Reach of Repulsion for Determinantal Point Processes in High Dimensions
aa r X i v : . [ m a t h . P R ] J u l REACH OF REPULSION FOR DETERMINANTAL POINTPROCESSES IN HIGH DIMENSIONS
FRANC¸ OIS BACCELLI, ∗ University of Texas at Austin
ELIZA O’REILLY, ∗ University of Texas at Austin
Abstract
Goldman [7] proved that the distribution of a stationary determinantal pointprocess (DPP) Φ can be coupled with its reduced Palm version Φ , ! such thatthere exists a point process η where Φ = Φ , ! ∪ η in distribution and Φ , ! ∩ η = ∅ .The points of η characterize the repulsive nature of a typical point of Φ. In thispaper, the first moment measure of η is used to study the repulsive behaviorof DPPs in high dimensions. It is shown that many families of DPPs havethe property that the total number of points in η converges in probability tozero as the space dimension n goes to infinity. It is also proved that for someDPPs there exists an R ∗ such that the decay of the first moment measure of η is slowest in a small annulus around the sphere of radius √ nR ∗ . This R ∗ canbe interpreted as the asymptotic reach of repulsion of the DPP. Examples ofclasses of DPP models exhibiting this behavior are presented and an applicationto high dimensional Boolean models is given. Keywords:
Laguerre-Gaussian Models; Normal variance mixture models;Bessel-type models; High dimensional geometry; Palm calculus; Pair corre-lation function; Stochastic ordering; Boolean model; Information theory; Errorexponent; Large deviations; Log-concave density2010 Mathematics Subject Classification: Primary 60G55Secondary 60D05; 60F10
1. Introduction
Determinantal point processes (DPPs) are useful models for point patterns wherethe points exhibit some repulsion from each other, resulting in a more regularly spaced ∗ Postal address: University of Texas at Austin, Department of Mathematics, RLM 8.100, 2515Speedway Stop C1200 Austin, Texas 78712-1202 pattern than a Poisson point process. These models originally appeared in randommatrix theory and the formalism was introduced by O. Macchi [17] who was motivatedby modeling Fermionic particles in quantum mechanics. They have since been usedin many applications, such as telecommunication networks, machine learning, ecology,etc. See [10], [12], [15], [16], and the references therein. This paper describes therepulsive behavior of stationary and isotropic DPPs as the space dimension goes toinfinity.In the following, a ball with center at the origin and radius r in R n is denoted B n ( r ).The ℓ vector norm will be denoted by | · | and the L -norm on the space L ( R n ) by || · || . Now, consider a sequence of point processes Φ n indexed by dimension, eachwith constant intensity ρ n . If ρ n = e nρ and R n = √ nR , with ρ ∈ R and R >
0, thenStirling’s formula givesVol( B n ( R n )) ∼ √ nπ (cid:18) πen (cid:19) n R nn , as n → ∞ . This implies there exists a threshold R ∗ = √ πee ρ such that as n → ∞ , E [Φ n ( B n ( R n ))] ∼ e n ( ρ + log 2 πe +log R )+ o ( n ) → , R < R ∗ ∞ , R > R ∗ . (1)This justifies the interest in considering this regime where the intensities grow exponen-tially with dimension and distances grow with the square root of the dimension. Thisregime also naturally arises in information theory, and following [1], it will be calledthe Shannon regime. In this paper, the effect of repulsion in this regime is studiedand the range and strength at which DPPs asymptotically exhibit repulsion betweenpoints is quantified.Mention of these issues appear in [24], where the authors characterize a certain classof DPPs by an effective “hard-core” diameter D that grows like √ n , aligning with ourobservations. They observe that for r < D , the number of points in a ball of radius r around a typical point will be zero with probability approaching one, and for r > D ,the number of points in a ball of radius r around a typical point is zero with probabilityapproaching zero as dimension n goes to infinity. The behavior for r < D is a resultof the natural separation due to dimensionality as exhibited in (1). However, theobservation that D is the maximal such separation is due to the ν -weakly sub-Poisson property of DPPs as defined in [3], and is a feature of all DPPs, not just those studiedin [24]. This behavior is the same as a sequence of Poisson point processes in the sameregime, and thus this separation of points in high dimensions is due to dimensionalityand not the repulsion of the DPP model. In this paper, a more precise descriptionof the repulsive behavior in high dimensions is given that is specific to the associatedkernel of the DPP.The measure of repulsiveness used in this paper is a refinement of the global measureof repulsiveness for stationary DPPs described in the on-line supplementary materialto [15] (see [14]). In that work, the authors consider the measure γ := ρ Z (1 − g ( x )) d x, (2)where ρ is the intensity, and ( x, y ) g ( x − y ) is the pair correlation function of thepoint process. A point process is considered more repulsive the farther g is away from1; g ≡ γ ≤ η such thatΦ = Φ , ! ∪ η in distribution, and Φ , ! ∩ η = ∅ , where Φ , ! denotes a point process with the reduced Palm distribution of Φ. Thus, η is the set of points that have to be removed from Φ due to repulsion when a point is“placed at” the origin. In the following, the first moment measure of η will be used asa measure of the repulsiveness of a DPP Φ, and the repulsive effect of a typical pointover a finite distance R is quantified by E [ η ( B n ( R ))]. Note also that E [ η ( B n ( R ))] = ρ Vol( B n ( R )) − E [Φ , ! ( B n ( R ))] = ρ [ K P oi ( R ) − K DP P ( R )] , where K P oi and K DP P are Ripley’s K-functions [20] for a Poisson point process andΦ, respectively. Finally, note that the measure of global repulsiveness (2) correspondsto η in the sense that γ = E [ η ( R n )]. In recent work [18], couplings of DPPs and theirreduced Palm distributions used to quantify repulsiveness of DPPs are studied further.Our main results describe the behavior of the first moment measure of η in theShannon regime. Consider a sequence of stationary DPPs { Φ n } , such that Φ n lies in Fran¸cois Baccelli, Eliza O’Reilly R n . For each n , let η n be the point process such that Φ n = Φ , ! n ∪ η n in distribution andΦ , ! n ∩ η n = ∅ . One can consider the quantity E [ η n ( R n )] and the probability measure E [ η n ( · )] E [ η n ( R n )] on R n that is defined to estimate the strength and reach of the repulsivenessof a DPP in any dimension.It is often the case that E [ η n ( R n )] → n → ∞ . In this case, Markov’s inequalityand the coupling inequality imply that, in high dimensions, the total variation distanceis small between Φ n and Φ , ! n . Indeed, || Φ n − Φ , ! n || T V ≤ P ( η n ( R n ) > ≤ E [ η n ( R n )] . (3)Since Φ n and Φ , ! n have the same distribution if and only if Φ n is Poisson by Slivnyak’stheorem [4], this says that such DPPs look increasingly like Poisson point processes asthe space dimension increases.However, the effect of the repulsion can still be observed by examining the prob-ability measure E [ η n ( · )] E [ η n ( R n )] on R n as seen in Propositions 3.1, 3.2, and 3.3. Letting X n be a random vector in R n with this probability distribution, it is shown that if | X n |√ n → R ∗ ∈ (0 , ∞ ) in probability, thenlim n →∞ E [ η n ( B n ( R √ n ))] E [ η n ( R n )] = , R < R ∗ , R > R ∗ . Here, R ∗ is interpreted as the asymptotic reach of repulsion in the Shannon regimefor these DPPs. This result implies that in high dimensions, a typical point has itsstrongest repulsive effect on points that are at a distance of √ nR ∗ away.The parametric families of DPP kernels presented in [2] and [15] provide examplesof DPPs exhibiting a reach of repulsion R ∗ and counterexamples where no finite R ∗ exists, as well as computational results on the rates of convergence when a thresholddoes occur. Four classes of DPPs are studied in Section 4: Laguerre-Gaussian DPPs,power exponential DPPs, Bessel-type DPPs, and normal-variance mixture DPPs. ForLaguerre-Gaussian DPPs, the sequence | X n | / √ n satisfies a large deviations principle(established later in Lemma 4.1). As a consequence, the reach of repulsion R ∗ becomesa phase transition for the exponential rate at which E [ η n ( B n ( R √ n ))] → n → ∞ (established later in Proposition 4.1). Power exponential DPPs are shown to havea finite reach of repulsion in the Shannon regime for certain parameters (established later in Proposition 4.2). Bessel-type DPPs are a more repulsive family that does notexhibit an R ∗ (established later in Proposition 4.3). Finally, normal-variance mixtureDPPs provide additional examples of DPPs that exhibit an R ∗ , including the Cauchyand Whittle-Mat´ern models (established later in Propositions 4.5 and 4.4).An application of these results is presented in Section 5. It can be shown that somethreshold results in [1] for Poisson Boolean models can be extended to generalizedLaguerre-Gaussian DPP Boolean models in the Shannon regime using the rates ofconvergence computed for these DPPs. Finally, concluding remarks and open questionsare stated in Section 6.
2. Preliminaries
Determinantal point processes are characterized by an integral operator K withkernel K , and can be defined in terms of their joint intensities, also known as correlationfunctions ([10], [15]). Definition 2.1.
A simple, locally finite, spatial point process Φ on R n is a deter-minantal point process with kernel K : R n × R n → R (Φ ∼ DP P ( K )) if its jointintensities exist for all order k and satisfy ρ ( k ) ( x , . . . , x k ) = det( K ( x i , x j )) ≤ i,j, ≤ k , k = 1 , , . . . . Note that the intensity function of Φ is given by ρ ( x ) = K ( x, x ). The degeneratecase where K ( x, y ) = δ { x = y } coincides with a Poisson point process with unit intensity.The following conditions on K are imposed to ensure Φ ∼ DP P ( K ) is well-defined.Let K : R n × R n → R be a continuous kernel and assume K is symmetric, i.e., K ( x, y ) = K ( y, x ). The kernel K then defines a self-adjoint integral operator K on L ( R n ) given by K f ( x ) = R K ( x, y ) f ( y )d y . For any compact set S ⊂ R n , the restrictedoperator K S given by K S f ( x ) = Z S K ( x, y ) f ( y )d y, x ∈ S, is a compact operator. By the spectral theory for self adjoint compact operators,the spectrum of K S consists solely of countably many eigenvalues { λ Sk } k ∈ N with anaccumulation point only possible at zero. See [21] for more on compact operators. Fran¸cois Baccelli, Eliza O’Reilly
These conditions imply that for any compact S ⊂ R n , the kernel K restricted to S × S has a spectral representation K ( x, y ) = ∞ X k =1 λ Sk φ Sk ( x ) φ Sk ( y ) , ( x, y ) ∈ S × S, where { φ Sk } k ∈ N are the eigenvectors of K S , and form an orthonormal basis of L ( S ). Theorem 2.1. (Macchi [17]) Under the conditions given above, a kernel K defines adeterminantal process on R n if and only if the spectrum of K is contained in [0 , . If K ( x, y ) = K ( x − y ), then Φ ∼ DP P ( K ) is stationary. In this case, the operator K is the convolution operator K ( f ) = K ⋆ f on L ( R n ). The intensity function ρ ( x )is then constant and satisfies ρ = K (0). For these stationary DPPs, there is a simplespectral condition for existence. Theorem 2.2. (Theorem 2.3 in [15]) Assume K is a symmetric continuous real-valued function in L ( R n ) . Let K ( x, y ) = K ( x − y ) . Then DPP( K ) exists if and onlyif ≤ ˆ K ≤ , where ˆ K denotes the Fourier transform of K . For the rest of this paper, when it is stated that Φ ∼ DP P ( K ) is stationary, it isassumed that K ( x, y ) = K ( x − y ) for a real-valued K ∈ L ( R n ), and K will be usedto mean K . There exist stationary DPPs with kernels that are not of this form (see[10, 4.3.7]), but they are complex-valued and not considered here. In addition, whenit is stated that Φ is isotropic, it is meant that K ( x ) = R ( | x | ) and the distributionof Φ is thus invariant under rotations about the origin in R n .The reduced Palm distribution of a stationary point process Φ can be interpretedas the distribution of Φ conditioned on there being a point at the origin with the pointat the origin removed (see [4, Chapter 4]) and will be denoted by P , ! . A point processwith the Palm distribution P , ! of Φ will be denoted Φ , ! . The following theorem is aspecial case of a useful result about the Palm distribution of DPPs. Theorem 2.3. (Theorem 6.5 in [23]). Let Φ ∼ DP P ( K ) in R n be stationary withintensity ρ = K (0) > . Then Φ , ! is a DPP with associated kernel K !0 ( x, y ) = K (0) det K ( x − y ) K ( x ) K ( y ) K (0) = K ( x − y ) − ρ K ( x ) K ( y ) . The nearest neighbor function of a stationary point process Φ in R n is defined as D ( r ) := P , ! (Φ( B n ( r )) > . (4)If Φ is Poisson, Slivnyak’s theorem gives that D ( r ) = 1 − e − E Φ( B n ( r )) . For Φ ∼ DP P ( K ), Theorem 2.3 implies that D ( r ) = P (Φ , ! ( B n ( r )) > , ! ∼ DP P ( K !0 ).As mentioned in the introduction, Goldman [7] proved the following result. Theorem 2.4. (Theorem 7 in [7]) Let Φ ∼ DP P ( K ) , where K is continuous, and thespectrum of the integral operator K with kernel K is contained in [0 , . Then, thereexists a point process η such that Φ = Φ , ! ∪ η in distribution, and Φ , ! ∩ η = ∅ . This theorem says that a point process with the distribution of Φ , ! can be obtainedfrom Φ by removing a subset of points η . This is a striking result, since the proceduredoes not include shifting any of the remaining points. The points in η characterize therepulsive nature of the DPP Φ, since these are the points that are “pushed out” bythe point at zero under the reduced Palm distribution. It also makes sense to comparethe repulsiveness of DPPs using η . For two stationary DPPs Φ and Φ with the sameintensity, Φ is defined to be more repulsive than Φ if E [ η ( R n )] > E [ η ( R n )]. Thiscorresponds to the definition in [15] using the measure γ defined in (2). Note that theassumptions for Theorem 2.4 excludes the interesting case where K has an eigenvalueof 1, which corresponds to when ˆ K ( x ) attains a value of one for some x .
3. Main Results
When considering the reach of repulsion of a DPP, it is natural to first considerthe nearest neighbor function (4). The following threshold behavior was observed forstationary DPPs in [24]. It is stated here for a sequence of DPPs in the Shannon regime.For each n , let Φ n ∼ DP P ( K n ) in R n be stationary with intensity K n (0) = e nρ forsome ρ ∈ R . Then, for ˜ R := (2 πe ) − e − ρ ,lim n →∞ P (Φ , ! n ( B n ( √ nR )) >
0) = , R < ˜ R , R > ˜ R. (5) Fran¸cois Baccelli, Eliza O’Reilly
A proof of this fact is given in Appendix A.This shows there is a separation of points as dimension tends to infinity for anystationary DPP. However, the same threshold behavior occurs if the elements of thesequence { Φ n } are stationary Poisson point processes, as a consequence of (1). Thisobservation shows that this separation is due purely to dimensionality and is not aresult of the repulsiveness of DPPs.The point process η n as defined in Theorem 2.4 gives an alternative characterizationof the repulsiveness of a DPP and can measure some consequence of repulsiveness inhigh dimensions that depends on the determinantal structure. Lemma 3.1.
Let Φ n ∼ DP P ( K n ) in R n be stationary and assume ≤ ˆ K n < . Let η n be the point process given in Theorem 2.4 and define the random vector X n in R n with probability density K n ( x ) || K n || . Then, P ( X n ∈ B ) = E [ η n ( B )] E [ η n ( R n )] , B ∈ B ( R n ) . The following result shows that under certain limit conditions on the kernels of asequence of DPPs, the repulsiveness measured by the first moment measure of η n isconcentrated at a distance of √ nR ∗ for some R ∗ ∈ (0 , ∞ ) as n goes to infinity. Proposition 3.1.
For each n , let Φ n ∼ DP P ( K n ) be a stationary and isotropic DPPin R n , and assume ≤ ˆ K n < . Let X n be a random vector in R n with probabilitydensity K n ( x ) || K n || . Assume that as n → ∞ , | X n |√ n → R ∗ in probability. (6) Then, lim n →∞ E [ η n ( B ( √ nR ))] E [ η n ( R n )] = , R < R ∗ , R > R ∗ . (7) Remark 3.1.
One way to show (6) is to show thatlim n →∞ Var( | X n | ) n = 0 and lim n →∞ (cid:18) E [ | X n | ] n (cid:19) / = R ∗ ∈ (0 , ∞ ) , and then apply Chebychev’s inequality. Remark 3.2.
For general vectors X n in R n , the concentration of | X n | for large n hasbeen well-studied (see [6], [9], [11]). Indeed, in [6, Proposition 3], it is proved that X n is concentrated in a “thin shell”, i.e., there exists a sequence { ε n } such that ε n → n → ∞ and for each n , P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) | X n | E [ | X n | ] − (cid:12)(cid:12)(cid:12)(cid:12) ≥ ε n (cid:19) ≤ ε n , (8)if and only if | X n | has a finite r th moment for r >
2, and for some 2 < p < r , (cid:12)(cid:12)(cid:12)(cid:12) E [ | X n | p ] /p E [ | X n | ] / − (cid:12)(cid:12)(cid:12)(cid:12) → n → ∞ . For random vectors with log-concave distributions, the deviation estimate can beimproved from the estimate obtained through Chebychev’s inequality (see Remark 3.1).The best known estimate is given by the following theorem in [9].
Theorem 3.1. (Gu´edon and Milman [9]) Let X denote a random vector in R n suchthat E X = 0 and E ( X ⊗ X ) = I n . Assume X has a log-concave density. Then, forsome C > and c > , P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) | X |√ n − (cid:12)(cid:12)(cid:12)(cid:12) ≥ t (cid:19) ≤ C exp (cid:0) − c √ n min( t , t ) (cid:1) . This gives the following result.
Proposition 3.2.
For each n , let Φ n ∼ DP P ( K n ) be a stationary and isotropic DPPin R n , and assume ≤ ˆ K n < . Let X n be a random vector with density K n ( x ) || K n || andlet σ n = E | X n | . If K n is log-concave for all n , then there exist positive constants C, c such that for all δ ∈ (0 , , E [ η n ( B n ( σ n (1 − δ )))] E [ η n ( R n )] ≤ Ce − c √ nδ , and for all δ > , E [ η n ( R n \ B n ( σ n (1 + δ )))] E [ η n ( R n )] ≤ Ce − c √ n min( δ ,δ ) . If, in addition, lim n →∞ σ n √ n = R ∗ ∈ (0 , ∞ ) , (9) then for this R ∗ , the threshold (7) occurs, and for all R < R ∗ , there exists a constant C ( R ) > such that lim inf n →∞ − √ n ln E [ η n ( B n ( √ nR ))] E [ η n ( R n )] ≥ C ( R ) . Remark 3.3.
The last conclusion of Proposition 3.2 about the rate also holds for
R > R ∗ if B n ( √ nR ) is replaced by R n \ B n ( √ nR ).The assumption of large deviation principle (LDP) concentration leads to an esti-mate of the exponential rate of convergence with speed n and an exact computationof the reach of repulsion R ∗ . Proposition 3.3.
For each n , let Φ n ∼ DP P ( K n ) be a stationary and isotropic DPPin R n , and assume ≤ ˆ K n < . Let X n be a random vector with density K n ( x ) || K n || andsuppose | X n |√ n satisfies a LDP with strictly convex rate function I . Then, for R ∗ suchthat I ( R ∗ ) = 0 , the threshold (7) occurs. Also, for R < R ∗ , − inf r The second conclusion of Proposition 3.3 about the rate also holds for R > R ∗ if B n ( √ nR ) is replaced by R n \ B n ( √ nR ).If a sequence of DPPs in increasing dimensions exhibits a reach of repulsion R ∗ , thissays that the points of η n are most likely to be near distance √ nR ∗ away from theorigin in high dimensions. If R ∗ is less than ˜ R from (5), points are most likely to beremoved at a distance where points of Φ n appear with probability decreasing to zeroas n increases due to dimensionality. If R ∗ can reach past ˜ R , the points “pushed out”by repulsion are most likely to lie at a distance where points of Φ n appear with highprobability. Thus it is of interest to check whether there exist DPP models such that R ∗ is greater than or equal to ˜ R , i.e., if P (Φ , ! n ( B n ( √ nR ∗ )) = 0) → n → ∞ . InSections 4.1 and 4.2 examples of DPP models with this reach are provided.The above results have strong assumptions, and open up additional questions. Thefirst question is whether the points of η n tend to lie at distances scaling with √ n , i.e., is the Shannon regime the right one to examine the repulsiveness between points of afamily of DPPs in high dimensions? By the radial symmetry of the density of each X n , the coordinates { X n,k } nk =1 are identically distributed, and the sequence | X n | isthe sequence of row sums of a triangular array of random variables with identicallydistributed rows. If the coordinate distributions depend on dimension in such a waythat E (cid:0) | X n | (cid:1) = O ( n ), then a different scaling is needed. 4. Examples In the following, specific families that were presented in [2] and [15] are examinedthat illustrate both examples of DPP models satisfying the above results, as wellas examples that do not. These examples provide a window into the wide scope ofrepulsive behavior that can be described using this framework.The first task will be to determine the behavior of E [ η n ( R n )] as n increases. Foreach of the examples provided in this section, lim n →∞ E [ η n ( R n )] = 0, but each classexhibits this convergence at different speeds. Then the goal is to determine if the DPPmodels satisfy the conditions of Propositions 3.1, 3.2, or 3.3. For each n , let Φ n ∼ DP P ( K n ) in R n be a Laguerre-Gaussian DPP as described in[2] with intensity K n (0) = e nρ , i.e., for some m ∈ N , α ∈ R + , let K n ( x ) = e nρ (cid:0) m − n m − (cid:1) L n/ m − (cid:18) m (cid:12)(cid:12)(cid:12) xα (cid:12)(cid:12)(cid:12) (cid:19) e − | x/α | m , x ∈ R n , (10)where L βm ( r ) = P mk =0 (cid:0) m + βm − k (cid:1) ( − r ) k k ! , for all r ∈ R , denote the Laguerre polynomials.From [2], the condition 0 ≤ ˆ K n < α n , α < e ρ ( mπ ) / (cid:18) m − n/ m − (cid:19) n . (11)Direct calculations give that the global measure of repulsiveness is E [ η n ( R n )] = e nρ α nn (cid:0) m − n m − (cid:1) (cid:16) mπ (cid:17) n m − X k,j =0 (cid:18) m − n m − − k (cid:19)(cid:18) m − n m − − j (cid:19) ( − k + j k ! j ! Γ (cid:0) n + k + j (cid:1) k + j Γ (cid:0) n (cid:1) . (12) By (11), E [ η ( R n )] < − n f ( n, m ), where f ( n, m ) = m − X k,j =0 (cid:0) m − n/ m − − k (cid:1)(cid:0) m − n/ m − − j (cid:1)(cid:0) m − n/ m − (cid:1) ( − k + j k ! j ! Γ (cid:0) n + k + j (cid:1) k + j Γ (cid:0) n (cid:1) = O ( n m − ) . It follows from [2, (5.7)] that for fixed n , lim m →∞ − n f ( n, m ) = 1, and as α → K n approaches the Poisson kernel. Thus, this class of DPPs covers a wide range ofrepulsiveness for fixed dimension n . However, for any fixed m , the dominant behavioras n increases is 2 − n .Since (cid:0) m − n/ m − (cid:1) n decreases to one as n goes to infinity, a sufficient condition for(11) to hold for all n is 0 < α < e − ρ ( mπ ) − . Note that this scaling for the intensity isthe right one for observing interactions between the parameters of the model becauseit provides a trade-off between how large the parameter α can be and the magnitudeof ρ . If the intensity did not grow as quickly with dimension, the upper bound on α would depend less and less on changes in ρ as dimension increased, and if the intensitygrew more quickly, the upper bound for α would tend to zero as n goes to infinity.Proposition 3.3 holds for this sequence of DPPs. Indeed, the next lemma shows thatthe sequence of R + -valued random variables | X n |√ n satisfies a LDP. Lemma 4.1. Fix m ∈ N , ρ ∈ R , and let α ∈ (0 , e − ρ ( mπ ) − / ) . For each n , let X n be a random vector in R n with probability density K n ( x ) || K n || , where K n is given by (10) .Then, the sequence { | X n |√ n } n satisfies an LDP with rate function Λ ∗ ( x ) = 2 x α m − 12 + 12 log (cid:18) α m x (cid:19) . Using this lemma, Proposition 3.3 implies that an R ∗ exists, and the exponentialrates can be determined. In addition, using (12), the exponential rate of decay of E [ η n ( B n ( √ nR ))] can be computed. Proposition 4.1. Fix m ∈ N , ρ ∈ R , and let α ∈ (0 , e − ρ ( mπ ) − / ) . For each n , let Φ n ∼ DP P ( K n ) where K n is given by (10). Then, for R ∗ := √ m α , lim n →∞ − n log E [ η n ( B n ( √ nR ))] = − ρ − log 2 πe + R α m − log R, < R < R ∗ − ρ − log α − log mπ , R > R ∗ . The rate decays as R increases to R ∗ := √ m α and then for R > R ∗ , the rate no longer depends on R . This coincides with our interpretation of R ∗ as the asymptoticreach of repulsion of the sequence of DPPs.For a fixed α , a larger m will give farther reach, and for a fixed m , a larger α willprovide a farther reach. However, by the bound α < e ρ ( mπ ) / , the following upperbound on the reach holds uniformly for all m : R ∗ := √ m α < e ρ π / . Note that the larger ρ is, the smaller the upper bound on R ∗ can be. This follows fromthe relationship between α and ρ : the higher the intensity, the smaller α must be forthe DPP to exist. Since a larger α implies a larger values of E [ η n ( R n )], the parameter α is associated with the strength of the repulsiveness. The relationship with ρ showcasesthe following tradeoff observed in [15]: the higher the intensity of the DPP, the lessrepulsive it can be.As mentioned in the previous section, it is of interest to know whether there is arange of parameters such that R ∗ is greater than ˜ R , the threshold for the convergenceof the nearest-neighbor function of Φ (5). For Laguerre-Gaussian models, R ∗ := √ mα is larger than ˜ R and α satisfies the condition of Lemma 4.1 if (cid:18) e (cid:19) / < e ρ √ mπα < . Since the lower bound is strictly less than one, there is a non-empty range for α suchthat the reach of repulsion reaches past ˜ R . The power exponential spectral models, introduced in [15], are defined through theFourier transform of the kernel. For almost all of these models, there is no closed formfor the kernel K . Using properties of the Fourier transform, a similar analysis of therepulsive behavior can still be performed.For each n , let Φ n ∼ DP P ( K n ) be a power exponential DPP with intensity K n (0) = e nρ and parapmeters ν > α n > 0, i.e., letˆ K n ( x ) = e nρ Γ( n + 1) α nn π n/ Γ( nν + 1) e −| α n x | ν , x ∈ R n . (13) When ν = 2, a closed form expression for K n exists and is called the Gaussian kernel.The condition 0 ≤ ˆ K n < α n : α n < Γ( nν + 1) n π / e ρ Γ (cid:0) n + 1 (cid:1) n , (14)and the asymptotic expansion for the upper bound on α n as n → ∞ is (cid:16) Γ( nν +1) π n/ e nρ Γ( n +1) (cid:17) /n ∼ (cid:18) √ πnν ( nνe ) n/ν π n/ e nρ √ πn ( n e ) n/ (cid:19) /n ∼ e − ρ n ν − (2 πe ) / ( νe ) /ν = O ( n ν − ) . By Parseval’s theorem and a change of variables, E [ η n ( R n )] = 1 e nρ || K n || = 1 e nρ || ˆ K n || = 1 e nρ e nρ Γ (cid:0) n + 1 (cid:1) α nn π n/ Γ (cid:0) nν + 1 (cid:1) ! Z R n e − | α n x | ν d x = e nρ (cid:18) Γ( n + 1) α nn π n/ Γ( nν + 1) (cid:19) nπ n/ Γ( n + 1) Z ∞ r n − e − αr ) ν d r = e nρ Γ( n + 1) α nn π n Γ( nν + 1) n nν να nn Z ∞ t nν − e − t d t = 2 − nν α nn e nρ Γ( n + 1) π n Γ( nν + 1) . (15)By the bound on α n (14), E [ η n ( R n )] < − nν . For fixed dimension n , the global measure of repulsion approaches its upper bound ofone for large ν . Thus, this class covers a wide range of repulsiveness similar to theLaguerre-Gaussian DPPs. However, for fixed ν , the measure decays exponentially as n goes to infinity. Note that for ν > 2, the rate is smaller than for the Laguerre-Gaussianmodels, i.e., the decay is slower.The following results show that if the parameters α n grow appropriately with n ,this sequence satisfies the assumptions of Proposition 3.1. Lemma 4.2. For each n , let X n be a vector in R n with density K n || K n || such that ˆ K n is given by (13). Assume α n ∼ αn ν − as n → ∞ for α ∈ (0 , ∞ ) , and α n < (cid:16) Γ( nν +1) π n/ e nρ Γ( n +1) (cid:17) /n for all n . Then, as n → ∞ , | X n |√ n → α (2 ν ) /ν π in probability . Now, applying Proposition 3.1, the following holds for a sequence of power expo-nential DPPs in the Shannon regime. Proposition 4.2. For each n , let Φ n ∼ DP P ( K n ) where ˆ K n satisfies the assumptionsin Lemma 4.2. Then, for R ∗ := α (2 ν ) /ν π , lim n →∞ E [ η n ( B n ( √ nR ))] E [ η n ( R n )] = , R < R ∗ , , R > R ∗ . For ν > 1, the reach of repulsion R ∗ for the power exponential models can alsoreach past the nearest neighbor threshold ˜ R . Indeed, for α n ∼ αn ν − , R ∗ := α (2 ν ) /ν π satisfies P [Φ n ( B n (0 , √ nR ∗ )) = 0] → n → ∞ if α (2 ν ) /ν π > √ πee ρ . By the asymptotic formula (14) for the upper bound of α n , α < √ πee ρ ( νe ) /ν . Thus, R ∗ reaches past ˜ R when α n ∼ αn ν − and4 π (2 ν ) /ν e ρ √ πe < α < √ πee ρ ( νe ) /ν . The interval is non-empty since the upper bound is strictly greater than the lowerbound for ν > Another class of DPP models presented in [2] is the Bessel-type. This class is morerepulsive than the previous two families of models. It is shown that while the Shannonregime is the right scaling to examine the repulsiveness of this class in high dimensions,a sequence of these DPPs does not satisfy the conditions of Proposition 3.1.For each n , let Φ n ∼ DP P ( K n ) be a Bessel-type DPP with parameters σ ≥ α > 0, and intensity K n (0) = e nρ , for ρ ∈ R . That is, let K n ( x ) = e nρ ( σ + n ) / Γ (cid:18) σ + n + 22 (cid:19) J ( σ + n ) / (2 | x/α | p ( σ + n ) / | x/α | p ( σ + n ) / ( σ + n ) / . (16)From [2], the bound 0 ≤ ˆ K n < α nn < ( σ + n ) n/ Γ (cid:0) σ + 1 (cid:1) e nρ (2 π ) n/ Γ (cid:0) σ + n + 1 (cid:1) . (17)Similarly to the previous examples, this family contains DPPs covering a wide rangeof repulsiveness measured by η n , and as n → ∞ , they are more repulsive in the sense that E [ η n ( R n )] decays slower. Indeed, E [ η n ( R n )] = 1 e nρ Z R n K n ( x ) dx = e nρ (2 π ) n/ α n ( σ + n ) n/ Γ (cid:0) n (cid:1) Γ (cid:0) σ + n +22 (cid:1) Γ (cid:0) n (cid:1) Γ( σ + 1)Γ (cid:0) σ + 1 (cid:1) Γ (cid:0) σ + n + 1 (cid:1) = e nρ (2 π ) n/ α n ( σ + n ) n/ Γ( σ + 1)Γ (cid:0) σ + n + 1 (cid:1) Γ (cid:0) σ + 1 (cid:1) Γ (cid:0) σ + n + 1 (cid:1) , and by the upper bound (17), E [ η n ( R n )] < Γ( σ + 1)Γ (cid:0) σ + n + 1 (cid:1) Γ (cid:0) σ + 1 (cid:1) Γ (cid:0) σ + n + 1 (cid:1) . By Stirling’s formula, as n → ∞ , Γ( σ +1)Γ ( σ + n +1 ) Γ ( σ +1 ) Γ ( σ + n +1 ) = O ( n − σ/ ).These DPPs do not satisfy the conditions of Proposition 3.1, and so the concen-tration of the first moment measure does not occur, contrary to the first two familiespresented. However, the repulsive measure does not reach past the √ n scale in thesense of the following proposition. Proposition 4.3. Let ρ ∈ R , α > , and σ > . For each n , let Φ n ∼ DP P ( K n ) in R n with K n given by (16) . Then, for any β > and R > , lim n →∞ E [ η n ( R n \ B n ( Rn β ))] E [ η n ( R n )] = 0 . Another class of DPPs described in [15] are those with normal-variance mixturekernels. Let Φ n ∼ DP P ( K n ) be a normal-variance mixture DPP in R n with intensity e nρ for ρ ∈ R , i.e., let K n ( x ) = e nρ E [ W − n/ e −| x | / (2 W ) ] E [ W − n/ ] , x ∈ R n , for some non-negative real-valued random variable W such that E [ W − n/ ] < ∞ . From[15], the bound 0 ≤ ˆ K < e nρ < E [ W − n/ ] / (2 π ) n/ . (18)If √ W = α , this is known as the Gaussian DPP model. If W ∼ Gamma( ν + n , α ),this is called the Whittle-Mat´ern model. The Cauchy model is given when W ∼ Gamma( ν, α − ). In both cases ν > α > This family of DPPs does not cover a wide range of repulsiveness like the previousfamilies. Indeed, for any random variable W in R + such that E [ W − n ] < ∞ , Parseval’stheorem, Jensen’s inequality, (18), and Fubini’s theorem imply E [ η n ( R n )] = 1 e nρ Z R n ˆ K n ( x ) d x = 1 e nρ Z R n (cid:18) e nρ (2 π ) n E h W − n i E h e − π | x | W i(cid:19) d x ≤ (2 π ) n E h W − n i Z R n E h e − π | x | W i d x = (2 π ) n E [ W − n ] E (cid:18) (4 πW ) − n E (cid:20) (4 πW ) n Z R n e − π | x | W d x (cid:12)(cid:12)(cid:12)(cid:12) W (cid:21)(cid:19) = 2 − n . Is it difficult to make further general statements about this class because the behav-ior of the sequence | X n |√ n depends greatly on the distribution of the R + -valued randomvariable W . The rest of the section will describe results for specific models in this class.Consider a sequence of normal-variance mixture DPPs all associated with the samerandom variable W . If W is a constant α , the random variables X n become multivariateGaussian vectors with mean zero and variance depending on α . The scaled norms ofthese vectors are well-known to satisfy a LDP since the coordinates are independent.This also corresponds to a Laguerre-Gaussian DPP with parameter m = 2.There is also a subclass of the normal-variance mixture models that satisfy Propo-sition 3.2. In [25], it is proved that if W has a log-concave density, then the normal-variance mixture distribution is log-concave. This implies that K n is log-concave, andthus if condition (9) holds, the conclusion of Proposition 3.2 holds. Since the Gammadistribution for shape parameter ν greater than 1 is log-concave and ν + n ≥ n , Whittle-Mat´ern DPPs are an example from this subclass and exhibit an R ∗ as shown in the following proposition. Proposition 4.4. For each n , let Φ n ∼ DP P ( K n ) be a Whittle-Mat´ern model in R n with intensity e nρ and parameters ν > and α > , i.e., let K n ( x ) = e nρ − ν Γ( ν ) | x | ν α ν K ν (cid:18) | x | α (cid:19) , x ∈ R n , (19) where α < Γ( ν ) n Γ ( ν + n ) n √ πe ρ and K ν is the modified Bessel kernel of the second kind.Then, for R ∗ := α , lim n →∞ E [ η n ( B n ( √ nR ))] E [ η n ( R n )] = , R < R ∗ , R > R ∗ . Remark 4.1. The upper bound on α needed for existence implies that for all ν , R ∗ := α < Γ( ν ) n Γ (cid:0) ν + n (cid:1) n √ πe ρ < √ πee ρ := ˜ R, since (cid:18) Γ( ν )Γ ( ν + n ) (cid:19) n ≤ > √ e . Thus, for these models, R ∗ never reaches pastthe nearest neighbor threshold ˜ R .Finally, the following proposition shows that the Cauchy models satisfy the condi-tions of Proposition 3.1 if the α parameter grows appropriately with n . Proposition 4.5. For each n , let Φ n ∼ DP P ( K n ) be a Cauchy model in R n withintensity e nρ and parameters ν > and α n > , i.e., let K n ( x ) = e nρ (1 + | xα n | ) ν + n , x ∈ R n . Assume α n ∼ αn / as n → ∞ for some α > such that α n < Γ( ν + n ) n √ πe ρ Γ( ν ) n for each n .Then, for R ∗ := α , lim n →∞ E [ η n ( B n ( √ nR ))] E [ η n ( R n )] = , R < R ∗ , R > R ∗ . Remark 4.2. The upper bound on α n has the following asymptotic expansion as n → ∞ : α n < Γ( ν + n ) n √ πe ρ Γ( ν ) n ∼ n / √ πee ρ . Thus, if α n ∼ αn , the reach of repulsion has the upper bound R ∗ := α < √ πee ρ . This upper bound is precisely the threshold ˜ R for the nearest neighbor function, andso unlike in the case of Laguerre-Gaussian DPPs and power exponential DPPs, thereach of repulsion R ∗ for a sequence of Cauchy models with fixed parameter ν cannotreach past ˜ R . 5. Application to determinantal Boolean models in the Shannon regime Poisson Boolean models in the Shannon regime were studied in [1], and the degreethreshold results can be extended to Laguerre-Gaussian DPPs using Proposition 4.1.The setting is the following: Consider a sequence of stationary DPPs Φ n , indexed bydimension, where Φ n ∼ DP P ( K n ) in R n . Assume that for each n , K n is continuous,symmetric, and 0 ≤ ˆ K n < 1. Let the intensity of Φ n be K n (0) = e nρ . Let Φ n = P k δ T ( k ) n and R > 0. Then, consider the sequence of particle processes [22], calleddeterminantal Boolean models, C n = [ k B n (cid:18) T ( k ) n , √ nR (cid:19) . The degree of each model is the expected number of balls that intersect the ball centeredat zero under the reduced Palm distribution, i.e., E [Φ , ! n ( B n ( √ nR ))]. In the case whenΦ n is Poisson, E , ! [Φ n ( B ( √ nR ))] = E [Φ n ( B ( √ nR ))] by Slivnyak’s theorem, andlim n →∞ n ln E , ! [Φ n ( B n ( √ nR ))] = ρ + 12 log 2 πe + log R. To extend this result to DPPs, it is needed that as n → ∞ , E [Φ , ! n ( B n ( √ nR ))] ∼ E [Φ n ( B n ( √ nR ))].Note that this would be impossible for a repulsive point process like the Mat´ernhardcore process, since E [Φ , ! n ( B n ( R n ))] = 0 for all R n less than the hardcore radius.However, for DPPs, notice that E [Φ , ! n ( B n ( √ nR ))] E [Φ n ( B n ( √ nR ))] = 1 − E [ η n ( B n ( √ nR ))] E [Φ n ( B n ( √ nR ))] . Thus, if E [ η n ( B n ( √ nR ))] E [Φ n ( B n ( √ nR ))] → n → ∞ , then the degree of the determinantal Booleanmodel has the same asymptotic behavior as the Poisson Boolean model.In the case of Laguerre-Gaussian kernels, this is the case, and the earlier resultseven provide the rate at which the quantity goes to zero, which exhibits a threshold at R ∗ as is expected. Proposition 5.1. Let m ∈ N and ρ ∈ R . For each n , let Φ n ∼ DP P ( K n ) in R n where K n ( x ) = e nρ (cid:16) m − n/ m − (cid:17) L n/ m − (cid:18) m (cid:12)(cid:12)(cid:12) xα (cid:12)(cid:12)(cid:12) (cid:19) e − | x/α | m , and α is a parameter such that < α < √ mπe ρ . Then, lim n →∞ − n ln E [ η n ( B n ( √ nR ))] E [Φ n ( B n ( √ nR ))] = R α m , < R < √ m α + log 2 − log α − log m + log R, R > √ m α . 6. Conclusion By examining a measure of repulsiveness of DPPs, this paper provides insight intothe high dimensional behavior of different families of DPP models. Most of the familiesof DPPs presented in this paper have a global measure of repulsion decreasing to zeroas dimension increases, indicating that they become more and more similar to Poissonpoint processes in high dimensions by (3). However, the reach of the small repulsiveeffect can still be quantified. By making a connection between the kernel of the DPPand the concentration in high dimensions of the norm of a random vector, we haveshown under certain conditions that there exists a distance on the √ n scale at whichthe repulsive effect of a point of the DPP model is strongest as n → ∞ . It has beenillustrated that some families of DPPs exhibit this reach of repulsion and some do not.The results are summarized in Table 1.Many questions remain concerning the range of possible repulsive behavior of DPPsin high dimensions. First, the results can be extended to scalings other than theShannon regime in the following way. Assumption (6) in Proposition 3.1 can begeneralized to the assumption that for some sequence b n , | X n | b n → R ∗ as n → ∞ .If b n = O ( n ), the result holds for a different scaling than the Shannon regime, andthe repulsiveness is strongest near R ∗ b n in high dimensions. While this is preciselywhat is shown not to happen for the Bessel-type DPPs if σ > 0, examples of thisgeneralization for b n = o ( n ) can be obtained from the power exponential DPPs when α n = o ( n ν − ). However, as noted in the introduction, any distance scaling smallerthan √ n will not reach the regime where the expected number of points goes to infinityas dimension grows. Thus, this scaling appears less interesting. It would be interestingto find a family of DPPs that exhibits the concentration for b n ≫ √ n .For all of the DPPs studied in this paper, E [ η n ( R n )] → n → ∞ . This is notalways the case. For instance, there exists a class of DPPs such that for c ∈ (0 , E [ η n ( R n )] = c for all n . Indeed, let K n ∈ L ( R n ) be such that its Fourier transform isˆ K n ( ξ ) = √ c B n ( r n ) ( ξ ) , ξ ∈ R n , (20)where r n ∈ R + is such that Vol( B n ( r n )) = K n (0). Then, E [ η n ( R n )] = 1 K n (0) Z R n K n ( x ) d x = 1 K n (0) Z R n ˆ K n ( ξ ) d ξ = cK n (0) Vol( B n ( r n )) = c. It would be useful to find a necessary and sufficient condition for E [ η n ( R n )] to convergeto zero. However, if E [ η n ( R n )] does not converge to zero, this does not necessarilyprevent P ( η n ( R n ) = 0) from approaching one as n goes to infinity. It would beinteresting to find a class of DPPs where P ( η n ( R n ) = 0) approaches some c < η is well-defined, it is assumed that the kernel K associated with Φ satisfies 0 ≤ ˆ K < 1. However, Φ still exists when ˆ K is allowedto attain the maximum value of one. For the models studied in this paper, it is thecase when the parameter achieves its upper bound. In this case, we can still define themeasure of repulsiveness (2) even though it may not be interpretable as the intensitymeasure of a point process η . Replacing E [ η ( B )] with R B (1 − g ( x ))d x for B ∈ B ( R n ),the main results (Propositions 3.1, 3.2, and 3.3) can be restated with the conditionthat 0 ≤ ˆ K ≤ 1. In this case, the reach of repulsion R ∗ is interpreted as the distanceon the √ n scale at which the measure of repulsion is strongest.A particularly interesting subclass of the DPPs described in the previous paragraphare the most repulsive stationary DPPs, introduced in the on-line supplementarymaterial to [15] (see [14]). These DPPs maximize the measure of repulsiveness γ ,and have a kernel K such that ˆ K is defined as in (20) but with c = 1. For themost repulsive DPPs, γ = 1 in any dimension. In addition, for a sequence of DPPs { Φ n } n ∈ N where Φ n is the most repulsive DPP in R n with intensity e nρ , X n as definedin Proposition 3.1 satisfies E [ | X n | ] = Z R n | x | K n ( x ) || K n || d x = Γ (cid:0) n + 1 (cid:1) π n/ Z R n | x | J n/ (cid:16) √ π Γ (cid:0) n + 1 (cid:1) /n e ρ | x | (cid:17) | x | n d x = n Z ∞ rJ n/ (cid:18) √ π Γ (cid:16) n (cid:17) /n e ρ r (cid:19) d r, where J ν is the Bessel function of the first kind of order ν (see [2]). By [19, Eq. 1.17.13],this integral does not converge, i.e., | X n | does not have a finite second moment. Table 1: Summary of Results DPP Class E [ η n ( R n )] R ∗ Rate type R ∗ > ˜ R Laguerre-Gaussian < − n O ( n m − ) √ m α LDP (cid:0) e (cid:1) < e ρ √ mπα < < − nν α (2 ν ) ν π Chebychev ν e < e ρ ν ν √ πe α < e ν Bessel-type < O ( n − σ/ ) N/A N/A N/AWhittle-Mat´ern < − n α Log-concave N/ACauchy < − n α Chebychev N/A Appendix A. Proof of (5) For each n , let Φ n ∼ DP P ( K n ) in R n be stationary with intensity K n (0) = e nρ .From (1), there exists ˜ R := √ πee ρ such thatlim n →∞ E [Φ n ( B n ( √ nR ))] = , R < ˜ R ∞ , R > ˜ R. By Theorem 2.3, E [Φ n ( B n ( √ nR ))] − E [Φ , ! n ( B n ( √ nR ))] = 1 e nρ Z B n ( √ nR ) K n ( x ) d x Then, by Parseval’s theorem and Theorem 2.2,1 e nρ Z B n ( √ nR ) K n ( x ) d x ≤ e nρ Z R n ˆ K n ( ξ ) d ξ ≤ e nρ Z R n ˆ K n ( ξ )d ξ = 1 . Also, since e nρ R B n ( √ nR ) K n ( x ) d x ≥ 0, the following bounds hold: E [Φ n ( B n ( √ nR ))] − ≤ E [Φ , ! n ( B n ( √ nR ))] ≤ E [Φ n ( B n ( √ nR ))] . Thus, the threshold remains the same for the reduced Palm expectation:lim n →∞ E [Φ , ! n ( B n ( √ nR ))] = , R < ˜ R ∞ , R > ˜ R. By the first moment inequality and Proposition 5.1 in [3], one has the following bounds:1 − E [Φ , ! n ( B ( √ nR ))] ≤ P (Φ , ! n ( B n ( √ nR )) = 0) ≤ exp (cid:0) − E [Φ , ! n ( B ( √ nR ))] (cid:1) . Thus, lim n →∞ P (Φ , ! n ( B n ( √ nR )) > 0) = , R < ˜ R R > ˜ R. Appendix B. Proof of Lemma 3.1 By Theorem 2.3, for any B ∈ B ( R n ), E [ η n ( B )] = E [Φ n ( B )] − E [Φ , ! n ( B )] = 1 K n (0) Z B K n ( x ) d x, (21)i.e. the first moment measure of η n has a density with respect to Lebesgue measureequal to K n (0) K n ( x ) . Then by the monotone convergence theorem, E [ η n ( R n )] = lim R →∞ E [ η n ( B n ( R ))] = 1 K n (0) Z R n K n ( x ) d x = || K n || K n (0) . Thus, for all B ∈ B ( R n ), P ( X n ∈ B ) = Z B K n ( x ) || K n || d x = E [ η n ( B )] E [ η n ( R n )] . Appendix C. Proof of Main ResultsC.1. Proof of Proposition 3.1 The assumption | X n |√ n → R ∗ in probability means that for all ε > P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) | X n |√ n − R ∗ (cid:12)(cid:12)(cid:12)(cid:12) > ε (cid:19) → , as n → ∞ . Now, assume R < R ∗ . Then, there exists ε > R = R ∗ − ε . Thus, P ( | X n | ≤ √ nR ) = P (cid:18) | X n |√ n ≤ R ∗ − ε (cid:19) ≤ P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) | X n |√ n − R ∗ (cid:12)(cid:12)(cid:12)(cid:12) > ε (cid:19) → n → ∞ . Second, assume R > R ∗ . Then, there exists ε > R = R ∗ + ε , and P ( | X n | ≤ √ nR ) = 1 − P (cid:18) | X n |√ n > R ∗ + ε (cid:19) ≥ − P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) | X n |√ n − R ∗ (cid:12)(cid:12)(cid:12)(cid:12) > ε (cid:19) → . Then, by Lemma 3.1, as n → ∞ , E [ η n ( B n ( √ nR ))] E [ η n ( R n )] = P (cid:0) | X n | ≤ √ nR (cid:1) → , R < R ∗ , R > R ∗ . C.2. Proof of Proposition 3.2 Since for all n , Φ n is isotropic, X n as defined in Proposition 3.1 has a radiallysymmetric density. Thus, X n has the same distribution as the product R n U n , where R n is equal in distribution to | X n | , U n is uniformly distributed on S n − , and R n and U n are independent. Letting σ n = E | X n | for each n , √ nσ n X n then satisfies the conditionsof Theorem 3.1 for each n . Then, by Theorem 3.1, for any δ > 0, there exist absoluteconstants C, c > P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) | X n | σ n − (cid:12)(cid:12)(cid:12)(cid:12) ≥ δ (cid:19) ≤ Ce − c √ n min( δ,δ ) . Now, let δ ∈ (0 , E [ η n ( B n ( σ n (1 − δ )))] E [ η n ( R n )] = P (cid:18) | X n | σ n ≤ − δ (cid:19) ≤ Ce − c √ nδ , since min( δ , δ ) = δ for δ ∈ (0 , δ > E [ η n ( R n \ B n ( σ n (1 + δ )))] E [ η n ( R n )] = P (cid:18) | X n | σ n ≥ δ (cid:19) ≤ Ce − c √ n min( δ ,δ ) . Now, assume σ n √ n → R ∗ ∈ (0 , ∞ ) as n → ∞ . For R < R ∗ , there exists ε ∈ (0 , 1) suchthat R = R ∗ (1 − ε ). Then, for all n large enough, √ nR ∗ σ n < − ε − ε and E [ η n ( B n ( √ nR ))] E [ η n ( R n )] = P (cid:0) | X n | ≤ √ nR (cid:1) = P (cid:18) | X n | σ n ≤ √ nRσ n (cid:19) = P (cid:18) | X n | σ n ≤ √ nR ∗ (1 − ε ) σ n (cid:19) ≤ P (cid:18) | X n | σ n ≤ − ε (cid:19) ≤ P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) | X n | σ n − (cid:12)(cid:12)(cid:12)(cid:12) ≥ ε (cid:19) ≤ Ce − c √ n ( ε/ . Thus for all R < R ∗ , there exists a constant C ( ε ( R )) = cε / such thatlim inf n →∞ − √ n ln E [ η n ( B n ( √ nR ))] E [ η n ( R n )] ≥ C ( ε ( R )) . A similar argument gives that for all R > R ∗ , there exists C ( ε ( R )) such thatlim inf n →∞ − √ n ln E [ η n ( R n \ B n ( √ nR ))] E [ η n ( R n )] ≥ C ( ε ( R )) . This implies the threshold (7). C.3. Proof of Proposition 3.3 If | X n |√ n satisfies a large deviations principle with convex rate function I , then bydefinition, − inf r R < R ∗ . If the rate function I is continuous at R , then the above inequalitiesbecome equalities and lim n →∞ − n ln E [ η n ( B n ( √ nR ))] E [ η n ( R n )] = I ( R ) . Appendix D. Proof of Lemma 4.1 The proof shows that the sequence of random variables satisfies the conditions ofthe G¨artner-Ellis theorem (see [5]). First, E [ e s | X n | ] = e nρ (cid:0) m − n/ m − (cid:1) || K n || 22 I(s) z }| {Z R n e − ( α m − s ) | x | (cid:16) L n/ m − (cid:16) m (cid:12)(cid:12) xα (cid:12)(cid:12) (cid:17)(cid:17) d x . Writing out the polynomial, the integral I above becomes I ( s ) = m − X k,j =0 (cid:18) m − n/ m − − k (cid:19)(cid:18) m − n/ m − − j (cid:19) ( − k + j k ! j !( mα ) k + j Z R n e − ( α m − s ) | x | | x | k +2 j d x. A quick calculation shows that for a > Z R n e − a | x | | x | b d x = π n/ a n + b Γ (cid:0) n + b (cid:1) Γ (cid:0) n (cid:1) . (22)Then, if s < α m , I ( s ) = π n/ (cid:0) α m − s (cid:1) n Γ (cid:0) n (cid:1) m − X k,j =0 (cid:18) m − n/ m − − k (cid:19)(cid:18) m − n/ m − − j (cid:19) ( − k + j Γ (cid:0) n + k + j (cid:1) k ! j ! (2 − smα ) k + j , and I ( s ) = ∞ otherwise. For each k, j ∈ N , (cid:18) m − n/ m − − k (cid:19)(cid:18) m − n/ m − − j (cid:19) Γ (cid:16) n k + j (cid:17) ∼ m − − k )!( m − − j )! (cid:16) n (cid:17) m − Γ (cid:16) n (cid:17) , (23)as n → ∞ . So, I ( s ) has the following asymptotic expansion for s < α m as n → ∞ : I ( s ) ∼ π n/ (cid:0) α m − s (cid:1) n (cid:16) n (cid:17) m − m − X k =0 m − X j =0 ( − k + j k ! j !( m − − k )!( m − − j )! 1(2 − smα ) k + j . By (12) and (23),1 e nρ || K n || ∼ α n (cid:0) m − n/ m − (cid:1) (cid:16) mπ (cid:17) n (cid:16) n (cid:17) m − m − X k,j =0 ( − k + j k ! j !( m − − k )!( m − − j )! 12 k + j , (24)and hence, E [ e s | X n | ] ∼ (cid:18) − sα m (cid:19) − n P m − k,j =0 ( − k + j k ! j !( m − − k )!( m − − j )! 1(2 − smα ) k + j P m − k,j =0 ( − k + j k ! j !( m − − k )!( m − − j )! 12 k + j , as n → ∞ . Thus,Λ( s ) = lim n →∞ n log E [ e s | X n | ] = − 12 log (cid:18) − sα m (cid:19) if s < α m , and is infinite otherwise. It is clear that 0 ∈ ( D (Λ)) ◦ , where D (Λ) = { s ∈ R : Λ( s ) < ∞} . Thus, the G¨artner-Ellis conditions are satisfied. The rate function for the LDPis computed with the optimizationΛ ∗ ( x ) = sup λ ∈ R [ xλ − Λ( λ )] = sup λ ∈ R (cid:20) xλ + 12 log (cid:18) − λα m (cid:19)(cid:21) . Then, since0 = ddλ (cid:20) xλ + 12 log (cid:18) − λα m (cid:19)(cid:21) = x − α m − α mλ if and only if λ = 2 α m − x , the rate function isΛ ∗ ( x ) = x (cid:18) α m − x (cid:19) + 12 log − (cid:0) α m − x (cid:1) α m ! = 2 xα m − 12 + 12 log (cid:18) α m x (cid:19) . Then by the contraction principle (see [5]), the sequence | X n |√ n satisfies an LDP withrate function Λ ∗ ( x ) = 2 x α m − 12 + 12 log (cid:18) α m x (cid:19) . Note that Λ ∗ ( x ) = 0 if and only if x = √ m α , implying | X n |√ n → √ m α in probability. Appendix E. Proof of Proposition 4.1 Proof. For each n , let X n be a random vector in R n with density K n || K n || . By Lemma4.1, for R < √ m α ,lim n →∞ − n log P (cid:18) | X n |√ n ≤ R (cid:19) = 2 R α m − 12 + 12 log (cid:18) α m R (cid:19) . Then by (24), as n → ∞ , E [ η n ( R n )] = 1 e nρ || K n || ∼ (cid:18) e ρ α mπ (cid:19) n m − X k,j =0 ( − k + j k ! j !( m − − k )!( m − − j )! 12 k + j , (25)Thus, by Lemma 3.1,lim n →∞ − n log E [ η n ( B n ( √ nR ))] = lim n →∞ − n log E [ η n ( R n )] + lim n →∞ − n log P (cid:18) | X n |√ n ≤ R (cid:19) = − ρ − log α − log (cid:0) mπ (cid:1) + (cid:16) R α m − + log (cid:16) α m R (cid:17)(cid:17) , < R < √ m α − ρ − log α − log (cid:0) mπ (cid:1) , R > √ m α = − ρ − log 2 πe + R α m − log R, < R < √ m α − ρ − log α − log mπ , R > √ m α . (cid:3) Appendix F. Proof of Lemma 4.2 Since for all n , ˆ K n ∈ C ( R n ), Parseval’s theorem implies E [ | X n | ] = 1 || K n || Z R n | x | K n ( x ) d x = 1 || ˆ K n || Z R n − △ ˆ K n ( ξ )(2 π ) ˆ K n ( ξ )d ξ. (26)To compute the Laplacian of ˆ K , we first see that for each i , ∂ ∂x i e −| αx | ν = ∂∂x i ( − να ν x i | x | ν − e −| αx | ν )= − να ν | x | ν − e −| αx | ν − να ν x i (cid:16) ∂∂x i | x | ν − (cid:17) e −| αx | ν + ( να ν x i | x | ν − ) e −| αx | ν = e −| αx | ν (cid:0) − να ν | x | ν − − ν ( ν − α ν x i | x | ν − + ν α ν x i | x | ν − (cid:1) = e −| αx | ν (cid:0) x i ( ν α ν | x | ν − − ν ( ν − α ν | x | ν − ) − να ν | x | ν − (cid:1) . Then, △ e −| αx | ν = n X i =1 ∂ ∂x i e −| αx | ν = n X i =1 e −| αx | ν (cid:0) x i ( ν α ν | x | ν − − ν ( ν − α ν | x | ν − ) − να ν | x | ν − (cid:1) = e −| αx | ν (cid:0) | x | ( ν α ν | x | ν − − ν ( ν − α ν | x | ν − ) − nνα ν | x | ν − (cid:1) = e −| αx | ν (cid:0) ν α ν | x | ν − − ( ν ( ν − α ν + nνα ν ) | x | ν − (cid:1) . Thus by (26) and (15), E [ | X n | ] = Γ( n + 1) α nn nν π π n/ Γ( nν + 1) Z R n e − | α n x | ν (cid:0) ( ν ( ν − α νn + nνα νn ) | x | ν − − ν α νn | x | ν − (cid:1) d x = Γ( n + 1) α n + νn nν ν π π n Γ( nν + 1) (cid:20) ( ν − n ) Z R n | x | ν − e − | α n x | ν d x − να νn Z R n e − | α n x | ν | x | ν − dx (cid:21) . Then, using (22), E [ | X n | ] = n α n + νn nν ν π Γ( nν + 1) " − να νn Γ( n +2 ν − ν ) ν ( n +2 ν − /ν α n +2 ν − n + ( ν − n )Γ( n + ν − ν ) ν ( n + ν − /ν α n + ν − n = n /ν α n π Γ( nν + 1) (cid:20) ( ν − n )2 Γ (cid:18) n − ν + 1 (cid:19) − ν (cid:18) n − ν + 2 (cid:19)(cid:21) = n /ν α n Γ (cid:0) n − ν + 1 (cid:1) π Γ( nν + 1) (cid:20) n ν − (cid:21) . By the asymptotic formula for the Gamma function, as n → ∞ , E [ | X n | ] ∼ n α n ν π (cid:18)r ν πn (cid:16) νen (cid:17) nν (cid:19) r π ( n − ν (cid:18) n − νe (cid:19) ( n − ν (cid:20) n ν − (cid:21) = n α n /ν π √ n − √ n (cid:18) − n (cid:19) nν (cid:18) n − νe (cid:19) − ν (cid:20) n ν − (cid:21) ∼ n − /ν α n (2 ν ) /ν π . By assumption, α n ∼ αn ν − for some constant α ∈ (0 , ∞ ). Thus,lim n →∞ E [ | X n | ] n = α (2 ν ) /ν π . For the second moment of | X n | , Parseval’s theorem is applied again and gives that E [( | X n | ) ] = 1 || K n || Z R n ( | x | K n ( x )) d x = 1 || K n || Z R n ( △ ˆ K n ( ξ )) (2 π ) d ξ. (27) Then, by the above computation of the Laplacian of ˆ K , (22), and (15), E [( | X n | ) ] = Γ( n + 1) α nn n/ν ν α νn (2 π ) π n/ Γ( nν + 1) Z R n e ( − | α n x | ν ) (cid:0) να νn | x | ν − − ( ν − n ) | x | ν − (cid:1) d x = Γ( n + 1) α nn n/ν ν α νn (2 π ) π n/ Γ( nν + 1) (cid:20) ( να νn ) Z R n e − | α n x | ν | x | ν − d x − να νn ( ν − n ) Z R n e − | α n x | ν | x | ν − dx + ( ν − n ) Z R n e − | α n x | ν | x | ν − d x (cid:21) = n α nn n/ν ν α νn (2 π ) Γ( nν + 1) (cid:20) ( να νn ) Γ( n +4 ν − ν ) ν ( n +4 ν − /ν α n +4 ν − n − να νn ( ν − n )Γ( n +3 ν − ν ) ν ( n +3 ν − /ν α n +3 ν − n + ( ν − n ) Γ( n +2 ν − ν ) ν ( n +2 ν − /ν α n +2 ν − n (cid:21) = n /ν ν α n (2 π ) Γ( nν + 1) (cid:20) ν Γ (cid:0) n − ν + 4 (cid:1) − ν − n )Γ (cid:0) n − ν + 3 (cid:1) + ( ν − n ) Γ (cid:0) n − ν + 2 (cid:1) ν (cid:21) = n /ν α n Γ (cid:0) n − ν + 1 (cid:1) (2 π ) Γ (cid:0) nν + 1 (cid:1) (cid:20) ν (cid:18) n − ν + 3 (cid:19) (cid:18) n − ν + 2 (cid:19) (cid:18) n − ν + 1 (cid:19) − ν ( n + ν − (cid:18) n − ν + 2 (cid:19) (cid:18) n − ν + 1 (cid:19) + ν ( n + ν − (cid:18) n − ν + 1 (cid:19) (cid:21) = n /ν α n (2 π ) Γ (cid:0) n − ν + 1 (cid:1) Γ (cid:0) nν + 1 (cid:1) (cid:18) n − n + n + o ( n ) (cid:19) = n /ν α n (2 π ) Γ (cid:0) n − ν + 1 (cid:1) Γ (cid:0) nν + 1 (cid:1) (cid:18) 116 + o (1) (cid:19) ∼ n /ν α n π ) r ν πn (cid:16) νen (cid:17) nν r π ( n − ν (cid:18) n − νe (cid:19) n − ν = n r n − n (cid:18) − n (cid:19) nν (cid:18) n − νe (cid:19) − ν α n /ν π ) ∼ n ( n − − ν α n (2 ν ) /ν π ) . Again, since α n ∼ αn ν − , E [( | X n | ) ] = O ( n ), andlim n →∞ E [( | X n | ) ] n = α (2 ν ) /ν π ) . Note that this limit is exactly the square of the limit of the expectation of | X n | n ,implying Var (cid:18) | X n | n (cid:19) = E [( | X n | ) ] n − (cid:18) E [ | X n | ] n (cid:19) → n → ∞ . Thus, by Chebychev’s inequality, | X n |√ n → α (2 ν ) /ν π in probability. Appendix G. Proof of Proposition 4.3 First, for k ≥ 0, we see that Z R n | x | k K ( x ) d x = Z R n | x | k e nρ ( σ + n ) / Γ (cid:18) σ + n + 22 (cid:19) J ( σ + n ) / (2 | x/α | p ( σ + n ) / | x/α | p ( σ + n ) / ( σ + n ) / ! d x = e nρ ( σ + n ) Γ (cid:18) σ + n + 22 (cid:19) Z R n | x | k J ( σ + n ) / (2 | x/α | p ( σ + n ) / (2 | x/α | p ( σ + n ) / ( σ + n ) d x = e nρ ( σ + n ) Γ (cid:18) σ + n + 22 (cid:19) π n/ Γ( n ) Z ∞ r n − r k J ( σ + n ) / (2( r/α ) p ( σ + n ) / (2( r/α ) p ( σ + n ) / ( σ + n ) d x, and by the change of variables y = (cid:16) α q σ + n (cid:17) r ,= e nρ σ + n π n/ Γ (cid:0) σ + n +22 (cid:1) Γ( n ) Z ∞ α r σ + n ! − k − n +1 J ( σ + n ) / ( y ) y σ +1 − k α r σ + n ! − d y = e nρ σ + n π n/ Γ (cid:0) σ + n +22 (cid:1) α k + n Γ( n )(2( σ + n )) k + n Z ∞ J ( σ + n ) / ( y ) y σ +1 − k d y. For σ + 1 − k > 0, from [19, 10.22.57], Z ∞ J ( σ + n ) / ( y ) y σ +1 − k d y = Γ (cid:0) n + k (cid:1) Γ( σ + 1 − k )2 σ − k +1 Γ (cid:0) σ − k + 1 (cid:1) Γ (cid:0) σ − k + n + 1 (cid:1) , and thus, Z R n | x | k K ( x ) d x = e nρ σ + n π n/ Γ (cid:0) σ + n +22 (cid:1) α k + n Γ( n )(2( σ + n )) k + n Γ (cid:0) n + k (cid:1) Γ( σ + 1 − k )2 σ − k +1 Γ (cid:0) σ − k + 1 (cid:1) Γ (cid:0) σ − k + n + 1 (cid:1) = e nρ (2 π ) n/ α k + n k/ Γ (cid:0) σ + n +22 (cid:1) ( σ + n ) k + n Γ( n ) Γ (cid:0) n + k (cid:1) Γ( σ + 1 − k )Γ (cid:0) σ − k + 1 (cid:1) Γ (cid:0) σ − k + n + 1 (cid:1) . Then, for σ > E [ | X n | ] = || K n || Z R n | x | K n ( x ) d x = (2 π ) n α n / Γ (cid:0) σ + n +22 (cid:1) Γ (cid:0) n + (cid:1) Γ( σ )( σ + n ) n Γ( n )Γ (cid:0) σ +12 (cid:1) Γ (cid:0) σ − + n + 1 (cid:1) ( σ + n ) n Γ (cid:0) σ + 1 (cid:1) Γ (cid:0) σ + n + 1 (cid:1) (2 π ) n α n Γ( σ + 1)Γ (cid:0) σ + n + 1 (cid:1) = α / ( σ + n ) / Γ( n ) Γ (cid:0) n + (cid:1) Γ( σ )Γ (cid:0) σ + 1 (cid:1) Γ (cid:0) σ + n + 1 (cid:1) Γ (cid:0) σ + (cid:1) Γ (cid:0) σ + n + (cid:1) Γ( σ + 1) ∼ α / ( σ + n ) / Γ( n ) Γ (cid:0) n (cid:1) (cid:0) n (cid:1) Γ( σ )Γ (cid:0) σ + 1 (cid:1) Γ (cid:0) n (cid:1) (cid:0) n (cid:1) σ +1 Γ (cid:0) σ +12 (cid:1) Γ (cid:0) n (cid:1) (cid:0) n (cid:1) σ + Γ( σ + 1) = α / ( σ + n ) / (cid:0) n (cid:1) Γ( σ )Γ (cid:0) σ + 1 (cid:1) Γ (cid:0) σ +12 (cid:1) Γ( σ + 1) ∼ n / α / Γ( σ )Γ (cid:0) σ + 1 (cid:1) Γ (cid:0) σ +12 (cid:1) Γ( σ + 1) = O ( n ) . Now, let β > . By Markov’s inequality,lim n →∞ E [ η n ( B n ( Rn β ) c )] E [ η n ( R n )] = lim n →∞ P (cid:0) | X n | ≥ Rn β (cid:1) ≤ lim n →∞ E | X n | Rn β = 0 . Appendix H. Proof of Proposition 4.4 First, from [8, 6.576.3], we have for all ν > k > ν − Z ∞ r k K ν (cid:16) rα (cid:17) dr = 2 − k α k +1 Γ(1 + k ) Γ (cid:18) k ν (cid:19) Γ (cid:18) k + 12 (cid:19) Γ (cid:18) k − ν (cid:19) , (28)where K ν is the modified Bessel function of the second kind.For the Whittle-Mat´ern Kernel (19), Z R n K n ( x ) dx = Z R n e nρ − ν Γ( ν ) | x | ν α ν K ν (cid:18) | x | α (cid:19) dx = 2 π n Γ( n ) e nρ − ν Γ( ν ) α ν Z ∞ r n − r ν K ν ( r ) dr = e nρ π n Γ( n ) 2 − ν Γ( ν ) α ν Z ∞ r n − ν K ν ( r ) dr. Then by (28), Z ∞ r n − ν K ν ( r ) dr = 2 − n +2 ν α n +2 ν Γ( n + 2 ν ) Γ (cid:18) n + 2 ν ν (cid:19) Γ (cid:18) n + 2 ν (cid:19) Γ (cid:18) n + 2 ν − ν (cid:19) = 2 − n +2 ν α n +2 ν Γ( n + 2 ν ) Γ (cid:16) n ν (cid:17) Γ (cid:16) n ν (cid:17) Γ (cid:16) n (cid:17) . Similarly, Z R n | x | K n ( x ) dx = e nρ π n Γ( n ) 2 − ν Γ( ν ) α ν Z ∞ r n +1+2 ν K ν ( r ) dr. and also by (28), Z ∞ r n +1+2 ν K n ( r ) dr = 2 − n +2 ν α n +2+2 ν Γ( n + 2 + 2 ν ) Γ (cid:16) n ν + 1 (cid:17) Γ (cid:16) n ν + 1 (cid:17) Γ (cid:16) n (cid:17) Then, E [ | X n | ] = R R n | x | K n ( x ) dx R R n K n ( x ) dx = (2 α ) Γ( n + 2 ν )Γ (cid:0) n + 2 ν + 1 (cid:1) Γ (cid:0) n + ν + 1 (cid:1) Γ (cid:0) n + 1 (cid:1) Γ( n + 2 + 2 ν )Γ (cid:0) n + 2 ν (cid:1) Γ (cid:0) n + ν (cid:1) Γ (cid:0) n (cid:1) = (2 α ) (cid:0) n + 2 ν (cid:1) (cid:0) n + ν (cid:1) (cid:0) n (cid:1) ( n + 1 + 2 ν )( n + 2 ν ) ∼ (cid:16) α (cid:17) n, as n → ∞ , and this implies E [ | X n | ] √ n → α , as n → ∞ . Thus, since the Whittle Mat´ern kernel is log-concave, the conclusion holds byTheorem 3.2. Appendix I. Proof of Proposition 4.5 First, recall the the beta function satisfies B ( x, y ) := Z t x − (1 − t ) y − d t = Z ∞ t x − (1 + t ) − ( x + y ) d t = Γ( x )Γ( y )Γ( x + y ) . Then, for any k ≥ Z R n | x | k K n ( x ) d x = Z R n | x | k e nρ (1 + | xα n | ) ν + n d x = e nρ π n/ Γ( n ) Z ∞ r n − k (cid:18) r α n (cid:19) − ν − n d r = e nρ π n/ Γ( n ) α n + kn Z ∞ t n − k (1 + t ) − (2 ν + n ) d t = e nρ π n/ Γ( n ) α n + kn B (cid:18) n k , ν + n − k (cid:19) . Thus, the expectation of | X n | is E [ | X n | ] = 1 || K n || Z R n | x | K n ( x ) d x = α n B ( n + 1 , ν + n − B ( n , ν + n )= α n Γ( n + 1)Γ(2 ν + n − n + 2 ν )Γ( n + 2 ν )Γ( n )Γ(2 ν + n ) = α n n n + 2 ν − 1) = α n nn + 4 ν − , and E [ | X n | ] = α n B ( n + 2 , ν + n − B ( n , ν + n ) = α n Γ( n + 2)Γ(2 ν + n − n + 2 ν )Γ( n + 2 ν )Γ( n )Γ(2 ν + n )= α n ( n + 1) n (2 ν + n − ν + n − 1) = α n n ( n + 2)( n + 4 ν − n + 4 ν − . Thus, by the assumption that α n ∼ αn as n → ∞ for some α > n →∞ E [ | X n | ] n = α and lim n →∞ Var( | X n | ) n = 0 . Thus, by Chebychev’s inequality, | X n |√ n → α in probability. Appendix J. Proof of Proposition 5.1 By Proposition 4.1,lim n →∞ − n ln E [ η n ( B n ( √ nR ))] = − ρ − log 2 πe + R α m − log R, < R < √ m α − ρ − log α − log mπ , R > √ m α . Recall that lim n →∞ n ln E [Φ n ( B n ( √ nR ))] = ρ + log 2 πe + log R . Thus,lim n →∞ − n ln E [ η n ( B n ( √ nR ))] E [Φ n ( B n ( √ nR ))]= lim n →∞ − n ln E [ η n ( B n ( √ nR ))] + 1 n ln E [Φ n ( B n ( √ nR ))]= − ρ − log 2 πe + R α m − log R + ρ + log 2 πe + log R, < R < √ m α − ρ − log α − log mπ + ρ + log 2 πe + log R, R > √ m α = R α m , < R < √ m α + log 2 − log α − log m + log R, R > √ m α . Acknowledgements The work of both authors was supported by a grant of the Simons Foundation( References [1] Anantharam, V. and Baccelli, F. (2016). The Boolean model in the Shannonregime: Three thresholds and related asymptotics. Journal of Applied Probability Biscio, C. and Lavancier, F. (2016). Quantifying repulsiveness of determinan-tal point processes. Bernoulli Blaszczyszyn, B. and Yogeshwaran, D. (2014). 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