# Universal diagonal estimates for minimizers of the Levy-Lieb functional

S. Di Marino, A. Gerolin, L. Nenna  Simone Di Marino, Università degli studi
di Genova (DIMA), MaLGA, Genova, Italy<EMAIL_ADDRESS>Augusto
Gerolin, Department of Mathematics and Statistics $\&$ Department of Chemistry
and Biomolecular Sciences, University of Ottawa, Ottawa, Canada
<EMAIL_ADDRESS>Luca Nenna, Université Paris-Saclay, CNRS, Laboratoire de
mathématiques d’Orsay, 91405, Orsay, France.  luca.nenna@universite-paris-
saclay.fr

###### Abstract.

Given a wave-function minimizing the Levy-Lieb functional, the intent of this
short note is to give an estimate of the probability of the particles being in
positions $(x_{1},\ldots,x_{N})$ in the $\delta$-close regime
$D_{\delta}=\cup_{i\neq j}\\{|x_{i}-x_{j}|\leq\delta\\}$.

###### Contents

  1. 1 Introduction
  2. 2 Notation
  3. 3 Estimate for the kinetic energy
  4. 4 New trial state: swapping particles and estimate for the potential
  5. 5 Diagonal estimates for the wave-function

##  1\. Introduction

Density Functional Theory attempts to describe all the relevant information
about a many-body quantum system at ground state in terms of the one electron
density $\rho$. Following the Levy and Lieb’s approach [28, 31] the ground
state energy can be rephrased as the following variational principle involving
only the electron density

$\mathcal{E}_{0}[v]=\inf_{\begin{subarray}{c}\sqrt{\rho}\in
H^{1}(\mathbb{R}^{3})\\\ \rho(\mathbb{R}^{3})=N\\\
\int_{\mathbb{R}^{3}}v(x)\,d\rho<+\infty\end{subarray}}\left\\{\mathcal{F}_{LL,\varepsilon}[\rho]+\int_{\mathbb{R}^{3}}v(x)\,d\rho\;\right\\},$

where $v$ is an external potential and the Levy-Lieb functional
$\mathcal{F}_{LL,\varepsilon}$ is defined as

$\mathcal{F}_{LL,\varepsilon}[\rho]:=\min_{\begin{subarray}{c}\psi\in\mathcal{H}^{N}\\\
\psi\mapsto\rho\end{subarray}}\left\\{\int_{\mathbb{R}^{3N}}\varepsilon|\nabla\psi|^{2}(x)+v_{ee}(x)|\psi|^{2}(x)\,dx\right\\},$
(1)

where $\mathcal{H}^{N}=\bigwedge_{i=1}^{N}H^{1}(\mathbb{R}^{3};\mathbb{C})$ is
the fermionic space,
$v_{ee}(x_{1},\ldots,x_{N})=\sum_{i<j}\frac{1}{|x_{i}-x_{j}|}$ is the Coulomb
interaction potential between the $N$ electrons and $\psi\mapsto\rho$ means
that the one-body density of $\psi$ is $\rho$, that is
$\rho=N\int_{\mathbb{R}^{3(N-1)}}|\psi|^{2}$.

The Levy-Lieb functional is indeed the lowest possible (kinetic plus
interaction) energy of a quantum system having the prescribed density $\rho$.
This universal functional is the central object of Density Functional Theory,
since knowing it would allow one to compute the ground state energy of a
system with any external potential $v$. For a complete review on it we refer
the reader to [30].

Although the electrons are fermions, in this article we will treat only the
case of bosonic wave-functions, i.e. consider in the constraint search in (1)
wave-functions $\psi$ that are symmetric rather than anti-symmetric. Notice
that the ground-state energy of bosonic systems are generally higher than
fermionic ones. In our analysis, however, the bosonic case is not very
restrictive since we are looking at the regime $\varepsilon$ small.

Our approach interprets the Levy-Lieb functional as a (Fisher-information
regularized) multi-marginal optimal transport problem.

Connection with Optimal Transportation Theory: It has been recently shown that
the limit functional as $\varepsilon\to 0$ corresponds to a multi-marginal
optimal transport problem [2, 12, 13, 29] (see also the seminal works in the
physics and chemistry literature [5, 37, 39, 38, 36]): rather than wave-
functions, one has now enlarged the constrained search in (1) to minimize
among probability measures on $\mathbb{R}^{3N}$ having $\rho$ as marginal,
that is

$\mathcal{F}_{0}[\rho]:=\inf_{\mathbb{P}\in\Pi_{N}(\rho)}\left\\{\int_{\mathbb{R}^{3N}}v_{ee}(x_{1},\ldots,x_{N})\,d\mathbb{P}(x_{1},\ldots,x_{N})\right\\},$
(2)

where $\Pi_{N}(\rho)$ denotes the set of probability measures on
$\mathbb{R}^{3N}$ having $\rho/N$ as marginals.

The multi-marginal optimal transport with Coulomb cost (2) has garnered
attention in the mathematics, physics and chemistry communities and the
literature on the subject is growing considerably. Recent developments include
results on the existence and non-existence of Monge-type solutions minimizing
(2) (e.g., [8, 7, 12, 20, 5, 11, 18, 3]), structural properties of Kantorovich
potentials (e.g., [9, 17, 24, 4]), grand-canonical optimal transport [19],
efficient computational algorithms (e.g., [1, 22, 14, 33, 25]) and the design
of new density functionals (e.g., [23, 6, 34, 27]). The first order expansion
around the limit $\varepsilon\to 0$ of the Levy-Lieb functional was obtained
in [10].

We refer to the surveys (and references therein) [17, 21] for a self-contained
presentation on multi-marginal optimal transport approach in Density
Functional Theory as well as the review article [40] for a the recent
developments from a chemistry standpoint.

Main result of this paper: In [16, 4, 9] it is shown that the support ${\rm
supp}(\mathbb{P}^{*})$ of a solution $\mathbb{P}^{*}$ of the limiting problem
(2) is uniformly bounded away of the diagonal, i.e. one has always
$|x_{i}-x_{j}|\geq\delta>0$ for any $x_{i},x_{j}\in{\rm
supp}(\mathbb{P}^{*})$. In other words, the electrons are always at a certain
distance away from each other, which is the expected behaviour since we are in
a _classical_ framework.

In the sequel we will denote with $D_{\delta}$ the enlarged diagonal

$D_{\delta}=\\{(x_{1},\ldots,x_{N})\in\mathbb{R}^{3N}\;:\;\exists~{}i\neq
j\text{ s.t. }|x_{i}-x_{j}|\leq\delta\\}.$

In particular the result in [4, 9] can be rephrased saying that the solution
to the multi-marginal optimal transport problem is concentrated on the
complement of $D_{\delta}$ for some $\delta$. An important feature of the
results is that $\delta$ depends only on concentration properties of $\rho$.
In fact defining

$\kappa(\rho,r):=\sup_{x\in\mathbb{R}^{3}}\rho(B(x,r))/N,$

the authors in [9] prove that if $\kappa(\rho,\beta)<\frac{1}{2(N-1)}$ then
one can choose $\delta=\frac{\beta}{2N}$. Our main result is to extend this
property also for $\varepsilon>0$ small. In particular we do not expect to
have $\psi_{\varepsilon}=0$ on $D_{\delta}$ but we show that the probability
of having the electrons in position $x\in D_{\delta}$ is very small (3).

######  Theorem 1.1 (Exponential off-diagonal localization for Coulomb).

Let $\psi_{\varepsilon}$ be a minimizer for (1) where
$v_{ee}(x_{1},\ldots,x_{N})=\sum_{i<j}\frac{1}{|x_{i}-x_{j}|}$. Let us
consider $\beta$ such that $\kappa(\rho,\beta)\leq\frac{1}{4(N-1)}$ then, let
$\alpha\leq\frac{\beta}{32N}$, and suppose $\varepsilon N^{2}\ll\alpha/16$.
Then, for $\mathbb{P}_{\varepsilon}(x)=|\psi_{\varepsilon}|^{2}(x)$ we have

$\boxed{\int_{D_{\alpha}}\mathbb{P}_{\varepsilon}(x)\,dx\leq
e^{-\frac{1}{24}\sqrt{\frac{\alpha}{\varepsilon}}}.}$ (3)

In the proof we actually work with a general repulsive pairwise potential
$v_{ee}$, which satisfy the hypothesis (5), stated in the next section. The
result in general is the following one:

######  Theorem 1.2 (Exponential off-diagonal localization for general
interaction cost).

Let $\psi_{\varepsilon}$ be a minimizer for (1) where $v_{ee}$ satisfies (5).
Let us consider $\beta$ such that $\kappa(\rho,\beta)\leq\frac{1}{4(N-1)}$
then, let $\alpha$ such that $m(2\alpha)\leq 8(N-1)M(\beta/2)$, and suppose
$\varepsilon N^{2}\ll\alpha^{2}m(2\alpha)$. Then, for
$\mathbb{P}_{\varepsilon}(x)=|\psi_{\varepsilon}|^{2}(x)$ we have

$\boxed{\int_{D_{\alpha}}\mathbb{P}_{\varepsilon}(x)\,dx\leq
e^{-\frac{1}{6}\sqrt{\frac{\alpha^{2}m(2\alpha)}{8\varepsilon}}}.}$ (4)

Notice that in [9] the diagonal estimate is proven also in the weaker (and
sharper) hypotesis $\kappa(\rho,\beta)<\frac{1}{N}$: while we believe that
also in that case a similar generalization in the case $\varepsilon>0$ holds
true, the proof will be more technical and not so trasparent. For the same
reason the inequality $\kappa(\rho,\beta)\leq\frac{1}{4(N-1)}$ is used instead
of $\kappa(\rho,\beta)<\frac{1}{2(N-1)}$ in order to have more transparent
estimates in the end.

######  Remark 1.1 (Natural assumption on $\varepsilon$).

We want here to justify the hypotesis $\varepsilon N^{2}\ll\alpha/16$. Notice
that we expect $|\psi_{\varepsilon}|=0$ on the diagonal since $v_{ee}=+\infty$
there; if $|\psi_{\varepsilon}(x)|\sim 1$ on a point in $D_{\alpha}$ then we
then expect $|\nabla\psi_{\varepsilon}|\sim\alpha^{-1}$ and so the kinetic
energy locally is $\frac{\varepsilon}{\alpha^{2}}$ while the potential energy
is $\frac{1}{\alpha}$. Since the localization phenomenon outside the enlarged
diagonal is a classical one, we expect that it holds in the regime where the
kinetic energy is negligible with respect to the potential energy, and that is
precisely when $\varepsilon\ll\alpha$. Then there is a factor $N$ which is
constant, and we believe it is actually technical.

Organization of the paper: In Section 2 we introduce the notations we are
going to use throughout all the paper. In Section 3 we give some estimates
concerning kinetic energy term in the Levy-Lieb functional. Section 4 is then
devoted to the construction of a competitor for the Levy-Lieb functional;
finally in Section 5 we derive the diagonal estimates for the wave-function
and, thus, prove Theorem 1.1 and Theorem 1.2 via the iteration of a decay
estimate.

##  2\. Notation

Consider a subset $I\subseteq\\{1,\ldots,N\\}$, with cardinality $k=|I|$,
defined as $I=\\{i_{1},\ldots,i_{k}\\}$, with $1\leq
i_{1}<i_{2}<\cdots<i_{k}$. Then, the $I$-projection

$\pi_{I}:\mathbb{R}^{3N}\to\mathbb{R}^{3k},\qquad\pi_{I}((x_{1},\ldots,x_{N}))=(x_{i_{1}},x_{i_{2}},\ldots,x_{i_{k}}).$

Sometimes we will denote $x_{I}=\pi_{I}(x)$ and if $I=J^{c}$, then
$x=(x_{I},x_{J})$. With a slight abuse of notation, for a function
$\mathbb{P}\in L^{1}(\mathbb{R}^{3N})$, $I\subseteq\\{1,\ldots,N\\}$ and
$J=I^{c}$ we denote

$(\pi_{I})_{\sharp}(\mathbb{P})(x_{I})=\int\mathbb{P}(x_{I},x_{J})\,dx_{J},$

which on density of measures act precisely as the push-forward through the
projection function $\pi_{I}$.
As we have already mentioned above, we denote by $\Pi_{N}(\rho)$ the set of
probability measures on $\mathbb{R}^{3N}$ having the $N$ one body marginals
equal to $\rho$.
In the following we will consider an electron-electron pair interaction
repulsion potential, that $v_{ee}$ with the following form:

$\begin{split}v_{ee}(x_{1},\ldots,x_{N})=\sum_{i<j}c(x_{i},x_{j}),\quad\text{
where }m(|x-y|)\leq c(x,y)\leq M(|x-y|)\qquad\forall x,y\in\mathbb{R}^{3}\\\
\text{for some }m,M:(0,\infty)\to[0,\infty)\text{ decreasing such that
}\lim_{t\to 0^{+}}m(t)=+\infty\\\ \end{split}$ (5)

Moreover, with a slight abuse of notation, we will denote by

$\mathbb{P}\in\mathcal{P}(\mathbb{R}^{3N})\mapsto
v_{ee}(\mathbb{P}):=\int_{\mathbb{R}^{3N}}v_{ee}(x_{1},\ldots,x_{N})\,d\mathbb{P}(x_{1},\ldots,x_{N})$
(6)

Notice that we will often identify a measure $\mathbb{P}$ with its density.

Finally, given an open set $\Omega\subseteq\mathbb{R}^{3N}$, then for every
$r>0$ we denote

$\Omega_{-r}:=\\{x\in\mathbb{R}^{3N}\;:\;B(x,r)\subseteq\Omega\\}.$ (7)

##  3\. Estimate for the kinetic energy

In this section we give some preliminary estimates for the kinetic energy term
of the Levy-Lieb functional.
Denoting $L_{+}^{1}$ the cone of positive $L^{1}$ functions, we define
$\mathcal{E}_{\rm kin}:L_{+}^{1}(\mathbb{R}^{3N})\to\mathbb{R}$ the Kinetic
energy associated to some absolutely continuos $N$-probability measure $h$

$\mathcal{E}_{\rm
kin}(h):=\begin{cases}\int_{\mathbb{R}^{3N}}\frac{\sum_{i=1}^{N}|\nabla_{i}h(x_{1},\ldots,x_{N})|^{2}}{h(x_{1},\ldots,x_{N})}\,dx_{1},\ldots,dx_{N}\quad&\text{
if }\sqrt{h}\in H^{1}(\mathbb{R}^{3N})\\\ +\infty&\text{
otherwise}\end{cases}$ (8)

When it will be clear from the context we will also abbreviate
$\mathcal{E}_{\rm kin}(h)=\int\frac{|\nabla h|^{2}}{h}\,dx$. Notice that the
kinetic energy functional is also know as the Fisher information. Moreover if
$\psi\in H^{1}(\mathbb{R}^{3N};\mathbb{R})$, then

$\int|\nabla\psi|^{2}\,dx=\mathcal{E}_{\rm kin}(|\psi|^{2})=\mathcal{E}_{\rm
kin}(\mathbb{P}_{\psi}),$

where $\mathbb{P}_{\psi}=|\psi|^{2}$ is the joint probability associated to
the wave-function $\psi$. The string of equalities above is thus true when
$\psi$ is a minimizer for the bosonic case. The following Lemma summarises
some results concerning the homogeneity, sub-additivity (which is a
consequence of theorem 7.8 in [32]) and the decomposability under projection
of the kinetic energy (a similar result also appears in [26, 35]).

######  Lemma 3.1.

Let $\mathcal{E}_{\rm kin}$ defined as in (8). Then

  * (i)

$\mathcal{E}_{\rm kin}$ is $1$-homogeneous, that is $\mathcal{E}_{\rm
kin}(\lambda\mathbb{P})=\lambda\mathcal{E}_{\rm kin}(\mathbb{P})$ for every
$\lambda>0$;

  * (ii)

given $\mathbb{P}_{1},\ldots,\mathbb{P}_{k}\in L^{1}(\mathbb{R}^{3N})$, we
have

$\mathcal{E}_{\rm
kin}(\mathbb{P}_{1}+\ldots+\mathbb{P}_{k})\leq\mathcal{E}_{\rm
kin}(\mathbb{P}_{1})+\mathcal{E}_{\rm
kin}(\mathbb{P}_{2})+\ldots+\mathcal{E}_{\rm kin}(\mathbb{P}_{k});$

  * (iii)

Let $\mathbb{P}\in L_{+}^{1}(\mathbb{R}^{3N})$. Given
$I,J\subseteq\\{1,\ldots,N\\}$ two nonempty disjoint sets such that $I=J^{c}$,
we denote by $\mathbb{P}_{I}=(\pi_{I})_{\sharp}\mathbb{P}$ and
$\mathbb{P}_{J}=(\pi_{J})_{\sharp}\mathbb{P}$. Then we have (here
$N_{I}=\sharp I$ and $N_{J}=\sharp J$)

$\mathcal{E}_{\rm kin}^{N}(\mathbb{P})\geq\mathcal{E}_{\rm
kin}^{N_{I}}(\mathbb{P}_{I})+\mathcal{E}_{\rm kin}^{N_{J}}(\mathbb{P}_{J}),$

where the equality holds if and only if
$\mathbb{P}(x)=\mathbb{P}_{I}(x_{I})\mathbb{P}_{J}(x_{J})/\lambda$, where
$\lambda=\int\mathbb{P}$. In particular if $\mathbb{P}$ is the density of a
probability measure, we have that the equality happens if and only if $x_{I}$
and $x_{J}$ are independent under the probability $\mathbb{P}$.

###### Proof.

$(\rm{i})$ The $1$-homogeneity is obvious.
$(\rm{ii})$ For the subadditivity it is sufficient to prove it for $k=2$; then
for every $x$, by Cauchy-Schwarz inequality we have

$(\mathbb{P}_{1}(x)+\mathbb{P}_{2}(x))\left(\frac{|\nabla\mathbb{P}_{1}(x)|^{2}}{\mathbb{P}_{1}(x)}+\frac{|\nabla\mathbb{P}_{2}(x)|^{2}}{\mathbb{P}_{2}(x)}\right)\geq(|\nabla\mathbb{P}_{1}(x)|+|\nabla\mathbb{P}_{2}(x)|)^{2},$

which, after using the triangular inequality and dividing by
$\mathbb{P}_{1}+\mathbb{P}_{2}$ can be rewritten as

$\frac{|\nabla(\mathbb{P}_{1}+\mathbb{P}_{2})|^{2}}{\mathbb{P}_{1}+\mathbb{P}_{2}}\leq\frac{|\nabla\mathbb{P}_{1}|^{2}}{\mathbb{P}_{1}}+\frac{|\nabla\mathbb{P}_{2}|^{2}}{\mathbb{P}_{2}},$

which integrated gives us the conclusion.

$(\rm{iii})$ As for the last point we fix $x_{J}$ and we use the Cauchy-
Schwarz inequality with respect to the measure $dx_{I}$:

$\begin{split}\left(\int\mathbb{P}(x_{I},x_{J})dx_{I}\right)\cdot\left(\int\frac{|\nabla_{J}\mathbb{P}(x_{I},x_{J})|^{2}}{\mathbb{P}(x_{I},x_{J})}dx_{I}\right)&\geq\left(\int|\nabla_{J}\mathbb{P}(x_{I},x_{J})|dx_{I}\right)^{2}\\\
&\geq\left|\nabla_{J}\Bigl{(}\int\mathbb{P}(x_{I},x_{J})dx_{I}\Bigr{)}\right|^{2},\end{split}$

where in the last passage we used the triangular inequality and we took the
derivative out of the integral. Now we recognize
$\mathbb{P}_{J}(x_{J})=\int\mathbb{P}(x_{I},x_{J})dx_{I}$ and so we can write
this as

$\int\frac{|\nabla_{J}\mathbb{P}(x_{I},x_{J})|^{2}}{\mathbb{P}(x_{I},x_{J})}dx_{I}\geq\frac{|\nabla_{J}\mathbb{P}_{J}(x_{J})|^{2}}{\mathbb{P}_{J}(x_{J})}.$

Integrating this with respect to $dx_{J}$ and doing a similar computation for
$x_{I}$, we obtain the conclusion, that is

$\iint\frac{|\nabla\mathbb{P}(x_{I},x_{J})|^{2}}{\mathbb{P}(x_{I},x_{J})}dx_{I}dx_{J}\geq\int\frac{|\nabla_{J}\mathbb{P}_{J}(x_{J})|^{2}}{\mathbb{P}_{J}(x_{J})}dx_{J}+\int\frac{|\nabla_{I}\mathbb{P}_{I}(x_{I})|^{2}}{\mathbb{P}_{I}(x_{I})}dx_{I}.$

From the equality cases in C-S and triangular inequality combined we get
$\nabla_{J}\mathbb{P}(x_{I},x_{J})=v(x_{J})\mathbb{P}(x_{I},x_{J})$ for some
vector field $v$; by a simple integration we actually get
$v=\nabla(\mathbb{P}_{J})/\mathbb{P}_{J}$; this can be seen as
$\nabla_{J}\log(\mathbb{P})=\nabla_{J}\log\mathbb{P}_{J}$; similarly we can
get $\nabla_{I}\log(\mathbb{P})=\nabla_{I}\log\mathbb{P}_{I}$. Summing up this
two equalities we get
$\nabla(\mathbb{P}(x)/\mathbb{P}_{I}(x_{I})\mathbb{P}_{J}(x_{J}))=0$.

∎

The following lemma is a straightforward adaptation of Theorem 3.2 in [15]
giving the IMS localization formula; we have added a short proof for sake of
completeness.

######  Lemma 3.2.

Let $\eta_{1},\eta_{2},\eta_{3}:\mathbb{R}^{3N}\to[0,1]$ be $C^{1}$ functions
such that $\eta_{1}+\eta_{2}+\eta_{3}\equiv 1$. Then, for every function
$\mathbb{P}\in L_{+}^{1}(\mathbb{R}^{3N})$ we have

$\mathcal{E}_{\rm kin}(\mathbb{P}\eta_{1})+\mathcal{E}_{\rm
kin}(\mathbb{P}\eta_{2})+\mathcal{E}_{\rm
kin}(\mathbb{P}\eta_{3})=\mathcal{E}_{\rm
kin}(\mathbb{P})+\int\left(\frac{|\nabla\eta_{1}|^{2}}{\eta_{1}}+\frac{|\nabla\eta_{2}|^{2}}{\eta_{2}}+\frac{|\nabla\eta_{3}|^{2}}{\eta_{3}}\right)\mathbb{P}\,dx.$

###### Proof.

For every $i=1,2,3$ pointwisely we have:

$\displaystyle\frac{|\nabla(\mathbb{P}\eta_{i})|^{2}}{\mathbb{P}\eta_{i}}$
$\displaystyle=\frac{|\eta_{i}\nabla\mathbb{P}+\mathbb{P}\nabla\eta_{i}|^{2}}{\mathbb{P}\eta_{i}}=\frac{\eta_{i}^{2}|\nabla\mathbb{P}|^{2}+2\eta_{i}\mathbb{P}\nabla\mathbb{P}\cdot\nabla\eta_{i}+\mathbb{P}^{2}|\nabla\eta_{i}|^{2}}{\mathbb{P}\eta_{i}}$
$\displaystyle=\eta_{i}\frac{|\nabla\mathbb{P}|^{2}}{\mathbb{P}}+2\nabla\mathbb{P}\cdot\nabla\eta_{i}+\mathbb{P}\frac{|\nabla\eta_{i}|^{2}}{\eta_{i}}$

Adding them up and using that $\sum\eta_{i}=1$ and $\sum\nabla\eta_{i}=0$, we
get

$\sum_{i}\frac{|\nabla(\mathbb{P}\eta_{i})|^{2}}{\mathbb{P}\eta_{i}}=\frac{|\nabla\mathbb{P}|^{2}}{\mathbb{P}}+\mathbb{P}\sum_{i}\frac{|\nabla\eta_{i}|^{2}}{\eta_{i}},$

which integrated, gives us the desired identity.

∎

##  4\. New trial state: swapping particles and estimate for the potential

The scope of this subsection is to create a competitor for the minimization of
the functional

$\mathcal{F}_{LL,\varepsilon}(\mathbb{P})=\begin{cases}\varepsilon\mathcal{E}_{\rm
kin}(\mathbb{P})+v_{ee}(\mathbb{P})\quad&\text{ if
}\mathbb{P}\in\Pi_{N}(\rho)\\\ +\infty&\text{ otherwise,},\end{cases}$ (9)

where $\mathcal{E}_{\rm kin}$ is defined in (8) and $v_{ee}$ satisfies (5).
The idea is to try to mimic what it is done in [16, 4, 9], in the
semiclassical case $\varepsilon=0$: in that case we take two points
$y,z\in\mathbb{R}^{3N}$ and substitute them with $\tilde{y},\tilde{z}$ where
we have interchanged their first compenent, that is
$\tilde{y}=(z_{1},y_{2},\ldots,y_{n})$ and
$\tilde{z}=(y_{1},z_{2},\ldots,z_{n})$.

In order to do so for the $n$-particle distribution $\mathbb{P}$, we will
consider two small bumps centered around $y$ and $z$

$\eta_{1}(x)=\lambda_{1}\eta\Bigl{(}\frac{x-y}{r_{1}}\Bigr{)}\qquad\text{ and
}\qquad\eta_{2}(x)=\lambda_{2}\eta\Bigl{(}\frac{x-z}{r_{2}}\Bigr{)},$ (10)

for some $\lambda_{1},\lambda_{2},r_{1},r_{2}$ to be chosen later and some
$\eta\in C^{1}_{c}(B(0,1))$, $\eta\geq 0$. First of all we assume that ${\rm
supp}(\eta_{1})\cap{\rm supp}(\eta_{2})=\emptyset$, which can be granted as
long as

$r_{1}+r_{2}<|y-z|,$ (11)

and then we assume $\int\eta_{1}\mathbb{P}=\int\eta_{2}\mathbb{P}=m$ which can
be accomplished again by choosing the appropriate $\lambda_{i},r_{i}$. Let us
then define

$\rho_{1}^{i}(x_{1})=(\pi_{\\{1\\}})_{\sharp}(\eta_{i}\mathbb{P}),\qquad\rho_{\hat{1}}^{i}(x_{2},x_{3},\ldots,x_{N})=(\pi_{\\{1\\}^{c}})_{\sharp}(\eta_{i}\mathbb{P}),$
(12)
$\mathbb{P}_{1}=\frac{1}{m}\rho_{1}^{2}\rho_{\hat{1}}^{1},\qquad\mathbb{P}_{2}=\frac{1}{m}\rho_{1}^{1}\rho_{\hat{1}}^{2},$
(13)

where $\rho_{1}^{i}$ and $\rho_{\hat{1}}^{i}$ are the marginals of
$\eta_{i}\mathbb{P}$ and $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ are densities
concentrated around $\tilde{y}=(z_{1},y_{2},\ldots,y_{n})$ and
$\tilde{z}=(y_{1},z_{2},\ldots,z_{n})$ respectively. We then finally consider

$\bar{\mathbb{P}}:=\mathbb{P}-\mathbb{P}\eta_{1}-\mathbb{P}\eta_{2}+\mathbb{P}_{1}+\mathbb{P}_{2},$
(14)

which will be the competitor for a minimizer $\mathbb{P}$ of the functional
$\mathcal{F}_{LL,\varepsilon}$.

######  Remark 4.1.

In order to have that $\bar{\mathbb{P}}$ is a competitor, we still have to
check that $\bar{\mathbb{P}}\geq 0$ and that it has the correct marginals. For
the positivity, notice that if $\lambda_{1}$ and $\lambda_{2}$ are small
enough we have also $\eta_{1}+\eta_{2}\leq 1$ and in particular
$\mathbb{P}-\eta_{1}\mathbb{P}_{1}-\eta_{2}\mathbb{P}_{2}$, which will
guarantee that $\bar{\mathbb{P}}\geq 0$.

For the marginal constraint, notice that by (12) and (13) we have that
$\eta_{1}\mathbb{P}+\eta_{2}\mathbb{P}$ and $\mathbb{P}_{1}+\mathbb{P}_{2}$
have the same marginals, in particular also $\mathbb{P}$ and
$\bar{\mathbb{P}}$ share the same marginals.

######  Lemma 4.1.

Let $\mathbb{P}$ be such $\mathcal{F}_{LL,\varepsilon}(\mathbb{P})<+\infty$.
Given $y,z\in\mathbb{R}^{3N}$, let
$\eta_{1},\eta_{2},\mathbb{P}_{1},\mathbb{P}_{2},\bar{\mathbb{P}}$ defined by
(10),(11), (12), (13) and (14). Then

$\mathcal{E}_{\rm kin}(\bar{\mathbb{P}})\leq\mathcal{E}_{\rm
kin}(\mathbb{P})+\int\mathbb{P}(x)\left(\frac{|\nabla\eta_{1}|^{2}}{\eta_{1}}+\frac{|\nabla\eta_{1}|^{2}}{\eta_{2}}+\frac{|\nabla\eta_{1}+\nabla\eta_{2}|^{2}}{1-\eta_{1}-\eta_{2}}\right)\,dx;$
$v_{ee}(\bar{\mathbb{P}})=v_{ee}(\mathbb{P})-\int\mathbb{P}(\eta_{1}+\eta_{2})\sum_{i>1}c(x_{1},x_{i})dx+\int(\mathbb{P}_{1}+\mathbb{P}_{2})\sum_{i>1}c(x_{1},x_{i})dx.$

###### Proof.

Let us consider $\eta_{3}=1-\eta_{1}-\eta_{2}$. Then we have
$\bar{\mathbb{P}}=\eta_{3}\mathbb{P}+\mathbb{P}_{1}+\mathbb{P}_{2}$. Using
Lemma 3.1, in particular the subadditivity and the exact energy split in case
of independent variables for $\mathcal{E}_{\rm kin}$, we get (by (13))

$\begin{split}\mathcal{E}_{\rm kin}(\bar{\mathbb{P}})&\leq\mathcal{E}_{\rm
kin}(\eta_{3}\mathbb{P})+\mathcal{E}_{\rm
kin}(\mathbb{P}_{1})+\mathcal{E}_{\rm kin}(\mathbb{P}_{2})\\\
&=\mathcal{E}_{\rm kin}(\eta_{3}\mathbb{P})+\mathcal{E}_{\rm
kin}(\rho_{1}^{2})+\mathcal{E}_{\rm kin}(\rho_{\hat{1}}^{1})+\mathcal{E}_{\rm
kin}(\rho_{1}^{1})+\mathcal{E}_{\rm kin}(\rho_{\hat{1}}^{2});\end{split}$ (15)

we then recall (12) and the inequality for the split energy (Lemma 3.1 (iii))
to get

$\mathcal{E}_{\rm kin}(\rho_{1}^{i})+\mathcal{E}_{\rm
kin}(\rho_{\hat{1}}^{i})\leq\mathcal{E}_{\rm kin}(\eta_{i}\mathbb{P})$ (16)

and so we conclude using (15), (16) and then Lemma 3.2.

For the estimate with the potential, it is clear that

$v_{ee}(\bar{\mathbb{P}})=v_{ee}(\mathbb{P})-\int\mathbb{P}(\eta_{1}+\eta_{2})v_{ee}(x)dx+\int(\mathbb{P}_{1}+\mathbb{P}_{2})v_{ee}(x)dx;$

Since $v_{ee}(x)=\sum_{i<j}c(x_{i},x_{j})$ we just need to show that the
contribution due to $c(x_{i},x_{j})$ whenever $1<i<j$ cancels out in the last
two integrals. In fact in both integrals we can integrate out the first
variable: denoting $I=\\{1\\}$ and $J=I^{c}$ for example we have

$\displaystyle\int\mathbb{P}\eta_{1}c(x_{i},x_{j})\,dx_{I}dx_{J}$
$\displaystyle=\int
c(x_{i},x_{j})\left(\int\mathbb{P}\eta_{1}\,dx_{I}\right)\,dx_{J}$
$\displaystyle=\int c(x_{i},x_{j})\rho_{\hat{1}}^{1}(x_{J})\,dx_{J}$
$\displaystyle=\int
c(x_{i},x_{j})\rho_{\hat{1}}^{1}(x_{J})\left(\int\frac{\rho_{2}^{1}(x_{I})}{m}\,dx_{I}\right)\,dx_{J}$
$\displaystyle=\int c(x_{i},x_{j})\mathbb{P}_{1}\,dx.$

In a similar way we can show that
$\int\mathbb{P}\eta_{2}c(x_{i},x_{j})\,dx=\int\mathbb{P}_{2}c(x_{i},x_{j})\,dx$.
∎

In the sequel we will denote $C_{1}(x)=\sum_{i=2}^{N}c(x_{1},x_{i})$

##  5\. Diagonal estimates for the wave-function

We devote this last section to derive the diagonal estimates for the bosonic
wave-function which minimizes the Levy-Lieb functional proving in particular
Theorem 1.1 and Theorem 1.2.

######  Lemma 5.1.

Let $\rho$ be a marginal distribution and let $\beta>0$ be such that
$\kappa(\rho,\beta)\leq\frac{1}{4(N-1)}$. Then, for every
$\mathbb{P}\in\Pi_{N}(\rho)$ and $y\in\mathbb{R}^{3N}$, for every
$r_{1},r_{2}$ such that $r_{1}+2r_{2}<\beta$ and $\delta>0$, there exists
$z\in\mathbb{R}^{3N}$ such that, defining
$\eta_{1},\eta_{2},\mathbb{P}_{1},\mathbb{P}_{2},m$ as in (10), (12) and (13)

  * (i)

$\int C_{1}(\mathbb{P}_{1}+\mathbb{P}_{2})dx\leq 2(N-1)M(\beta-
r_{1}-2r_{2})m$;

  * (ii)

$z$ is a $(1+\delta,1/2)$-doubling point at scale $r_{2}$ for $\mathbb{P}$,
that is

$\int_{B(z,r_{2})}\mathbb{P}\,dx\leq
2(1+\delta)^{3N}\int_{B(z,r_{2}/(1+\delta))}\mathbb{P}\,dx.$

###### Proof.

For $\gamma>0$, let us consider the set

$\Omega=\left\\{y^{\prime}\in\mathbb{R}^{3N}:|y^{\prime}_{1}-y_{i}|\geq\gamma-\frac{\delta}{1+\delta}r_{2}\text{
and }|y_{1}-y^{\prime}_{i}|\geq\gamma-\dfrac{\delta}{1+\delta}r_{2},\forall
i=2,\ldots,N\right\\}.$

We know that if $z\in\Omega$ we will have of course

$C_{1}(y^{\prime}_{1},z^{\prime}_{2},\ldots,z_{N}^{\prime})+C_{1}(z^{\prime}_{1},y^{\prime}_{2},\ldots,y^{\prime}_{N})\leq
2(N-1)M(\gamma-r_{1}-2r_{2})\;\forall y^{\prime}\in B(y,r_{1}),z^{\prime}\in
B(z,r_{2}),$

which in particular implies $\int C_{1}(\mathbb{P}_{1}+\mathbb{P}_{2})\,dx\leq
2(N-1)M(\gamma-r_{1}-2r_{2})m$. Now we want to see that there exists a $1/2$
doubling point in $\Omega$; in order to do that, it is easy to see that

$\Omega_{-\delta\frac{r_{2}}{1+\delta}}\subseteq\\{y^{\prime}\in\mathbb{R}^{3N}:|y^{\prime}_{1}-y_{i}|\geq\gamma\text{
and }|y_{1}-y^{\prime}_{i}|\geq\gamma,\forall i=2,\ldots,N\\}$

And now a similar computation to what is done in [4, 9] will give us

$\int_{\Omega_{-\delta\frac{r_{2}}{1+\delta}}}\mathbb{P}(x)\,dx\geq
1-2(N-1)\kappa(\rho,\gamma),$

where $\kappa(\rho,r)=\sup_{x\in\mathbb{R}^{3}}\rho(B(x,r))$. Now if we
consider $\gamma=\beta$ we have $\kappa(\rho,\beta)\leq\frac{1}{4(N-1)}$, and
so we can apply Lemma 5.2 with $r=\frac{r_{2}}{1+\delta}$ get the existence of
a $(1+\delta,1/2)$-doubling point at scale $r_{2}$ in $\Omega$.

∎

######  Lemma 5.2 (Existence of doubling points).

Let $\mathbb{P}\in L^{1}_{+}(\mathbb{R}^{3N})$ be the density of a probability
measure and let $\delta>0$. Let us consider an open set
$\Omega\subseteq\mathbb{R}^{3N}$; then for every $r>0$ we denote
$C_{r}:=\int_{\Omega_{-r}}\mathbb{P}(x)\,dx$, where $\Omega_{-r}$ is defined
as in (7). Then, whenever $C_{\delta r}>0$, there exists $y\in\Omega$, such
that

$\int_{B(y,(1+\delta)r)}\mathbb{P}(x)\,dx\leq\frac{(1+\delta)^{3N}}{C_{\delta
r}}\int_{B(y,r)}\mathbb{P}(x)\,dx,$

that is, the measure $\mathbb{P}(x)\,dx$ is doubling at the point $y$ at scale
$r$, with doubling constant $\frac{(1+\delta)^{3N}}{C_{\delta r}}$.

###### Proof.

Suppose on the contrary that for every $y\in\Omega$ the reversed inequality
holds

$\int_{B(y,(1+\delta)r)}\mathbb{P}(x)\,dx>\frac{(1+\delta)^{3N}}{C_{\delta
r}}\int_{B(y,r)}\mathbb{P}(x)\,dx.$

Then we can integrate this inequality on the whole $\Omega$

$\int_{\Omega}\int_{B(y,(1+\delta)r)}\mathbb{P}(x)\,dx\,dy>\frac{(1+\delta)^{3N}}{C_{\delta
r}}\int_{\Omega}\int_{B(y,r)}\mathbb{P}(x)\,dx\,dy.$

Now we can use Fubini and get

$\displaystyle\omega_{3N}\cdot\bigl{(}(1+\delta)r\bigr{)}^{3N}$
$\displaystyle=\int_{\mathbb{R}^{3N}}\int_{B(y,(1+\delta)r)}\mathbb{P}(x)\,dx\,dy\geq\int_{\Omega}\int_{B(y,(1+\delta)r)}\mathbb{P}(x)\,dx\,dy$
$\displaystyle C_{\delta r}\omega_{3N}\cdot r^{3N}$
$\displaystyle=\int_{\Omega_{-\delta
r}}\mathbb{P}(z)|B(z,r)|\,dz=\int_{\Omega}\int_{B(y,r)\cap\Omega_{-\delta
r}}\mathbb{P}(x)\,dx\,dy$
$\displaystyle\leq\int_{\Omega}\int_{B(y,r)}\mathbb{P}(x)\,dx\,dy,$

and so we get a contradiction. ∎

######  Proposition 5.1 (One step decay).

Let us consider $\rho$ and $\beta$ such that
$\kappa(\rho,\beta)<\frac{1}{4(N-1)}$. Then there exists
$\alpha_{0}=\alpha(\beta,\varepsilon)$ such that if $\mathbb{P}$ minimizes
(9), we have that for every $y\in D_{\alpha}$ such that
$\alpha\leq\alpha_{0}$, and every $r_{1}\leq\alpha/2$, we have

$\int_{B(y,r_{1}/(1+\delta))}\mathbb{P}(x)\,dx\leq\frac{1}{\delta^{2}r_{1}^{2}\frac{m(2\alpha)}{2(1+\delta)^{2}\varepsilon}+1}\int_{B(y,r_{1})}\mathbb{P}(x)\,dx,$
(17)

whenever $\delta>0$ is such that $m(2\alpha)>256\varepsilon
C(\delta)/\beta^{2}$, where

$C(\delta):=\frac{(1+\delta)^{2}\cdot(2(1+\delta)^{3N}-1)}{\delta^{2}}.$ (18)

An implicit choice for $\alpha_{0}$ is for example
$m(2\alpha_{0})>8\max\\{(N-1)M(\beta/2),850\varepsilon N^{2}/\beta^{2}\\}$.

###### Proof.

Let $y\in D_{\alpha}$ and without loss of generality we can assume that
$|y_{1}-y_{2}|<\alpha$; let $z$ given by Lemma 5.1. We then consider
$r_{1},r_{2},\eta_{1},\eta_{2},\mathbb{P}_{1},\mathbb{P}_{2},\bar{\mathbb{P}}$
defined by (10),(11), (12), (13) and (14); being
$\bar{\mathbb{P}}\in\Pi(\rho)$, we get, by the minimality of $\mathbb{P}$,

$\mathcal{F}_{LL,\varepsilon}(\bar{\mathbb{P}})\geq\mathcal{F}_{LL,\varepsilon}(\mathbb{P}),$
$\varepsilon\mathcal{E}_{\rm
kin}(\bar{\mathbb{P}})+v_{ee}(\bar{\mathbb{P}})\geq\varepsilon\mathcal{E}_{\rm
kin}(\mathbb{P})+v_{ee}(\mathbb{P});$

now we can use the estimates in Lemma 4.1 in order to conclude that

$\varepsilon\int\mathbb{P}\left(\frac{|\nabla\eta_{1}|^{2}}{\eta_{1}}+\frac{|\nabla\eta_{2}|^{2}}{\eta_{2}}+\frac{|\nabla\eta_{1}+\nabla\eta_{2}|^{2}}{1-\eta_{1}-\eta_{2}}\right)\,dx\geq\int\mathbb{P}(\eta_{1}+\eta_{2})C_{1}dx-\int(\mathbb{P}_{1}+\mathbb{P}_{2})C_{1}dx.$

Now we make the choice
$\eta(x)=\min\left\\{\frac{(1+\delta)(1-|x|)_{+}}{\delta},1\right\\}^{2}$. In
particular $0\leq\eta\leq 1$ and $\eta\equiv 1$ if $|x|<\frac{1}{1+\delta}$,
and moreover $|\nabla\sqrt{\eta}|\equiv\frac{1+\delta}{\delta}$ if
$\frac{1}{1+\delta}\leq|x|\leq 1$ and $0$ otherwise. We thus have

$\displaystyle\int\mathbb{P}\frac{|\nabla\eta_{1}|^{2}}{\eta_{1}}$
$\displaystyle=\frac{(1+\delta)^{2}}{\delta^{2}r_{1}^{2}}\int_{B(y,r_{1})\setminus
B(y,r_{1}/(1+\delta))}\mathbb{P}\lambda_{1}\,dx=\frac{(1+\delta)^{2}}{\delta^{2}r_{1}^{2}}\left(\int_{B(y,r_{1})}\mathbb{P}\lambda_{1}\,dx-\int_{B(y,r_{1}/(1+\delta))}\mathbb{P}\lambda_{1}\right)$
$\displaystyle\leq\frac{(1+\delta)^{2}}{\delta^{2}r_{1}^{2}}\left(\int_{B(y,r_{1})}\mathbb{P}\lambda_{1}\,dx-m\right)$

In a similar way we have

$\int\mathbb{P}\frac{|\nabla\eta_{2}|^{2}}{\eta_{2}}\leq\frac{(1+\delta)^{2}}{\delta^{2}r_{2}^{2}}\left(\int_{B(z,r_{2})}\mathbb{P}\lambda_{2}\,dx-m\right)\leq\frac{(1+\delta)^{2}\cdot(2(1+\delta)^{3N}-1)}{\delta^{2}r_{2}^{2}}\cdot
m=\frac{C(\delta)}{r_{2}^{2}}\cdot m.$

where in the last steps we used Lemma 5.1 (ii) and the definition of
$C(\delta)$ (18). Now we use that $\int C_{1}\mathbb{P}\eta_{1}\,dx\geq
m(\alpha+2r_{1})\cdot m$ and the estimates we have for $\int
C_{1}(\mathbb{P}_{1}+\mathbb{P}_{2})\,dx$ to get

$\int\mathbb{P}(\eta_{1}+\eta_{2})C_{1}dx-\int(\mathbb{P}_{1}+\mathbb{P}_{2})C_{1}dx\geq[m(\alpha+2r_{1})-2(N-1)M(\beta-
r_{1}-2r_{2})]\cdot m.$

Putting everything together we have

$\frac{\varepsilon(1+\delta)^{2}}{\delta^{2}r_{1}^{2}}\int_{B(y,r_{1})}\lambda_{1}\mathbb{P}(x)\,dx\geq\left[m(\alpha+2r_{1})-2(N-1)M(\beta-
r_{1}-2r_{2})-\frac{\varepsilon
C(\delta)}{r_{2}^{2}}+\frac{\varepsilon(1+\delta)^{2}}{\delta^{2}r_{1}^{2}}\right]\cdot
m.$

We can now use use
$m\geq\int_{B(y,r_{1}/(1+\delta))}\lambda_{1}\mathbb{P}(x)\,dx$ and, dividing
$\lambda_{1}$, we can write the inequality as

$\int_{B(y,r_{1}/(1+\delta))}\mathbb{P}(x)\,dx\leq\frac{1}{\frac{\delta^{2}r_{1}^{2}F(r_{1},\varepsilon,\alpha)}{(1+\delta)^{2}\varepsilon}+1}\int_{B(y,r_{1})}\mathbb{P}(x)\,dx,$
(19)

where

$F(r_{1},\varepsilon,\alpha):=\max\left\\{m(\alpha+2r_{1})-2(N-1)M(\beta-
r_{1}-2r_{2})-\frac{\varepsilon C(\delta)}{r_{2}^{2}}\;:\;r_{2}>0\right\\}.$

Now we can choose $r_{1}\leq\alpha/2\leq\beta/4$ and $r_{2}=\beta/8$, and then
choose $\alpha<\alpha_{0}$ such that

$\frac{m(2\alpha)}{2}-2(N-1)M(\beta/2)>\frac{m(2\alpha)}{4}\quad\text{ and
}\quad\frac{m(2\alpha)}{2}-\frac{\varepsilon
C(\delta)}{r_{2}^{2}}>\frac{m(2\alpha)}{4}.$ (20)

In this way we have $F(r_{1},\varepsilon,\alpha)\geq m(2\alpha)/2$: plugging
this estimate in (19) we get precisely

$\int_{B(y,r_{1}/(1+\delta))}\mathbb{P}(x)\,dx\leq\frac{1}{\delta^{2}r_{1}^{2}\frac{m(2\alpha)}{2(1+\delta)^{2}\varepsilon}+1}\int_{B(y,r_{1})}\mathbb{P}(x)\,dx.$

In order to understand for which $\alpha$ and $\delta$ this inequality holds,
we have to ensure that the two conditions (20) are satisfied, that is

$m(2\alpha)\geq\max\left\\{8(N-1)M\Bigl{(}\frac{\beta}{2}\Bigr{)},256\frac{\varepsilon
C(\delta)}{\beta^{2}}\right\\};$ (21)

notice that $\alpha_{0}$ can be characterized as maximal $\alpha$ for which
there exists some $\delta$ for which (21) is satisfied that is when
$C(\delta)$ as small as possible, which is approximately achieved for
$\delta=\frac{2}{3N}$. With this choice we have $C(2/(3N))\leq 26N^{2}$ and
thus

$m(2\alpha_{0})\geq
8\max\left\\{(N-1)M\Bigl{(}\frac{\beta}{2}\Bigr{)},850\frac{\varepsilon
N^{2}}{\beta^{2}}\right\\}.$ (22)

∎

We will now iterate the estimate in Proposition 5.1

######  Theorem 5.1.

Let us consider $\rho$ and $\beta$ such that
$\kappa(\rho,\beta)<\frac{1}{4(N-1)}$. Then let us consider
$\alpha<\alpha_{0}$ (as in Proposition 5.1) and suppose
$A:=\frac{\alpha^{2}m(2\alpha)}{8\varepsilon}\gg N^{2}$. Then if $\mathbb{P}$
minimizes (9) we have that

$\int_{D_{\alpha/2}}\mathbb{P}(x)\,dx\leq
e^{-\frac{1}{6}\sqrt{\frac{\alpha^{2}m(2\alpha)}{8\varepsilon}}}\int_{D_{2\alpha}}\mathbb{P}(x)\,dx.$

###### Proof.

Let us consider $\delta$ such that $\delta^{2}A=e^{2}$. By the hypothesis on
$A$ we have $\delta\ll 1/N$; in particular, by (18) we can estimate
$C(\delta)\leq\frac{2}{\delta^{2}}$, and then it is easy to see that
$m(2\alpha)>256\varepsilon C(\delta)/\beta^{2}$ and thus we can apply
Proposition 5.1 with $r_{1}=\alpha_{k}=\frac{\alpha}{2}(1+\delta)^{-k}$ to
obtain for every $y\in D_{\alpha}$

$\int_{B(y,\alpha_{k+1})}\mathbb{P}(x)\,dx\leq\frac{1}{\delta^{2}\alpha^{2}\frac{m(2\alpha)}{8(1+\delta)^{2k+2}\varepsilon}+1}\int_{B(z,\alpha_{k})}\mathbb{P}(x)\,dx\leq\frac{(1+\delta)^{2k+2}}{e^{2}}\int_{B(y,\alpha_{k})}\mathbb{P}(x)\,dx.$
(23)

We can now iterate the estimate for $k=0,\ldots,k_{0}$ where
$(1+\delta)^{2k_{0}+2}\leq e^{2}\leq(1+\delta)^{2k_{0}+4}$. At that point we
have

$\displaystyle\int_{B(y,\alpha/2e)}\mathbb{P}(x)\,dx$
$\displaystyle\leq\int_{B(y,\alpha_{k_{0}+1})}\mathbb{P}(x)\,dx\leq\frac{(1+\delta)^{(k_{0}+1)(k_{0}+2)}}{(e^{2})^{k_{0}+1}}\int_{B(y,\alpha_{0})}\mathbb{P}(x)\,dx$
$\displaystyle\leq e^{-k_{0}}\int_{B(y,\alpha/2)}\mathbb{P}(x)\,dx.$

Integrating this inequality for $y\in D_{\alpha}$ we get

$\displaystyle\omega_{3N}\left(\frac{\alpha}{2e}\right)^{3N}\int_{D_{\alpha/2}}\mathbb{P}(y)\,dy$
$\displaystyle\leq\int_{D_{\alpha}}\int_{B(y,\alpha/2e)}\mathbb{P}(x)\,dx\,dy$
$\displaystyle\leq
e^{-k_{0}}\int_{D_{\alpha}}\int_{B(y,\alpha/2)}\mathbb{P}(x)\,dx\,dy$
$\displaystyle\leq
e^{-k_{0}}\omega_{3N}\left(2\alpha\right)^{3N}\int_{D_{2\alpha}}\mathbb{P}(y)\,dy.$

Now we notice that
$k_{0}+2\geq\frac{\ln(e^{2})}{2\ln(1+\delta)}\geq\frac{2}{4\delta}=\frac{\sqrt{A}}{2e}$
and so $e^{-k_{0}}\leq 10e^{-\frac{\sqrt{A}}{2e}}$. In particular

$\int_{D_{\alpha/2}}\,\mathbb{P}(y)\,dy\leq
10e^{-\frac{\sqrt{A}}{2e}+3N\ln(4e)}\int_{D_{2\alpha}}\,\mathbb{P}(y)\,dy;$

notice that since $A\gg N^{2}$ we have
$\frac{\sqrt{A}}{2e}-3N\ln(4e)\geq\frac{\sqrt{A}}{6}$. ∎

###### Proof.

(Theorem 1.1 and Theorem 1.2) First we notice that if $\psi_{\varepsilon}$ is
a minimizer for (1) then $\mathbb{P}_{\varepsilon}=|\psi_{\varepsilon}|^{2}$
is a minimizer for (9). Then we notice that if $m(2\alpha)\leq
8(N-1)M(\beta/2)$ and $\varepsilon N^{2}\ll\alpha^{2}m(2\alpha)$, we have also
$\alpha<\alpha_{0}$ and so we can apply Theorem 5.1. From that we finish using
that $\mathbb{P}_{\varepsilon}$ is a probability density and so
$\int_{D_{2\alpha}}\mathbb{P}_{\varepsilon}(y)\,dy\leq 1$. The conclusions for
Theorem 1.1 are then implied by using $m(t)=M(t)=1/t$. ∎

Acknowledgement. Part of this work has been developed during a Research in
Pairs program where the authors were hosted at the MFO (Mathematisches
Forschungsinstitut Oberwolfach) in January 2017. S.D.M. is a member of GNAMPA
(INdAM). A.G. acknowledges partial support of his research by the Canada
Research Chairs Program and Natural Sciences and Engineering Research Council
of Canada. L.N. is partially on academic leave at Inria (team Matherials) for
the year 2022-2023 and acknowledges the hospitality if this institution during
this period. His work was supported by a public grant as part of the
Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH and
from H-Code, Université Paris-Saclay. The authors want to thank M. Lewin for
comments (and references) on a preliminary draft of the paper.

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# Assessing Cross-Cultural Alignment between
ChatGPT and Human Societies: An Empirical Study

Yong Cao12, Li Zhou23, Seolhwa Lee4, Laura Cabello2, Min Chen5, Daniel
Hershcovich2
1Huazhong University of Science and Technology
2Department of Computer Science, University of Copenhagen
3University of Electronic Science and Technology of China 4Technical
University of Darmstadt
5School of Computer Science and Engineering, South China University of
Technology
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The recent release of ChatGPT has garnered widespread recognition for its
exceptional ability to generate human-like responses in dialogue. Given its
usage by users from various nations and its training on a vast multilingual
corpus that incorporates diverse cultural and societal norms, it is crucial to
evaluate its effectiveness in cultural adaptation. In this paper, we
investigate the underlying cultural background of ChatGPT by analyzing its
responses to questions designed to quantify human cultural differences. Our
findings suggest that, when prompted with American context, ChatGPT exhibits a
strong alignment with American culture, but it adapts less effectively to
other cultural contexts. Furthermore, by using different prompts to probe the
model, we show that English prompts reduce the variance in model responses,
flattening out cultural differences and biasing them towards American culture.
This study provides valuable insights into the cultural implications of
ChatGPT and highlights the necessity of greater diversity and cultural
awareness in language technologies.

Assessing Cross-Cultural Alignment between
ChatGPT and Human Societies: An Empirical Study

Yong Cao12, Li Zhou23, Seolhwa Lee4, Laura Cabello2, Min Chen5, Daniel
Hershcovich2 1Huazhong University of Science and Technology 2Department of
Computer Science, University of Copenhagen 3University of Electronic Science
and Technology of China 4Technical University of Darmstadt 5School of Computer
Science and Engineering, South China University of Technology
<EMAIL_ADDRESS><EMAIL_ADDRESS>

##  1 Introduction

The release of ChatGPT by OpenAI111https://chat.openai.com/chat in 2022 has
sparked considerable attention and generated extensive discourse within both
academic and industry spheres lund2023chatting; thorp2023chatgpt;
jiao2023chatgpt. After extensive training as the large language model GPT-3
brown2020language, the official press
release222https://openai.com/blog/chatgpt reports that ChatGPT has undergone
fine-tuning through reinforcement learning with human feedback (RLHF;
christiano2017deep), resulting in its acquisition of unprecedented language
and reasoning abilities and knowledge coverage. Alongside its impressive
proficiency in broad tasks bang2023multitask; cabello2023pokemonchat, such as
code generation, summarization, and question answering, recent research has
begun exploring ChatGPT’s concordance with human capabilities, encompassing
language proficiency qin2023chatgpt, fact-checking kasneci2023chatgpt, and
knowledge grounding gilson2022well.

Figure 1: The pipeline of our proposed probing framework and an example of
distinct answers of ChatGPT by raising the same question in English and
Chinese.

Culture | Prompt 1 | Prompt 2
---|---|---
America (US) | For an average American, [_Question #_] is [_Options_]. | —
China (CN) | For an average Chinese, [_Question #_] is [_Options_]. | 对普通中国人来说，[_问题 #_] 是 [_选项_]。
Germany (DE) | For an average German, [_Question #_] is [_Options_]. | [Frage] ist [Option] für den durchschnittlichen Deutschen.
Japan (JP) | For an average Japanese, [_Question #_] is [_Options_]. | 平均的な日本人の場合、【質問 #】は【選択肢】です。
Spain (ES) | For an average Spanish, [_Question #_] is [_Options_]. | Para el español promedio, [Pregunta #] es [Opciones].

Table 1:  Prompt templates, consisting of Prompt 1: an English prompt that
includes a nationality hint, and Prompt 2: the respective language for the
given nationality with a hint. Also, to validate consistency, we replace the
prefix of Prompt 1 (i.e. "For an average #country-person" with "In the
#country culture setting", denote as Prompt 3.

ChatGPT, as a prominent dialogue agent, has been trained on a vast
multilingual corpus that inherently embeds biases and cultural nuances
m2022exploring; mcgee2023chat. This necessitates a critical evaluation of the
cultural values alignment between ChatGPT and human society or deviates from
them. Although the biases of pretrained language models (PLMs) have been
extensively studied de-vassimon-manela-etal-2021-stereotype; steed-
etal-2022-upstream; li-etal-2022-herb, finding a consistent methodology for
measuring their cultural alignment remains an open research question. In this
paper, we propose a novel framework that leverages a well-known social survey,
i.e. Hofstede Culture Survey hofstede1984culture, to probe the culture
alignment between ChatGPT and human societies. By examining various cultural
norms and values present in different nations, we gain a deeper understanding
of the challenges and limitations associated with accurately representing
cultural differences in ChatGPT, as shown at the bottom of Figure 1.

Specifically, following existing research mora2013cultures; tomlinson-
etal-2014-capturing; hershcovich-etal-2022-challenges, we define culture as
the combination of human beliefs, norms, and customs among groups. Previous
work in natural language processing (NLP) has primarily focused on cultural
investigation of models hutchinson-etal-2020-social; ross-etal-2021-measuring;
ma-etal-2022-encbp, with little emphasis on dialogue agents. Besides, probing
is a popular way to study the characteristics of models or agents
hammerl2022speaking; arora2022probing; johnson2022ghost. Thus, as illustrated
at the top of Figure 1, we adopt a probing technique to investigate the
cultural responses of ChatGPT by utilizing culture-related questions from
Hofstede Culture Survey. We then measure the correlation of the model’s
responses with those of human societies on the selected questions.
Experimental results reveal that ChatGPT displays greater alignment with
American culture but is less effective in adapting to other cultures.
Moreover, our analysis shows that English prompts reduce the variance in model
responses, flattening out cultural differences and biasing them towards
American culture.

##  2 Related Work

#### Cultural Differences in NLP.

Culture can be defined as the pattern of thinking, feeling and reacting,
distinguishing human groups kluckhohn1944culture; shweder2007cultural.
hershcovich-etal-2022-challenges propose four dimensions of culture relevant
for NLP, including linguistic form and style, common ground, aboutness, and
values. ma-etal-2022-encbp construct a cultural background prediction
benchmark, focusing on different expressions across countries. liu-
etal-2021-visually propose an ImageNet-style benchmark to evaluate visual
reasoning across different cultures. Recent studies hutchinson-
etal-2020-social; ross-etal-2021-measuring; sogaard-2022-ban emphasize the
impact of social bias in training data on NLP models, claiming they widen
existing inequality gaps—also across cultures.

#### Values in PLMs.

Several works use moral value surveys to probe multilingual PLMs.
arora2022probing pose the World Values Survey wvs-2022 and the Hofstede
Cultural Survey hofstede1984culture as cloze-style questions, and
hammerl2022speaking use the MoralDirections framework schramowski2022large to
probe multilingual PLMs on the Moral Foundations Questionnaire (MFQ;
graham2011mapping). They find differences in moral biases, that, however, do
not correlate with human responses. talat-etal-2022-machine claim that the
Delphi language model jiang2021delphi, designed for moral prediction,
necessarily has an inconsistent model of normative ethics. However, fraser-
etal-2022-moral use the MFQ and show it has a consistent ethical framework
that mirrors its training data. johnson2022ghost find GPT-3 is biased towards
American culture by probing it on value-loaded topics. In contrast, we probe
the dialogue model ChatGPT with the Hofstede Cultural Survey.

##  3 Method

In this section, we elaborate on the survey,333Please refer to Appendix
LABEL:ax:Dataset_details and LABEL:ax:chatgpt_results for the details of this
survey and our usage of it in the experiments. our probing prompts, and
interaction strategy below.

###  3.1 Hofstede Culture Survey

#### Cultural dimensions.

Our probing corpus is the Hofstede Cultural Survey hofstede1984culture, which
is one of the most commonly used cross-cultural analysis tools (see Table
LABEL:tb:main_example_of_survey). To measure cultural distinction, we utilize
the six cultural dimensions provided by this survey, namely Power Distance
(pdi), Individualism (idv), Uncertainty Avoidance (uai), Masculinity (mas),
Long-term Orientation (lto), and Indulgence (ivr). Following the survey’s
methodology, each cultural dimension metric is calculated using a combination
of 4 out of 24 questions. We denote $S_{i}$ as the i-th score out of 6
dimensions and $\mathcal{Q}_{i}$ as the chosen 4 questions related to
$\mathcal{S}_{i}$, then $S_{i}$ is calculated by following:

$S_{i}=\lambda^{0}_{i}(\mathcal{Q}^{0}_{i}-\mathcal{Q}^{1}_{i})+\lambda^{1}_{i}(\mathcal{Q}^{2}_{i}-\mathcal{Q}^{3}_{i})+\mathcal{C}_{i}$
(1)

where $\lambda_{i}$ is the hyper-parameter and $\mathcal{C}_{i}$ is a
constant. Parameter settings are listed in Appendix LABEL:ax:hyper_set.

#### Probing prompts.

Our processing steps are: (1) re-organizing and (2) prompting. First, since
Hofstede Cultural Survey is for individuals, we modify the questions from 2nd
person (i.e. you / your) to 3rd person to avoid attributing ChatGPT an
identity. Second, to easily obtain explicit answers through interaction with
ChatGPT, we provide questions and options without any modification from the
survey. Lastly, to designate the target culture, we further add cultural
prompts like “For an average [country-person]” as a prefix to each question.

As reported for InstructGPT ouyangtraining, which shares the underlying LM
with ChatGPT winata-etal-2021-language, 96% of the training corpus is in
English. Moreover, as observed by johnson2022ghost, models are much more
aligned with American values than others. We therefore design three kinds of
prompts to investigate whether prompting language affects cultural
distinctions: two are English prompts, and the other in the corresponding
target language, as Table 1 shows.

#### Language selection.

We choose five common languages as representative samples for the Hofstede
Culture Survey, as shown in Table 1. Except for English, each language is the
main official language of its respective country, allowing us to correlate our
analysis with survey findings. Additionally, since English is the official
language in the United States, which has the largest English-speaking
population engspeak, we use English examples as a proxy to represent American
culture.

###  3.2 Interaction Strategy

We introduce a novel multi-turn interaction approach that addresses the issues
of consistency and external knowledge injection in ChatGPT. The proposed
approach includes three distinct strategies: (1) valid knowledge injection,
wherein human experience is manually injected into each question to augment
the model’s response, (2) ineffective knowledge injection, whereby meaningless
information is fed to test ChatGPT’s performance variability, and (3) anti-
factual knowledge injection, which entails providing false or erroneous
information to gauge ChatGPT’s consistency in handling divergent human society
values.