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BC TYPE Z-MEASURES AND DETERMINANTAL POINT PROCESSES CESARCUENCA Abstract. The (BC type) z-measures are a family of four parameter z,z′,a,b probability measures on the path space of the nonnegative Gelfand-Tsetlin graph with Jacobi-edge multiplicities. We can 7 interpret the z-measures as random point processes Pz,z′,a,b on the punctured positive real line X = 1 R>0\{1}. Ourmainresultisthattheserandomprocessesaredeterminantalandmoreoverwecompute 0 theircorrelationkernelsexplicitlyintermsofhypergeometricfunctions. 2 For very special values of the parameters z,z′, the processes Pz,z′,a,b on X are essentially scaling n limitsofRacahorthogonal polynomialensemblesandtheircorrelationkernelscanbecomputedsimply a fromsomelimitsoftheRacahpolynomials. Thus,inthelanguageofrandommatrices,westudycertain J analytic continuations of processes that are limits of Racah ensembles, and such that they retain the determinantal structure. Another interpretation ofour results, and the mainmotivation ofthis paper, 4 istherepresentationtheoryofbiggroups. Inrepresentation-theoreticterms,thispapersolvesanatural 2 problemofharmonicanalysisforseveralinfinite-dimensionalsymmetricspaces. ] T R . h t Contents a m 1. Introduction 1 [ 2. Notation and Terminology 7 1 3. BC type z-measures 8 v 4. The discrete orthogonalpolynomial ensemble (N) 10 O 0 5. The L-ensemble (N) 16 6 L 6. Discrete Riemann-Hilbert Problem and connection to L-ensembles 19 0 7. Correlation kernel of (N) 22 07 8. The continuous pointLprocess on R>0 1 33 P \{ } . 9. Main result: An explicit correlation kernel for the point process 38 1 P Appendix A. Identities for generalized hypergeometric functions 46 0 7 References 52 1 : v i X r a 1. Introduction A wide range of probabilistic and statistical mechanics models are integrable in the sense that observables of the system admit closed forms that can be analyzed in different limit regimes. The tools that are used to obtain closed form formulas for the observables, or correlation functions, are typically combinatorial or algebraic in nature and very often they originate from representation theory. Some notable examples of integrable probability ensembles come from Schur measures, [27, 29]. The firstkindofSchurmeasuresthatappearedinthe literatureareknownasz-measuresandthey originated from the problem of harmonic analysis for the infinite symmetric group, see [30] for an introduction. A more sophisticated class of probability measures, known as zw-measures, originated from harmonic analysisofthe infinite unitary group;various aspects ofthe z-andzw-measureswere studiedin the past decade, see for example [9, 10, 8]. All of the theory we have mentioned above is based on representation theory of type A, that is, the algebraic machinery behind the scenes comes from the symmetric and unitary groups. In this paper we work with the ‘type BC’ analogues of zw-measures that we call BC type z-measures; they come from harmonicanalysisofthe infinite symplectic, orthogonalgroupsandother infinite-dimensionalsymmetric 1 2 CESARCUENCA spaces. In a degenerate case, the BC type z-measures are equivalent to a scaling limit of the Racah or- thogonalpolynomialensembles,whichareatthetopofthehierarchyofalldiscreteorthogonalpolynomial ensembles of the classical Askey scheme. Our analysis of the BC type z-measures is based on a map from the space where they are defined to the space of simple point configurations in R 1 . Under this map, the BC type z-measures >0 \ { } define a four-parameter family of stochastic point processes , with infinitely many points, in z,z′,a,b P the punctured positive real line. We began a study on these processes in [13], where we constructed a continuous-time Markov chain that preserves the z-measures. In this paper, the random processes themselves are our subject of study: our main result states that they are determinantal point processes and moreover we compute explicit correlation kernels in terms of special functions. The resulting kernel has appeared in the literature before, in [9], and is known as the hypergeometric kernel in view of the fact that it can be expressed in terms of the hypergeometric function F . The previous appearance of 2 1 the hypergeometric kernel was in the problem of harmonic analysis of the infinite-dimensional unitary group (‘type A’ analogue of our problem), but it is far from obvious that the same kernel should show up in our context; the author cannot offer at this point a conceptual explanation for it. The principal theme of this paper is the ‘approximation’of the point processes onthe contin- z,z′,a,b P uous state space R 1 by a sequence (N) of point processes, with a finite number of points, \{ } {Lz,z′,a,b}N≥1 (N) on a quadratic lattice. For any fixed N, the point processes are determinantal and a correlation Lz,z′,a,b kernel for it can be expressed in terms of a finite system of orthogonal polynomials on a quadratic lattice, known as the Wilson-Neretin polynomials. In addition, we need to have suitable expressions of the correlation kernels of (N) for the limit transition N . To obtain such expressions, we make use Lz,z′,a,b →∞ of a discrete Riemann-Hilbert problem, which has been used before in other problems from asymptotic representation theory. Finally, the expressions that admit a limit as N are obtained after massive →∞ calculations and use of recent formulas[24] that relate severalgeneralizedhypergeometricfunctions F . 4 3 Below we give a more detailed account on the results of this paper. 1.1. BC type z-measures. Let us fix two real parameters a b 1/2, and momentarily fix also a positive integer N N. In this introduction, we only discuss B≥C ty≥pe−z-measures of level N (or simply ∈ z-measures of level N), which are certain probability measures on the countable space of N-positive signatures GT+ = λ = (λ ,...,λ ) : λ ... λ . The BC type z-measures depend on a,b and N { 1 N 1 ≥ ≥ N} on two additional complex parameters z,z′ subject to certain constraints, see Definition 3.2 below. For example, the readercan think of the case z′ =z C R, or of the degenerate case z =n N, z′ >n 1 ∈ \ ∈ − which gives rise to the Racah orthogonalpolynomial ensemble. Explicitly, z-measures of level N are N 1 P (λz,z′,a,b)= ∆ (l ,...,l )2 W(l z,z′,a,b;N), λ GT+, N | Z (z,z′,a,b) · N 1 N · i| ∈ N N i=1 Y b where we denoted by l = λ +N i, 1 i N, the shifted coordinates of λ = (λ ,...,λ ); also i i 1 N − ≤ ≤ ∆ (l ,...,l )= l + a+b+1 2 l + a+b+1 2 , the weight function is N 1 N 1≤i<j≤N i 2 − j 2 b QW(xz,z′,(cid:16)a(cid:0),b;N)= x(cid:1)+ a(cid:0)+b+1 Γ(cid:1)(x(cid:17)+a+b+1)Γ(x+a+1) | 2 Γ(x+b+1)Γ(x+1) (cid:18) (cid:19) 1 , ×Γ(z x+N)Γ(z′ x+N)Γ(z+x+N +a+b+1)Γ(z′+x+N +a+b+1) − − and Z (z,z′,a,b) > 0 is a normalization constant that makes P a probability measure. Observe that N N theweightfunctionW( z,z′,a,b;N)isnotnecessarilynonnegativeonZ forallcomplexvaluesofz,z′, ≥0 but it is for some specia·|l paremeters, for example when z′ =z and z =n N, z′ >n 1 as before. ∈ − In what follows, we construct several point processes associated to the z-measures of level N. The reader can refer to [14, 15, 21] for generalities on point processes; see also [1, Sec. 4.2] and [5] for generalities on determinantal point processes. 1.2. Point processes in the quadratic half-lattice. Let ǫ= a+b+1 0 and Zǫ = ǫ2,(1+ǫ)2,(2+ 2 ≥ + { ǫ)2,... be aquadratichalf-lattice. LetConf (Zǫ )be the spaceofmultiplicity-freepointconfigurations } fin + BC TYPE Z-MEASURES AND DETERMINANTAL POINT PROCESSES 3 onZǫ with finitely many points. The space Conf (Zǫ ) has a canonicalsigmaalgebraand a probability + fin + measure on it determines a stochastic point process on Zǫ . + Under any (measurable) map (N) : GT+ Conf (Zǫ ), the pushforwards of z-measures of level N P N → fin + yield random point processes (N) = (N) on Zǫ that depend on the parameters z,z′,a,b, and that P Pz,z′,a,b + we denote by the same letter (N). Our goal is to obtain a point process with infinitely many points on the state spaceR 1 asaPnaturalscalinglimit ofthe processes (N) withfinitely manypoints inthe >0 quadratic lattice Zǫ\({th}e scaling is of order N2). The most obviousPmap that we can consider is + (N)(λ)= (λ +N 1+ǫ)2,(λ +N 2+ǫ)2,...,(λ +ǫ)2 , λ GT+. O { 1 − 2 − N } ∀ ∈ N It turns out that the processes (N) do not have the desired scaling limit as N tends to infinity. The O point processes that we need arise from a different map that we now describe. An N-positive signature λ GT+ can be described uniquely by its Frobenius coordinates. If ∈ N we let d = d(λ) be the largest positive integer such that λ d, then the Frobenius coordinates d ≥ (p ,...,p q ,...,q ) of λ are given by 1 d 1 d | p =λ i, q =λ′ i, 1 i d, i i− i i− ≤ ≤ where λ′ = n Z :λ i for all i. It is clear that p >...> p 0 and q > >q 0. Now i |{ ∈ >0 n ≥ }| 1 d ≥ 1 ··· d ≥ consider the map (N) :GT+ Conf (Zǫ ) given by L N → fin + (N)(λ)= (N +p +ǫ)2 >...>(N +p +ǫ)2 >(N 1 q +ǫ)2 >...>(N 1 q +ǫ)2 . 1 d 1 d L { − − − − } The point processes (N) have a natural scaling limit that we describe next. L 1.3. Point processes on the continuous state space R 1 . Let X=R 1 =(0,1) (1, ) >0 >0 \{ } \{ } ⊔ ∞ and Conf(X) be the space of multiplicity free point configurations on X, with its canonical σ-algebra. The point processes (N) described above are probability measures on the space Conf (Zǫ ). Consider L fin + the composition of measurable maps 1 j :Zǫ Zǫ ֒ X N + −→ N +ǫ 1 2 + → − 2 x x (cid:0) (cid:1) , 7−→ N +ǫ 1 2 − 2 which induces a map j : Conf (Zǫ ) Conf((cid:0)X), denoted(cid:1) by the same letter (the normalization by N +ǫ 1 2 is chosenNso that fin Z+ǫ+ → X). Denote by (N) the pushforward of (N) under the − 2 (N+ǫ−1/2)2 ⊂ P L (cid:0)map jN abo(cid:1)ve; then (N) is a probability measure on Conf(X) and determines a point process onX that P e we also denote by (N). P Then the point preocesses (N) converge to certain point process = in the sense that all z,z′,a,b P P P the correlation funcetions of (N) converge to the corresponding correlation functions of , as N tends P P to infinity. The process is oeur hero: the main result of this paper is a complete characterizationof the P stochastic point process foer very general parameters z,z′,a,b. P Let us remark that the point process has an intrinsic definition, which is independent of the finite P point processes (N). However we are able to thoroughly study because we can realize it as a scaling P P limit of the processes (N), see Section 8 for details. P e 1.4. The main theoreem. Let a b 1/2 and (z,z′) C2 be an admissible pair (see Definition 3.2 below,or simply think ofthe cases≥z′ =≥z− C Rorz =n∈ N, z′ >n 1for now). The pointprocesses ∈ \ ∈ − = on X are determinantal. Moreover an explicit correlation kernel KP(x,y) for is given by z,z′,a,b P P P the expressions in (9.1) and (9.2) below. For example, if x,y >1, x=y, then 6 R(x)S(y) R(y)S(x) (1.1) KP(x,y)= ψ(x)ψ(y) − , · x y − p 4 CESARCUENCA where sin(πz)sin(πz′) ψ(x) = xb(x 1)−z−z′, 2π2 − 1 z′ z′+b z′ 1 R(x) = 1 F ; , − x 2 1 z+z′+b x (cid:18) (cid:19) (cid:20) (cid:21) 2 1 z′ Γ(1+z)Γ(1+z′)Γ(1+z+b)Γ(1+z′+b) z′+b+1 z′+1 1 S(x) = 1 F ; . x − x Γ(1+z+z′+b)Γ(2+z+z′+b) 2 1 z+z′+b+2 x (cid:18) (cid:19) (cid:20) (cid:21) The kernel KP : X X R is known as the hypergeometric kernel, the reason being that KP can be × → expressed in terms of the hypergeometric functions F , see [9] for another problem in which this kernel 2 1 appears. Observe that the expression for KP in (1.1) is only defined for x = y due to the singularities 6 on the diagonal. However it admits the following analytic continuation: KP(x,x)=ψ(x) (R′(x)S(x) R(x)S′(x)) x>1. · − ∀ 1.5. Harmonic analysis on big groups. We follow the exposition of [28]. Aninfinitesymmetric space G/K isaninductivelimitofRiemanniansymmetricspacesG(n)/K(n)of rankn,withrespecttoanaturalchainofmapsG(1) G(n) G(n+1) ... thatiscompatible →···→ → → with the inclusions K(n) G(n). Some examples of such infinite symmetric spaces are the following ⊂ (1) G/K =U(2 )/U( ) U( )=lim (U(2n)/U(n) U(n)). n→∞ ∞ ∞ × ∞ × (2) G/K =O( ) O( )/O( )=lim (O(n) O(n)/O(n)). n→∞ ∞ × ∞ ∞ × (3) G/K =Sp( ) Sp( )/Sp( )=lim (Sp(n) Sp(n)/Sp(n)). n→∞ ∞ × ∞ ∞ × (In(2), we wroten to indicate that the rankoftheeRiemanenianesymmetric space in questionis n/2 , ⌊ ⌋ and n can be either 2n or 2n+1; either one leads to the same G/K, up to isomorphism.) The natural problem of noncommutative harmonic analysis for finite or compact groups asks for the e e decompositionofthe regularrepresentationintoirreducible representations. Inthis context,the solution e is given by the well known Peter-Weyl theorem. In the infinite-dimensional context, it is not even clear what the problem of harmonic analysis should be. The difficulty is that there is no analogueof the Haar measure. The problem of harmonic analysis for the infinite-dimensional unitary group U( ) was posed ∞ by G. Olshanski in [33]. He used certain completion U of U( ), which admits an analogue of the Haar ∞ measure. We hope to explain the representation theoretic picture for the symmetric spaces listed above elsewhere, but for now let us only describe the problem in terms of spherical functions. A spherical function of the infinite symmetric space G/K is a function φ:G R satisfying → φ is a K-bi-invariantfunction, i.e., φ(kgk−1)=φ(g) for all g G, k K. • ∈ ∈ φ is normalized by φ(I)=1, where I G is the unit element. • ∈ φ is positive definite, i.e., for any g ,...,g G, the matrix [φ(g g−1)]n is positive definite. • 1 n ∈ i j i,j=1 It is clear that the space of spherical functions of G/K is convex. The extreme points of the set of sphericalfunctionsaretheirreducible (or extreme) spherical functions. Thespaceofirreduciblespherical functions for all three spaces (1), (2), (3) above are isomorphic; that space will be denoted by Ω . In ∞ fact, it was shown in [28] that Ω can be embedded as the subspace of R∞ R∞ R consisting of ∞ + × + × + triples ω =(α;β;δ) satisfying α α 0, 1 β β 0, 1 2 1 2 ≥ ≥···≥ ≥ ≥ ≥···≥ ∞ >δ (α +β ). i i ∞ ≥ i=1 X Let us denote by φω the irreducible spherical function corresponding to ω Ω . ∞ ∈ In analogy to [33] it is possible, for each of the three infinite symmetric spaces above, to construct a family of natural generalized quasi-regular representations of G/K, each with a distinguished cyclic K-invariant vector. The members of the family of generalized quasi-regular representations (Tz,vz) z { } (we let vz be the distinguished vector of the representation Tz) of G/K are parameterized by complex numbers z satisfying z > b, where b=0, 1,1 for (1), (2) and (3), respectively. Eachrepresentation ℜ −2 −2 2 Tz is determined completely by its spherical function φz defined by φz() = (Tz()vz,vz). As it turns out, for our three cases of interest, the spherical functions φz : G R a·dmit anal·ytic continuations in → BC TYPE Z-MEASURES AND DETERMINANTAL POINT PROCESSES 5 the variables z and z′ = z to the domain (z+z′) > b , where b = 0, 1,1 for the spaces (1), (2), {ℜ − } −2 2 (3) above. Furthermore, one can write down a spherical function φz,z′,a,b : G R depending on four → parameters z,z′,a,b (subject to some constraints) that reduces to the representation-theoretic spherical functions of G/K upon specialization of the parameters a,b as follows: (1) (a,b)=(0,0). (2) (a,b)=( 1, 1). ±2 −2 (3) (a,b)=(1,1). 2 2 (In (2), a= 1 if n=2n and a= 1 if n=2n+1.) −2 2 A general result from functional analysis implies that all spherical functions φz,z′,a,b of G/K are mixtures of the irredeucible spherical repreesentations {φω}ω∈Ω∞, in the sense that there exists a Borel probability measure π on Ω such that z,z′,a,b ∞ φz,z′,a,b = φωdπ (ω). z,z′,a,b ZΩ∞ The probability measure π on Ω is the z-measure with parameters z,z′,a,b and also known as z,z′,a,b ∞ the spectral measure of φz,z′,a,b. The probability measures π describe how the spherical function φz,z′,a,b is ‘decomposed’ into z,z′,a,b irreducible spherical functions. A (far reaching) generalization of the problem of harmonic analysis for the infinite symmetric spaces (1), (2), (3) above is therefore the following: Problem: Describe the spectral measures π on Ω for the most general set of parameters z,z′,a,b ∞ z,z′,a,b possible. The content of the present article solves the problem just posed. In fact, the pushforwards of the measures π under the map z,z′,a,b i:Ω Conf(X) ∞ (1.2) −→ ω =(α;β;δ) (1+α )2 (1 β )2 0,1 . i i≥1 j j≥1 7→ { } ⊔{ − } \{ } define the point processes on (cid:0)X, and our main result is a com(cid:1)plete characterization of these z,z′,a,b P processes, see Section 8 below for more details. Inparticular,Theorem9.1withb=0, 1,1 solvesthe problemofharmonicanalysis,statedabovefor −2 2 the infinite symmetric spaces in (1), (2) and (3), respectively. The reader is referred to [11] for a self-contained survey of the problem of harmonic analysis of the infinite symmetric space (G,K)=(U( ) U( ),U( )) and some applications. ∞ × ∞ ∞ 1.6. Racah orthogonal polynomial ensemble. We have mentioned that BC type z-measures are esentially analytic continuations of scaling limits of Racah orthogonal polynomial ensembles. Let us clarify the meaning of that statement and briefly explain how the main result of this paper can be deduced from limits of Racah polynomials in the degenerate case. Let z = k N and z′ > k 1 (similar considerations hold if z′ = k and z > k 1). For any N 1, ∈ − − ≥ the corresponding z-measure P =P ( n,z′,a,b) of level N is supported on the set N N ·| GT+(k)d=ef λ GT+ :k λ λ 0 . N { ∈ N ≥ 1 ≥···≥ N ≥ } The process (N) = (N) on Zǫ coming from P lives in the finite space Zǫ d=ef ǫ2,(1 + O Ok,z′,a,b + N N+k−1 { ǫ)2,...,(N+k 1+ǫ)2 . Thecorrespondingz-measureπ =π issupportedonthefinitedimensional k,z′,a,b − } simplex def Ω (k)= ω Ω :α=δ =0, β =β = =0 . ∞ ∞ k+1 k+2 { ∈ ··· } The point process = that corresponds to π under the map i given in (8.4) lives in the open k,z′,a,b P P (N) interval (0,1). Such process is the scaling limit of the processes , as described previously. In this Lk,z′,a,b special case, it will also be the scaling limit of some processes (N) = (N) , which come from the Y Yk,z′,a,b 6 CESARCUENCA pushforwards of (N) under the maps O Conf(Zǫ ) Conf(Zǫ ) ≤N+k−1 → ≤N+k−1 X Zǫ X. 7→ ≤N+k−1\ Clearly (N) isapointprocessonZǫ withexactlyk points. Simplecalculationsshowthat (N) Y ≤N+k−1 Y is a Racah orthogonalpolynomial ensemble associated to the parameters (1.3) α= k N, β = z′ N a, γ =b, δ =a. − − − − − In other words, if we denote y+ a+b+1 2 by y, for any y 0,1,...,N +k 1 , then 2 ∈{ − } (cid:0) (cid:1) k Prob (N) = y ,...,y =const b (y y )2 w (y (y +a+b+1) α,β,δ,γ), Yk,z′,a,b { 1 k} · i− j · R i i | (cid:16) (cid:17) 1≤iY<j≤k iY=1 bdef b γ+δ+1 Γ(y+γb+δ+b1)Γ(y+γ+1)Γ(y+α+1)Γ(β γ y) w (y(y+a+b+1))= y+ − − , R 2 Γ(y+1)Γ(y+δ+1)Γ(y α+γ+δ+1)Γ( β δ y) (cid:18) (cid:19) − − − − where α,β,δ,γ are related to k,z′,a,b via the identification of parameters in (1.3). The weight function w (y(y+a+b+1) α,β,γ,δ) on the quadratic lattice n(n+a+b+1): n Z corresponds to the R + | { ∈ } classical Racah orthogonal polynomials R (y(y+a+b+1)α,β,γ,δ), see [20, Ch. 9.2] and [26, Ch. 3]. n | One deduces that (N) is a determinantal process with correlationfunctions of the form Y Prob y ,...,y (N) =const 1 det KY(N)(y ,y ) , { 1 p}⊆Yk,z′,a,b · {p≤k}·1≤i,j≤p i j (cid:16) (cid:17) h i w (x)w (y) R (x)R (y) R (x)R (y) KY(bN)(x,y)b= R R k k−1 − k−1 kb b, H · x y p k−1 − where x=x(x+a+b+1),byb=y(y+a+eb+1e), x=(xe+(a+be+1)/2)2,ey =(ye+(a+b+1)/2)2 and H is the squared norm of R =R ( α,β,γ,δ). b b k−1 k−1 k−1 ·| Theepoint process = ke,z′,a,b has exactly k pboints and is a scaling limbit of the processes (N), all P P Y of which have k points. It is easily justified in this case that is a determinantal point process and a P calculation of its correlationkernel KP would follow from a suitable limit of the Racah polynomials and of the kernel KY(N) above. Alternatively the same kernelKP canbe readfromTheorem9.1 after setting z =k (in that theorem, note ψ (x) = 0 for z = k N, which implies that there is no particles in (1, )). The general point >1 ∈ ∞ processes that we consider (z,z′ not integers) can be considered as analytic continuations of the z,z′,a,b P limits of Racah ensembles (N), but we need much more work to study them. k,z′,a,b P Y 1.7. Other results. In addition to the main result described above, other results in this article are: The point process (N) is a discrete orthogonal polynomial ensemble. Moreover we find a cor- • O relation kernel KO(N) in terms of the generalized hypergeometric function F , see Theorem 4 3 4.5. The point process (N) is an L-ensemble. Moreover we find an explicit correlation kernel KL(N) • L in terms of the generalized hypergeometric function F , see Theorem 7.8. 4 3 1.8. Organization of the paper. The present section is the introduction. In the next section, we introduce some terminology that is used throughout the paper. In section 3, we introduce the BC type z-measuresoflevelN foralargesetofparametersz,z′,a,b,andintroducethepointprocesses (N), (N) onthestatespaceZǫ . Insection4,weprovethat (N) isadiscreteorthogonalpolynomialenOsembleLand + O find a correlationkernel. Insection5, we provethat (N) is anL-ensemble. After recallingthe theory of L thediscreteRiemann-Hilbertprobleminsection6,wefindacorrelationkernelfor (N) insection7. The L point processes (N) yield a point process with infinitely many particles in a continuous state space, L P under a suitable scaling limit. We construct these point processesas scalinglimits of the point processes (N) in section8. In section 9,we provethe main theoremof this paper,which is anexplicit formula for L the correlationkernel of . Finally, the appendix contains severaltechnical proofs of various statements P throughout the paper. BC TYPE Z-MEASURES AND DETERMINANTAL POINT PROCESSES 7 1.9. Acknowledgments. I would like to thank Alexei Borodin and Grigori Olshanski for many helpful discussions, e-mail correspondence and comments on a draft of this paper. 2. Notation and Terminology We collect here some of the terminology and notation that is used throughout the paper, for the reader’s convenience. The parameters a,b in this paper are real numbers, and we define • defa+b+1 ǫ= . 2 We assume throughout that a,b satisfy a b 1/2, ≥ ≥− which implies ǫ 0. This restriction allows us to use the main results from [28, 31]. InSection3belo≥w,wedefineseveralsubsetsofC2 fromwhichwetakepairs(z,z′)asparametersofthe • z-measures and for some proofs involving analytic continuations. Some of these sets are the following domains of C2: d=ef (z,z′) C2 : (z+z′+b)> 1 , U { ∈ ℜ − } d=ef (z,z′) : z,z′,z+b,z′+b ..., 3, 2, 1 = . 0 U { ∈U { }∩{ − − − } ∅} Moreover we have the more complicated set of admissible parameters defined by d=ef (z,z′) :(z,z′),(z+2ǫ,z′+2ǫ) , adm 0 U { ∈U ∈Z} where is defined below in (3.7). The set is not a domain of C2. The following inclusions hold, adm Z U see Lemma 3.3 below: . adm 0 U ⊂U ⊂U Given any (z,z′) , we denote Σd=efz+z′+b. Observe that if (z,z′) , then Σ ( 1,+ ). adm We write Z and∈NUfor the set of nonnegative and positive integers, res∈peUctively. We al∈so w−rite∞ + • Zǫ d=ef (n+ǫ)2 :n Z + { ∈ +} for the quadratic half-lattice. For any x Z, we let ∈ xd=ef(x+ǫ)2. Very often, starting in Section 4, we shall need the splitting Zǫ = Zǫ Zǫ into the sets Zǫ d=ef b + ≥N ⊔ <N ≥N (N +ǫ)2,(N +1+ǫ)2,... and Zǫ d=ef ǫ2,(1+ǫ)2,...,(N 1+ǫ)2 , for some N N. { } <N { − } ∈ For any N N, the set GT+ of positive signatures of length N (also known as the set of N-positive • ∈ N signatures) is defined as GT+ d=ef λ=(λ ,...,λ ) ZN :λ λ 0 . N { 1 N ∈ + 1 ≥···≥ N ≥ } To each positive N-signature λ = (λ λ ) GT+ we can associate the N-tuples l = (l > 1 ≥ ··· ≥ N ∈ N 1 >l ) ZN and l =(l > >l ) (Zǫ )N via the equations ··· N ∈ + 1 ··· N ∈ + l =λ +N i, l =(l +ǫ)2, 1 i N. b b i i b − i i ∀ ≤ ≤ BelowweshalloftenbeworkingwithapositivesignatureoflengthN denotedbyλ GT+. Wedenote b ∈ N its two associated N-tuples given as above by l =(l >...>l ) and (l >...>l ). 1 N 1 N We use the following simplified notation for products of (ratios of) Gamma functions • b b def Γ[a , a ,..., a ] = Γ(a )Γ(a ) Γ(a ), 1 2 m 1 2 m ··· a , a , , a 1 2 ··· m def Γ(a1)Γ(a2) Γ(am) Γ = ··· . b1, b2, ···, bn Γ(b1)Γ(b2)···Γ(bn)   8 CESARCUENCA If x=(x ,...,x ) is any N-tuple of complex numbers, then the Vandermonde determinant is 1 N • def ∆ (x)=∆ (x ,...,x )= (x x ). N N 1 N i j − 1≤i<j≤N Y 3. BC type z-measures For any N N, the set of N-positive signatures is GT+ d=ef λ = (λ ... λ ) ZN . Let us ∈ N { 1 ≥ ≥ N ∈ } beginbyconsideringthe followingfunctiononN-positivesignaturesλ GT+,whichdepends oncertain parameters z,z′ C, a b 1/2,and is given by ∈ N ∈ ≥ ≥− N 2 (3.1) P′ (λz,z′,a,b)= ∆ (l) w(l z,z′,a,b;N), N | N · i| (cid:16) (cid:17) Yi=1 where ∆ (l) = (l l ) is the Vanderbmonde determinant on the variables l = (l + ǫ)2, N 1≤i<j≤N i− j i i l =λ +N i, 1 i N, and i i − Q≤ ≤ b b b x+2ǫ, x+a+1 b w(xz,z′,a,b;N)=(x+ǫ) Γ | ×  x+1, x+b+1 (3.2) 1   . ×Γ[z x+N, z′ x+N, z+x+N +2ǫ, z′+x+N +2ǫ] − − When thereis no riskofconfussion,we writew(xz,z′,a,b;N)simply asw(x), forsimplicity ofnotation. | Observethattheconditionsontherealparametersa,bensurethat(x+ǫ)Γ(x+2ǫ)/Γ(x+1)andΓ(x+a+1) are well defined for all x Z and therefore P′ (λz,z′,a,b) is well defined for all λ GT+. ∈ + N | ∈ N The following equality (3.3) P′ (λz,z′,a,b)=S (z,z′,a,b), N | N λ∈XGT+N where N Γ[b+z+z′+i, a+i, i] S (z,z′,a,b)= , N Γ[z+i, z+b+i, z′+i, z′+b+i, z+z′+a+b+N +i] i=1 Y was proved in [31] for all z,z′ u C : u > (1+b) and extended by analytic continuation, in [13], ∈ { ∈ ℜ − 2 } for all pairs (z,z′) in the complex domain (3.4) d=ef (z,z′) C2 : (z+z′+b)> 1 . U ∈ ℜ − Define also the subdomain 0 (cid:8) consisting of those pairs (z,(cid:9)z′) for which SN(z,z′,a,b) never U ⊂ U vanishes; clearly such set is (3.5) d=ef (z,z′) : z,z′,z+b,z′+b ..., 3, 2, 1 = . 0 U { ∈U { }∩{ − − − } ∅} For any pair (z,z′) , we can therefore define the (complex) measure on GT+ given by ∈U0 N P′ (λz,z′,a,b) (3.6) P (λz,z′,a,b)= N | . N | S (z,z′,a,b) N From(3.3), the measureP has finite totalmeasureequalto 1. Next, we find verygeneralconditions on N complex pairs (z,z′) which guarantee that P (λz,z′,a,b) is a probability measure on GT+. { N | }λ∈GT+N N By following common terminology on zw-measures, see for example [9], define the sets def (3.7) = princ compl degen Z Z ⊔Z ⊔Z d=ef (z,z′) C2 R2 :z′ =z princ Z { ∈ \ } d=ef (z,z′) R2 :n<z,z′<n+1 for some n Z compl Z { ∈ ∈ } ∞ def = degen degen,n Z Z n=1 G d=ef (z,z′) R2 :z =n, z′ >n 1, or z′ =n, z >n 1 . degen,n Z { ∈ − − } BC TYPE Z-MEASURES AND DETERMINANTAL POINT PROCESSES 9 The subscripts ‘princ’, ‘compl’ and ‘degen’ stand for principal, complementary and degenerate. The reason why the definitions above are important for us is the following statement. Lemma 3.1 ([33, Lemma 7.9]). Let (z,z′) C2, then ∈ (Γ(z k)Γ(z′ k))−1 0 for all k Z iff (z,z′) . • (Γ(z−k)Γ(z′−k))−1 >≥0 for all k ∈Z iff (z,z′)∈Z . princ compl • If(z,−z′) − ,then(Γ(z k)Γ∈(z′ k))−1 =0∈fZork ⊔ZZ,k nand(Γ(z k)Γ(z′ k))−1 >0 degen,n • for k Z∈, kZ n 1. − − ∈ ≥ − − ∈ ≤ − Fromthelemmaabove,wecandefineasetofcomplexpairs(z,z′)suchthatthenumbersP′ (λz,z′,a,b) N | are nonnegative real values for all positive N-signatures. Definition 3.2. The admisible domain C2 is defined as the set of complex pairs (z,z′) C2 adm U ⊂ ∈ suchthat (z+z′+b)> 1andboth (z,z′), (z+2ǫ,z′+2ǫ)belongto . Apair (z,z′) is called adm ℜ − Z ∈U an admissible pair. Thus for (z,z′) , then P′ (λz,z′,a,b) 0 and not all of them are zero. Then (3.3) shows ∈Uadm∩U0 N | ≥ S (z,z′,a,b)>0 and therefore the ratios (3.6) define a probability measure on GT+. We actually only N N need to require (z,z′) , which automatically implies (z,z′) , as the following lemma implies. adm 0 ∈U ∈U Lemma 3.3. The inclusion holds, where is defined in (3.5) and in Definition 3.2. adm 0 0 adm U ⊂U U U Proof. Let (z,z′) . Then (z+z′+b) > 1, which implies (z,z′) . By the definition of , adm 0 ∈ U ℜ − ∈ U U we simply need to show that none of the four complex numbers z,z′,z+b,z′+b is a negative integer. We argue in three cases. If (z,z′) , then z,z′ / R which implies that none of z,z′,z+b,z′+b is a real number. princ If (z,z′)∈Z , then ther∈e exists n Z for which n < z,z′ < n+1. Coupled with the conditions compl ∈ Z ∈ b 1/2 and (z+z′+b)> 1, we have n 1. Therefore z,z′> 1 and so z,z′ cannot be negative ≥− ℜ − ≥− − integers. Moreoverz+b,z′+b> 1+b 3/2,so z+b,z′+b cannot take on integer values less than − ≥− 2. Ifz+b= 1,thenn= 1inthe analysisabove,so 1<z,z′<0. Thenz+z′+b= 1+z′< 1, − − − − − − a contradiction. Similarly z′+b= 1 and we are done with this case. If (z,z′) for some n6 −N, then we can assume without loss of generality that z = n and degen,n ∈ Z ∈ z′ >n 1. In particular, z′ >z 1=n 1 0, thus implying that z and z′ are both positive and not − − − ≥ negativeintegers. Moreoverz′+b>z+b 1=n+b 1 1 1/2 1= 1/2,thus implying thatz+b and z′+b are both larger than 1/2 and−therefore no−t ne≥gat−ive int−egers.− (cid:3) − In view of Lemmas 3.1 and 3.3, it follows that the numbers P (λz,z′,a,b) : λ GT+ define { N | ∈ N} a probability measure on GT+, whenever (z,z′) . We call such probability measures the BC type z-measures of level N (aNssociated to the par∈amUeatdemrs z,z′,a,b). Let Conf (Zǫ ) be the space of fin + simple (multiplicity free) finite point configurations on Zǫ , equipped with its canonical Borel structure. + Any measurable map GT+ Conf (Zǫ ) pushforwards the BC type z-measures of level N into point processes on the discrete sNta→te spacefinZǫ .+We consider two such maps. + The most natural map GT+ Conf (Zǫ ) determines a random point configuration with exactly N N → fin + points. In fact, consider the map :GT+ Conf (Zǫ ) (3.8) O N −→ fin + λ=(λ λ ) (λ)=l=(l > >l ), 1 N 1 N ≥···≥ −→O ··· where we do not write N in the definition of for simplicity, and recall that l is one of the N-tuples O b b b associatedtoλ,seeSection2. Underthemapabove,thepushforwardofthez-measureP isaprobability N measure on simple N-point configurations in Zǫ , therefore determining a pointbprocess on Zǫ that we + + denote by (N). Next weOdescribe a different map GT+ Conf (Zǫ ) that gives rise to a different point process. N → fin + Recall that λ GT+ can be described by its Frobenius coordinates. If we let d=d(λ) be the side-length ∈ N of the largest square that can be inscribed in the Young diagram of λ, then the Frobenius coordinates (p ,...,p q ,...,q ) of λ are defined by 1 d 1 d | (3.9) p (λ)=p d=efλ i, q (λ)=q d=efλ′ i, i=1,2,...,d, i i i− i i i− ∀ 10 CESARCUENCA where λ′ is the cardinality of the set j 0,1,...,N : λ i , or equivalently the length of the i-th i { ∈ { } j ≥ } column in the Young diagram of λ. Observe that p > ... > p 0 and q > ... > q 0. A balanced 1 d 1 d point configuration in Zǫ with respect to Zǫ is a subset ofZǫ w≥ith an equalnumber of≥points in either + <N + of the sets Zǫ d=ef ǫ2,(1+ǫ)2,...,(N 1+ǫ)2 , (3.10) <N { − } Zǫ d=ef (N +ǫ)2,(N +1+ǫ)2,... =Zǫ Zǫ . ≥N { } +\ <N For example, the empty configuration (the one with no points) is balanced (with respect to Zǫ ), and <N any balanced point configuration in Zǫ has at most 2N points. Consider the map + :GT+ Conf (Zǫ ) (3.11) L N −→ fin + λ=(p ,...,p q ,...,q ) (λ)= (N +p +ǫ)2 (N 1 q +ǫ)2 , 1 d 1 d i 1≤i≤d i 1≤i≤d | −→L { } ⊔{ − − } where, again, we do not write N in the definition of for simplicity. Evidently (λ) is a balanced point configurationwith d=d(λ) points in each of the setsLZǫ and Zǫ . The pushfLorwardof the z-measure <N ≥N P ,under themapabove,determinesaprobabilitymeasureonbalancedpointconfigurationsinZǫ that N + we also denote by (N). L The connection between the point configurations (N) and (N), as well as a summary of our defini- O L tions, is given in the following statement. Definition-Proposition 3.4. Let N N, and consider parameters a b 1/2, (z,z′) . The adm ∈ ≥ ≥ − ∈ U probability measure P =P ( z,z′,a,b) defined in (3.6) is called the BC type z-measure of level N (or N N ·| simply z-measure of level N) associated to the parameters z,z′,a,b. The pushforwards of P under the maps and , defined in (3.8) and (3.11) respectively, are point N O L processesonZǫ thatwedenote by (N) and (N) ,orsimply by (N) and (N). The process (N) + Oz,z′,a,b Lz,z′,a,b O L O is an N-point configuration, while (N) is a balanced configuration on Zǫ with respect to Zǫ . L + <N The following relation holds (3.12) (λ)= (λ) Zǫ =( (λ) Zǫ ) (Zǫ (λ)), L O △ <N O \ <N ∪ <N \O for any λ GT+. We denote this relation by ∈ N (N) =( (N))△. L O Proof. The only statement to proveis (3.12), for any λ GT+. If λ is the “empty”signature (0N), then evidently (λ)= ǫ2,(1+ǫ)2,...,(N 1+ǫ)2 =Zǫ , ∈(λ)=N ,andtherefore(3.12)isevident. Assume O { − } <N L ∅ otherwise that λ=(0N). Then d=d(λ) can be characterizedas the largestinteger in 1,2,...,N such 6 { } thatλ d, thusimplyingthatλ d. Inthis case,wehavethatthe set λ +N 1,...,λ +N d d d+1 1 d ≥ ≤ { − − } is a subset of N,N+1,... of size d and also the set λ +N d 1,λ +N d 2,...,λ is a d+1 d+2 N { } { − − − − } subset of 0,1,...,N 1 of size N d. It is easy to check the equalities of sets { − } − λ +N 1,...,λ +N d = N +p ,...,N +p 1 d 1 d { − − } { } λ +N d 1,λ +N d 2,...,λ N 1 q ,...,N 1 q = 0,1,...,N 1 , d+1 d+2 N 1 d { − − − − }⊔{ − − − − } { − } where (p ,...,p q ,...,q ) are the Frobenius coordinates of λ. From the definitions of the sets (λ) 1 d 1 d and (λ), identity| (3.12) is now obvious. O (cid:3) L 4. The discrete orthogonal polynomial ensemble (N) O In this section, we provethat the N-point process (N) is a discrete orthogonalpolynomialensemble, O and therefore a determinantal point process. The main result of this section is Theorem 4.5, where we compute explicitly a correlation kernel KO(N) : Zǫ Zǫ R for the point process (N) in terms of + × + −→ O generalized hypergeometric functions. As we shall do hereinafter, we consider four parameters z,z′,a,b satisfying a b 1/2 and (z,z′) . adm ≥ ≥− ∈U