On Bose–Einstein Condensation in One-Dimensional Noninteracting Bose Gases in the Presence of Soft Poisson Obstacles

Abstract

We study Bose–Einstein condensation (BEC) in one-dimensional noninteracting Bose gases in Poisson random potentials on $\mathbb R$ with single-site potentials that are nonnegative, compactly supported, and bounded measurable functions in the grand-canonical ensemble at positive temperatures and in the thermodynamic limit. For particle densities larger than a critical one, we prove the following: With a probability arbitrarily close to one when choosing the fixed strength of the random potential sufficiently large, BEC where only the ground state is macroscopically occupied occurs. If the strength of the Poisson random potential converges to infinity in a certain sense but arbitrarily slowly, then this kind of BEC occurs in probability and in the rth mean, $r \ge 1$. Furthermore, in Poisson random potentials of any fixed strength a probability arbitrarily close to one for type-I g-BEC to occur is also obtained, but our upper bound for the number of macroscopically occupied one-particle states may be large in this case.

Appendices

Appendix A: Generalized Bose–Einstein Condensation

In this appendix, we prove Theorem 2.4 part (ii), that is, the ${\mathbb {P}}$-almost sure occurrence of g-BEC in the case of a Poisson random potential with a strength that converges to infinity, see Theorem A.7. For this, we firstly state several lemmata (Lemmata A.1–A.6). For the proof of Lemma A.1, we refer to [15, Corollary 3.5] in combination with [10, Lemma A.5]. We prove Lemmata A.2–A.6, since these statements are more general than the ones that can be found in [15] and [10]. Also, note that we use these Lemmata in our proofs in Sect. 5.

Recall Remark 2.2 and the fact that the chemical potential $\mu _{N,V_N}^{\omega } \in (-\infty , E_{N,V_N}^{1,\omega })$ is for ${\mathbb {P}}$-almost all $\omega \in \Omega$ and for all $N \in {\mathbb {N}}$ uniquely determined by the equation (2.15).

Lemma A.1

Let V be a Poisson random potential of fixed strength. If $\mu < 0$, then ${\mathbb {P}}$-almost surely

$$\begin{aligned} \lim \limits _{N \rightarrow \infty } \int _{(0,\infty )} {\mathcal {B}}(E - \mu ) \, {\mathcal {N}}_{N,V}^{\omega }(\mathrm{d} E) = \int _{(0,\infty )} {\mathcal {B}}(E - \mu ) \, {\mathcal {N}}_{\infty ,V}(\mathrm{d} E). \end{aligned}$$

(A.1)

For the proofs of the next two lemmata, we define the function

$$\begin{aligned} \chi _{{\mathcal {E}}_1, {\mathcal {E}}_2} : {\mathbb {R}}&\rightarrow [0,1], \nonumber \\ E&\mapsto \dfrac{E - \frac{{\mathcal {E}}_1}{2}}{\frac{{\mathcal {E}}_1}{2}} {\mathbf {1}}_{(\frac{{\mathcal {E}}_1}{2},{\mathcal {E}}_1)}(E) + {\mathbf {1}}_{[{\mathcal {E}}_1, {\mathcal {E}}_2]}(E) + [1 - (E - {\mathcal {E}}_2)] {\mathbf {1}}_{({\mathcal {E}}_2, {\mathcal {E}}_2 + 1)}(E) \end{aligned}$$

(A.2)

where ${\mathcal {E}}_1, {\mathcal {E}}_2 > 0$ are two constants with $0<{\mathcal {E}}_1<{\mathcal {E}}_2$. The reason for the introduction of this continuous cutoff function is the ${\mathbb {P}}$-almost sure convergence of the density of states $({\mathcal {N}}_{N,V}^{\omega })_{N \in {\mathbb {N}}}$ in the vague rather than weak sense.

Lemma A.2

Let V be a Poisson random potential of fixed strength. Furthermore, let $\varepsilon >0$ and ${\mathcal {E}}_2 > \varepsilon$ be given, and $\chi _{\varepsilon , {\mathcal {E}}_2}$ as in (A.2). Then for ${\mathbb {P}}$-almost all $\omega \in \Omega$ we have: If $(\hat{\mu }_N^{\omega })_{N \in {\mathbb {N}}} \subset {\mathbb {R}}$ is a sequence with $\lim _{N \rightarrow \infty } \hat{\mu }_N^{\omega } = 0$, then

$$\begin{aligned} \lim \limits _{N \rightarrow \infty } \int _{{\mathbb {R}}} \chi _{\varepsilon , {\mathcal {E}}_2}(E) \, {\mathcal {B}}(E - \hat{\mu }_N^{\omega }) \, {\mathcal {N}}_{N,V}^{\omega } (\mathrm{d} E) = \int _{{\mathbb {R}}} \chi _{\varepsilon , {\mathcal {E}}_2}(E) \, {\mathcal {B}}(E) \, {\mathcal {N}}_{\infty ,V} (\mathrm{d} E). \end{aligned}$$

(A.3)

Proof

For ${\mathbb {P}}$-almost all $\omega \in \Omega$ and all $N \in {\mathbb {N}}$ we have

$$\begin{aligned}&\left| \, \int _{{\mathbb {R}}} \chi _{\varepsilon , {\mathcal {E}}_2}(E) \, {\mathcal {B}}(E - \hat{\mu }_N^{\omega }) \, {\mathcal {N}}_{N,V}^{\omega } (\mathrm{d} E) - \int _{{\mathbb {R}}} \chi _{\varepsilon , {\mathcal {E}}_2}(E) \, {\mathcal {B}}(E) \, {\mathcal {N}}_{\infty ,V} (\mathrm{d} E) \, \right| \\&\quad \le \, \left| \, \int _{{\mathbb {R}}} \chi _{\varepsilon , {\mathcal {E}}_2}(E) \big [ {\mathcal {B}}(E - \hat{\mu }_N^{\omega }) - {\mathcal {B}}(E) \big ] \, {\mathcal {N}}_{N,V}^{\omega }(\mathrm{d} E) \, \right| \\&\qquad + \, \left| \, \int _{{\mathbb {R}}} \chi _{\varepsilon , {\mathcal {E}}_2}(E) {\mathcal {B}}(E) \, {\mathcal {N}}_{N,V}^{\omega }(\mathrm{d} E) - \int _{{\mathbb {R}}} \chi _{\varepsilon , {\mathcal {E}}_2}(E) {\mathcal {B}}(E) \, {\mathcal {N}}_{\infty ,V}(\mathrm{d} E) \, \right| . \end{aligned}$$

The last term ${\mathbb {P}}$-almost surely converges to zero in the limit $N \rightarrow \infty$, because ${\mathcal {N}}_{N,V}^{\omega }$ ${\mathbb {P}}$-almost surely convergences in the vague sense to ${\mathcal {N}}_{\infty ,V}$ and $\chi _{\varepsilon , {\mathcal {E}}_2} \, {\mathcal {B}}$ is a continuous, compactly supported function. In addition, using the fact that ${\mathcal {B}}(E - \hat{\mu }_N^{\omega }) - {\mathcal {B}}(E) = (\mathrm{e}^{\beta E} - \mathrm{e}^{\beta (E - \hat{\mu }_N^{\omega })}) {\mathcal {B}}(E ) {\mathcal {B}}(E - \hat{\mu }_N^{\omega } )$ and inequality (5.1), we conclude that the first term in the limit $N \rightarrow \infty$ converges to zero as well. $\square$

Lemma A.3

Suppose V is a Poisson random potential of fixed strength. Let an $\varepsilon > 0$ and an $\omega \in \widetilde{\Omega }$ be arbitrarily given, and let $(\hat{\mu }_N^{\omega })_{N \in {\mathbb {N}}} \subset {\mathbb {R}}$ be a sequence that converges to zero. Then

$$\begin{aligned} \limsup \limits _{N \rightarrow \infty } \int _{(\varepsilon ,\infty )} {\mathcal {B}}(E - \hat{\mu }_N^{\omega }) \, {\mathcal {N}}_{N,V}^{\omega } (\mathrm{d} E) \le \int _{(\varepsilon ,\infty )} {\mathcal {B}}(E) \, {\mathcal {N}}_{\infty ,V}(\mathrm{d} E) + \dfrac{2}{\beta \varepsilon } {\mathcal {N}}_{\infty ,V}^{\mathrm{I}}(\varepsilon ) \end{aligned}$$

(A.4)

and

$$\begin{aligned} \liminf \limits _{N \rightarrow \infty } \int _{(\varepsilon ,\infty )} {\mathcal {B}}(E - \hat{\mu }_N^{\omega }) \, {\mathcal {N}}_{N,V}^{\omega } (\mathrm{d} E) \ge \int _{(\varepsilon ,\infty )} {\mathcal {B}}(E) \, {\mathcal {N}}_{\infty ,V}(\mathrm{d} E) - \dfrac{4}{\beta \varepsilon } {\mathcal {N}}_{\infty ,V}^{\mathrm{I}}(2\varepsilon ). \end{aligned}$$

(A.5)

Proof

We begin by showing (A.4). Let $E_2 > \varepsilon$ be arbitrarily given. Using the fact that the function ${\mathcal {B}}$ is monotonically decreasing, integration by parts, and the inequality ${\mathcal {N}}_{N,V}^{\mathrm{I},\omega }(E) \le \pi ^{-1} E^{1/2}$ for ${\mathbb {P}}$-almost all $\omega \in \Omega$, for all $E \ge 0$, and all $N \in {\mathbb {N}}$, we conclude

$$\begin{aligned} \int _{(E_2,\infty )} {\mathcal {B}}(E - \hat{\mu }_N^{\omega }) \, {\mathcal {N}}_{N,V}^{\omega } (\mathrm{d} E) \le \beta \pi ^{-1} \int _{E_2}^{\infty } E^{1/2} \big [ {\mathcal {B}}(E- \varepsilon /2) \big ]^{2} \mathrm{e}^{\beta (E - \varepsilon /2)} \, \mathrm{d} E \end{aligned}$$

for all but finitely many $N \in {\mathbb {N}}$, which converges to zero in the limit $E_2 \rightarrow \infty$. Thus, with $\chi _{\varepsilon , E_2}$ as in (A.2), with Lemma A.2, and since ${\mathcal {B}}(\varepsilon /2) \le 2/(\beta \varepsilon )$,

$$\begin{aligned} \limsup \limits _{N \rightarrow \infty } \int _{(\varepsilon ,\infty )} {\mathcal {B}}(E - \hat{\mu }_N^{\omega }) \, {\mathcal {N}}_{N,V}^{\omega } (\mathrm{d} E)&\le \lim \limits _{E_2 \rightarrow \infty } \int _{{\mathbb {R}}} \chi _{\varepsilon , E_2}(E) \, {\mathcal {B}}(E) \, {\mathcal {N}}_{\infty ,V}(\mathrm{d} E) \\&\le \int _{(\varepsilon ,\infty )} {\mathcal {B}}(E) \, {\mathcal {N}}_{\infty ,V}(\mathrm{d} E) + \dfrac{2}{\beta \varepsilon } {\mathcal {N}}_{\infty ,V}^{\mathrm{I}}(\varepsilon ). \end{aligned}$$

Inequality (A.5) similarly follows by using Lemma A.2 and inequality (5.1). $\square$

Lemma A.4

Let V be a Poisson random potential of fixed strength. Moreover, let $\omega \in \widetilde{\Omega }$ and a sequence $(\hat{\mu }_N^{\omega })_{N \in {\mathbb {N}}} \subset {\mathbb {R}}$ be given. If $(\hat{\mu }_N^{\omega })_{N \in {\mathbb {N}}}$ converges to zero, then we have

$$\begin{aligned}&\lim \limits _{\varepsilon \searrow 0} \limsup \limits _{N \rightarrow \infty } \int _{(\varepsilon ,\infty )} {\mathcal {B}}(E - \hat{\mu }_N^{\omega }) \, {\mathcal {N}}_{N,V}^{\omega }(\mathrm{d} E) \nonumber \\&\qquad = \lim \limits _{\varepsilon \searrow 0} \liminf \limits _{N \rightarrow \infty } \int _{(\varepsilon ,\infty )} {\mathcal {B}}(E - \hat{\mu }_N^{\omega }) \, {\mathcal {N}}_{N,V}^{\omega }(\mathrm{d} E) = \rho _{c,V}. \end{aligned}$$

(A.6)

Proof

This follows directly from Definition (2.16) and Lemma A.3. $\square$

Lemma A.5

Let $(V_N)_{N \in {\mathbb {N}}}$ be a Poisson random potential of fixed strength or with a strength that converges to infinity, and $\omega \in \widetilde{\Omega }$. Then the sequence $( \mu _{N,V_N}^{\omega } )_{N \in {\mathbb {N}}}$ has at least one accumulation point, and all accumulation points of $( \mu _{N,V_N}^{\omega } )_{N \in {\mathbb {N}}}$ are smaller than or equal to zero.

Proof

We follow [15, Theorem 4.1] in large parts. For all $N \in {\mathbb {N}}$ and $\omega \in \widetilde{\Omega }$, we define $\Phi _N^{\omega } := |\Lambda _N|^{-1} \sum _{j \in {\mathbb {N}}} \mathrm{e}^{-\beta E_{N,V_N}^{j, \omega }}$. Let $N \in {\mathbb {N}}$ and $\omega \in \widetilde{\Omega }$ be arbitrarily given. Using integration by parts and the fact that ${\mathcal {N}}_{N,V_N}^{\mathrm{I},\omega }(E) \le \pi ^{-1} E^{1/2}$ for ${\mathbb {P}}$-almost all $\omega \in \Omega$, for all $E \ge 0$, and all $N \in {\mathbb {N}}$, we obtain $0< \Phi _N^{\omega } \le \beta \int _{0}^{\infty } \pi ^{-1} E^{1/2} \mathrm{e}^{-\beta E} \, \mathrm{d} E < \infty .$ Furthermore, we conclude $\rho \le \Phi _N^{\omega } \mathrm{e}^{\beta \mu _{N,V_N}^{\omega }}/[1 - \mathrm{e}^{-\beta (E_{N,V_N}^{1,\omega } - \mu _{N,V_N}^{\omega })}]$ and $\beta ^{-1} \ln ( \rho /[\Phi _N^{\omega } + \rho \mathrm{e}^{-\beta E_{N,V_N}^{1,\omega }}]) \le \mu _{N,V_N}^{\omega }$. In addition, $\mu _{N,V_N}^{\omega }< E_{N,V_N}^{1,\omega }$ for ${\mathbb {P}}$-almost all $\omega \in \Omega$ and for all $N \in {\mathbb {N}}$ and the ground-state energy $E_{N,V_N}^{1,\omega }$ ${\mathbb {P}}$-almost surely converges to zero, see Theorem 4.1 and an appropriate version of Lemma B.2 (in combination with the Borel–Cantelli lemma). Thus, the sequence $( \mu _{N,V_N}^{\omega } )_{N \in {\mathbb {N}}}$ ${\mathbb {P}}$-almost surely is bounded and accordingly has at least one accumulation point. In addition, this shows that ${\mathbb {P}}$-almost surely every accumulation point of $( \mu _{N,V_N}^{\omega } )_{N \in {\mathbb {N}}}$ is equal to or smaller than zero. $\square$

Lemma A.6

Let $(V_N)_{N \in {\mathbb {N}}}$ be a Poisson random potential of fixed strength or with a strength that converges to infinity. If $\rho \ge \rho _{c,V_1}$, then $\lim _{N \rightarrow \infty } \mu _{N,V_N}^{\omega } = 0$ ${\mathbb {P}}$-almost surely.

Proof

Let an $\omega \in \widetilde{\Omega }$ be arbitrarily given. According to Lemma A.5, $\limsup _{N \rightarrow \infty } \mu _{N,V_N}^{\omega } \le 0$. Suppose $\liminf _{N \rightarrow \infty } \mu _{N,V_N}^{\omega } < 0$. Then there is a subsequence $(N_m)_{m \in {\mathbb {N}}}$ of $(N)_{N \in {\mathbb {N}}}$ and a constant $\mu ^{\omega }_{\infty } < 0$ such that $\lim _{m \rightarrow \infty } \mu _{N_m,V_{N_m}}^{\omega } = \mu ^{\omega }_{\infty }$. Because we have $\rho = \int _{(0,\infty )} {\mathcal {B}}(E - \mu _{N, V_N}^{\omega }) \, {\mathcal {N}}_{N, V_N}^{\omega }(\mathrm{d} E)$ for all $N \in {\mathbb {N}}$, by integration by parts, and by Lemma A.1,

$$\begin{aligned} \rho&\le \lim \limits _{m \rightarrow \infty } \lim \limits _{a \rightarrow 0} \lim \limits _{b \rightarrow \infty } \Big [ \int _{[a,b]} {\mathcal {N}}_{N_m, V_{N_m}}^{\mathrm{I}, \omega }(E-) \, ( - {\mathcal {B}}'(E - \mu _{N_m, V_{N_m}}^{\omega })\, \mathrm{d} E \\&\qquad \qquad \qquad \qquad \qquad \qquad + {\mathcal {N}}_{N_m, V_{N_m}}^{\mathrm{I}, \omega }(b+) {\mathcal {B}}(b - \mu _{N_m, V_{N_m}}^{\omega }) \Big ]\\&\le \lim \limits _{m \rightarrow \infty } \lim \limits _{a \rightarrow 0} \lim \limits _{b \rightarrow \infty } \Big [ \int _{[a,b]} {\mathcal {B}}(E - \mu _{N_m, V_{N_m}}^{\omega }) \, {\mathcal {N}}_{N_m, V_1}^{\omega }(\mathrm{d} E) \\&\qquad \qquad \qquad \qquad \qquad \qquad + {\mathcal {N}}_{N_m, V_1}^{\mathrm{I}, \omega }(a-) {\mathcal {B}}(a - \mu _{N_m, V_{N_m}}^{\omega }) \Big ]\\&\le \lim \limits _{m \rightarrow \infty } \int _{(0,\infty )} {\mathcal {B}}(E - \mu _{\infty }^{\omega }/2) \, {\mathcal {N}}_{N_m, V_1}^{\omega }(\mathrm{d} E) \\&= \int _{(0,\infty )} {\mathcal {B}}(E - \mu _{\infty }^{\omega }/2) \, {\mathcal {N}}_{\infty ,V_1}(\mathrm{d} E) \\&< \rho _{c,V_1}. \end{aligned}$$

$\square$

We finally prove the ${\mathbb {P}}$-almost sure occurrence of g-BEC for sufficiently large particle densities.

Theorem A.7

Let $(V_N)_{N \in {\mathbb {N}}}$ be a Poisson random potential of fixed strength or with a strength that converges to infinity. Then we have

$$\begin{aligned} {\mathbb {P}} \Bigg ( \lim \limits _{\varepsilon \searrow 0} \liminf \limits _{N \rightarrow \infty } \dfrac{1}{N} \sum \limits _{j \in {\mathbb {N}} : E_{N,V_N}^{j,\omega } \le \varepsilon } n_{N,V_N}^{j,\omega } \ge \dfrac{\rho - \rho _{c,V_1}}{\rho } \Bigg ) = 1. \end{aligned}$$

(A.7)

Proof

Suppose $\rho \ge \rho _{c,V_1}$. Using Lemmas A.4 and A.6 and proceeding similarly as in the proof of Proposition 5.2, we ${\mathbb {P}}$-almost surely conclude

$$\begin{aligned}&\lim \limits _{\varepsilon \searrow 0} \limsup \limits _{N \rightarrow \infty } \int _{(\varepsilon ,\infty )} {\mathcal {B}}(E - \mu _{N,V_N}^{\omega }) \, \, {\mathcal {N}}_{N,V_N}^{\omega }(\mathrm{d} E) \\&\quad \le \lim \limits _{\varepsilon \searrow 0} \limsup \limits _{N \rightarrow \infty } \lim \limits _{M \rightarrow \infty } \Big [ \int _{[\varepsilon ,M]} {\mathcal {N}}_{N,V_N}^{\mathrm{I},\omega }(E-) \, ( - {\mathcal {B}}'(E - \mu _{N,V_N}^{\omega })) \, \mathrm{d}E \\&\qquad \qquad \qquad \qquad \qquad \qquad + {\mathcal {N}}_{N,V_N}^{\mathrm{I},\omega }(M+) \, {\mathcal {B}}(M - \mu _{N,V_N}^{\omega })\Big ] \\&\quad \le \lim \limits _{\varepsilon \searrow 0} \limsup \limits _{N \rightarrow \infty } \Big [ \lim \limits _{M \rightarrow \infty } \int _{[\varepsilon ,M]} {\mathcal {B}}(E - \mu _{N,V_N}^{\omega }) \, \, {\mathcal {N}}_{N,V_1}^{\omega }(\mathrm{d} E) + {\mathcal {N}}_{N,V_1}^{\mathrm{I},\omega }(\varepsilon -) \, {\mathcal {B}}(\varepsilon - \mu _{N,V_N}^{\omega })\Big ] \\&\quad = \rho _{c,V_1}. \end{aligned}$$

$\square$

Appendix B: Further Properties of the Poisson Point Process on ${\mathbb {R}}$

In this section, we collect further properties of the Poisson point process on ${\mathbb {R}}$ that we used in Sect. 2. For the first lemma, recall that $\kappa _N^{\omega }$ is the number of atoms of ${\mathcal {M}}_{\nu }^{\omega }$ within $\Lambda _N$.

Lemma B.1

For all $0< \varepsilon < 1/2$ and for all but finitely many $N \in {\mathbb {N}}$ we have

$$\begin{aligned} {\mathbb {P}} \Big ( (1-L_N^{-\varepsilon }) L_N \le \kappa _N^{\omega } \le (1+L_N^{-\varepsilon }) L_N \Big ) \ge 1 - 2 \mathrm{e}^{ - (\nu /3) L_N^{1-2\varepsilon }}. \end{aligned}$$

(B.1)

Proof

Let $0< \varepsilon < 1/2$ be arbitrarily given. For all $N \in {\mathbb {N}}$ we have

$$\begin{aligned} {\mathbb {P}}\left( \kappa _N^{\omega } \ge (1 + L_N^{-\varepsilon }) \nu L_N \right)&= \sum \limits _{m = (1 + L_N^{-\varepsilon }) \nu L_N}^{\infty } \mathrm{e}^{-\nu L_N} \dfrac{(\nu L_N)^m}{m!} \\&\le \mathrm{e}^{\nu L_N[L_N^{-\varepsilon } - (1 + L_N^{-\varepsilon }) \ln (1 + L_N^{-\varepsilon })]}, \end{aligned}$$

see also [23, Sect. 3.3.2]. Next, by using the inequality $(1+x) \ln (1+x) - x \ge x^2/(\frac{2}{3} x + 2)$ for all $x > -1$, we obtain

$$\begin{aligned} {\mathbb {P}}\left( \kappa _N^{\omega } \ge (1 + L_N^{-\varepsilon }) \nu L_N \right) \le \mathrm{e}^{- \nu L_N L_N^{-2\varepsilon } (\tfrac{2}{3} L_N^{-\varepsilon } + 2)^{-1}} \end{aligned}$$

for all $N \in {\mathbb {N}}$. Similarly,

$$\begin{aligned} {\mathbb {P}} ( \kappa _N^{\omega } \le (1 - L_N^{-\varepsilon }) \nu L_N ) \le \mathrm{e}^{- \nu L_N L_N^{-2\varepsilon } (-\tfrac{2}{3} L_N^{-\varepsilon } + 2)^{-1}} \end{aligned}$$

for all $N \in {\mathbb {N}}$ such that $L_N^{-\varepsilon } < 1$. $\square$

Lemma B.2

For all $\zeta > 0$,

$$\begin{aligned} \lim \limits _{N \rightarrow \infty } {\mathbb {P}} \left( (1-\zeta ) \nu ^{-1} \ln (L_N) \le l_{N,>}^{(1),\omega }\le (1+\zeta ) \nu ^{-1} \ln (L_N) \right) = 1. \end{aligned}$$

(B.2)

Proof

Let $\zeta > 0$ be arbitrarily given. Since ${\mathbb {P}}$-almost surely $\lim _{N \rightarrow \infty } \kappa _N^{\omega } / L_N = \nu$ and $(\hat{l}^{j})_{j \in {\mathbb {Z}} \backslash \{0\}}$ are independent and identically distributed random variables with common probability density $\nu \mathrm{e}^{-\nu l} \mathbf {1}_{(0,\infty )}(l)$, $l \in {\mathbb {R}}$, we have

$$\begin{aligned} \limsup \limits _{N \rightarrow \infty } {\mathbb {P}} \left( l_{N,>}^{(1),\omega }> (1+\zeta ) \nu ^{-1} \ln (L_N) \right) \le \limsup \limits _{N \rightarrow \infty } \left( 1 - (1 - L_N^{-1-\zeta })^{(1+\varepsilon )\nu L_N} \right) \end{aligned}$$

for any $\varepsilon > 0$. By using the inequality $1-x^{-1} \le \ln (x) \le x-1$ for all $x > 0$, one shows that this probability converges to zero in the limit $N \rightarrow \infty$. Similarly, one has

$$\begin{aligned} \limsup \limits _{N \rightarrow \infty } {\mathbb {P}} \left( l_{N,>}^{(1),\omega }< (1-\zeta ) \nu ^{-1} \ln (L_N) \right) \le \limsup \limits _{N \rightarrow \infty } \big ( 1 - L_N^{-1+\zeta } \big )^{(1-\varepsilon ) \nu L_N} \end{aligned}$$

for any $0< \varepsilon < 1$ where the right-hand side converges to zero in the limit $N \rightarrow \infty$ as well. For more details, see also [25, Chap. 3, Lemma 3.4] or [11, Lemma A.5]. $\square$

Let $( \tilde{l}^{j})_{j \in {\mathbb {N}}}$ be a sequence of independent and identically distributed random variables with common probability density $\nu \mathrm{e}^{-\nu l} \mathbf {1}_{(0,\infty )}(l)$, $l \in {\mathbb {R}}$. For ${\mathbb {P}}$-almost all $\omega \in \Omega$, we define $\tilde{l}_{k,>}^{(1),\omega }$ and $\tilde{l}_{k,>}^{(2),\omega }$ as the largest and the second largest element of the set $\{ \tilde{l}^{j,\omega }\}_{j=1}^{k}$, respectively.

Lemma B.3

For any bounded sequence $(c_N)_{N \in {\mathbb {N}}} \subset [0,\infty )$ and any $0< \varepsilon < 1/2$ we have

$$\begin{aligned}&\liminf \limits _{N \rightarrow \infty } {\mathbb {P}}\Big ( l_{N,>}^{(1),\omega }- l_{N,>}^{(2),\omega }> c_N \Big ) \nonumber \\&\quad \ge \liminf \limits _{N \rightarrow \infty } {\mathbb {P}} \Big ( \tilde{l}_{\lfloor (1 - L_N^{-\varepsilon }) \nu L_N \rfloor - 2,>}^{(1),\omega } - \tilde{l}_{\lfloor (1 - L_N^{-\varepsilon }) \nu L_N \rfloor - 2,>}^{(2),\omega } > c_N \Big ). \end{aligned}$$

(B.3)

Proof

Let a bounded sequence $(c_N)_{N \in {\mathbb {N}}} \subset [0,\infty )$ and an $0< \varepsilon < 1/2$ be arbitrarily given. For each $\omega \in \widetilde{\Omega }$ and $N \in {\mathbb {N}}$, let $\kappa _N^{(\text {L}),\omega }$ and $\kappa _N^{(\text {R}),\omega }$ be the number of atoms of ${\mathcal {M}}_{\nu }^{\omega }$ within $(-L_N/2,0)$ and within $(0,L_N/2)$, respectively. Proceeding almost identically as in the proof of Lemma B.1 one can show that

$$\begin{aligned} \lim _{N \rightarrow \infty } {\mathbb {P}} \Big ( \big \{ (1-L_N^{-\varepsilon }) L_N/2 \le \kappa _N^{(\text {L}),\omega }, \kappa _N^{(\text {R}),\omega } \le (1+L_N^{-\varepsilon }) L_N/2 \big \} \Big ) = 1. \end{aligned}$$

For convenience, we define the sets $J_k:= \{ -k, -k+1, \ldots , k-1, k \}\backslash \{0\}$, $k \in {\mathbb {N}}$ as well as, for all $N \in {\mathbb {N}}$ and $\omega \in \widetilde{\Omega }$,

$$\begin{aligned} \ell _N^{1,\omega }&:= \big \{\hat{l}^{j,\omega } : j \in J_{\lfloor (1 - L_N^{-\varepsilon })L_N/2 \rfloor - 2} \big \}, \\ \ell _N^{2,\omega }&:= \big \{\hat{l}^{j,\omega } : j \in J_{\lceil (1+ L_N^{-\varepsilon }) L_N/2 \rceil } \big \}, \\ \end{aligned}$$

and

$$\begin{aligned} \ell _N^{3,\omega } := \big \{\hat{l}^{j,\omega } : j \in J_{\lceil (1+ L_N^{-\varepsilon }) L_N/2 \rceil } \backslash J_{\lfloor (1 - L_N^{-\varepsilon })L_N/2 \rfloor - 2} \big \}. \end{aligned}$$

Using the inequality $-x/(1-x) \le \ln (1-x) \le -x$ for all $0<x<1$ we have

$$\begin{aligned} \lim \limits _{N \rightarrow \infty } {\mathbb {P}} \Big ( \max (\ell _N^{3,\omega }) \le (1 - \varepsilon /2) \nu ^{-1} \ln (L_N) \Big ) = \lim \limits _{N \rightarrow \infty } (1 -L_N^{-1+\varepsilon /2})^{2\nu L_N^{1-\varepsilon }} = 1 \end{aligned}$$

and

$$\begin{aligned} \lim \limits _{N \rightarrow \infty } {\mathbb {P}} \Big (\max (\ell _N^{1,\omega }) \ge (1-\varepsilon /3)\nu ^{-1} \ln (L_N) \Big ) = 1. \end{aligned}$$

Furthermore, the two outer intervals within $\Lambda _N$ and the interval that contains the zero are relatively small, that is, $\lim \limits _{N \rightarrow \infty } {\mathbb {P}}(\Omega _N^{(\text {R})} \cap \Omega _N^{(\text {L})} \cap \Omega _N^{(0)}) = 1$ where $\Omega _N^{(\text {R})} := \{ \omega \in \widetilde{\Omega } : |I_N^{(\text {R}),\omega }| \le (2\nu )^{-1} \ln (L_N) \}$, $\Omega _N^{(\text {L})} := \{ \omega \in \widetilde{\Omega } : |I_N^{(\text {L}),\omega }| \le (2\nu )^{-1} \ln (L_N) \}$, and $\Omega _N^{(0)} := \{ \omega \in \widetilde{\Omega } : |I_N^{0,\omega }| \le (2\nu )^{-1} \ln (L_N) \}$ with $I_N^{(\text {R}),\omega } := I_N^{j_N^{\text {max},\omega },\omega }$, $j_N^{\text {max},\omega } := \max \{ j \in {\mathbb {Z}}: I_N^{j,\omega } \ne \emptyset \}$, and $I_N^{(\text {L}),\omega } := I_N^{j_N^{\text {min},\omega },\omega }$, $j_N^{\text {min},\omega } := \min \{ j \in {\mathbb {Z}} : I_N^{j,\omega } \ne \emptyset \}$ for all $N \in {\mathbb {N}}$ and all $\omega \in \widetilde{\Omega }$. Also, for any $\omega \in \widetilde{\Omega }$ and any $2 \le k \in {\mathbb {N}}$ the second-largest element of the set $\{ \hat{l}^{j,\omega } \in J_k\}$ is given by $\max \big ( \{ \hat{l}^{j,\omega } \in J_k \} \backslash \big \{ \max ( \{ \hat{l}^{j,\omega } \in J_k \} ) \big \} \big )$. For convenience, we define for all $N \in {\mathbb {N}}$

$$\begin{aligned} \widehat{\Omega }_N := \Omega _N^{(\text {R})}&\cap \Omega _N^{(\text {L})} \cap \Omega _N^{(0)} \\&\cap \big \{ (1-L_N^{-\varepsilon })L_N/2 \le \kappa _N^{(\text {L}),\omega }, \kappa _N^{(\text {R}),\omega } \le (1+L_N^{-\varepsilon }) L_N/2 \big \}. \end{aligned}$$

Let $N \in {\mathbb {N}}$ be given. We have

$$\begin{aligned} {\mathbb {P}} \Big ( l_{N,>}^{(1),\omega }- l_{N,>}^{(2),\omega }> c_N \Big )&\ge {\mathbb {P}} \Big ( \big \{ l_{N,>}^{(1),\omega }- l_{N,>}^{(2),\omega }> c_N \big \} \cap \widehat{\Omega }_N \\&\quad \cap \big \{ \max (\ell _N^{3,\omega } ) \le (1 - \varepsilon /2) \nu ^{-1} \ln (L_N) \big \} \\&\quad \cap \big \{ \max (\ell _N^{1,\omega } ) \ge (1 - \varepsilon /3) \nu ^{-1} \ln (L_N) \big \} \Big ). \end{aligned}$$

Since the second largest element of the set $\ell _N^{2,\omega }$ is either in the set $\ell _N^{1,\omega }$ or in the set $\ell _N^{3,\omega }$, we conclude

$$\begin{aligned}&{\mathbb {P}} \Big ( l_{N,>}^{(1),\omega }- l_{N,>}^{(2),\omega }> c_N \Big ) \\&\quad \ge {\mathbb {P}} \Big ( \Big \{ \max ( \ell _N^{1,\omega } ) - \max \Big ( \ell _N^{1,\omega } \backslash \big \{ \max ( \ell _N^{1,\omega } ) \big \} \Big )> c_N \Big \} \\&\qquad \cap \Big \{ \max \Big ( \ell _N^{2,\omega } \backslash \big \{ \max \big ( \ell _N^{2,\omega } \big ) \big \} \Big ) \in \ell _N^{1,\omega } \Big \} \cap \widehat{\Omega }_N \\&\qquad \cap \big \{ \max \ell _N^{3,\omega } \le (1 - \varepsilon /2) \nu ^{-1} \ln (L_N) \big \} \\&\qquad \cap \big \{ \max \ell _N^{1,\omega } \ge (1 - \varepsilon /3) \nu ^{-1} \ln (L_N) \big \} \Big ) \\&\qquad + {\mathbb {P}} \Big ( \Big \{ \max (\ell _N^{1,\omega }) - \max \Big ( \ell _N^{2,\omega } \backslash \big \{ \max ( \ell _N^{1,\omega } ) \big \} \Big ) > c_N \Big \} \\&\qquad \cap \Big \{ \max \Big ( \ell _N^{2,\omega } \backslash \big \{ \max \big ( \ell _N^{2,\omega } \big ) \big \} \Big ) \in \ell _N^{3,\omega } \Big \} \cap \widehat{\Omega }_N \\&\qquad \cap \big \{ \max \ell _N^{3,\omega } \le (1 - \varepsilon /2) \nu ^{-1} \ln (L_N) \big \} \\&\qquad \cap \big \{ \max \ell _N^{1,\omega } \ge (1 - \varepsilon /3) \nu ^{-1} \ln (L_N) \big \} \Big ). \end{aligned}$$

Furthermore, since ${\mathbb {P}}(A \cap B) \ge {\mathbb {P}}(A) + {\mathbb {P}}(B) - 1$ for any events $A,B \subset \Omega$,

$$\begin{aligned}&{\mathbb {P}} \Big ( l_{N,>}^{(1),\omega }- l_{N,>}^{(2),\omega }> c_N \Big ) \\&\quad \ge {\mathbb {P}} \Big ( \tilde{l}_{\lfloor (1 - L_N^{-\varepsilon }) \nu L_N \rfloor - 2,>}^{(1),\omega } - \tilde{l}_{\lfloor (1 - L_N^{-\varepsilon }) \nu L_N \rfloor - 2,>}^{(2),\omega } > c_N \Big ) \\&\qquad + {\mathbb {P}} \Big ( \max \Big ( \ell _N^{2,\omega } \backslash \big \{ \max \big ( \ell _N^{2,\omega } \big ) \big \} \Big ) \in \ell _N^{1,\omega } \Big \} \Big ) - 1\\&\qquad + {\mathbb {P}} \Big ( \max \Big ( \ell _N^{2,\omega } \backslash \big \{ \max \big ( \ell _N^{2,\omega } \big ) \big \} \Big ) \in \ell _N^{3,\omega } \Big ) + 2 {\mathbb {P}} ( \widehat{\Omega }_N ) - 2\\&\qquad + 2 {\mathbb {P}} \big ( \max \ell _N^{3,\omega } \le (1 - \varepsilon /2) \nu ^{-1} \ln (L_N) \big ) - 2\\&\qquad + 2 {\mathbb {P}} \big ( \max \ell _N^{1,\omega } \ge (1 - \varepsilon /3) \nu ^{-1} \ln (L_N) \big \} \big ) - 2 \end{aligned}$$

for all but finitely many $N \in {\mathbb {N}}$. Lastly, by also using Lemmata B.1 and B.2 as well as the fact that ${\mathbb {P}}(A) + {\mathbb {P}}(A^c) =1$ for any event $A \subset \Omega$,

$$\begin{aligned}&\liminf \limits _{N \rightarrow \infty } {\mathbb {P}} \Big ( l_{N,>}^{(1),\omega }- l_{N,>}^{(2),\omega }> c_N \Big ) \\&\quad \ge \liminf \limits _{N \rightarrow \infty } {\mathbb {P}} \Big ( \tilde{l}_{\lfloor (1 - L_N^{-\varepsilon }) \nu L_N \rfloor - 2,>}^{(1),\omega } - \tilde{l}_{\lfloor (1 - L_N^{-\varepsilon }) \nu L_N \rfloor - 2,>}^{(2),\omega } > c_N \Big ). \end{aligned}$$

$\square$

Appendix C: The Luttinger–Sy Model

In Sects. 2 and 3 we mentioned the Luttinger–Sy model. For the convenience of the reader, we provide more details regarding these models in this section.

The Luttinger–Sy model is a system of noninteracting bosonic particles corresponding to the sequence of one-particle random Schrödinger operators $(H_{N,\mathrm{LS}})_{N \in {\mathbb {N}}}$ that are defined, for ${\mathbb {P}}$-almost all $\omega \in \Omega$ and all $N \in {\mathbb {N}}$, as the strong resolvent limit $\gamma \rightarrow \infty$ of the self-adjoint operators defined by the quadratic form $q^{\omega }_{N,\gamma }$, see (C.3). Informally, it consists of noninteracting bosons on the real line in a Poisson random potential on ${\mathbb {R}}$ with the single-site potential $\gamma \delta$ where $\delta$ is the Dirac delta function and $\gamma = \infty$ [14, Sect. 2]. ${\mathbb {P}}$-almost surely, the one-particle Schrödinger operator $H_{N,\mathrm{LS}}^{\omega }$ has, for each $N \in {\mathbb {N}}$, a purely discrete spectrum. We denote its eigenvalues, arranged in increasing order and each eigenvalue repeated according to its multiplicity, by $E_{N,\mathrm{LS}}^{j,\omega }$, $j \in {\mathbb {N}}$. The sequence of the integrated densities of states $({\mathcal {N}}_{N, \mathrm{LS}}^{\mathrm{I},\omega })_{N \in {\mathbb {N}}}$ of the Luttinger–Sy model ${\mathbb {P}}$-almost surely converges pointwise to the nonrandom limiting integrated density of states

$$\begin{aligned} {\mathcal {N}}_{\infty , \mathrm{LS}}^{\mathrm{I}} : {\mathbb {R}} \rightarrow [0,\infty ), \ E \mapsto {\mathcal {N}}_{\infty , \mathrm{LS}}^{\mathrm{I}}(E) = \nu \dfrac{\mathrm{e}^{-\nu \pi E^{-1/2}}}{1 - \mathrm{e}^{-\nu \pi E^{-1/2}}} \mathbf {1}_{(0,\infty )}(E). \end{aligned}$$

(C.1)

In addition, the critical density $\rho _{c,\mathrm{LS}} = \int _{{\mathbb {R}}} {\mathcal {B}}(E) \, {\mathcal {N}}_{\infty ,\mathrm{LS}}(\mathrm{d} E)$ is finite, for any $\beta > 0$.

The Luttinger–Sy model of finite strength is a one-dimensional noninteracting Bose gas in a Poisson random potential on ${\mathbb {R}}$ with, informally, a single-site potential

$$\begin{aligned} u : {\mathbb {R}} \rightarrow {\mathbb {R}}, \ x \mapsto u(x) := \gamma \delta (x) \end{aligned}$$

(C.2)

where $\gamma > 0$ and $\delta$ is the Dirac delta function. Rigorously, the corresponding one-particle Schrödinger operator $H_{N,\gamma }^{\omega }$ is, for all $N \in {\mathbb {N}}$ and ${\mathbb {P}}$-almost all $\omega \in \Omega$, the self-adjoint operator on $\mathrm{L}^2(\Lambda _N)$ that is uniquely defined via the quadratic form

$$\begin{aligned} q^{\omega }_{N,\gamma }: \mathrm{H}^1_0(\Lambda _N) \times \mathrm{H}^1_0(\Lambda _N)&\rightarrow {\mathbb {C}} \nonumber \\ (\varphi ,\psi )&\mapsto \int _{\Lambda _N}\overline{\varphi ^{\prime }(x)}\psi ^{\prime }(x) \, \mathrm{d}x + \gamma \sum _{j \in {\mathbb {Z}}: \, \hat{x}_j^{\omega } \in \Lambda _N}\overline{\varphi (\hat{x}_j^{\omega })} \psi (\hat{x}_j^{\omega }) \end{aligned}$$

(C.3)

on $\mathrm{L}^2(\Lambda _N)$. For more details regarding this model, see, e.g., [19, pp. 146–149] and [10].