On the spectral gap in the Kac-Luttinger model and Bose-Einstein condensation

We consider the Dirichlet eigenvalues of the Laplacian among a Poissonian cloud of hard spherical obstacles of fixed radius in large boxes of $\mathbb{R}^d$, $d \ge 2$. In a large box of side-length $2l$ centered at the origin, the lowest eigenvalue is known to be typically of order $(\log l)^{-2/d}$. We show here that with probability arbitrarily close to $1$ as $l$ goes to infinity, the spectral gap stays bigger than $\sigma (\log l)^{-(1 + 2/d)}$, where the small positive number $\sigma$ depends on how close to $1$ one wishes the probability. Incidentally, the scale $(\log l)^{-(1+ 2/d)}$ is expected to capture the correct size of the gap. Our result involves the proof of new deconcentration estimates. Combining this lower bound on the spectral gap with the results of Kerner-Pechmann-Spitzer, we infer a type-I generalized Bose-Einstein condensation in probability for a Kac-Luttinger system of non-interacting bosons among Poissonian spherical impurities, with the sole macroscopic occupation of the one-particle ground state when the density exceeds the critical value.


Introduction
In this article we are interested in the Kac-Luttinger model [12], [13], and consider the Dirichlet eigenvalues of the Laplacian among a Poissonian cloud of hard spherical obstacles in large boxes of R d , d ≥ 2. Our main result states an asymptotic lower bound for the spectral gap, that is, for the difference between the second lowest and the lowest Dirichlet eigenvalues. In a large box of side-length 2ℓ centered at the origin, the lowest eigenvalue is known to be typically of order (log ℓ) −2 d . We show here that with probability arbitrarily close to 1 as ℓ goes to infinity, the spectral gap stays bigger than σ(log ℓ) −(1+2 d) , where the small positive number σ depends on how close to 1 one wishes the probability. Incidentally, (log ℓ) −(1+2 d) is expected to capture the correct size of the spectral gap. Whereas detailed information on the statistics of eigenvalues in large boxes is known for various kinds of random potentials, see [3], [11], Chapter 6 §3 of [17], or Section 6 of [2], much less seems to be known in the case of hard or soft Poissonian obstacles. Part of the difficulty stems from the delicate nature of the competition between the so-called "clearings". Loosely speaking, these are nearly spherical pockets of rough size (log ℓ) 1 d with a rarefied presence of the obstacles that underpin low eigenvalues. Various facets of these questions emerge in related studies concerning Brownian motion in a Poissonian potential and kindred models, see [28], [17], [2], and references therein, as well as [6], [5], and [25] for some recent developments. In the present work we bring new deconcentration estimates. We also have a different purpose. Combining the lower bound on the spectral gap, the known Lifshitz tail behavior of the model, and the results of Kerner, Pechmann and Spitzer in [16], we infer a so-called type-I generalized Bose-Einstein condensation in probability for a Kac-Luttinger system of non-interacting bosons among Poissonian spherical impurities in the thermodynamic limit, which appears to be novel, see [19], [16], [24].
We now describe the results in more detail, and refer to Section 1 for additional notation and references. We consider R d , d ≥ 2, and the canonical law P on the space Ω of locally finite simple point measures on R d , of a Poisson cloud of constant intensity ν > 0. To each point of the cloud we attach a spherical obstacle corresponding to a closed ball of radius a > 0 centered at the point. Given ℓ > 0 and the open box of side-length 2ℓ centered at the origin B ℓ = (−ℓ, ℓ) d , we denote by B ℓ,ω the complement of the obstacle set in B ℓ . We write for the successive Dirichlet eigenvalues of − 1 2 ∆ in B ℓ,ω , repeated according to their multiplicity, with the convention that λ i,ω (B ℓ ) = ∞ for all i ≥ 1, when B ℓ,ω = ∅ (otherwise all λ i,ω (B ℓ ) are finite).
We let ω d stand for the volume of an open ball of radius 1, λ d for the corresponding principal Dirichlet eigenvalue of − 1 2 ∆, and define , the radius of a ball of volume d ν , (0.2) , the principal Dirichlet eigenvalues of − 1 2 ∆ in an open ball (0. 3) of radius R 0 .
One knows from Theorem 4.6, p. 191 of [28] (which treats the more delicate case of soft obstacles, see (1.18), but remains valid in the present context) that there exists χ ∈ (0, d) and γ > 0 such that on a set of full P-measure, for large ℓ (0.4) The main result of the present article is Theorem 6.1. It proves an asymptotic lower bound on the spectral gap. Namely, it shows that As an aside, it is plausible that (log ℓ) −(1+2 d) captures the correct size of the spectral gap, so that its product with (log ℓ) 1+2 d remains tight as ℓ → ∞: the bounds on the fluctuations of λ 1,ω (B ℓ ) derived in Section 3 of [27] and their link with the behavior of the spectral gap studied here, make a compelling case, see Remark 6. 5 1). It should also be pointed out that the discrepancy between the upper and lower bounds in (0.4) is much bigger than (log ℓ) −(1+2 d) , and the proof of (0.5) does not consist in getting a "good lower bound" on λ 2,ω (B ℓ ), and a "good upper bound" on λ 1,ω (B ℓ ). Perhaps, to illustrate the significance of the scale (log ℓ) −(1+2 d) , one can mention that when λ(ℓ) ∼ c 0 (log ℓ) −2 d , as ℓ → ∞, and λ ′ (ℓ) = λ(ℓ) + σ(log ℓ) −(1+2 d) , the balls with principal Dirichlet eigenvalues for − 1 2 ∆ corresponding to λ and λ ′ have volumes equivalent to d ν (log ℓ), as ℓ → ∞, but the difference of their volumes ω d (λ d λ) d 2 − ω d (λ d λ ′ ) d 2 tends to the constant d 2 σ (2c 0 ν), see also (0.9).
With this in mind let us describe the strategy of the proof of Theorem 6.1 (see (0.5)). It first involves showing in Theorem 4.1, with the help of the method of enlargement of obstacles, see Chapter 4 of [28], and quantitative Faber-Krahn inequalities, see [4] and [10], that for large ℓ on most of the event in (0.5) there are distant ballsB andB ′ in B ℓ with same radiusR slightly bigger than R 0 (log ℓ) 1 d , see (0.2), such that the principal Dirichlet eigenvalues λ 1,ω (B) and λ 1,ω (B ′ ), see (1.9), are both within close range (slightly bigger than σ(log ℓ) −(1+2 d) ) of λ 1,ω (B ℓ ).
The control of (0.7) and the resulting lower bound on the spectral gap (0.5), i.e. Theorem 6.1, hinges on the main deconcentration estimate (6.2) in Theorem 6.2, which embodies a central new aspect of this work. It allows to dominate up to a multiplicative constant the probability that appears in (0.7) by P[λ 1,ω (D 0 ) ∈ J i ] for any i, for a suitable (large) collection J 1 , . . . , J m of pairwise disjoint sub-intervals of (0, t ℓ ). Whereas increasing λ 1,ω (D 0 ) is a comparatively easier task (it can be achieved by the addition of one single obstacle in a location where ϕ 1,D 0 ,ω is not too small), decreasing λ 1,ω (D 0 ) in a controlled fashion, as required by the constraint that the many disjoint J 1 , . . . , J m remain in (0, t ℓ ), is more delicate. For this task we do not leverage the geometric information concerning the underlying near spherical clearings of Theorem 4.1. This information is anyway too coarse. We instead perform a "gentle expansion" of the Poisson cloud by homotheties of ratio exp{u i D 0 } for suitable u 1 < ⋅ ⋅ ⋅ < u m in (0, 1) in the proof of Theorem 6.2, see (6.49). They dilute the Poisson point process and tendentially decrease eigenvalues. This is tailored so that the expansion corresponding to u i takes λ 1,ω (D 0 ) from [t, t + ε ℓ ] to J i . Incidentally, the constraint on ϕ 1,D 0 ,ω in (0.7) is important and ensures a proper centering of the underlying clearing in D 0 , shielding it from damage at the boundary of D 0 under the gentle expansion. We also refer to Lemma 3.3 of [5] for other kinds of transformations involving the removal of obstacles, which however do not seem adequate for the task of proving (6.2), and to the proof of Proposition 1.12 of [8] for a recent instance of deconcentration estimates in a percolation context. Let us also mention that unlike the results of the previous sections, which rather straightforwardly could be adapted to the case of Poissonian soft obstacles, see (1.18), the proof of the deconcentration estimates in Theorem 6.2 makes a genuine use of the hard spherical Poissonian obstacles (and the continuity of the space, as in [8]).
Owing to the work Kerner-Pechmann-Spitzer [16], our results have a natural application to a version of the problem investigated by Kac and Luttinger in [12], [13] concerning the Bose-Einstein condensation of a gas of non-interacting bosons among hard obstacles corresponding to balls of radius a > 0 centered at the points of a Poisson cloud of intensity ν > 0 on R d , d ≥ 2. In the one-dimensional case we refer to the results in [15], [16] for the related Luttinger-Sy model, and in [24] for soft Poissonian obstacles.
In the model under consideration here, one knows, see (7.9), (7.10), that the density of states m(dλ) is a measure on R + with Laplace transform: where E t 0,0 denotes the expectation for a Brownian bridge in time t in R d from the origin to itself, and W a t stands for the Wiener sausage in time t and radius a of that bridge (i.e. the closed a-neighborhood of the bridge trajectory). The density of states has a so-called Lifshitz tail behavior, see for instance Corollary 3.5 of [26], the original proof going back to the work of Donsker-Varadhan [7], see also Chapter 10.B of [22]: One then has (see Section 2 of [16]) a finite critical density for the system at inverse temperature β > 0: Thus, in the thermodynamic limit, for a fixed density ρ ∈ (0, ∞), one lets the particle number N ≥ 1 and the length scales ℓ N , such that N = ρ B ℓ N tend to infinity. One then defines suitably truncated λ j,ω (B ℓ N ), j ≥ 1, see (7.5), which coincide with the λ j,ω (B ℓ N ), j ≥ 1, when B ℓ N ,ω = ∅ (an increasing sequence of events with P-probability tending to 1), and corresponding occupation numbers n j,ω N of the j-th eigenstate for a grand-canonical version of non-interacting bosons with density ρ in B ℓ N in the presence of the spherical impurities attached to ω, see (7.7). As an application of the lower bound on the spectral gap in Theorem 6.1 and the results in [16], we show in Theorem 7.1 that when ρ > ρ c (β) a so-called type-I Bose-Einstein condensation in probability in a single mode takes place: as N → ∞, n 1,ω N N tends to ρ − ρ c (β) ρ in P-probability, and (0.11) for all j ≥ 2, n j,ω N tends to 0 in P-probability.
This seems to be the first multi-dimensional example in the natural class of random Poissonian obstacles for which type-I condensation has been established, see [16].
We will now describe the organization of this article. Section 1 collects notation as well as some basic results concerning the Dirichlet eigenvalues and the eigenfunctions under consideration. It also states the quantitative Faber-Krahn inequality of [4], see (1.17). Section 2 briefly recalls the results of the method of enlargement of obstacles from Chapter 4 of [28] that will be used in Sections 3 and 4. Section 3 introduces a certain event T in (3.24), encapsulating typical configurations in scale ℓ, of probability tending to 1 as ℓ goes to infinity. In Section 4, the main result is Theorem 4.1, which in particular reduces the analysis to the consideration of two distant balls of radiusR slightly bigger than R 0 (log ℓ) 1 d within B ℓ , with principal Dirichlet eigenvalues within close range (almost σ(log ℓ) −(1+2 d) ) of λ 1,ω (B ℓ ). In Section 5, the proof of the lower bound on the spectral gap is reduced to proving (0.7) in Proposition 5.3. Section 6 proves the deconcentration estimates in Theorem 6.2 from which the lower bound on the spectral gap, see (0.5), follows in Theorem 6.1. Section 7 contains the application to the Bose-Einstein condensation for the version of the Kac-Luttinger model, which we consider.
Finally, throughout the article we denote by c,c, c ′ , . . . positive constants changing from place to place, which depend on d. From Section 3 onwards they will also implicitly depend, unless otherwise stated, on the parameters selected there in the context of the method of enlargement of obstacles. As for numbered constants such as c 0 , c 1 , c 2 , . . . , they refer to the value corresponding to their first appearance in the text (for instance see (0.3) for c 0 ).

Acknowledgements:
The author wishes to thank Joachim Kerner, Maximilian Pechmann, and Wolfgang Spitzer for several stimulating discussions during the conference "On mathematical aspects of interacting systems in low dimension" that took place in Hagen in June 2019.

Set-up and some useful facts
In this section we collect further notation as well as some results concerning eigenvalues and eigenfunctions. We also state the quantitative Faber-Krahn Inequality at the end of the section. Throughout we assume that d ≥ 2.
When I is a finite set we let I stand for the number of elements of I. We write N = {0, 1, . . . } for the set of non-negative integers. For a, b real numbers we write a ∧ b, resp. a ∨ b, for the minimum, resp. the maximum, of a and b. Given r ≥ 0 we denote by [r] the integer part of r and by ⌈r⌉ the ceiling of r. For (a n ) n≥1 and (b n ) n≥1 positive sequences, the notation a n ≫ b n or b n = o(a n ) means that lim n b n a n = 0. We write ⋅ and ⋅ ∞ for the Euclidean and the supremum norms on R d . We denote by B(x, r) and ○ B(x, r) the closed and open Euclidean balls with center x ∈ R d and radius r ≥ 0. We write B ∞ (x, r) and ○ B ∞ (x, r) in the case of the supremum norm, and also refer to them as the closed and open boxes with center x and side-length 2r. Given A, B ⊆ R d , we denote by d(A, B) = inf{ x − y ; x ∈ A, y ∈ B} the mutual Euclidean distance between A and B, and by diam(A) = sup{ x − y ; x, y ∈ A} the diameter of A. We define d ∞ (A, B) and diam ∞ (A) in an analogous fashion with ⋅ ∞ in place of ⋅ . We write B(R d ) for the collection of Borel subsets of R d , and for A ∈ B(R d ) we let A stand for the Lebesgue measure of A (hopefully this causes no confusion with the notation for the cardinality of A). When f is a function f + = max{f, 0}, f − = max{−f, 0} stand for the positive and negative parts of f , and f ∞ for the supremum norm of f . When 1 ≤ p < ∞ and A ∈ B(R d ) we denote by L p (A) the L p -space of p-integrable functions for the Lebesgue measure that vanish outside of A, and write ⋅ p for the L p -norm. Given U an open subset of R d , we write H 1 (U) and H 1 0 (U) for the usual Sobolev spaces (corresponding to W 1,2 (U) and W 1,2 0 (U) in [1], p. 45).
We turn to the description of the random medium. The canonical space Ω consists of locally finite, simple point measures on R d , endowed with the canonical σ-algebra G generated by the applications ω ∈ Ω → ω(A) ∈ N ∪ {∞}, for A = B(R d ). We routinely write ω = Σ i δ x i for a generic ω ∈ Ω, and x ∈ ω to denote that x belongs to supp ω (the support of ω). On Ω endowed with the above σ-algebra, we let (see also [18]): (1.1) P stand for the law of the Poisson point process on R d with constant intensity ν > 0.
The radius of the obstacles is given by For U an open subset of R d and ω ∈ Ω, we write (1.4) U ω = U Obs ω for the possibly empty open subset remaining after deletion of the obstacle set. When U is bounded, U ω has finitely many connected components (since ω is locally finite). Of special interest for us is the case when U is an open box of side-length 2ℓ centered at the origin: Note that B ℓ,ω is non-decreasing in ℓ, and that by a routine Borel-Cantelli type argument P-a.s., B ℓ,ω is not empty for large ℓ, i.e.
We then proceed with some notation concerning Brownian motion. We denote by the transition density for Brownian motion. When x ∈ R d , we let P x stand for the Wiener measure starting from x, i.e. the canonical law of Brownian motion starting at x on the space W = C(R + , R d ) of continuous R d -valued trajectories. We write (X t ) t≥0 for the canonical process, (F t ) t≥0 for the canonical right-continuous filtration, and (θ t ) t≥0 for the canonical shift. Given an open subset U of R d and w ∈ W , we denote by For U open subset of R d and ω ∈ Ω, we denote by the transition kernel of Brownian motion killed outside U ω , see (3.4), p. 13 of [28]. It is jointly measurable, symmetric in x, y, it vanishes if x or y does not belong to U ω , satisfies the Chapman-Kolmogorov equations, see pp. 13,14 of [28], and it is a continuous function of (t, x, y) in (0, ∞) × U ω × U ω , see Proposition 3.5, p. 18 of [28]. If U satisfies an exterior cone condition, so does U ω , and the proof of Proposition 3.5, p. 18 of [28] can be adapted (see (3.20) on p. 18 of this reference) to show that r U,ω (t, x, y) is a continuous function on (0, ∞) × R d × R d . We will use this fact when U is an open box in R d .
We now proceed with the discussion of the eigenvalues and eigenfunctions. Given a bounded open subset U of R d , we denote by the successive Dirichlet eigenvalues of − 1 2 ∆ in U ω , repeated according to their multiplicity, with the convention that λ i,ω (U) = ∞ for all i ≥ 1, if U ω = ∅. They are measurable in ω (as follows for instance from the min-max principles, see Version 3 in Theorem 12.1, p. 301 of [20]). Note that the open set U ω need not be connected (even when U is connected) and the λ i,ω (U), i ≥ 1, correspond to the non-decreasing reordering of the collection of Dirichlet eigenvalues of − 1 2 ∆ in the finitely many connected components of U ω . Also some of the λ i,ω (U), i ≥ 1, may not be simple, for instance in the case U = B ℓ , and no obstacle falls into B ℓ , i.e. when B ℓ,ω = B ℓ , an event of positive probability.
In particular, we will implicitly choose the version of ϕ, which is continuous in U ω and identically equal to 0 outside U ω . When U satisfies an exterior cone condition and U ω = ∅, such an eigenfunction ϕ will be continuous on R d by the remark above (1.9). The next lemma will be used repeatedly.
As already mentioned, when U is a bounded open set and ω ∈ Ω, the open set U ω need not be connected, and the eigenvalue λ 1,ω (U) need not be simple. To take care of this feature, when U ω = ∅, we denote by ϕ 1,U,ω the L 2 -normalized orthogonal projection of the function 1 on the eigenspace attached to λ 1,ω (U): Note that ϕ 1,U,ω (x) is positive exactly when x belongs to a connected component of U ω with principal Dirichlet eigenvalue for − 1 2 ∆ equal to λ 1,ω (U). By convention, when U ω = ∅, we simply set ϕ 1,U,ω = 0. From time to time for U bounded open subset of R d , we will use the notation so that in (0.3) with the notation from the beginning of this section We proceed with the statement of the quantitative Faber-Krahn inequality. The classical Faber-Krahn inequality states that for U bounded open subset of R d , λ − 1 2 ∆ (U) is bigger or equal to the principal Dirichlet eigenvalue of − 1 2 ∆ in an open ball of same volume as U, that is λ d (ω d U ) 2 d . We will use in the proofs of Theorems 4.1 and 4.2 the following quantitative version of Faber-Krahn's inequality, see the Main Theorem on p. 1781 in [4]: One has a dimension dependent constant c 2 such that for any bounded non-empty open subset U of R d : where A(U) = inf{ U ∆B B ; B a ball with B = U } is the Fraenkel asymmetry of U (and ∆ stands for the symmetric difference).
The Theorem 1.1 of [10] states a similar inequality with A(U) 2 replaced by A(U) 4 (and a different constant), which would also suffice for our purpose in Section 4.
Finally, in several places we refer to soft obstacles. This corresponds to the case when we have a function W (⋅), non-negative, bounded, measurable, compactly supported, and not a.e. equal to zero, and for each ω = ∑ i δ x i in Ω we consider the non-negative locally bounded function (the random potential ): The objects of study are now the Dirichlet eigenvalues and corresponding eigenfunctions of

Inputs from the method of enlargement of obstacles
In this section we collect various facts from the method of enlargement of obstacles, see Chapter 4, Sections 1 to 3 of [28], which will be employed or underpin some of the results in the next two sections.
We begin with an informal description. In a nutshell the method of enlargement of obstacles is a procedure, which for ℓ positive real (say bigger than 10) and a configuration ω ∈ Ω attaches in a measurable fashion two disjoint subsets of R d the density set D ℓ (ω) and the bad set B ℓ (ω), so that ω has no point outside D ℓ (ω) ∪ B ℓ (ω): In the presentation made here (log ℓ) 1 d corresponds to the unit scale in Chapter 4 of [28] and ε = (log ℓ) −1 d to the small parameter in the same reference. The statements recalled below will hold in the large ℓ limit (i.e. small ε limit) but uniformly in ω.
In essence, adding Dirichlet boundary conditions on D ℓ (ω) (the closure of D ℓ (ω)) does not increase too much Dirichlet eigenvalues of type λ 1,ω (U) when they are below M(log ℓ) −2 d (see Theorem 2.1 below), and the bad set B ℓ (ω) has small relative volume on each box of side-length (log ℓ) 1 d in R d , see Theorem 2.4. In addition, the sets D ℓ (ω) and B ℓ (ω) have "low combinatorial complexity": their restriction to each cube is a union of disjoint cubes in an L-adic decomposition of C q , of size larger than (and of order) This feature constrains the number of possible shapes of the restriction of D ℓ (ω) and of B ℓ (ω) to any such cube C q , to at most 2 (log ℓ) γ in the case of D ℓ (ω), and at most 2 (log ℓ) β in the case of B ℓ (ω), see (2.12). This reduced combinatorial complexity of the density set and of the bad set underpins the coarse graining aspect of the method, and its power when bounding the probability of events of a large deviation nature.
We now turn to the precise statements that will be helpful for us in the next two sections. One first selects parameters that fulfill the requirements in (3.66), p. 181 of [28] (see also (3.27), p. 173 and (3.64), p. 180). These parameters are 0 < α < γ < β < 1, an integer L ≥ 2 (entering the L-adic decomposition of the boxes C q , q ∈ Z d in (2.2)), δ > 0 (entering the definition of the density set), ρ > 0 (governing the quality of the eigenvalue estimates), κ > 0 (governing the local volume of the bad set). As mentioned above they are chosen so as to satisfy (3.66), p. 181 of [28]. With this choice performed, the results in Chapter 4, Sections 2 and 3 of [28] apply in the context of the hard spherical obstacles considered here (see (1.3)). They yield the following statements: For any M > 0, where in the supremum ω runs over Ω and U over all bounded open sets in R d (and ∧ refers to the minimum, see the beginning of Section 1). See Theorem 2.3, p. 158 of [28] for the proof. In the next sections we will only need the value M = 2c 0 (with c 0 from (0.3)). The statement above can actually be extended to arbitrary open sets U, but here in (1.9) we have only defined λ 1,ω (U) for bounded open sets U, a general enough set-up for our purpose.
One also has an estimate, which provides a lower bound on the probability that Brownian motion enters the obstacle set before moving at distance L(log ℓ) (1−α) d : For large ℓ one has that for any ω ∈ Ω and x ∈ D ℓ (ω), where for u > 0, τ u = inf{s ≥ 0; X s − X 0 ∞ ≥ u} and we recall the notation (1.3), and H Obsω is the entrance time in Obs ω , see above (1.8).
For the proof, see Lemma 2.1, p. 154 (and (2.19)', p. 157) of [28]. One actually has a much stronger estimate in the quoted reference, but (2.4) will suffice for our purpose.
The next result that we quote corresponds to the case of (2.5) so that outside of A ∪ D ℓ (ω), U 2 has "small relative volume" in all boxes C q , q ∈ Z d , namely, one has r > 0 with and in addition, one has R > 0 so that U 1 contains the trace on U 2 of an R-neighborhood of A ∩ U 2 for the supremum distance, that is The next theorem provides a setting in which λ 1,ω (U 1 ) is not much bigger than λ 1,ω (U 2 ).
Once again, only the choice M = 2c 0 will be used in Sections 3 and 4.
where sup denotes the supremum over all ω ∈ Ω, For the proof we refer to Theorem 2.6, p. 164 of [28]. Concerning the volume estimate for the bad set, one has Theorem 2.4.
As for the combinatorial complexity of the density set and the bad set, one has (2.12) for each ℓ ≥ 10, q ∈ Z d and ω ∈ Ω, the sets C q ∩ D ℓ (ω) and C q ∩ B ℓ (ω) take at most 2 (log ℓ) β possible shapes.
These shapes corresponds to the various unions of L-adic subboxes of C q of the same size, which is bigger or equal to (log ℓ) (1−γ) d in the case of the density set, see (2.7) and (2.13), pp. 151-152 of [28], and bigger or equal to (log ℓ) (1−β) d in the case of the bad set, see (3.43) -(3.46), p. 177 of the same reference.

Setting up typical configurations
In this section we collect some first consequences of the method of enlargement of obstacles. We will introduce an event depending on ℓ, of high probability as ℓ → ∞, see (3.24), (3.25), which will encapsulate the nature of "typical configurations", and will be very convenient in the analysis of the first and second Dirichlet eigenvalues λ 1,ω (B ℓ ) and λ 2,ω (B ℓ ) for large ℓ, see (1.9), (1.5) for notation. With the exception of Lemma 3.1, we mainly collect here results from Section 4 in Chapter 4 of [28], which although written in the context of soft obstacles remains valid in the (simpler) context of hard spherical obstacles. We recall that a > 0 denotes the radius of these spheres, ν > 0 the intensity of the Poisson point process, and we have selected a fixed choice of admissible parameters 0 < α < γ < β < 1, L ≥ 2 integer, δ > 0, ρ > 0, κ > 0 that satisfy the requirements in (3.66), p. 181 of [28] so that the results stated in the previous section apply. We further choose as in (4.41), p. 189 of [28] (3.1) β ′ ∈ (β, 1).
Unless otherwise specified, the positive constants will implicitly depend on the dimension d and the above parameters, as explained at the end of the Introduction. We recall that throughout we assume d ≥ 2 and ℓ > 10. Corresponding to r in (4.22), p. 186 of [28], we have a ("small enough", see the quoted reference) (with r 0 as in Theorem 2.3) and we define R 1 (d, ℓ, ν) corresponding to R in (4.23), p. 186 of [28], as the smallest positive integer for which, in the notation of Theorem 2.3, We then define the random open set (3.4) (In the terminology of [28] the boxes C q satisfying the above condition are the so-called clearing boxes).
Then, by (4.25), (4.26) in Proposition 4.2, p. 186 of [28], we have a constant Moreover, see Proposition 4.3, p. 188 of [28], one can choose a constant γ 2 (d, ν, a) > 0 such that the event Further, see (4.42), p. 189 of [28], we introduce the event (see Section 2 for notation): Then, by Lemma 4.4, p. 189 of [28], we have We also wish to discard the boxes in C ℓ that are too close to the boundary of B ℓ . To this effect, with γ 1 , γ 2 as in (3.5) and (3.9) above, we define the event and introduce the sub-collections of interior boxes and boundary boxes in C ℓ : One then has Lemma 3.1.
Proof. We define the event We will show that for large ℓ, the eventẼ has high probability, and one has the inclusioñ E ⊆ E. With this in mind, we first note that as in (4.44), p. 189 of [28], for large ℓ and any B ∈ C ℓ , one has: Then, by a union bound and the fact that C bound ℓ ≤ 2d ℓ d−1 for large ℓ, we find that As we now explain, Indeed, when ℓ is large, then for any ω ∈Ẽ E, one can find B ∈ C bound a contradiction. This proves (3.19).
. The main Theorem 4.1 of this section shows that for large ℓ on T ∩R, one can find two distant sub-boxes of side-length of order (log ℓ) 1 d contained in B ℓ , with principal Dirichlet eigenvalues, which are close to In addition, the principal Dirichlet eigenfunctions attached to these boxes are well localized in balls of (1)) with centers close to the respective centers of these boxes. These results will in essence follow from the application of the method of enlargement of obstacles recalled in Section 2, and the quantitative Faber-Krahn inequality (1.17). The reduction to the analysis of boxes with size of order (log ℓ) 1 d (unlike the boxes in C ℓ , see (3.6)) will be important in Section 6 for the quality of the deconcentration estimates in Theorem 6.2. We refer to the beginning of Section 3 concerning the choices of parameters (notably for the method of enlargement of obstacles), which remain in force, and for the convention concerning positive constants.
We first need some additional notation. We introduce the length and consider the open boxes of side-length L 0 and center in (log ℓ) 1 d Z d : We will typically write D 0 or speak of an L 0 -box to refer to a generic box of the form D 0,q . Given such a box D 0 , we will refer to To define the resonance set, we first pick We will eventually let σ tend to 0 in Section 6. The resonance event is then We will show in Theorem 6.1 that lim σ→0 lim sup ℓ→∞ P[R] = 0 (this will be the lower bound on the spectral gap). For the time being the main object of this section is the proof of Theorem 4.1. The simpler Theorem 4.2 is also of interest. We recall from (3.9), (4.5) that , and γ 2 is the constant from (3.9). The likely event T is defined in (3.24).
and there are open ballsB,B ′ with centers having rational coordinates, belonging to the respective central boxes (see (4 and (with the notation (1.14)) We stated (4.7) for completeness but our main interest in view of Sections 5 and 6 lies in (4.8). The rational coordinates of the centers of the balls are mentioned to highlight the measurability of the events under consideration. One also has the simpler Theorem 4.2. Withη 1 ,η as in Theorem 4.1, there exists ℓ 1 > 10 such that for ℓ ≥ ℓ 1 , on the event T , one has a box D 0 contained in B ℓ such that and an open ball B # with center having rational coordinates, belonging to the central box of D 0 with radiusR, such that the first lines of (4.7) and (4.8) The proof of Theorem 4.2 is simpler than that of Theorem 4.1 and is also quite similar. We mainly focus on the proof of Theorem 4.1 in the remainder of this section. In Remark 4.7 1) we briefly sketch the main steps in the proof of Theorem 4.2.
It may be appropriate to describe here the general line of the arguments, which we use. We begin with two lemmas, which for large ℓ on T ∩ R provide us with two distant boxes B, B ′ in C int ℓ (see (3.14)) with principal Dirichlet eigenvalues, essentially within ρ ℓ (= σ(log ℓ) −(1+2 d) ) from λ 1,ω (B ℓ ), and such that after deletion of the closure of the density set, the corresponding principal Dirichlet eigenvalues do not increase too much (but may well be much bigger than λ 1,ω (B ℓ ) + ρ ℓ ). In the Proposition 4.5 we combine the volume estimates and the eigenvalue estimates for B D ℓ (ω) and B ′ D ℓ (ω) with the quantitative Faber-Krahn inequality (1.17) to bring into play two balls of same respective volumes as the above two sets, and symmetric differences with these respective sets of small volume. We also find two suitable boxes ofC ℓ (see (3.21)) each containing one of the above balls close to their center, and such that after deletion of the ball and D ℓ (ω), the volume of the remaining set in each box is small compared to log ℓ. Once Proposition 4.5 is proved, we can apply Theorem 2.3, and also use the representation formula (1.12) for eigenfunctions combined with Lemma 2.2, to establish the existence of two distant L 0 -boxes (concentric with the above boxes ofC ℓ ) having the desired principal Dirichlet eigenvalue estimates, and adequately small principal Dirichlet eigenfunctions outside balls of deterministic radiusR (see above (4.7)) with suitable centers in the central boxes (see (4.3)) of these L 0 -boxes.
With this plan in mind, the first step is a deterministic statement. Proof. When λ 1,ω (B ℓ ) < ∞ (or equivalently when B ℓ,ω = ∅, see (1.4)), as noted below (1.9), the eigenvalues λ i,ω , i ≥ 1, correspond to the reordering of the union (with multiplicities) of the Dirichlet eigenvalues of − 1 2 ∆ in each of the finitely many connected components of B ℓ,ω . Thus, at least one of the items below occurs: and λ 2,ω (B ℓ ) correspond to principal Dirichlet eigenvalues of − 1 2 ∆ in distinct connected components of B ℓ,ω , ii) λ 1,ω (B ℓ ) and λ 2,ω (B ℓ ) are the first two Dirichlet eigenvalues of − 1 2 ∆ in one of the connected components of B ℓ,ω .
When (4.11) i) occurs, the claim (4.10) is immediate: one simply chooses U and U ′ as the connected components mentioned in (4.11) i).
We thus assume that (4.11) ii) occurs. We denote by W a connected component of B ℓ,ω such that the first two Dirichlet eigenvalues of − 1 2 ∆ in W respectively coincide with λ 1,ω (B ℓ ) and λ 2,ω (B ℓ ) and by ψ an L 2 -normalized Dirichlet eigenfunction in W corresponding to λ 2,ω (B ℓ ). Note that W satisfies an exterior cone condition (each boundary point of W belongs to B c ℓ or to a closed ball of radius a in W c ). As explained below (1.10), the function ψ is continuous and it equals 0 outside W . Since ψ is attached to the second Dirichlet eigenvalue of − 1 2 ∆ (and orthogonal to the principal Dirichlet eigenfunction on the connected open set W ), it changes sign, and we can choose non-empty connected components U of {ψ > 0} and U ′ of {ψ < 0}. As we now explain (4.12) λ 1,ω (U) and λ 1,ω (U ′ ) are equal to λ 2,ω (B ℓ ).
Next, as a result of Theorem 2.1 (with M = 2c 0 ), when ℓ is large so that in particular On the other hand, it follows from the inclusion T ⊆ H, see (3.22), (3.24), that the left member of (4.22) is at most d ν (log ℓ) + (log ℓ) β ′ . So when ℓ is large, we can in addition assume that B ∩ B ′ = ∅ and coming back to (3.22) that diam(B ∪ B ′ ) ≥ ℓ 3 4 (log ℓ) 1 d so that (4.14) holds.
The claim (4.15) ii) and the corresponding claim for B ′ now follow from the application of Theorem 2.1, the bound λ 1,ω (B) + (log ℓ) −(2+ρ) d < 2c 0 (log ℓ) −2 d and the similar bound for B ′ , which are consequences of (4.15) i) and the corresponding bound for B ′ (ℓ being large). This concludes the proof of Lemma 4.4.
We will now gather upper bounds on eigenvalues and on volume, and combine them with the quantitative Faber-Krahn inequality (1.17) in the course of the proof of the next proposition. We recall the definition ofC ℓ in (3.21), and we define the central box ofB ℓ inC ℓ similarly as in (4.3), namely as the closed concentric box ofB ℓ with sidelength 2(log ℓ) 1 d . The parameter α appeared in the selection made for the method of enlargement of obstacles, see Lemma 2.2 and Remark 4.6 below.
which respectively contain open ballsB andB ′ with centers having rational coordinates that belong to the respective central boxes ofB andB ′ , and have same radiusR = R 0 (log ℓ) 1 d + 2(log ℓ)η 1 . These balls have the property that denoting byB int andB ′ int the smaller closed concentric balls with radiusR int = R 0 (log ℓ) 1 d + (log ℓ)η 0 , one has and a similar inequality withB ′ andB ′ int in place ofB andB int , as well as (in the notation of (4.5) and (3.9)) and similar inequalities withB ′ in place ofB.
Proof. We first choose (recall that β ′ , κ, ρ are among the parameters selected at the beginning of Section 3) Then by Lemma 4.4, for large ℓ, on T ∩ R we have two boxes B, B ′ ∈ C int ℓ , which satisfy (4.14), (4.15). By the inclusion G ⊆ T , see (3.11), (3.24), and the volume bound on the bad set from Theorem 2.4, we can further assume that By (4.15) i) and ii) we additionally know that and a similar inequality with B ′ in place of B.
By the quantitative Faber-Krahn inequality (1.17) one can then find ballsB andB ′ with centers having rational coordinates, with same respective volumes as B D ℓ (ω) and B ′ D ℓ (ω) and so that the ratio of the volumes of the symmetric differences (B D ℓ (ω)) ∆B and (B ′ D ℓ (ω)) ∆B ′ to the respective volumes B and B′ is smaller than 2 √ c 2 (log ℓ) −χ ′ 2 < 1 2 . In particular, the centers ofB andB ′ respectively belong to B and B ′ (otherwise the above mentioned ratios would be at least 1 2 ). In addition, by (4.27) and the Faber-Krahn inequality, due to the value of c 0 recalled above (4.28), and the choice χ ′ < (1 ∧ ρ) d in (4.25), we can additionally assume that and a similar inequality with B ′ in place of B.
Thus, combining (4.26) and (4.29), we have upper and lower bounds on the volumes of B D ℓ (ω) and B ′ D ℓ (ω) which respectively coincide with B and B′ . By the bound on the volumes of the symmetric differences stated below (4.28), we see that, ℓ being large, we can assume that with similar inequalities for B ′ ,B ′ , so that with a similar lower bound with B ′ andB ′ in place of B and B ′ .
We now turn to the second point, namely the proof of (4.8). Recall thatη 0 >μ d > (1 − α) d. Using Lemma 2.2, we assume from now on that ℓ is large enough so that for any y ∈ D ℓ (ω), the Brownian motion starting at y enters the obstacle set before moving at ⋅ ∞ -distance 1 2 (log ℓ)η 0 with probability at least 1 2 , see (2.4): , for all y ∈ D ℓ (ω).
We prove (4.8) in the case of D 0 andB (the case of D ′ 0 andB ′ is handled in the same fashion). By (4.23) for any x ∈B B int the ⋅ ∞ -ball with center x and volume 2(log ℓ)μ has at least half of its volume occupied byB c ∪B int ∪ D ℓ (ω). Thus, Brownian motion starting at x entersB c ∪B int ∪ D ℓ (ω) before exiting the concentric box of double radius with a probability at least c(d) > 0. Sinceμ d <η 0 and ℓ is large, the strong Markov property and (4.36) shows that the Brownian motion starting at x exitsB ω B int (i.e. entersB c ∪B int ∪ Obs ω ) before moving at ⋅ ∞ -distance (log ℓ)η 0 with a non-degenerate probability, namely: where we have set τ = τ (log ℓ)η0 in the notation of (2.4).
2) As already mentioned, we will mainly use (4.6) and (4.8) of Theorem 4.1 in what follows. The statement (4.7) is there for clarity and completeness. ◻

Tuning and resonance control
In this section we derive an asymptotic upper bound on the resonance event R, see (4.5), in the large ℓ limit, in terms of a quantity, which measures the deconcentration of the law of λ 1,ω (D 0 ) in a suitably tuned regime of low values, with an additional information on the corresponding eigenfunction ϕ 1,D 0 ,ω , see Proposition 5.3. In the next section we will prove deconcentration estimates, which will bound the above quantity, and lead to a lower bound on the spectral gap. We recall the definition of D 0 -boxes in (4.2). Their side-length is L 0 = 10(⌈R 0 ⌉ + 1)(log ℓ) 1 d , see (4.1).
Our first task in this section is to suitably tune the "low level" of λ 1,ω (D 0 ) that is pertinent for our purpose. We first need some notation. For ℓ > 10 we definê C ℓ the collection of D 0 -boxes included in B ℓ , (5.1)Ĉ * ℓ the sub-collection ofĈ ℓ consisting of boxes D 0,q ⊆ B ℓ (5.2) such that q ∈ 20(⌈R 0 ⌉ + 1) Z d (see (4.2) for notation).
Note that for large ℓ, in the terminology of (4.3), withR as above (4.7), (5.6) any ball with center in the central box of D 0 and radiusR is contained in D int 0 .
We will eventually let Γ tend to infinity in the next section.
Thus, splitting between the case when min D ′′ 0 ∈Ĉ ℓ λ 1,ω (D ′′ 0 ) is strictly smaller, or is equal to t ℓ , we now find that for large ℓ Using the independence of random variables corresponding to D 0 -boxes at mutual distance bigger than 2a, one finds with a union bound, independent variables ω 1 and ω 2 , and hopefully obvious notation We then bound the product probability in the right member of (5.23) with the help of (5.9) ii), (5.4), as well as (5.10), and find that for large ℓ (5.24) Since lim ℓ P[T ] = 0 by (3.25), the claim (5.18) now follows from the above inequality and the definition of Σ in (5.19). This proves Proposition 5.3.

Lower bound on the spectral gap via deconcentration
In this section we prove the main asymptotic lower bound on the spectral gap in Theorem 6.1. The scale (log ℓ) −(1+2 d) that appears in Theorem 6.1 is expected to capture the correct size of the spectral gap, see Remark 6.5 1) at the end of the section. The main ingredient in the proof of Theorem 6.1 lies in the deconcentration estimates shown in Theorem 6.2. Whereas the results of the previous sections can be adapted with the techniques of Chapter 4 of [28] to the case of soft obstacles, see (1.18), the proof of Theorem 6.2 uses in a substantial manner the hard sphere obstacles considered here.
We first explain how Theorem 6.1 follows from Theorem 6.2.
Proof of Theorem 6.1 based on Theorem 6.2: We consider Γ > 0, m = ⌈K Γ 3 ⌉, and a corresponding σ 0 > 0 as in Theorem 6.2. Then, for all σ ∈ (0, σ 0 ), for large ℓ, for all t ∈ [s ℓ , t ℓ ], there are pairwise disjoint compact intervals J 1 , . . . , J m in (0, t ℓ ) such that (6.2) holds. Adding these m inequalities, we see that the left member of (6.2) is at most Thus, coming back to (5.19), we see that for all σ < σ 0 , It then follows from (5.18) that for all σ < σ 0 (which depends on Γ) Letting Γ tend to infinity, (6.1) follows. This proves Theorem 6.1. ◻ We will now turn to the proof of Theorem 6.2. To establish the deconcentration estimate in a large deviation regime of low values of λ 1,ω (D 0 ) corresponding to (6.2), we will bring into play transformations of the obstacle configurations which do not change too much probabilities (see the multiplicative factor K in (6.2)), and induce a controlled decrease of λ 1,ω (D 0 ). In this features lies a difficulty. Whereas increasing λ 1,ω (D 0 ) is not difficult (for instance by inserting an additional obstacle in a spot where the principal Dirichlet eigenfunction ϕ 1,D 0 ,ω is not too small, see Theorem 2.3, p. 109 of [28]), decreasing λ 1,ω (D 0 ) in a controlled (to have many disjoint intervals J i , 1 ≤ i ≤ m, in (6.2)) and probabilistically economical fashion is a more delicate endeavor. To this end, we will exploit the effect of a "gentle expansion" of the Poisson cloud ω to lower eigenvalues. The constraint on ϕ 1,D 0 ,ω in the left member of (6.2) will ensure a "proper centering of the underlying clearing" in the box (so that it does not get damaged by the expansion). We also refer to Lemma 3.3 of [5] for other kind of transformations, which however do not seem adequate for the task of proving (6.2).
Proof of Theorem 6.2: Throughout the proof, using translation invariance, without loss of generality, we let D 0 stand for the L 0 -box centered at the origin, i.e. corresponding to q = 0 in (4.2). We then consider Γ > 0, m ≥ 1 and σ > 0 to be later chosen small, see (6.49). We further introduce the "expansion ratio" ≤ d 10 d < log 2 , and the homothety centered at the origin of ratio λ: Given an ω = ∑ i δ x i in Ω, we write (i.e. the principal Dirichlet eigenvalue of − 1 2 ∆ in U ⋃ x∈ω B(x, λ a)), and (6.10)φ 1,U,ω for the corresponding principal Dirichlet eigenfunction (defined similarly to (1.14) and below (1.14) with a replaced by λ a).
Then, by Brownian scaling, for U bounded open set and ω ∈ Ω, one has λ 1,ω (λU) = on Ω generated by the random variables ω(C) with C Borel subset of λD a 0 .
We also note that just as in (1.11) (replacing a by λa in the proof of (1.11)), with the same dimension dependent constant c 1 , one has (6.20) for all ω ∈ Ω (one could also use the scaling identities (6.11), (6.12) and (1.11) to infer (6.20)).
Proof. Without loss of generality, we assume thatλ 1,ω (λD 0 ) < ∞ and writeφ as a shorthand forφ 1,λD 0 ,ω . As noted above (6.13),φ is a continuous function. If t > t 1 is close to t 1 and 4t D 0 1 2 < 1, the functionψ = (φ − t) + is continuous, compactly supported and vanishes on a neighborhood of D c 0 . In addition, one has Moreover, by the Cauchy-Schwarz inequality and ∫ D 0ψ 2 dx ≤ ∫ D 0φ 2 dx ≤ 1, one also has ∫ D 0ψ dx ≤ D 0 1 2 . So, coming back to (6.23), we find that Since 4t D 0 1 2 < 1, it follows thatψ is not identically 0. It vanishes on a neighborhood of D c 0 and on ⋃ y∈ω B(y, λa) and is thus compactly supported in D 0,ω , which is not empty. By construction it also belongs to H 1 (R d ) and hence to H 1 0 (D 0,ω ). Its Dirichlet integral is at most that ofφ (see for instance Corollary 6.18 on p. 153 of [20]) so that (6.25) Letting t decrease to t 1 , we find (6.22).
Thus, for large ℓ, we see that for all u ∈ (0, 1) and t ∈ [s ℓ , t ℓ ] on the eventÃ in (6.18) (recall that J = [t, t + ε ℓ ]) one has We then turn to the choice of t 2 .
We consider ω ∈Ã (see (6.18)) and an arbitrary y ∈ ω such that B(y, λa) intersects D 0 and ϕ = ϕ 1,D 0 ,ω is not identically 0 on B(y, λa). Since ϕ = 0 on B(y, a) we consider x ∈ B(y, λa) B(y, a) such that ϕ(x) > 0. We write U for the connected component of D 0,ω containing x and U ′ for the intersection of U with ○ B(y, 100a), the open ball with center y and radius 100a. With ℓ large and (6.30) we can assume that (see (1.15), (1.16) for notation) . Then, by (1.12) (in a rather similar fashion to (4.39)), we have Note that when Brownian motion enters B(y, a), it exits U, so that using scaling and translation invariance, we find that To bound the above expectation, we consider the functions w(z) = z −b where b ∈ (0, d−2]. The first and second radial derivatives are ∂ r w = −b z −(b+1) and ∂ 2 . We then introduce the stopping time S = T ○ B(0,100) ∧ H B(0,1) , so that for z ∈ R d with 1 < z ≤ λ, under P z . So for such b, after P z -integration and letting t tend to infinity, we have By a classical formula corresponding to b = d − 2 in the first line of (6.35), one also has ) (or see (6) on p. 29 of [9]). We now pick and infer from (6.35) that As a result, when ℓ is large enough so that 3c 0 a 2 (log ℓ) −2 d < 1 2 b(d − 2 − b) 100 −2 , the expectation in (6.36) is bigger or equal to the expectation in (6.33), and hence (6.37) ϕ(x) ≤ c ′′ (d)(t + ε ℓ ) d 4 u D 0 (6.30),(4.1) ≤ c 7 (d, ν) u D 0 3 2 , for all x ∈ ⋃ y∈ω B(y, λa).
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 (⋅) in (1.18) would be transformed into W (⋅) = 1 λ 2 W ( ⋅ λ ) (the case under present consideration formally corresponds to W (⋅) = ∞ 1{ ⋅ ≤ a} and W (⋅) = ∞ 1{ ⋅ ≤ λa}). ◻

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 ν > 0. The dimension of space is d ≥ 2. The radius a although fixed can be arbitrarily small, and the complement in 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) ρ > 0, we consider the positive sequence ℓ N , N ≥ 1, indexed by the particle number N, which tends to infinity and satisfies (see (1.5) for notation) (7.2) ρ B ℓ N = N, for N ≥ 1 (in the notation of [16], 2ℓ N = L N ).
We recall some known facts. As N → ∞, the random measures on R + (7.9) δ λ i,ω (B ℓ N ) , for ω ∈ Ω N , 0, for ω ∈ Ω c N , are known to P-a.s. converge vaguely to a deterministic measure, the density of states (7.10) m on [0, ∞), characterized by its Laplace transform where E t 0,0 denotes the expectation with respect to the Brownian bridge in time t from 0 to 0, and W a t 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 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): (The quantity exp{−νω d (λ d λ) d 2 } is the probability that the Poisson point process places no point in the open ball of radius (λ d λ) 1 2 centered at the origin. This ball has a principal Dirichlet eigenvalue for − 1 2 ∆ equal to λ.) We then introduce the critical density for our system (see (2.6) of [16]), namely (7.13) ρ c (β) = ∞ 0 1 e βλ − 1 dm(λ) (< ∞ by (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 ρ > ρ c (β), 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 ρ > ρ c (β): Theorem 7.1. When ρ > ρ c (β), then as N tends to infinity, (7.14) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ i) n 1,ω N N tends to ρ − ρ c (β) ρ in P-probability, and ii) n j,ω N tends to 0 in P-probability, for any j ≥ 2.
As for iv) of Assumptions 2.2 in [16], by the Lifshitz tail asymptotics (7.12), one has, picking η 1 ∈ (0, 1) and settingC 1 = ν ω d λ Together with (0.4) (or (3.10)) and the fact that log B ℓ N ∼ d log ℓ N , as N → ∞, this is more than enough to show that forC 2 = d c 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 R d , d ≥ 2, in a suitable nonpercolative regime for the vacant set, and suitably strong short-range repulsive pairinteractions, 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 ℓ N ,ω (in the notation of (1.4)). ◻