Pfaffian Stochastic Dynamics of Strict Partitions

We study a family of continuous time Markov jump processes on strict partitions (partitions with distinct parts) preserving the distributions introduced by Borodin (1997) in connection with projective representations of the infinite symmetric group. The one-dimensional distributions of the processes (i.e., the Borodin's measures) have determinantal structure. We express the dynamical correlation functions of the processes in terms of certain Pfaffians and give explicit formulas for both the static and dynamical correlation kernels using the Gauss hypergeometric function. Moreover, we are able to express our correlation kernels (both static and dynamical) through those of the z-measures on partitions obtained previously by Borodin and Olshanski in a series of papers. The results about the fixed time case were announced in the author's note arXiv:1002.2714. A part of the present paper contains proofs of those results.

We introduce and study a family 1 of continuous time Markov jump processes on the 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 Transmission Problems, Bolshoy Karetny per. 19, Moscow, 127994, Russia E-mail address: lenia.petrov@gmail.com.
The author was partially supported by the RFBR-CNRS grant 10-01-93114, by A. Kuznetsov's graduate student scholarship and by the Dynasty foundation fellowship for young scientists. 1 The whole picture depends on two continuous parameters α > 0 and 0 < ξ < 1.
Borodin [Bor99] in connection with the harmonic analysis of projective representations of the infinite symmetric group. The construction of our dynamics is similar to that of Borodin and Olshanski [BO06a] and is based on a special coherency property 2 of the measures on strict partitions introduced in [Bor99]. Regarding each strict partition λ = (λ 1 > · · · > λ ℓ > 0), λ j ∈ Z, as a point configuration {λ 1 , . . . , λ ℓ } on the half-lattice Z >0 := {1, 2, . . . }, one can say that the state space of our Markov processes is the space of all finite point configurations on Z >0 . The fixed time distributions of our dynamics are probability measures on this configuration space. In other words, in the static (fixed time) picture one sees a random point process on Z >0 . The main result of the paper is the computation of the dynamical (or spacetime) correlation functions for our family of Markov processes. We show that these correlation functions have certain Pfaffian form, and compute the corresponding kernel. 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 Gauss hypergeometric 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 Markov processes can be reduced to a determinantal one. Thus, in the fixed time picture we have a determinantal point process on Z >0 . Its kernel K α,ξ has integrable form and is also expressed through the Gauss hypergeometric function. (About integrable operators, e.g., see [IIKS90], [Dei99], discrete integrable operators are discussed in [Bor00] and [BO00,§6].) We call this kernel the hypergeometric-type kernel. The results about the static case were announced in the note [Pet10a]. A part of the present paper ( §4- §8) contains complete proofs of those results.
Comparison with results for the z-measures. Our model of random strict partitions and associated stochastic dynamics is very similar to the one of the zmeasures on ordinary partitions. 3 The structure of static and dynamical correlation functions in that case was investigated in [BO00], [Oko01b], [BO06a], [BO06b]. Let us discuss the relationship of our results with the ones from those papers.
• The main feature of our model is that its dynamical correlation functions are expressed in terms of Pfaffians and not determinants, as it is for the z-measures.
2 which has a representation-theoretic meaning. 3 The z-measures originated from the problem of harmonic analysis for the infinite symmetric group S∞ [KOV93], [KOV04] and were studied in detail by Borodin, Okounkov, Olshanski, and other authors, e.g., see the bibliography in [BO09].
• Determinantal (static) correlation kernels of random point processes often appear to be projection operators. In particular, this holds for the z-measures. In contrast, in our situation the kernel K α,ξ (x, y) (x, y ∈ Z >0 ) is not a projection operator in the corresponding coordinate Hilbert space ℓ 2 (Z >0 ). • On the other hand, the static Pfaffian kernel Φ α,ξ (s, x; s, y) (where x, y ∈ Z) in our model has a structure which is very similar to that of the determinantal kernel of the z-measures on semi-infinite point configurations on the lattice. Viewed as an operator in the Hilbert space ℓ 2 (Z), Φ α,ξ (s, ·; s, ·), is a rank one perturbation of an orthogonal projection operator. • Furthermore, our extended Pfaffian kernel Φ α,ξ (s, x; t, y) is obtained from the static kernel Φ α,ξ (s, x; s, y) in a way which is common for Markov processes on configuration spaces with determinantal dynamical correlation functions. In particular, the same extension happens in the case of the z-measures. • Markov processes of [BO06a], as well as many dynamical determinantal models that arise in the theory of random matrices and random tilings (e.g., see [NF98], [War07], [JN06], [ANvM10], [Joh02], [Joh05], [BGR10]), are closely related to orthogonal polynomials. Moreover, connections with orthogonal polynomials also arise in static Pfaffian models of random-matrix and representation-theoretic origin [NW91a], [NW91b], [TW96], [KNT04], [Kat05], [Nag07], [BS06], [BS09], [Str10a]. For our model there also exists a connection with orthogonal polynomials (namely, the Krawtchouk polynomials), but this connection does not help us to 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 kernels in the case of the z-measures. These functions first appeared in the works of Vilenkin and Klimyk [VK88], [VK95]. In particular, certain degenerations of them lead to the classical Meixner and Krawtchouk orthogonal polynomials. Using this fact, we are able to express our kernels directly through the corresponding kernels for the z-measures. These expressions seem to have no direct probabilistic meaning at the level of random point processes, but in particular they 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 kernels in an explicit form is different from those of [BO00], [BO06b], [BO06a], and is based on computations in the fermionic Fock space involving so-called Kerov's operators which span a certain sl(2, C)-module. Both the static and dynamical correlation kernels 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. 4 In computations in this paper we use the ordinary fermionic Fock space instead of the (closely related, but different) infinite wedge space of [Oko01b]. Moreover, our situation also requires to deal with a Clifford algebra (acting in the fermionic Fock space) of a different type. One can say that our Clifford algebra is an infinite-dimensional generalization of the Clifford algebra over an odd-dimensional quadratic space. Similar Clifford algebras were used in [DJKM82], [Mat05], [Vul07]. In the latter two papers 4 A possibility of use of this method in studying the dynamical model related to the z-measures was pointed out in [BO06a], and later this approach was carried out by Olshanski [Ols08b]. the fermionic Fock space is also used for computations of certain correlation functions. That approach is analogous to the formalism of Schur measures and Schur processes [Oko01a], [OR03] and differs from the one used in the present paper.
Organization of the paper. In §2 we give main definitions and state main results about our model. In §3 we discuss combinatorial constructions from which our model arises. We also give an argument why the corresponding fixed time random point processes on Z >0 are determinantal.
In §4 we study Kerov's operators on strict partitions. These operators provide us with a convenient way of writing expectations with respect to our point processes. Such formulas are used in the computation of both static and dynamical correlation functions. In §5 we recall the formalism of the fermionic Fock space and define an action of a Clifford algebra in it. These structures are extensively used in our computations.
In §6 we discuss functions (matrix elements of a certain sl(2, C)-module) which are used in explicit expressions for our correlation kernels. These functions are eigenfunctions of a certain second order difference operator on the lattice Z. This fact allows later to interpret our kernels through orthogonal spectral projections related to that operator on the lattice. In §6 we also recall the results of [BO00], [BO06b], and [BO06a] about the z-measures on ordinary partitions which we use in the study of our model.
In §7 we prove that the static correlation functions of our Markov dynamics can be written as certain Pfaffians. We express the Pfaffian kernel through matrix elements related to Kerov's operators, and through the functions discussed in §6. In §8 we write the static correlation functions as determinants and express the determinantal correlation kernel in various forms (including a so-called integrable form).
The Markov processes on strict partitions are defined in §9 in terms of their jump rates. In §10 we show that the dynamical correlation functions of our Markov processes have Pfaffian form, and give an explicit expression for the dynamical (Pfaffian) correlation kernel in terms of the functions discussed in §6. We consider the asymptotic behavior of our dynamical Pfaffian kernel under a degeneration and in two limits regimes in §11.
Acknowledgements. The author is very grateful to Grigori Olshanski for permanent attention to the work, fruitful discussions, and access to his unpublished manuscript [Ols08b]. I would also like to thank Alexei Borodin and Vadim Gorin for useful comments on my work.

Model and results
2.1. Point processes on the half-lattice. Let us first describe the fixed time picture, that is, the random point processes on the half-lattice Z >0 that we study. They arise from a model of random strict partitions introduced in [Bor99].
The description of the model of [Bor99] starts with the Plancherel measures on strict partitions of a fixed weight: λ ∈ S n (2.1) (by Pl n (λ) we denote the measure of a singleton {λ}, and the same agreement for other measures on strict partitions is used throughout the paper). The measure Pl n is a probability measure on S n . The set S n parametrizes irreducible truly projective representations of the symmetric group S n [Sch11], [HH92], and the measures Pl n on S n are analogues (in the theory of projective representations of S n ) of the wellknown Plancherel measures on ordinary partitions. The system of measures {Pl n } possesses the coherency property (3.3) that has a representation-theoretic meaning, see §3.2 below. The Plancherel measures on strict partitions were studied in, e.g., [Bor99], [Iva99], [Iva06], [Pet10b]. We consider the poissonized Plancherel measure on the set S := ∞ n=0 S n of all strict partitions: where θ > 0 is a parameter. In other words, we mix the measures Pl n on S n using the Poisson distribution on the set Z ≥0 := {0, 1, 2, . . . } of indices n. Regarding each strict partition λ = (λ 1 , . . . , λ ℓ(λ) ) as a point configuration λ 1 , . . . , λ ℓ(λ) on Z >0 (to the empty partition ∅ corresponds the empty configuration), we view Pl θ as a random point process on the half-lattice Z >0 . 6 Like for the Plancherel measures on ordinary partitions [Joh01], [Oko00], [BOO00] (see also [BDJ99], [BDJ00]), the poissonization (2.2) of the measures Pl n on strict partitions leads to a determinantal point process, see Theorem 1 below.
Our first result is the computation of the correlation functions of the point processes M α,ξ and Pl θ . Recall that the correlation functions of a random point process on Z >0 are defined as ρ (n) (x 1 , . . . , x n ) := Prob {the random configuration contains x 1 , . . . , x n } , (2.9) where n = 1, 2, . . . and x 1 , . . . , x n are pairwise distinct points of Z >0 . Under mild assumptions, the correlation functions determine the point process uniquely. A point process on Z >0 is called determinantal, if there exists a function K on Z >0 × Z >0 (called the (determinantal ) correlation kernel ) such that the correlation functions ρ (n) , n = 1, 2, . . . , have the following form: . About determinantal point processes see, e.g., the surveys [Sos00], [HKPV06], [Bor09].
Theorem 1. Both the point processes M α,ξ and Pl θ on the half-lattice Z >0 are determinantal. The correlation kernel K α,ξ of M α,ξ is expressed through the Gauss hypergeometric function (8.1), (8.2). The correlation kernel K θ of Pl θ can be written in terms of the Bessel function of the first kind (8.5), (8.6).
We call the kernel K α,ξ the hypergeometric-type kernel. The kernel K θ is obtained 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 hypergeometric kernel introduced in [BO00], [BO06b]. This is done in the same spirit as is explained in §2.3 below.
2.2. Dynamical model. Let us now describe a family of continuous time Markov jump processes (λ α,ξ (t)) t∈[0,+∞) on the space of all strict partitions S (which is the same as the set of all finite configurations on Z >0 ). These processes preserve the measures 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 process on Z >0 denoted by (n α,ξ (t)) t∈[0,+∞) . It depends on our parameters α and ξ and has the following jump rates: The process n α,ξ preserves the negative binomial distribution π α,ξ (2.4) on Z >0 and is reversible with respect to it. About birth and death processes in general see, e.g., [KM57], [KM58].
We describe the dynamics λ α,ξ on strict partitions in terms of jump rates. The jumps are of two types: one can either add a box to the random shifted Young diagram, or remove a box from it (of course, the result must still be a shifted Young diagram). The events of adding and removing a box are governed by the birth 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), the choice 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 the jump n → n − 1 of 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 − 1) λ,µ , where µ ∈ S n−1 .
The fact that the process n α,ξ preserves the mixing distribution π α,ξ together with (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≥0 we mean the equilibrium process (that is, the process starting from the invariant distribution M α,ξ ).
The nth up/down Markov chain on S n can be reconstructed from (λ α,ξ (t)) t≥0 as follows. Condition the process λ α,ξ to stay in the set S n × S n+1 , and take its embedded Markov chain, that is, consider the continuous time process only at the times 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 reconstruct the Markov chain on S n with the transition operator p ↑ α (n, n + 1) • p ↓ (n + 1, n).
The dynamical (or space-time) correlation functions of the Markov process λ α,ξ are defined as The notion of dynamical correlation functions is a combination of finite-dimensional distributions of a stochastic dynamics and correlation functions of a random point process. Indeed, the finite-dimensional distribution of the process λ α,ξ at times t 1 , . . . , t n (let these times be distinct for simplicity) is a probability measure on configurations on the space Z >0 ⊔ · · · ⊔ Z >0 (n copies), and ρ (n) α,ξ (t 1 , x 1 ; . . . ; t n , x n ) (t j 's fixed) are just the correlation functions of this measure on configurations. The dynamical 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 correlation functions of λ α,ξ .
To formulate the result, we need a notation. By Z =0 denote the set of all nonzero integers, and for x 1 , . . . , x n ∈ Z >0 put, by definition, x −k := −x k , k = 1, . . . , n.
Remark 2.3. 1. Observe that it is enough for Φ α,ξ (s, x; t, y) to be defined only for 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 we always let Φ α,ξ (s, x; t, y) to be defined for all x, y ∈ Z. The same is applicable to the static Pfaffian kernel Φ α,ξ (x, y) := Φ α,ξ (s, x; s, y) (see 7.16 below). 2. In (2.12) we require that the time moments t j are ordered. However, Theorem 2 allows to compute the correlation functions ρ (n) α,ξ (t 1 , x 1 ; . . . ; t n , x n ) with arbitrary order of time moments: one should simply permute the space-time points (t 1 , x 1 ), . . . , (t n , x n ) (this does not change the value of ρ (n) α,ξ (t 1 , x 1 ; . . . ; t n , x n )) in such a way that the time moments become nondecreasing, and then apply (2.12).
In §11.2 we consider the Plancherel degeneration (2.8) of the dynamical kernel Φ α,ξ . The resulting kernel Φ θ is expressed through the Bessel function of the first kind (Theorem 11.2). The dynamical kernel Φ θ has analogues related to the Plancherel measures on ordinary partitions and associated dynamics, see [PS02], [BO06c]. The asymptotic behavior of the dynamical kernel Φ α,ξ in two other limit regimes (corresponding to studying smallest resp. largest components of a random strict partition) is considered in §11.3- §11.4. In the static case these two limit regimes were described in [Pet10a, §3], where limit static (determinantal) correlation kernels were written out.
Remark 2.4 (Hidden determinantal structure in Pfaffian processes). If in Theorem 2 we set t 1 = · · · = t n , then the dynamical correlation functions turn into the (static) correlation functions of the point process M α,ξ on Z >0 . Thus, Theorem 2 implies that the point process M α,ξ on Z >0 is Pfaffian. To show that it is in fact determinantal requires some work (see Theorem 8.1 and Proposition A.2 from Appendix). Thus, one can say that in the static case the determinantal structure of correlation functions is hidden under the Pfaffian one. In particular, in this way we discover the determinantal structure of the poissonized Plancherel measure Pl θ (2.2) on strict partitions, thus strengthening a result of Matsumoto [Mat05] who gave a Pfaffian formula for the correlation functions of Pl θ , see §8.3.
On the other hand, numerical computations suggest that the dynamical correlation functions of the Markov process λ α,ξ cannot be written as determinants. We plan to give a rigorous proof of this fact in a subsequent work.
2.3. Expression through the kernel for the z-measures. The dynamical Pfaffian kernel Φ α,ξ (s, x; t, y) of Theorem 2 has a relatively simple structure we are about to describe.
In §6.2- §6.3 we define a system of functions ϕ m (x; α, ξ), where α, ξ are our parameters, and the argument x and the index m range over Z. Each ϕ m can be expressed through the Gauss hypergeometric function (6.13), (6.8). For fixed α, ξ, the functions { ϕ m } m∈Z form an orthonormal basis of the Hilbert space ℓ 2 (Z).
Our kernel then has the form (x, y ∈ Z and s ≤ t): Here δ(m) = δ m,0 is the Kronecker delta. The functions ϕ m are particular cases of the functions used by Borodin and Olshanski to describe the static and dynamical correlation kernels for the z-measures on partitions [BO06b], [BO06a] (see §6.1- §6.2). From this fact it follows that our dynamical kernel Φ α,ξ (s, x; t, y) can be expressed through the extended discrete hypergeometric kernel of [BO06a]: 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 + 1 2 . Here and below for any two numbers a and b, by a ∨ b and a ∧ b we denote the maximum and the minimum of a and b, respectively. The identity (2.14) seems to be only formal and have no direct probabilistic consequences (such as probabilistic relations between our random point processes or dynamics on Z >0 and the corresponding objects for the z-measures). However, (2.14) can serve as a useful tool in studying the asymptotics of our dynamical kernel Φ α,ξ (s, x; t, y) simply by a direct application of the results of [BO06a] (see §11.3- §11.4 below).
The set S = ∞ n=0 S n of all strict partitions is equipped with a structure of a graded graph: for µ ∈ S n−1 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. We assume the edges to be oriented from S n−1 to S n . In this way S becomes a graded graph. It is called the Schur graph. 8 This graph describes the branching of (suitably normalized) 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 the initial vertex ∅ to the vertex λ. This number is given by [Mac95, Ch. III, §8, Observe that if the components of λ are not distinct, then dim S λ vanishes. The numbers dim S λ satisfy the recurrence relations The number dim S λ can also be interpreted as the number of shifted standard tableaux of the form λ [Sag87], [Wor84], and as the (suitably normalized) dimension of the irreducible truly projective representation of the symmetric group S |λ| corresponding to the shifted diagram λ [HH92], [Iva99].
Similarly, by dim S (µ, λ) denote the total number of paths from µ to λ in the graph 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 general formalism (e.g., see [KOO98]), one can define coherent systems of measures on the Schur graph. This definition starts from the notion of the down transition probabilities. For λ, µ ∈ S, set By (3.2), the restriction of p ↓ to S n+1 ×S n for all n = 0, 1, . . . is a Markov transition kernel. We denote it by p ↓ (n + 1, n) = {p ↓ (n + 1, n) λ,µ } λ∈Sn+1, µ∈Sn , and call it the down transition kernel.
Definition 3.1. Let M n be a probability measure on S n , n = 0, 1, . . . . We call {M n } a coherent system of measures iff 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) > 0 for all n and λ ∈ S n ), we can define the corresponding up transition probabilities. They depend on a choice of a coherent system. For λ, κ ∈ S, set if λ ∈ S n , κ ∈ S n+1 and λ ր κ, 0, otherwise.
By (3.3), the restriction of p ↑ to S n ×S n+1 for all n = 0, 1, . . . is a Markov transition kernel. We denote it by p ↑ (n, n + 1) = {p ↑ (n, n + 1) λ,κ } λ∈Sn, κ∈Sn+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 coherency relation (3.3). The set of all coherent systems of measures on the Schur graph is a convex set. Its extreme points are identified with the points of the infinitedimensional ordered simplex This is the so-called Martin boundary of the Schur graph. It was first described by Nazarov [Naz92]. Another proof of this result can be obtained using the general methods of [KOO98] together with the formulas of [Iva99] for dimensions of skew shifted Young diagrams.
Moreover, the following characterization of the coherent systems holds: Naz92]). There is a bijection between coherent systems of measures on the Schur graph S and Borel probability measures on the simplex Ω + .
In the opposite direction, the points of Ω + are in one-to-one correspondence with the indecomposable normalized projective characters of the infinite symmetric group, and any Borel probability measure P on Ω + can be viewed as a (possibly decomposable) projective character χ of S ∞ . This character χ can be restricted to the finite symmetric group S n ⊂ S ∞ (of any order n) and expressed as a linear combination of (suitably normalized) irreducible truly projective characters of S n . These characters are parametrized by the set S n [Sch11], [HH92]. The coefficients of the expansion of χ| Sn are the numbers {M n (λ)} λ∈Sn , where {M n } is the coherent system corresponding to the measure P on Ω + by Theorem 3.2. The coherency condition (3.3) for the measures {M n } arises naturally in this context because the restrictions of the character χ to symmetric groups S n for different n must be consistent with each other.

Multiplicative measures.
There is a distinguished coherent system on the Schur graph, namely, the Plancherel measures {Pl n } ∞ n=0 (2.1). This coherent system corresponds (in the sense of Theorem 3.2) to the delta measure at the point (0, 0, . . . ) ∈ Ω + . Using the function dim S λ defined by (3.1), one can write The Plancherel measures on strict partitions are analogues (in the theory of projective representations of symmetric groups) of the well-known Plancherel measures on 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 characterization of the measures {M α,n } from [Bor99].
for all n and all λ ∈ S n .
Here c n , n = 0, 1, . . . , are normalizing constants. The product above is taken over all boxes = (i, j) of the shifted Young diagram λ, where i and j are the row and column numbers of the box , respectively. (Note that for shifted Young diagrams we always have j ≥ i.) Bor99]). Let {M n } be a nondegenerate coherent system of measures on the Schur graph. It is multiplicative iff the function f has the form for some parameter α ∈ (0, +∞]. If f (i, j) is given by (3.4), then c n = α(α + 2) . . . (α + 2n − 2). The case α = +∞ is understood in the limit sense: lim Recall that the number (j − i) is called the content of the box = (i, j). For shifted Young diagrams all contents are nonnegative.
3.4. 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, and by 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) ∼ = {0, 1} X , and we take the standard coordinatewise topology on {0, 1} X which turns it into a compact space. In more detail about random point processes, e.g., see [Sos00].
As explained in §2.1, we mix the measures M α,n (2.3) using the negative binomial distribution π α,ξ (2.4) on the set {0, 1, . . . } of indices n. As a result we get a probability measure M α,ξ (2.6) on the set S of all strict partitions. Identifying strict partitions with point configurations in a natural way ( §2.1), we see that the set S is the same as Conf fin (Z >0 ). Thus, M α,ξ can be viewed as a random point process on Z >0 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 >0 are determinantal. Observe that both these processes have a general structure described in the following definition: Definition 3.6. Let w be a nonnegative function on Z >0 such that ∞ x=1 w(x) < ∞. (3.5) By P (w) denote the point process on Z >0 that lives on finite configurations and assigns the probability The process M α,ξ has the form P (w) with w(x) = w α,ξ (x) given by (2.7), and for Pl θ we have Let L (w) be the following Z >0 × Z >0 matrix: The condition (3.5) implies that the operator in ℓ 2 (Z >0 ) corresponding to L (w) is of trace class, and, in particular, the Fredholm determinant det(1 + L (w) ) is well defined.
Note that the normalizing constant in (3.6) is equal to 1 det(1+L (w) ) , so the condition (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, because it has the form K (w) = L (w) (1 + L (w) ) −1 , where L (w) is symmetric. However, the operator of the form L (w) (1 + L (w) ) −1 cannot be a projection operator in ℓ 2 (Z >0 ). This aspect discriminates our processes from many other determinantal processes appearing in, e.g., random matrix models (see the references given in Introduction).
On the other hand, the static Pfaffian kernel in our model resembles the structure of 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 >0 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 3.7 does not give any suggestions on how to calculate them. Below we compute the correlation kernel K α,ξ using a fermionic Fock space technique. The kernel K θ is obtained from K α,ξ via the Plancherel degeneration ( §8.3).

Kerov's operators
4.1. Definition, characterization and properties. The main tool that we use in the present paper to compute the correlation functions of the point processes M α,ξ (and also of the associated dynamical models, see §9- §10) is a representation of the Lie algebra sl(2, C) in the pre-Hilbert space ℓ 2 fin (S) given by the so-called Kerov's operators. This approach was introduced by Okounkov [Oko01b] for the z-measures on ordinary partitions.
By ℓ 2 fin (S) we denote the space of all finitely supported functions on S with the inner product This is a pre-Hilbert space whose Hilbert completion is the usual space ℓ 2 (S) of all functions on S which are square integrable with respect to the counting measure on S. The standard orthonormal basis in ℓ 2 (S) is denoted by {λ} λ∈S , that is, Definition 4.1. The Kerov's operators in ℓ 2 fin (S) depend on our parameter α > 0 and are defined as Hλ := 2|λ| + α 2 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 where Z n is a normalizing constant. See also the end of this subsection for more connections between the Kerov's operators and the measures M α,n . The Kerov's operators (4.2) satisfy the following: defines a representation of the Lie algebra sl(2, C) in ℓ 2 fin (S). That is, the operators U, D, and H satisfy the commutation relations 2. The operators U and D are adjoint to each other in the space ℓ 2 fin (S). 3. For any λ ∈ S, the vector Uλ 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 and column numbers). Likewise, the vector Dλ is a linear combination of vectors µ, where µ ր λ, and the coefficient of µ depends only on the box λ/µ. 4. Each basis vector λ, λ ∈ S, is an eigenvector of the operator H, and the eigenvalue of λ depends only on |λ|.
The only property above that is not obvious is the first one: Proof. Denote This is a linear combination of vectors ρ, where ρ ∈ S n and either ρ = λ, or ρ = λ + 1 − 2 for some boxes 1 = 2 . In the second case the coefficient by the vector ρ with ρ = λ + 1 − 2 is Thus, in (4.7) it remains to consider only the terms with ρ = λ. Therefore, one must establish the combinatorial identity The proof of this identity (using Kerov's interlacing coordinates of shifted Young diagrams) is essentially contained in §3.1 of the paper [Pet10b] (the arXiv version).
In fact, the Kerov's operators (4.2) are completely characterized by the above four properties: Proposition 4.4. If three operators U, D, and H in the space ℓ 2 fin (S) satisfy the four 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) we take the same branches of the square roots α + c(c + 1), c = 0, 1, 2, . . . . In fact, it is the square of the function q α (·, ·) (4.6) that really plays the role in the definition of the Kerov's operators.
To find q(n, n) 2 for n ≥ 3, use the vector λ with λ = (n, n − 1, . . . , 1): Finally, to find q(i, j) 2 for arbitrary j > i > 1 (these are the remaining unknown values of q(·, ·) 2 ), we apply the relation [D, U] = H to the vector λ with λ = (j, j − 1, . . . , j − i + 1): Putting all together, we see that the function q is identical to the function q α defined by (4.6) (but note the remark after the formulation of the present proposition). The fact that the commutation relation [D, U] λ = Hλ (with the above choice of q = q α ) holds for all shifted Young diagrams λ ∈ S follows from Lemma 4.3. This concludes the proof.
Remark 4.5. One can also prove a statement analogous to Proposition 4.4 without property 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: where f is some nonzero function on the set of boxes. One possible choice of such operators (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 Young graph whose vertices are ordinary partitions. As a result we will get operators similar to those considered in [Oko01b]. This allows one to give a purely combinatorial characterization of the z-measures on the Young graph. Another characterization of 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 to the z-measures on the Young graph.
First, using commutation relations (4.5) for the Kerov's operators, one can compute the normalizing constants Z n in (4.3) that are defined as 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 By the commutation relations (4.5), Using the fact that D∅ = 0, we get Taking into account the initial value Z 0 = (U 0 D 0 ∅, ∅) = 1, we see that Z n = n!(α/2) n . Thus, the (complex-valued) measures M α,n are well-defined by (4.3) for all α ∈ C \ {0, −2, −4, . . . } because for such α the normalizing constants Z n are nonzero for all n. Moreover, under this assumption the measures M α,n are nondegenerate in the sense that M α,n (λ) = 0 for all n and all λ ∈ S n . Many formulas in the present paper hold in a purely algebraic sense for α ∈ C \ {0, −2, −4, . . . }. Now let us present an alternative proof of the coherency condition (3.3) of the measures {M α,n }. Another proof can be found in [Bor99]. Here we also assume that α ∈ C \ {0, −2, −4, . . . }. Consider the following operators which are slightly different from U and D: for all n and λ ∈ S n (4.9) (here Z n = n!(α/2) n is the same as in (4.3)). Fix n = 1, 2, . . . and µ ∈ S n−1 . Write 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 fall into the framework of the paper by Fulman [Ful09]. Namely, consider the operators U n : CS n → CS n+1 and D n : CS n → CS n−1 defined by U n λ := 1 n+α/2 Uλ and D n λ := Dλ (where λ ∈ S n ). Then the operators U n and D n satisfy the commutation relations in the form of [Ful09, (1.1)]: D n+1 U n = a n U n−1 D n + b n I n , where I n : CS n → CS n is the identity operator and a n = 1 − 1 n+α/2 , b n = 1 + n n+α/2 . 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) (which is a mixture of the measures M α,n ) can be written for small enough ξ as follows: Here e √ ξD λ is clearly an element of ℓ 2 fin (S). The fact that the vector e √ ξU λ belongs to ℓ 2 (S) (for small enough ξ) requires a justification (see the proof of Proposition 4.8), because the operator U in ℓ 2 (S) is unbounded. This makes the above formula for M α,ξ (λ) not very convenient for taking averages with respect to the measure M α,ξ . 9 In this subsection we overcome this difficulty and give a convenient way of writing expectations with respect to M α,ξ . Our approach here is similar to that of Olshanski [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)) a representation of the complex Lie algebra sl(2, C) in the (complex) pre-Hilbert space ℓ 2 fin (S). Consider the real form su(1, 1) ⊂ sl(2, C) spanned by the matrices U − D, i(U + D), and iH (here i = √ −1). The corresponding operators U − D, i(U + D), and iH act skew-symmetrically in ℓ 2 fin (S). Now we prove that the representation of the Lie algebra su(1, 1) can be lifted to a representation of a corresponding Lie group: Proposition 4.8. All vectors of the space ℓ 2 fin (S) are analytic for the described above action of the Lie algebra su(1, 1) in ℓ 2 fin (S). Consequently, this action of su(1, 1) gives rise to a unitary representation of the universal covering group SU (1, 1) ∼ in the Hilbert space ℓ 2 (S).

Proof. Recall [Nel59] that a vector h is analytic for an operator
in s has a positive radius of convergence. We can use Lemma 9.1 in [Nel59] that guarantees the existence of the desired unitary representation of SU (1, 1) ∼ in ℓ 2 (S) if we first prove that for some constant s 0 > 0 we have for any h ∈ ℓ 2 fin (S), all sufficiently large n (the bound on n depends on h), and any indices i 1 , . . . , i n taking values 1, 2, 3, where A 1 = U − D, A 2 = i(U + D), and A 3 = iH. Note that this in fact implies that any vector in ℓ 2 fin (S) is analytic for the action of su(1, 1).
It suffices to prove the estimate (4.10) forÂ 1 := U,Â 2 := D, andÂ 3 := H, this can only affect the value of the constant s 0 . Moreover, we can consider only the cases when h = κ for an arbitrary κ ∈ S. Because all the matrix elements of the operators U, D, and H are nonnegative in the standard basis {λ} λ∈S , we have The desired estimate would follow if we show that the power series expansion of exp (s(U + D + H)) κ converges for some small enough s > 0. For matrices in SL(2, C) (see (4.4)) we have 9 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 of Since the operator D is locally nilpotent and the operator H acts on each λ as multiplication by (2|λ| + α/2), to obtain the desired estimate (4.10) it remains to show that the series ∞ n=0 U n µ s n n! converges for all µ ∈ S for sufficiently small s > 0 (the bound on s must not depend on µ). Let us fix µ with |µ| = k. We can write by definition of U: where q α is defined by (4.6). Here the product is taken over all boxes of the skew shifted diagram λ/µ (see the end of §3.1). Since dim S (µ, λ) ≤ dim S λ, we can estimate The factor ∈µ q α ( ) −2 is just a constant depending on µ, and the normalizing constants Z n = n!(α/2) n were computed in the previous subsection. Putting all together, we get (4.11) Using [Erd53, 1.18.(5)], we see that so the series (4.11) converges for small enough s > 0. This concludes the proof of the proposition.
To formulate the central statement of this section, we need some preparation. By G ξ denote the matrix Clearly, (G ξ ) 0≤ξ<1 is a continuous curve in SU (1, 1) starting at the unity. By ( G ξ ) 0≤ξ<1 denote the lifting of this curve to SU (1, 1) ∼ , again starting at the unity. The unitary operators in ℓ 2 (S) corresponding (by Proposition 4.8) to G ξ are denoted by G ξ . The next thing we need is the weighted ℓ 2 space ℓ 2 (S, M α,ξ ) -the space of functions on S that are square summable with the weight M α,ξ . This is a Hilbert space with the inner product There is an isometry map I α,ξ from ℓ 2 (S, M α,ξ ) to ℓ 2 (S): The standard orthonormal basis {λ} λ∈S (4.1) of the space ℓ 2 (S) corresponds to the orthonormal basis (M α,ξ (λ)) − 1 2 λ λ∈S of ℓ 2 (S, M α,ξ ). To any operator A in ℓ 2 (S, M α,ξ ) corresponds the operator I α,ξ AI −1 α,ξ acting in ℓ 2 (S). Now we can formulate and prove the main statement of this section: Proposition 4.9. Let A be a bounded operator in ℓ 2 (S, M α,ξ ). Then (4.14) Here 1 ∈ ℓ 2 (S, M α,ξ ) is the constant identity function. On the left the inner product is in ℓ 2 (S, M α,ξ ), while on the right it is taken in ℓ 2 (S).
Proof. Let us first show that In the matrix group SL(2, C) we have The vector ∅ ∈ ℓ 2 fin (S) is analytic for the action of su(1, 1) (Proposition 4.8), so on this vector the representation of SU (1, 1) ∼ can be extended to a representation of the local complexification of the group SU (1, 1) ∼ (see, e.g., the beginning of §7 in [Nel59]). This means that for small enough ξ (when G ξ is close to the unity of the group SU (1, 1) ∼ ) we have The operator e − √ ξD preserves ∅, and thus We have established (4.15) for small ξ. The left-hand side of (4.15) is analytic in ξ 10 because ∅ is an analytic vector for the operator G ξ by Proposition 4.8. The right-hand side of (4.15) is also analytic in ξ by definition of M α,ξ , see §2.1. Thus, (4.15) holds for all ξ ∈ (0, 1).
, because the operator G ξ is unitary and has real matrix elements. We have This concludes the proof. Remark 4.10. The left-hand side of (4.14) can be regarded as an expectation with respect to the measure M α,ξ of the function (A1)(·) on S. In the special case when 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: This case is used in the computation of the static correlation functions, and for the dynamical correlation functions we need to use the more general statement of Proposition 4.9.

Fermionic Fock space
In this section we realize the Hilbert space ℓ 2 (S) as a fermionic Fock space over ℓ 2 (Z >0 ), 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 over an odd-dimensional space (similar Clifford algebras and their Fock representations were 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 an even-dimensional space. This difference, in particular, leads to the fact that in our case a priori the use of this algebra provides us only with a Pfaffian formula for the correlation 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 §3.4 and Theorem 8.1) which in fact do not work in the case of dynamical correlation functions. 5.1. Wick's theorem. We begin with the definition of a certain Clifford algebra over the Hilbert space V := ℓ 2 (Z). Denote the standard orthonormal basis of the Let V + and V − be the spans of {v x } x∈Z>0 and {v x } x∈Z<0 , respectively, and let V 0 denote the space Cv 0 . Note that the spaces V + and V − are maximal isotropic subspaces for the form ·, · , and By Cl (V ) denote the Clifford algebra over the quadratic space (V, ·, · ), that is, Cl (V ) is the quotient of the tensor algebra ∞ n=0 V ⊗n of the space V by the two-sided ideal generated by the elements The tensor product of v and v ′ in Cl (V ) is denoted simply by vv ′ . Thus, Theorem 5.1. Let F be a linear functional on Cl (V ) such that F(1) = 1 and for any p, q, r ∈ Z ≥0 , f + 1 , . . . , f + p ∈ V + , and f − 1 , . . . , f − q ∈ V − , we have if at least one of the numbers p, q is nonzero. Then for any n ≥ 1 and any 2n elements f 1 , . . . , f 2n ∈ V we have where F f 1 , . . . , f 2n is the skew-symmetric 2n × 2n matrix in which the kj-th entry above the main diagonal is F(f k f j ), 1 ≤ k < j ≤ 2n. Proof.
Step 1. Consider decompositions where each s j is a sign, s j ∈ {−, 0, +}, and the sum is taken over all 3 2n possible sequences of signs.
Step 2. Fix any particular sequence of signs (s 1 , . . . , s 2n ). Consider first the case when all of the s j 's are nonzero. We aim to prove that is the 2n × 2n skew-symmetric matrix in which the kjth entry above the main diagonal is F(f s k k f sj j ). First, note that if in the sequence (s 1 , . . . , s 2n ) all the "+" signs are on the left and all the "−" signs are on the right, 11 then by (5.2) we get (5.3), because in the Pfaffian in the right-hand side of (5.3) each entry is zero.
Next, observe that (5.3) is equivalent to this is just the standard Pfaffian expansion (here f s k k means the absence of f s k k ). It can be readily verified that the right-hand side and the left-hand side of (5.4) vary in the same way under the interchange f sr r ↔ f sr+1 r+1 for any r = 1, . . . , 2n − 1. This implies that (5.4) holds because one can always move the "+" signs to the left and the "−" signs to the right. This argument is similar to the proof of Lemma 2.3 in [Vul07].
In the next subsection we describe the structure of Fock(Z >0 ) in more detail.
5.3. Creation and annihilation operators. Vacuum average. Let φ k , k = 1, 2, . . . , be the creation operators in Fock(Z >0 ), that is, Let φ * k , k = 1, 2, . . . , be the operators that are adjoint to φ k with respect to the inner product in Fock(Z >0 ). They are called the annihilation operators and act as follows: We also need the operator φ 0 = φ * 0 acting as To simplify certain formulas below, we organize the operators φ k , φ 0 and φ * k into a single family: It can be readily checked that the operators φ m satisfy the following anti-commutation relations: otherwise. (5.7) In agreement to these definitions, let {v x } x∈Z be another orthonormal basis in x ∈ Z, and extended to the whole Cl (V ) by (5.1) and by linearity. The fact that T is indeed a representation follows from (5.7) and (5.9). It can be readily verified that the functional F vac on Cl (V ) satisfies the hypotheses of Wick's Theorem 5.1. 5.4. The representation R. The space ℓ 2 fin (S) is isometric to Fock fin (Z >0 ), and thus the Kerov's operators U, D, and H (4.2) in ℓ 2 fin (S) give rise to certain operators in Fock fin (Z >0 ). We obtain a representation of the Lie algebra sl(2, C) in Fock(Z >0 ), 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 >0 ) (this subspace of Fock(Z >0 ) is invariant for the representation R of sl(2, C)) can be expressed in terms of the creation and annihilation operators as follows: Proposition 4.8 can be reformulated for the representation R. Namely, the representation R of sl(2, C) restricted to the real form su(1, 1) ⊂ sl(2, C) gives rise to a unitary representation of the universal covering group SU (1, 1) ∼ in the Hilbert space Fock(Z >0 ). Denote this representation also by R.
Under the identification of ℓ 2 (S) with Fock(Z >0 ), we say that the map I α,ξ (4.13) is an isometry between ℓ 2 (S, M α,ξ ) and Fock(Z >0 ). By Proposition 4.9, for any bounded operator A in ℓ 2 (S, M α,ξ ) we have Here G ξ ∈ SU (1, 1) ∼ , 0 ≤ ξ < 1 is defined in §4.2, and 1 ∈ ℓ 2 (S, M α,ξ ) is the constant identity function. The inner products on the left and on the right are taken in the spaces ℓ 2 (S, M α,ξ ) and Fock(Z >0 ), respectively. Formula (4.16) for the expectation of a bounded function f (·) on S with respect to the measure M α,ξ is rewritten as where A f is the operator of multiplication by f . As we will see below, for averages expressing the correlation functions, the righthand side of (5.11) (and (5.12)) can be written as a vacuum average. That is, the ) has the form T (w) for a certain w ∈ Cl (V ).
6. Z-measures and an orthonormal basis in ℓ 2 (Z) In this section we examine functions on the lattice which are used in our expressions for correlation kernels (both static and dynamical). They form an orthonormal basis in the Hilbert space ℓ 2 (Z) and are eigenfunctions of a certain second order difference operator on the lattice. These functions arise as a particular case of the functions used to describe correlation kernels in the model of the z-measures on ordinary partitions, and we begin this section by recalling some of the results of the papers [BO06b], [BO06a] which we will use below.
Consider the following 3-parameter family of measures on the set of all ordinary partitions: σi is a generalization of the Pochhammer symbol. Here the parameter ξ ∈ (0, 1) is the same as our parameter ξ (e.g., in §2.1), and the parameters 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. To any ordinary partition σ = (σ 1 , . . . , σ ℓ(σ) , 0, 0, . . . ) is associated an infinite point configuration (sometimes called the Maya diagram) on the lattice Z ′ = Z + 1 2 : One can see that the correspondence σ → X(σ) is a bijection between ordinary partitions and those (infinite) configurations X ⊂ Z ′ for which the symmetric difference 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 on the 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 are determinantal with a correlation kernel K z,z ′ ,ξ (x', y') (where x', y' ∈ Z ′ ) which is called the discrete hypergeometric kernel [BO00], [BO06b].
There are explicit formulas for the discrete hypergeometric kernel K z,z ′ ,ξ which we will use. We proceed to describe them, but first we need to recall certain functions defined in [BO06b, (2.1)]: Here z, z ′ , ξ are the parameters of the z-measures, and the index a' of the functions runs over the lattice Z ′ . As usual, 2 F 1 is the Gauss hypergeometric function, (A)n(B)n (C)nn! w n . As is explained in [BO06b,§2], the expression (6.3) makes sense for all a', x' ∈ Z ′ due to the assumptions on the parameters z, z ′ (see p. 26) and the fact that ξ ∈ (0, 1). Moreover, the functions ψ a' (x'; z, z ′ , ξ) are real-valued. Let us summarize their properties for future use: Proposition 6.2 ([BO06b, §2]). 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 2 (Z ′ ). 2) Consider the following second order difference operator D(z, z ′ , ξ) in ℓ 2 (Z ′ ) (acting on functions f (x'), where x' ranges over Z ′ ): . The operator D(z, z ′ , ξ) is symmetric. The functions ψ a' are eigenfunctions of this operator: 3) The functions ψ a' satisfy the following symmetry relations: 4) The functions ψ a' satisfy the following three-term relation (a' ∈ Z ′ ): (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 kernel given by From Proposition 6.2, one readily sees that the discrete hypergeometric kernel K z,z ′ ,ξ (viewed as an operator in ℓ 2 (Z ′ )) is an orthogonal spectral projection operator corresponding to the positive part of the spectrum of the difference operator 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 dynamical model associated to the z-measures) below in §11.1 while describing how our dynamical Pfaffian kernel is related to it. 6.2. An orthonormal basis {ϕ m } in the Hilbert space ℓ 2 (Z). For the study of our model, we need the following family of functions: where ν(α) is given in Definition 2.1. Here the argument x and the index m range over the lattice Z. Because α > 0, we have Γ( 1 2 + ν(α) + k)Γ( 1 2 − ν(α) + k) > 0 for any k ∈ Z. Thus, the expression in (6.8) which is taken to the power 1 2 is positive. Note also that while the hypergeometric function 2 F 1 (A, B; C; w) is not defined if C is a negative integer, the ratio 2 F1(A,B;C;w)

Γ(C)
(occurring in (6.8)) is well-defined for all C ∈ C. Thus, we see that the functions ϕ m (x; α, ξ) are well-defined.
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 2 (Z) with the canonical orthonormal basis {k} k∈Z (that is, k(x) = δ k,x ) defined as follows: This representation depends on our parameter α > 0. For it one can prove an analogue of Proposition 4.8: Proposition 6.7. All vectors of the space ℓ 2 fin (Z) (consisting of finite linear combinations of the basis vectors {k}) are analytic for the action S of sl(2, C) (6.15). The representation S of the Lie algebra su(1, 1) ⊂ sl(2, C) in ℓ 2 fin (Z) lifts to a unitary representation of the Lie group P SU (1, 1) in the Hilbert space ℓ 2 (Z).
If 0 < α ≤ 1 4 , the above irreducible representation of P SU (1, 1) in ℓ 2 (Z) belongs to complementary series, and for α > 1 4 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, 1) again by S. For notational reasons (e.g., see Proposition 6.8 below), also by S let us denote the corresponding representations of SU (1, 1) and SU (1, 1) ∼ in ℓ 2 (Z) that are 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 ) −1 (where G ξ ∈ SU (1, 1) is defined in (4.12)) in the basis {k} k∈Z . These matrix elements will be used below in formulas for our correlation kernels.
Proposition 6.8. For all x, k ∈ Z we have S(G ξ ) −1 x, k ℓ 2 (Z) = ϕ −k (x; α, ξ). (6.16) Proof. Fix x, k ∈ Z. By Proposition 6.7, the function ξ → S(G ξ ) −1 x, k ℓ 2 (Z) is analytic. The right-hand side of (6.16) is also analytic in ξ, see (6.8). Thus, it suffices to prove (6.16) for small ξ. Also by Proposition 6.7, on x ∈ ℓ 2 fin (Z) the representation S can be extended to a representation of the local complexification of P SU (1, 1). This means that for small ξ (when G ξ is close to the unity of the group) we can write: (this follows from the corresponding identity for matrices in SL(2, C), see also the proof of Proposition 4.9). Denote c y := y(y + 1) + α, so that S(U )y = c y ·y + 1 and S(D)y = c y−1 ·y − 1. Note that c 2 y = y(y + 1) + α = (y + ν(α) + 1 2 )(y − ν(α) + 1 2 ). Also set a := − √ ξ Clearly, (x − l + r, k) ℓ 2 (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 ≥ 0 and set l = r + x − k. For x ≤ k we sum over l and set r = l + k − x. After direct calculations we obtain (we omit the argument in ν(α)): It is known that the expression 2F1 (A,B;C;w)
Let us apply this identity in the case x ≤ k above: for m ∈ Z, see (2.5)). Thus, we get the desired result (6.16) for small ξ, and hence for all ξ ∈ (0, 1) by analyticity. This concludes the proof.
Remark 6.9. The operator D α,ξ acting in ℓ 2 (Z), can be written through the operators of the representation S of sl(2, C) as follows: where H ξ := 1 2 G ξ HG −1 ξ ∈ sl(2, C). Indeed, this is verified by a simple matrix computation (see (4.12)): Operators corresponding to the matrix H ξ under other representations of sl(2, C) appear in our model twice more, see §9.4. 6.5. Connection with Meixner and Krawtchouk polynomials. The z-measures M z,z ′ ,ξ ( §6.1) for 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, 1), and one of the numbers z and z ′ (say, z) is a nonzero integer while z ′ has the same sign and, moreover, |z ′ | > |z| − 1.
Here if z = N = 1, 2, . . . , then the measure M z,z ′ ,ξ (σ) vanishes unless • (second degenerate series) ξ < 0, and z = N and z ′ = −N ′ , where N and N ′ are positive integers. In this case, the measure M z,z ′ ,ξ is supported by the (finite) set of all ordinary Young 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,§1.9]). In the second degenerate series these functions are related to the Krawtchouk orthogonal polynomials [KS96, §1.10].
For our measures M α,ξ on strict partitions there exists only one degenerate series of parameters: α = −N (N + 1) for some N = 1, 2, . . . , and ξ < 0 (Remark 3.5). In this case, the measure M α,ξ is supported by the set of all shifted Young diagrams which are contained inside the staircase shifted shape (N, N − 1, . . . , 1). This case corresponds to the second degenerate series of the 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 on the finite lattice {1, . . . , N }, and one could also define a suitable dynamics for these measures as is done below in §9. The results of the present paper about the structure of the static and dynamical correlation kernels also hold for the degenerate model, and the Krawtchouk polynomials enter formulas for these correlation kernels.

Static correlation functions
In this section we obtain a Pfaffian formula for the correlation functions of the point 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 integers. For x 1 , . . . , x n ∈ Z >0 we put, by definition, 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 §5 for definitions of objects below) where the inner product is taken in Fock(Z >0 ). (For now Φ α,ξ (x, y) is defined for x, y ∈ Z =0 , but in §7.2 we extend the definition of 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: Theorem 7.1. The correlation functions ρ (n) α,ξ (2.9) of the measures M α,ξ (2.6) are given by the following Pfaffian formula: where X = {x 1 , . . . , x n } ⊂ Z >0 (here x j 's are distinct), andΦ α,ξ X is the skew-symmetric 2n × 2n matrix with rows and columns indexed by the numbers 1, 2, . . . , n, −n, . . . , −2, −1, and the kj-th entry inΦ α,ξ X above the main diagonal is Φ α,ξ (x k , x j ), where k, j = 1, . . . , n, −n, . . . , −1. 12 Below in (7.16) we write Φ α,ξ (x, y) in terms of the Gauss hypergeometric function. For this reason, we call Φ α,ξ the Pfaffian hypergeometric-type kernel. Another form of a 2n × 2n 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 §10.4. The above formΦ α,ξ X is most useful when rewriting the Pfaffian in (7.3) as a determinant, see Theorem 8.1 and Proposition A.2 from Appendix.
The rest of this subsection is devoted to proving Theorem 7.1. Consider the following operators in ℓ 2 (S, M α,ξ ): Fix a finite subset X = {x 1 , . . . , x n } ⊂ Z >0 and set ∆ X := ∆ x1 . . . ∆ xn . This is a diagonal operator of multiplication by a function which is the indicator of the event {λ : λ ⊇ X}. We view ∆ X as an operator acting in ℓ 2 (S, M α,ξ ). Since this operator is diagonal, it does not change under the isometry I α,ξ : ℓ 2 (S, M α,ξ ) → Fock(Z >0 ) (4.13). Thus, ∆ X also acts in Fock(Z >0 ) (in the same way). The correlation functions ρ where 1 ∈ ℓ 2 (S, M α,ξ ) is the constant identity function. Using (5.11) (or (5.12)), we can rewrite the correlation functions as In this formula the operator ∆ X acts in Fock(Z >0 ). Clearly, ∆ X is expressed through the creation and annihilation operators in the Fock space Fock(Z >0 ) as It is more convenient for us to rewrite ∆ X using the anti-commutation relations for φ x and φ * x (see (5.7)) as follows: (after moving all the φ k 's to the left and φ * k 's to the right there is no change of sign). Our next step is to write (7.5) with ∆ X given by (7.6) as the vacuum average functional F vac applied to a certain element of the Clifford algebra Cl (V ) ( §5.1).

Recall that in §5.3 we have defined the representation T of Cl
where {v x } x∈Z is the basis of V = ℓ 2 (Z) defined by (5.8). Using the anti-commutation relations (5.7), one can readily compute the commutators between the operators T (v x ) and the operators of the representation R (5.10): x ∈ Z.
These formulas motivate the following definition: Definition 7.2. LetŠ 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: (7.7) The representationŠ is chosen in such a way that for all matrices M ∈ sl(2, C) and vectors v ∈ V fin we have (the equality of operators in Fock fin (Z >0 )). This follows from definitions of R, T , andŠ. Comparing (7.7) and (6.15), we see that the representationŠ is conjugate to the representation S discussed in §6.4 above. Namely,Š = Z −1 SZ, where Z is an operator in V = ℓ 2 (Z) defined by Zv x := 2 δ(x)/2 x, x ∈ Z. This means that Proposition 6.7 also holds for the representationŠ. In particular,Š lifts to a representation of the group P SU (1, 1) in the Hilbert space V . 13 Note that due to the conjugation by Z, the representationŠ is not unitary (but we do not need this property).
The next proposition (due to Olshanski [Ols08b]) is a "group level" version of the identity (7.8).
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 This is an equality of operators in the Hilbert space Fock(Z >0 ). It is enough to show that these operators agree on Fock fin (Z >0 ), which is true if x ∈ Z, and λ ∈ S. (7.11) 13 Also byŠ we denote the corresponding representations of SU (1, 1) and SU (1, 1) ∼ in V that are obtained from the representationŠ of P SU (1, 1) by a trivial lifting procedure.
Step 3. Now let us prove that both sides of (7.11) are analytic functions in g ∈ SU (1, 1) ∼ with values in Fock(Z >0 ): • (left-hand side) The vector T (v x )λ belongs to Fock fin (Z >0 ), and hence is analytic for 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 proposition), the function g →Š(g)v x is an analytic function with values in the Hilbert space V . Since T is continuous in the norm topology, T Š (g)v x is an analytic function with values in the Banach space End Fock(Z >0 ) of bounded operators in the space Fock(Z >0 ). On the other hand, the function R(g)λ is also analytic (with values in Fock(Z >0 )). Therefore, the function g → T Š (g)v x R(g)λ is analytic, too.
Step 4. Now it remains to compare the Taylor series expansions of both sides of (7.10) at g = e, the unity element of SU (1, 1) ∼ . That is, we need to establish that for any M ∈ sl(2, C) and any x ∈ Z: This should be understood as an equality of formal power series in s with coefficients being operators in Fock fin (Z >0 ). Let us divide both sides by the last formal sum, . After that it can be readily verified that the identity (7.12) of formal power series is a corollary of the "Lie algebra level" commutation identity (7.8).
This last step concludes the proof of the proposition.
Define (the second equality holds becauseŠ is a representation of P SU (1, 1)) Putting this together with Proposition 7.3, we can rewrite the correlation functions (7.5) as the vacuum average (see Definition 5.3): (7.14) Observe that for x, y ∈ Z =0 we have as in (7.2). Therefore, formula (7.14) together with Wick's Theorem 5.1 immediately implies our Theorem 7.1.

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) (where the vectors v x,ξ , v y,ξ are defined by (7.13)). By definitions of §5.3, we have Proposition 7.4. For any r, k ∈ Z we have where the functions ϕ m are defined in §6.2.
Proof. By (7.13) and then by (5.8), Using the fact thatŠ = Z −1 SZ (see the discussion before Proposition 7.3) and Proposition 6.8, we conclude the proof.
Therefore, since Φ α,ξ (x, y) is defined for x, y ∈ Z =0 , we have (in our derivation we also used (6.11)): (7.16) In the rest of the paper we agree that by this formula the kernel Φ α,ξ (x, y) is defined for arbitrary x, y ∈ Z (see also Remark 2.3.1). This is needed to view Φ α,ξ as an operator in ℓ 2 (Z). One also has (see §6.3) 7.3. Interpretation through spectral projections. One can interpret the kernel Φ α,ξ through orthogonal spectral projections related to the difference operator D α,ξ defined by (6.14). Namely, the projection onto the positive part of the spectrum of D α,ξ has the form (see §6.3) We also need the projection onto the zero eigenspace, which is simply Proj =0 ( D α,ξ )(x, y) = ϕ 0 (x; α, ξ) ϕ 0 (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 2 (Z), is the operator corresponding to the reflection of the lattice Z with respect to 0: Since R 2 is the identity operator, we see that the operator Φ α,ξ R is a rank one perturbation of the orthogonal spectral projection operator corresponding to the positive part of the spectrum of D α,ξ . 7.4. Expression through the discrete hypergeometric kernel. Recall that the discrete hypergeometric kernel K z,z ′ ,ξ (x', y') (where x', y' ∈ Z ′ = Z + 1 2 ) serves as a determinantal kernel for the z-measures on ordinary partitions ( §6.1). Under a suitable choice of parameters 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 could express our Pfaffian kernel Φ α,ξ through the discrete hypergeometric kernel: Proposition 7.6. For all x, y ∈ Z we have , where K is the discrete hypergeometric kernel described in §6.1, and ν(α) is given by Definition 2.1.
Proof. Using (6.7) and (6.9) with d = −1 and d = 0, we observe that for x, y ∈ Z: Taking half sum and using (7.16), we conclude the proof.
A time-dependent version of (7.18) is (2.14), which is obtained and used in §11. A similar identity for the (static) determinantal kernels in (8.3) below. 7.5. Reduction formulas. It is possible to rewrite the Pfaffian hypergeometrictype kernel Φ α,ξ (x, y) in a closed form (without the sum): Proposition 7.7. For any x, y ∈ Z we have For x = −y there is a singularity in the numerator (this is seen using (6.11)) as well as in the denominator. In this case the value of Φ α,ξ (x, y) is understood according to 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 representation-theoretic arguments as in the proof of Theorem 3 in [Oko01b]. Another way is to argue directly using the three-term relations for the functions ϕ m (6.12) to simplify the sum (7.16) similarly to the standard derivation of the Christoffel-Darboux formula for orthogonal polynomials. We use Proposition 7.6 together with the existing closed form expression for K z,z ′ ,ξ [BO06b, Proposition 3.10]: 14 (of course, the parameters of the functions ψ above are z, z ′ , ξ). We plug this formula into (7.18), and then express each function ψ a' through ϕ m using (6.9) with d = −1 and d = 0. Observe that for such d we have z(α)z ′ (α) = α. After that we apply (6.11) to simplify the resulting expression. This concludes the proof.

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 >0 . 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 proof of Theorem 1 from §2.
(4) Viewed as an operator in ℓ 2 (Z >0 ), K α,ξ can be interpreted in terms of orthogonal spectral projections corresponding to the difference operator D α,ξ (6.14) as follows (we restrict the operator below to ℓ 2 (Z >0 ) ⊂ ℓ 2 (Z)): where I is the identity operator and R is the reflection, see Proposition 7.5.
The expression K α,ξ (x, y) is a so-called gauge transformation of the original correlation kernel K α,ξ , that is, K α,ξ is related to K α,ξ by a conjugation by a diagonal matrix. This means that the Z >0 × Z >0 matrix K α,ξ can also serve as a correlation 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 (Corollary 7.8) allow us to apply Proposition A.2 from Appendix. This implies that where X = {x 1 , . . . , x n } ⊂ Z >0 (with pairwise distinct x j 's),Φ α,ξ X is the skewsymmetric 2n × 2n matrix introduced in Theorem 7.1, and K α,ξ is related to Φ α,ξ as 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 the kernel K α,ξ . Namely, claims 1 and 2 of the present theorem directly follow from the 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 (the last equality is by Corollary 7.8.(3)), so Now we see that claim 3 follows from (7.16) and (7.19)-(7.20), and claim 4 is due to Proposition 7.5. This concludes the proof. 3. The form (8.2) of the kernel K α,ξ is called integrable because the operator (8.2) in ℓ 2 (Z >0 ) can be viewed as a discrete analogue of an integrable operator (if we take x 2 and y 2 as variables). About integrable operators, e.g., see [IIKS90], [Dei99]. Discrete integrable operators are discussed in [Bor00] and [BO00,§6]. This remark is also applicable to the kernel (8.6) below.
4. 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 point processes. About the behavior of (8.3) in a scaling limit as ξ ր 1 see Remark 11.5 below.
5. Consider the Z >0 × Z >0 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 Theorem 3.3 in [BO00] (and also using the identities from Appendix in that paper) that K α,ξ = L α,ξ (1 + L α,ξ ) −1 . That is, the kernel K α,ξ is precisely the one given by Lemma 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. Denote by J k the Bessel function (of the first kind) of order k and argument 2 √ θ: Theorem 8.2. Under the Plancherel degeneration (2.8), the point processes M α,ξ on Z >0 converge to the poissonized Plancherel measure Pl θ . This is a determinantal point process on Z >0 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 a series and in an integrable form 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 the corresponding 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 also use the three-term relations for the Bessel functions (e.g., see [Erd53,7.2
The three-term relations for the Bessel functions (8.7) are obtained via the Plancherel degeneration from Proposition 6.5.2) (or, equivalently, from (6.12), by the self-duality of ϕ m 's). This agrees with the general approach described in [Ols08a] of studying limits of determinantal point processes via corresponding limits of self-adjoint operators which "control" the processes (in the sense that the correlation kernels of the processes are spectral projections corresponding to these operators).
Proof of Theorem 8.2. II. Another way of proof is to observe that the point process Pl θ on Z >0 is again an L-ensemble (see Lemma 3.7). Denote by L θ the corresponding Z >0 × Z >0 matrix which is given by (3.7) with w(x) = w θ (x) = θ x 2(x!) 2 . To prove the theorem it suffices (by Lemma 3.7) to show that the kernel K θ has the form K θ = L θ (1 + L θ ) −1 . This is equivalent to a certain identity for the Bessel functions which is readily verified using, e.g., Lemma 2.4 in [BOO00]. Equivalently, one may say that this identity is the Plancherel degeneration of the one in §8.2.5.   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 Proposition A.2 it follows that the poissonized Plancherel measure is a determinantal point process with the correlation kernel given by K θ (x, y) = 2 √ xy x+y Φ θ (x, −y), which is exactly formula (8.5) for K θ .

Markov processes
In this section we explain in detail the construction of the dynamical model on strict partitions described in §2.2.
The construction of our Markov processes on the Schur graph is similar to Borodin-Olshanski's construction [BO06a] of the Markov processes on the Young graph which preserve the z-measures. In contrast to [BO06a], we restrict our attention to the stationary (time homogeneous) case, that is, we assume that the parameter ξ does not vary in time. The introduction of the non-stationary processes in [BO06a] was motivated by the technique of handling the stationary case (in particular, by the method of the computation of the dynamical correlation functions). The technique that we use in the present paper does not require dealing with non-stationary processes. 9.1. Defining Markov processes in terms of jump rates. Let us first recall some basic notions and facts concerning Markov processes. Let E be a finite or countable space. Assume that we have a continuous time homogeneous Markov process on E with the time parameter t ∈ R ≥0 . By (P(t)) t≥0 denote the transition probabilities 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≥0 have the following properties: (P1) P ab (t) ≥ 0 for all t ≥ 0 and P ab (0) = δ ab for all a, b ∈ E; (P2) 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 matrix form, P ab (t + s) = c∈E P ac (t)P cb (s), where a, b ∈ E.
Assume that there exists a E × E matrix É such that The elements of the matrix É are called the jump rates. Note that (9.1) implies that each P ab (t) is continuous at t = 0: A family of matrices satisfying (P1)-(P4) is a (continuous) stochastic matrix semigroup. One can say that É is the infinitesimal matrix of this semigroup, that is, É = d dt P(t) t=0 . From (9.1) it is clear that (Q1) É ab ≥ 0 for a = b and É aa ≤ 0.
We assume that the jump rates also have the property The property (Q2) implies (e.g., see [KT99,Ch. 14.2]) that the jump rates É and the transition probabilities P(t) are related to each other via the system of Kolmogorov's backward equations: with the initial conditions We would like to start with the jump rates É satisfying properties (Q1)-(Q2) and obtain a stochastic matrix semigroup (P(t)) t≥0 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 b∈E P ab (t) ≤ 1) always exists. Among all possible substochastic solutions there is a distinguished minimal solution (P(t)) t≥0 , that is, P ab (t) ≥P ab (t) for t ≥ 0 and a, b ∈ E, where (P(t)) t≥0 is any substochastic solution. 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 a unique solution of (9.2)-(9.3) in the class of substochastic matrices. About solving Kolmogorov'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≥0 (or, which is equivalent, the minimal solution of this system is stochastic), we say that the jump rates É define a continuous time homogeneous Markov process on E (with transition probabilities P(t)) that can start from any point and any probability distribution. A common sufficient condition for this is sup a∈E |É aa | < +∞, which however 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 be derived 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 exit from X of the process starting at a. Though we do not know yet if the process itself is uniquely determined by its jump rates É, the random variable τ a,X can be constructed from É as follows. Contract all the states b ∈ E \ X into one absorbing stateb with Éb ,c = 0 for all c ∈ X ∪ {b}. On the finite set X ∪ {b} the backward equations have a unique solution (P(t)) t≥0 , 15 whereP(t) are matrices with rows and 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 É. In other words, the hypotheses of this proposition mean that the Markov process does not make infinitely many jumps in finite time.
9.2. Birth and death processes. Here we discuss underlying birth and death processes on Z ≥0 involved in the construction of the Markov processes on strict partitions (see §2.2).
A general birth and death process on E = Z ≥0 is a continuous time homogeneous Markov process with jump rates {Õ k,j } k,j∈Z ≥0 satisfying conditions (Q1)-(Q2) from the previous subsection, with an additional property that Õ k,j = 0 if |k − j| > 1.
This means that from any point n ≥ 1 of Z ≥0 the process can jump only to the neighbor points n − 1 and n + 1, and that from 0 it can jump only to 1.
The following necessary and sufficient condition is well-known and can be deduced, e.g., from [GSK04, Ch. III.2, Thm. 4].
This means that the processes on S "extend" the birth and death processes n α,ξ .
Proof. Let n, K be nonnegative integers, n ≤ K. Consider the random variable τ n,{0,...,K−1} for the birth and death process n α,ξ , that is, the time of the first exit from {0, 1, . . . , K − 1} of the process with jump rates Õ (9.5) starting at n.
Let λ ∈ S n . Observe that the time of the first exit from S 0 ∪ · · · ∪ S K−1 of the process on strict partitions with jump rates É (9.8) starting at λ has the same distribution as τ n,{0,...,K−1} . Applying Remark 9.3 and Proposition 9.1, we conclude the proof.
Thus, the jump rates É (9.8) uniquely define a continuous time Markov process on S that can start from any point and any probability distribution.
The operator É acts as where É λ,µ (9.8) are the jump rates of the process λ α,ξ . The operator É is symmetric with respect to the inner product (·, ·) M α,ξ . Moreover, it is closable in ℓ 2 (S, M α,ξ ), and its closure generates the semigroup (P(t)) t≥0 (see Remark 9.8 below). That is, É is the pre-generator of the process λ α,ξ . Remark 9.5. As a wider domain for the operator É (9.10) one can take the space of all functions f on S such that both f and Éf (defined by (9.10)) belong to ℓ 2 (S, M α,ξ ). This space clearly includes finitely supported functions.
Proposition 9.6. We have Here q α is the function of a box defined by (4.6).
Proof. Fix λ ∈ S, and for any ν ∈ S one has (É is given in (9.8)), and Proposition follows from a direct computation.
Corollary 9.7. The operator B in Fock fin (Z >0 ) has the form B = −R(H ξ ) + α 4 I, where I is the identity operator, the unitary representation R of sl(2, C) in the Hilbert space Fock fin (Z >0 ) is defined in §5.4, and H ξ is given in Remark 6.9.
Proof. This is a straightforward consequence of Proposition 9.6 (where, of course, we identify ℓ 2 (S) and Fock(Z >0 )) and the matrix computation in Remark 6.9. 16 We denote, e.g., the operators B and É by different symbols only to indicate in what spaces they act. Essentially, these operators are the same.
Proposition 10.2. The correlation functions of λ α,ξ have the form Note that now the expectation is taken in Fock(Z >0 ).
Note that in contrast to the static case (7.5), the operator ∆ T,X is not diagonal (see also Remark 4.10). It is worth noting that formula (10.2) does not hold if t j 's are not ordered as t 1 ≤ · · · ≤ t n .
10.2. Pre-generator and Kerov's operators. Our next aim is to extend the definition of the semigroup (V(t)) t≥0 from real nonnegative values of t to complex values of t with ℜt ≥ 0. This will be needed in the next subsection for computation of the dynamical correlation functions.
Observe that the matrix iH ξ (where H ξ is defined in Remark 9.9 and here and below i = √ −1) belongs to the real form su(1, 1) ⊂ sl(2, C). Denote The family {W ξ (τ )} τ ∈R for any fixed ξ ∈ [0, 1) is a continuous curve in SU (1, 1) passing through the unity at τ = 0. By { W ξ (τ )} τ ∈R denote the lifting of this curve to the universal covering group SU (1, 1) ∼ . 17 For real τ one can consider unitary operators Here the operator R( W 0 (τ )) (corresponding to ξ = 0) acts on Fock fin (Z >0 ) as Informally speaking, for s ∈ R ≥0 , the operator V(s) means e sB , and for τ ∈ R, the operator R( W ξ (τ ))e iτ α 4 I means e iτ B (here B is the generator of the semigroup (V(s)) s≥0 , see §9.4). Thus, it is natural to give the following definition: Definition 10.3. For t = s+iτ ∈ C + := {w ∈ C : ℜw ≥ 0} let V(t) be the following operator in Fock(Z >0 ): For real nonnegative t the operator V(t) is self-adjoint and bounded, it was defined in §9.4. For purely imaginary t, the operator V(t) is unitary. Thus, the operators V(t) are bounded for all t ∈ C + . Moreover, V(t 1 + t 2 ) = V(t 1 )V(t 2 ) for any t 1 , t 2 ∈ C + , so {V(t)} t∈C+ is a semigroup (with complex parameter) that can be viewed as an analytic continuation of the semigroup {V(s)} s∈R ≥0 . In particular, the operators V(t) commute with each other. Moreover, it is clear that the function t → V(t)h is bounded and continuous in C + and holomorphic in the interior {w ∈ C + : ℜw > 0} of C + for any vector h ∈ Fock(Z >0 ) which is analytic for the operator B. where the function Φ α,ξ (s, x; t, y) (x, y ∈ Z, s ≤ t) is given by (see (6.8) for definition of ϕ m ). In (10.4), (t 1 , x 1 ), . . . , (t n , x n ) ∈ R ≥0 × Z >0 are pairwise distinct space-time points such that 0 ≤ t 1 ≤ · · · ≤ t n , and Φ α,ξ T, X is the 2n × 2n skew-symmetric matrix with rows and columns indexed by the numbers 1, −1, . . . , n, −n, such that the kj-th entry in Φ α,ξ T, X above the main diagonal is Φ α,ξ (t |k| , x k ; t |j| , x j ), where k, j = 1, −1, . . . , n, −n (thus, |k| ≤ |j|). 18 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 + which is 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 : ℜw = 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 and continuous up to the boundary (with possible exception of one point corresponding to w = ∞ ∈ C + ). For any fixed ζ 0 with |ζ 0 | < 1, the value G(ζ 0 ) is represented by Cauchy's integral over the circle |ζ| = r, for |ζ 0 | < r < 1. By our hypotheses, this Cauchy's integral has a limit as r → 1, which gives an expression of G(ζ 0 ) through the boundary values.
(10.6) Denote the right-hand side of (10.6) by F (t 2,1 , . . . , t n,n−1 ; x 1 , . . . , x n ). As a function in n − 1 variables t 2,1 , . . . , t n,n−1 , F is initially defined for t j,j−1 taking real nonnegative values. However, as explained in §10.2, the definition of each operator V(t j,j−1 ) can be extended to t j,j−1 ∈ C + , so F is defined on (C + ) n−1 ⊂ C n−1 . Moreover, F is continuous and bounded in (t 2,1 , . . . , t n,n−1 ) belonging to the closed domain (C + ) n−1 and is holomorphic in the interior of this domain. Therefore, by Lemma 10.5, F (t 2,1 , . . . , t n,n−1 ; x 1 , . . . , x n ) is uniquely determined by its values when all the variables t j,j−1 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−1 ) is unitary, and, moreover, (here by agreement t 1,1 = 0, and V(0) = I, the identity operator). For purely imaginary t j , we want to rewrite F (t 2,1 , . . . , t n,n−1 ; x 1 , . . . , x n ) as a certain Pfaffian. First, we need a notation. Recall that in §7.1 we have defined a representationŠ of SU (1, 1) in the Hilbert space V = ℓ 2 (Z) with the standard orthonormal basis {v For t = 0 this vector becomes v x,ξ defined by (7.13).
Proof. The operators ∆ x have the form x ∈ Z >0 .
On the space V fin ⊂ V = ℓ 2 (Z) consisting of finite linear combinations of the basis vectors {v x } x∈Z , the operatorŠ(W 0 (u)) acts as e −iuŠ(H)/2 (where u ∈ R). From this fact and (7.15), we see that Note that here s and t are still purely imaginary. However, one can view the righthand side of (10.10) as a function in (t − s) ∈ C + . This function is bounded and continuous 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− 1 2 ξ m 2 (this can be readily observed from the analytic expression (6.8)). We are interested in the restriction of the right-hand side of (10.10) to real nonnegative values of (t − s). Observe that 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 2,1 , . . . , t n,n−1 , that is, for 0 ≤ t 1 ≤ · · · ≤ t n . This fact together with Proposition 10.2 implies Theorem 10.4.
Thus, we have finished the proof of Theorem 2 from §2.
10.4. Skew-symmetric matrices in Pfaffian formulas. In the right-hand sides of our Pfaffian formulas (7.3) and (10.4) for static and dynamical correlation functions we see certain skew-symmetric 2n × 2n matrices constructed using Pfaffian kernels 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. (In fact, this is the reason why we use the same notation for these kernels.) However, the matrix of (10.4) for t 1 = · · · = t n does not become the one from (7.3). Let us explain how one can transform (10.4) to get the expected behavior in the static picture.
When t = s, (11.1) reduces to the static version (7.18). Observe that the parameters z, z ′ (depending on α) in (11.1) are admissible (see p. 26). However, similarly to the static identity, (11.1) seems to imply no probabilistic connections between the dynamics related to the z-measures [BO06a] and our Markov processes λ α,ξ . 11.2. Plancherel degeneration.
Proof. This is a consequence of (10.5) and the fact that under the Plancherel degeneration (2.8) one has ϕ m (x; α, ξ) → J x+m .
In the static case (t = s), the above theorem reduces to a Pfaffian formula for correlation 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. 11.3. "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 largest rows in a strict partition. We embed the half-lattice Z >0 into the half-line R >0 as x → (1 − ξ)x, where x ∈ Z >0 , and then pass to the limit as ξ ր 1. For the static picture this limit transition was described in [Pet10a, §3.2], where the corresponding limit of the hypergeometric-type kernel K α,ξ (x, y) was given.
Proof. This is a consequence of results of [BO06a,§9]. Formulas for Φ W α are obtained using formula (10.5) for the pre-limit kernel Φ α,ξ . One should consider various cases of signs of x, y, and with the help of (6.11) make the arguments of the functions ϕ m (·; α, ξ) positive. After that one should replace each ϕ m (·; α, ξ) by the corresponding w m (·; α). All limit transitions can be justified using expression (11.1) of our kernel through the dynamical kernel of [BO06a].
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 in the limit described in the above theorem. For this kernel to have a limit one should perform a certain particle-hole involution, see [BO06a,§8]. However, we see that a combination of the values of the kernel 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.

Appendix A. Reduction of Pfaffians to determinants
Let us first recall basic definitions and properties related to Pfaffians. We use the following notations for matrices. Let X be an abstract finite space of indices and a = (a 1 , . . . , a 2n ) be a sequence of length 2n of points of X. Let F : X × X → C be some function. Form a 2n × 2n skew-symmetric matrix  1 , a 2 ) . . . F (a 1 , a 2n−1 ) F (a 1 , a 2n ) −F (a 1 , a 2 ) 0 . . . Denote this matrix by F a . This skew-symmetric matrix has rows and columns indexed by a 1 , . . . , a 2n , such that the ijth element above the main diagonal is equal to F (a i , a j ) (here 1 ≤ i < j ≤ 2n).
Definition A.1. Let a = (a 1 , . . . , a 2n ) and F a be as defined above. The determinant det(F a ) is a perfect square as a polynomial in F (a i , a j ) (where i < j).
The Pfaffian of F a , denoted by Pf F a , is defined to be the square root of det F a having the "+" sign by the monomial F (a 1 , a 2 ) . . . F (a 2n−1 , a 2n ).
The following properties of Pfaffians are well known: • Let A be a skew-symmetric 2n × 2n matrix and B be any 2n × 2n matrix, then Now we give a sufficient condition under which a 2n × 2n Pfaffian can be reduced to 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 →â we denote the corresponding involution of the space X that interchanges X + and X − . Let a := (a 1 , . . . , a n ,â n , . . . ,â 1 ), and a i ∈ X + (soâ i ∈ X − ), i = 1, . . . , n.
Proposition A.2. Suppose that the function F on X × X satisfies the following properties: 19 (1) F (a,b) = F (b,â) for any a, b ∈ X.
(2) F (a, b) = −F (b, a) for any a, b ∈ X such that a =b. for any a, b ∈ X + .