A method of defining central and Gibbs measures and the ergodic method
aa r X i v : . [ m a t h . D S ] F e b A method of defining central and Gibbs measuresand the ergodic method
A. M. Vershik ∗ Abstract
We formulate a general statement of the problem of defining invari-ant measures with certain properties and suggest an ergodic methodof perturbations for describing such measures.
In the second half of the last century, a new method of defining proba-bility measures in infinite-dimensional systems, alternative to the classical(Kolmogorov’s) one, was gradually developed. Instead of a system of consis-tent finite-dimensional distributions, which uniquely defines a measure viaprojections, the new method involves another data system, roughly speak-ing, a consistent system of conditional measures. The method appearedindependently in the theory of Markov processes (E. B. Dynkin), in statis-tical physics (R. L. Dobrushin), etc. We present an abstract version of themethod, regarding it simultaneously as a far-reaching generalization of thetheory of measurable partitions of Lebesgue spaces and Rokhlin’s systems ofconditional measures and as the problem of describing invariant measures inthe theory of dynamical systems and Gibbs measures. The presentation isin terms of equipped equivalence relations ( e.e.r. ), or, in other words, Borelpartitions in a standard Borel space and “projective conditional measures”on elements of these partitions. We could also use the language of groupoidsor the language of the theory of extensions of measures from special alge-bras (but not σ -algebras) of sets to a σ -algebra (not the whole σ -algebra) of ∗ St. Petersburg Department of Steklov Institute of Mathematics, St. Petersburg StateUniversity, and Institute for Information Transmission Problems. X, A ) (isomorphic to the interval [0 , σ -algebra A of all Borel sets. Let τ be a Borel equivalence relation ( e.r. ) (i.e., partition) on this space withcountable classes and ρ be a Borel 2-cocycle with nonnegative real values onthis relation, i.e., a Borel function on pairs of equivalent points satisfyingthe conditions ρ ( x, x ) = ρ ( x, y ) ρ ( y, x ) = 1, ρ ( x, y ) ρ ( y, z ) = ρ ( x, z ). Thecocycle ρ uniquely (up to a positive coefficient) defines a finite or σ -finitenonnegative measure (in short, “conditional projective measure”) on eachequivalence class. The pair ( τ, ρ ) will be called an equipped equivalencerelation on the space ( X, A ). If all classes are finite, then the e.e.r. is nothingelse than a Borel measurable partition, and the cocycle defines conditionalprobability measures on all classes.On the other hand, it is well known that if µ is a probability measureon X (i.e., on a Lebesgue space) and τ is an e.r., then µ uniquely defines anequipment of this e.r., that is, a 2-cocycle, or a conditional projective mea-sure on almost every equivalence class. In the case where the e.r. is the par-tition into the orbits of an action of a countable group with a quasi-invariantmeasure, this cocycle is the so-called Radon–Nikodym cocycle RN µ . If thecocycle is identically equal to 1, then the measure is said to be invariant. Ifthe cocycle is not indicated, it is assumed to be identically equal to 1. Wenow formulate the main problem. Problem 1.
Let ( τ, ρ ) be an e.e.r. on a standard Borel space ( X, A ) ; findall Borel probability measures µ for which the Radon–Nikodym cocycle RN µ coincides with ρ almost everywhere with respect to the measure µ , or, inother words, find all probability measures with given conditional projectivemeasures on equivalence classes. In the case where such a measure is unique (it is this case that corre-sponds to Kolmogorov’s system of finite-dimensional distributions), we mayspeak of extending the measure to the σ -algebra of all measurable sets; inthe general case, there can be no uniqueness. It would be natural, extending Rokhlin’s terminology, to introduce the term “semimea-surable partition with a system of conditional projective measures.” X, A ) determined by Problem 1 is welldefined and forms, in a natural way, a Choquet simplex. The set of itsextreme points (Choquet boundary) is called the absolute of the e.e.r. ( τ, ρ )and denoted by Ab( X, τ, ρ ). Any two different measures from Ab(
X, τ, ρ )are mutually singular and defined on different complete σ -algebras.The traditional construction of Gibbs measures, as well as the problemof describing invariant measures of group actions in dynamics, obviously, fitsin this scheme. The notion of absolute is closely related to various notionsof boundary.If all classes of an e.e.r. are finite, then we have a well-defined Borelquotient space X/τ , which coincides with Ab(
X, τ, ρ ), and describing all(not ergodic) measures from M ( τ,ρ ) reduces to indicating a measure on thisquotient space. But if the e.r. does not define a Borel quotient space (spaceof classes), then studying the structure of the absolute is a difficult prob-lem and depends significantly on the geometry of classes. The solution ofProblem 1 can be “wild,” i.e., it may happen that the absolute has no reason-able parametrization, but in many (for example, combinatorial) problems,a parametrization can be found.It is not difficult to extend all these definitions to e.r.’s τ whose equiv-alence classes are not countable, but endowed with a well-defined locallycompact topology.It is important to emphasize that an e.r. can be studied only togetherwith a cocycle, i.e., a system of conditional projective measures (even if thecocycle is identically equal to 1). The main role in the further analysis ofthe subject (uniqueness, special properties of measures, etc.) must be playedby the geometry of classes of e.e.r.’s, but it is still poorly studied.We now state the inverse problem. Problem 2.
Let M be a family of probability measures defined on the σ -algebra A of a standard Borel space X . Find the minimal e.r. τ for whichall measures µ ∈ M define the same cocycle ρ ≡ RN µ . This problem is a generalization of the traditional problem related tosufficient statistics (cf. [1]), in which one usually seeks only measurable (forexample, finite) equivalence relations. In the above setting, there are norestrictions on the e.r. For example, consider the set of all Bernoulli measures Q ∞ ( p, − p ), p ∈ [0 , Q ∞ {
0; 1 } . The requirede.e.r. is the de Finetti partition with cocycle identically equal to 1: twosequences are equivalent if they coincide from some index n on and have thesame number of zeros among the first n coordinates.3 Hyperfinite and tame e.e.r.’s, the universal Markovmodel
We now consider Problem 1 for a special case whose importance is due toa large number of applications. Namely, it includes the problem of describingcharacters of locally finite groups, or, more generally, describing traces onAF-algebras, as well as the problem of describing central measures on pathspaces of graded graphs.
An equipped equivalence relation τ is said to be hyperfinite if it is a mono-tonely increasing limit of a sequence of finite equivalence relations: τ = S n ξ n . Thus, a hyperfinite e.e.r. can be defined by a sequence of its finiteapproximations, i.e., a decreasing sequence of measurable partitions { ξ n } n with finite elements and conditional measures on them. Such sequences arecalled filtrations . For details, see [8].By a number of well-known theorems, the orbit partition for a groupaction with an invariant measure is hyperfinite if and only if the group isamenable. However, orbit partitions with nontrivial cocycles can be hyper-finite also for nonamenable groups. Note that the hyperfiniteness conditionfor an e.e.r. is a condition on the cocycle, i.e., on the conditional projectivemeasures, but apparently it has not been stated in this form. For Lebesguespaces, a hyperfinite e.e.r. is unique up to isomorphism (generalized H. Dye’stheorem).We impose a slightly stronger (than hyperfiniteness) condition on theapproximating sequence of measurable partitions { ξ n } : an e.r. is said to be tame , or locally hyperfinite , if for every n the number of types of conditionalmeasures of the partition ξ n is finite. This condition singles out a class ofhyperfinite e.e.r.’s that is of most interest for applications. Now we describea universal model of a tame e.e.r. Definition 1.
Let X n be a finite or compact space and { π n } be a set of“transition operators” that send a point x ∈ X n to a subset π n ( x ) ⊂ X n +1 .The corresponding Markov (nonstationary) compactum Mar is the space ofsequences
Mar ⊂ {{ x n } ∞ n =1 : x n ∈ X n , n = 1 , , . . . } where { x n } ∈ Mar ⇐⇒ x n +1 ∈ π n ( x n ) for every n ≥ . Elements of Mar are called trajectories, or paths. The tail equivalencerelation τ on the Markov compactum Mar is the following relation on tra-4ectories: { x n } ∼ τ { y n } ⇐⇒ there exists N such that x n = y n for every n > N . The Markov compactum Mar is endowed with the weak topology andBorel structure. A Markov Borel measure P is defined by an initial dis-tribution µ ( · ) of the coordinate x and a collection of transition proba-bilities, i.e., a family of measures { P n,x } , n = 1 , , . . . , x ∈ X n , where P n,x ( y ) = Prob( x n +1 = y | x n = x ).But we will need another data system on a Markov compactum, a sys-tem of cotransition probabilities . It is a family of measures { P n,x } on X n , n = 1 , , . . . , x ∈ X n +1 , where P n,x ( y ) = Prob( x n = y | x n +1 = x ). Sucha system does not yet define a global measure on the entire Markov com-pactum. Lemma 1.
Every system of cotransition measures defines a cocycle on thetail e.r. of the Markov compactum: the quotient of the conditional measuresof two paths { x n } and { y n } coinciding for n > N is equal to the quotient ofthe products of the corresponding transition probabilities Y ≤ i ≤ N Prob( x i | x i +1 )Prob( y i | y i +1 ) . Such cocycles will be called Markov cocycles; a Markov compactumequipped with a Markov cocycle, i.e., a system of cotransitions, will becalled an equipped Markov compactum .Obviously, every Markov measure on Mar uniquely determines a Markovcocycle, but, in general, a system of contransitions, i.e., a Markov cocycle,does not uniquely determine a Markov measure. It is equally clear that ona Markov compactum there can exist non-Markov cocycles.
Theorem 1 (universal model) . For every standard Borel space X anda tame equipped equivalence relation τ on X with cocycle ρ there exists anequipped Markov compactum Mar and a Borel isomorphism T : X → Mar that sends τ to the tail e.r. on Mar and sends ρ to a Markov cocycle.Thus, Problem of finding all invariant measures for a tame e.e.r. re-duces to the problem of finding all Markov probability measures P on a com-pactum Mar with a given system of cotransition probabilities. In other words,to the problem of describing Markov chains with given cotransitions.
If the cocycle is identically equal to 1, i.e., all conditional measures of allorders are uniform, then we obtain the problem of describing all measuresof maximal entropy on a given Markov compactum.5he absolute of a Markov compactum Mar is denoted by Ab(Mar). Theproof of the theorem essentially follows from the results of [8].Instead of the language of Markov compacta, one can use the languageof N -graded graphs (Bratteli diagrams): the path space of such a graph isa Markov compactum, which ensures the parallelism. In many situations(mainly of combinatorial nature), the language of graphs is more preferable.The author does not know models similar to the universal model for generale.e.r.’s. By the ergodic method of solving Problem 1 about invariant measures forhyperfinite equivalence relations we mean the method of finding invariantdistributions and invariant measures based on the pointwise ergodic the-orem or, more exactly, on the pointwise martingale convergence theorem,applied to the characteristic functions of sets from a basis of the σ -algebraon which the required measure is defined. In this meaning, the term isused in the author’s paper [7] and in earlier papers (see, e.g., [4]). But thepractical task of finding invariant measures, i.e., probabilities of cylinders,or transition probabilities, as limits of some conditional expectations canbe quite difficult. A very important factor is the choice of a basis of setswhose measures are being calculated. But, on the other hand, the problemof finding all ergodic measures can be “wild,” so complete calculations canbe essentially infeasible. A reasonable classification of hyperfinite absolutes(i.e., systems of conditional projective measures) is hardly possible; whilethe Borel classification of e.e.r.’s is, on the contrary, too rough (see [2]); theauthor does not know any other, intermediate, classification criteria. Thatis why, it is important to have feasible solvability criteria for the problem ofdescribing invariant measures, as well as methods of reduction of problemsto a few canonical problems.One of these fundamental problems, whose solution is obtained by a canon-ical application of the ergodic method, is the problem of describing all er-godic measures on the infinite product X ∞ = Q ∞ X (where X is a Borelspace) invariant under the group S ∞ of all finite permutations of coordinates.Denote by τ F the e.r. on X ∞ generated by the partition into the orbits ofthe action of S ∞ . The answer to Problem 1 is given by de Finetti’s theo-rem and says that every ergodic invariant measure is a Bernoulli measurewith an arbitrary one-dimensional distribution (= measure on X ). Thus,6b( X ∞ , τ F ) = Meas( X ).If we regard this answer up to a measure-theoretic isomorphism, it turnsout that the absolute consists of a unique purely continuous measure on X ,a continuum of discrete measures, and their mixtures, i.e., Ab( X ∞ , τ F ) = {{ α n } : α n ≥ , P n α n ≤ } . In many recent examples of problems related to absolute in combinatorialand algebraic situations, the answer (hypothetical or otherwise) has a sim-ilar structure: the absolute is a Choquet simplex (which can be called thesecondary simplex), i.e., the ergodic measures themselves also admit a de-composition.
That is why, it is natural to conjecture that a proof of this factshould be sought not in a direct calculation but in analyzing the reductionto the fundamental de Finetti problem described above. We suggest thefollowing method, which can be called the method of perturbations. Foran unperturbed problem, we take the de Finetti problem about τ F . Thefirst step is to construct a homomorphism T of the space where we seek theabsolute for some e.r. τ to the space X ∞ such that T ( τ ) is a subpartitionof τ F . At the second step, we must verify that the absolute of T ( τ ) is similarto, or even coincides with, the absolute of τ F . To find T , if it exists, is themost nontrivial part of the method.The second step is related to a problem about the infinite product X ∞ which is of independent interest. Problem 3.
For what e.r.’s τ satisfying the condition τ ≻ τ F does theabsolute Ab( X ∞ , τ ) consist of all Bernoulli measures? The following partial answer is useful:
Lemma 2.
Let τ, τ ′ be two e.e.r.’s on a Borel space X with cocycles iden-tically equal to , and let Ab(
X, τ ′ ) ⊂ Ab(
X, τ ) . These absolutes coincideif and only if the following condition is satisfied: for every ergodic measure µ ∈ Ab(
X, τ ′ ) , the e.e.r. τ is ergodic with respect to µ . In turn, proving ergodicity reduces to verifying that a certain sequence offunctionals converges to a constant in measure, and not to the more difficultproblem of finding weak limits, as in the general ergodic method.An illustrative example of the usefulness of the method of perturba-tions is given by the problem about central measures on the Young graph.Thoma’s theorem about characters, more exactly, a rephrasing of this the-orem as an assertion about the absolute of the Young graph, leaves nodoubt that this problem should be regarded in connection with de Finetti’stheorem. The fact is that the answers to these problems are remarkably7imilar. Namely, the absolute is stratified: a stratum of discrete measuresparametrized by one-dimensional frequencies with sum 1 and a stratum ofmeasures with zero frequencies. However, all proofs known to date (see [9])are not elementary and do not reveal the closeness of these problems. Thisrelation is indeed nontrivial, and the main role here is played by the dynamicproperties of the RSK algorithm, which makes it possible to construct a re-quired lifting of the graph of Q -tableaux to the Schur–Weyl graph.Using Q -tableaux arising in the RSK algorithm to cover central measureson the Young graph by Bernoulli measures was first suggested in [3]; thefact that this correspondence is an isomorphism was proved in [5]. But theembedding discussed above has not been noted; at the same time, a carefulanalysis shows that the method of perturbations allows one also to provethe theorem about the absolute, i.e., to prove that in this way we obtain allergodic central measures. For Young tableaux with finitely many rows, thisresult is implicitly contained in [11]. References [1] P. Diaconis and D. Freedman, Partial exchangeability and sufficiency,in:
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