2. The model

We study interacting Bose gases in $\mathbb{R}^d$ , $2 \le d \in \mathbb{N}$ , and in an external Poisson random potential $V(\omega,x)$ . Denoting the underlying probability space by $(\Omega,\Sigma,\mathbb{P})$ , on an informal level the external potential reads

(2.1) \begin{equation} V(\omega,x) \,:\!=\,\sum_{j}u\big(\| x-x_j^{\omega}\|_{\mathbb{R}^d}\big) , \qquad x \in \mathbb{R}^d , \ \omega \in \Omega,\end{equation}

where $\{x_j^{\omega}\}_{j}$ is a set of random points generated by a Poisson point process on $\mathbb{R}^d$ with intensity $\nu > 0$ . For more details regarding Poisson point processes, we refer the reader to [Reference Kingman20, Reference Last and Penrose22]. Furthermore, we assume that the single-site potential $u\,:\,\mathbb{R}^d \rightarrow \overline{\mathbb{R}}$ is given by

\begin{equation*} u(x)\,:\!=\,\begin{cases} 0 \quad & \text{if} \ x > R , \\[3pt] \infty & \text{otherwise} , \end{cases}\end{equation*}

where $R > 0$ is a constant. This means that we place hard balls $\overline{B_R(x_j^{\omega})}$ with radius $R > 0$ at each random point $x_j^{\omega}$ . Note that such a random potential appears in well-known models such as the Kac–Luttinger model in the area of BEC [Reference Kac and Luttinger15, Reference Kac and Luttinger16] and the Poisson Boolean model in stochastic geometry [Reference Gouéré13].

Also, we will investigate BEC in the thermodynamic limit. In this limit, N bosons are placed in the cube $\Lambda_N\,:\!=\,({-}L_N/2,+L_N/2)^d \subset \mathbb{R}^d$ of side length $L_N >0 $ such that the particle density,

(2.2) \begin{equation} \rho\,:\!=\,\frac{N}{L_N^d} ,\end{equation}

remains constant in the limit $N \rightarrow \infty$ . The N-particle configuration space in the external random potential (2.1) is given by $( \Lambda_N^{\omega} )^N$ , with

(2.3) \begin{equation} \Lambda_N^{\omega}\,:\!=\, ({-}L_N/2,+L_N/2)^d\setminus \bigcup_{j}\overline{B_{R}(x_j^{\omega})} \end{equation}

representing the one-particle configuration space.

Hardcore pair interactions are then introduced by further reducing the configuration space $(\Lambda_N^{\omega} )^N$ . For this, we define the set

\begin{equation*} \Lambda^{(\mathrm{HC}), \omega}_{N} \,:\!=\, \big\{x=(x_j) \in (\Lambda_N^{\omega})^N \,:\, \|x_i -x_j\|_{\mathbb{R}^d} > a_N ,\ i,j=1,\ldots,N,\ i \neq j \big\} ,\end{equation*}

where $(a_N)_{N \in \mathbb{N}} \subset (0,\infty)$ denotes the sequence of radii describing the range of the pair interaction. On a rigorous level, the N-particle Hamiltonian with hardcore pair interactions is the self-adjoint dN-dimensional Dirichlet Laplacian defined on $\mathrm{L}^2_s\big(\Lambda^{(\mathrm{HC}), \omega}_{N}\big)$ ; here, the index s refers to the totally symmetric subspace of $\mathrm{L}^2\big(\Lambda^{(\mathrm{HC}), \omega}_{N}\big)$ . On an informal level, the N-particle Hamiltonian with hardcore pair interaction is given by

(2.4) \begin{equation} H_N^{\omega}\,:\!=\,\sum_{i=1}^{N}\left({-} \mathbin\bigtriangleup_i+V(\omega,x_i)\right)+\sum_{1 \le i < j \le N}w_N^{\textrm{hc}}(\|x_i-x_j\|_{\mathbb{R}^d}) ,\end{equation}

where

\begin{equation*} w_N^{\textrm{hc}}(x)\,:\!=\,\begin{cases} 0 \quad & \text{if}\ x > a_N , \\[4pt] \infty & \text{otherwise} . \end{cases}\end{equation*}

We study hardcore pair interactions in Section 4.1.

In Section 4.2 we also consider a class of soft repulsive pair interactions: for all $N \in \mathbb{N}$ , with $w_N \in L^{\infty}(\mathbb{R}) \cap L^{1}(\mathbb{R},x^{d-1}\mathrm{d} x)$ non-negative, we introduce the N-particle Hamiltonian

(2.5) \begin{equation} H_N^{\omega}\,:\!=\,\sum_{i=1}^{N}\left({-}\mathbin\bigtriangleup_i+V(\omega,x_i)\right)+\sum_{1 \le i < j \le N}w_N(\|x_i-x_j\|_{\mathbb{R}^d})\end{equation}

on the Hilbert space $L_{\mathrm{s}}^2\big((\Lambda_N^{\omega})^N\big)$ . Finally, we write

(2.6) \begin{equation} h_0^{\omega}\,:\!=\, -\! \mathbin\bigtriangleup +V(\omega,x) \end{equation}

for the underlying one-particle Hamiltonian on $L^2(\Lambda_N^{\omega})$ . Note that $h^{\omega}_0$ can be defined rigorously as a direct sum of Dirichlet Laplacians, each defined over a connected component of $\Lambda_N^{\omega}$ . In this way we obtain a canonical set of eigenfunctions, namely those from each component, continued by zero to the rest of $\Lambda_N^{\omega}$ .

3. Probabilistic results

First, we note that the volume of the vacancy set $\Lambda_N^{\omega}$ , see (2.3), tends to be a constant fraction of $\Lambda_N$ in the limit $N \to \infty$ . More precisely, for any $\varepsilon > 0$ we have $\lim_{N \to \infty} \mathbb{P}\big(\big| |\Lambda_N^{\omega}|/|\Lambda_N| - \mathrm{e}^{-\nu \omega_d R^d} \big| < \varepsilon \big) = 1$ , where $\omega_d$ is the volume of the unit ball in d dimensions; see, for example, [Reference Sznitman32, p. 147]. Consequently, for any $0 < c < \mathrm{e}^{-\nu \omega_d R^d}$ there is $\mathbb{P}$ -almost surely a subsequence $(N_j)_{j \in \mathbb{N}} \subseteq \mathbb{N}$ such that $|\Lambda_{N_j}^{\omega}| > c |\Lambda_{N_j}|$ for all but finitely many $j \in \mathbb{N}$ . Also, $\Lambda_N^{\omega}$ is possibly divided into components (regions), but $\mathbb{P}$ -almost surely has only finitely many components for each $N \in \mathbb{N}$ [Reference Meester and Roy28, Proposition 4.1]. We denote the component of $\Lambda_N^{\omega}$ with the largest volume by $\Lambda_{N,>}^{(1), \omega}$ and its volume by $\big|\Lambda_{N,>}^{(1), \omega}\big|$ .

Next, we estimate the volume of the largest component of the vacancy set $\Lambda_N^{\omega}$ . Note that, $\mathbb{P}$ -almost surely, a ball free of Poisson points with radius $({d}/({\nu \omega_d}))^{1/d} (\!\ln L_N)^{1/d} - c$ (for an arbitrary constant $c > 0$ ) occurs within $\Lambda_N$ for all but finitely many $N \in \mathbb{N}$ and for dimensions $d \ge 2$ ; recall that $\omega_d$ is the volume of the unit ball in d dimensions. This has been shown in [Reference Sznitman32, Proof of Proposition 4.4.3] but, for the convenience of the reader, we provide more details on this fact. We set $\hat R \,:\!=\, ({d}/({\nu \omega_d}))^{1/d}$ and $R_N \,:\!=\, \hat R (\!\ln L_N)^{1/d} - c$ for a constant $c > 0$ . For each $N \in \mathbb{N}$ , we place $\hat c L_N^d/\ln L_N$ disjoint boxes with side length $2R_N$ in the box $\Lambda_N$ , where $\hat c = \hat c (\nu,d) > 0$ is a constant independent of N. The probability that, in any of these smaller boxes, the centered ball with radius $R_N$ is free of Poisson points is given by $\mathrm{e}^{-\nu \omega_d R_N^d}$ . Thus, the probability that none of the $\hat c L_N^d/\ln L_N$ disjoint boxes has such a centered ball free of Poisson points is (using the inequality $0 \le 1 -x \le \mathrm{e}^{-x}$ for $0 \le x \le 1$ )

\begin{align*} \big(1- \mathrm{e}^{-\nu \omega_d R_N^d}\big)^{\hat c L_N^d / \ln L_N} & \le \exp\!\big[{-}\hat c (L_N^d / \ln L_N) \mathrm{e}^{-\nu \omega_d R_N^d}\big] \\ & \le \exp\!\big[{-}\hat c (L_N^d / \ln L_N) \mathrm{e}^{-\nu \omega_d \hat R^d \ln\!(L_N)(1- (const.) (\!\ln L_N)^{-1/d})}\big] \\ & \le \exp\!\big[{-}\hat c (1 / \ln L_N) \mathrm{e}^{d (const.) (\!\ln L_N)^{1/2}}\big] \\ & \le \mathrm{e}^{- (\!\ln L_N)^2} \le L_N^{-2} = \rho^2 N^{-2}\end{align*}

for all but finitely many $N \in \mathbb{N}$ . The statement then follows with the Borel–Cantelli lemma. On the other hand, we have the following result.

Proof. We partition $\mathbb{R}^d$ into the boxes $\big[s j_1-\frac{s}{2}, s j_1+\frac{s}{2}\big) \times \big[s j_2-\frac{s}{2}, s j_2+\frac{s}{2}\big) \times \cdots \times $ $\big[s j_d-\frac{s}{2}, s j_d+\frac{s}{2}\big)$ , where $j = (j_1, j_2, \ldots, j_d) \in \mathbb{Z}^d$ and $s \,:\!=\, R / \sqrt{d}$ . The centers of the boxes are then given by the points $(s j_1, s j_2, \ldots, s j_d)$ , and we shall call them vertices. Vertices with a Euclidean distance of s are called adjacent and consequently we obtain a discrete graph $\mathcal G$ . A sequence $(v_i)_{j=1}^J$ , $J \in \{ 1,2,\ldots,\infty\}$ , of vertices in $\mathcal G$ such that $v_i$ and $v_{i+1}$ are adjacent for all $j \in \{1,2,\ldots,J-1\}$ is called a path in $\mathcal G$ . A path in $\mathcal G$ is called finite if $J < \infty$ and infinite whenever $J=\infty$ .

It is important to note that if a Poisson point $x_j^{\omega}$ is contained in such a box, then the box is not contained in $\Lambda_N^{\omega}$ (informally, this is equivalent to saying that the external potential is infinitely high across the box). We call a vertex vacant if the corresponding box does not contain any Poisson point, and occupied if the corresponding box contains at least one Poisson point. In the same way we call a path in $\mathcal G$ vacant if the path contains only vacant vertices.

Furthermore, we shall assume that the intensity of the Poisson point process $\nu > 0$ is larger than the critical intensity $\nu_\mathrm{c} \,:\!=\, \inf\{ \nu > 0 \,:\, \theta^0(\nu) = 0 \}$ where $\theta^{0}(\nu) = \mathbb{P}($ there exists an infinite, self-avoiding, vacant path starting at 0). Note that $0 < \nu_\mathrm{c} < \infty$ , due to a Peierls argument and since the graph $\mathcal G$ is of finite degree; see [Reference Hammersley14], [Reference Kesten19, p. 349].

Now, let $W^{\omega}(v)$ , $v \in s\mathbb{Z}^d = (s j_1, s j_2, \ldots, s j_d)$ , $j = (j_1, j_2, \ldots, j_d) \in \mathbb{Z}^d$ , be the union of all vertices that can be reached by a vacant path on $\mathcal G$ from v, and let $\#W^{\omega}(v)$ denote the number of vertices in $W^{\omega}(v)$ . Due to [Reference Kesten19, Theorem 2], [Reference Aizenman and Barsky2] and [Reference Menshikov29], there are constants $0 < C_1,C_2 < \infty$ such that, for any $n \in \mathbb{N}$ ,

(3.1) \begin{equation} \mathbb{P}(\#W^{\omega}(0) \ge n) \le C_1 \mathrm{e}^{-C_2 n} . \end{equation}

We choose a $C > 2 C_2^{-1}$ and set $n = C \ln\!((L_N +2)^d)$ . Using inequality (3.1), we conclude that, for any $N \in \mathbb{N}$ , the probability that the number of boxes $\big[s j_1-\frac{s}{2}, s j_1+\frac{s}{2}\big) \times \big[s j_2-\frac{s}{2}, s j_2+\frac{s}{2}\big) \times \ldots \times \big[s j_d-\frac{s}{2}, s j_d+\frac{s}{2}\big)$ intersecting any component of the vacancy set $\Lambda_{N,>}^{(1), \omega}$ is equal to or larger than n is bounded from above by

\begin{align*} \sum\limits_{v \in s\mathbb{Z}^d \cap ({-}\lceil L_N/2 \rceil , + \lceil L_N/2 \rceil)^d} \mathbb{P}(\#W^{\omega}(v) \ge n) & \le s^{-d} (L_N + 2)^d \mathbb{P}(\#W^{\omega}(0) \ge n) \\ & \le C_1 s^{-d} [(L_N + 2)^d]^{1 - C C_2} , \end{align*}

which converges to zero in the limit $N \to \infty$ . In addition, using the Borel–Cantelli lemma, we conclude that for $\mathbb{P}$ -almost all $\omega \in \Omega$ there exists an $\widetilde N \in \mathbb{N}$ such that, for all $N \ge \widetilde N$ , the number of these boxes intersecting any component of the vacancy set within $\Lambda_N$ is smaller than $C \ln\!((L_N +2)^d)$ .

Remark 3.1. This theorem implies the following. Suppose that the intensity of the Poisson random potential is sufficiently large. Then the probability that the volume of the largest component $\Lambda_{N,>}^{(1), \omega}$ is bounded by $C\ln\!(L_N)$ , for a sufficiently large constant $C > 0$ converges to 1; i.e. there is a $\widetilde C > 0$ such that, for all $C > \widetilde C$ ,

\begin{align*} \lim\limits_{N \to \infty} \mathbb{P}\big(\big|\Lambda_{N,>}^{(1), \omega}\big| < C \ln\!(L_N)\big) = 1 . \end{align*}

In addition, there is a $\widetilde C > 0$ such that, for all $C > \widetilde C$ and for $\mathbb{P}$ -almost all $\omega \in \Omega$ , there is an $\widetilde N \in \mathbb{N}$ such that, for all $N \ge \widetilde N$ , $\big|\Lambda_{N,>}^{(1), \omega}\big| < C \ln\!(L_N)$ .

For the proof of Lemma 4.1 in Section 4.2, we need the following lemma. It is a statement about the number of disjoint balls with a given constant radius within $\Lambda_N$ that are free of Poisson points.

Lemma 3.1. Let $d \ge 2$ and $\nu > 0$ be given. Also, let $(c_N)_{N \in \mathbb{N}}$ be a sequence that goes to infinity. Then, for $\mathbb{P}$ -almost all $\omega \in \Omega$ , there exists an $\widetilde N \in \mathbb{N}$ such that, for all $N \ge \widetilde N$ , the number $B_N^{\omega}$ of disjoint balls with diameter $\hat R>0$ that are completely within $\Lambda_N$ and are free of Poisson points is at least $L_N^d/(c_N \ln\!(N))$ , that is,

(3.2) \begin{align} B_N^{\omega} & \ge \dfrac{L_N^d}{c_N \ln\!(N)} . \end{align}

Proof. We shall put $(2\lfloor (({L_N}/{2}) - ({\hat R}/{2}))/\hat R \rfloor + 1)^d$ disjoint balls, each with diameter $\hat R > 0$ , in $\Lambda_N$ . More specifically, the balls shall have the centers $(\hat R j_1, \hat R j_2, \ldots, \hat R j_d)$ with $(j_1, j_2, \ldots, j_d) \in \mathbb{Z}^d$ and $j_i \in [{-}\lfloor L_N/(2 \hat R) - \frac{1}{2} \rfloor, \lfloor L_N/(2 \hat R) - \frac{1}{2} \rfloor]$ , $i=1,\ldots, d$ .

Next, we derive an upper bound on the probability that less than $\lceil L_N^d/(c_N \ln\!(L_N)) \rceil $ of these balls are free of Poisson points. We denote the probability that one given ball is free of Poisson points by c. Notice that $0 < c < 1$ . Furthermore,

\begin{align*} \sum\limits_{i=0}^{\lceil L_N^d/(c_N \ln\!(L_N)) \rceil -1} & \binom{(2\lfloor {L_N}/({2\hat R}) - \frac{1}{2} \rfloor + 1)^d}{i} c^i (1-c)^{(2\lfloor {L_N}/({2\hat R}) - \frac{1}{2} \rfloor + 1)^d-i} \\ & \le \dfrac{L_N^d}{c_N \ln\!(L_N)} \bigg( \frac{L_N}{\hat R} \bigg)^{\tfrac{d L_N^d}{c_N \ln\!(L_N)}} (1-c)^{(2\lfloor {L_N}/({2\hat R}) - \frac{1}{2} \rfloor + 1)^d-\tfrac{L_N^d}{c_N \ln\!(L_N)}} \\ & \le \exp\!\bigg\{d \ln\!(L_N) + \frac{dL_N^d}{c_N} \Big( 1-\frac{\ln\!(\hat R)}{\ln\!(L_N)} \Big) \\ & \qquad \qquad + \bigg[ \bigg( 2 \bigg\lfloor \frac{L_N}{2\hat R} - \frac{1}{2} \bigg\rfloor + 1 \bigg)^d -\frac{L_N^d}{c_N \ln\!(L_N)} \bigg] \ln\!(1-c) \bigg\} \\ & \le \mathrm{e}^{3^{-d} L_N^d \hat R^{-d} \ln\!(1-c)} \end{align*}

for all but finitely many $N \in \mathbb{N}$ . Since, using relation (2.2),

\begin{align*} \sum\limits_{N \in \mathbb{N}} \mathrm{e}^{3^{-d} L_N^d \hat R^{-d} \ln\!(1-c)} = \sum\limits_{N \in \mathbb{N}} ((1-c)^{3^{-d} \rho^{-1} \hat R^{-d}} )^N \le \dfrac{1}{1 - (1-c)^{3^{-d} \rho^{-1} \hat R^{-d}}} < \infty , \end{align*}

the claim follows with the Borel–Cantelli lemma.

We would like to comment on the main difference between Lemma 3.1 and the corresponding one-dimensional result [Reference Kerner and Pechmann17, Lemma A.1]. In the one-dimensional case, the lengths of the intervals that are introduced by a Poisson point process on the real line are independent, identically distributed random variables with exponential distribution. This fact was used in the proof of [Reference Kerner and Pechmann17, Lemma A.1]. In higher dimensions, however, we know less about the distribution of the volume of the components of the vacancy set. To offset this, we require that the denominator in (3.2) converges to infinity at a sufficient speed, and needed to use a different strategy here compared to the corresponding one-dimensional case.

4. Results on Bose–Einstein condensation

In this section we apply the probabilistic results derived in Section 3 in order to say something about BEC in a system of interacting bosons placed in a random environment. In fact, we consider two kinds of pair interactions: hardcore interactions, where each particle has to keep a certain distance to all other particles, and soft interactions. Physically, hardcore interactions are described by the informal Hamiltonian (2.4) and soft interactions by the Hamiltonian (2.5). As mentioned before, based on methods presented in [Reference Aonghusa and Pulé3], our aim is to generalize the results from [Reference Kerner and Pechmann17] to the higher-dimensional setting.

In the canonical ensemble, the N-particle state of the system (the density matrix) at inverse temperature $\beta=1/T \in (0,\infty)$ is given by $\varrho_N^{\beta,\omega} = {\mathrm{e}^{-\beta H_{N}^{\omega}}} / {\text{Tr}(\mathrm{e}^{-\beta H_{N}^{\omega}})}$ . Here, $\text{Tr}({\cdot})$ refers to the trace of a (trace-class) operator on the associated N-particle Hilbert space. Regarding $\mathrm{e}^{-\beta H_N^{\omega}}$ being trace-class, we note that $\text{Tr}(\mathrm{e}^{-\beta H_N^{\omega}}) \le \sum_{j \in \mathbb{N}_0} \mathrm{e}^{-\beta E_N^{j}}$ , where $(E_N^{j})_{j \in \mathbb{N}_0}$ are the eigenvalues of the Dirichlet Laplacian on $\Lambda_N^N$ . The latter series is finite due to Weyl’s law. Moreover, let $\varrho_N^{\beta,\omega}(\cdot,\cdot)$ denote the kernel of $\varrho_N^{\beta,\omega}$ . In order to calculate the density of particles in a given one-particle state, we use the reduced one-particle density matrix which acts as a trace-class operator on the underlying one-particle Hilbert space $\mathrm{L}^2(\Lambda_{N}^{\omega})$ ; see [Reference Michelangeli30, Chapter 4]. The kernel of the corresponding reduced one-particle density matrix is then obtained as

(4.1) \begin{equation} \varrho_N^{\beta,(1),\omega}(x,y) = N \int_{\Lambda_{N}^{\omega}} \mathrm{d} z_1 \cdots \int_{\Lambda_{N}^{\omega}} \mathrm{d} z_{N-1}\, \varrho_N^{\beta,\omega}(x,z_1,\ldots,z_{N-1},y,z_1,\ldots,z_{N-1}),\end{equation}

with $x,y \in \Lambda_{N}^{\omega}$ . Here, with a slight abuse of notation and whenever we consider hardcore interactions, we understand the kernel $\varrho_N^{\beta,\omega}(\cdot,\cdot)$ to be extended by zero such that the integration in (4.1) makes sense. The average particle density in a one-particle state $\varphi \in \mathrm{L}^2(\Lambda_{N}^{\omega})$ can be calculated as

\begin{equation*} \rho_N^{\beta,\omega}(\varphi) \,:\!=\, \frac{1}{L^d_N}\int_{\Lambda_{N}^{\omega}} \int_{\Lambda_{N}^{\omega}} \overline{\varphi(x)}\varrho_N^{\beta,(1),\omega}(x,y) \varphi(y)\, \mathrm{d} y \, \mathrm{d} x ;\end{equation*}

see, for example, [Reference Aonghusa and Pulé3] or [Reference Michelangeli30, Chapter 4]. This leads to the following definition.

For related definitions of BEC we refer to [Reference Dimonte, Falconi and Olgiati9, Reference Michelangeli30].

4.1. Hardcore interactions

We first consider the N-particle Hamiltonian with hardcore pair interaction,

\begin{equation*} H_N^{\omega}=\sum_{i=1}^{N}\left({-} \mathbin\bigtriangleup_i+V(\omega,x_i)\right)+\sum_{1 \le i < j \le N}w_N^{\textrm{hc}}(\|x_i-x_j\|_{\mathbb{R}^d}) ,\end{equation*}

where

\begin{equation*} w_N^{\textrm{hc}}(x)\,:\!=\,\begin{cases} 0 \quad & \text{if}\quad x > a_N , \\ \infty & \text{otherwise} . \end{cases}\end{equation*}

We decompose $\mathbb{R}^d$ into the boxes $\Lambda^{(n)}_N \,:\!=\, \{x \in \mathbb{R}^d \,:\, r_Nn_j \leq x_j < r_N(n_j+1) , \ j=1,\ldots,d \}$ , where $n = (n_1, n_2, \ldots, n_d) \in \mathbb{Z}^d$ and $N \in \mathbb{N}$ . If the side length of these boxes satisfies $r_N \leq a_N/\sqrt{d}$ , then any box $\Lambda^{(n)}_N$ can be occupied by at most one particle. Consequently, for a normalized one-particle state $\varphi_N^{\omega} \in L^2(\Lambda_N^{\omega})$ , $N \in \mathbb{N}$ , $\omega \in \Omega$ , we have that, $\mathbb{P}$ -almost surely, for all $\beta \in (0,\infty)$ , and for all but finitely many $N \in \mathbb{N}$ ,

(4.2) \begin{equation} \rho_N^{\beta,\omega}(\varphi_N^{\omega}) \leq \frac{1}{L^d_N}\Bigg(\sum_{n \in \mathbb{Z}^d} \bigg( \int_{\Lambda^{(n)}_N}|\varphi_N^{\omega}(x)|^2\, \mathrm{d} x \bigg)^{1/2} \Bigg)^2 ; \end{equation}

see [Reference Aonghusa and Pulé3, Lemma 2]. Note that each $\varphi_N^{\omega}$ in (4.2) is understood to be extended by zero to all of $\mathbb{R}^d$ . We can now give and prove the main statement of this subsection.

Theorem 4.1 (Absence of BEC I.) Let $\beta, \rho > 0$ be arbitrarily given. We assume that $R > 0$ and $\nu > 0$ are such that Theorem 3.1 holds. Suppose that the bounded sequence of radii $(a_N)_{N \in \mathbb{N}}$ is such that

\begin{equation*} \lim\limits_{N \to \infty} \dfrac{1}{N} \left( \dfrac{\ln\!(N)}{a_N^d} \right)^2 = 0 . \end{equation*}

Then, for $\mathbb{P}$ -almost $\omega \in \Omega$ , if $(\varphi_N^{\omega})_{N \in \mathbb{N}}$ , $\varphi_N^{\omega} \in L^2(\Lambda_N^{\omega})$ for all $N \in \mathbb{N}$ , is a sequence of normalized one-particle states for which the number $A_N^{\omega}$ of components of $\Lambda_N^{\omega}$ intersecting $\mathrm{supp}(\varphi_N^{\omega})$ satisfies

\begin{equation*} \lim_{N \to \infty} \dfrac{1}{N} \bigg( \dfrac{A_N^{\omega}\ln\!(N)}{a_N^d}\bigg)^2 = 0 , \end{equation*}

then $(\varphi_N^{\omega})_{N \in \mathbb{N}}$ is not macroscopically occupied, that is, there cannot exist a subsequence $(N_j)_{j \in \mathbb{N}} \subseteq \mathbb{N}$ such that $\lim_{j \rightarrow \infty} \rho_{N_j}^{\beta,\omega}(\varphi_{N_j}^{\omega}) > 0$ .

Proof. The proof is obtained from a suitable adaptation of the proof of [Reference Kerner and Pechmann17, Theorem 3.3]. Using inequality (4.2) and Theorem 3.1, we obtain, for a constant $C >0$ and $\mathbb{P}$ -almost surely,

\begin{align*} \lim_{N \to \infty} \rho_N^{\beta,\omega}(\varphi_N^{\omega}) & \leq \lim_{N \to \infty} \frac{1}{L^d_N} \Bigg(\sum_{n \in \mathbb{Z}^d}\bigg(\int_{\Lambda^{(n)}_N}|\varphi_N^{\omega}(x)|^2\, \mathrm{d} x\bigg)^{1/2}\Bigg)^2 \\ & \leq \lim_{N \to \infty} \frac{1}{L^d_N}\Bigg(\sum_{n \in \mathbb{Z}^d:\mathrm{supp}(\varphi_N^{\omega})\cap \Lambda^{(n)}_N \neq \emptyset}1\Bigg)^2 \\ & \leq C \lim_{N \to \infty} \frac{1}{L^d_N } \bigg(\dfrac{A_N^{\omega} \ln\!(N)}{a_N^d} \bigg)^2 = 0 . \end{align*}

Note here that $\Lambda^{(n)}_N$ is the grid defined with $r_N=a_N/\sqrt{d}$ . Hence, comparing this N-dependent grid with the fixed grid used in Theorem 3.1 eventually leads to the factor $a^{-d}_N$ in the third line.

In particular, Theorem 4.1 shows that $\mathbb{P}$ -almost surely all canonical eigenstates of the underlying one-particle Hamiltonian (2.6) are not macroscopically occupied if the interaction strength is large enough since they are supported on one component only. In this context, it would be desirable to know if the ground state (or all eigenstates for that matter) is $\mathbb{P}$ -almost surely simple for all $N \in \mathbb{N}$ large enough. Although this seems to be true, to the best of our knowledge it has not been proved for the Poissonian model with hard obstacles so far; we refer to [Reference Klein, Germinet and Hislop21] for related results on the Poissonian model with soft obstacles.

4.2. Soft interactions

Finally, we study the case when a soft pair interaction is present. Namely, we now consider the N-particle Hamiltonian (2.5), that is,

\begin{equation*} H_N^{\omega}\,:\!=\,\sum_{i=1}^{N}\left({-} \mathbin\bigtriangleup_i+V(\omega,x_i)\right)+\sum_{1 \le i < j \le N}w_N(\|x_i-x_j\|_{\mathbb{R}^d}) ,\end{equation*}

where $w_N \in L^{\infty}(\mathbb{R}) \cap L^{1}(\mathbb{R},x^{d-1}\mathrm{d} x)$ is a non-negative function. We furthermore assume that for every $N \in \mathbb{N}$ there exist two numbers $a_N , b_N > 0$ (recall that $(a_N)_{N\in\mathbb{N}}$ is assumed to be bounded) such that

(4.3) \begin{equation} w_N(x) \ge b_N \qquad \text{for almost every} \ x \in [{-}a_N,+a_N] .\end{equation}

Note that similar ‘volume-dependent’ interactions were also considered in [Reference Bolte and Kerner6]. The main result of this section is the following theorem, which says that any normalized one-particle state that is sufficiently localized, such as any canonical eigenstate of the corresponding one-particle Hamiltonian (2.6), is $\mathbb{P}$ -almost surely not macroscopically occupied, given that the pair interactions are sufficiently strong.

Theorem 4.2. (Absence of BEC II) Let $\nu > 0$ and $R > 0$ be given such that Theorem 3.1 holds. Let $\varphi_{N}^{\omega} \in \mathrm{L}^2(\Lambda_{N}^{\omega})$ be a normalized one-particle state and $A_N^{\omega}$ the number of components of $\Lambda_N^{\omega}$ with non-empty intersection with $\mathrm{supp}(\varphi_N^{\omega})$ , $N \in \mathbb{N}$ , $\omega \in \Omega$ . Furthermore, suppose that

\begin{equation*} \lim\limits_{N \to \infty} \frac{b_N a_N^{3d} N}{(A_N^{\omega})^3 \ln^3\!(N)} = \infty , \qquad \lim\limits_{N \to \infty} \frac{a_N^{3d} N}{(A_N^{\omega})^3 \ln^3\!(N)} = \infty , \end{equation*}

and $\lim_{N \to \infty} \ln^2\! (N)\|w_N(\|\cdot\|_{\mathbb{R}^d})\|_{\mathrm{L}^1(\mathbb{R}^{d})} =0$ . Then, for all $\beta \in (0,\infty)$ , $(\varphi_{N}^{\omega})_{N \in \mathbb{N}}$ is $\mathbb{P}$ -almost surely not macroscopically occupied.

In the remainder of this work we prove Theorem 4.2. We proceed similarly to [Reference Kerner and Pechmann17]. For the convenience of the reader and to be able to point out the differences between the higher-dimensional case discussed here and the one-dimensional case discussed in [Reference Kerner and Pechmann17], we present the main steps of the proof.

In the first step, we show that the expected energy density with respect to the canonical ensemble is bounded in the thermodynamic limit. However, in contrast to the one-dimensional setting in [Reference Kerner and Pechmann17], we have to assume that the pair interaction is weak enough in a suitable sense. This is due to the fact that, unlike in [Reference Kerner and Pechmann17, Lemma A.1], we require the denominator in (3.2) to converge to infinity at a certain speed.

Lemma 4.1. (Bound energy density) Let $\beta \in (0,\infty)$ be arbitrarily given. Assume that $\lim_{N \to \infty} \ln^2\! (N) \|w_N(\|\cdot\|_{\mathbb{R}^d})\|_{L^1(\mathbb{R}^d)} =0$ . Then there exists a constant $C>0$ such that, $\mathbb{P}$ -almost surely,

(4.4) \begin{equation} \limsup_{N \rightarrow \infty} \frac{\langle {H_N^{\omega}} \rangle_{\varrho_N^{\beta,\omega}}}{L^d_N} < C . \end{equation}

Proof. Let a typical $\omega \in \Omega$ be given. As in [Reference Kerner and Pechmann17], we prove (4.4) by showing that the right-hand side of the inequality

(4.5) \begin{equation} \frac{\beta}{2} \dfrac{\langle {H_N^{\omega}} \rangle_{\varrho_N^{\beta,\omega}} }{L_N^d} \le \dfrac{\ln\!(\text{Tr}(\mathrm{e}^{-(\beta/2) H_N^{\omega}}))}{L_N^d} - \dfrac{\ln\!(\text{Tr}(\mathrm{e}^{-\beta H_N^{\omega}}))}{L_N^d} \end{equation}

is bounded by a constant in the limit $N \to \infty$ . We note that the inequality (4.5) holds because $\langle {H_N^{\omega}} \rangle_{\varrho_N^{\beta,\omega}} = - \frac{ \mathrm{d}}{\mathrm{d} \beta} \ln\!(\text{Tr}(\mathrm{e}^{-\beta H_N^{\omega}}))$ and $\ln\!(\text{Tr}(\mathrm{e}^{-\beta H_N^{\omega}}))$ is a convex function in $\beta$ .

Regarding the first term in (4.5), we compare the eigenvalues of $H_N^{\omega}$ with the eigenvalues $(E_N^{j})_{j \in \mathbb{N}_0}$ of the Dirichlet Laplacian $- \mathbin\bigtriangleup$ on $\Lambda_N^N$ and conclude that $\text{Tr}(\mathrm{e}^{-(\beta/2) H_N^{\omega}}) \le \sum_{j \in \mathbb{N}_0} \mathrm{e}^{-(\beta/2) E_N^{j}}$ . Then we use the fact that there is a constant $C_1 = C_1(\beta) > 0$ such that $\lim_{N \to \infty} L_N^{-d} \ln\big(\!\sum_{j \in \mathbb{N}_0} \mathrm{e}^{-(\beta/2) E_N^{j}}\big) = C_1$ ; see, for example, [Reference Ruelle31, Theorem 3.5.8]. Thus, there is a constant $C_1 > 0$ such that, $\mathbb{P}$ -almost surely,

\begin{align*} \limsup_{N \rightarrow \infty} \dfrac{\ln\!(\text{Tr}(\mathrm{e}^{-(\beta/2) H_N^{\omega}}))}{L_N^d} \le C_1 . \end{align*}

Next, we show that the second term in (4.5), including the minus sign, is also bounded from above by a constant in the limit $N \to \infty$ . We use the inequality [Reference Lieb and Seiringer26, Lemma 14.1 and Remark 14.2]

\begin{equation*} - \ln\!(\text{Tr}(\mathrm{e}^{-\beta H_N^{\omega}})) \le -\ln\!\big( \mathrm{e}^{-\beta \| \Psi_N^{\omega}\|_{\mathbb{R}^d}^{-2} \langle \Psi_N^{\omega}, H_N^{\omega} \Psi_N^{\omega} \rangle} \big) = \beta \| \Psi_N^{\omega}\|_{\mathbb{R}^d}^{-2} \langle \Psi_N^{\omega}, H_N^{\omega} \Psi_N^{\omega} \rangle , \end{equation*}

$N \in \mathbb{N}$ , for a state $\Psi_N^{\omega}$ in the domain of $H_N^{\omega}$ . We choose the N-particle state $\Psi_N^{\omega}$ to be a product state $\prod_{j=1}^{N} \psi_N^{\omega}(x_j)$ . The one-particle state $\psi_N^{\omega}(x)$ , $x \in \mathbb{R}^d$ , is constructed as follows. We consider a rotational symmetric function $f(\|x\|_{\mathbb{R}^d})$ that is equal to one for $\|x\|_{\mathbb{R}^d} \le 1/4$ and smoothly decreases to zero for $1/4\le \|x\|_{\mathbb{R}^d} \le 1/2$ . Then $\psi_N^{\omega}$ is the sum of all such functions placed at the center of each disjoint ball with diameter $(1 + 2R)$ that are within $\Lambda_N$ and free of Poisson points. As in Lemma 3.1, we denote the number of such disjoint balls by $B_N^{\omega}$ . Then,

\begin{align*} \|\Psi_N^{\omega}\|_{\mathbb{R}^d}^{-2} \langle \Psi_N^{\omega}, H_N^{\omega} \Psi_N^{\omega} \rangle & = N \|\psi_N^{\omega}\|_{L^2(\mathbb{R}^d)}^{-2} \int\limits_{\Lambda_N} | \nabla \psi_N^{\omega}(x)|^2 \, \mathrm{d} x \\ & \quad + \binom{N}{2} \|\psi_N^{\omega}\|_{L^2(\mathbb{R}^d)}^{-4} \int\limits_{\Lambda_N} \int\limits_{\Lambda_N} w_N(\|x-y\|_{\mathbb{R}^d}) |\psi_N^{\omega}(x)|^2 |\psi_N^{\omega}(y)|^2 \, \mathrm{d} y \, \mathrm{d} x . \end{align*}

Now, by construction there are positive constants $c_1,c_2 > 0$ independent of N such that $\int_{\Lambda_N} |\nabla \psi_N^{\omega}(x)|^2 \, \mathrm{d} x \le c_1 B_N^{\omega}$ , $\|\psi_N^{\omega}\|_{L^2(\mathbb{R}^d)}^{2} \ge c_2 B_N^{\omega}$ , and $|\psi_N^{\omega}(x)| \le 1$ for all $x \in \mathbb{R}^d$ . Employing Lemma 3.1 in combination with our condition on $\|w_N(\|\cdot\|_{\mathbb{R}^d})\|_{L^1(\mathbb{R}^d)}$ , we finally conclude that there is a constant $C_2> 0$ such that, $\mathbb{P}$ -almost surely,

\begin{equation*} \limsup\limits_{N \to \infty} \dfrac{-\ln\!(\text{Tr}(\mathrm{e}^{-\beta H_N^{\omega}}))}{L_N^d} \le \beta C_2 . \end{equation*}

Proof of Theorem 4.2. Suppose there was a set $\widetilde \Omega \subset \Omega$ with $\mathbb{P}(\widetilde \Omega) >0$ and such that, for all $\omega \in \widetilde \Omega$ , there was a sequence of normalized one-particle states $(\varphi_N^{\omega})_{N \in \mathbb{N}} \in L^2(\Lambda_N^{\omega})$ that are macroscopically occupied and that fulfill the properties described in Theorem 4.2. Then, for all such $\omega \in \widetilde \Omega$ , we show that the expected energy density with respect to the canonical ensemble would diverge in the thermodynamic limit. However, since this is in contradiction to Lemma 4.1, Theorem 4.2 follows.

It remains to prove divergence of the energy density (recall that the proof is a suitable adaptation of the corresponding ones from [Reference de Smedt8, Reference Bolte and Kerner6, Reference Kerner and Pechmann17]). Using second quantization (see, for example, [Reference Martin and Rothen27] for an introduction), an equivalent definition for a sequence of one-particle states to be macroscopically occupied reads

\begin{equation*} \limsup_{N \rightarrow \infty}\frac{\langle {a^\ast(\varphi_{N}^{\omega})a(\varphi_{N}^{\omega})} \rangle_{\varrho_N^{\beta,\omega}}}{L^d_N} > 0 . \end{equation*}

Here, the creation operator $a^{\ast}(\varphi_N^{\omega}) = \int_{\Lambda_N^{\omega}} \varphi^{\omega}_N(x)a^{\ast}(x)\, \mathrm{d} x$ and the annihilation operator $a(\varphi_N^{\omega}) = \int_{\Lambda_N^{\omega}} \varphi_N^{\omega}(x)a (x)\, \mathrm{d} x$ fulfill the canonical commutation relations $[a(x),a^{\ast}(y)] = \delta(x-y)$ and $[a(x),a(y)] = [a^{\ast}(x),a^{\ast}(y)]=0$ , $x,y \in \Lambda_N^{\omega}$ . Also, $\delta({\cdot})$ is the Dirac $\delta$ distribution.

The key ingredient now is a lower bound for the expected energy density. Let $\{G_j\}$ be a partition of $\mathbb{R}^d$ into boxes of side length $a_N/\sqrt{d} > 0$ similar to that constructed in Theorem 3.1. By $K_N^{\omega}$ we denote the number of boxes in $\{G_j\}$ with a non-empty intersection with $\mathrm{supp}(\varphi_N^{\omega})$ . Employing (4.3), we now obtain

\begin{equation*} \frac{\langle {H_N^{\omega}} \rangle_{\varrho_N^{\beta,\omega}}}{L^d_N} \geq \frac{b_N}{2L^d_N}\sum_{j}^{K_N^{\omega}} \int_{G_j}\mathrm{d} x\int_{G_j}\mathrm{d} y \, \langle {a^{\ast}(x)a^{\ast}(y)a(x)a(y)} \rangle_{\varrho_N^{\beta,\omega}}\,:\!=\,\frac{b_N}{2L^d_N}\sum_{j}^{K_N^{\omega}} \mathcal{C}^{(j)}_N , \end{equation*}

where the summation is over all the boxes in $\{G_j\}$ that have a non-empty intersection with $\mathrm{supp}(\varphi_N^{\omega})$ . Introducing the functions $\varphi^{(j),\omega}_{N}\,:\!=\,\varphi_{N}^{\omega}\textbf{1}_{G_j}$ , $j=1,\ldots,K_N^{\omega}$ , as well as using [Reference de Smedt8, (14)–(16b)], we get

\begin{equation*} \sum_{i,j}^{K_N^{\omega}}\Big(\langle {a^{\ast}(\varphi^{(i),\omega}_{N})a(\varphi^{(j),\omega}_{N})} \rangle_{\varrho_N^{\beta,\omega}}\Big)^4 \leq \Bigg(\sum_{j}^{K_N^{\omega}} \mathcal{C}^{(j)}_N+\rho L^d_N\Bigg)^2 . \end{equation*}

From this, using the inequality $\big|\!\sum_{i=1}^{n}x_j \big|^2 \leq n \sum_{i=1}^{n}|x_i|^2$ , we obtain

\begin{equation*} \langle {a^{\ast}(\varphi^{\omega}_{N})a(\varphi^{\omega}_{N})} \rangle_{\varrho_N^{\beta,\omega}}^4 = \Bigg( \sum_{i,j}^{K_N^{\omega}}\langle {a^{\ast}(\varphi^{(i),\omega}_{N})a(\varphi^{(j),\omega}_{N})} \rangle_{\varrho_N^{\beta,\omega}}\Bigg)^4 \leq (K_N^{\omega})^6 \Bigg(\sum_{j}^{K_N^{\omega}} \mathcal{C}^{(j)}_N+\rho L^d_N\Bigg)^2 , \end{equation*}

and therefore

\begin{equation*} \frac{\langle {H_N^{\omega}} \rangle_{\varrho_N^{\beta,\omega}}}{L^d_N} \geq \frac{b_N}{2 L^d_N (K_N^{\omega})^3} \langle a^{\ast}(\varphi^{\omega}_{N})a(\varphi^{\omega}_{N}) \rangle^{2}_{\varrho_N^{\beta,\omega}}-\frac{b_N\rho}{2} . \end{equation*}

Finally, by Theorem 3.1, $\mathbb{P}$ -almost surely and for all but finitely many $N \in \mathbb{N}$ we have $K_N^{\omega} \le C A_N^{\omega}a^{-d}_N \ln\!(N)$ for some constant $C>0$ .