Two-dimensional Coulomb gases with multiple outposts

T w o-dimensional Coulom b gases with m ultiple outp osts K ohei No da F ebruary 26, 2026 Abstract W e study t w o-dimensional Coulom b gases in the presence of m ∈ N > 0 outp osts. An outpost is a connected comp onen t of the coincidence set that lies outside the droplet. The case m = 1 w as previously inv estigated by Ameur, Charlier, and Cron v all. They show ed that, as the total n umber of particles in the Coulom b gas tends to infinity , the num b er of particles accum ulating near the outp ost remains of order one and con verges in distribution to the Heine distribution. In this w ork, w e extend this analysis to the case of an arbitrary but fixed num b er m of outp osts. W e prov e that the join t distribution of the n umbers of particles near the outp osts con verges to a m ultidimensional Heine distribution. Our results reveal a interesting phenomenon: although the outp osts are geometrically disconnected, the particle coun t near eac h outp ost is strongly correlated with the particle counts near all other outp osts, not only the nearest ones (pro vided the outp osts are not separated by a comp onen t of the droplet). AMS Subject Classifica tion (2020) : 60G55, 60F05, 31A99. Keywords : Coulomb gases, random normal matrices, Heine distribution 1 In tro duction and statemen t of results F or n ∈ N > 0 , consider the tw o-dimensional determinantal Coulomb gas { z j } n j =1 ⊂ C n defined by d P n ( z 1 , . . . , z n ) := 1 Z n Y 1 ≤ j 1 to ensure that Z n < + ∞ . Assuming that Q is lo wer semi-contin uous and finite on some set of p ositiv e capacity , F rostman’s theorem [ 33 ] implies the existence of a unique e quilibrium me asur e σ , whic h minimizes the w eighted logarithmic energy I Q [ µ ] := Z C 2 log 1 | z − w | dµ ( z ) dµ ( w ) + Z C Q ( z ) dµ ( z ) , (1.2) 1 o ver all compactly supp orted Borel probability measures µ . The supp ort of the equilibrium measure is called the dr oplet and is denoted S ≡ S Q := supp( σ ). If Q is C 2 -smo oth in a neighbourho od of S , then σ is absolutely contin uous with resp ect to the Leb esgue measure, and takes the form dσ ( z ) := ∆ Q ( z ) 1 S ( z ) d 2 z π , (1.3) where ∆ is the quarter Laplacian defined b y ∆ := ∂ ∂ = 1 4 ( ∂ 2 x + ∂ 2 y ) and 1 S for the characteristic function of S . Let us define the obstacle function ˇ Q ( z ) to b e the p oin t wise suprem um of s ( z ), where s runs through the clas s of subharmonic functions s : C → R whic h satisfy s ≤ Q on C and s ( z ) ≤ 2 log | z | + O (1) as | z | → + ∞ . Clearly , ˇ Q ( z ) is a sub-harmonic function such that ˇ Q ≤ Q and ˇ Q ( z ) = 2 log | z | + O (1) as | z | → + ∞ . The c oincidenc e set S ∗ = S ∗ Q for the obstacle problem is defined by S ∗ = { z ∈ C : Q ( z ) = ˇ Q ( z ) } . (1.4) W e assume throughout the rest of this work that Q is C 6 -smo oth in a neighborho o d of S ∗ . F ollowing [ 7 ], we refer to p oin ts of S ∗ \ S as shal low p oints , and call a connected comp onen t of S ∗ \ S an outp ost of the droplet. In our setting, i.e., in the case where the function Q is rotation-inv arian t, a connected comp onen t of S ∗ is either a disk D r := { z : | z | ≤ r } , an annulus A ( a, b ) := { z ∈ C : a ≤ | z | ≤ b } , the singleton { 0 } , or a circle { z ∈ C : | z | = t } . As a mild restriction, w e will assume that S ∗ has only finitely man y connected comp onen ts. Thus, the droplet S is of the form S = N [ ν =0 A ( a ν , b ν ) , (1.5) where 0 ≤ a 0 < b 0 < a 1 < b 1 < · · · < a N < b N , and S ∗ is obtained by p ossibly adjoining finitely man y outp osts { z ∈ C : | z | = t p } , where t p ≥ 0. In this work, we consider the following tw o cases (see Figure 1 ): Case 1: N = 0 and 0 ≤ a 0 < b 0 < t 1 < · · · < t m , S ∗ = A ( a 0 , b 0 ) ∪ m [ p =1 { z ∈ C : | z | = t p } . (1.6) Case 2: N = 1 and 0 ≤ a 0 < b 0 < t 1 < · · · < t m < a 1 < b 1 , S ∗ = A ( a 0 , b 0 ) ∪ m [ p =1 { z ∈ C : | z | = t p } ∪ A ( a 1 , b 1 ) . (1.7) T o explain wh y we fo cus on the cases N ∈ { 0 , 1 } , let us first consider the situation S = S ∗ as in ( 1.5 ), that is, when the droplet has no outp osts. Let f b e a smo oth and rotation-inv arian t test function such that f ≡ 1 on an open neigh b orhoo d of S and f ≡ 0 otherwise. Then, b y [ 7 , Corollary 1.3], as n → + ∞ , E h e s P n j =1 f ( z j ) i = E h e s N ( e f ,v f ) i · N − 1 Y ν =0 E h e s P n j =1 f ν ( z j ) i · (1 + o (1)) , (1.8) 2 (a) Case 1 (with m = 3) (b) Case 2 (with m = 3) Figure 1: T w o cases of S ∗ uniformly for s in compact subsets of C , where f ν for ν = 0 , 1 , . . . , N − 1 are smooth, rotation- in v ariant test functions suc h that f ν ≡ 1 in neigh b orhoo ds of the b oundaries | z | = b ν of A ( a ν , b ν ) and | z | = a ν +1 of A ( a ν +1 , b ν +1 ) and f ν ≡ 0 otherwise. Here, N ( e f , v f ) denotes a real Gaussian random v ariable with mean e f and v ariance v f ; see [ 7 , (1.14), (1.15)] for explicit expressions. F orm ula ( 1.8 ) has the in terpretation that the fluctuations of smo oth linear statistics for S = S ∗ as in ( 1.5 ) are decoupled into a Gaussian part and contributions coming from neighborho ods of the b oundaries of eac h gap, which are asymptotically indep enden t. Therefore, it suffices to consider tw o cases: case 1 , where the outp osts are outside the outermost connected comp onen t of the droplet, and case 2 , where the outp osts are located inside a spectral gap b et ween t wo connected comp onen ts of the droplet. (The case where a 0 > 0 and the outp osts are inside the innermost connected comp onen t is similar to case 1 and is therefore not considered in this work.) F or concreteness, we no w provide an explicit example of a p oten tial Q satisfying case 1 . (Explicit examples of p oten tials Q satisfying case 2 can b e constructed similarly .) Example 1.1. Consider the quadratic (Ginibre) p oten tial Q g ( z ) = | z | 2 . Its obstacle function is given b y ˇ Q g ( z ) = 1 + 2 log | z | . W e define a mo dified p oten tial Q ( z ) by Q ( z ) :=        Q g ( z ) , if | z | ≤ 1 , Υ( z ) , if 1 < | z | < 3 , Q g ( z ) , if | z | ≥ 3 , (1.9) where Υ( z ) is a smo oth, rotation-in v ariant function satisfying the following prop erties: 3 • Υ( z ) matches Q g ( z ) smo othly at | z | = 1 and | z | = 3; • ˇ Q g ( z ) ≤ Υ( z ) ≤ Q g ( z ) for all 1 < | z | < 3; • Υ( z ) = ˇ Q g ( z ) holds only for | z | = t 1 , . . . , t m , for some m ∈ N > 0 , and 1 < t 1 < · · · < t m < 3. Then ˇ Q = ˇ Q g and S ∗ is of the form ( 1.6 ). Hence, the function Q in ( 1.9 ) is an example of a case 1 p oten tial. Case 1 when m = 1 has b een previously studied in [ 7 ]. W e recall the one-dimensional Heine distribution, see [ 7 , Subsection 1.3.2]. Definition 1.1 (One-dimensional Heine distribution) . Let θ ∈ R > 0 and q a num b er with 0 < q < 1. A N := { 0 , 1 , 2 , . . . } -v alued random v ariable X is said to hav e a Heine distribution with parameters ( θ , q ), denoted X d ∼ He( θ , q ), if P ( { X = k } ) = 1 ( − θ , q ) ∞ q 1 2 k ( k − 1) θ k ( q ; q ) k , k ∈ N , (1.10) where ( z ; q ) k = k − 1 Y i =0 (1 − z q i ) , ( z ; q ) ∞ = + ∞ Y i =0 (1 − z q i ) . (1.11) In this setting, it is natural to ask for the limiting distribution of the num b er of particles lying in a small neighborho od of the circle {| z | = t 1 } . This question w as answ ered in [ 7 , Corollary 1.10]. Theorem 1.2. ([ 7 , Corollary 1.10]) Consider the c ase 1 when m = 1 . L et N n b e the numb er of p articles lying in a smal l but fixe d neighb orho o d of {| z | = t 1 } . A s n → + ∞ , the r andom variables N n c onver ge in distribution to He( θρ, ρ 2 ) , wher e θ = q ∆ Q ( b 0 ) ∆ Q ( t 1 ) and ρ = b 0 t 1 . In particular, the num b er of particles near the outp ost is finite in probability . Indeed, if X d ∼ He( θ ρ, ρ 2 ), then we hav e E [ X ] = + ∞ X j =0 θ ρ 2 j +1 1 + θ ρ 2 j +1 , whic h sho ws that the exp ected num b er of particles near the outp ost is finite. By ( 1.8 ) and the discussion b elo w it, it is natural to ask whether the n umbers of particles lying in small but fixed neighborho o ds of outp osts within a sp ectral gap exhibit asymptotic indep endence. The aim of this work is to answer this question b y going b ey ond Theorem 1.2 and extend the results from [ 7 ] to the m ultiple-outp ost case 1 and case 2 . Our results sho w that the particle coun t near each outp ost is strongly correlated with the particle counts near all other outp osts, not only the nearest ones (for b oth case 1 and case 2 ). Note that more complicated case 2 was not considered in [ 7 ], ev en for m = 1. 1.1 Main results W e start by defining the multi-dimensional Heine distribution, which generalizes Definition 1.1 . Definition 1.3 (Multi-dimensional Heine distribution) . Let ( θ 1 , . . . , θ m ) ∈ R m > 0 and ( q 1 , . . . , q m ) ∈ (0 , 1) m . F or ( α 1 , . . . , α m ) ∈ N m , a N m -v alued random v ariable X m = ( X 1 , . . . , X m ) is said to 4 ha ve a m ulti-dimensional Heine distribution with parameters ( θ 1 , . . . , θ m ; q 1 , . . . , q m ), denoted X m d ∼ He( θ 1 , . . . , θ m ; q 1 , . . . , q m ), if P  { X 1 = α 1 , X 2 = α 2 , . . . , X m = α m }  = θ α 1 1 · · · θ α m m P J 1 ,...,J m ⊆ N , k  = ℓ ; | J k | = α k ,J k ∩ J ℓ = ∅ Q m k =1 q P j ∈ J k j k Q + ∞ j =0 (1 + P m k =1 θ k q j k ) , (1.12) where | A | denotes the num b er of elements of a set A . F or an y ( α 1 , . . . , α m ) ∈ N m , it is obvious that 0 < P  { X 1 = α 1 , X 2 = α 2 , . . . , X m = α m }  , and + ∞ X α 1 ,...,α m =0 P  { X 1 = α 1 , X 2 = α 2 , . . . , X m = α m }  = P J 1 ,...,J m ⊆ N , k  = ℓ ; J k ∩ J ℓ = ∅  Q m k =1 q P j ∈ J k j k θ | J k | k  Q + ∞ j =0 (1 + P m k =1 θ k q j k ) = 1 , whic h follows by extracting the coefficients θ α 1 1 θ α 2 2 · · · θ α m m from Q + ∞ j =0 (1 + P m k =1 θ k q j k ). Therefore, ( 1.12 ) is indeed a probability mass function. Remark 1.4 (Consistency with the one-dimensional Heine distribution) . The m ulti-dimensional Heine distribution recov ers the Heine distribution when m = 1. Indeed, ( 1.12 ) with m = 1 yields P  { X = α }  = θ α P J ⊆ N , | J | = α q P j ∈ J j Q + ∞ j =0 (1 + θ q j ) , where we set X ≡ X 1 , θ ≡ θ 1 , J ≡ J 1 , and α = α 1 . Any sets J ⊂ N such that | J | = α , can b e written in the form J = { j 1 , . . . , j α } for some non-negativ e integers j 1 < · · · < j α . Let us denote i r := j r − ( r − 1) for r = 1 , 2 , . . . , α . Since j 1 + · · · + j α = P α r =1 ( i r + ( r − 1)) = P α r =1 i r + 1 2 α ( α − 1), w e hav e θ α X J ⊆ N , | J | = α q P j ∈ J j = θ α X 0 ≤ j 1 < ··· 0 and ( q 1 , . . . , q m ) ∈ (0 , 1) m . F or each j ∈ N , we define a random v ariable Y j ∈ { 0 , 1 , . . . , m } , (1.13) whose discrete probability distribution is giv en b y p j, 0 = P  Y j = 0  = 1 1 + P m ℓ =1 θ ℓ q j ℓ p j,k = P  Y j = k  = θ k q j k 1 + P m ℓ =1 θ ℓ q j ℓ , k = 1 , 2 , . . . , m. (1.14) Assume moreov er that the random v ariables { Y j } are indep enden t, and define X k := ∞ X j =0 1 ( Y j = k ) , k = 1 , 2 , . . . , m. (1.15) 5 Note that by the Borel-Cantelli theorem, X k < + ∞ with probabilit y one. W e no w show that the multi-dimensional Heine distribution can b e realized as the joint distribu- tion of ( X 1 , . . . , X m ), where X k = P j ≥ 0 1 ( Y j = k ) and { Y j } j ≥ 0 are independent { 0 , 1 , . . . , m } -v alued random v ariables. Lemma 1.5. The joint distribution of the r andom variables X 1 , . . . , X m define d in ( 1.15 ) is ( X 1 , . . . , X m ) d ∼ He( θ 1 , . . . , θ m ; q 1 , . . . , q m ) . Pr o of. W e define I k := { j ∈ N : Y j = k } . F or any j ∈ N , the even t { j ∈ I k ∩ I ℓ } would imply Y j = k and Y j = ℓ simultaneously , which is imp ossible for k  = ℓ . Therefore, I k ∩ I ℓ = ∅ with probability one. Note that X k = ∞ X j =0 1 ( Y j = k ) = | I k | . (1.16) Note that F or disjoint subsets I 1 , . . . , I m ⊂ N , P  { Y j = k if j ∈ I k , and Y j = 0 if j / ∈ ∪ m k =1 I k }  = Q m k =1 Q j ∈ I k θ k q j k Q ∞ j =0 (1 + P m ℓ =1 θ ℓ q j ℓ ) = Q m k =1 θ | I k | k q P j ∈ I k j k Q ∞ j =0 (1 + P m ℓ =1 θ ℓ q j ℓ ) , where w e ha ve used the fact that the family { Y j } j is independent and ( 1.14 ) to mak e the ab o v e expression well-defined. Note that the even t { X 1 = α 1 , . . . , X m = α m } for an y ( α 1 , . . . , α m ) ∈ N m is the disjoint union o ver all families of subsets ( I 1 , . . . , I m ) such that I k ⊂ N , | I k | = α k , and I k ∩ I ℓ = ∅ for k  = ℓ for k , ℓ = 1 , 2 , . . . , m of the even ts { Y j = k if j ∈ I k , and Y j = 0 if j / ∈ ∪ m k =1 I k } . Therefore, P  { X 1 = α 1 , . . . , X m = α m }  = X I 1 ,...,I m ⊂ N I k ∩ I ℓ = ∅ ,k  = ℓ, | I k | = α k P  { Y j = k if j ∈ I k , and Y j = 0 if j / ∈ ∪ m k =1 I k }  = θ α 1 1 · · · θ α m m P I 1 ,...,I m ⊂ N I k ∩ I ℓ = ∅ ,k  = ℓ, | I k | = α k Q m k =1 q P j ∈ I k j k Q ∞ j =0 (1 + P m ℓ =1 θ ℓ q j ℓ ) . This coincides with Definition 1.3 . Remark 1.6. W e remark that the marginal distribution of the m ulti-dimensional Heine distribution is completely different from the one-dimensional Heine distribution. Define B j,k := 1 ( Y j = k ) ∈ { 0 , 1 } . Then, by ( 1.16 ), X k = ∞ X j =0 B j,k is the sum of indep enden t Bernoulli random v ariables with probabilities ( 1.14 ). Therefore, X k follo ws a P oisson-binomial distribution. In particular, since we assumed that the random v ariables { Y j } j are indep enden t, { B j,k } j are also indep enden t. Hence, we hav e E  s X k  = ∞ Y j =0 E  s B j,k  = ∞ Y j =0 (1 − p j,k + p j,k s ) = ∞ Y j =0 1 + P ℓ  = k θ ℓ q j ℓ + θ k q j k s 1 + P m ℓ =1 θ ℓ q j ℓ . 6 By Definition 1.3 and Remark 1.4 , since the generating function of the one-dimensional Heine distri- bution He( θ k ; q k ) with parameters ( θ k ; q k ) is given by ∞ Y j =0 1 + θ k q j k s 1 + θ k q j k , it is only when all other parameters are set to b e zero, i.e., θ ℓ = 0 for all ℓ  = k , that we reco ver the one-dimensional Heine distribution. This shows that the marginal distribution of the multi- dimensional Heine distribution do es not degenerate to the one-dimensional Heine distribution. As a consequence, the multi-dimensional Heine distribution should not b e identified with a multinomial P oisson distribution. Moreov er, each marginal follo ws a Poisson–binomial distribution. W e now state the main results of this work, which extend those obtained in [ 7 , Theorem 1.2]. Belo w, we use the following smo oth test function. Define η ( u ) := ( e − 1 /u , u > 0 , 0 , u ≤ 0 , b η ( u ) := η ( u ) η ( u ) + η (1 − u ) . Then b η ∈ C ∞ ( R ), and it satisfies b η ( u ) = 0 for u ≤ 0 and b η ( u ) = 1 for u ≥ 1. Let ε > 0 b e sufficiently small so that the 2 ε -neighborho ods of distinct outp osts do not ov erlap. F or each k = 1 , . . . , m , define the radial cutoff χ k ( r ) := b η  r − ( t k − ε ) ε/ 2  b η  ( t k + ε ) − r ε/ 2  , r ≥ 0 , where t k for k = 1 , 2 , . . . , m are given by ( 1.6 ) or ( 1.7 ), and set the rotation-inv ariant bump function h k ( z ) := χ k ( | z | ) , z ∈ C . (1.17) Then h k ∈ C ∞ ( C ) is rotation-inv arian t, and h k ( z ) = 1 for t k − ε 2 ≤ | z | ≤ t k + ε 2 , h k ( z ) = 0 for | z | / ∈ [ t k − ε, t k + ε ] . W e first consider case 1 when S ∗ is given by ( 1.6 ). W e derive the asymptotic b eha vior of the m ultiv ariate moment generating function of the particle n umbers at each outp ost. Theorem 1.7. ( Case 1 ) L et S ∗ b e as in ( 1.6 ) . Define ϑ k = s ∆ Q ( b 0 ) ∆ Q ( t k ) and ρ k = b 0 t k , k = 1 , 2 , . . . , m. L et h k b e as in ( 1.17 ) for k = 1 , 2 , . . . , m , and define N n,k = n X j =1 h k ( z j ) , k = 1 , 2 , . . . , m. Then lim n → + ∞ E " m Y k =1 e s k N n,k # = + ∞ Y j =0 1 + P m k =1 e s k ϑ k ρ 2 j +1 k 1 + P m k =1 ϑ k ρ 2 j +1 k , (1.18) uniformly for | s k | ≤ log n , k = 1 , 2 , . . . , m . In p articular, by Definition 1.3 , as n → + ∞ , we have ( N n, 1 , N n, 2 , . . . , N n,m ) d − → He  ϑ 1 ρ 1 , . . . , ϑ m ρ m ; ρ 2 1 , . . . , ρ 2 m  . (1.19) 7 Using Theorem 1.7 , we obtain the exp ectation, v ariance, and co v ariance of the num b ers of particles lying in small but fixed neigh b orho ods of an outp ost. Corollary 1.8. ( Case 1 : exp ectation, v ariance, and cov ariance function) L et p, q ∈ { 1 , 2 , . . . , m } b e such that p  = q . With the same setting as in The or em 1.7 , as n → + ∞ , we have E  N n,p  → + ∞ X j =0 ϑ p ρ 2 j +1 p 1 + P m k =1 ϑ k ρ 2 j +1 k , (1.20) V ar  N n,p  → + ∞ X j =0 ϑ p ρ 2 j +1 p  1 + P m k =1 ϑ k ρ 2 j +1 k  2  1 + m X k =1; k  = p ϑ k ρ 2 j +1 k  , (1.21) Co v  N n,p , N n,q  → − + ∞ X j =0 ( ρ p ρ q ) 2 j +1  1 + P m k =1 ϑ k ρ 2 j +1 k  2 ϑ p ϑ q . (1.22) This leads to the following probabilistic interpretations. • F ormula ( 1.20 ) implies that the num b er of particles lying in a small neighborho o d of {| z | = t k } for k = 1 , 2 , . . . , m is finite in probability . • The exp ected num b er ( 1.20 ) of particles at a given outp ost decreases when the outp ost is lo cated farther aw ay from the connected comp onen t of the droplet. More precisely , when t k increases, then ρ k = b 0 /t k decreases, which leads to a smaller limiting exp ected o ccupation n umber. • By ( 1.22 ), the particle num b ers associated with different outp osts are negativ ely correlated, re- flecting an intrinsic competition b et ween outposts outside the outermost connected comp onen t of the droplet. Finally , w e provid e the fluctuations for case 2 , i.e., when S ∗ is given by ( 1.7 ). Theorem 1.9. ( Case 2 ) L et S ∗ b e as in ( 1.7 ) . F or x n = M 0 n − ⌊ M 0 n ⌋ , M 0 = σ ( {| z | ≤ b 0 } ) , define e ρ 0 ≡ b ρ m +1 ≡ ρ 0 := b 0 a 1 ∈ (0 , 1) e ρ k = t k a 1 ∈ (0 , 1) , b ρ k = b 0 t k ∈ (0 , 1) , k = 1 , 2 , . . . , m, and e ϑ 0 ,n := s ∆ Q ( a 1 ) ∆ Q ( b 0 ) ρ − 2 x n 0 , b ϑ m +1 ,n := e ϑ − 1 0 ,n , e ϑ k,n := s ∆ Q ( a 1 ) ∆ Q ( t k ) e ρ − 2 x n k , b ϑ k,n := s ∆ Q ( b 0 ) ∆ Q ( t k ) b ρ 2 x n k , k = 1 , 2 , . . . , m. L et h 0 b e a smo oth, r otation-invariant test function such that h 0 ≡ 1 in neighb orho o ds of the b ound- aries | z | = b 0 of A ( a 0 , b 0 ) and | z | = a 1 of A ( a 1 , b 1 ) and h 0 ≡ 0 otherwise. L et h k b e as in ( 1.17 ) for k = 1 , 2 , . . . , m , and set N n,k = n X j =1 h k ( z j ) , k = 0 , 1 , 2 , . . . , m. 8 Then as n → + ∞ , we have E " m Y k =0 e s k N n,k # = + ∞ Y j =0  1 + P m k =0 e s k e ϑ k,n e ρ 2 j +1 k  1 + P m +1 k =1 e s k b ϑ k,n b ρ 2 j +1 k   1 + P m k =0 e ϑ k,n e ρ 2 j +1 k  1 + P m +1 k =1 b ϑ k,n b ρ 2 j +1 k  + o (1) , (1.23) uniformly for | s k | ≤ log n , k = 0 , 1 , 2 , . . . , m with s 0 ≡ s m +1 . In p articular, by Definition 1.3 , as n → + ∞ , ( N n, 0 , N n, 1 , N n, 2 , . . . , N n,m ) − ( X (1) m +1 ,n + X (2) m +1 ,n ) d − → 0 , (1.24) wher e X (1) m +1 ,n and X (2) m +1 ,n ar e indep endent, and X (1) m +1 ,n d ∼ He  e ϑ 0 ,n e ρ 0 , . . . , e ϑ m,n e ρ m ; e ρ 2 0 , e ρ 2 1 , . . . , e ρ 2 m  , X (2) m +1 ,n d ∼ He  b ϑ 1 ,n b ρ 1 , . . . , b ϑ m +1 ,n b ρ m +1 ; b ρ 2 1 , . . . , b ρ 2 m +1  . (1.25) In p articular, when m = 1 , as n → + ∞ , we have ( N n, 0 , N n, 1 ) − ( X (1) 2 ,n + X (2) 2 ,n ) d − → 0 , (1.26) wher e X (1) 2 ,n and X (2) 2 ,n ar e indep endent, and X (1) 2 ,n d ∼ He  e ϑ 0 ,n e ρ 0 , e ϑ 1 ,n e ρ 1 ; e ρ 2 0 , e ρ 2 1  , X (2) 2 ,n d ∼ He  b ϑ 1 ,n b ρ 1 , b ϑ 0 ,n b ρ 0 ; b ρ 2 1 , b ρ 2 2  . (1.27) Using Theorem 1.9 , we obtain the exp ectation, v ariance, and cov ariance of the counting statistics near the outp osts. Corollary 1.10. ( Case 2 : exp ectation, v ariance, and cov ariance function) L et p, q ∈ { 1 , 2 , . . . , m } b e such that p  = q . With the same setting as in The or em 1.9 , as n → + ∞ , we have E  N n,p  = + ∞ X j =0 e ϑ p,n e ρ 2 j +1 p 1 + P m k =0 e ϑ k,n e ρ 2 j +1 k + + ∞ X j =0 b ϑ p,n b ρ 2 j +1 p 1 + P m +1 k =1 b ϑ k,n b ρ 2 j +1 k + o (1) , (1.28) V ar  N n,p  = + ∞ X j =0 e ϑ p,n e ρ 2 j +1 p 1 + P m k =0 e ϑ k,n e ρ 2 j +1 k  1 + m X k =0; k  = p e ϑ k,n e ρ 2 j +1 k  + + ∞ X j =0 b ϑ p,n b ρ 2 j +1 p 1 + P m +1 k =1 b ϑ k,n b ρ 2 j +1 k  1 + m +1 X k =1; k  = p b ϑ k,n b ρ 2 j +1 k  + o (1) , (1.29) Co v  N n,p , N n,q  = − + ∞ X j =0 e ϑ p,n e ρ 2 j +1 p e ϑ q ,n e ρ 2 j +1 q  1 + P m +1 k =1 e ϑ k,n e ρ 2 j +1 k  2 − + ∞ X j =0 b ϑ p,n b ρ 2 j +1 p b ϑ q ,n b ρ 2 j +1 q  1 + P m +1 k =1 b ϑ k,n b ρ 2 j +1 k  2 + o (1) . (1.30) In p articular, when m = 1 , as n → + ∞ , we have E  N n, 1  = + ∞ X j =0 e ϑ 1 ,n e ρ 2 j +1 1 1 + e ϑ 0 ,n e ρ 2 j +1 0 + e ϑ 1 ,n e ρ 2 j +1 1 + + ∞ X j =0 b ϑ 1 ,n b ρ 2 j +1 1 1 + b ϑ 1 ,n b ρ 2 j +1 1 + b ϑ 2 ,n b ρ 2 j +1 2 + o (1) , V ar  N n, 1  = + ∞ X j =0 e ϑ 1 ,n e ρ 2 j +1 1 1 + e ϑ 0 ,n e ρ 2 j +1 0 + e ϑ 1 ,n e ρ 2 j +1 1  1 + e ϑ 0 ,n e ρ 2 j +1 0  + + ∞ X j =0 b ϑ 1 ,n b ρ 2 j +1 1 1 + b ϑ 1 ,n b ρ 2 j +1 1 + b ϑ 2 ,n b ρ 2 j +1 2  1 + b ϑ 2 ,n b ρ 2 j +1 2  + o (1) , 9 This leads to the following in terpretations. • In the degenerate case m = 0 (no outp osts in the gap), the result reduces to the displacement phenomenon for a single sp ectral gap: the fluctuation of N n, 0 is giv en by the sum of tw o indep enden t one-dimensional Heine distributions from the tw o sides of the gap, see [ 7 , Theorem 1.7]. • The decomp osition into tw o indep enden t random v ariables X (1) m +1 ,n = ( X (1) 0 ,n , X (1) 1 ,n , . . . , X (1) m,n ) and X (2) m +1 ,n = ( X (2) 1 ,n , . . . , X (2) m,n , X (2) m +1 ,n ) in the limit shows that the n umber of particles in small neighborho ods of the outp osts within a sp ectral gap is indep enden tly influenced by the inner and outer boundaries of the gap. Moreov er, this decomp osition also captures the dis- placemen t phenomenon of particles o ccurring b et w een the inner and outer b oundaries across the sp ectral gap. • In contrast with case 1 , case 2 is driv en by tw o indep enden t sources whose contributions add up in the limit. The limiting co v ariance b et ween the num b ers of particles lying in small but fixed neigh b orhoo ds of distinct outp osts is the sum of tw o negativ e terms ( 1.30 ) induced by the inner and outer droplets. Since X (1) m +1 ,n and X (2) m +1 ,n giv en by ( 1.25 ) are indep enden t, the mixed cov ariances v anish, Co v  X (1) p,n , X (2) q ,n  = o (1) , p  = q ∈ { 0 , 1 , . . . , m, m + 1 } , n → + ∞ , and b oth exp ectations and v ariances decomp ose additively: for p ∈ { 1 , 2 , . . . , m } , E [ N n,p ] = E [ X (1) p,n ]+ E [ X (2) p,n ]+ o (1) , V ar( N n,p ) = V ar( X (1) p,n )+V ar( X (2) p,n )+ o (1) , n → + ∞ . Note how ever that Co v ( X ( j ) p,n , X ( j ) q ,n ) for p  = q ∈ { 0 , 1 , . . . , m, m + 1 } and j = 1 , 2 remains of order 1 as n → + ∞ . Commen ts and related work In this work, we extend the results of Ameur, Charlier, and Cronv all to the case of an arbitrary but fixed num b er of outp osts, and we prov e that the joint distribution of the num b ers of particles near the outp osts conv erges to a multi-dimensional Heine distribution. Belo w w e discuss related w orks and p ossible directions for future research. The pap ers [ 14 , 17 , 18 , 20 , 21 , 22 ] provide free energy expansions and precise asymptotics for the momen t generating functions of coun ting statistics in tw o-dimensional Coulomb gases. It would b e p ossible and interesting to adapt the analysis in [ 7 ], as well as the metho ds developed in the presen t w ork, to this setting. In the setting of [ 9 ], Ameur and Cronv all wen t b ey ond the results of [ 7 ] and studied fluctuations of outp osts for non-radially symmetric p oten tials. They also pro vided compatibilit y conditions for the case of a single outp ost in such a general setting. It would b e v ery interesting to identify analo- gous compatibilit y conditions in the presence of several outp osts, and to analyze the corresp onding fluctuations. In the recent work [ 3 ], the case where the equilibrium densit y ∆ Q v anishes along a circle inside the droplet is in vestigated. This can b e viewed as an analogue of [ 13 , 25 , 26 , 32 ] in the theory of Hermitian random matrices. As mentioned in [ 7 ], it would also b e in teresting to study the situation where ∆ Q v anishes along a outp ost. Concerning correlation functions, Ameur and Jahic in forthcoming work [ 12 ] provide the asymp- totic analysis of the correlation kernel near an outp ost. The counterpart of the Szeg¨ o kernel arising 10 near the edge in the absence of outp osts has b een studied in [ 8 , 5 , 6 ], while the limiting kernel corresp onding to the case of a single outp ost is analyzed in [ 12 ]. Finally , it is an interesting question whether the multi-dimensional Heine distribution app ears in other applications. In the one-dimensional setting, related results for the Heine distribution can b e found in the work of Kemp [ 29 ]. Plan of this pap er In Section 2 , we recap sev eral results in [ 7 ]. In Section 3 , we prov e Theorem 1.7 and 1.9 and Corollary 1.8 and 1.10 . A c knowledgmen t The author is grateful to Joakim Cronv all for insightful discussions during the XXI Brunel–Bielefeld W orkshop, and to Y acin Ameur for useful comments on an earlier draft. The author ackno wledges supp ort from the Europ ean Research Council (ERC), Gran t Agreement No. 101115687. 2 Preliminaries In this section, we collect sev eral results from [ 7 ]. Consider the L 2 -space ov er C with norm ∥ f ∥ 2 := R C | f ( z ) | 2 dA ( z ). The monic weigh ted orthogonal p olynomial of degree j in p oten tial Q is denoted p j,sf ( z ) = z j e − n 2 e Q ( z ) , e Q ( z ) := Q ( z ) − s n f ( z ) , (2.1) where f ( z ) := f ( | z | ) is a rotation in v ariant and smo oth test function. When f ≡ 0, we simply write p j ( z ) ≡ p j, 0 ( z ). Then, by Andr ´ reief ’s identit y (see e.g., [ 27 ]), we hav e E h e s P n j =1 f ( z j ) i = n − 1 Y j =0 ∥ p j,sf ∥ 2 ∥ p j ∥ 2 . (2.2) A crucial metho d to compute the ab o ve right hand side is the Laplace metho d developed in [ 7 ]. W e recall the facts of the lo cal p eak sets. F or a fixed num b er τ with 0 ≤ τ ≤ 1, put g τ ( r ) := q ( r ) − 2 τ log r. If r = r τ is a solution to g ′ τ ( r ) = 0, and if Q is smo oth ar r , then r q ′ ( r ) = 2 τ , q ′′ τ ( r ) = 4∆ Q ( r ) . The solutions r = r τ , whic h give lo cal minima for g τ , are called lo c al p e ak p oints . W e take N small enough so that Q is C 6 -smo oth and strictly subharmonic on the set { z = r e iθ : r ∈ N , 0 ≤ θ ≤ 2 π } . W e denote the totality of lo cal p eak p oin ts in N by LP( τ ) := { r ∈ N : g ′ τ ( r ) = 0 } , and we set P ( τ ) of global p eak p oin ts by P( τ ) := { r ≥ 0 : g τ ( r ) = B τ } . 11 Giv en the assumption on g ′′ τ ( r τ ) = 4∆ Q ( r τ ) > 0, all p oin ts in LP( τ ) are strict lo cal minima. As a consequence, there is at most one lo cal p eak p oin t r τ in the vicinit y of a given connected comp onen t C of S ∗ ∩ [0 , + ∞ ). W e set δ n := C log n n , ϵ n := p δ n , (2.3) where C is a large constant, and define the set of signific ant lo c al p e ak p oints to b e SLP( τ ) := { r ∈ LP( τ ) : g τ ( r ) < B τ + δ n } . (2.4) In the follo wing, we call a num b er τ ∈ [0 , 1] is a br anching value if the p eak set P( τ ) consists of at least t wo p oin ts. The v alues M 0 , M 1 , . . . , M N − 1 are branc hing v alues, and these are all in the op en in terv al (0 , 1). The v alue τ 0 is a branching v alue if there is a outp ost | z | = c with c < a 0 and τ = 1 is a branching v alue if there is an outp ost with c > b N . With these sets, we recall the following tw o facts. Lemma 2.1 (Lemma 2.8 in [ 7 ]) . SLP( τ ) c onsists of a single p oint r = r τ when τ is sufficiently for away fr om the br anching values, in the sense that | τ − M ν | ≥ c > 0 for al l ν . If τ is close to M ν , | τ − M ν | < c , ther e might b e sever al signific ant lo c al p e aks (the end-p oints b ν , a ν +1 and p ossibly some shal low p oints in b etwe en if 0 ≤ ν ≤ N − 1 ). Lemma 2.2 (Lemma 2.9 in [ 7 ]) . If | τ − M ν | ≥ C log n n for al l br anching values M ν , wher e C is lar ge enough, then SLP( τ ) c onsists of a single p oint in the interior of S . As summary , the global p eak set P( τ ) describ es the radial lo cations where outp osts may o ccur from the viewpoint of the obstacle problem. It consists of the relev an t b oundary points of the droplet and, at branching v alues, p ossibly additional shallow p oin ts in the sp ectral gaps. The lo cal p eak set LP( τ ) = { r ∈ N : g ′ τ ( r ) = 0 } consists of the lo cal minima of g τ . These p oin ts gov ern the Laplace-t yp e asymptotics of the integrals and pro vide the candidates for exp onen tially dominant con tributions. The set of significant lo cal p eaks SLP( τ ) ⊂ LP( τ ) collects those lo cal p eaks whose v alues of g τ are within order n − 1 log n of the global minimum. These are precisely the p oin ts that con tribute at leading order to the asymptotics for finite n . W e write I τ ( n ) := 2 Z + ∞ 0 r 1+2 α e sh ( r ) e − ng τ ( r ) dr . (2.5) Lemma 2.3 (Lemma 2.10 in [ 7 ]) . F or e ach 0 ≤ τ ≤ 1 , define J τ ≡ J n,τ := { r ≥ 0 : dist( r, SLP( τ )) < ϵ n } . (2.6) A lso, write I # τ ( n ) := 2 Z J τ r 1+2 α e sh ( r ) e − ng τ ( r ) dr . (2.7) Then if C is lar ge enough, the inte gr al ( 2.5 ) satisfies I τ ( n ) = I # τ ( n ) · (1 + O ( n − 100 )) , (2.8) wher e the err or term is uniform for 0 ≤ τ ≤ 1 and al l r e al s with | s | ≤ log n . Therefore, to show Theorem 1.7 and 1.9 , Lemma 2.3 allows us to fo cus on fluctuations for m ultiple outp osts when we compute ( 2.2 ) as n → + ∞ . 12 3 Pro ofs of Theorem 1.7 and 1.9 3.1 Pro of of Theorem 1.7 W e imitate [ 7 , Pro of of Theorem 1.8]. W e consider S ∗ as ( 1.6 ). Lemma 3.1. L et h b e a smo oth test function to b e 1 on neighb orho o d of e ach outp ost {| z k | = t k } for k = 1 , 2 , . . . , m and to b e 0 otherwise. F or k = 1 , 2 , . . . , m , write ρ k := b t k , θ k := s ∆ Q ( b ) ∆ Q ( t k ) , c k := h ( t k ) − h ( b ) , µ ( s ) := e c k s θ k . (3.1) Then, as n → + ∞ , we have lim n → + ∞ E h e s P n j =1 h ( z j ) i = + ∞ Y j =0 1 + P m k =1 µ k ( s ) ρ 2 j +1 k 1 + P m k =1 µ k (0) ρ 2 j +1 k . (3.2) uniformly for | s | ≤ log n . Pr o of. Let L n := C log n , where C is large enough. W e write h k,j := Z {| r − r k,j | <ϵ n } 2 r e sh ( r ) e − ng τ ( j ) ( r ) dr , (3.3) where k ∈ { 0 , 1 , . . . , m } if j ≥ n − L n while k = 0 if j < n − L n . By Lemma 2.3 , as n → + ∞ , w e ha ve h j = ( h 0 ,j · (1 + O ( n − 100 )) , if j < n − L n , ( h 0 ,j + h 1 ,j + · · · + h m,j ) · (1 + O ( n − 100 )) , if j ≥ n − L n , whic h give rise to n − 1 X j =0 log h j = n − 1 X j =0 log h 0 ,j + n − 1 X j = n − L n log  1 + h 1 ,j + · · · + h m,j h 0 ,j  + O  1 n 99  . Let log E h e s P n j =1 h ( z j ) i = n − 1 X j = n − L n log  1 + h 1 ,j + · · · + h m,j h 0 ,j  . Then, we hav e log E h e s P n j =1 h ( z j ) i = n − 1 X j = n − L n log  1 + m X k =1 ρ 2( n − j ) − 1 k µ k ( s )  + O  1 + | s | n  , = L n − 1 X j =0 log  1 + m X k =1 ρ 2 j +1 k µ k ( s )  + O  1 + | s | n  → + ∞ X j =0 log  1 + m X k =1 ρ 2 j +1 k µ k ( s )  , n → + ∞ , where ρ k = b t k and µ k ( s ) = e s ( h ( t k ) − h ( b )) q ∆ Q ( b ) ∆ Q ( t k ) . 13 F or k = 1 , 2 , . . . , m , let us pick a smo oth test function h k as ( 1.17 ). Let us define N n,k := n X j =1 h k ( z j ) , k = 1 , 2 , . . . , m, whic h corresp onds to the random v ariable of the n umber of particles which are found in a vicinity of the one outp ost. F or  s = ( s 1 , . . . , s m ) ∈ R m , define the multiv ariate moment generating function G n,m (  s ) := E h m Y k =1 e s k N n,k i , (3.4) Then, by Lemma 3.1 , as n → + ∞ , we hav e G m (  s ) := lim n → + ∞ G n,m (  s ) = + ∞ Y j =0 1 + P m k =1 µ k ( s k ) ρ 2 j +1 k 1 + P m k =1 µ k (0) ρ 2 j +1 k . Here, the con vergence is uniform for  s in compact subsets of R m . In particular, the random v ariable N n := ( N n, 1 , N n, 2 , . . . , N n,m ) conv erges in distribution to He( ϑ 1 ρ 1 , ϑ 2 ρ 2 , . . . , ϑ m ρ m ; ρ 2 1 , ρ 2 2 , . . . , ρ 2 m ) = ( N 1 , N 2 , . . . , N m ), where ϑ k := q ∆ Q ( b ) ∆ Q ( t k ) for k = 1 , 2 , . . . , m . Note that ∂ s p log G m (  s )     s =  0 = + ∞ X j =0 ϑ p ρ 2 j +1 p 1 + P m k =1 ϑ k ρ 2 j +1 k , whic h leads to the exp ectation E [ N p ] for p ∈ { 1 , 2 , . . . , m } . Note also that ∂ 2 s p log G m (  s )     s =  0 = + ∞ X j =0 ϑ p ρ 2 j +1 p  1 + P m k =1 ϑ k ρ 2 j +1 k  2  1 + m X k =1; k  = p ϑ k ρ 2 j +1 k  , whic h leads to the v ariance V ar[ N p ] for p ∈ { 1 , 2 , . . . , m } . Finally , the cov ariance b et ween ( N p , N q ) for p  = q , which corresp onds to the limiting co v ariance function of the distinct outp osts, can b e computed as ∂ s p ∂ s q log G m (  s )     s =  0 = − + ∞ X j =0 ϑ p ϑ q ( ρ p ρ q ) 2 j +1  1 + P m k =1 ϑ k ρ 2 j +1 k  2 . 3.2 Pro of of Theorem 1.9 W e consider S ∗ as ( 1.6 ). Next we consider the case of several outp osts inside the droplet. By [ 7 , Subsection 3.3 for N = 1], n − 1 X j =0 log h j = m 0 − 1 X j =0 log h 0 ,j + n − 1 X j = m 0 log h n,j + T n , (3.5) where T n := m 0 + L n X j = m 0 log  1 + h 0 ,j + h 1 ,j + · · · + h m,j h n,j  + m 0 − 1 X j = m 0 − L n log  1 + h 1 ,j + h 2 ,j + · · · + h m,j + h n,j h 0 ,j  . As in the case of several outp osts outside the droplet, by lo calizing the p oten tial Q to e Q in a prop er w ay , it suffices to compute the large- n asymptotics of T n . 14 Lemma 3.2. L et µ 0 ( s ) := s ∆ Q ( b 0 ) ∆ Q ( a 1 ) e s ( h ( a 1 ) − h ( b 0 ))  b 0 a 1  2 x 0 , ρ 0 := b 0 a 1 , η k ( s ) := s ∆ Q ( a 1 ) ∆ Q ( t k ) e s ( h ( t k ) − h ( a 1 ))  a 1 t k  2 x 0 , ξ k := t k a 1 e η k ( s ) := s ∆ Q ( b 0 ) ∆ Q ( t k ) e s ( h ( t k ) − h ( b 0 ))  b 0 t k  2 x 0 , e ξ k := b 0 t k . Then, for | s | ≤ log n , as n → + ∞ , we have T n =  + ∞ X j =0 log  1 + m X k =1 η k ( s ) ξ 2 j +1 k + µ 0 ( s ) − 1 ρ 2 j +1 0  + + ∞ X j =0 log  1 + m X k =1 e η k ( s ) e ξ 2 j +1 k + µ 0 ( s ) ρ 2 j +1 0    1 + O  1 + | s | n  . Pr o of. The pro of is done in a similar wa y of [ 7 , Lemma 3.10]. W e omit the details. Let h k b e ( 1.17 ). W e define N n,k := n X j =1 h k ( z j ) , k = 0 , 1 , . . . , m + 1 , and for  s := ( s 0 , s 1 , . . . , s m , s m +1 ) ∈ R m +2 , let us denote G (an) n,m (  s ) := E h m +1 Y k =0 e s k N n,k i . By Lemma 3.2 , we hav e G (an) m (  s ) = + ∞ Y j =0  1 + P m k =0 e s k − s 1 e ϑ k e ρ 2 j +1 k  1 + P m +1 k =1 e s k − s 0 b ϑ k b ρ 2 j +1 k   1 + P m k =0 e ϑ k e ρ 2 j +1 k  1 + P m +1 k =1 b ϑ k b ρ 2 j +1 k  + o (1) , uniformly for  s in compact subsets of R m +2 . F or p = 1 , 2 , . . . , m , we hav e ∂ s p log G (an) n,m (  s )     s =  0 = + ∞ X j =0 e ϑ p e ρ 2 j +1 p 1 + P m k =0 e ϑ k e ρ 2 j +1 k + + ∞ X j =0 b ϑ p b ρ 2 j +1 p 1 + P m +1 k =1 b ϑ k b ρ 2 j +1 k + o (1) . F or p = 1 , 2 , . . . , m , we hav e ∂ 2 s p log G (an) n,m (  s )     s =  0 = + ∞ X j =0 e ϑ p e ρ 2 j +1 p 1 + P m k =0 e ϑ k e ρ 2 j +1 k  1 + m X k =0; k  = p e ϑ k e ρ 2 j +1 k  + + ∞ X j =0 b ϑ p b ρ 2 j +1 p 1 + P m +1 k =1 b ϑ k b ρ 2 j +1 k  1 + m +1 X k =1; k  = p b ϑ k b ρ 2 j +1 k  + o (1) . 15 F or p  = q ∈ { 1 , 2 , . . . , m } , we hav e ∂ s p ∂ s q log G (an) n,m (  s )     s =  0 = − + ∞ X j =0 e ϑ p e ρ 2 j +1 p e ϑ q e ρ 2 j +1 q  1 + P m +1 k =1 e ϑ k e ρ 2 j +1 k  2 − + ∞ X j =0 b ϑ p b ρ 2 j +1 p b ϑ q b ρ 2 j +1 q  1 + P m +1 k =1 b ϑ k b ρ 2 j +1 k  2 + o (1) , whic h provide s the cov ariance b et ween the outp osts. References [1] G. Akemann, S.-S. Byun and M. Ebke, Universalit y of the n umber v ariance in rotational inv ari- an t tw o-dimensional Coulomb gases, J. Stat. Phys. 190 (2023), no.1, Paper No. 9, 34 pp. [2] G. Akemann, S.-S. Byun, M. Ebke and G. 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