V}$. By (3.10) for large $k_{0}$ and $k$ as in (6.51), one has

(6.52) ${\mathbb{P}}[M_{\ell=2^{k},\varepsilon}]\geq
1-(7^{d}+1)\,\varepsilon.$

Then, on $M_{\ell=2^{k},\varepsilon}$, one of the $B\in{\cal C}_{(4\ell)}$,
see (3.9), is such that $\lambda_{1,\omega}(B\cap B_{4\ell})$ lies between
$\lambda_{1,\omega}(B_{4\ell})$ and $\lambda_{1,\omega}(B_{4\ell})+(\log
4\ell)^{-(2+2/d)}$, and the same holds true for
$\lambda_{1,\omega}(B_{\ell,v})$ if $B_{\ell,v}$ contains $B\cap B_{4\ell}$.
Thus, $k_{0}$ (and hence $k$) being large; on $M_{\ell=2^{k},\varepsilon}$, we
can consider disjoint boxes $B_{\ell,v}$ and $B_{\ell,v^{\prime}}$ in
$B_{4\ell}$ such that both $\lambda_{1,\omega}(B_{\ell,v})$ and
$\lambda_{1,\omega}(B_{\ell,v^{\prime}})$ lie in
$[\lambda_{1,\omega}(B_{4\ell})$, $\lambda_{1,\omega}(B_{4\ell})+(\log
4\ell)^{-(2+2/d)}+|I_{\ell,\varepsilon}|]$. Since $B_{\ell,v}$ and
$B_{\ell,v^{\prime}}$ are disjoint, this implies by the min-max principles,
see Version 3 of Theorem 12.1, p. 301 of [20], that the spectral gap
$\lambda_{2,\omega}(B_{4\ell})-\lambda_{1,\omega}(B_{4\ell})$ is at most
$(\log
4\ell)^{-(2+2/d)}+|I_{\ell,\varepsilon}|\stackrel{{\scriptstyle(\ref{6.51})}}{{\leq}}(\log
4\ell)^{(-2+2/d)}+A_{\varepsilon}(\log\ell)^{-(1+2/d)}$. This shows that for
any $\varepsilon>0$, for large $k_{0}$ (with $2^{k}$ playing the role of
$4\ell$)

(6.53) $\begin{array}[]{l}\mbox{for more than half of $k_{0}\leq
k<k_{0}+[k_{0}^{\zeta}]$,
${\mathbb{P}}[\lambda_{2,\omega}(B_{2^{k}})-\lambda_{1,\omega}(B_{2^{k}})\leq$}\\\
\mbox{$(A_{\varepsilon}+1)(\log\ell^{k})^{-(1+2/d)}]\geq
1-(7^{d}+1)\,\varepsilon$}.\end{array}$

Incidentally, in the above argument “half” could be replaced by any number
less than $1$. In any case, from (6.53) one sees that it is plausible that
(6.50) holds.

2) One can naturally wonder whether the strategy in the proof of Theorem 6.2
can be adapted to derive deconcentration estimates in the context of
Poissonian soft obstacles, see (1.18), and whether the corresponding lower
bound on the spectral gap corresponding to Theorem 6.1 can be established in
this context as well. Incidentally, in the scaling identities corresponding to
(6.11), (6.12), the original bump function $W(\cdot)$ in (1.18) would be
transformed into
$\widetilde{W}(\cdot)=\frac{1}{\lambda^{2}}\,W(\frac{\cdot}{\lambda})$ (the
case under present consideration formally corresponds to
$W(\cdot)=\infty\,1\\{|\cdot|\leq a\\}$ and
$\widetilde{W}(\cdot)=\infty\,1\\{|\cdot|\leq\lambda a\\}$). $\square$

##  7 Bose-Einstein condensation

In this section we combine the lower bound on the spectral gap obtained in the
main Theorem 6.1 with the results of Kerner-Pechmann-Spitzer in [16]. We prove
a so-called type-I generalized Bose-Einstein condensation in probability for a
model in the spirit of Kac-Luttinger [12], [13] consisting of a non-
interacting Bose gas among a Poisson cloud of hard obstacles made of closed
balls of radius $a>0$ centered at the points of the cloud, which has intensity
$\nu>0$. The dimension of space is $d\geq 2$. The radius $a$ although fixed
can be arbitrarily small, and the complement in ${\mathbb{R}}^{d}$ of the
spherical impurities may possibly percolate, see Chapter 4 of [21]. Our main
result is Theorem 7.1.

Following [16], we consider a thermodynamic limit in a grand-canonical set-up.
Given a fixed particle density (not to be confused with the parameter of
Section 2).

(7.1) $\rho>0,$

we consider the positive sequence $\ell_{N},N\geq 1$, indexed by the particle
number $N$, which tends to infinity and satisfies (see (1.5) for notation)

(7.2) $\rho\,|B_{\ell_{N}}|=N,\;\mbox{for $N\geq 1$}$

(in the notation of [16], $2\ell_{N}=L_{N}$).

The events (see (1.9), (1.4) for notation)

(7.3)
$\Omega_{N}=\\{\omega\in\Omega;\lambda_{1,\omega}(B_{\ell_{N}})<\infty\\}=\\{\omega\in\Omega;B_{\ell_{N},\omega}\not=\phi\\},\;N\geq
1$

are non-decreasing, have positive probability, and

(7.4) $\mbox{$\bigcup_{N\geq 1}\,\Omega_{N}=\Omega_{\infty}$ (the event of
full ${\mathbb{P}}$-measure in (\ref{1.6}))}.$

To stay within the framework of [16], on $\Omega^{c}_{N}$ we replace the
variables $\lambda_{1,\omega}(B_{\ell_{N}}),j\geq 1$, which are all infinite
on $\Omega^{c}_{N}$, by the ordered sequence of Dirichlet eigenvalues of
$-\frac{1}{2}\,\Delta+1$ in $B_{\ell_{N}}$ (any positive constant in place of
$1$ would do). We thus define the modified single particle eigenvalues (see
(1.9) for notation)

(7.5)
$\overline{\lambda}_{j,\omega}(B_{\ell_{N}})=\left\\{\begin{array}[]{l}\mbox{$\lambda_{j,\omega}(B_{\ell_{N}})$,
when $j\geq 1$ and $\omega\in\Omega_{N}$},\\\\[4.30554pt] \mbox{the $j$-th
Dirichlet eigenvalue of $-\frac{1}{2}\,\Delta+1$ in $B_{\ell_{N}}$, when
$j\geq 1$ and}\\\ \mbox{$\omega\notin\Omega_{N}$}.\end{array}\right.$

Note that

(7.6) $\mbox{on $\Omega_{\infty}$, for large $N$,
$\lambda_{j,\omega}(B_{\ell_{N}})=\overline{\lambda}_{j,\omega}(B_{\ell_{N}})$,
for all $j\geq 1$},$

i.e. outside a ${\mathbb{P}}$-negligible set, the original sequence of
eigenvalues agrees with the modified sequences of eigenvalues, when $N$ is
large. We denote by $\beta\in(0,\infty)$ the inverse temperature. Given a
chemical potential
$\mu\in(-\infty,\overline{\lambda}_{1,\omega}(B_{\ell_{N}})$), the quantity
$(e^{\beta(\overline{\lambda}_{1,\omega}(B_{\ell_{N}})-\mu)}-1)^{-1}$ is the
number of particles occupying the $j$-the eigenstate in the grand-canonical
ensemble, see (2.7) of [16], and we choose
$\mu=\mu^{\omega}_{N}\in(-\infty,\overline{\lambda}_{1,\omega}(B_{\ell_{N}}))$
so that

(7.7)
$\mbox{\footnotesize$\displaystyle\frac{1}{|B_{\ell_{N}}|}$}\;\sum\limits_{j\geq
1}\;\overline{n}\,^{j,\omega}_{N}=\rho,\;\mbox{with}\;\overline{n}\,^{j,\omega}_{N}=1\,/\,(e^{\beta(\overline{\lambda}_{j,\omega}(B_{\ell_{N}})-\mu^{\omega}_{N})}-1).$

In view of (7.6) and setting
$n^{j,\omega}_{N}=1\,/\,(e^{\beta(\lambda_{j,\omega}(B_{\ell_{N}})-\mu^{\omega}_{N})}-1)$
on $\Omega_{N}$, we also see that

(7.8) $\mbox{on $\Omega_{\infty}$, for large $N$,
$\overline{n}\,^{j,\omega}_{N}=n^{j,\omega}_{N}$, for all $j\geq 1$},$

(i.e. the modification (7.5) is immaterial on $\Omega_{\infty}$ for large
$N$).

We recall some known facts. As $N\rightarrow\infty$, the random measures on
${\mathbb{R}}_{+}$

(7.9)
$m_{N,\omega}=\left\\{\begin{array}[]{l}\mbox{\footnotesize$\displaystyle\frac{1}{|B_{\ell_{N}}|}$}\;\sum\limits_{i\geq
1}\;\delta_{\lambda_{i,\omega}(B_{\ell_{N}})},\;\mbox{for
$\omega\in\Omega_{N}$},\\\\[8.61108pt] 0,\;\;\mbox{for
$\omega\in\Omega^{c}_{N}$},\end{array}\right.$

are known to ${\mathbb{P}}$-a.s. converge vaguely to a deterministic measure,
the density of states

(7.10) $\begin{array}[]{l}\mbox{$m$ on $[0,\infty)$, characterized by its
Laplace transform}\\\
\displaystyle\int_{[0,\infty)}e^{-t\lambda}dm(\lambda)=\mbox{\footnotesize$\displaystyle\frac{1}{(2\pi
t)^{d/2}}$}\;E^{t}_{0,0}[\exp\\{-\nu\,|W_{t}^{a}|\\}],\;\mbox{for
$t>0$},\end{array}$

where $E^{t}_{0,0}$ denotes the expectation with respect to the Brownian
bridge in time $t$ from $0$ to $0$, and $W_{t}^{a}$ is the Wiener sausage of
radius $a$ in time $t$, i.e. the closed $a$-neighborhood of the Brownian
bridge trajectory. (The proof is similar to that of Theorem 5.18, p. 99 of
[22]). In view of (7.6) we also see that

(7.11) $\begin{array}[]{l}\mbox{${\mathbb{P}}$-a.s., the measures
$\overline{m}_{N,\omega}=\mbox{\footnotesize$\displaystyle\frac{1}{|B_{\ell_{N}}|}$}\;\displaystyle\sum\limits_{i\geq
1}\;\delta_{\overline{\lambda}_{i,\omega}(B_{\ell_{N}})}$ on $[0,\infty)$
converge vaguely}\\\ \mbox{as $N\rightarrow\infty$ to the measure $m$ in
(\ref{7.10})}.\end{array}$

One also knows that $m$ has a so-called Lifshitz tail close to $0$. More
precisely, see Corollary 3.5 of [26] or as in Theorem 10.2, p. 221 of [22],
one has (see above (0.2) for notation):

(7.12)
$m([0,\lambda])=\exp\big{\\{}-\nu\omega_{d}(\lambda_{d}/\lambda)^{d/2}\big{(}1+o(1)\big{)}\big{\\}},\;\mbox{as
$\lambda\rightarrow 0$}.$

(The quantity $\exp\\{-\nu\omega_{d}(\lambda_{d}/\lambda)^{d/2}\\}$ is the
probability that the Poisson point process places no point in the open ball of
radius $(\lambda_{d}/\lambda)^{1/2}$ centered at the origin. This ball has a
principal Dirichlet eigenvalue for $-\frac{1}{2}\,\Delta$ equal to $\lambda$.)
We then introduce the critical density for our system (see (2.6) of [16]),
namely

(7.13)
$\rho_{c}(\beta)=\displaystyle\int^{\infty}_{0}\;\mbox{\footnotesize$\displaystyle\frac{1}{e^{\beta\lambda}-1}$}\;dm(\lambda)\;(<\infty\;\mbox{by
(\ref{7.12}))}.$

In our model a generalized Bose-Einstein condensation, with a macroscopic
occupation of an arbitrary small energy band of one-particle states, is known
to occur when $\rho>\rho_{c}(\beta)$, see Theorem 2.5 of [16] or Theorem 4.1
of [19]. The main result of this section is the following theorem, which pins
down the nature of the condensation. It shows a type-I generalized Bose-
Einstein condensation in probability, when $\rho>\rho_{c}(\beta)$:

######  Theorem 7.1.

When $\rho>\rho_{c}(\beta)$, then as $N$ tends to infinity,

(7.14) $\left\\{\begin{array}[]{rl}{\rm
i)}&\mbox{$\overline{n}^{1,\omega}_{N}/N$ tends to
$\big{(}\rho-\rho_{c}(\beta)\big{)}/\rho$ in ${\mathbb{P}}$-probability,
and}\\\\[8.61108pt] {\rm ii)}&\mbox{$\overline{n}^{j,\omega}/N$ tends to $0$
in ${\mathbb{P}}$-probability, for any $j\geq 2$.}\end{array}\right.$

Equivalently, for every fixed $N_{1}\geq 1$, as $N\geq N_{1}$ tends to
infinity,

(7.15) $\left\\{\begin{array}[]{rl}{\rm i)}&\mbox{$n^{1,\omega}_{N}/N$ tends
to $\big{(}\rho-\rho_{c}(\beta)\big{)}/\rho$ in
${\mathbb{P}}_{N_{1}}$-probability, and}\\\\[8.61108pt] {\rm
ii)}&\mbox{$n^{j,\omega}_{N}/N$ tends to $0$ in
${\mathbb{P}}_{N_{1}}$-probability,}\end{array}\right.$

where ${\mathbb{P}}_{N_{1}}$ stands for the conditional probability
${\mathbb{P}}(\cdot\,|\,\Omega_{N_{1}})$, see (7.3).

(When $\rho\leq\rho_{c}(\beta)$, the above quantities in (7.14), respectively
in (7.15), are known to converge to $0$ in ${\mathbb{P}}$-probability,
respectively ${\mathbb{P}}_{N_{1}}$-probability, as $N\rightarrow\infty$, see
for instance Theorem 3.2.4, p. 30 of [23].)

###### Proof.

The claim (7.15) is a simple restatement of (7.14) since
$n^{j,\omega}_{N},j\geq 1,N\geq N_{1}$ coincide with
$\overline{n}^{1,\omega}_{N},j\geq 1,N\geq N_{1}$ on $\Omega_{N_{1}}$ and
$\lim_{N_{1}\rightarrow\infty}{\mathbb{P}}(\Omega_{N_{1}})=1$, see (7.4),
(7.5), (7.7). We only need to prove (7.14).

We want to apply Theorem 2.9 of [16]. We first need to check the Assumptions
2.2 of the above reference. We have ${\mathbb{P}}$-a.s.,
$\overline{\lambda}_{1,\omega}(B_{\ell_{N}})\rightarrow 0$, as $N\rightarrow
0$, by (0.4) and (7.6), and ${\mathbb{P}}$-a.s., $\overline{m}_{N,\omega}$
converges vaguely to the deterministic measure $m$, as $N\rightarrow\infty$,
by (7.11). This shows that i) and ii) of Assumptions 2.2 of [16] hold. As we
now show

(7.16) ${\mathbb{E}}[\overline{m}_{N,\omega}([0,\lambda])]\leq
m([0,\lambda])\;\mbox{for $\lambda<1$ and $N\geq 1$}.$

This will imply that iii) of Assumptions 2.2 of the above reference holds as
well. To prove (7.16), we first observe that by Dirichlet bracketing and
translation invariance, for $N,k\geq 1,\lambda\geq 0$,
${\mathbb{E}}[m_{N,\omega}([0,\lambda])]\leq{\mathbb{E}}[m_{k^{d}N}([0,\lambda])]$
(we recall that by (7.2) $\ell_{k^{d}N}=k\ell_{N}$). Then, on a set of full
${\mathbb{P}}$-measure $m_{k^{d}N,\omega}$ converges vaguely to $m$ as $k$
goes to infinity (see (7.9), (7.10)), and hence on that set for every point
$\lambda$ of continuity of $m([0,\cdot])$,
$\lim_{k}m_{k^{d}N,\omega}([0,\lambda])=m([0,\lambda])$. For any such fixed
$\lambda$ we also have $m_{k^{d}N,\omega}([0,\lambda])\leq
m_{k^{d}N,\omega=0}([0,\lambda])$ which is deterministic and bounded in $k$.
Thus, by dominated convergence, we see that for any $\lambda$ point of
continuity of $m([0,\cdot])$, one has

(7.17) ${\mathbb{E}}[m_{N,\omega}([0,\lambda])]\leq m([0,\lambda])$

and this inequality extends to any $\lambda\geq 0$ by right-continuity and
monotonicity. Since
$\overline{m}_{N,\omega}([0,\lambda])=m_{N,\omega}([0,\lambda])$ for all
$\lambda<1$ and $\omega\in\Omega$ by (7.5), the claim (7.16) follows.

As for iv) of Assumptions 2.2 in [16], by the Lifshitz tail asymptotics
(7.12), one has, picking $\eta_{1}\in(0,1)$ and setting
$\widetilde{C}_{1}=\nu\,\omega_{d}\,\lambda_{d}^{d/2}(1+\eta_{1}/2)\stackrel{{\scriptstyle(\ref{0.3})}}{{=}}d\,c_{0}^{d/2}(1+\eta_{1}/2)$,

(7.18)
$\lim\limits_{N}\;N^{1-\eta_{1}}\;m([0,\\{\widetilde{C}_{1}/\log(|B_{\ell_{N}}|)\\}^{2/d}])=0.$

This shows that Assumptions 2.2 of [16] are satisfied.

Now by the lower bound on the spectral gap in Theorem 6.1 and (7.6),

(7.19)
$\lim\limits_{N}\;{\mathbb{P}}[\overline{\lambda}_{2,\omega}(B_{\ell_{N}})-\overline{\lambda}_{1,\omega}(B_{\ell_{N}})\geq(\log\ell_{N})^{-(1+2/d+\varepsilon)}]=1,\;\mbox{for
any $\varepsilon>0$}.$

Together with (0.4) (or (3.10)) and the fact that $\log|B_{\ell_{N}}|\sim
d\log\ell_{N}$, as $N\rightarrow\infty$, this is more than enough to show that
for $\widetilde{C}_{2}=d\,c_{0}^{d/2}(1+\eta_{1}/4)$ ($<\widetilde{C}_{1}$
above), one has

(7.20)
$\lim\limits_{N}\;{\mathbb{P}}[\overline{\lambda}_{2,\omega}(B_{\ell_{N}})-\overline{\lambda}_{1,\omega}(B_{\ell_{N}})\geq
N^{-(1-\eta_{1})}\;\mbox{and}\;\overline{\lambda}_{1,\omega}(B_{\ell_{N}})\leq\\{\widetilde{C}_{2}/\log(|B_{\ell_{N}}|)\\}^{2/d}]=1.$

The assumptions of Theorem 2.9 of [16] are thus fulfilled (with $c_{2}=1$ and
$c_{3}=1$), and the claim (7.14) follows. This concludes the proof of Theorem
7.1. ∎

######  Remark 7.2.

1) One can wonder whether the above results extend to the context of soft
Poissonian obstacles, see (1.18). In the case of dimension $1$ we refer to
[24] for results in the case when the strength of the soft obstacle tends to
infinity with $N$, see Corollary 5.8 of this reference, and a less specific
companion statement in the case of a fixed strength, see Corollary 5.6 of
[24].

2) In the case of hard spherical Poissonian obstacles in
${\mathbb{R}}^{d},d\geq 2$, in a suitable non-percolative regime for the
vacant set, and suitably strong short-range repulsive pair-interactions, we
refer to [14], which, among other results, shows the absence of Bose-Einstein
condensation into the normalized eigenstates of the Dirichlet Laplacian in
$B_{\ell_{N},\omega}$ (in the notation of (1.4)). $\square$

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