aa r X i v : . [ m a t h . P R ] A p r PFAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS
LEONID PETROV
Abstract.
We study a family of continuous time Markov jump processes onstrict partitions (partitions with distinct parts) preserving the distributionsintroduced by Borodin [Bor99] in connection with projective representationsof the infinite symmetric group. The one-dimensional distributions of the pro-cesses (i.e., the Borodin’s measures) have determinantal structure. We expressthe dynamical correlation functions of the processes in terms of certain Pfaf-fians and give explicit formulas for both the static and dynamical correlationkernels using the Gauss hypergeometric function. Moreover, we are able toexpress our correlation kernels (both static and dynamical) through those ofthe z-measures on partitions obtained previously by Borodin and Olshanski ina series of papers.The results about the fixed time case were announced in the note [Pet10a].A part of the present paper contains proofs of those results.
Contents
1. Introduction 12. Model and results 43. Schur graph and multiplicative measures 104. Kerov’s operators 145. Fermionic Fock space 226. Z-measures and an orthonormal basis in ℓ ( Z ) 267. Static correlation functions 328. Static determinantal kernel 389. Markov processes 4110. Dynamical correlation functions 4611. Dynamical Pfaffian kernel 51Appendix A. Reduction of Pfaffians to determinants 54References 551. Introduction
We introduce and study a family of continuous time Markov jump processes onthe set of all strict partitions (that is, partitions in which nonzero parts are distinct).Our Markov processes preserve the family of probability measures introduced by Dobrushin Mathematics Laboratory, Kharkevich Institute for Information Trans-mission Problems, Bolshoy Karetny per. 19, Moscow, 127994, Russia
E-mail address : [email protected] .The author was partially supported by the RFBR-CNRS grant 10-01-93114, by A. Kuznetsov’sgraduate student scholarship and by the Dynasty foundation fellowship for young scientists. The whole picture depends on two continuous parameters α > < ξ < Borodin [Bor99] in connection with the harmonic analysis of projective representa-tions of the infinite symmetric group. The construction of our dynamics is similarto that of Borodin and Olshanski [BO06a] and is based on a special coherencyproperty of the measures on strict partitions introduced in [Bor99]. Regardingeach strict partition λ = ( λ > · · · > λ ℓ > λ j ∈ Z , as a point configuration { λ , . . . , λ ℓ } on the half-lattice Z > := { , , . . . } , one can say that the state spaceof our Markov processes is the space of all finite point configurations on Z > . Thefixed time distributions of our dynamics are probability measures on this configu-ration space. In other words, in the static (fixed time) picture one sees a randompoint process on Z > .The main result of the paper is the computation of the dynamical (or space-time) correlation functions for our family of Markov processes. We show that thesecorrelation functions have certain Pfaffian form, and compute the correspondingkernel. Here the kernel is a function Φ α,ξ ( s, x ; t, y ) of two space-time variables,where x, y ∈ Z and s, t ∈ R , which is explicitly expressed through the Gausshypergeometric function. Following the common terminology (e.g., see [NF98],[Joh05], [BO06a]), we call Φ α,ξ the extended ( Pfaffian ) hypergeometric-type kernel .In the static case the Pfaffian formula for the correlation functions of our Markovprocesses can be reduced to a determinantal one. Thus, in the fixed time picture wehave a determinantal point process on Z > . Its kernel K α,ξ has integrable form andis also expressed through the Gauss hypergeometric function. (About integrableoperators, e.g., see [IIKS90], [Dei99], discrete integrable operators are discussed in[Bor00] and [BO00, § hypergeometric-type kernel . Theresults about the static case were announced in the note [Pet10a]. A part of thepresent paper ( § §
8) contains complete proofs of those results.Models with correlation functions of Pfaffian form first appeared in theory of ran-dom matrices, e.g., see [Dys70], [MP83a], [MP83b], [NW91a], [NW91b], [TW96],[Nag07], and the book by Mehta [Meh04]. An essentially time-inhomogeneous Pfaf-fian dynamical model of random-matrix type was considered by Nagao, Katori andTanemura [KNT04], [Kat05]. Static Pfaffian random point processes of various ori-gins have also been studied, e.g., see [Rai00], [Fer04], [Mat05], [BR05], [Vul07], and §
10 of the survey [Bor09]. Borodin and Strahov [BS06], [BS09], [Str10a] consideredstatic models which are discrete analogues of Pfaffian models of random-matrixtype, they involve random ordinary (i.e., not necessary strict) partitions and havea representation-theoretic interpretation (see [Str10b]). The dynamical model thatwe study in the present paper seems to be a first example of a stationary (in contrastto the model of [KNT04], [Kat05]) Pfaffian dynamics.
Comparison with results for the z -measures. Our model of random strictpartitions and associated stochastic dynamics is very similar to the one of the z -measures on ordinary partitions. The structure of static and dynamical correlationfunctions in that case was investigated in [BO00], [Oko01b], [BO06a], [BO06b]. Letus discuss the relationship of our results with the ones from those papers. • The main feature of our model is that its dynamical correlation functions areexpressed in terms of Pfaffians and not determinants, as it is for the z -measures. which has a representation-theoretic meaning. The z -measures originated from the problem of harmonic analysis for the infinite symmetricgroup S ∞ [KOV93], [KOV04] and were studied in detail by Borodin, Okounkov, Olshanski, andother authors, e.g., see the bibliography in [BO09]. FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 3 • Determinantal (static) correlation kernels of random point processes often appearto be projection operators. In particular, this holds for the z -measures. Incontrast, in our situation the kernel K α,ξ ( x, y ) ( x, y ∈ Z > ) is not a projectionoperator in the corresponding coordinate Hilbert space ℓ ( Z > ). • On the other hand, the static Pfaffian kernel Φ α,ξ ( s, x ; s, y ) (where x, y ∈ Z ) inour model has a structure which is very similar to that of the determinantal kernelof the z -measures on semi-infinite point configurations on the lattice. Viewed asan operator in the Hilbert space ℓ ( Z ), Φ α,ξ ( s, · ; s, · ), is a rank one perturbationof an orthogonal projection operator. • Furthermore, our extended Pfaffian kernel Φ α,ξ ( s, x ; t, y ) is obtained from thestatic kernel Φ α,ξ ( s, x ; s, y ) in a way which is common for Markov processeson configuration spaces with determinantal dynamical correlation functions. Inparticular, the same extension happens in the case of the z -measures. • Markov processes of [BO06a], as well as many dynamical determinantal modelsthat arise in the theory of random matrices and random tilings (e.g., see [NF98],[War07], [JN06], [ANvM10], [Joh02], [Joh05], [BGR10]), are closely related to or-thogonal polynomials. Moreover, connections with orthogonal polynomials alsoarise in static Pfaffian models of random-matrix and representation-theoreticorigin [NW91a], [NW91b], [TW96], [KNT04], [Kat05], [Nag07], [BS06], [BS09],[Str10a]. For our model there also exists a connection with orthogonal polynomi-als (namely, the Krawtchouk polynomials), but this connection does not help usto compute the correlation kernels as it was for the z -measures [BO06a], [BO06b]. • The expressions for our correlation kernels involve the same special functions(expressed through the Gauss hypergeometric function) which arise for the ker-nels in the case of the z -measures. These functions first appeared in the worksof Vilenkin and Klimyk [VK88], [VK95]. In particular, certain degenerations ofthem lead to the classical Meixner and Krawtchouk orthogonal polynomials.Using this fact, we are able to express our kernels directly through the corre-sponding kernels for the z -measures. These expressions seem to have no directprobabilistic meaning at the level of random point processes, but in particularthey allow to study asymptotics of our kernels with the help of results of [BO00],[BO06a]. Method.
Our technique of obtaining both static and dynamical correlation kernelsin an explicit form is different from those of [BO00], [BO06b], [BO06a], and is basedon computations in the fermionic Fock space involving so-called Kerov’s operatorswhich span a certain sl (2 , C )-module. Both the static and dynamical correlationkernels in our model are expressed through matrix elements related to this module.This approach is similar to the one invented by Okounkov [Oko01b] to calculate the(static) correlation kernel of the z -measures on ordinary partitions. In computa-tions in this paper we use the ordinary fermionic Fock space instead of the (closelyrelated, but different) infinite wedge space of [Oko01b]. Moreover, our situation alsorequires to deal with a Clifford algebra (acting in the fermionic Fock space) of adifferent type. One can say that our Clifford algebra is an infinite-dimensional gen-eralization of the Clifford algebra over an odd-dimensional quadratic space. SimilarClifford algebras were used in [DJKM82], [Mat05], [Vul07]. In the latter two papers A possibility of use of this method in studying the dynamical model related to the z -measureswas pointed out in [BO06a], and later this approach was carried out by Olshanski [Ols08b]. LEONID PETROV the fermionic Fock space is also used for computations of certain correlation func-tions. That approach is analogous to the formalism of Schur measures and Schurprocesses [Oko01a], [OR03] and differs from the one used in the present paper.
Organization of the paper. In § § Z > are determinantal.In § § § sl (2 , C )-module) whichare used in explicit expressions for our correlation kernels. These functions areeigenfunctions of a certain second order difference operator on the lattice Z . Thisfact allows later to interpret our kernels through orthogonal spectral projectionsrelated to that operator on the lattice. In § z -measures on ordinary partitions which we usein the study of our model.In § § § § §
10 we show that the dynamical correlation functions of our Markovprocesses have Pfaffian form, and give an explicit expression for the dynamical(Pfaffian) correlation kernel in terms of the functions discussed in §
6. We considerthe asymptotic behavior of our dynamical Pfaffian kernel under a degeneration andin two limits regimes in § Acknowledgements.
The author is very grateful to Grigori Olshanski for per-manent attention to the work, fruitful discussions, and access to his unpublishedmanuscript [Ols08b]. I would also like to thank Alexei Borodin and Vadim Gorinfor useful comments on my work.2.
Model and results
Point processes on the half-lattice.
Let us first describe the fixed timepicture, that is, the random point processes on the half-lattice Z > that we study.They arise from a model of random strict partitions introduced in [Bor99].By a strict partition we mean a partition in which nonzero parts are distinct, thatis, λ = ( λ > · · · > λ ℓ ( λ ) > λ j ∈ Z > . The number | λ | := λ + · · · + λ ℓ ( λ ) iscalled the weight of the partition, and the number of nonzero parts ℓ ( λ ) is the length of the partition. By S n denote the set of all strict partitions of weight n = 0 , , . . . . By agreement, the set S consists of the empty partition ∅ . FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 5
Throughout the paper we identify strict partitions and corresponding shifted Youngdiagrams as in [Mac95, Ch. I, §
1, Ex. 9].The description of the model of [Bor99] starts with the
Plancherel measures onstrict partitions of a fixed weight: Pl n ( λ ) := 2 n − ℓ ( λ ) · n !( λ ! . . . λ ℓ ( λ ) !) Y ≤ k
To simplify certain formulas, instead of the parameter α we willsometimes use another parameter ν ( α ) := √ − α . If 0 < α ≤ , then ν ( α ) isreal, 0 ≤ ν ( α ) < . If α > , then ν ( α ) can take arbitrary purely imaginary values.The whole picture is symmetric with respect to the replacement of ν ( α ) by − ν ( α ).Sometimes the argument α in ν ( α ) is omitted.Similarly to the poissonization of the Plancherel measures (2.2), we consider acertain mixing of the measures M α,n . But now as the mixing distribution we take Throughout the paper we use this identification of strict partitions with point configurationson Z > whenever we speak about random point processes and their correlation functions. The mixing of Plancherel measures Pl n and the measures M α,n over n can also be viewed asa passage to the grand canonical ensemble , cf. [Ver96]. LEONID PETROV a special case of the negative binomial distribution (on Z ≥ ) π α,ξ ( n ) := (1 − ξ ) α/ α/ n n ! ξ n , n = 0 , , , . . . , (2.4)with an additional parameter ξ ∈ (0 , a ) k := a ( a + 1) . . . ( a + k −
1) = Γ( a + k ) / Γ( a ) (2.5)is the Pochhammer symbol , and Γ( · ) is the Euler gamma function. As a result ofthe mixing, we obtain a random point process M α,ξ on Z > : M α,ξ ( λ ) := π α,ξ ( | λ | ) · M α, | λ | ( λ ) , λ ∈ S . The process M α,ξ is supported by finite configurations. The probability of eachconfiguration λ = { λ , . . . , λ ℓ } ⊂ Z > has the form M α,ξ ( λ ) = (1 − ξ ) α/ · ℓ Y k =1 w α,ξ ( λ k ) · Y ≤ k
Theorem 1.
Both the point processes M α,ξ and Pl θ on the half-lattice Z > aredeterminantal. The correlation kernel K α,ξ of M α,ξ is expressed through the Gausshypergeometric function (8.1), (8.2). The correlation kernel K θ of Pl θ can be writ-ten in terms of the Bessel function of the first kind (8.5), (8.6). FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 7
We call the kernel K α,ξ the hypergeometric-type kernel. The kernel K θ is ob-tained from K α,ξ via the Plancherel degeneration (2.8).In (8.3) we are able to express the kernel K α,ξ through the discrete hypergeo-metric kernel introduced in [BO00], [BO06b]. This is done in the same spirit as isexplained in § Dynamical model.
Let us now describe a family of continuous time Markovjump processes ( λ α,ξ ( t )) t ∈ [0 , + ∞ ) on the space of all strict partitions S (which is thesame as the set of all finite configurations on Z > ). These processes preserve themeasures M α,ξ . The construction of the processes λ α,ξ uses the same ideas as in[BO06a]. The first key ingredient is the continuous time birth and death processon Z > denoted by ( n α,ξ ( t )) t ∈ [0 , + ∞ ) . It depends on our parameters α and ξ andhas the following jump rates: Prob { n α,ξ ( t + dt ) = n + 1 | n α,ξ ( t ) = n } = (1 − ξ ) − ξ ( n + α/ dt, Prob { n α,ξ ( t + dt ) = n − | n α,ξ ( t ) = n } = (1 − ξ ) − ndt. The process n α,ξ preserves the negative binomial distribution π α,ξ (2.4) on Z > and is reversible with respect to it. About birth and death processes in general see,e.g., [KM57], [KM58].The second key ingredient is the collection of Markov transition kernels p ↑ α ( n, n +1) from S n to S n +1 and p ↓ ( n + 1 , n ) from S n +1 to S n , n = 0 , , , . . . , such that M α,n ◦ p ↑ α ( n, n + 1) = M α,n +1 and M α,n +1 ◦ p ↓ ( n + 1 , n ) = M α,n . (2.10)These kernels are canonically associated with the system of measures { M α,n } ∞ n =0 (see § p ↑ α ( n, n + 1) dependon the parameter α , and the kernels p ↓ ( n + 1 , n ) do not depend on any parameter.The values p ↑ α ( n, n + 1) µ, κ and p ↓ ( n + 1 , n ) κ ,µ , where µ ∈ S n and κ ∈ S n +1 (theseare the individual transition probabilities), vanish unless the shifted Young diagram κ is obtained from µ by adding a box. In other words, the transition kernels p ↑ α ( n, n + 1) and p ↓ ( n + 1 , n ) describe random procedures of adding and deletingone box, respectively.We describe the dynamics λ α,ξ on strict partitions in terms of jump rates. Thejumps are of two types: one can either add a box to the random shifted Youngdiagram, or remove a box from it (of course, the result must still be a shiftedYoung diagram). The events of adding and removing a box are governed by thebirth and death process n α,ξ = | λ α,ξ | . Conditioned on λ α,ξ ( t ) = λ and the jump n → n +1 (where n = | λ | ) of the process n α,ξ during the time interval ( t, t + dt ), thechoice of the box to be added to the diagram λ is made according to the probabilities p ↑ α ( n, n + 1) λ, κ , where κ ∈ S n +1 . Similarly, conditioned on λ α,ξ ( t ) = λ and thejump n → n − n α,ξ during ( t, t + dt ), the choice of the box to be removed from λ is made according to the probabilities p ↓ ( n, n − λ,µ , where µ ∈ S n − .The fact that the process n α,ξ preserves the mixing distribution π α,ξ togetherwith (2.10) implies that the measure M α,ξ on S is invariant for the process λ α,ξ .Moreover, the process is reversible with respect to M α,ξ . In this paper by ( λ α,ξ ( t )) t ≥ we mean the equilibrium process (that is, the process starting from the invariantdistribution M α,ξ ). LEONID PETROV
Remark 2.2.
A closely related dynamical model was considered in [Pet10b], namely,a sequence of discrete time Markov chains on the sets S n , n = 0 , , . . . , with transi-tion operators p ↑ α ( n, n + 1) ◦ p ↓ ( n + 1 , n ). These chains (called the up/down Markovchains ) preserve the measures M α,n (2.3). Similar models on ordinary partitionswith various up and down transition kernels were studied in [Ful05], [Ful09] (spec-tral properties of the chains), and [BO09], [Pet09], [Ols10] (large n limits).The n th up/down Markov chain on S n can be reconstructed from ( λ α,ξ ( t )) t ≥ as follows. Condition the process λ α,ξ to stay in the set S n × S n +1 , and take itsembedded Markov chain, that is, consider the continuous time process only at thetimes of jumps. We get a Markov chain on S n × S n +1 that belongs to S n at, say,even discrete time moments. Taking this chain at these moments, we reconstructthe Markov chain on S n with the transition operator p ↑ α ( n, n + 1) ◦ p ↓ ( n + 1 , n ).Let ( t , x ) , . . . ( t n , x n ) ∈ R ≥ × Z > be pairwise distinct space-time points. Thedynamical (or space-time) correlation functions of the Markov process λ α,ξ aredefined as ρ ( n ) α,ξ ( t , x ; . . . ; t n , x n ) (2.11):= Prob { the configuration λ α,ξ ( t ) at time t = t j contains x j , j = 1 , . . . , n } . The notion of dynamical correlation functions is a combination of finite-dimensionaldistributions of a stochastic dynamics and correlation functions of a random pointprocess. Indeed, the finite-dimensional distribution of the process λ α,ξ at times t , . . . , t n (let these times be distinct for simplicity) is a probability measure onconfigurations on the space Z > ⊔ · · · ⊔ Z > ( n copies), and ρ ( n ) α,ξ ( t , x ; . . . ; t n , x n )( t j ’s fixed) are just the correlation functions of this measure on configurations. Thedynamical correlation functions uniquely determine the dynamics ( λ α,ξ ( t )) t ∈ [0 , + ∞ ) .The main result of the present paper is the computation of the dynamical cor-relation functions of λ α,ξ .To formulate the result, we need a notation. By Z =0 denote the set of all nonzerointegers, and for x , . . . , x n ∈ Z > put, by definition, x − k := − x k , k = 1 , . . . , n . Theorem 2.
The equilibrium continuous time dynamics ( λ α,ξ ( t )) t ≥ is Pfaffian,that is, there exists a function Φ α,ξ ( s, x ; t, y ), x, y ∈ Z , s ≤ t , such that the dynam-ical correlation functions of λ α,ξ have the form ρ ( n ) α,ξ ( t , x ; . . . ; t n , x n ) = Pf ( Φ α,ξ J T, X K ) , ≤ t ≤ · · · ≤ t n , (2.12)where Φ α,ξ J T, X K is the 2 n × n skew-symmetric matrix with rows and columnsindexed by 1 , − , . . . , n, − n , and the kj -th entry in Φ α,ξ J T, X K above the maindiagonal is Φ α,ξ ( t | k | , x k ; t | j | , x j ), where k, j = 1 , − , . . . , n, − n (thus, | k | ≤ | j | ). Thekernel Φ α,ξ can be expressed through the Gauss hypergeometric function (10.5). Remark 2.3. 1.
Observe that it is enough for Φ α,ξ ( s, x ; t, y ) to be defined onlyfor x, y ∈ Z =0 because only such values of Φ α,ξ ( s, x ; t, y ) are used in the theorem.However, our kernel Φ α,ξ ( s, x ; t, y ) extends to x, y ∈ Z in a very natural way, so wealways let Φ α,ξ ( s, x ; t, y ) to be defined for all x, y ∈ Z . The same is applicable tothe static Pfaffian kernel Φ α,ξ ( x, y ) := Φ α,ξ ( s, x ; s, y ) (see 7.16 below). In (2.12) we require that the time moments t j are ordered. However, The-orem 2 allows to compute the correlation functions ρ ( n ) α,ξ ( t , x ; . . . ; t n , x n ) with ar-bitrary order of time moments: one should simply permute the space-time points FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 9 ( t , x ) , . . . , ( t n , x n ) (this does not change the value of ρ ( n ) α,ξ ( t , x ; . . . ; t n , x n )) insuch a way that the time moments become nondecreasing, and then apply (2.12).In § Φ α,ξ . The resulting kernel Φ θ is expressed through the Bessel function of thefirst kind (Theorem 11.2). The dynamical kernel Φ θ has analogues related to thePlancherel measures on ordinary partitions and associated dynamics, see [PS02],[BO06c]. The asymptotic behavior of the dynamical kernel Φ α,ξ in two other limitregimes (corresponding to studying smallest resp. largest components of a randomstrict partition) is considered in § § § Remark 2.4 (Hidden determinantal structure in Pfaffian processes) . If in Theo-rem 2 we set t = · · · = t n , then the dynamical correlation functions turn into the(static) correlation functions of the point process M α,ξ on Z > . Thus, Theorem 2implies that the point process M α,ξ on Z > is Pfaffian. To show that it is in factdeterminantal requires some work (see Theorem 8.1 and Proposition A.2 from Ap-pendix). Thus, one can say that in the static case the determinantal structure ofcorrelation functions is hidden under the Pfaffian one. In particular, in this waywe discover the determinantal structure of the poissonized Plancherel measure Pl θ (2.2) on strict partitions, thus strengthening a result of Matsumoto [Mat05] whogave a Pfaffian formula for the correlation functions of Pl θ , see § λ α,ξ cannot be written as determinants. Weplan to give a rigorous proof of this fact in a subsequent work.2.3. Expression through the kernel for the z -measures. The dynamical Pfaf-fian kernel Φ α,ξ ( s, x ; t, y ) of Theorem 2 has a relatively simple structure we areabout to describe.In § § e ϕ m ( x ; α, ξ ), where α, ξ are ourparameters, and the argument x and the index m range over Z . Each e ϕ m can beexpressed through the Gauss hypergeometric function (6.13), (6.8). For fixed α, ξ ,the functions { e ϕ m } m ∈ Z form an orthonormal basis of the Hilbert space ℓ ( Z ).Our kernel then has the form ( x, y ∈ Z and s ≤ t ): Φ α,ξ ( s, x ; t, y ) = X ∞ m =0 − δ ( m ) e − m ( t − s ) e ϕ m ( x ) e ϕ m ( − y ) . (2.13)Here δ ( m ) = δ m, is the Kronecker delta.The functions e ϕ m are particular cases of the functions used by Borodin and Ol-shanski to describe the static and dynamical correlation kernels for the z -measureson partitions [BO06b], [BO06a] (see § § Φ α,ξ ( s, x ; t, y ) can be expressed through the extended discretehypergeometric kernel of [BO06a]: Φ α,ξ ( s, x ; t, y ) = ( − x ∧ y ∨ h e − ( t − s ) K ν ( α ) − , − ν ( α ) − ,ξ ( t, x + ; s, − y + )+ e ( t − s ) K ν ( α )+ , − ν ( α )+ ,ξ ( t, x − ; s, − y − ) i , (2.14)where ν ( α ) is given by Definition 2.1, x, y ∈ Z , and the kernel K z,z ′ ,ξ [BO06b],[BO06a] lives on the lattice of (proper) half-integers Z ′ := Z + . Here and below for any two numbers a and b , by a ∨ b and a ∧ b we denote the maximum and theminimum of a and b , respectively.The identity (2.14) seems to be only formal and have no direct probabilisticconsequences (such as probabilistic relations between our random point processesor dynamics on Z > and the corresponding objects for the z -measures). However,(2.14) can serve as a useful tool in studying the asymptotics of our dynamicalkernel Φ α,ξ ( s, x ; t, y ) simply by a direct application of the results of [BO06a] (see § § Schur graph and multiplicative measures
Schur graph.
We identify strict partitions λ = ( λ > · · · > λ ℓ ( λ ) > λ j ∈ Z > , and corresponding shifted Young diagrams as in [Mac95, Ch. I, § λ consists of ℓ ( λ ) rows. Each k th row ( k = 1 , . . . , ℓ ( λ )) has λ k boxes, and for j = 1 , . . . , ℓ ( λ ) − j + 1)th row is right under the second box of the j th row. For example, theshifted Young diagram corresponding to the strict partition λ = (6 , , ,
1) looks asfollows:Let µ and λ be strict partitions. If | λ | = | µ | + 1 and the shifted diagram λ isobtained from the shifted diagram µ by adding a box, then we write µ ր λ , or,equivalently, λ ց µ . The box that is added is denoted by λ/µ .The set S = F ∞ n =0 S n of all strict partitions is equipped with a structure of agraded graph: for µ ∈ S n − and λ ∈ S n we draw an edge between µ and λ iff µ ր λ . Thus, the edges in S are drawn only between consecutive floors. Weassume the edges to be oriented from S n − to S n . In this way S becomes a gradedgraph. It is called the Schur graph . This graph describes the branching of (suitablynormalized) irreducible truly projective characters of symmetric groups, e.g., see[Iva99].Let dim S λ be the total number of oriented paths in the Schur graph from theinitial vertex ∅ to the vertex λ . This number is given by [Mac95, Ch. III, § S λ = | λ | ! λ ! . . . λ ℓ ( λ ) ! Y ≤ k Similarly, by dim S ( µ, λ ) denote the total number of paths from µ to λ in thegraph S . Clearly, dim S ( µ, λ ) vanishes unless µ ⊆ λ , that is, unless µ k ≤ λ k for all k . If µ ⊆ λ , by λ/µ denote the corresponding skew shifted Young diagram, that is,the set difference of λ and µ . We have dim S λ = dim S ( ∅ , λ ).3.2. Coherent systems of measures on the Schur graph. Following the gen-eral formalism (e.g., see [KOO98]), one can define coherent systems of measureson the Schur graph. This definition starts from the notion of the down transitionprobabilities . For λ, µ ∈ S , set p ↓ ( λ, µ ) := (cid:26) dim S µ/ dim S λ, if µ ր λ ;0 , otherwise . By (3.2), the restriction of p ↓ to S n +1 × S n for all n = 0 , , . . . is a Markov transitionkernel. We denote it by p ↓ ( n + 1 , n ) = { p ↓ ( n + 1 , n ) λ,µ } λ ∈ S n +1 , µ ∈ S n , and call it the down transition kernel . Definition 3.1. Let M n be a probability measure on S n , n = 0 , , . . . . We call { M n } a coherent system of measures iff M n ( λ ) = X κ : κ ց λ M n +1 ( κ ) p ↓ ( κ , λ ) for all n and λ ∈ S n . (3.3)In other words, M n +1 ◦ p ↓ ( n + 1 , n ) = M n for all n (cf. (2.10)).Having a nondegenerate coherent system { M n } (that is, M n ( λ ) > n and λ ∈ S n ), we can define the corresponding up transition probabilities . They dependon a choice of a coherent system. For λ, κ ∈ S , set p ↑ ( λ, κ ) := (cid:26) M n +1 ( κ ) p ↓ ( κ , λ ) /M n ( λ ) , if λ ∈ S n , κ ∈ S n +1 and λ ր κ , , otherwise . By (3.3), the restriction of p ↑ to S n × S n +1 for all n = 0 , , . . . is a Markov transitionkernel. We denote it by p ↑ ( n, n + 1) = { p ↑ ( n, n + 1) λ, κ } λ ∈ S n , κ ∈ S n +1 and call it the up transition kernel . We have M n ◦ p ↑ ( n, n + 1) = M n +1 (cf. (2.10)).Let us make a comment on the representation-theoretic meaning of the coherencyrelation (3.3). The set of all coherent systems of measures on the Schur graph isa convex set. Its extreme points are identified with the points of the infinite-dimensional ordered simplexΩ + := n ( ω , ω , . . . ) : ω ≥ ω ≥ · · · ≥ , X ∞ k =1 ω k ≤ o . This is the so-called Martin boundary of the Schur graph. It was first described byNazarov [Naz92]. Another proof of this result can be obtained using the generalmethods of [KOO98] together with the formulas of [Iva99] for dimensions of skewshifted Young diagrams.Moreover, the following characterization of the coherent systems holds: Theorem 3.2 ([Naz92]) . There is a bijection between coherent systems of measureson the Schur graph S and Borel probability measures on the simplex Ω + . Let us explain how this bijection works. Consider embeddings S n → Ω + , S n ∋ λ = ( λ , . . . , λ ℓ ) (cid:0) λ n , . . . , λ ℓ n , , , . . . (cid:1) ∈ Ω + . For a coherent system { M n } on S , the corresponding measure P on Ω + can be reconstructed as the weak limit (as n → + ∞ ) of the push-forwards of the measures M n under these embeddings. In the opposite direction, the points of Ω + are in one-to-one correspondencewith the indecomposable normalized projective characters of the infinite symmetricgroup, and any Borel probability measure P on Ω + can be viewed as a (possiblydecomposable) projective character χ of S ∞ . This character χ can be restrictedto the finite symmetric group S n ⊂ S ∞ (of any order n ) and expressed as a linearcombination of (suitably normalized) irreducible truly projective characters of S n .These characters are parametrized by the set S n [Sch11], [HH92]. The coefficients ofthe expansion of χ | S n are the numbers { M n ( λ ) } λ ∈ S n , where { M n } is the coherentsystem corresponding to the measure P on Ω + by Theorem 3.2. The coherencycondition (3.3) for the measures { M n } arises naturally in this context because therestrictions of the character χ to symmetric groups S n for different n must beconsistent with each other.3.3. Multiplicative measures. There is a distinguished coherent system on theSchur graph, namely, the Plancherel measures { Pl n } ∞ n =0 (2.1). This coherent sys-tem corresponds (in the sense of Theorem 3.2) to the delta measure at the point(0 , , . . . ) ∈ Ω + . Using the function dim S λ defined by (3.1), one can write Pl n ( λ ) = 2 n − ℓ ( λ ) n ! (dim S λ ) , n ∈ Z > , λ ∈ S n . The Plancherel measures on strict partitions are analogues (in the theory of projec-tive representations of symmetric groups) of the well-known Plancherel measureson ordinary partitions.Borodin [Bor99] has introduced a deformation M α,n (2.3) of the measures Pl n on S n depending on one real parameter α > 0. Here let us recall the characterizationof the measures { M α,n } from [Bor99]. Definition 3.3. A system of probability measures M n on S n is called multiplicative if there exists a function f : { (i , j) : j ≥ i ≥ } → C such that M n ( λ ) = c n · Pl n ( λ ) · Y (cid:3) =(i , j) ∈ λ f (i , j) for all n and all λ ∈ S n . Here c n , n = 0 , , . . . , are normalizing constants. The product above is taken overall boxes (cid:3) = (i , j) of the shifted Young diagram λ , where i and j are the row andcolumn numbers of the box (cid:3) , respectively. (Note that for shifted Young diagramswe always have j ≥ i.) Theorem 3.4 ([Bor99]) . Let { M n } be a nondegenerate coherent system of measureson the Schur graph. It is multiplicative iff the function f has the form f (i , j) = (j − i)(j − i + 1) + α (3.4) for some parameter α ∈ (0 , + ∞ ] . If f (i , j) is given by (3.4), then c n = α ( α + 2) . . . ( α + 2 n − α = + ∞ is understood in the limit sense: lim α → + ∞ c n Q (cid:3) =(i , j) ∈ λ f (i , j) = 1 for all n . This casecorresponds to the Plancherel measures { Pl n } .Recall that the number (j − i) is called the content of the box (cid:3) = (i , j). Forshifted Young diagrams all contents are nonnegative.We denote by { M α,n } ∞ n =0 the multiplicative coherent system corresponding tothe parameter α ∈ (0 , + ∞ ). We see that M α,n converges to Pl n as α → + ∞ . The FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 13 up transition kernel on S n × S n +1 ( § { M α,n } is denotedby p ↑ α ( n, n + 1). Remark 3.5. For certain negative values of α one can also define the measures M α,n by Definition 3.3 with f given by (3.4). Namely, for α = − N ( N + 1) (where N = 1 , , . . . ) the measures M α,n are well-defined and nonnegative for 0 ≤ n ≤ N ( N + 1) / 2. Moreover, M α,n ( λ ) > λ is inside the staircase-shaped shifteddiagram ( N, N − , . . . , Mixing of measures. Point configurations on the half-lattice. For a set X , by Conf ( X ) denote the space of all (locally finite) point configurations on X , andby Conf fin ( X ) ⊂ Conf ( X ) denote the subset consisting only of finite configurations.A Borel probability measure (with respect to a certain natural topology) on Conf ( X )is called a random point process on X . If X is discrete, then Conf ( X ) ∼ = { , } X ,and we take the standard coordinatewise topology on { , } X which turns it into acompact space. In more detail about random point processes, e.g., see [Sos00].As explained in § M α,n (2.3) using the negative binomialdistribution π α,ξ (2.4) on the set { , , . . . } of indices n . As a result we get aprobability measure M α,ξ (2.6) on the set S of all strict partitions. Identifying strictpartitions with point configurations in a natural way ( § S isthe same as Conf fin ( Z > ). Thus, M α,ξ can be viewed as a random point process on Z > supported by finite configurations. Under the Plancherel degeneration (2.8),the measures M α,ξ become the poissonized Plancherel measure Pl θ (2.2).Let us now prove that the point processes M α,ξ and Pl θ on Z > are determinan-tal. Observe that both these processes have a general structure described in thefollowing definition: Definition 3.6. Let w be a nonnegative function on Z > such that X ∞ x =1 w ( x ) < ∞ . (3.5)By P ( w ) denote the point process on Z > that lives on finite configurations andassigns the probability P ( w ) ( λ ) := const · ℓ Y k =1 w ( λ k ) Y ≤ k Let λ = { λ , . . . , λ ℓ } ⊂ Z > be a point configuration. We have P ( w ) ( λ ) = det L ( w ) ( λ )det(1 + L ( w ) ) , where L ( w ) ( λ ) denotes the submatrix (cid:2) L ( w ) ( λ k , λ j ) (cid:3) ℓk,j =1 of L ( w ) . (2) The point process P ( w ) is determinantal with the correlation kernel K ( w ) = L ( w ) (1 + L ( w ) ) − .Proof. The first claim directly follows from the Cauchy determinant identity [Mac95,Ch. I, § 4, Ex. 6].This means that the point process P ( w ) is a so-called L-ensemble correspondingto the matrix L ( w ) defined above (e.g., see [BO00, Prop. 2.1] or [Bor09, § (cid:3) Note that the normalizing constant in (3.6) is equal to L ( w ) ) , so the condi-tion (3.5) is necessary for the point process P ( w ) to be well defined. Remark 3.8. The correlation kernel K ( w ) of the process P ( w ) is symmetric, be-cause it has the form K ( w ) = L ( w ) (1 + L ( w ) ) − , where L ( w ) is symmetric. How-ever, the operator of the form L ( w ) (1 + L ( w ) ) − cannot be a projection operator in ℓ ( Z > ). This aspect discriminates our processes from many other determinantalprocesses appearing in, e.g., random matrix models (see the references given inIntroduction).On the other hand, the static Pfaffian kernel in our model resembles the structureof a spectral projection operator, see Proposition 7.5.Lemma 3.7 implies, in particular, that our point processes M α,ξ and Pl θ on Z > are determinantal. Denote their correlation kernels (given by Lemma 3.7(2)) by K α,ξ and K θ , respectively. These kernels are symmetric. However, Lemma 3.7does not give any suggestions on how to calculate them. Below we compute thecorrelation kernel K α,ξ using a fermionic Fock space technique. The kernel K θ isobtained from K α,ξ via the Plancherel degeneration ( § Kerov’s operators Definition, characterization and properties. The main tool that we usein the present paper to compute the correlation functions of the point processes M α,ξ (and also of the associated dynamical models, see § § 10) is a representationof the Lie algebra sl (2 , C ) in the pre-Hilbert space ℓ ( S ) given by the so-calledKerov’s operators. This approach was introduced by Okounkov [Oko01b] for the z -measures on ordinary partitions.By ℓ ( S ) we denote the space of all finitely supported functions on S with theinner product ( f, g ) := X λ ∈ S f ( λ ) g ( λ ) . This is a pre-Hilbert space whose Hilbert completion is the usual space ℓ ( S ) of allfunctions on S which are square integrable with respect to the counting measureon S . The standard orthonormal basis in ℓ ( S ) is denoted by { λ } λ ∈ S , that is, λ ( µ ) := (cid:26) , if µ = λ ;0 , otherwise . (4.1) FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 15 Definition 4.1. The Kerov’s operators in ℓ ( S ) depend on our parameter α > U λ := X κ : κ ց λ − δ (j − i) / p (j − i)(j − i + 1) + α · κ , (i , j) = κ /λ ; D λ := X µ : µ ր λ − δ (j − i) / p (j − i)(j − i + 1) + α · µ, (i , j) = λ/µ ; (4.2) H λ := (cid:0) | λ | + α (cid:1) λ. We denote a box by (i , j) iff its row number is i and its column number is j.The Kerov’s operators are closely related to the measures M α,n (2.3) on S n .Namely, it is clear that( U n ∅ , λ ) = ( D n λ, ∅ ) = dim S λ · − ℓ ( λ ) / Y (cid:3) =(i , j) ∈ λ p (j − i)(j − i + 1) + α for all n and λ ∈ S n , so M α,n ( λ ) = Z − n ( U n ∅ , λ )( D n λ, ∅ ) , (4.3)where Z n is a normalizing constant. See also the end of this subsection for moreconnections between the Kerov’s operators and the measures M α,n .The Kerov’s operators (4.2) satisfy the following: Properties 4.2. 1. The map U := (cid:20) (cid:21) → U , D := (cid:20) − (cid:21) → D , H := (cid:20) − (cid:21) → H (4.4)defines a representation of the Lie algebra sl (2 , C ) in ℓ ( S ). That is, the oper-ators U , D , and H satisfy the commutation relations[ H , U ] = 2 U , [ H , D ] = − D , [ D , U ] = H . (4.5) The operators U and D are adjoint to each other in the space ℓ ( S ). For any λ ∈ S , the vector U λ is a linear combination of vectors κ , where κ ց λ ,and the coefficient of κ depends only on the box κ /λ (through its row andcolumn numbers). Likewise, the vector D λ is a linear combination of vectors µ ,where µ ր λ , and the coefficient of µ depends only on the box λ/µ . Each basis vector λ , λ ∈ S , is an eigenvector of the operator H , and the eigenvalueof λ depends only on | λ | .The only property above that is not obvious is the first one: Lemma 4.3. The Kerov’s operators U , D , and H (4.2) satisfy the commutationrelations (4.5).Proof. Denote q α ( (cid:3) ) = q α (i , j) := 2 − δ (j − i) / p (j − i)(j − i + 1) + α, (4.6)where (cid:3) = (i , j). The relation [ H , U ] = 2 U is straightforward:[ H , U ] λ = H X κ ց λ q α ( κ /λ ) κ − (cid:16) | λ | + α (cid:17) X κ ց λ q α ( κ /λ ) κ = 2 ( | λ | + 2 − | λ | ) U λ = 2 U λ, and the same for the relation [ H , D ] = − D . It remains to prove that [ D , U ] = H . The vector [ D , U ] λ has the form X κ ց λ X ρ ր κ q α ( κ /λ ) q α ( κ /ρ ) ρ − X µ ր λ X ρ ց µ q α ( λ/µ ) q α ( ρ/µ ) ρ. (4.7)This is a linear combination of vectors ρ , where ρ ∈ S n and either ρ = λ , or ρ = λ + (cid:3) − (cid:3) for some boxes (cid:3) = (cid:3) . In the second case the coefficient by thevector ρ with ρ = λ + (cid:3) − (cid:3) is q α ( (cid:3) ) q α ( (cid:3) ) − q α ( (cid:3) ) q α ( (cid:3) ) = 0 . Thus, in (4.7) it remains to consider only the terms with ρ = λ . Therefore, onemust establish the combinatorial identity X κ : κ ց λ q α ( κ /λ ) − X µ : µ ր λ q α ( λ/µ ) = 2 | λ | + α for all λ ∈ S . The proof of this identity (using Kerov’s interlacing coordinates of shifted Youngdiagrams) is essentially contained in § (cid:3) In fact, the Kerov’s operators (4.2) are completely characterized by the abovefour properties: Proposition 4.4. If three operators U , D , and H in the space ℓ ( S ) satisfy thefour properties 4.2, then they have the form (4.2) with some parameter α ∈ C . By agreement, for arbitrary complex α , in the definition of U and D in (4.2) wetake the same branches of the square roots p α + c ( c + 1), c = 0 , , , . . . . In fact, itis the square of the function q α ( · , · ) (4.6) that really plays the role in the definitionof the Kerov’s operators. Proof. By properties 2 and 3, there exists a (complex-valued) function q on the setof all boxes, that is, on the set { (i , j) : j ≥ i ≥ } (where i and j are the row andcolumn numbers of the box, respectively), such that the operators U and D havethe form U λ = X κ ց λ q ( κ /λ ) κ , D λ = X µ ր λ q ( λ/µ ) µ. By property 4, for all n = 0 , , . . . and all λ ∈ S n we have H λ = h n λ for some(complex) numbers h n . Using the commutation relation [ H , U ] = 2 U (property 1),it is easy to see that h n +1 = h n + 2 for all n = 0 , , . . . . Set α := 2 h (this is somecomplex parameter). Thus, we have h n = 2 n + α .The function q ( · , · ) can be found by applying the commutation relation [ D , U ] = H (property 1) to various vectors λ , λ ∈ S . First, applying this relation to ∅ and (cid:3) (the latter vector corresponds to the one-box shifted diagram), we get q (1 , = α and q (1 , = 2 + α. Next, apply [ D , U ] = H to ( n ) for n ≥ q (1 , n + 1) − q (1 , n ) + q (2 , = 2 n + α . Solving this recurrence and taking into account the initial value q (1 , , we get q (1 , n ) = − ( n − q (2 , + n (cid:0) n − α (cid:1) , n = 2 , , . . . . To find q (2 , , we apply [ D , U ] = H to (2 , q (1 , − q (2 , = 6 + α ⇒ q (2 , = α , FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 17 and so q (1 , n ) = n ( n − 1) + α, n = 2 , , . . . . To find q ( n, n ) for n ≥ 3, use the vector λ with λ = ( n, n − , . . . , q (1 , n + 1) − q ( n, n ) = 2 · n ( n + 1)2 + α ⇒ q ( n, n ) = α . Finally, to find q (i , j) for arbitrary j > i > q ( · , · ) ), we apply the relation [ D , U ] = H to the vector λ with λ =(j , j − , . . . , j − i + 1): q (1 , j + 1) + q (i + 1 , i + 1) − q (i , j) = 2 · i(2j − i + 1)2 + α . We thus have q (i , j) = (j − i)(j − i + 1) + α for all j > i > q is identical to the function q α defined by (4.6) (but note the remark after the formulation of the present proposi-tion). The fact that the commutation relation [ D , U ] λ = H λ (with the above choiceof q = q α ) holds for all shifted Young diagrams λ ∈ S follows from Lemma 4.3.This concludes the proof. (cid:3) Remark 4.5. One can also prove a statement analogous to Proposition 4.4 withoutproperty 4.2.2. The operator H is still defined uniquely up to a parameter α ∈ C .The two other operators are equal to U and D (4.2) up to a “gauge transformation”that is written in terms of matrix elements in the basis { λ } λ ∈ S as:( U λ, κ ) f ( κ /λ ) · ( U λ, κ ) , ( D λ, µ ) f ( λ/µ ) − · ( D λ, µ ) , where f is some nonzero function on the set of boxes. One possible choice of suchoperators (which are not adjoint to each other) is (4.8) below. Remark 4.6. A statement parallel to Proposition 4.4 can be proved for the Younggraph whose vertices are ordinary partitions. As a result we will get operators sim-ilar to those considered in [Oko01b]. This allows one to give a purely combinatorialcharacterization of the z -measures on the Young graph. Another characterizationof the z -measures similar to Theorem 3.4 (of [Bor99]) is given in [Roz99].Let us give some remarks on how deep is the connection between the measures M α,n (2.3) and the Kerov’s operators (4.2). This discussion is also applicable tothe z -measures on the Young graph.First, using commutation relations (4.5) for the Kerov’s operators, one can com-pute the normalizing constants Z n in (4.3) that are defined as Z n = X λ ∈ S n ( U n ∅ , λ )( D n λ, ∅ ) . In the above sum the parameter α is hidden in the definition of the operators U and D (4.2), and one can assume α to be an arbitrary complex number. Write Z n = X λ ∈ S n ( U n ∅ , λ )( D n λ, ∅ ) = (cid:16) D n X λ ∈ S n ( U n ∅ , λ ) · λ, ∅ (cid:17) = ( D n U n ∅ , ∅ ) . By the commutation relations (4.5), DU n = U n D + X n − k =0 U n − k − HU k . Using the fact that D ∅ = 0, we get Z n = n − X k =0 ( D n − U n − k − HU k ∅ , ∅ ) = Z n − n − X k =0 (cid:16) k + α (cid:17) = n (cid:16) n − α (cid:17) Z n − . Taking into account the initial value Z = ( U D ∅ , ∅ ) = 1, we see that Z n = n !( α/ n .Thus, the (complex-valued) measures M α,n are well-defined by (4.3) for all α ∈ C \ { , − , − , . . . } because for such α the normalizing constants Z n are nonzerofor all n . Moreover, under this assumption the measures M α,n are nondegenerate inthe sense that M α,n ( λ ) = 0 for all n and all λ ∈ S n . Many formulas in the presentpaper hold in a purely algebraic sense for α ∈ C \ { , − , − , . . . } .Now let us present an alternative proof of the coherency condition (3.3) of themeasures { M α,n } . Another proof can be found in [Bor99]. Here we also assumethat α ∈ C \ { , − , − , . . . } . Consider the following operators which are slightlydifferent from U and D :ˆ U λ := X κ : κ ց λ − δ (j − i) (cid:0) (j − i)(j − i + 1) + α (cid:1) · κ , (i , j) = κ /λ ;ˆ D λ := X µ : µ ր λ µ. (4.8)Clearly, [ ˆ D , ˆ U ] = H (see also Remark 4.5), and ( ˆ D λ, ∅ ) = dim S λ . Moreover, M α,n ( λ ) = 1 Z n (ˆ U n ∅ , λ )( ˆ D n λ, ∅ ) for all n and λ ∈ S n (4.9)(here Z n = n !( α/ n is the same as in (4.3)). Fix n = 1 , , . . . and µ ∈ S n − . Write X λ : λ ց µ S λ (ˆ U n ∅ , λ )( ˆ D n λ, ∅ ) = X λ : λ ց µ (ˆ U n ∅ , λ ) = (ˆ U n ∅ , ˆ D ∗ µ ) = ( ˆ D ˆ U n ∅ , µ )= n − X k =0 (ˆ U n − k − H ˆ U k ∅ , µ ) = Z n Z n − (ˆ U n − ∅ , µ )= Z n Z n − · S µ (ˆ U n − ∅ , µ )( ˆ D n − µ, ∅ ) . The identity that we have obtained is clearly equivalent to the coherency condition(3.3) for the measures { M α,n } written in the form (4.9). Remark 4.7. The Kerov’s operators (4.2) with certain minor modifications fallinto the framework of the paper by Fulman [Ful09]. Namely, consider the opera-tors U n : CS n → CS n +1 and D n : CS n → CS n − defined by U n λ := n + α/ U λ and D n λ := D λ (where λ ∈ S n ). Then the operators U n and D n satisfy the commu-tation relations in the form of [Ful09, (1.1)]: D n +1 U n = a n U n − D n + b n I n , where I n : CS n → CS n is the identity operator and a n = 1 − n + α/ , b n = 1 + nn + α/ .Another similar operators were earlier considered by Stanley [Sta88] and Fomin[Fom94].4.2. Kerov’s operators and averages with respect to our point processes. The probability assigned to a strict partition λ by the measure M α,ξ (2.6) (whichis a mixture of the measures M α,n ) can be written for small enough ξ as follows: M α,ξ ( λ ) = (1 − ξ ) α/ ( e √ ξ U ∅ , λ )( e √ ξ D λ, ∅ ) . FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 19 Here e √ ξ D λ is clearly an element of ℓ ( S ). The fact that the vector e √ ξ U λ belongsto ℓ ( S ) (for small enough ξ ) requires a justification (see the proof of Proposition4.8), because the operator U in ℓ ( S ) is unbounded. This makes the above formulafor M α,ξ ( λ ) not very convenient for taking averages with respect to the measure M α,ξ . In this subsection we overcome this difficulty and give a convenient way ofwriting expectations with respect to M α,ξ . Our approach here is similar to that ofOlshanski [Ols08b] and is also based on the ideas of [Oko01b].Recall that the Kerov’s operators U , D , and H (4.2) define (via the map (4.4)) arepresentation of the complex Lie algebra sl (2 , C ) in the (complex) pre-Hilbert space ℓ ( S ). Consider the real form su (1 , ⊂ sl (2 , C ) spanned by the matrices U − D , i ( U + D ), and iH (here i = √− U − D , i ( U + D ),and i H act skew-symmetrically in ℓ ( S ). Now we prove that the representationof the Lie algebra su (1 , 1) can be lifted to a representation of a corresponding Liegroup: Proposition 4.8. All vectors of the space ℓ ( S ) are analytic for the described aboveaction of the Lie algebra su (1 , in ℓ ( S ) . Consequently, this action of su (1 , gives rise to a unitary representation of the universal covering group SU (1 , ∼ inthe Hilbert space ℓ ( S ) .Proof. Recall [Nel59] that a vector h is analytic for an operator A if the powerseries ∞ X n =0 k A n h k n ! s n in s has a positive radius of convergence.We can use Lemma 9.1 in [Nel59] that guarantees the existence of the desiredunitary representation of SU (1 , ∼ in ℓ ( S ) if we first prove that for some constant s > k A i . . . A i n h k ≤ n ! s n (4.10)for any h ∈ ℓ ( S ), all sufficiently large n (the bound on n depends on h ), andany indices i , . . . , i n taking values 1 , , 3, where A = U − D , A = i ( U + D ), and A = i H . Note that this in fact implies that any vector in ℓ ( S ) is analytic for theaction of su (1 , A := U , ˆ A := D , and ˆ A := H , thiscan only affect the value of the constant s . Moreover, we can consider only thecases when h = κ for an arbitrary κ ∈ S . Because all the matrix elements of theoperators U , D , and H are nonnegative in the standard basis { λ } λ ∈ S , we have k ˆ A i . . . ˆ A i n κ k ≤ k ( U + D + H ) n κ k . The desired estimate would follow if we show that the power series expansion ofexp ( s ( U + D + H )) κ converges for some small enough s > 0. For matrices in SL (2 , C ) (see (4.4)) we haveexp( s ( U + D + H )) = exp (cid:18) s − s U (cid:19) exp (cid:18) log (cid:18) − s (cid:19) H (cid:19) exp (cid:18) s − s D (cid:19) . Static correlation functions are readily expressed as averages with respect to M α,ξ (see (7.4)below), so we need good tools for computing such averages. Thus, the power series expansion of exp ( s ( U + D + H )) κ is the same as that ofexp (cid:18) s − s U (cid:19) exp (cid:18) log (cid:18) − s (cid:19) H (cid:19) exp (cid:18) s − s D (cid:19) κ . Since the operator D is locally nilpotent and the operator H acts on each λ asmultiplication by (2 | λ | + α/ X ∞ n =0 k U n µ k s n n !converges for all µ ∈ S for sufficiently small s > s must not dependon µ ). Let us fix µ with | µ | = k . We can write by definition of U : k U n µ k = X λ ∈ S k + n ( U n µ, λ ) = X λ ∈ S k + n dim S ( µ, λ ) Y (cid:3) ∈ λ/µ q α ( (cid:3) ) , where q α is defined by (4.6). Here the product is taken over all boxes of the skewshifted diagram λ/µ (see the end of § S ( µ, λ ) ≤ dim S λ , we canestimate k U n µ k ≤ (cid:16) Y (cid:3) ∈ µ q α ( (cid:3) ) − (cid:17) · X λ ∈ S k + n (dim S λ ) Y (cid:3) ∈ λ q α ( (cid:3) ) = (cid:16) Y (cid:3) ∈ µ q α ( (cid:3) ) − (cid:17) · X λ ∈ S n + k ( U n ∅ , λ ) = Z n + k · (cid:16) Y (cid:3) ∈ µ q α ( (cid:3) ) − (cid:17) . The factor Q (cid:3) ∈ µ q α ( (cid:3) ) − is just a constant depending on µ , and the normalizingconstants Z n = n !( α/ n were computed in the previous subsection. Putting alltogether, we get ∞ X n =0 s n n ! k U n µ k ≤ (cid:16) Y (cid:3) ∈ µ q α ( (cid:3) ) − (cid:17) · ∞ X n =0 s n n ! p ( n + k )!( α/ n + k . (4.11)Using [Erd53, 1.18.(5)], we see that p ( n + k )!( α/ n + k n ! ∼ s n k + α/ − Γ( α/ , so the series (4.11) converges for small enough s > 0. This concludes the proof ofthe proposition. (cid:3) To formulate the central statement of this section, we need some preparation.By G ξ denote the matrix G ξ := √ − ξ √ ξ √ − ξ √ ξ √ − ξ √ − ξ = (cid:18) √ ξ − √ ξ (cid:19) U − D ∈ SU (1 , , ≤ ξ < . (4.12)Clearly, ( G ξ ) ≤ ξ< is a continuous curve in SU (1 , 1) starting at the unity. By( e G ξ ) ≤ ξ< denote the lifting of this curve to SU (1 , ∼ , again starting at the unity.The unitary operators in ℓ ( S ) corresponding (by Proposition 4.8) to e G ξ are denotedby e G ξ .The next thing we need is the weighted ℓ space ℓ ( S , M α,ξ ) — the space offunctions on S that are square summable with the weight M α,ξ . This is a Hilbertspace with the inner product( f, g ) M α,ξ := X λ ∈ S f ( λ ) g ( λ ) M α,ξ ( λ ) . FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 21 There is an isometry map I α,ξ from ℓ ( S , M α,ξ ) to ℓ ( S ): I α,ξ := multiplication of f ∈ ℓ ( S , M α,ξ ) by the function λ p M α,ξ ( λ ) . (4.13)The standard orthonormal basis { λ } λ ∈ S (4.1) of the space ℓ ( S ) corresponds tothe orthonormal basis (cid:8) ( M α,ξ ( λ )) − λ (cid:9) λ ∈ S of ℓ ( S , M α,ξ ). To any operator A in ℓ ( S , M α,ξ ) corresponds the operator I α,ξ AI − α,ξ acting in ℓ ( S ).Now we can formulate and prove the main statement of this section: Proposition 4.9. Let A be a bounded operator in ℓ ( S , M α,ξ ) . Then ( A , ) M α,ξ = (cid:0)e G − ξ ( I α,ξ AI − α,ξ ) e G ξ ∅ , ∅ (cid:1) . (4.14) Here ∈ ℓ ( S , M α,ξ ) is the constant identity function. On the left the inner productis in ℓ ( S , M α,ξ ) , while on the right it is taken in ℓ ( S ) .Proof. Let us first show that e G ξ ∅ = X λ ∈ S ( M α,ξ ( λ )) λ. (4.15)In the matrix group SL (2 , C ) we have G ξ = exp (cid:16)p ξU (cid:17) exp (cid:18) 12 log(1 − ξ ) H (cid:19) exp (cid:16) − p ξD (cid:17) . The vector ∅ ∈ ℓ ( S ) is analytic for the action of su (1 , 1) (Proposition 4.8), so onthis vector the representation of SU (1 , ∼ can be extended to a representation ofthe local complexification of the group SU (1 , ∼ (see, e.g., the beginning of § ξ (when e G ξ is close to the unity of thegroup SU (1 , ∼ ) we have e G ξ ∅ = exp (cid:16)p ξ U (cid:17) exp (cid:18) 12 log(1 − ξ ) H (cid:19) exp (cid:16) − p ξ D (cid:17) ∅ . The operator e −√ ξ D preserves ∅ , and thus e G ξ ∅ = (1 − ξ ) α/ X λ ∈ S ξ | λ | | λ | ! dim S λ · (cid:16)Y (cid:3) ∈ λ q α ( λ ) (cid:17) λ = X λ ∈ S ( M α,ξ ( λ )) λ. We have established (4.15) for small ξ . The left-hand side of (4.15) is analytic in ξ because ∅ is an analytic vector for the operator e G ξ by Proposition 4.8. Theright-hand side of (4.15) is also analytic in ξ by definition of M α,ξ , see § ξ ∈ (0 , I − α,ξ e G ξ ∅ = ∈ ℓ ( S , M α,ξ ), see (4.13). Therefore,( e G − ξ I α,ξ AI − α,ξ e G ξ ∅ , ∅ ) = ( e G − ξ I α,ξ ( A ) , ∅ ) = ( I α,ξ ( A ) , e G ξ ∅ ) , because the operator e G ξ is unitary and has real matrix elements. We have( I α,ξ ( A ) , e G ξ ∅ ) = (cid:18) I α,ξ ( A ) , X λ ∈ S ( M α,ξ ( λ )) λ (cid:19) = X λ ∈ S ( I α,ξ ( A ) , λ ) ( M α,ξ ( λ )) = X λ ∈ S ( I α,ξ ( A ) , I α,ξ ( λ )) = (cid:16) A , X λ ∈ S λ (cid:17) M α,ξ = ( A , ) M α,ξ . This concludes the proof. (cid:3) Throughout the paper, when speaking about analytic functions in ξ , we assume that ξ liesin the unit open disc { z ∈ C : | z | < } . Remark 4.10. The left-hand side of (4.14) can be regarded as an expectationwith respect to the measure M α,ξ of the function ( A )( · ) on S . In the special casewhen the operator A is diagonal, say, A = A f is the multiplication by a (bounded)function f ( · ) on S , (4.14) is rewritten as the following formula for an expectation: E α,ξ f := X λ ∈ S f ( λ ) M α,ξ ( λ ) = (cid:0)e G − ξ A f e G ξ ∅ , ∅ (cid:1) . (4.16)This case is used in the computation of the static correlation functions, and forthe dynamical correlation functions we need to use the more general statement ofProposition 4.9. 5. Fermionic Fock space In this section we realize the Hilbert space ℓ ( S ) as a fermionic Fock space over ℓ ( Z > ), and also define a representation of a Clifford algebra in this Fock space.This Clifford algebra is an infinite-dimensional analogue of a Clifford algebra overan odd-dimensional space (similar Clifford algebras and their Fock representationswere considered in, e.g., [DJKM82], [Mat05], [Vul07]). Note that in the case of the z -measures [Oko01b] one should work with an analogue of a Clifford algebra over aneven-dimensional space. This difference, in particular, leads to the fact that in ourcase a priori the use of this algebra provides us only with a Pfaffian formula for thecorrelation functions of the point processes M α,ξ (the static case). The proof that M α,ξ is actually a determinantal process requires additional considerations (see § Wick’s theorem. We begin with the definition of a certain Clifford algebraover the Hilbert space V := ℓ ( Z ). Denote the standard orthonormal basis of thespace V by { v x } x ∈ Z . Define a symmetric bilinear form h· , ·i on V by h v x , v y i := , if x = − y = 0;2 , if x = y = 0;0 , otherwise . Let V + and V − be the spans of { v x } x ∈ Z > and { v x } x ∈ Z < , respectively, and let V denote the space C v . Note that the spaces V + and V − are maximal isotropicsubspaces for the form h· , ·i , and V = V − ⊕ V ⊕ V + . By Cl ( V ) denote the Clifford algebra over the quadratic space ( V, h· , ·i ), thatis, Cl ( V ) is the quotient of the tensor algebra L ∞ n =0 V ⊗ n of the space V by thetwo-sided ideal generated by the elements { v ⊗ v ′ + v ′ ⊗ v − h v, v ′ i : v, v ′ ∈ V } . The tensor product of v and v ′ in Cl ( V ) is denoted simply by vv ′ . Thus, vv ′ + v ′ v = h v, v ′ i for all v, v ′ ∈ V . (5.1)Now let us prove a version of Wick’s theorem that allows to write certain func-tionals on Cl ( V ) as Pfaffians. (In § Cl ( V ) calledthe vacuum average to which this version of Wick’s theorem is applicable.) FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 23 Theorem 5.1. Let F be a linear functional on Cl ( V ) such that F (1) = 1 and forany p, q, r ∈ Z ≥ , f +1 , . . . , f + p ∈ V + , and f − , . . . , f − q ∈ V − , we have F ( f +1 . . . f + p v r f − . . . f − q ) = 0 (5.2) if at least one of the numbers p, q is nonzero.Then for any n ≥ and any n elements f , . . . , f n ∈ V we have F ( f . . . f n ) = Pf( F J f , . . . , f n K ) , where F J f , . . . , f n K is the skew-symmetric n × n matrix in which the kj -th entryabove the main diagonal is F ( f k f j ) , ≤ k < j ≤ n .Proof. Step 1. Consider decompositions f j = f − j + f j + f + j , j = 1 , . . . , n, where f ± j ∈ V ± and f j ∈ V = C v . Thus, F ( f . . . f n ) = X s ,...,s n F ( f s . . . f s n n ) , where each s j is a sign, s j ∈ {− , , + } , and the sum is taken over all 3 n possiblesequences of signs. Step 2. Fix any particular sequence of signs ( s , . . . , s n ). Consider first thecase when all of the s j ’s are nonzero. We aim to prove that F ( f s . . . f s n n ) = Pf( F J f s , . . . , f s n n K ) , (5.3)where F J f s , . . . , f s n n K is the 2 n × n skew-symmetric matrix in which the kj thentry above the main diagonal is F ( f s k k f s j j ).First, note that if in the sequence ( s , . . . , s n ) all the “+” signs are on the leftand all the “ − ” signs are on the right, then by (5.2) we get (5.3), because in thePfaffian in the right-hand side of (5.3) each entry is zero.Next, observe that (5.3) is equivalent to F ( f s . . . f s n n ) = X n − k =1 ( − k +1 F ( f s . . . d f s k k . . . f s n − n − ) F ( f s k k f s n n ) , (5.4)this is just the standard Pfaffian expansion (here d f s k k means the absence of f s k k ). Itcan be readily verified that the right-hand side and the left-hand side of (5.4) varyin the same way under the interchange f s r r ↔ f s r +1 r +1 for any r = 1 , . . . , n − 1. Thisimplies that (5.4) holds because one can always move the “+” signs to the left andthe “ − ” signs to the right. This argument is similar to the proof of Lemma 2.3 in[Vul07]. Step 3. Now assume that among the sequence of signs ( s , . . . , s n ) there canbe zeroes. It is not hard to see that both sides of (5.3) vanish unless the numberof zeroes is even. Let the positions of zeroes be j < · · · < j k . Thus, moving all f j , . . . , f j k to the left, we have F ( f s . . . f s n n ) = ( − k P m =1 ( j m − m ) F ( f j . . . f j k ) F ( f s . . . c f j . . . d f j k . . . f s n n ) . (5.5)By (5.3), the factor F ( f s . . . c f j . . . d f j k . . . f s n n ) is written as the correspondingPfaffian of order (2 n − k ). Assume that f j m = c m v (where m = 1 , . . . , k ), then F ( f j . . . f j k ) = c . . . c k . Including the case when there are only “+” or only “ − ” signs. Since for any f ∈ V + ⊕ V − we have (using (5.2)) F ( v f ) = F ( f v ) = 0, theright-hand side of (5.5) can be interpreted as the Pfaffian of the block 2 n × n matrix with blocks formed by rows and columns with numbers j , . . . , j k and { , . . . , n } \ { j , . . . , j k } , respectively. This skew-symmetric 2 n × n matrix isexactly F J f s , . . . , f s n n K for our sequence ( s , . . . , s n ).This implies that (5.3) holds for any choice of signs ( s , . . . , s n ), s j ∈ {− , , + } . Step 4. Let us now deduce the claim of the theorem from (5.3). We must provethat X s ,...,s n Pf( F J f s , . . . , f s n n K ) = Pf( F J f , . . . , f n K ) . This is done by induction on n . The base is n = 1: F ( f − f +2 ) + F ( f f ) = F ( f f )(all other combinations of signs in the left-hand side give zero contribution). Theinduction step is readily verified using the Pfaffian expansion (5.4). This concludesthe proof of the theorem. (cid:3) Fermionic Fock space. Consider the space ℓ ( Z > ) with the standard or-thonormal basis { ε k } k ∈ Z > . The exterior algebra ∧ ℓ ( Z > ) is the vector space withthe basis { vac } ∪ { ε i ∧ · · · ∧ ε i ℓ : ∞ > i > . . . > i ℓ ≥ , ℓ = 1 , , . . . } , (5.6)where vac ≡ vacuum vector . Define an inner product ( · , · ) in theexterior algebra ∧ ℓ ( Z > ) with respect to which the basis (5.6) is orthonormal.This inner product turns ∧ ℓ ( Z > ) into a pre-Hilbert space. Its Hilbert comple-tion is called the ( fermionic ) Fock space and is denoted by Fock ( Z > ). The space( ∧ ℓ ( Z > ) , ( · , · )) consisting of finite linear combinations of the basis vectors (5.6) isdenoted by Fock fin ( Z > ).Clearly, the map λ ε λ ∧ · · · ∧ ε λ ℓ ( λ ) , λ ∈ S (in particular, ∅ maps to vac ) defines an isometry between the pre-Hilbert spaces ℓ ( S ) and Fock fin ( Z > ), and also between their Hilbert completions ℓ ( S ) and Fock ( Z > ). Below we identify ℓ ( S ) and Fock ( Z > ), and by λ we mean the vector ε λ ∧ · · · ∧ ε λ ℓ ( λ ) .In the next subsection we describe the structure of Fock ( Z > ) in more detail.5.3. Creation and annihilation operators. Vacuum average. Let φ k , k =1 , , . . . , be the creation operators in Fock ( Z > ), that is, φ k λ := ε k ∧ λ, λ ∈ S . Let φ ∗ k , k = 1 , , . . . , be the operators that are adjoint to φ k with respect to theinner product in Fock ( Z > ). They are called the annihilation operators and act asfollows: φ ∗ k λ = X ℓ ( λ ) j =1 ( − j +1 δ k,λ j · ε λ ∧ · · · ∧ c ε λ j ∧ · · · ∧ ε λ ℓ ( λ ) . We also need the operator φ = φ ∗ acting as φ λ := ( − ℓ ( λ ) λ. FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 25 To simplify certain formulas below, we organize the operators φ k , φ and φ ∗ k intoa single family: φ m := (cid:26) φ m , if m ≥ − m φ ∗− m , otherwise , where m ∈ Z . It can be readily checked that the operators φ m satisfy the following anti-commu-tation relations: φ k φ l + φ l φ k = (cid:26) , if k = l = 0;( − l δ k, − l , otherwise . (5.7)In agreement to these definitions, let { v x } x ∈ Z be another orthonormal basis inthe space V = ℓ ( Z ) defined as v x := (cid:26) v x , if x ≥ − x v x , if x < , where x ∈ Z . (5.8)In other words, v x = ( − x ∧ v x . In the Clifford algebra Cl ( V ) we have v x v y + v y v x = h v x , v y i = (cid:26) , x = y = 0;( − x δ x, − y , otherwise . (5.9) Definition 5.2. Let T be a representation of the Clifford algebra Cl ( V ) in Fock ( Z > )defined on V by T ( v x ) := φ x , x ∈ Z , and extended to the whole Cl ( V ) by (5.1) and by linearity. The fact that T isindeed a representation follows from (5.7) and (5.9). Definition 5.3. The representation T allows to consider the following functionalon the Clifford algebra Cl ( V ): F vac ( w ) := ( T ( w ) vac , vac ) , w ∈ Cl ( V )called the vacuum average . Here the inner product on the right is taken in Fock ( Z > ).It can be readily verified that the functional F vac on Cl ( V ) satisfies the hypothe-ses of Wick’s Theorem 5.1.5.4. The representation R . The space ℓ ( S ) is isometric to Fock fin ( Z > ), andthus the Kerov’s operators U , D , and H (4.2) in ℓ ( S ) give rise to certain operatorsin Fock fin ( Z > ). We obtain a representation of the Lie algebra sl (2 , C ) in Fock ( Z > ),denote this representation by R .It can be readily verified that the action of the operators R ( U ), R ( D ), and R ( H )in Fock fin ( Z > ) (this subspace of Fock ( Z > ) is invariant for the representation R of sl (2 , C )) can be expressed in terms of the creation and annihilation operators asfollows: R ( U ) = X ∞ k =0 − δ ( k ) / ( − k p k ( k + 1) + α · φ k +1 φ − k ,R ( D ) = X ∞ k =0 − δ ( k ) / ( − k +1 p k ( k + 1) + α · φ k φ − k − , (5.10) R ( H ) = α + 2 X ∞ k =1 ( − k k φ k φ − k . Proposition 4.8 can be reformulated for the representation R . Namely, the rep-resentation R of sl (2 , C ) restricted to the real form su (1 , ⊂ sl (2 , C ) gives rise toa unitary representation of the universal covering group SU (1 , ∼ in the Hilbertspace Fock ( Z > ). Denote this representation also by R . Under the identification of ℓ ( S ) with Fock ( Z > ), we say that the map I α,ξ (4.13) is an isometry between ℓ ( S , M α,ξ ) and Fock ( Z > ). By Proposition 4.9, forany bounded operator A in ℓ ( S , M α,ξ ) we have( A , ) M α,ξ = (cid:0) R ( e G ξ ) − ( I α,ξ AI − α,ξ ) R ( e G ξ ) vac , vac (cid:1) . (5.11)Here e G ξ ∈ SU (1 , ∼ , 0 ≤ ξ < § ∈ ℓ ( S , M α,ξ ) is theconstant identity function. The inner products on the left and on the right aretaken in the spaces ℓ ( S , M α,ξ ) and Fock ( Z > ), respectively.Formula (4.16) for the expectation of a bounded function f ( · ) on S with respectto the measure M α,ξ is rewritten as E α,ξ f = (cid:0) R ( e G ξ ) − A f R ( e G ξ ) vac , vac (cid:1) , (5.12)where A f is the operator of multiplication by f .As we will see below, for averages expressing the correlation functions, the right-hand side of (5.11) (and (5.12)) can be written as a vacuum average. That is, theoperator R ( e G ξ ) − ( I α,ξ AI − α,ξ ) R ( e G ξ ) (respectively, R ( e G ξ ) − A f R ( e G ξ )) has the form T ( w ) for a certain w ∈ Cl ( V ).6. Z-measures and an orthonormal basis in ℓ ( Z )In this section we examine functions on the lattice which are used in our expres-sions for correlation kernels (both static and dynamical). They form an orthonormalbasis in the Hilbert space ℓ ( Z ) and are eigenfunctions of a certain second orderdifference operator on the lattice. These functions arise as a particular case of thefunctions used to describe correlation kernels in the model of the z -measures onordinary partitions, and we begin this section by recalling some of the results ofthe papers [BO06b], [BO06a] which we will use below.6.1. Results about the z-measures on ordinary partitions. For an ordinary(i.e., not necessary strict) partition σ = ( σ , . . . , σ ℓ ( σ ) ), let dim σ denote the numberof standard Young tableaux of shape σ (we identify partitions with ordinary Youngdiagrams as usual, e.g., see [Mac95, Ch. I, § | σ | be the number of boxes inthe Young diagram σ .Consider the following 3-parameter family of measures on the set of all ordinarypartitions: M z,z ′ ,ξ ( σ ) = (1 − ξ ) zz ′ ξ | σ | ( z ) σ ( z ′ ) σ (cid:18) dim σ | σ | ! (cid:19) , (6.1)where ( a ) σ := Q ℓ ( σ ) i =1 ( a ) σ i is a generalization of the Pochhammer symbol. Herethe parameter ξ ∈ (0 , 1) is the same as our parameter ξ (e.g., in § z, z ′ are in one of the following two families (we call such parameters admissible ): • ( principal series ) The numbers z, z ′ are not real and are conjugate to each other. • ( complementary series ) Both z, z ′ are real and are contained in the same openinterval of the form ( m, m + 1), where m ∈ Z .To any ordinary partition σ = ( σ , . . . , σ ℓ ( σ ) , , , . . . ) is associated an infinitepoint configuration (sometimes called the Maya diagram ) on the lattice Z ′ = Z + : σ X ( σ ) := { σ i − i + } ∞ i =1 ⊂ Z ′ . (6.2) FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 27 One can see that the correspondence σ X ( σ ) is a bijection between ordinarypartitions and those (infinite) configurations X ⊂ Z ′ for which the symmetric dif-ference X △ Z ′− is a finite subset containing equally many points in Z ′ + and Z ′− (Here Z ′ + and Z ′− denote the sets of all positive resp. negative half-integers.)Using the above identification of ordinary partitions with point configurations onthe lattice Z ′ , it is possible to speak about the correlation functions of the measures M z,z ′ ,ξ (6.1) in the same way as in (2.9). The resulting random point processes aredeterminantal with a correlation kernel K z,z ′ ,ξ ( x ‘ , y ‘ ) (where x ‘ , y ‘ ∈ Z ′ ) which iscalled the discrete hypergeometric kernel [BO00], [BO06b]. Remark 6.1. Whenever speaking about points in the shifted lattice Z ′ = Z + ,we denote them by x ‘ , y ‘ , . . . , because we want to reserve the letters x, y, . . . for thenon-shifted integers: x, y, . . . ∈ Z .There are explicit formulas for the discrete hypergeometric kernel K z,z ′ ,ξ whichwe will use. We proceed to describe them, but first we need to recall certainfunctions defined in [BO06b, (2.1)]: ψ a ‘ ( x ‘ ; z, z ′ , ξ ) := (cid:18) Γ( x ‘ + z + )Γ( x ‘ + z ′ + )Γ( − a ‘ + z + )Γ( − a ‘ + z ′ + ) (cid:19) ξ ( x ‘ + a ‘ ) (1 − ξ ) ( z + z ′ ) − a ‘ ×× F ( − z + a ‘ + , − z ′ + a ‘ + ; x ‘ + a ‘ + 1; ξξ − )Γ( x ‘ + a ‘ + 1) , x ‘ ∈ Z ′ . (6.3)Here z, z ′ , ξ are the parameters of the z -measures, and the index a ‘ of the func-tions runs over the lattice Z ′ . As usual, F is the Gauss hypergeometric function, F ( A, B ; C ; w ) := P ∞ n =0 ( A ) n ( B ) n ( C ) n n ! w n . As is explained in [BO06b, § a ‘ , x ‘ ∈ Z ′ due to the assumptions on the parameters z, z ′ (see p. 26) and the fact that ξ ∈ (0 , ψ a ‘ ( x ‘ ; z, z ′ , ξ )are real-valued. Let us summarize their properties for future use: Proposition 6.2 ([BO06b, § . 1) The functions ψ a ‘ ( x ‘ ; z, z ′ , ξ ) , as the index a ‘ ranges over Z ′ , form an orthonormal basis in the Hilbert space ℓ ( Z ′ ) . Consider the following second order difference operator D ( z, z ′ , ξ ) in ℓ ( Z ′ ) (act-ing on functions f ( x ‘ ) , where x ‘ ranges over Z ′ ): D ( z, z ′ , ξ ) f ( x ‘ ) = q ξ ( z + x ‘ + )( z ′ + x ‘ + ) f ( x ‘ + 1)+ q ξ ( z + x ‘ − )( z ′ + x ‘ − ) f ( x ‘ − − (cid:0) x ‘ + ξ ( z + z ′ + x ‘ ) (cid:1) f ( x ‘ ) . The operator D ( z, z ′ , ξ ) is symmetric. The functions ψ a ‘ are eigenfunctions ofthis operator: D ( z, z ′ , ξ ) ψ a ‘ ( x ‘ ; z, z ′ , ξ ) = a ‘ (1 − ξ ) ψ a ‘ ( x ‘ ; z, z ′ , ξ ) , a ‘ , x ‘ ∈ Z ′ . The functions ψ a ‘ satisfy the following symmetry relations: ψ a ‘ ( x ‘ ; z, z ′ , ξ ) = ψ x ‘ ( a ‘ ; − z, − z ′ , ξ ); (6.4) ψ a ‘ ( x ‘ ; z, z ′ , ξ ) = ( − x ‘ + a ‘ ψ − a ‘ ( − x ‘ ; − z, − z ′ , ξ ) , a ‘ , x ‘ ∈ Z ′ . (6.5) The functions ψ a ‘ satisfy the following three-term relation ( a ‘ ∈ Z ′ ): (1 − ξ ) x ‘ ψ a ‘ = q ξ ( z − a ‘ + )( z ′ − a ‘ + ) ψ a ‘ − 18 LEONID PETROV + q ξ ( z − a ‘ − )( z ′ − a ‘ − ) ψ a ‘ +1 + ( − a ‘ + ξ ( z + z ′ − a ‘ )) ψ a ‘ . (6.6) Theorem 6.3 ([BO00], [BO06b]) . Under the correspondence σ X ( σ ) (6.2), the z -measures become a determinantal point process on Z ′ with the correlation kernelgiven by K z,z ′ ,ξ ( x ‘ , y ‘ ) = X a ‘ ∈ Z ′ + ψ a ‘ ( x ‘ ; z, z ′ , ξ ) ψ a ‘ ( y ‘ ; z, z ′ , ξ ) , x ‘ , y ‘ ∈ Z ′ . (6.7)From Proposition 6.2, one readily sees that the discrete hypergeometric ker-nel K z,z ′ ,ξ (viewed as an operator in ℓ ( Z ′ )) is an orthogonal spectral projection operator corresponding to the positive part of the spectrum of the difference op-erator D ( z, z ′ , ξ ). We will discuss the extended discrete hypergeometric kernel K z,z ′ ,ξ ( s, x ‘ ; t, y ‘ ) (which serves as a correlation kernel in a determinantal dynam-ical model associated to the z -measures) below in § An orthonormal basis { ϕ m } in the Hilbert space ℓ ( Z ) . For the studyof our model, we need the following family of functions: ϕ m ( x ; α, ξ ) := (cid:18) Γ( + ν ( α ) + x )Γ( − ν ( α ) + x )Γ( + ν ( α ) − m )Γ( − ν ( α ) − m ) (cid:19) ξ 12 ( x + m ) (1 − ξ ) − m ×× F ( + ν ( α ) + m, − ν ( α ) + m ; x + m + 1; ξξ − )Γ( x + m + 1) , (6.8)where ν ( α ) is given in Definition 2.1. Here the argument x and the index m rangeover the lattice Z . Because α > 0, we have Γ( + ν ( α ) + k )Γ( − ν ( α ) + k ) > k ∈ Z . Thus, the expression in (6.8) which is taken to the power is positive.Note also that while the hypergeometric function F ( A, B ; C ; w ) is not defined if C is a negative integer, the ratio F ( A,B ; C ; w )Γ( C ) (occurring in (6.8)) is well-definedfor all C ∈ C . Thus, we see that the functions ϕ m ( x ; α, ξ ) are well-defined.It can be readily verified that ϕ m ’s arise as a particular case of the functions ψ a ‘ : described in § ϕ m ( x ; α, ξ ) = ψ m + + d ( x − − d ; ν ( α ) + + d, − ν ( α ) + + d ; ξ ) (6.9)for any d ∈ Z . For x, m, d ∈ Z , the numbers m + + d and x − − d belong to Z ′ ,as it should be. Observe that the parameters z = z ( α ) := ν ( α ) + + d, z ′ = z ′ ( α ) := − ν ( α ) + + d for any d ∈ Z are admissible (i.e., of principal or complementary series, see p. 26).By Definition 2.1, for 0 < α ≤ these parameters belong to the complementaryseries, and for α > they are of principal series. Remark 6.4. The fact that (6.9) holds for any d is a reflection of a certain trans-lation invariance property of the z -measures, see [BO98, § 10, 11].From Proposition 6.2 one can readily deduce the corresponding properties of ourfunctions ϕ m : Proposition 6.5. 1) The functions ϕ m ( x ; α, ξ ) , as the index m ranges over Z ,form an orthonormal basis in the Hilbert space ℓ ( Z ) : X x ∈ Z ϕ m ( x ; α, ξ ) ϕ l ( x ; α, ξ ) = δ ml , m, l ∈ Z . FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 29 The functions ϕ m are eigenfunctions of the following second order differenceoperator in ℓ ( Z ) (acting on functions f ( x ) , where x ranges over Z ): D α,ξ f ( x ) = p ξ ( α + x ( x + 1)) f ( x + 1)+ p ξ ( α + x ( x − f ( x − − x (1 + ξ ) f ( x ) . This operator is symmetric in ℓ ( Z ) . We have D α,ξ ϕ m ( x ; α, ξ ) = m (1 − ξ ) ϕ m ( x ; α, ξ ) , m, x ∈ Z . The functions ϕ m satisfy the following symmetry relations: ϕ m ( x ; α, ξ ) = ϕ x ( m ; α, ξ ); (6.10) ϕ m ( x ; α, ξ ) = ( − x + m ϕ − m ( − x ; α, ξ ) , x, m ∈ Z . (6.11) The functions ϕ m satisfy the three-term relation: (1 − ξ ) x ϕ m = p ξ ( m ( m + 1) + α ) ϕ m − (6.12)+ p ξ ( m ( m − 1) + α ) ϕ m +1 − m (1 + ξ ) ϕ m , m ∈ Z . Note that the property (6.10) here means that the functions ϕ m ( x ; α, ξ ) areself-dual (in contrast to the more general functions ψ a ‘ ( x ‘ ; z, z ′ , ξ ), cf. (6.4)). Proof. Every property is a straightforward consequence of (6.9) and the correspond-ing claim of Proposition 6.2. (cid:3) “Twisting”. To simplify certain formulas in the paper (in particular, (2.13)above), we will also need certain versions of our functions ϕ m ( x ; α, ξ ) which differfrom the original ones by multiplying by ( − x ∧ : e ϕ m ( x ; α, ξ ) := ( − x ∧ ϕ m ( x ; α, ξ ) , x, m ∈ Z . (6.13)These functions also form an orthonormal basis in ℓ ( Z ). They are eigenfunctionsof a difference operator e D α,ξ in ℓ ( Z ) which is conjugate to D α,ξ :( e D α,ξ f )( x ) := ( − x< p ξ ( α + x ( x + 1)) f ( x + 1) (6.14)+( − x ≤ p ξ ( α + x ( x − f ( x − − x (1 + ξ ) f ( x ) , x ∈ Z (here means the indicator), e D α,ξ e ϕ m ( x ; α, ξ ) = m (1 − ξ ) e ϕ m ( x ; α, ξ ) , m ∈ Z . The functions { e ϕ m } m ∈ Z also satisfy certain symmetry relations similar to (6.10)–(6.11). Moreover, they clearly satisfy the same three-term relations (6.12) as thenon-twisted functions ϕ m ( x ; α, ξ ).6.4. Matrix elements of sl (2 , C ) -modules. Here we interpret the functions { ϕ m } (6.8) introduced in this section through certain matrix elements of irreducible uni-tary representations of the Lie group P SU (1 , 1) = SU (1 , / {± I } ( I is the identitymatrix) in the Hilbert space ℓ ( Z ). Remark 6.6. The more general functions ψ a ‘ (6.3) first appeared in the works ofVilenkin and Klimyk [VK88], [VK95] as matrix elements of unitary representationsof the universal covering group SU (1 , ∼ . In a context similar to ours they wereobtained by Okounkov [Oko01b] in a computation of the discrete hypergeometrickernel K z,z ′ ,ξ (6.7) using the fermionic Fock space. Let S be the representation of the Lie algebra sl (2 , C ) (spanned by the operators U, D , and H (4.4)) in the Hilbert space ℓ ( Z ) with the canonical orthonormal basis { k } k ∈ Z (that is, k ( x ) = δ k,x ) defined as follows: S ( U ) k = p k ( k + 1) + α · k + 1; S ( D ) k = p k ( k − 1) + α · k − 1; (6.15) S ( H ) k = 2 k · k. This representation depends on our parameter α > 0. For it one can prove ananalogue of Proposition 4.8: Proposition 6.7. All vectors of the space ℓ ( Z ) (consisting of finite linear combi-nations of the basis vectors { k } ) are analytic for the action S of sl (2 , C ) (6.15). Therepresentation S of the Lie algebra su (1 , ⊂ sl (2 , C ) in ℓ ( Z ) lifts to a unitaryrepresentation of the Lie group P SU (1 , in the Hilbert space ℓ ( Z ) .Proof. The operators S ( U ) − S ( D ), i (cid:0) S ( U ) + S ( D ) (cid:1) , and iS ( H ) act skew-symmet-rically in the (complex) pre-Hilbert space ℓ ( Z ). It is known (e.g., see [Puk64]or [Lan85, Ch. VI, § α > S of su (1 , ℓ ( Z ) is irreducible (this is an irreducible Harish–Chandra module) and liftsto a unitary representation of the Lie group SU (1 , 1) in ℓ ( Z ). Moreover, since S ( H ) k = 2 k · k , this is in fact a representation of the group P SU (1 , § 3, Thm. 7]. (cid:3) If 0 < α ≤ , the above irreducible representation of P SU (1 , 1) in ℓ ( Z ) belongsto complementary series, and for α > it is of principal series, e.g., see [Puk64] (cf.the series of the parameters ( z, z ′ ) in (6.9)). Denote this representation of P SU (1 , S . For notational reasons (e.g., see Proposition 6.8 below), also by S let usdenote the corresponding representations of SU (1 , 1) and SU (1 , ∼ in ℓ ( Z ) thatare obtained from the representation of P SU (1 , 1) by a trivial lifting procedure.Now let us compute the matrix elements of the operator S ( G ξ ) − (where G ξ ∈ SU (1 , 1) is defined in (4.12)) in the basis { k } k ∈ Z . These matrix elements will beused below in formulas for our correlation kernels. Proposition 6.8. For all x, k ∈ Z we have (cid:0) S ( G ξ ) − x, k (cid:1) ℓ ( Z ) = ϕ − k ( x ; α, ξ ) . (6.16) Proof. Fix x, k ∈ Z . By Proposition 6.7, the function ξ (cid:0) S ( G ξ ) − x, k (cid:1) ℓ ( Z ) isanalytic. The right-hand side of (6.16) is also analytic in ξ , see (6.8). Thus, itsuffices to prove (6.16) for small ξ . Also by Proposition 6.7, on x ∈ ℓ ( Z ) therepresentation S can be extended to a representation of the local complexificationof P SU (1 , ξ (when G ξ is close to the unity of thegroup) we can write: S ( G ξ ) − x = exp (cid:16) − p ξS ( U ) (cid:17) exp (cid:18) √ ξ − ξ S ( D ) (cid:19) exp (cid:18) 12 log(1 − ξ ) S ( H ) (cid:19) x (this follows from the corresponding identity for matrices in SL (2 , C ), see also theproof of Proposition 4.9).Denote c y := p y ( y + 1) + α , so that S ( U ) y = c y · y + 1 and S ( D ) y = c y − · y − c y = y ( y + 1) + α = ( y + ν ( α ) + )( y − ν ( α ) + ). Also set a := −√ ξ FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 31 and b := √ ξ/ (1 − ξ ). We have (cid:0) S ( G ξ ) − x, k (cid:1) ℓ ( Z ) = (1 − ξ ) x (cid:16) e aS ( U ) e bS ( D ) x, k (cid:17) ℓ ( Z ) = (1 − ξ ) x ∞ X r =0 ∞ X l =0 a r b l r ! l ! c x − l + r − . . . c x − l c x − l . . . c x − ( x − l + r, k ) ℓ ( Z ) . Clearly, ( x − l + r, k ) ℓ ( Z ) = δ x − l + r,k . There are two cases: x ≥ k , and x ≤ k . For x ≥ k we perform the above summation over r ≥ l = r + x − k . For x ≤ k we sum over l and set r = l + k − x . After direct calculations we obtain (we omitthe argument in ν ( α )): (cid:0) S ( G ξ ) − x, k (cid:1) ℓ ( Z ) = (1 − ξ ) x b x − k (cid:18) Γ( x + ν + )Γ( x − ν + )Γ( k + ν + )Γ( k − ν + ) (cid:19) ×× F ( − k − ν, − k + ν ; x − k + 1; ab )( x − k )! , if x ≥ k ; (cid:0) S ( G ξ ) − x, k (cid:1) ℓ ( Z ) = (1 − ξ ) x a k − x (cid:18) Γ( k + ν + )Γ( k − ν + )Γ( x + ν + )Γ( x − ν + ) (cid:19) ×× F ( − x − ν, − x + ν ; k − x + 1; ab )( k − x )! , if x ≤ k. It is known that the expression F ( A,B ; C ; w )Γ( C ) is well-defined for all C ∈ C , and by[Erd53, 2.1.(3)] we see that F ( A, B ; n + 1; w )Γ( n + 1) = F ( A − n, B − n ; − n + 1; w )( A − n ) n ( B − n ) n Γ( − n + 1) w − n , n = 1 , , . . . . Let us apply this identity in the case x ≤ k above: F ( − x − ν, − x + ν ; k − x + 1; ab )( k − x )!= F ( − k − ν, − k + ν ; x − k + 1; ab )Γ( x − k + 1) · Γ( + x − ν )Γ( + x + ν )Γ( + k − ν )Γ( + k + ν ) ( ab ) x − k (we have also used the fact that ( − m Γ( + m ± ν ) = Γ( + ν )Γ( − ν )Γ( ∓ ν − m ) for m ∈ Z , see(2.5)). Thus, we get the desired result (6.16) for small ξ , and hence for all ξ ∈ (0 , (cid:3) Remark 6.9. The operator D α,ξ acting in ℓ ( Z ), can be written through theoperators of the representation S of sl (2 , C ) as follows:(1 − ξ ) − D α,ξ = √ ξ − ξ ( S ( U ) + S ( D )) − 12 1 + ξ − ξ S ( H ) = − S ( H ξ ) , where H ξ := 12 G ξ HG − ξ ∈ sl (2 , C ) . Indeed, this is verified by a simple matrix computation (see (4.12)):12 G ξ HG − ξ = 12(1 − ξ ) (cid:20) √ ξ √ ξ (cid:21) (cid:20) − (cid:21) (cid:20) −√ ξ −√ ξ (cid:21) = 12 1+ ξ − ξ − √ ξ − ξ √ ξ − ξ − 12 1+ ξ − ξ . Operators corresponding to the matrix H ξ under other representations of sl (2 , C )appear in our model twice more, see § Connection with Meixner and Krawtchouk polynomials. The z -me-asures M z,z ′ ,ξ ( § ξ ∈ (0 , 1) and ( z, z ′ ) of principal or complementary series(see p. 26) are supported by the set of all ordinary partitions. As is known (e.g.,see [BO06b]), the z -measures admit two degenerate series of parameters: • ( first degenerate series ) ξ ∈ (0 , z and z ′ (say, z ) is anonzero integer while z ′ has the same sign and, moreover, | z ′ | > | z | − z = N = 1 , , . . . , then the measure M z,z ′ ,ξ ( σ ) vanishes unless ℓ ( σ ) ≤ N . Likewise, if z = − N , M z,z ′ ,ξ ( σ ) = 0 if ℓ ( σ ′ ) = σ exceeds N ( σ ′ denotes thetransposed Young diagram). • ( second degenerate series ) ξ < 0, and z = N and z ′ = − N ′ , where N and N ′ arepositive integers.In this case, the measure M z,z ′ ,ξ is supported by the (finite) set of all ordinaryYoung diagrams which are contained in the rectangle N × N ′ (that is, ℓ ( σ ) ≤ N and ℓ ( σ ′ ) ≤ N ′ ).As is explained in the paper [BO06b], in the first degenerate series the functions ψ a ‘ ( x ‘ ; z, z ′ , ξ ) are expressed through the classical Meixner orthogonal polynomials(about their definition, e.g., see [KS96, § § M α,ξ on strict partitions there exists only one degenerate seriesof parameters: α = − N ( N + 1) for some N = 1 , , . . . , and ξ < M α,ξ is supported by the set of all shifted Young diagramswhich are contained inside the staircase shifted shape ( N, N − , . . . , z -measures, and our functions ϕ m are expressed through the Krawtchouk orthogonal polynomials.The measures M α,ξ in this case are interpreted as random point processes onthe finite lattice { , . . . , N } , and one could also define a suitable dynamics for thesemeasures as is done below in § 9. The results of the present paper about the structureof the static and dynamical correlation kernels also hold for the degenerate model,and the Krawtchouk polynomials enter formulas for these correlation kernels.7. Static correlation functions In this section we obtain a Pfaffian formula for the correlation functions of thepoint process M α,ξ , and discuss the resulting Pfaffian kernel.7.1. Pfaffian formula. Recall that by Z =0 we denote the set of all nonzero inte-gers. For x , . . . , x n ∈ Z > we put, by definition, x − k := − x k , k = 1 , . . . , n. (7.1)We use this convention in the formulation of the next theorem. Let the function Φ α,ξ on Z =0 × Z =0 be defined by (see § Φ α,ξ ( x, y ) := ( − x ∧ y ∧ (cid:16) R ( e G ξ ) − φ x φ y R ( e G ξ ) vac , vac (cid:17) , (7.2)where the inner product is taken in Fock ( Z > ). (For now Φ α,ξ ( x, y ) is defined for x, y ∈ Z =0 , but in § Φ α,ξ ( x, y ) to zero values of x, y in a natural way. See also Remark 2.3.1.) In this subsection we prove the following: FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 33 Theorem 7.1. The correlation functions ρ ( n ) α,ξ (2.9) of the measures M α,ξ (2.6) aregiven by the following Pfaffian formula: ρ ( n ) α,ξ ( x , . . . , x n ) = Pf( ˆ Φ α,ξ J X K ) , (7.3) where X = { x , . . . , x n } ⊂ Z > (here x j ’s are distinct), and ˆ Φ α,ξ J X K is theskew-symmetric n × n matrix with rows and columns indexed by the numbers , , . . . , n, − n, . . . , − , − , and the kj -th entry in ˆ Φ α,ξ J X K above the main diago-nal is Φ α,ξ ( x k , x j ) , where k, j = 1 , . . . , n, − n, . . . , − . Below in (7.16) we write Φ α,ξ ( x, y ) in terms of the Gauss hypergeometric func-tion. For this reason, we call Φ α,ξ the Pfaffian hypergeometric-type kernel . Anotherform of a 2 n × n skew-symmetric matrix (constructed using the kernel Φ α,ξ ( x, y ))which can be put in the right-hand side of (7.3) is discussed below in § Φ α,ξ J X K is most useful when rewriting the Pfaffian in (7.3) as a deter-minant, see Theorem 8.1 and Proposition A.2 from Appendix.The rest of this subsection is devoted to proving Theorem 7.1. Consider thefollowing operators in ℓ ( S , M α,ξ ):∆ x λ := (cid:26) λ, if x ∈ λ ;0 , otherwise , x ∈ Z > . Fix a finite subset X = { x , . . . , x n } ⊂ Z > and set ∆ J X K := ∆ x . . . ∆ x n . This is adiagonal operator of multiplication by a function which is the indicator of the event { λ : λ ⊇ X } . We view ∆ J X K as an operator acting in ℓ ( S , M α,ξ ). Since this operatoris diagonal, it does not change under the isometry I α,ξ : ℓ ( S , M α,ξ ) → Fock ( Z > )(4.13). Thus, ∆ J X K also acts in Fock ( Z > ) (in the same way).The correlation functions ρ ( n ) α,ξ (2.9) of the measures M α,ξ (2.6) clearly have theform ρ ( n )( α,ξ ) ( x , . . . , x n ) = M α,ξ ( λ : λ ⊇ { x , . . . , x n } ) = (∆ J X K , ) M α,ξ , (7.4)where ∈ ℓ ( S , M α,ξ ) is the constant identity function. Using (5.11) (or (5.12)),we can rewrite the correlation functions as ρ ( n )( α,ξ ) ( x , . . . , x n ) = (cid:16) R ( e G ξ ) − ∆ J X K R ( e G ξ ) vac , vac (cid:17) . (7.5)In this formula the operator ∆ J X K acts in Fock ( Z > ). Clearly, ∆ J X K is expressedthrough the creation and annihilation operators in the Fock space Fock ( Z > ) as∆ J X K = Y nk =1 φ x k φ ∗ x k . It is more convenient for us to rewrite ∆ J X K using the anti-commutation relationsfor φ x and φ ∗ x (see (5.7)) as follows:∆ J X K = φ x . . . φ x n φ ∗ x n . . . φ ∗ x (7.6)(after moving all the φ k ’s to the left and φ ∗ k ’s to the right there is no change ofsign).Our next step is to write (7.5) with ∆ J X K given by (7.6) as the vacuum averagefunctional F vac applied to a certain element of the Clifford algebra Cl ( V ) ( § Theorem 7.1 is the same as Proposition 2 in [Pet10a], the only difference is that in [Pet10a]the factor ( − P nk =1 x k is put in front of the Pfaffian, and thus in the definition of the Pfaffiankernel in [Pet10a] there is no factor of the form ( − x ∧ y ∧ . Recall that in § T of Cl ( V ) in Fock ( Z > )such that T ( v x ) = φ x , x ∈ Z , where { v x } x ∈ Z is the basis of V = ℓ ( Z ) definedby (5.8). Using the anti-commutation relations (5.7), one can readily compute thecommutators between the operators T ( v x ) and the operators of the representation R (5.10):[ R ( U ) , T ( v x )] = 2 ( δ ( x ) − δ ( x +1)) / p x ( x + 1) + α · T ( v x +1 );[ R ( D ) , T ( v x )] = 2 ( δ ( x ) − δ ( x − / p x ( x − 1) + α · T ( v x − );[ R ( H ) , T ( v x )] = 2 x · T ( v x ) , x ∈ Z . These formulas motivate the following definition: Definition 7.2. Let ˇ S be the representation of the Lie algebra sl (2 , C ) in the (pre-Hilbert) space V fin (consisting of of all finite linear combinations of the basis vectors { v x } x ∈ Z ) defined as:ˇ S ( U ) v x := 2 ( δ ( x ) − δ ( x +1)) / p x ( x + 1) + α · v x +1 ;ˇ S ( D ) v x := 2 ( δ ( x ) − δ ( x − / p x ( x − 1) + α · v x − ;ˇ S ( H ) v x := 2 x · v x , x ∈ Z . (7.7)The representation ˇ S is chosen in such a way that for all matrices M ∈ sl (2 , C )and vectors v ∈ V fin we have[ R ( M ) , T ( v )] = T ( ˇ S ( M ) v ) (7.8)(the equality of operators in Fock fin ( Z > )). This follows from definitions of R , T ,and ˇ S .Comparing (7.7) and (6.15), we see that the representation ˇ S is conjugate tothe representation S discussed in § S = Z − SZ , where Z isan operator in V = ℓ ( Z ) defined by Z v x := 2 δ ( x ) / x , x ∈ Z . This means thatProposition 6.7 also holds for the representation ˇ S . In particular, ˇ S lifts to arepresentation of the group P SU (1 , 1) in the Hilbert space V . Note that due tothe conjugation by Z , the representation ˇ S is not unitary (but we do not need thisproperty).The next proposition (due to Olshanski [Ols08b]) is a “group level” version ofthe identity (7.8). Proposition 7.3. For all g ∈ SU (1 , ∼ and all v ∈ V we have R ( g ) T ( v ) R ( g ) − = T ( ˇ S ( g ) v ) (7.9) (the equality of operators in Fock ( Z > ) ).Proof. Step 1. Since the representation T is norm preserving, it suffices to take v ∈ V from the dense subspace V fin . Without loss of generality, we can assume that v = v x for some x ∈ Z . Step 2. Rewrite the claim (7.9) as R ( g ) T ( v x ) = T ( ˇ S ( g ) v x ) R ( g ) . (7.10)This is an equality of operators in the Hilbert space Fock ( Z > ). It is enough toshow that these operators agree on Fock fin ( Z > ), which is true if R ( g ) T ( v x ) λ = T (cid:0) ˇ S ( g ) v x (cid:1) R ( g ) λ for all g ∈ SU (1 , ∼ , x ∈ Z , and λ ∈ S . (7.11) Also by ˇ S we denote the corresponding representations of SU (1 , 1) and SU (1 , ∼ in V thatare obtained from the representation ˇ S of P SU (1 , 1) by a trivial lifting procedure. FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 35 Step 3. Now let us prove that both sides of (7.11) are analytic functions in g ∈ SU (1 , ∼ with values in Fock ( Z > ): • (left-hand side) The vector T ( v x ) λ belongs to Fock fin ( Z > ), and hence is analyticfor the representation R , see Proposition 4.8. This means that the function g R ( g ) T ( v x ) λ is analytic. • (right-hand side) By Proposition 6.7 (and remarks before the present proposi-tion), the function g ˇ S ( g ) v x is an analytic function with values in the Hilbertspace V . Since T is continuous in the norm topology, T (cid:0) ˇ S ( g ) v x (cid:1) is an analyticfunction with values in the Banach space End (cid:0) Fock ( Z > ) (cid:1) of bounded operatorsin the space Fock ( Z > ). On the other hand, the function R ( g ) λ is also analytic(with values in Fock ( Z > )). Therefore, the function g 7→ T (cid:0) ˇ S ( g ) v x (cid:1) R ( g ) λ isanalytic, too. Step 4. Now it remains to compare the Taylor series expansions of both sidesof (7.10) at g = e , the unity element of SU (1 , ∼ . That is, we need to establishthat for any M ∈ sl (2 , C ) and any x ∈ Z : ∞ X k =0 R ( M ) k s k k ! T ( v x ) = ∞ X l =0 T (cid:0) ˇ S ( M ) l v x (cid:1) s l l ! ! ∞ X r =0 R ( M ) r s r r ! ! . (7.12)This should be understood as an equality of formal power series in s with coefficientsbeing operators in Fock fin ( Z > ). Let us divide both sides by the last formal sum, P ∞ r =0 R ( M ) r s r r ! . After that it can be readily verified that the identity (7.12) of formalpower series is a corollary of the “Lie algebra level” commutation identity (7.8).This last step concludes the proof of the proposition. (cid:3) Define (the second equality holds because ˇ S is a representation of P SU (1 , v x,ξ := ˇ S ( e G ξ ) − v x = ˇ S ( G ξ ) − v x ∈ V, x ∈ Z . (7.13)Putting this together with Proposition 7.3, we can rewrite the correlation functions(7.5) as the vacuum average (see Definition 5.3): ρ ( n ) α,ξ ( x , . . . , x n ) = F vac ( v x ,ξ . . . v x n ,ξ v − x n ,ξ . . . v − x ,ξ ) . (7.14)Observe that for x, y ∈ Z =0 we have F vac ( v x,ξ v y,ξ ) = ( − x ∧ y ∧ (cid:16) R ( e G ξ ) − φ x φ y R ( e G ξ ) vac , vac (cid:17) = Φ α,ξ ( x, y ) , as in (7.2). Therefore, formula (7.14) together with Wick’s Theorem 5.1 immedi-ately implies our Theorem 7.1.7.2. Static Pfaffian kernel. Let us express our static Pfaffian kernel Φ α,ξ ( x, y )through the functions ϕ m defined by (6.8). This kernel is defined for x, y ∈ Z =0 and has the form (see the previous subsection) Φ α,ξ ( x, y ) = F vac ( v x,ξ v y,ξ ) = X k,l ∈ Z ( v x,ξ , v k ) ℓ ( Z ) ( v y,ξ , v l ) ℓ ( Z ) F vac ( v k v l )(where the vectors v x,ξ , v y,ξ are defined by (7.13)). By definitions of § F vac ( v k v l ) = (cid:26) , if l = − k ≥ , , otherwise . (7.15)Therefore, Φ α,ξ ( x, y ) = X ∞ m =0 ( v x,ξ , v − m ) ℓ ( Z ) ( v y,ξ , v m ) ℓ ( Z ) . Proposition 7.4. For any r, k ∈ Z we have ( v r,ξ , v k ) ℓ ( Z ) = ( − r ∧ k ∧ ( δ ( r ) − δ ( k )) / ϕ − k ( r ; α, ξ ) , where the functions ϕ m are defined in § By (7.13) and then by (5.8),( v r,ξ , v k ) ℓ ( Z ) = ( ˇ S ( G ξ ) − v r , v k ) ℓ ( Z ) = ( − r ∧ k ∧ ( ˇ S ( G ξ ) − v r , v k ) ℓ ( Z ) . Using the fact that ˇ S = Z − SZ (see the discussion before Proposition 7.3) andProposition 6.8, we conclude the proof. (cid:3) Therefore, since Φ α,ξ ( x, y ) is defined for x, y ∈ Z =0 , we have (in our derivationwe also used (6.11)): Φ α,ξ ( x, y ) = ( − x ∧ y ∨ X ∞ m =0 − δ ( m ) ϕ m ( x ; α, ξ ) ϕ m ( − y ; α, ξ ) . (7.16)In the rest of the paper we agree that by this formula the kernel Φ α,ξ ( x, y ) is definedfor arbitrary x, y ∈ Z (see also Remark 2.3.1). This is needed to view Φ α,ξ as anoperator in ℓ ( Z ). One also has (see § Φ α,ξ ( x, y ) = X ∞ m =0 − δ ( m ) e ϕ m ( x ; α, ξ ) e ϕ m ( − y ; α, ξ ) . (7.17)7.3. Interpretation through spectral projections. One can interpret the ker-nel Φ α,ξ through orthogonal spectral projections related to the difference operator e D α,ξ defined by (6.14). Namely, the projection onto the positive part of the spec-trum of e D α,ξ has the form (see § > ( e D α,ξ )( x, y ) = X ∞ m =1 e ϕ m ( x ; α, ξ ) e ϕ m ( y ; α, ξ ) . We also need the projection onto the zero eigenspace, which is simplyProj =0 ( e D α,ξ )( x, y ) = e ϕ ( x ; α, ξ ) e ϕ ( y ; α, ξ ) . From (7.17) we get the following interpretation of our kernel: Proposition 7.5. Viewing the static Pfaffian kernel Φ α,ξ as an operator in ℓ ( Z ) ,we have Φ α,ξ = (cid:0) Proj > ( e D α,ξ ) + Proj =0 ( e D α,ξ ) (cid:1) R , where R : ℓ ( Z ) → ℓ ( Z ) is the operator corresponding to the reflection of the lattice Z with respect to : (R f )( x ) := f ( − x ) , f ∈ ℓ ( Z ) . Since R is the identity operator, we see that the operator Φ α,ξ R is a rank oneperturbation of the orthogonal spectral projection operator corresponding to thepositive part of the spectrum of e D α,ξ .7.4. Expression through the discrete hypergeometric kernel. Recall thatthe discrete hypergeometric kernel K z,z ′ ,ξ ( x ‘ , y ‘ ) (where x ‘ , y ‘ ∈ Z ′ = Z + ) servesas a determinantal kernel for the z -measures on ordinary partitions ( § z, z ′ , the functions involved in the formula for K z,z ′ ,ξ ( x ‘ , y ‘ ) turn into our functions ϕ m , see (6.9). This means that one couldexpress our Pfaffian kernel Φ α,ξ through the discrete hypergeometric kernel: FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 37 Proposition 7.6. For all x, y ∈ Z we have Φ α,ξ ( x, y ) = ( − x ∧ y ∨ h K ν ( α ) − , − ν ( α ) − ( x + , − y + ) (7.18)+ K ν ( α )+ , − ν ( α )+ ( x − , − y − ) i , where K is the discrete hypergeometric kernel described in § ν ( α ) is givenby Definition 2.1.Proof. Using (6.7) and (6.9) with d = − d = 0, we observe that for x, y ∈ Z : X ∞ m =1 ϕ m ( x ; α, ξ ) ϕ m ( y ; α, ξ ) = K ν ( α ) − , − ν ( α ) − ,ξ ( x + , y + ); (7.19) X ∞ m =0 ϕ m ( x ; α, ξ ) ϕ m ( y ; α, ξ ) = K ν ( α )+ , − ν ( α )+ ,ξ ( x − , y − ) . (7.20)Taking half sum and using (7.16), we conclude the proof. (cid:3) A time-dependent version of (7.18) is (2.14), which is obtained and used in § Reduction formulas. It is possible to rewrite the Pfaffian hypergeometric-type kernel Φ α,ξ ( x, y ) in a closed form (without the sum): Proposition 7.7. For any x, y ∈ Z we have Φ α,ξ ( x, y ) = ( − x ∧ y ∧ √ αξ − ξ ) ×× ϕ ( x ) (cid:0) ϕ ( y ) − ϕ − ( y ) (cid:1) − ϕ ( y ) (cid:0) ϕ ( x ) − ϕ − ( x ) (cid:1) x + y . For x = − y there is a singularity in the numerator (this is seen using (6.11)) as wellas in the denominator. In this case the value of Φ α,ξ ( x, y ) is understood accordingto the L’Hospital’s rule using the analytic expression for ϕ m (6.8).Proof. There are several ways of establishing this fact. One could use representa-tion-theoretic arguments as in the proof of Theorem 3 in [Oko01b]. Another way isto argue directly using the three-term relations for the functions ϕ m (6.12) to sim-plify the sum (7.16) similarly to the standard derivation of the Christoffel–Darbouxformula for orthogonal polynomials.We use Proposition 7.6 together with the existing closed form expression for K z,z ′ ,ξ [BO06b, Proposition 3.10]: K z,z ′ ,ξ ( x ‘ , y ‘ ) = √ zz ′ ξ − ξ ψ − ( x ‘ ) ψ ( y ‘ ) − ψ ( x ‘ ) ψ − ( y ‘ ) x ‘ − y ‘ , x ‘ , y ‘ ∈ Z ′ . (of course, the parameters of the functions ψ above are z, z ′ , ξ ). We plug thisformula into (7.18), and then express each function ψ a ‘ through ϕ m using (6.9)with d = − d = 0. Observe that for such d we have z ( α ) z ′ ( α ) = α . After thatwe apply (6.11) to simplify the resulting expression. This concludes the proof. (cid:3) Corollary 7.8 (Reduction formulas for Φ α,ξ ) . For all x, y ∈ Z we have: (1) Φ α,ξ ( x, − y ) = Φ α,ξ ( y, − x ) ; (2) Φ α,ξ ( x, − y ) = − Φ α,ξ ( − x, y ) if x = y ; In fact, [BO06b, Proposition 3.10] itself is established using the three-term relations for thefunctions ψ a ‘ (6.6). (3) ( x + y ) Φ α,ξ ( x, y ) = ( x − y ) Φ α,ξ ( x, − y ) (note that Φ α,ξ ( x, x ) = 0 for all x = 0 ).Proof. Claim (1) is best seen from (7.18), because the kernel K is symmetric.Claims (2) and (3) follow from Proposition 7.7 and (6.11). (cid:3) Static determinantal kernel In this section we compute and discuss the determinantal correlation kernel K α,ξ of the point process M α,ξ on Z > . We also consider the Plancherel degeneration(2.8) of the measures M α,ξ and of the kernel K α,ξ . This section completes the proofof Theorem 1 from § Hypergeometric-type kernel K α,ξ .Theorem 8.1. For all α > and < ξ < , the point process M α,ξ on Z > isdeterminantal. Its correlation kernel K α,ξ can be expressed in several ways (here x, y ∈ Z > ): (1) As an infinite sum K α,ξ ( x, y ) = 2 √ xyx + y ∞ X m =0 − δ ( m ) ϕ m ( x ; α, ξ ) ϕ m ( y ; α, ξ ) (8.1) (the functions ϕ m are defined in § (2) In an integrable form K α,ξ ( x, y ) = √ αξxy − ξ · P ( x ) Q ( y ) − Q ( x ) P ( y ) x − y , (8.2) where P ( x ) := ϕ ( x ; α, ξ ) and Q ( x ) := ϕ ( x ; α, ξ ) − ϕ − ( x ; α, ξ ) . (3) In terms of the discrete hypergeometric kernel of the z -measures ( § e K α,ξ ( x, y ) = K ν ( α )+ , − ν ( α )+ ,ξ ( x − , y − ) (8.3)+ ( − y K ν ( α ) − , − ν ( α ) − ,ξ ( x + , − y + ) , where we have denoted e K α,ξ ( x, y ) := r xy · K α,ξ ( x, y ) . (4) Viewed as an operator in ℓ ( Z > ) , e K α,ξ can be interpreted in terms of orthogo-nal spectral projections corresponding to the difference operator e D α,ξ (6.14) asfollows (we restrict the operator below to ℓ ( Z > ) ⊂ ℓ ( Z ) ): e K α,ξ = (cid:0) Proj > ( e D α,ξ ) + Proj =0 ( e D α,ξ ) (cid:1)(cid:0) I + R (cid:1) , where I is the identity operator and R is the reflection, see Proposition 7.5. The expression e K α,ξ ( x, y ) is a so-called gauge transformation of the originalcorrelation kernel K α,ξ , that is, e K α,ξ is related to K α,ξ by a conjugation by adiagonal matrix. This means that the Z > × Z > matrix e K α,ξ can also serve as acorrelation kernel for the point process M α,ξ . Proof. The fact that the process M α,ξ is determinantal is guaranteed by Lemma 3.7.On the other hand, the reduction formulas for the Pfaffian kernel Φ α,ξ (Corollary7.8) allow us to apply Proposition A.2 from Appendix. This implies that ρ ( n ) α,ξ ( X ) = Pf( ˆ Φ α,ξ J X K ) = det [ K α,ξ ( x k , x j )] nk,j =1 , FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 39 where X = { x , . . . , x n } ⊂ Z > (with pairwise distinct x j ’s), ˆ Φ α,ξ J X K is the skew-symmetric 2 n × n matrix introduced in Theorem 7.1, and K α,ξ is related to Φ α,ξ as K α,ξ ( x, y ) = 2 √ xyx + y Φ α,ξ ( x, − y ) , x, y ∈ Z > . This gives an argument (independently of Lemma 3.7) that the process M α,ξ is determinantal. Moreover, this also provides us with explicit formulas for thekernel K α,ξ . Namely, claims 1 and 2 of the present theorem directly follow fromthe expressions of Φ α,ξ as a series (7.16) and in a closed form (Proposition 7.7).To prove claims 3 and 4, observe that K α,ξ ( x, y ) = r yx (cid:20) x − yx + y + 1 (cid:21) Φ α,ξ ( x, − y ) = r yx [ Φ α,ξ ( x, y ) + Φ α,ξ ( x, − y )](the last equality is by Corollary 7.8.(3)), so e K α,ξ ( x, y ) = Φ α,ξ ( x, y ) + Φ α,ξ ( x, − y ) . Now we see that claim 3 follows from (7.16) and (7.19)–(7.20), and claim 4 is dueto Proposition 7.5. This concludes the proof. (cid:3) Comments to Theorem 8.1. 1. Formulas (8.1) and (8.2) for the correlationkernel K α,ξ are the same as the statements of Theorems 2.1 and 2.2 in [Pet10a].This can be seen from the expression (6.8) for the functions ϕ m . It is possible to obtain double contour integral expressions for the kernel K α,ξ ( x, y ) (given in [Pet10a, Propositions 3 and 4]). They can be derived from(8.1) in the same way as in the proof of [BO06b, Theorem 3.3]. The form (8.2) of the kernel K α,ξ is called integrable because the operator(8.2) in ℓ ( Z > ) can be viewed as a discrete analogue of an integrable operator(if we take x and y as variables). About integrable operators, e.g., see [IIKS90],[Dei99]. Discrete integrable operators are discussed in [Bor00] and [BO00, § Relation (8.3) between the (determinantal) correlation kernels of the measures M α,ξ on strict partitions and the z -measures on ordinary partitions, respectively,seems to be purely formal and have no consequences at the level of random pointprocesses. About the behavior of (8.3) in a scaling limit as ξ ր Consider the Z > × Z > matrix L α,ξ which is defined by (3.7), where w ( x ) = w α,ξ ( x ) is given by (2.7). Then one can show similarly to the proof of Theorem3.3 in [BO00] (and also using the identities from Appendix in that paper) that K α,ξ = L α,ξ (1 + L α,ξ ) − . That is, the kernel K α,ξ is precisely the one given byLemma 3.7.8.3. Plancherel degeneration. Here we consider the Plancherel degeneration(2.8) of the hypergeometric-type kernel K α,ξ studied above in this section. De-note by J k the Bessel function (of the first kind) of order k and argument 2 √ θ : J k := J k (2 √ θ ) = ∞ X r =0 ( − r θ r + k r !Γ( r + k + 1) , k ∈ Z . (8.4) Theorem 8.2. Under the Plancherel degeneration (2.8), the point processes M α,ξ on Z > converge to the poissonized Plancherel measure Pl θ . This is a determinantalpoint process on Z > supported by finite configurations. The correlation kernel K θ ( x, y ) of Pl θ can be expressed through the Bessel function in two ways: as aseries K θ ( x, y ) = 2 √ xyx + y ∞ X m =0 − δ ( m ) J m + x J m + y , (8.5) and in an integrable form K θ ( x, y ) = 2 √ xyx − y (cid:16) √ θJ x − J y − √ θJ y − J x − ( x − y ) J x J y (cid:17) . (8.6)In principal, one can express the kernel K θ through the discrete Bessel kernel of[BOO00] similarly to (8.3) above, but we do not focus on this expression.We will discuss three ways of proving Theorem 8.2. The fact that formulas (8.5)and (8.6) are equivalent can be obtained as in [BOO00, Proposition 2.9]. Proof of Theorem 8.2. I. Formulas (8.5) and (8.6) for K θ can be obtained from thecorresponding formulas (8.1) and (8.2) for K α,ξ via the Plancherel degeneration(2.8). Namely, under the Plancherel degeneration we have ϕ m ( x ; α, ξ ) → J m + x .(This can be obtained by a termwise limit from the hypergeometric series for ϕ m ,this series converges rapidly for fixed x and m .) From this one can readily derive(8.5). To obtain (8.6) from the Plancherel degeneration of (8.2), one should alsouse the three-term relations for the Bessel functions (e.g., see [Erd53, 7.2.(56)]): J k +1 − k √ θ J k + J k − = 0 , k ∈ Z . (8.7)This concludes the proof. (cid:3) The three-term relations for the Bessel functions (8.7) are obtained via thePlancherel degeneration from Proposition 6.5.2) (or, equivalently, from (6.12), bythe self-duality of ϕ m ’s). This agrees with the general approach described in[Ols08a] of studying limits of determinantal point processes via corresponding lim-its of self-adjoint operators which “control” the processes (in the sense that thecorrelation kernels of the processes are spectral projections corresponding to theseoperators). Proof of Theorem 8.2. II. Another way of proof is to observe that the point process Pl θ on Z > is again an L-ensemble (see Lemma 3.7). Denote by L θ the corresponding Z > × Z > matrix which is given by (3.7) with w ( x ) = w θ ( x ) = θ x x !) . To provethe theorem it suffices (by Lemma 3.7) to show that the kernel K θ has the form K θ = L θ (1 + L θ ) − . This is equivalent to a certain identity for the Bessel functionswhich is readily verified using, e.g., Lemma 2.4 in [BOO00]. Equivalently, one maysay that this identity is the Plancherel degeneration of the one in § (cid:3) Proof of Theorem 8.2. III. This way of proving the theorem uses the result of Mat-sumoto [Mat05, Thm. 3.1] that states that the correlation functions ρ ( n ) θ ( x , . . . , x n )of the poissonized Plancherel measure Pl θ have Pfaffian form similar to (7.3) withthe Pfaffian kernel Φ θ given by Φ θ ( x, y ) = 12 ( − x ∧ y ∧ [ z x w y ] (cid:26) e √ θ ( z − /z ) e √ θ ( w − /w ) z − wz + w (cid:27) , x, y ∈ Z , where [ z x w y ] { . . . } denotes the coefficient by z x w y . The function e √ θ ( z − /z ) is thegenerating series for the Bessel functions J k (2 √ θ ) [Erd53, 7.2.(25)], and thus Φ θ ( x, y ) = ( − x ∧ y ∨ X ∞ m =0 − δ ( m ) J m + x J m − y . FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 41 For the Pfaffian kernel Φ θ we have the same reduction formulas as in Corollary 7.8.They can be verified independently, or obtained from the reduction formulas for Φ α,ξ , because Φ θ is the Plancherel degeneration of Φ α,ξ . Thus, from PropositionA.2 it follows that the poissonized Plancherel measure is a determinantal pointprocess with the correlation kernel given by K θ ( x, y ) = √ xyx + y Φ θ ( x, − y ), which isexactly formula (8.5) for K θ . (cid:3) Theorems 8.1 and 8.2 constitute Theorem 1 from § Markov processes In this section we explain in detail the construction of the dynamical model onstrict partitions described in § z -measures. In contrast to [BO06a], we restrict our at-tention to the stationary (time homogeneous) case, that is, we assume that theparameter ξ does not vary in time. The introduction of the non-stationary pro-cesses in [BO06a] was motivated by the technique of handling the stationary case(in particular, by the method of the computation of the dynamical correlation func-tions). The technique that we use in the present paper does not require dealingwith non-stationary processes.9.1. Defining Markov processes in terms of jump rates. Let us first recallsome basic notions and facts concerning Markov processes. Let E be a finite orcountable space. Assume that we have a continuous time homogeneous Markovprocess on E with the time parameter t ∈ R ≥ . By ( P ( t )) t ≥ denote the transitionprobabilities of this Markov process. That is, each P ( t ) is a E × E matrix, and P ab ( t ) (where a, b ∈ E ) is the probability that the process starting from the state a will be at the state b after time t . The matrices ( P ( t )) t ≥ have the followingproperties:(P1) P ab ( t ) ≥ t ≥ P ab (0) = δ ab for all a, b ∈ E ;(P2) P b ∈ E P ab ( t ) = 1 for all a ∈ E ;(P3) ( Chapman-Kolmogorov equation ) P ( t + s ) = P ( t ) P ( s ) for t, s ≥ 0, or, in matrixform, P ab ( t + s ) = P c ∈ E P ac ( t ) P cb ( s ), where a, b ∈ E .Assume that there exists a E × E matrix Q such that P ab ( t ) = δ ab + Q ab · t + o ( t ) , t → , a, b ∈ E. (9.1)The elements of the matrix Q are called the jump rates . Note that (9.1) impliesthat each P ab ( t ) is continuous at t = 0:(P4) lim t ↓ P ab ( t ) = δ ab , a, b ∈ E .A family of matrices satisfying (P1)–(P4) is a ( continuous ) stochastic matrixsemigroup . One can say that Q is the infinitesimal matrix of this semigroup, thatis, Q = ddt P ( t ) (cid:12)(cid:12) t =0 . From (9.1) it is clear that(Q1) Q ab ≥ a = b and Q aa ≤ Q aa = − P b = a Q ab for all a ∈ E . The property (Q2) implies (e.g., see [KT99, Ch. 14.2]) that the jump rates Q and the transition probabilities P ( t ) are related to each other via the system of Kolmogorov’s backward equations : d P ab ( t ) dt = X c ∈ E Q ac P cb ( t ) , a, b ∈ E, (9.2)with the initial conditions P ab (0) = δ ab , a, b ∈ E. (9.3)We would like to start with the jump rates Q satisfying properties (Q1)–(Q2) andobtain a stochastic matrix semigroup ( P ( t )) t ≥ by solving backward equations (9.2)–(9.3). It is known that a solution in a wider class of substochastic matrix semigroups (when the condition (P2) is replaced by P b ∈ E P ab ( t ) ≤ 1) always exists. Among allpossible substochastic solutions there is a distinguished minimal solution (¯ P ( t )) t ≥ ,that is, P ab ( t ) ≥ ¯ P ab ( t ) for t ≥ a, b ∈ E , where ( P ( t )) t ≥ is any substochasticsolution. A minimal solution can be constructed using an approximation method(e.g., see [KT99, Ch. 14.3]). If the minimal solution is stochastic, then it is aunique solution of (9.2)–(9.3) in the class of substochastic matrices. About solvingKolmogorov’s backward equations see also [GSK04, Ch. III.2].If the system of backward equations (9.2)–(9.3) has a unique solution ( P ( t )) t ≥ (or, which is equivalent, the minimal solution of this system is stochastic), we saythat the jump rates Q define a continuous time homogeneous Markov process on E (with transition probabilities P ( t )) that can start from any point and any probabilitydistribution. A common sufficient condition for this is sup a ∈ E | Q aa | < + ∞ , whichhowever does not hold in our case.Let us recall another useful sufficient condition for the minimal solution of (9.2)–(9.3) to be stochastic. We formulate it as in [BO06a, Prop. 4.3], it also can bederived from the discussion of [GSK04, Ch. III.2].Let X ⊂ E be a finite set and a ∈ X . By τ a,X denote the time of the first exitfrom X of the process starting at a . Though we do not know yet if the processitself is uniquely determined by its jump rates Q , the random variable τ a,X can beconstructed from Q as follows. Contract all the states b ∈ E \ X into one absorbingstate ˜ b with Q ˜ b,c = 0 for all c ∈ X ∪ { ˜ b } . On the finite set X ∪ { ˜ b } the backwardequations have a unique solution (˜ P ( t )) t ≥ , where ˜ P ( t ) are matrices with rowsand columns indexed by the set X ∪ { ˜ b } . The distribution of the random variable τ a,X then has the form Prob { τ a,X ≤ t } = ˜ P a, ˜ b ( t )and is defined only in terms of the jump rates Q . Proposition 9.1 ([BO06a, Prop. 4.3]) . If for any a ∈ E , any t ≥ , and any ǫ > there exists a finite set X ( ǫ ) ⊂ E such that Prob (cid:8) τ a,X ( ǫ ) ≤ t (cid:9) ≤ ǫ, then the minimal solution of the system of Kolmogorov’s backward equations (9.2)–(9.3) is stochastic. In other words, the hypotheses of this proposition mean that the Markov processdoes not make infinitely many jumps in finite time. Indeed, because the state space is finite, one can readily see that the minimal solution isstochastic. FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 43 Birth and death processes. Here we discuss underlying birth and deathprocesses on Z ≥ involved in the construction of the Markov processes on strictpartitions (see § E = Z ≥ is a continuous time homogeneousMarkov process with jump rates { q k,j } k,j ∈ Z ≥ satisfying conditions (Q1)–(Q2) fromthe previous subsection, with an additional property that q k,j = 0 if | k − j | > n ≥ Z ≥ the process can jump only to theneighbor points n − n + 1, and that from 0 it can jump only to 1.The following necessary and sufficient condition is well-known and can be de-duced, e.g., from [GSK04, Ch. III.2, Thm. 4]. Proposition 9.2. The minimal solution of the system of Kolmogorov’s backwardequations (9.2)–(9.3) for a birth and death process is stochastic iff ∞ X n =1 (cid:20) q n,n +1 + q n,n − q n − ,n q n,n +1 + · · · + q n,n − . . . q , q , q , q , . . . q n − ,n q n,n +1 (cid:21) = + ∞ . (9.4)Now let us turn to our concrete situation and define the birth and death processthat we use in the present paper. Its jump rates depend on our parameters α > < ξ < q n,n +1 := (1 − ξ ) − ξ ( n + α/ q n,n − := (1 − ξ ) − n ; (9.5) q n,n := − ( q n,n +1 + q n,n − ) = − (1 − ξ ) − { ξ ( n + α/ 2) + n } , where n = 0 , , , . . . . All other jump rates are zero. It is not hard to see thatjump rates (9.5) satisfy condition (9.4). Thus, Proposition 9.2 together with thefacts from § Z ≥ with jump rates (9.5) that can start from any point and any probabilitydistribution. This process preserves the negative binomial distribution π α,ξ (2.4)on Z ≥ because π α,ξ ◦ q = 0, or, in matrix form, X ∞ k =0 π α,ξ ( k ) q k,j = 0 for all j ∈ Z ≥ . (9.6)Denote the equilibrium version of this process (that is, starting from the distribution π α,ξ ) by ( n α,ξ ( t )) t ≥ . Since π α,ξ ( n ) q n,n +1 = π α,ξ ( n + 1) q n +1 ,n for all n ∈ Z ≥ , (9.7)the process n α,ξ is reversible with respect to π α,ξ .The transition probabilities of the process n α,ξ can be expressed through theMeixner orthogonal polynomials, see [BO06a, § Remark 9.3. From, e.g., the discussion of [BO06a, § 4] it follows that the process n α,ξ satisfies the hypotheses of Proposition 9.1. This fact will be used in the nextsubsection in construction of Markov processes on strict partitions that “extend”the processes n α,ξ .9.3. Markov processes on strict partitions. Here we define continuous timeMarkov processes on the set S of all strict partitions. They depend on our param-eters α > < ξ < λ ∈ S n , n = 0 , , . . . ): Q λ, κ := (1 − ξ ) − ξ ( n + α/ p ↑ α ( n, n + 1) λ, κ , where κ ց λ ; Q λ,µ := (1 − ξ ) − np ↓ ( n, n − λ,µ , where µ ր λ ; Q λ,λ := − X κ : κ ց λ Q λ, κ − X µ : µ ր λ Q λ,µ (9.8)= − (1 − ξ ) − { ξ ( α/ n ) + n } . All other jump rates are zero. Here p ↓ ( n, n − 1) and p ↑ α ( n, n + 1) are the downand up transition kernels, respectively. Recall that (see § 3) if λ ∈ S n , µ ր λ , and κ ց λ , we have p ↓ ( n, n − λ,µ = dim S µ dim S λ , p ↑ α ( n, n + 1) λ, κ = dim S λ dim S κ · M α,n +1 ( κ ) M α,n ( λ ) , where dim S ( · ) is given by (3.1) and { M α,n } is the multiplicative coherent systemof measures on the Schur graph ( § S → Z ≥ , λ 7→ | λ | , the S × S matrix Q (9.8) turns into the Z ≥ × Z ≥ matrix q of jump rates of the birth and death process n α,ξ from § S “extend” the birth and death processes n α,ξ . Proposition 9.4. The minimal solution of the system of Kolmogorov’s backwardequations (9.2)–(9.3) for the matrix Q (9.8) is stochastic.Proof. Let n, K be nonnegative integers, n ≤ K . Consider the random variable τ n, { ,...,K − } for the birth and death process n α,ξ , that is, the time of the first exitfrom { , , . . . , K − } of the process with jump rates q (9.5) starting at n .Let λ ∈ S n . Observe that the time of the first exit from S ∪ · · · ∪ S K − ofthe process on strict partitions with jump rates Q (9.8) starting at λ has the samedistribution as τ n, { ,...,K − } . Applying Remark 9.3 and Proposition 9.1, we concludethe proof. (cid:3) Thus, the jump rates Q (9.8) uniquely define a continuous time Markov processon S that can start from any point and any probability distribution.Recall that the measures M α,ξ and { M α,n } n ∈ Z ≥ are related as M α,ξ ( λ ) = π α,ξ ( n ) M α,n ( λ ) , λ ∈ S n , (9.9)where π α,ξ is the negative binomial distribution (2.4). Moreover, the measure π α,ξ isinvariant for the birth and death process n α,ξ on Z ≥ (see (9.6)), and the measures M α,n are consistent with the up and down transition kernels (see § M α,n ◦ p ↑ α ( n, n + 1) = M α,n +1 and M α,n +1 ◦ p ↓ ( n + 1 , n ) = M α,n . This implies that M α,ξ ◦ Q = 0, and hence the process with jump rates (9.8)preserves the measure M α,ξ on S . By ( λ α,ξ ( t )) t ∈ [0 , + ∞ ) we denote the equilibriumversion of this process.The process λ α,ξ is reversible with respect to M α,ξ because M α,ξ ( λ ) Q λ,µ = M α,ξ ( µ ) Q µ,λ for all µ, λ ∈ S . Indeed, it suffices to consider µ ր λ . Let | λ | = n .Then M α,ξ ( λ ) Q λ,µ = (cid:16) π α,ξ ( n ) n − ξ (cid:17) (cid:0) M α,n ( λ ) p ↓ ( n, n − λ,µ (cid:1) = (cid:16) π α,ξ ( n − ξ ( n − α )1 − ξ (cid:17)(cid:16) M α,n − ( µ ) p ↑ α ( n − , n ) µ,λ (cid:17) = M α,ξ ( µ ) Q µ,λ by (9.7), (9.9), and the definition of the up transition kernel p ↑ α ( n − , n ), see § FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 45 Pre-generator. Here we discuss the pre-generator of the Markov process λ α,ξ . We now regard the S × S matrices ( P λ,µ ( t )) t ≥ of transition probabilitiesof the process λ α,ξ as operators acting on functions on S (from the left):( P ( t ) f )( λ ) := X µ ∈ S P λ,µ ( t ) f ( µ ) . The family ( P ( t )) t ≥ is a Markov semigroup of self-adjoint contractive operators inthe weighted space ℓ ( S , M α,ξ ) (see § ℓ ( S , M α,ξ )).The semigroup ( P ( t )) t ≥ in ℓ ( S , M α,ξ ) has a generator which is an unboundedoperator. By Q let us denote the restriction of this generator to ℓ ( S , M α,ξ ) ⊂ ℓ ( S , M α,ξ ), the dense subspace of all finitely supported functions in ℓ ( S , M α,ξ ).The operator Q acts as( Q f )( λ ) = X µ ∈ S Q λ,µ f ( µ ) , f ∈ ℓ ( S , M α,ξ ) , (9.10)where Q λ,µ (9.8) are the jump rates of the process λ α,ξ .The operator Q is symmetric with respect to the inner product ( · , · ) M α,ξ . More-over, it is closable in ℓ ( S , M α,ξ ), and its closure generates the semigroup ( P ( t )) t ≥ (see Remark 9.8 below). That is, Q is the pre-generator of the process λ α,ξ . Remark 9.5. As a wider domain for the operator Q (9.10) one can take the spaceof all functions f on S such that both f and Q f (defined by (9.10)) belong to ℓ ( S , M α,ξ ). This space clearly includes finitely supported functions.Using the isometry I α,ξ : ℓ ( S , M α,ξ ) → ℓ ( S ) (4.13), we get a symmetric operator B in ℓ ( S ) and a Markov semigroup ( V ( t )) t ≥ of self-adjoint contractive operatorsin ℓ ( S ) corresponding to Q and ( P ( t )) t ≥ , respectively. Let us compute thematrix elements of the operator B in the standard orthonormal basis { λ } λ ∈ S . Proposition 9.6. We have B λ = X ν ∈ S ( B λ, ν ) ν = − (1 − ξ ) − {| λ | + ξ ( | λ | + α } λ + √ ξ − ξ X µ : µ ր λ q α ( λ/µ ) µ + √ ξ − ξ X κ : κ ց λ q α ( κ /λ ) κ . Here q α is the function of a box defined by (4.6).Proof. Fix λ ∈ S , and for any ν ∈ S one has( B λ, ν ) = (cid:0) ( M α,ξ ( λ )) − Q λ, ( M α,ξ ( ν )) − ν (cid:1) M α,ξ = ( M α,ξ ( λ ) M α,ξ ( ν )) − ( Q λ, ν ) M α,ξ = M α,ξ ( ν ) M α,ξ ( λ ) − Q ν,λ ( Q is given in (9.8)), and Proposition follows from a direct computation. (cid:3) Corollary 9.7. The operator B in Fock fin ( Z > ) has the form B = − R ( H ξ ) + α I , where I is the identity operator, the unitary representation R of sl (2 , C ) in theHilbert space Fock fin ( Z > ) is defined in § H ξ is given in Remark 6.9.Proof. This is a straightforward consequence of Proposition 9.6 (where, of course,we identify ℓ ( S ) and Fock ( Z > )) and the matrix computation in Remark 6.9. (cid:3) We denote, e.g., the operators B and Q by different symbols only to indicate in what spacesthey act. Essentially, these operators are the same. Remark 9.8. From the above corollary it follows that the operator B (with domain Fock fin ( Z > )) is essentially self-adjoint because, by Proposition 4.8, all vectors ofthe space Fock fin ( Z > ) are analytic for the operator R ( H ξ ). The same also holds forthe operator R ( H ) (corresponding to the case ξ = 0). Moreover, the closure of B looks as B = α I − R ( H ξ ) = α I − R ( e G ξ ) R ( H ) R ( e G ξ ) − , and this operator generatesthe semigroup ( V ( t )) t ≥ .These properties of B in fact imply (using the isometry I α,ξ (4.13)) that the op-erator Q is closable in ℓ ( S , M α,ξ ), and its closure generates the semigroup ( P ( t )) t ≥ of the Markov process λ α,ξ on strict partitions. Remark 9.9. Apart from the situations described in Remarks 6.9 and 9.8, thematrix H ξ ∈ sl (2 , C ) appears also in connection with the birth and death process( n α,ξ ( t )) t ≥ defined in § n α,ξ which acts in ℓ ( Z ≥ , π α,ξ )(with jump rates given by (9.5)). Then use the isometry between ℓ ( Z ≥ , π α,ξ ) and ℓ ( Z ≥ ) which is similar to (4.13) to rewrite this pre-generator in the latter Hilbertspace. We arrive at the following operator acting on f ∈ ℓ ( Z ≥ ): f ( x ) (1 − ξ ) − q ξ ( x + 1)( x + α ) f ( x + 1) + (1 − ξ ) − q ξx ( x − α ) f ( x − − (1 − ξ ) − { ξ ( x + α ) + x } f ( x ) . (9.11)Now consider the lowest weight representation T of the Lie algebra sl (2 , C ) in ℓ ( Z ≥ ) under which the matrices U, D , and H (see (4.4)) act in the standard basis { k } k ∈ Z ≥ as T ( U ) k = p ( k + 1)( k + α ) · k + 1, T ( D ) k = p k ( k − α ) · k − 1, and T ( H ) k = (2 k + α ) k . Then the above operator (9.11) in ℓ ( Z ≥ ) is just α I − T ( H ξ ),i.e., the same as in Corollary 9.7, but under a different representation of sl (2 , C ).10. Dynamical correlation functions In this section we prove a Pfaffian formula for the dynamical correlation functions ρ ( n ) α,ξ (2.11) of the Markov processes λ α,ξ on strict partitions, thus finishing the proofof Theorem 2 from § Dynamical correlation functions and Markov semigroups. Let us fix n ≥ t , x ) , . . . , ( t n , x n ) ∈ R ≥ × Z > .We assume that the time moments are ordered as 0 ≤ t ≤ · · · ≤ t n . Recall theoperators ∆ x (where x ∈ Z > ) in the Hilbert space ℓ ( S , M α,ξ ) defined in § Lemma 10.1. The dynamical correlation functions of λ α,ξ have the form ρ ( n ) α,ξ ( t , x ; . . . ; t n , x n ) = (cid:0) ∆ x P ( t − t )∆ x . . . ∆ x n − P ( t n − t n − )∆ x n , (cid:1) M α,ξ , where ( P ( t )) t ≥ is the semigroup of the process λ α,ξ in the space ℓ ( S , M α,ξ ) and ∈ ℓ ( S , M α,ξ ) is the constant identity function.Proof. This is a simple consequence of the Markov property of the process λ α,ξ .Indeed, let us assume (for simplicity) that t j ’s are distinct. The n -dimensionaldistribution of the process λ α,ξ at time moments t < · · · < t n is a probabilitymeasure on S × · · · × S ( n copies) which assigns the probability M α,ξ ( λ (1) ) P λ (1) ,λ (2) ( t − t ) . . . P λ ( n − ,λ ( n ) ( t n − t n − ) (10.1) FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 47 to every point ( λ (1) , . . . , λ ( n ) ), λ ( i ) ∈ S . By definition, ρ ( n ) α,ξ ( t , x ; . . . ; t n , x n ) isexactly the mass of the set { λ (1) ∋ x , . . . , λ ( n ) ∋ x n } under the measure (10.1).This proves the claim for distinct t j ’s. It can be readily verified that the claim alsoholds if some of t j ’s coincide. This concludes the proof. (cid:3) Let us consider the following operator in Fock ( Z > ):∆ J T,X K := ∆ x V ( t − t )∆ x . . . ∆ x n − V ( t n − t n − )∆ x n . Here ( V ( t )) t ≥ is the semigroup in ℓ ( S ) defined in § ℓ ( S ) with Fock ( Z > ) as in § x , x ∈ Z > , are now acting in Fock ( Z > ). Proposition 10.2. The correlation functions of λ α,ξ have the form ρ ( n ) α,ξ ( t , x ; . . . ; t n , x n ) = (cid:16) R ( e G ξ ) − ∆ J T,X K R ( e G ξ ) vac , vac (cid:17) . (10.2) Note that now the expectation is taken in Fock ( Z > ) .Proof. Since V ( t ) = I α,ξ P ( t ) I − α,ξ , the claim is a direct consequence of Lemma 10.1and formula (5.11) with A = ∆ x P ( t − t )∆ x . . . ∆ x n − P ( t n − t n − )∆ x n . (cid:3) Note that in contrast to the static case (7.5), the operator ∆ J T,X K is not diagonal(see also Remark 4.10). It is worth noting that formula (10.2) does not hold if t j ’sare not ordered as t ≤ · · · ≤ t n .10.2. Pre-generator and Kerov’s operators. Our next aim is to extend thedefinition of the semigroup ( V ( t )) t ≥ from real nonnegative values of t to complexvalues of t with ℜ t ≥ 0. This will be needed in the next subsection for computationof the dynamical correlation functions.Observe that the matrix iH ξ (where H ξ is defined in Remark 9.9 and here andbelow i = √− 1) belongs to the real form su (1 , ⊂ sl (2 , C ). Denote W ξ ( τ ) := e − iτH ξ = G ξ (cid:20) e − iτ/ e iτ/ (cid:21) G − ξ ∈ SU (1 , , τ ∈ R . The family { W ξ ( τ ) } τ ∈ R for any fixed ξ ∈ [0 , 1) is a continuous curve in SU (1 , τ = 0. By { f W ξ ( τ ) } τ ∈ R denote the lifting of this curveto the universal covering group SU (1 , ∼ . For real τ one can consider unitary operators R (cid:0)f W ξ ( τ ) (cid:1) = R ( e G ξ ) R (cid:0)f W ( τ ) (cid:1) R ( e G ξ ) − in the Fock space Fock ( Z > ). Here the operator R ( f W ( τ )) (corresponding to ξ = 0)acts on Fock fin ( Z > ) as R ( f W ( τ )) λ = e − iτR ( H ) / λ = e − iτ ( | λ | + α ) λ, λ ∈ S , τ ∈ R . (10.3)Informally speaking, for s ∈ R ≥ , the operator V ( s ) means e s B , and for τ ∈ R ,the operator R ( f W ξ ( τ )) e iτ α I means e iτ B (here B is the generator of the semigroup( V ( s )) s ≥ , see § Compare this with the definition of e G ξ in § Definition 10.3. For t = s + iτ ∈ C + := { w ∈ C : ℜ w ≥ } let V ( t ) be the followingoperator in Fock ( Z > ): V ( t ) := V ( s ) R ( f W ξ ( τ )) e iτ α I For real nonnegative t the operator V ( t ) is self-adjoint and bounded, it wasdefined in § t , the operator V ( t ) is unitary. Thus, theoperators V ( t ) are bounded for all t ∈ C + . Moreover, V ( t + t ) = V ( t ) V ( t ) forany t , t ∈ C + , so { V ( t ) } t ∈ C + is a semigroup (with complex parameter) that canbe viewed as an analytic continuation of the semigroup { V ( s ) } s ∈ R ≥ . In particular,the operators V ( t ) commute with each other. Moreover, it is clear that the func-tion t V ( t ) h is bounded and continuous in C + and holomorphic in the interior { w ∈ C + : ℜ w > } of C + for any vector h ∈ Fock ( Z > ) which is analytic for theoperator B .10.3. Pfaffian formula for dynamical correlation functions.Theorem 10.4. The dynamical correlation functions of the equilibrium Markovprocess ( λ α,ξ ( t )) t ≥ have the form ρ ( n ) α,ξ ( t , x ; . . . ; t n , x n ) = Pf( Φ α,ξ J T, X K ) , (10.4) where the function Φ α,ξ ( s, x ; t, y ) ( x, y ∈ Z , s ≤ t ) is given by Φ α,ξ ( s, x ; t, y ) := ( − x ∧ y ∨ ∞ X m =0 − δ ( m ) e − m ( t − s ) ϕ m ( x ; α, ξ ) ϕ m ( − y ; α, ξ ) (10.5) (see (6.8) for definition of ϕ m ). In (10.4), ( t , x ) , . . . , ( t n , x n ) ∈ R ≥ × Z > arepairwise distinct space-time points such that ≤ t ≤ · · · ≤ t n , and Φ α,ξ J T, X K isthe n × n skew-symmetric matrix with rows and columns indexed by the numbers , − , . . . , n, − n , such that the kj -th entry in Φ α,ξ J T, X K above the main diagonalis Φ α,ξ ( t | k | , x k ; t | j | , x j ) , where k, j = 1 , − , . . . , n, − n (thus, | k | ≤ | j | ). The rest of this subsection is devoted to proving Theorem 10.4. Lemma 10.5 ([Ols08b]) . Let F ( z ) be a function on the right half-plane C + whichis bounded and continuous in C + and is holomorphic in the interior of C + . Then F is uniquely determined by its values on the imaginary axis { w ∈ C : ℜ w = 0 } .Proof. Conformally transforming C + to the unit disc | ζ | < 1, we get a function G on the disc which is holomorphic in the interior of the disc and bounded andcontinuous up to the boundary (with possible exception of one point correspondingto w = ∞ ∈ C + ).For any fixed ζ with | ζ | < 1, the value G ( ζ ) is represented by Cauchy’s integralover the circle | ζ | = r , for | ζ | < r < 1. By our hypotheses, this Cauchy’s integralhas a limit as r → 1, which gives an expression of G ( ζ ) through the boundaryvalues. (cid:3) Let us fix pairwise distinct space-time points ( t , x ) , . . . , ( t n , x n ) ∈ R ≥ × Z > such that 0 ≤ t ≤ · · · ≤ t n . For convenience, set t kj := t k − t j . Above we haveexpressed the dynamical correlation functions as (10.2), that is, ρ ( n ) α,ξ ( t , x ; . . . ; t n , x n )= (cid:16) R ( e G ξ ) − ∆ x V ( t , )∆ x . . . ∆ x n − V ( t n,n − )∆ x n R ( e G ξ ) vac , vac (cid:17) . (10.6) Here and below we use convention (7.1). FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 49 Denote the right-hand side of (10.6) by F ( t , , . . . , t n,n − ; x , . . . , x n ). As a func-tion in n − t , , . . . , t n,n − , F is initially defined for t j,j − taking realnonnegative values. However, as explained in § V ( t j,j − ) can be extended to t j,j − ∈ C + , so F is defined on ( C + ) n − ⊂ C n − .Moreover, F is continuous and bounded in ( t , , . . . , t n,n − ) belonging to the closeddomain ( C + ) n − and is holomorphic in the interior of this domain. Therefore,by Lemma 10.5, F ( t , , . . . , t n,n − ; x , . . . , x n ) is uniquely determined by its valueswhen all the variables t j,j − are purely imaginary.From now on in the computation we will assume that the variables t j = iτ j (where τ j ∈ R , j = 1 , . . . , n ) are purely imaginary. This implies that the differences t k,j = i ( τ k − τ j ) are also purely imaginary. For such t j , each operator V ( t j,j − ) isunitary, and, moreover, V ( t j,j − ) = V ( t j − , ) − V ( t j, ) , j = 1 , . . . , n (10.7)(here by agreement t , = 0, and V (0) = I , the identity operator).For purely imaginary t j , we want to rewrite F ( t , , . . . , t n,n − ; x , . . . , x n ) as acertain Pfaffian. First, we need a notation. Recall that in § S of SU (1 , 1) in the Hilbert space V = ℓ ( Z ) with the standardorthonormal basis { v x } x ∈ Z . Denote v ( t ) x,ξ := ˇ S ( W ( τ )) ˇ S ( G ξ ) − v x ∈ V, x ∈ Z , t = iτ ∈ i R . (10.8)For t = 0 this vector becomes v x,ξ defined by (7.13). Lemma 10.6. For t j = iτ j ∈ i R and distinct x j ∈ Z > ( j = 1 , . . . , n ) we have F ( t , , . . . , t n,n − ; x , . . . , x n ) = Pf( F J T, X K ) , (10.9) where F J T, X K is the n × n skew-symmetric matrix with rows and columns in-dexed by the numbers , − , . . . , n, − n , such that the kj -th entry in F J T, X K abovethe main diagonal is F vac (cid:0) v ( t | k | , ) x k ,ξ , v ( t | j | , ) x j ,ξ (cid:1) , where k, j = 1 , − , . . . , n, − n (and thus | k | ≤ | j | ). Here F vac is the vacuum average on the Clifford algebra Cl ( V ) , see § The operators ∆ x have the form∆ x = T ( v x ) T ( v − x ) = T ( v x v − x ) , x ∈ Z > . By (10.8) and Proposition 7.3, we have R ( f W ( τ )) R ( e G ξ ) − ∆ x R ( e G ξ ) R ( f W ( τ )) − = T ( v ( t ) x,ξ v ( t ) − x,ξ ) , x ∈ Z , t = iτ ∈ i R . A straightforward computation using Definition 10.3 and (10.7) allows us to rewritethe operator in the right-hand side of (10.6) as R ( e G ξ ) − ∆ x V ( t , )∆ x . . . ∆ x n − V ( t n,n − )∆ x n R ( e G ξ )= T ( v ( t , ) x ,ξ v ( t , ) − x ,ξ . . . v ( t n, ) x n ,ξ v ( t n, ) − x n ,ξ ) R ( f W ( τ n, )) e iτ n, α I . Observe that from (10.3) it follows that R ( f W ( τ n, )) e iτ n, α I vac = vac , so F ( t , , . . . , t n,n − ; x , . . . , x n ) = (cid:0) T ( v ( t , ) x ,ξ v ( t , ) − x ,ξ . . . v ( t n, ) x n ,ξ v ( t n, ) − x n ,ξ ) vac , vac (cid:1) = F vac (cid:16) v ( t , ) x ,ξ v ( t , ) − x ,ξ . . . v ( t n, ) x n ,ξ v ( t n, ) − x n ,ξ (cid:17) . An application of Wick’s Theorem 5.1 concludes the proof. (cid:3) Now that we have established a Pfaffian formula for purely imaginary time vari-ables t j,j − ( j = 2 , . . . , n ), we want to extend it to the case when all t j,j − ’s arereal nonnegative. Let us look closer at the function F vac ( v ( s ) x,ξ v ( t ) y,ξ ), where s = iσ and t = iτ are purely imaginary. We have v ( s ) x,ξ = ˇ S ( W ( σ )) v x,ξ , v ( t ) y,ξ = ˇ S ( W ( τ )) v y,ξ , where v x,ξ and v y,ξ are defined by (7.13). Therefore, we get F vac ( v ( s ) x,ξ v ( t ) y,ξ ) = X k,l ∈ Z ( v x,ξ , v k ) ℓ ( Z ) ( v y,ξ , v l ) ℓ ( Z ) F vac (cid:0)(cid:0) ˇ S ( W ( σ )) v k (cid:1)(cid:0) ˇ S ( W ( τ )) v l (cid:1)(cid:1) . On the space V fin ⊂ V = ℓ ( Z ) consisting of finite linear combinations of the basisvectors { v x } x ∈ Z , the operator ˇ S ( W ( u )) acts as e − iu ˇ S ( H ) / (where u ∈ R ). Fromthis fact and (7.15), we see that F vac ( v ( s ) x,ξ v ( t ) y,ξ ) = X ∞ m =0 e − m ( t − s ) ( v x,ξ , v − m ) ℓ ( Z ) ( v y,ξ , v m ) ℓ ( Z ) = ( − x ∧ y ∨ X ∞ m =0 − δ ( m ) e − m ( t − s ) ϕ m ( x ) ϕ m ( − y ) . (10.10)Note that here s and t are still purely imaginary. However, one can view the right-hand side of (10.10) as a function in ( t − s ) ∈ C + . This function is bounded andcontinuous in C + and is holomorphic in the interior of C + , because the functions ϕ m ( x ) for fixed x and m → + ∞ decay as Const · m − x − ξ m (this can be readilyobserved from the analytic expression (6.8)). We are interested in the restrictionof the right-hand side of (10.10) to real nonnegative values of ( t − s ). Observethat this is exactly the kernel Φ α,ξ ( s, x ; t, y ) (10.5). By application of Lemma 10.5,we see that formula (10.9) holds for real nonnegative t , , . . . , t n,n − , that is, for0 ≤ t ≤ · · · ≤ t n . This fact together with Proposition 10.2 implies Theorem 10.4.Thus, we have finished the proof of Theorem 2 from § Skew-symmetric matrices in Pfaffian formulas. In the right-hand sidesof our Pfaffian formulas (7.3) and (10.4) for static and dynamical correlation func-tions we see certain skew-symmetric 2 n × n matrices constructed using Pfaffiankernels Φ α,ξ ( x, y ) and Φ α,ξ ( s, x ; t, y ), respectively. It is clear from (7.16) and (10.5)that for t = s , the dynamical kernel Φ α,ξ ( s, x ; t, y ) turns into the static one. (Infact, this is the reason why we use the same notation for these kernels.) However,the matrix of (10.4) for t = · · · = t n does not become the one from (7.3). Letus explain how one can transform (10.4) to get the expected behavior in the staticpicture.One can readily verify that conjugating the matrix Φ α,ξ J T, X K from (10.4) bythe matrix C of the permutation (1 , n, , n − , . . . , n, n + 1), we get the following2 n × n skew-symmetric matrix:( C Φ α,ξ J T, X K C T ) i,j (10.11)= Φ α,ξ ( t i , x i ; t j , x j ) , if 1 ≤ i < j ≤ n ; Φ α,ξ ( t i , x i ; t j ′ , − x j ′ ) , if 1 ≤ i ≤ n < j ≤ n and i ≤ j ′ ; − Φ α,ξ ( t j ′ , − x j ′ ; t i , x i ) , if 1 ≤ i ≤ n < j ≤ n and i > j ′ ; − Φ α,ξ ( t j ′ , − x j ′ ; t i ′ , − x i ′ ) , if n < i < j ≤ n, where 1 ≤ i < j ≤ n , and i ′ := 2 n + 1 − i , j ′ := 2 n + 1 − j . The permutationmatrix C has determinant one (cf. how (7.6) is obtained), so the Pfaffian does notchange under such a conjugation. It is worth noting that matrices similar to (10.11) FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 51 appeared in [Mat05, Thm. 3.1] and [Vul07, Thm. 2.2]). Using Corollary 7.8, it isclear that for t = · · · = t n , (10.11) becomes the matrix ˆ Φ α,ξ J X K . Observe thatCorollary 7.8.(2) does not hold in the dynamical case, so one cannot put a matrixof the form ˆ Φ α,ξ J T, X K in the right-hand side of (10.4).11. Dynamical Pfaffian kernel In this section we discuss some properties of the extended Pfaffian hypergeo-metric-type kernel Φ α,ξ ( s, x ; t, y ) (obtained in the previous section) of our Markovprocesses ( λ α,ξ ( t )) t ≥ on strict partitions.11.1. Expression through the extended discrete hypergeometric kernel. The extended discrete hypergeometric kernel introduced in [BO06a] serves as adeterminantal kernel for a Markov dynamics preserving the z -measures on ordinarypartitions. It is given by [BO06a, Thm. A (Part 2)]: K z,z ′ ,ξ ( t, x ‘ ; s, y ‘ ) = ± X a ‘ ∈ Z ′ + e − a ‘ | t − s | ψ ± a ‘ ( x ‘ ; z, z ′ , ξ ) ψ ± a ‘ ( y ‘ ; z, z ′ , ξ ) , where x ‘ , y ‘ ∈ Z ′ = Z + , the “+” sign is taken for t ≥ s , and the “ − ” sign is takenfor t < s . Our kernel Φ α,ξ ( s, x ; t, y ) can be expressed through K z,z ′ ,ξ ( t, x ‘ ; s, y ‘ ): Proposition 11.1. For all x, y ∈ Z and s ≤ t , we have Φ α,ξ ( s, x ; t, y ) = ( − x ∧ y ∨ h e − ( t − s ) K ν ( α ) − , − ν ( α ) − ,ξ ( t, x + ; s, − y + )+ e ( t − s ) K ν ( α )+ , − ν ( α )+ ,ξ ( t, x − ; s, − y − ) i , (11.1) Proof. This is established similarly to (7.18): using the above formula for the kernel K z,z ′ ,ξ ( t, x ‘ ; s, y ‘ ) and (6.9), one can write two identities analogous to (7.19) and(7.20), then take their half sum and use (10.5). (cid:3) When t = s , (11.1) reduces to the static version (7.18). Observe that the param-eters z, z ′ (depending on α ) in (11.1) are admissible (see p. 26). However, similarlyto the static identity, (11.1) seems to imply no probabilistic connections betweenthe dynamics related to the z -measures [BO06a] and our Markov processes λ α,ξ .11.2. Plancherel degeneration.Theorem 11.2. Under the Plancherel degeneration (2.8), the extended hypergeo-metric-type kernel Φ α,ξ has a pointwise limit. The limiting kernel is expressedthrough the Bessel function (8.4): Φ θ ( s, x ; t, y ) = ( − x ∧ y ∨ X ∞ m =0 − δ ( m ) e − m ( t − s ) J m + x (2 √ θ ) J m − y (2 √ θ ) (here x, y ∈ Z and ≤ s ≤ t ).Proof. This is a consequence of (10.5) and the fact that under the Plancherel de-generation (2.8) one has ϕ m ( x ; α, ξ ) → J x + m . (cid:3) In the static case ( t = s ), the above theorem reduces to a Pfaffian formula forcorrelation functions of the poissonized Plancherel measure with the kernel Φ θ ( x, y ).This Pfaffian formula then can be written as a determinantal one, see Theorem 8.2. “Whittaker” limit. Here we consider the limit of our dynamical kernel Φ α,ξ ( s, x ; t, y ) which corresponds to studying the dynamics of the scaled largestrows in a strict partition. We embed the half-lattice Z > into the half-line R > as x (1 − ξ ) x , where x ∈ Z > , and then pass to the limit as ξ ր 1. For the staticpicture this limit transition was described in [Pet10a, § K α,ξ ( x, y ) was given.Let w m ( u ; α ) be the functions indexed by m ∈ Z with argument u ∈ R > , whichare expressed through the classical Whittaker functions W κ,µ ( u ) (e.g., see [Erd53,Ch. 6.9]) as follows: w m ( u ; α ) := (cid:0) Γ( − m − ν ( α ))Γ( − m + ν ( α )) (cid:1) − u − W − m,ν ( α ) ( u ) . These functions w m ( u ; α ) are related to the functions w a ‘ ( u ; z, z ′ ) [BO06a, (9.1)]in the same manner as in (6.9): w m ( u ; α ) = w m + + d ( u ; ν ( α ) + + d, − ν ( α ) + + d ) , m ∈ Z , u > , (11.2)where d ∈ Z is arbitrary. Thus, from [BO06a, Prop. 9.1] it follows that if ξ ր x ∈ Z goes to + ∞ so that (1 − ξ ) x → u > 0, then ϕ m ( x ; α, ξ ) ∼ (1 − ξ ) w m ( u ; α ). Theorem 11.3. As ξ ր , x, y → ∞ (inside Z ), and (1 − ξ ) x → u , (1 − ξ ) y → v ,where u, v ∈ R =0 , there exists a limit of the extended Pfaffian kernel: Φ Wα ( s, u ; t, v ) = lim ξ ր (1 − ξ ) − Φ α,ξ ( s, x ; t, y ) , ≤ s ≤ t (the prefactor (1 − ξ ) − is due to the rescaling of the space Z > ). For s < t , thelimiting kernel Φ Wα ( s, u ; t, v ) is given for u, v > by Φ Wα ( s, u ; t, v ) = X ∞ m =0 − δ ( m ) e − m ( t − s ) ( − m w m ( u ; α ) w − m ( v ; α ) , Φ Wα ( s, u ; t, − v ) = X ∞ m =0 − δ ( m ) e − m ( t − s ) w m ( u ; α ) w m ( v ; α ) , Φ Wα ( s, − u ; t, v ) = X ∞ m =0 − δ ( m ) e − m ( t − s ) w − m ( u ; α ) w − m ( v ; α ) , Φ Wα ( s, − u ; t, − v ) = X ∞ m =0 − δ ( m ) e − m ( t − s ) ( − m w − m ( u ; α ) w m ( v ; α ) . For s = t , one should take limits in the expression for the static kernel given byProposition 7.7 (the above expressions for Φ Wα ( s, u ; s, v ) and Φ Wα ( s, − u ; s, − v ) donot converge). This leads to an “integrable” expression for the limiting static Pfaf-fian kernel Φ Wα ( s, u ; s, v ) . This kernel is reduced to the determinantal Macdonaldkernel given in [Pet10a, Theorem 3.2] .Proof. This is a consequence of results of [BO06a, § Φ Wα are ob-tained using formula (10.5) for the pre-limit kernel Φ α,ξ . One should considervarious cases of signs of x, y , and with the help of (6.11) make the arguments ofthe functions ϕ m ( · ; α, ξ ) positive. After that one should replace each ϕ m ( · ; α, ξ ) bythe corresponding w m ( · ; α ). All limit transitions can be justified using expression(11.1) of our kernel through the dynamical kernel of [BO06a]. (cid:3) Remark 11.4. It is known [BO06a, § 8] that the discrete hypergeometric kernel K z,z ′ ,ξ ( s, x ; t, y ) itself (even in the fixed time s = t picture) does not converge inthe limit described in the above theorem. For this kernel to have a limit one shouldperform a certain particle-hole involution, see [BO06a, § K z,z ′ ,ξ (with suitably chosen parameters)as in (11.1) does have a limit in this regime, which is given in the above theorem. FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 53 Remark 11.5. It is interesting to compare the limit behavior of the relation (8.3)in the above scaling limit as ξ ր § r uv K Mac α ( u, v ) = K ++ ν ( α )+ , − ν ( α )+ ( u, v ) − K + − ν ( α ) − , − ν ( α ) − ( u, v ) , u, v > , where K Mac α is the Macdonald kernel of [Pet10a, Thm. 3.2], and K ++ z,z ′ and K + − z,z ′ are the blocks of the matrix Whittaker kernel of [BO00, § K Mac α ( u, v ) = K ++ ν ( α ) − , − ν ( α ) − ( u, v ) − i K + − ν ( α ) − , − ν ( α ) − ( u, v ) , u, v > . The advantage of the first relation is that it involves admissible parameters z, z ′ (seep. 26) in contrast to the second relation. One can show that the two above relationsare equivalent by manipulations with the confluent hypergeometric function usingformulas from [Erd53, Ch. VI].11.4. “Gamma” limit. The second limit regime we consider corresponds to study-ing the dynamics of the smallest rows in a strict partition. We stay on the lattice Z > and pass to the limit as ξ ր 1. An appropriate scaling of time is needed. Forthe static picture the corresponding limit of the determinantal kernel K α,ξ ( x, y )was given in [Pet10a, § Theorem 11.6. Let x, y ∈ Z and s = (1 − ξ ) σ , t = (1 − ξ ) τ , where σ, τ ∈ R , σ ≤ τ .As ξ ր , there exists a limit of the extended Pfaffian kernel: Φ gamma α ( σ, x ; τ, y ) = lim ξ ր Φ α,ξ (cid:0) (1 − ξ ) σ, x ; (1 − ξ ) τ, y (cid:1) . The limiting kernel has the form Φ gamma α ( σ, x ; τ, y ) = ( − x ∧ y ∨ Z + ∞ e − u ( τ − σ ) w x ( u ; α ) w − y ( u ; α ) du. For σ = τ , the static Pfaffian kernel Φ gamma α ( σ, x ; σ, y ) admits a simpler “inte-grable” form which corresponds to the determinantal kernel given in [Pet10a, The-orem 3.1] .Proof. After a simple computation, this follows from the result of [BO06a, Thm.10.1] together with (11.1) and (11.2). (cid:3) Observe that in contrast to the scaling limit regime of the previous subsection(see Remark 11.4), here the kernel K z,z ′ ,ξ ( s, x ; t, y ) itself has a limit in the regimedescribed in the above theorem. Remark 11.7. One could also prove Theorems 11.3 and 11.6 by writing doublecontour integrals for the pre-limit kernel Φ α,ξ ( s, x ; t, y ), and passing to the corre-sponding limits. In this way one can obtain double contour integral expressions forthe kernels Φ Wα ( s, u ; t, v ) and Φ gamma α ( σ, x ; τ, y ) similar to the ones of Theorems 9.4and 10.1 in [BO06a], respectively. On the other hand, these double contour integralexpressions for Φ Wα and Φ gamma α readily follow from our formulas of Theorems 11.3and 11.6. See also [BO06a, § z -measures. Appendix A. Reduction of Pfaffians to determinants Let us first recall basic definitions and properties related to Pfaffians. We usethe following notations for matrices. Let X be an abstract finite space of indicesand a = ( a , . . . , a n ) be a sequence of length 2 n of points of X . Let F : X × X → C be some function. Form a 2 n × n skew-symmetric matrix F ( a , a ) . . . F ( a , a n − ) F ( a , a n ) − F ( a , a ) 0 . . . F ( a , a n − ) F ( a , a n ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . − F ( a , a n − ) − F ( a , a n − ) . . . F ( a n − , a n ) − F ( a , a n ) − F ( a , a n ) . . . − F ( a n − , a n ) 0 . Denote this matrix by F J a K . This skew-symmetric matrix has rows and columnsindexed by a , . . . , a n , such that the ij th element above the main diagonal is equalto F ( a i , a j ) (here 1 ≤ i < j ≤ n ). Definition A.1. Let a = ( a , . . . , a n ) and F J a K be as defined above. The deter-minant det( F J a K ) is a perfect square as a polynomial in F ( a i , a j ) (where i < j ).The Pfaffian of F J a K , denoted by Pf (cid:0) F J a K (cid:1) , is defined to be the square root ofdet F J a K having the “+” sign by the monomial F ( a , a ) . . . F ( a n − , a n ).The following properties of Pfaffians are well known: • Let A be a skew-symmetric 2 n × n matrix and B be any 2 n × n matrix, thenPf( BAB T ) = det B · Pf( A ) . (A.1)where B T means the transposed matrix; • If M is any n × n matrix, thenPf (cid:18) M − M T (cid:19) = ( − n ( n − / det M. (A.2)Now we give a sufficient condition under which a 2 n × n Pfaffian can be reducedto a certain n × n determinant. Assume that the set X is divided into two parts X = X + ⊔ X − , and there exists a bijection between X + and X − . By a ˆ a wedenote the corresponding involution of the space X that interchanges X + and X − .Let a := ( a , . . . , a n , ˆ a n , . . . , ˆ a ), and a i ∈ X + (so ˆ a i ∈ X − ), i = 1 , . . . , n . Proposition A.2. Suppose that the function F on X × X satisfies the followingproperties: (1) F ( a, ˆ b ) = F ( b, ˆ a ) for any a, b ∈ X . (2) F ( a, b ) = − F ( b, a ) for any a, b ∈ X such that a = ˆ b . (3) There exists a strictly positive function f : X + → R with the property f ( a ) = f ( b ) if a = b , such that ( f ( a ) − f ( b )) F ( a, ˆ b ) = ( f ( a ) + f ( b )) F ( a, b ) for any a, b ∈ X + . Then Pf (cid:0) F J a , . . . , a n , ˆ a n , . . . , ˆ a K (cid:1) = det[ K ( a r , a s )] nr,s =1 , where K has the form K ( u, v ) = 2 F ( u, ˆ v ) p f ( u ) f ( v ) f ( u ) + f ( v ) , u, v ∈ X + . (A.3) cf. Corollary 7.8. FAFFIAN STOCHASTIC DYNAMICS OF STRICT PARTITIONS 55 Note that the third property above implies that F ( a, a ) = 0 for all a ∈ X + . Proof. In this proof we denote the matrix F J a , . . . , a n , ˆ a n , . . . , ˆ a K simply by F .We act on F by SL (2 , C ) n : each j th copy of SL (2 , C ) acts as F C j F C Tj , where C j is the 2 n × n identity matrix except for the 2 × j and 2 n + 1 − j . By (A.1), this actionof SL (2 , C ) n does not change the Pfaffian of F . We want to choose C ∈ SL (2 , C ) n such that the matrix CF C T becomes a block matrix as in (A.2).Define g ( a ) := log f ( a ), a ∈ X + . As the j th element in C ∈ SL (2 , C ) n we takethe hyperbolic rotation (cid:18) cosh g ( a j ) sinh g ( a j )sinh g ( a j ) cosh g ( a j ) (cid:19) . The whole matrix C looks as C = cosh g ( a ) . . . . . . sinh g ( a ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 . . . cosh g ( a n ) sinh g ( a n ) . . . . . . sinh g ( a n ) cosh g ( a n ) . . . 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .sinh g ( a ) . . . . . . cosh g ( a ) . It can be readily verified using the properties of F that CF C T = (cid:18) M − M T (cid:19) ,where the rows of M are indexed by i = 1 , , . . . , n , and columns are indexed by j = n + 1 , . . . , n , and M ij = F ( a i , a n +1 − j ) p f ( a i ) f ( a n +1 − j ) f ( a i ) − f ( a n +1 − j ) , if i + j = 2 n,F ( a i , ˆ a i ) , otherwise . Set, for r, s = 1 , . . . , n , K ( a r , a s ) := M r, n + s − , (A.4)and note that det [ K ( a r , a s )] nr,s =1 = ( − n ( n − / det M . 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