq
2}\pi(C_{i})\left[M_{p}+d_{i}^{p}\right]\right)^{1/p}$

We bound separately the left and right term in the right-hand side of (34),
starting with the right one.

_Step 2_.

Choose some $q>p$ and use Chebyshev’s inequality to bound the sum on the right
by

(35) $\sum_{i\geq 2}\frac{M_{q}}{d_{i-1}^{q}}\left[M_{p}+d_{i}^{p}\right]$

Take $d_{i}=\rho^{i}M_{p}^{1/p}$, (35) becomes

$\displaystyle M_{q}M_{p}^{1-q/p}\rho^{q}\sum_{i\geq
2}[\rho^{-qi}+\rho^{(p-q)i}]$ $\displaystyle=$ $\displaystyle
M_{q}M_{p}^{1-q/p}\left[\frac{\rho^{-q}}{1-\rho^{-q}}+\frac{\rho^{2p-q}}{1-\rho^{p-q}}\right].$

Assume for example that $\rho\geq 2$ : this implies

$\sum_{i\geq 2}\pi(C_{i})\left[M_{p}+d_{i}^{p}\right]\leq
4M_{q}M_{p}^{1-q/p}\rho^{2p-q}.$

For later use, we set $\zeta=q/p-2$ and the above yields

(36) $2^{1-1/p}\left(\sum_{i\geq
2}\pi(C_{i})\left[M_{p}+d_{i}^{p}\right]\right)^{1/p}\leq
4M_{(\zeta+2)p}^{1/p}M_{p}^{-(1+\zeta)/p}\rho^{-\zeta}.$

_Step 3._

We now turn our attention to the term on the left in (34).

Once again, we apply (15) to obtain

$W_{p}(1/m_{1}L_{n}^{1},\pi)\lesssim
4^{-k}d_{1}+\sum_{j=1}^{k}4^{-j}\left(\sum_{l=1}^{m(j)}|((1/m_{1})L_{n}-\pi)(X_{j,l})|\right)^{1/p}$

Multiply by $m_{1}^{1/p}$ and pass to expectations :

$\displaystyle\mathbb{E}\left[m_{1}^{1/p}W_{p}(m_{1}^{-1}L_{n}^{1},\pi)\right]\lesssim$
$\displaystyle\,\sum_{j=1}^{k}4^{-j}\left(\sum_{l=1}^{m(j)}\mathbb{E}|(L_{n}-m_{1}\pi)(X_{j,l})|\right)^{1/p}$
$\displaystyle+4^{-k}d_{1}\mathbb{E}(m_{1}^{1/p}).$

First, notice that $0\leq m_{1}\leq 1$ a.s. so that
$\mathbb{E}(m_{1}^{1/p})\leq 1$. Next, write

$\displaystyle\sum_{l=1}^{m(j)}\mathbb{E}|(L_{n}-m_{1}\pi)(X_{j,l})|$
$\displaystyle\leq\sum_{l=1}^{m(j)}\mathbb{E}\left(|(L_{n}-\pi)(X_{j,l})|+|(m_{1}\pi-\pi)(X_{j},l)|\right)$
$\displaystyle\leq\sum_{l=1}^{m(j)}\mathbb{E}|(L_{n}-\pi)(X_{j,l})|+\mathbb{E}(|m_{1}-1|)\pi(C_{1})$
$\displaystyle\leq\sum_{l=1}^{m(j)}\mathbb{E}|(L_{n}-\pi)(X_{j,l})|+\mathbb{E}|L_{n}(C_{1})-1|.$

The first of these two terms is controlled using (30) : we have

$\sum_{l=1}^{m(j)}\mathbb{E}|(L_{n}-\pi)(X_{j,l})|\leq\frac{1}{\sqrt{n}}\frac{2\sqrt{2C}}{\sqrt{1-\lambda}}\|\frac{d\nu}{d\pi}\|_{r}^{1/2}m(j)^{1/2+1/2r}$

And on the other hand,

$\displaystyle\mathbb{E}|L_{n}(C_{1})-1|$
$\displaystyle\leq\mathbb{E}|(L_{n}-\pi)(C_{1})|+\pi(C_{1}^{c})$
$\displaystyle\leq\frac{1}{\sqrt{n}}\frac{2\sqrt{2C}}{\sqrt{1-\lambda}}\|\frac{d\nu}{d\pi}\|_{r}^{1/2}+\pi(C_{1}^{c}).$

Here we have used (30) again.

We skip over details here as they are similar to those in previous proofs.
Choosing an appropriate value for $k$ and using the estimates above allows us
to recover the following :

(37)
$\displaystyle\mathbb{E}\left[m_{1}^{1/p}W_{p}(1/m_{1}L_{n}^{1},\pi)\right]\lesssim$
$\displaystyle\left(\frac{C}{(1-\lambda)n}\|\frac{d\nu}{d\pi}\|_{r}\right)^{1/2p}\int_{t}^{d_{1}/4}N(C_{1},\delta)^{1/2p(1+1/r)}d\delta$
$\displaystyle+\pi(C_{1}^{c})+t.$

The term $\pi(C_{1}^{c})$ is bounded by the Chebyshev inequality :

$\pi(C_{1}^{c})\leq\int x^{\zeta}d\pi/d_{1}^{\zeta}=\int
x^{\zeta}d\pi\left(\int x^{p}d\pi\right)^{-\zeta/p}\rho^{-\zeta}.$

_Step 4._

Use (36) and (37), along with assumption (31) : this yields

$\mathbb{E}(W_{p}(L_{n},\pi))\lesssim
K(\zeta)\left(\rho^{-\zeta}+t+A_{n}\rho^{\alpha/2p(1+1/r)}t^{1-\alpha/2p(1+1/r)}\right)$

where
$A_{n}=\left(\frac{C}{(1-\lambda)n}\|\frac{d\nu}{d\pi}\|_{r}\right)^{1/2p}$,
and

$K(\zeta)=\frac{m_{\zeta}}{m_{p}^{\zeta/p}}\vee\frac{m_{\zeta+2p}}{m_{p}^{1+\zeta/p}}\vee
k_{E}^{1/2p(1+1/r)}\frac{2p}{\alpha(1+1/r)}m_{p}^{\alpha/(2p^{2})(1+1/r)}.$

The remaining step is optimization in $t$ and $\rho$. We obtain the following
result : there exists a constant $C(p,r,\zeta)$ depending only on the values
of $p$, $r$, $\zeta)$, such that

$\mathbb{E}(W_{p}(L_{n},\pi))\lesssim
C(p,r,\zeta)K(\zeta)A_{n}^{2p/(\alpha(1+1/r)(1+1/\zeta))}.$

There is a caveat : we have used the condition $\rho\geq 2$ at some point, and
with this restriction the optimization above is valid only when $A_{n}\leq
C^{\prime}(p,r,\zeta)$, where the constant $C^{\prime}(p,r,\zeta)$ only
depends on the values of $p$, $r$, $\zeta$.

∎

##  Appendix A Transportation inequalities for Gaussian measures on a Banach
space

Transportation inequalities, also called transportation-entropy inequalities,
have been introduced by K. Marton [23] to study the phenomenon of
concentration of measure. M. Talagrand showed that the finite-dimensional
Gaussian measures satisfy a $\mathbf{T}_{2}$ inequality. The following
appendix contains a simple extension of this result to the infinite-
dimensional case. For much more on the topic of transportation inequalities,
the reader may refer to the survey [14] by N. Gozlan and C. Léonard.

For $\mu\in\mathcal{P}(E)$, let $H(.|\mu)$ denote the relative entropy with
respect to $\mu$ :

$H(\nu|\mu)=\int_{E}\frac{d\nu}{d\mu}\log\frac{d\nu}{d\mu}d\mu$

if $\nu\ll\mu$, and $H(\nu|\mu)=+\infty$ otherwise.

We say that $\mu\in\mathcal{P}_{p}(E)$ satisfies a $\mathbf{T}_{p}(C)$
transportation inequality when

$W_{p}(\nu,\mu)\leq\sqrt{CH(\nu|\mu)}\quad\forall\nu\in\mathcal{P}_{p}(E)$

We identify what kind of transport inequality is satisfied by a Gaussian
measure on a Banach space. We remind the reader of the following definition :
let $(E,\mu)$ be a Gaussian Banach space and $X\sim\mu$ be a $E$-valued r.v..
The weak variance of $\mu$ or $X$ is defined by

$\sigma^{2}=\sup_{f\in E^{*},|f|\leq 1}\mathbb{E}(f^{2}(X)).$

The lemma below is optimal, as shown by the finite-dimensional case.

######  Lemma A.1.

Let $(E,\mu)$ be a Gaussian Banach space, and let $\sigma^{2}$ denote the weak
variance of $\mu$. Then $\mu$ satisfies a $\mathbf{T}_{2}(2\sigma^{2})$
inequality.

###### Proof.

According e.g. to [20], there exists a sequence $(x_{i})_{i\geq 1}$ in $E$ and
an orthogaussian sequence $(g_{i})_{i\geq 1}$ (meaning a sequence of i.i.d.
standard normal variables) such that

$\sum_{i\geq 1}g_{i}x_{i}\sim\mu,$

where convergence of the series holds a.s. and in all the $L^{p}$’s. In
particular, the laws $\mu_{n}$ of the partial sums $\sum_{i=1}^{n}g_{i}x_{i}$
converge weakly to $\mu$.

As a consequence of the stability result of Djellout-Guillin-Wu (Lemma 2.2 in
[9]) showing that $\mathbf{T}_{2}$ is stable under weak convergence, it thus
suffices to show that the measures $\mu_{n}$ all satisfy the
$\mathbf{T}_{2}(2\sigma^{2})$ inequality.

First, by definition of $\sigma$, we have

$\sigma^{2}=\sup_{f\in E^{*},|f|\leq
1}\mathbb{E}(\sum_{i=1}^{+\infty}f(x_{i})g_{i})^{2}$

and since $(g_{i})$ is an orthogaussian sequence, the sum is equal to
$\sum_{i=1}^{+\infty}f^{2}(x_{i})$.

Consider the mapping

$\displaystyle T:$ $\displaystyle(\mathbb{R}^{n},N)\rightarrow(E,\|.\|)$
$\displaystyle(a_{1},\ldots,a_{n})\mapsto\sum_{i=1}^{n}a_{i}x_{i}.$

(here $\mathbb{R}^{n}$ is equipped with the Euclidean norm $N$). With the
remark above it is easy to check that $\|T(a)\|\leq\sigma N(a)$ for
$a\in\mathbb{R}^{n}$. Consequently, $T$ is $\sigma$-Lipschitz, and we can use
the second stability result of Djellout-Guillin-Wu (Lemma 2.1 in [9]) : the
push forward of a measure satisfying $\mathbf{T}_{2}(C)$ by a $L$-Lipschitz
function satisfies $\mathbf{T}_{2}(L^{2}C)$. As is well-known, the standard
Gaussian measure $\gamma^{n}$ on $\mathbb{R}^{n}$ satisfies
$\mathbf{T}_{2}(2)$ and thus $T_{\\#}\gamma^{n}$ satisfies
$\mathbf{T}_{2}(2\sigma^{2})$. But it is readily checked that
$T_{\\#}\gamma^{n}=\mu_{n}$, which concludes this proof.

∎

######  Remark.

M.Ledoux indicated to us another way to obtain this result. First, one shows
that the Gaussian measure satisfies a $\mathbf{T}_{2}(2)$ inequality when
considering the cost function $c=d_{H}^{2}$, where $d_{H}$ denotes the
Cameron-Martin metric on $E$ inherited from the scalar product on the Cameron-
Martin space. This can be done in a number of ways, for example by
tensorization of the finite-dimensional $\mathbf{T}_{2}$ inequality for
Gaussian measures or by adapting the Hamilton-Jacobi arguments of Bobkov-
Gentil-Ledoux [3] in the infinite-dimensional setting. It then suffices to
observe that this transport inequality implies the one we are looking for
since we have the bound $d\leq\sigma d_{H}$ (here $d$ denotes the metric
inherited from the norm of the Banach space).

Let $L_{n}$ denote the empirical measure associated with $\mu$. As a
consequence of Lemma A.1, we can give an inequality for the concentration of
$W_{2}(L_{n},\mu)$ around its mean, using results from transportation
inequalities. This is acutally a simple case of more general results of N.
Gozlan and C. Léonard ([13], [14]), we reproduce a proof here for convenience.

######  Corollary A.2.

Let $\mu$ be as above. The following holds :

$\mathbb{P}(W_{2}(L_{n},\mu)\geq\mathbb{E}[W_{2}(L_{n},\mu)]+t)\leq
e^{-nt^{2}/(2\sigma^{2})}.$

###### Proof.

The proof relies on the property of dimension-free tensorization of the
$\mathbf{T}_{2}$ inequality, see [14]. Since $\mu$ satisfies
$\mathbf{T}_{2}(2\sigma^{2})$, the product measure $\mu^{\otimes n}$ on the
product space $E^{n}$ endowed with the $l_{2}$ metric

$d_{2}((x_{1},\ldots,x_{n}),(y_{1},\ldots,y_{n}))=\sqrt{|x_{1}-y_{1}|^{2}+\ldots+|x_{n}-y_{n}|^{2}}$

also satisfies a $\mathbf{T}_{2}(2\sigma^{2})$ inequality ([14], Corollary
4.4). Therefore, it also satisfies a $\mathbf{T}_{1}$ inequality by Jensen’s
inequality, and this implies that we have the concentration inequality

$\mu^{\otimes n}(f\geq\int fd\mu^{\otimes n}+t)\leq e^{-t^{2}/(2\sigma^{2})}$

for all $1$-Lipschitz functions $f:(E^{n},d_{2})\rightarrow\mathbb{R}$ ([14],
Theorem 1.7). For $x=(x_{1},\ldots,x_{n})\in E^{n}$, denote
$L_{n}^{x}=1/n\sum_{i=1}^{n}\delta x_{i}$. To conclude it suffices to notice
that $(x_{1},\ldots,x_{n})\rightarrow W_{2}(L_{n}^{x},\mu)$ is
$\sqrt{n}$-Lipschitz from $(E^{n},d_{2})$ to $\mathbb{R}$. ∎

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