The quasi-invariance property for the Gamma kernel determinantal measure
The Gamma kernel is a projection kernel of the form , where and are certain functions on the one-dimensional lattice expressed through Euler’s -function. The Gamma kernel depends on two continuous parameters; its principal minors serve as the correlation functions of a determinantal probability measure defined on the space of infinite point configurations on the lattice. As was shown earlier (Borodin and Olshanski, Advances in Math. 194 (2005), 141-202; arXiv:math-ph/0305043), describes the asymptotics of certain ensembles of random partitions in a limit regime.
Theorem: The determinantal measure is quasi-invariant with respect to finitary permutations of the nodes of the lattice.
This result is motivated by an application to a model of infinite particle stochastic dynamics.
- 0.1 Preliminaries: a general problem
- 0.2 The Gamma kernel measure
- 0.3 The main result
- 0.4 Scheme of proof of Main Theorem
- 0.5 Organization of the paper
- 0.6 Acknowledgment
- 1 Z–measures and related objects
- 2 Multiplicative functionals and Fredholm determinants
- 3 Radon–Nikodým derivatives
- 4 Main result: Formulation and beginning of proof
- 5 Convergence of diagonal blocks in the topology of the trace class norm
- 6 Convergence of off–diagonal blocks in Hilbert–Schmidt norm
0.1. Preliminaries: a general problem
Recall a few well-known notions from measure theory. Let be a Borel space (that is, a set with a distinguished sigma-algebra of subsets). Two Borel measures on are said to be equivalent if has a density with respect to and vice versa. They are said to disjoint or mutually singular if there exist disjoint Borel subsets and such that is supported by and is supported by (that is, ). Assume is a group acting on by Borel transformations; then a Borel measure is said to be -quasi-invariant if is equivalent to its transform by any element .
In practice, especially for measures living on “large” spaces, verifying the property of equivalence, disjointness or quasi-invariance, and explicit computation of densities (Radon–Nikodým derivatives) for equivalent measures can be a nontrivial task. There exist nice general results for particular classes of measures: infinite product measures (Kakutani’s theorem [Ka]), Gaussian measures on infinite-dimensional spaces (Feldman–Hajek’s theorem and related results, see [Kuo, Ch. II]), Poisson measures (see [Bro]).
Assume that is a locally compact space, take as the “large” space the space of locally finite point configurations on , and assume that the measures under consideration are probability measures on ; they are also called point processes on (for fundamentals of point processes, see, e.g., [Le]). Poisson measures are just the simplest yet important example of point processes. The next by complexity example is the class of determinantal measures (processes). Determinantal measures are specified by their correlation kernels which are functions on . Note an analogy with covariation kernels of Gaussian measures which are also functions in two variables. Note also that, informally, Poisson measures can be viewed as a degenerate case of determinantal measures corresponding to kernels concentrated on the diagonal .
Many concrete examples of determinantal measures are furnished by random matrix theory and other sources, see, e.g., the surveys [So] and [Bor]. The interest to determinantal measures especially increased in the last years. However, to the best of my knowledge, the following problem was never discussed in the literature:
Assume we are given two determinantal measures, and on a common space . How to test their equivalence (or, on the contrary, disjointness)? Is it possible to decide this by inspection of the respective correlation kernels and ?
One could imagine that equivalence holds if the kernels are close to each other in an appropriate sense. However, there is a subtlety here, see Subsection 1.6 below.
Let be a group of homeomorphisms . Then also acts, in a natural way, on the space and hence on the space of probability measures on . Observe that the latter action preserves the determinantal property: If is a determinantal measure on with correlation kernel , then the transformed measure is determinantal, too, and serves as its correlation kernel. Thus, the question of -quasi-invariance of becomes a special instance of Problem 1:
Let and be as above. How to test whether is -quasi-invariant? Is it possible to decide this by comparing the correlation kernels and for ?
0.2. The Gamma kernel measure
In the present paper we are dealing with a concrete model of determinantal measures, introduced in [BO2]. The space is assumed to be discrete and countable; it is convenient to identify it with the lattice of half–integers. Then the space is simply the space of all subsets of . We consider a two-parameter family of kernels on . Following [BO2], we denote them as ; here and are some continuous parameters, and are the arguments, which range over . Each kernel is real-valued and symmetric. Moreover, it is a projection kernel meaning that it corresponds to a projection operator in the Hilbert space . Like many examples of kernels from random matrix theory, our kernels can be written in the so-called integrable form [IIKS], [De]
resembling Christoffel–Darboux kernels associated to orthogonal polynomials. In our situation and are certain functions on the lattice , which are expressed through Euler’s -function. For this reason we call the Gamma kernel. In [BO2] we conjectured that the Gamma kernel might be a universal microscopic limit of the Christoffel–Darboux kernels for generic discrete orthogonal polynomials, in an appropriate asymptotic regime.
The Gamma kernel serves as the correlation kernel for a determinantal measure on , called the Gamma kernel measure and denoted as . According to the general definition of determinantal measures (see [So], [Bor]), the measure is characterized by its correlation functions
which in turn are equal to principal minors of the kernel:
Here and is an arbitrary -tuple of pairwise distinct points from .
As shown in [BO2], the Gamma kernel measure arises from several models of representation–theoretic origin, through certain limit transitions.
0.3. The main result
We take as the group of permutations of the set fixing all but finitely many points. Such permutations are said to be finitary. Clearly, is a countable group. It is generated by the elementary transpositions of the lattice : Here and transposes the points and of . Each permutation induces, in a natural fashion, a transformation of the space , which in turn results in a transformation of probability measures on .
The main result of the paper says that the Gamma kernel measure is quasi-invariant with respect to the action of the group :
Main Theorem. For any , the measures and are equivalent. Moreover, the Radon–Nikodým derivative can be explicitly computed.
This result gives a solution to the first question of Problem 2 in a concrete situation. As will be shown in another paper, the quasi-invariance property established in the theorem makes it possible to construct an equilibrium Markov process on with determinantal dynamical correlation functions and equilibrium distribution . This application is one of the motivations of the present work. 111A connection between quasi-invariance and existence of Markov dynamics, sometimes in hidden form, is present in various situations. See, e.g., [AKR], [SY].
It seems plausible that is not quasi-invariant with respect to the transformations of generated by the translations of the lattice. Note that the translation with amounts to the shift of the parameters (see Theorem 1.4). One can ask, more generally, whether any two Gamma kernel measures with distinct parameters are disjoint. 222The pair should be viewed as an unordered pair of parameters, because the transposition does not affect the measure, see Theorem 1.4.
0.4. Scheme of proof of Main Theorem
The proof relies on the fact that for fixed , the measure can be approximated by simpler measures which are –quasiinvariant and whose Radon–Nikodým derivatives (with respect to the action of the group ) are readily computable.
The approximating measures depend on an additional parameter and are denoted as . These are purely atomic probability measures supported by a single –orbit. They come from certain probability distributions on Young diagrams, and are called the z–measures (Kerov–Olshanski–Vershik [KOV], Borodin–Olshanski [BO1]). As goes to , the measures weakly converge to : this is simply the initial definition of given in [BO2].
What we actually need to prove is that the convergence of the measures holds not only in the weak topology (that is, on bounded continuous test functions) but also in a much stronger sense: Namely,
for certain test functions which, like the Radon–Nikodým derivatives, may be unbounded and not everywhere defined. Here and in the sequel the angular brackets denote the pairing between functions and measures.
To explain this point more precisely we need some preparation.
First of all, it is convenient to transform all the measures in question by means of an involutive homeomorphism of the compact space . This homeomorphism, denoted as “”, assigns to a configuration its symmetric difference with the set .
An equivalent description is the following. Regard as a configuration of charged particles occupying some of the sites of the lattice , while the holes (that is, the unoccupied sites of ) are interpreted as anti–particles with opposite charge. Now, the new configuration is formed by the particles sitting to the right of 0 and the anti–particles to the left of 0. We call “” the particle/hole involution on .
For instance, if then , the empty configuration. The configuration plays a distinguished role because the –orbit of this configuration is the support of the pre–limit measures . The map “” transforms this distinguished orbit into the set of all finite balanced configurations, that is, finite configurations with equally many points to the right and to the left of .
Note that the transform by “” leaves intact the action of all the elementary transpositions with , only the action of is perturbed.
Note also that if is a determinantal measure on then so is its push–forward , and there is a simple relation between the correlation kernels of and ([BOO, Appendix]). If the kernel of is symmetric then that of has a different kind of symmetry: it is symmetric with respect to an indefinite inner product (see [BO1, Proposition 2.3 and Remark 2.4]), which reflects the presence of two kinds of particles.
Instead of the measures and we will deal with their transforms by “”, denoted as and . Clearly, the transform does not affect the formulation of the theorem, only the initial action of the group on has to be conjugated by the involution: an element now acts as the transformation
An advantage of the transformed measures as compared to the initial ones is that the pre–limit measures live on finite configurations. In a weaker form, this property is inherited by the limit measures. Namely, let us say that a configuration is sparse if
Denote the set of all sparse configurations as . There is a natural embedding assigning to a sparse configuration its characteristic function multiplied by the function , and we equip with the “–topology”, that is, the one induced by the norm of the Banach space . The –topology is finer than the topology induced from the ambient space .
Now we are in a position to describe the scheme of proof.
The limit measures are concentrated on the set of sparse configurations.
The claim makes sense because the set is a Borel subset in .
Given a function on the lattice such that , we define a function on the set of sparse configurations by the formula
(the product is convergent). Such functions will be called multiplicative functionals on configurations. Any multiplicative functional is continuous in the –topology.
Given a permutation and a measure on , we denote by the push–forward of under the transformation , see (0.1).
Let be fixed and range over . For any , let be the Radon–Nikodým derivative of the measure with respect to the measure . That is,
here belongs to the countable set of finite balanced configurations.
Fix an arbitrary .
(i) The function has a unique extension to a continuous function on .
(ii) As , the extended functions obtained in this way converge pointwise to a continuous function on .
(iii) The limit function can be written as a finite linear combination of multiplicative functionals of the form (0.2).
Here continuity is assumed with respect to the –topology. Actually, a somewhat stronger claim holds, see Proposition 3.1 and the subsequent discussion.
Claim 2 suggests that the limit function might serve as the Radon–Nikodým derivative for the limit measure, that is,
This relation is indeed true. We reduce it to the following claim.
Let be an arbitrary function on such that . Then the multiplicative functional given by (0.2) is absolutely integrable with respect to both the pre–limit and limit measures, and we have
This claim is stronger than the assertion about the weak convergence of measures which was known previously. Indeed, weak convergence of measures on means convergence on continuous test functions, while multiplicative functionals are, generally speaking, unbounded functions on and thus cannot be extended to continuous functions on the compact space .
To prove Claim 3 we use the well–known fact that the expectation of a multiplicative functional with respect to a determinantal measure can be expressed as a Fredholm determinant involving the correlation kernel. This makes it possible to reformulate the claim in terms of the correlation operators and (these are operators in the Hilbert space whose matrices are the correlation kernels of the measures and , respectively).
The reformulation is given in Claim 4 below. Represent the Hilbert space as the direct sum of two subspaces according to the splitting (positive and negative half–integers). Then any bounded operator in can be written as a matrix with operator entries (or “blocks”). Let denote the set (actually, algebra) of bounded operators in whose two diagonal blocks are trace class operators and two off–diagonal blocks are Hilbert–Schmidt operators. If then the Fredholm determinant makes sense ([BOO, Appendix]). We equip with the combined topology determined by the trace class norm on the diagonal blocks and the Hilbert–Schmidt norm on the off–diagonal ones.
Let stand for the operator of pointwise multiplication by the function in the space . The operator lies in , and, as goes to , the operators approach the operator in the combined topology of .
Note that the operator is not in , because the series taken over all is divergent. This is the source of difficulties. For instance, the assertion that belongs to is not a formal consequence of the boundedness of .
Claim 4 is the key technical result of the paper. The proof relies on explicit expressions for the correlation kernels in terms of contour integrals and requires a considerable computational work.
I do not know whether the operators approach simply in the trace class norm. The point is that the diagonal blocks are Hermitian nonnegative operators while the operators themselves are not. For nonnegative operators, one can use the fact that the convergence in the trace class norm is equivalent to the weak convergence together with the convergence of traces. For non–Hermitian operators, dealing with the trace class norm is difficult, while the Hilbert–Schmidt norm turns out to be much easier to handle. Fortunately, for the off–diagonal blocks, the convergence in the Hilbert–Schmidt norm already suffices.
0.5. Organization of the paper
Section 1 contains the basic notation and definitions related to the measures under consideration and their correlation kernels. Section 2 starts with basic facts related to multiplicative functionals and their connection to Fredholm determinants; then the proof of Claim 1 follows; it is readily derived from the explicit expression for the first correlation function of the measure . Section 3 is devoted to the proof of Claim 2. In Section 4, we formulate the main result (Theorem 4.1). Then we reduce to it to Theorem 4.2 and next to Theorem 4.3; they correspond to Claims 3 and 4, respectively. The main technical work is done in Sections 5 and 6, where we prove Theorem 4.3 (or Claim 4) separately for diagonal and off–diagonal blocks.
I am very much indebted to Alexei Borodin for a number of important suggestions which helped me in dealing with asymptotics of contour integral representations. I am also grateful to Leonid Petrov and Sergey Pirogov for valuable remarks.
1. Z–measures and related objects
1.1. Partitions and lattice point configurations
A partition is an infinite sequence of nonnegative integers such that and only finitely many ’s are nonzero. We set . Let denote the set of all partitions; it is a countable set. Following [Ma], we identify partitions and Young diagrams.
Let denote the set of all half–integers; that is, . By and we denote the subsets of positive and negative half–integers, so that is the disjoint union of and .
Subsets of are viewed as configurations of particles occupying the nodes of the lattice . The unoccupied nodes are called holes. Let denote the space of all particle configurations on . The space can be identified with the infinite product space and we equip it with the product topology. In this topology, is a totally disconnected compact space.
Recall (see Subsection 0.4) that the particle/hole involution on is the involutive map keeping intact particles and holes on and changing particles by holes and vice versa on . We denote the particle/hole involution by the symbol “”. In a more formal description, “” assigns to a configuration its symmetric difference with . In particular, .
To a partition we assign the semi–infinite point configuration
Note that among ’s some terms may repeat while the numbers are all pairwise distinct. Clearly, the correspondence is one–to–one. The configuration is sometimes called the Maya diagram of , see Miwa–Jimbo–Date [MJD].
For instance, the Maya diagram of the zero partition is . Any Maya diagram can be obtained from this one by finitely many elementary moves consisting in shifting one particle to the neighboring position on the right provided that it is unoccupied.
A finite configuration is called balanced if . An important fact is that “” establishes a bijective correspondence between the Maya diagrams and the balanced configurations. We set
An alternative interpretation of the balanced configuration is as follows: with
(recall that ), where the positive half–integers and are the modified Frobenius coordinates (Vershik–Kerov [VK]) of the Young diagram . They differ from the conventional Frobenius coordinates [Ma] by the additional summand . A slight divergence with the conventional notation is that we arrange the coordinates in the ascending order.
A direct explanation: is the number of diagonal boxes in and
where is the transposed diagram.
Thus, we have defined two embeddings of the countable set into the space , namely, and . These two embeddings are related to each other by the particle/hole involution on .
Note that each of the two embeddings maps onto a dense subset in .
1.2. Z-measures on partitions
Here we introduce a family of probability measures on , called the z–measures. The subscripts , , and are continuous parameters. Their range is as follows: parameter belongs to the open unit interval , and parameters and should be such that for any integer . Detailed examination of this condition shows that either and (the principal series of values), or both and are real numbers contained in an open interval with (the complementary series of values).
We shall need the generalized Pochhammer symbol :
where is the number of nonzero coordinates and
is the conventional Pochhammer symbol. Note that
where the product is taken over the boxes of the Young diagram , and and stand for the row and column numbers of a box.
In this notation, the weight of assigned by the z–measure is written as
where is the dimension of the irreducible representation of the symmetric group of degree indexed by .
Note that and enter the formula symmetrically, so that their interchange does not affect the z-measure.
For the origin of formula (1.1) and the proof that is indeed a probability measure, see Borodin–Olshanski [BO1], [BO2], [BO3] and references therein. Note that all the weights are strictly positive: this follows from the conditions imposed on parameters and . The z–measures form a deformation of the poissonized Plancherel measure and are a special case of Schur measures (see Okounkov [Ok]).
1.3. Limit measures
Throughout the paper the parameters and are assumed to be fixed. If the third parameter approaches 0, then the z–measures converge to the Dirac measure at the zero partition: this is caused by the factor .
A much more interesting picture arises as approaches 1. Then the factor forces each of the weights to tend to 0 (note that ). This means that the z–measures on the discrete space escape to infinity. However, the situation changes when we embed into . Recall that we have two embeddings, one producing semi–infinite configurations and the other producing finite balanced configurations . Denote by and the push–forwards of the z–measure under these two embeddings. Then the following result holds, see [BO2]:
In the space of probability measures on the compact space , there exist weak limits
Of course, and are transformed to each other under the particle/hole involution on , and the same holds for the limit measures.
1.4. Projection correlation kernels
The key object is a second order difference operator on the lattice . This operator acts on a test function , , according to
Since is an integer for , the expressions under the square root are strictly positive, due to the conditions imposed on the parameters and .
As shown in [BO4], determines an unbounded selfadjoint operator in the Hilbert space . This operator has simple, purely discrete spectrum filling the subset .
In the sequel we will freely pass from bounded operators in to their kernels and vice versa using the natural orthonormal basis in indexed by points : If is an operator in then its kernel (or simply matrix) is defined as .
Let denote the projection in onto the positive part of the spectrum of , and let denote the corresponding kernel. (Here and below all projection operators are assumed to be orthogonal projections.)
is the correlation kernel of the measure .
The operator corresponding to a correlation kernel of a determinantal measure will be called its correlation operator. Thus, the projection is the correlation operator of .
Let denote the difference operator on which is obtained by setting in the above formula defining . One can show that still determines a selfadjoint operator in . Its spectrum is simple, purely continuous, filling the whole real line. Let denote the projection onto the positive part of the spectrum.
(i) As goes to , the projection operators weakly converge to a projection operator .
(ii) serves as the correlation operator of the limit measure , that is, the kernel is the correlation kernel of .
Note that the weak convergence of operators in whose norms are uniformly bounded is the same as the pointwise convergence of the corresponding kernels. Note also that on the set of projections, the weak operator topology coincides with the strong operator topology.
The above definition of the operators and through the difference operators and is nice and useful but one often needs explicit expressions for the correlation kernels. Various such expressions are available:
Assume . For and outside the diagonal ,
and is Euler’s –function.
On the diagonal ,
where is the logarithmic derivative of the –function.
See [BO2] for a proof. In that paper, we called the kernel the Gamma kernel.
In the case (then necessarily ) an explicit expression can be obtained by taking the limit (see [BO2]), and the result is expressed through the function (outside the diagonal) or its derivative (on the diagonal):
Assume . For and outside the diagonal ,
On the diagonal ,
Here is a simple corollary of the above formulas, which we will need later on:
Let denote the density function of . We have
Recall that is related to by the particle/hole involution transformation on . It follows that the density functions of the both measures coincide on . By the very definition of determinantal measures, the density function of is given by the values of the correlation kernel on the diagonal . The formulas of Theorem 1.4 and Theorem 1.5 express through the psi–function and its derivative. The asymptotic expansion of as is given by formula 1.18(7) in Erdelyi [Er], which implies
Using this we readily get
1.5. –Symmetric kernels and block decomposition
For technical reasons, it will be more convenient for us to deal, instead of and , with the correlation kernels for the measures and . The latter kernels will be denoted as and , respectively. The link between two kinds of kernels, the “ kernels” and the “ kernels”, is given by the following relation (see [BOO, Appendix] for a proof):
Note that the factor does not affect the correlation functions (see Subsection 1.6). This factor becomes important in the limit regime considered in [BO1] and [BO3, §8], but for the purpose of the present paper, it is inessential and could be omitted; I wrote it only to keep the notation consistent with that of the previous papers [BO1], [BO2], [BO3].
Decompose the Hilbert space into the direct sum , where . Then every operator in can be written in a block form,
where acts from to , acts from to , etc.
In terms of the block form, (1.2) can be rewritten as follows (below denotes the operator of multiplication by ):
It follows that if is an Hermitian operator in then is also Hermitian, but with respect to an indefinite inner product in :
Such operators are called –Hermitian or –symmetric operators. Thus, the operators and are –symmetric.
The pre–limit operators belong to the trace class.
This claim is not obvious from the definition of the operators nor from the explicit expressions for the kernels, but can be easily derived from the results of [BO1] (it is immediately seen that the “–operator” related to through the formula is of trace class). The trace class property of is related to the fact that the measure lives on finite configurations (note that the trace of a correlation operator equals the expected total number of particles).
As for the limit measure , it lives on infinite configurations, and the limit operator is not of trace class.
1.6. Gauge transformation of correlation kernels
An arbitrary transformation of correlation kernels of the form
with a nonvanishing function does not affect the minors giving the values of the correlation functions. We call this a gauge transformation.
Thus, the correlation kernel is not a canonical object attached to a determinantal measure. This circumstance must be taken into account in attempting to solve Problem 1.
Recall that by we denote transposition of Young diagrams. Return to formula (1.1) for the z–measure weights and observe that and . This implies the important symmetry relation
Next, observe that under transposition , the modified Frobenius coordinates interchange: