Asymptotic Formulas for Macdonald Polynomials and the Boundary of the $(q,t)$-Gelfand-Tsetlin Graph

We introduce Macdonald characters and use algebraic properties of Macdonald polynomials to study them. As a result, we produce several formulas for Macdonald characters, which are generalizations of those obtained by Gorin and Panova in [Ann. Probab. 43 (2015), 3052-3132, arXiv:1301.0634], and are expected to provide tools for the study of statistical mechanical models, representation theory and random matrices. As first application of our formulas, we characterize the boundary of the $(q,t)$-deformation of the Gelfand-Tsetlin graph when $t=q^{\theta}$ and $\theta$ is a positive integer.


Introduction
Macdonald polynomials are remarkable two-parameter q, t generalizations of Schur polynomials. They were first introduced by Ian G. Macdonald in [31]; the canonical reference is his classical book [32]. The Macdonald polynomials are very interesting objects for representation theory and integrable systems, due to their connections with quantum groups, e.g., [20,33], double affine Hecke algebras, e.g., [13,28], etc. More recently, and releveant for us, Macdonald polynomials have been heavily used to study probabilistic models arising in mathematical physics and random matrix theory. The important work [4] of Borodin and Corwin showed how to use the algebraic properties of these polynomials to obtain analytic formulas that allow asymptotic analysis of the so-called Macdonald processes. Remarkably, by specialization or degeneration of the parameters q, t defining Macdonald processes, the paper [4] yields tools that can be used to analyze interacting particle systems [5], beta Jacobi corners processes [6], probabilistic models from asymptotic representation theory [3], among others; see the survey [9] and references therein.
It should be noted that the special case t = q of Macdonald processes are known as Schur processes and they have the special property of being determinantal point processes, thus allowing much more control over their asymptotics. Schur processes were introduced by Okounkov and Reshetikhin, as generalizations of the classical Plancherel measures, several years before the work of Borodin and Corwin [35,38]. The Schur processes, though a very special case of Macdonald processes, produced various applications to statistical models of plane partitions and random matrices, see for example [27,39]. However, most of the physical models that were studied with the Macdonald processes do not have a determinantal structure, and therefore they could not have been analyzed solely by means of the Schur processes machinery.
The conclusion from the story of Schur and Macdonald processes is that studying the more complicated object can allow one to tackle more complicated questions, despite losing some integrability (such as the determinantal structure in the case of Schur processes). We follow this philosophy in our work, by introducing and studying Macdonald characters, two-parameter q, t generalizations of normalized characters of unitary groups. The normalized characters of the unitary groups are expressible in terms of Schur polynomials, reason why we will call them Schur characters, whereas the generalization we present involves Macdonald polynomials.
Our main results are asymptotic formulas for Macdonald characters, which are generalizations of those for Schur characters, proved in [24] by different methods. As it is expected, the asymptotic formulas for Schur characters are simpler and they involve certain determinantal structure, whereas the formulas for Macdonald polynomials are more complicated and the determinantal structure is no longer present. However the advantage of our work on Macdonald polynomials, much like the advantage of Macdonald processes over Schur processes, is that we are able to access a number of asymptotic questions that are more general than those given in [24]. The tools we obtain in this paper are therefore very exciting, given that the formulas for Schur characters have already produced several applications to stochastic discrete particle systems, lozenge and domino tiling models, and asymptotic representation theory [10,11,12,22,24,44].
The paper [15] is this article's companion, in which the author studies Jack characters, a natural degeneration of Macdonald characters and obtains their asymptotics in the Vershik-Kerov limit regime. The approach to the study of asymptotics of Jack characters is different from the approach we use here to study the asymptotics of Macdonald characters; in particular, it relies heavily on the Pieri integral formula, see [15] for further details. The tools from this paper and [15] afford us a very strong control over the asymptotics of Macdonald characters, Jack characters and Bessel functions [36,43], if the number of variables remains fixed and the rank tends to infinity. As another application of the developed toolbox, we have studied a Jack-Gibbs model of lozenge tilings in the spirit of [7,25]. The author was able to prove the weak convergence of statistics of the Jack-Gibbs lozenge tilings model near the edge of the boundary to the well-known Gaussian beta ensemble, see, e.g., Forrester [1,Chapter 20] and references therein. This result, and its rational limit concerning corner processes of Gaussian matrix ensembles, will appear in a forthcoming publication.
We proceed with a more detailed description of the results of the present paper.

Description of the formulas
The main object of study in this paper are the Macdonald characters, which we define as follows.
For integers 1 ≤ m ≤ N , a Macdonald character of rank N and m variables is a polynomial, with coefficients in C(q, t), of the form P λ (x 1 , . . . , x m ; N, q, t) def = P λ x 1 , . . . , x m , 1, t, . . . , t N −m−1 ; q, t P λ 1, t, t 2 , . . . , t N −1 ; q, t , where P λ (x 1 , . . . , x N ; q, t) is the Macdonald polynomial of N variables parametrized by the signature λ = (λ 1 ≥ λ 2 ≥ · · · ≥ λ N ) ∈ Z N . Macdonald characters, under the specialization t = q, turn into q-Schur characters, which have appeared previously in [21,24]. The reason behind the use of the word "character" is that q-Schur characters turn into normalized characters of the irreducible rational representations of unitary groups, after the degeneration q → 1. We make one further comment about terminology. Macdonald characters, as defined here, are two-parameter q, t degenerations of normalized and irreducible characters of unitary groups. One could also consider a two-parameter degeneration of the characters of the symmetric groups, in the spirit of Lassalle's work [29], where a one-parameter degeneration of symmetric group characters was considered. Thus a better name for our object would be Macdonald unitary character. For convenience, we simply will use the name Macdonald character. We remark that Macdonald symmetric group characters have not been considered yet, to the author's best knowledge. However, there have been many articles studying the structural theory and asymptotics on Jack symmetric group characters, notably several recent works by Maciej Do lȩga, Valentin Féray and PiotrŚniady, see, e.g., [16,17,18,46].
The main theorems of this paper fall into two categories: (A) Integral representations for Macdonald characters of one variable and arbitrary rank N .
The initial idea that led to the integral representations in this paper is due to Andrei Okounkov, see [24,remark following Theorem 3.6]. An example of the integral formulas we prove is the following theorem. Observe that the integrand is a simple expression in terms of q-Gamma functions and can be analyzed by well known methods of asymptotic analysis, such as the method of steepest descent or the saddle-point method [14]. In this paper, we study the regime in which the signatures grow to infinity, whereas the remaining parameters are fixed. Theorem 1.1 (consequence of Theorem 3.2). Assume q ∈ (0, 1) and θ > 0. Let N ∈ N, λ ∈ GT N and x ∈ C \ {0}, |x| ≤ q θ(1−N ) . The integral below converges absolutely and the identity holds where C + is a certain contour described in Theorem 3.2, and which looks as in Fig. 2. In the formula above, we used the q-Pochhammer symbol (z; q) ∞ and the q-Gamma function Γ q (z); see Appendix A.
(B) Formulas expressing Macdonald characters of m variables (and rank N ) in terms of Macdonald characters of one variable (and rank N ).
These formulas will involve certain q-difference operators. Formulas of this kind will be called multiplicative formulas. 1 One of the simplest multiplicative formulas we prove is the one below that expresses a Macdonald character of two variables in terms of those of one variable; the general formula is given below in Theorem 4.1.
Observe that the multiplicative formula above requires θ ∈ N. It is somewhat surprising that all the identities we prove in this paper, even the integral representations, behave better for θ ∈ N. 1 The reason for the name is that analogous formulas for Schur characters were used to prove statements of the form lim 1.2 The boundary of the (q, t)-Gelfand-Tsetlin graph As an application of our formulas, we characterize the space of central probability measures in the path-space of the (q, t)-Gelfand-Tsetlin graph, when t = q θ and θ is a positive integer. To state our result, we first introduce a few notions. Assume that q, t ∈ (0, 1) are generic real parameters for the moment.
The Gelfand-Tsetlin graph, or simply GT graph, is an undirected graph whose vertices are the signatures of all lengths GT = N ≥0 GT N ; we also include the empty signature ∅ as the only element of GT 0 for convenience. The set of edges is determined by the interlacing constraints, namely the edges in the GT graph can only join signatures whose lengths differ by 1 and µ ∈ GT N is joined to λ ∈ GT N +1 if and only if λ N +1 ≤ µ N ≤ λ N ≤ · · · ≤ λ 2 ≤ µ 1 ≤ λ 1 .
If the above inequalities are satisfied, we write µ ≺ λ. If µ ∈ GT N and λ ∈ GT N +1 are joined by an edge, then we consider the expression Λ N +1 N (λ, µ) given by Λ N +1 N (λ, µ) = ψ λ/µ (q, t) P µ t N , . . . , t 2 , t; q, t P λ t N , . . . , t, 1; q, t , where ψ λ/µ (q, t) is given in the branching rule for Macdonald polynomials, see Theorem 2.5 below. If µ ∈ GT N is not joined to λ ∈ GT N +1 , set Λ N +1 N (λ, µ) = 0. One can easily show, see, e.g., Theorem 2.3 below, Thus, for any N ∈ N, λ ∈ GT N +1 , Λ N +1 N (λ, ·) is a probability measure on GT N . For this reason, we will call the expressions Λ N +1 N (λ, µ) cotransition probabilities. Next we define the path-space T of the GT graph as the set of infinite paths in the GT graph that begin at ∅ ∈ GT 0 : T = {τ = (∅ = τ (0) ≺ τ (1) ≺ τ (2) ≺ · · · ) : τ (n) ∈ GT n ∀ n ∈ Z ≥0 }. Each finite path of the form φ = ∅ = φ (0) ≺ φ (1) ≺ · · · ≺ φ (n) defines a cylinder set S φ = τ ∈ T : τ (1) = φ (1) , . . . , τ (n) = φ (n) ⊂ T . We equip T with the σ-algebra generated by the cylinder sets S φ , over all finite paths φ. Equivalently, the σ-algebra of T is its Borel σ-algebra if we equip T with the topology it inherits as a subspace of the product n≥0 GT n . Each probability measure M on T admits a pushforward to a probability measure on GT m via the obvious projection map We say that a probability measure M on T is a (q, t)-central measure if for all N ≥ 0, all finite paths φ (0) ≺ · · · ≺ φ (N ) , and for some probability measures M N on GT N . It then automatically follows that M N = (Proj N ) * M are the pushforwards of M ; moreover, they satisfy the coherence relations We denote by M prob (T ) the set of (q, t)-central (probability) measures on T ; it is clearly a convex subset of the Banach space of all finite and signed measures on T . Let us denote by Ω q,t = Ex(M prob (T )) the set of its extreme points. From a general theorem, we can deduce that Ω q,t ⊂ M prob (T ) is a Borel subset. We call Ω q,t , with its inherited topology, the boundary of the (q, t)-Gelfand-Tsetlin graph. The theorem stated below, which is our main application, completely characterizes the topological space Ω q,t . Before stating it, let us make a couple of relevant definitions.
For any k ∈ Z, one can easily show that µ ∈ GT m interlaces with λ ∈ GT m+1 iff A k µ ∈ GT m interlaces with A k λ ∈ GT m+1 , that is, µ ≺ λ iff A k µ ≺ A k λ. This allows us to define automorphisms A k of T by One can similarly obtain maps A k on the set of finite paths of length n by φ = (φ (0) ≺ φ (1) ≺ · · · ≺ φ (n) ) → A k φ = (A k φ (0) ≺ A k φ (1) ≺ · · · ≺ A k φ (n) ). Consequently we can also define automorphisms on cylinder sets by in the natural way.
We named several maps above by the same letter A k , but there should be no risk of confusion. For our main theorem, we make the assumption θ ∈ N, t = q θ . We believe the theorem can be generalized for any θ > 0, but we do not have a proof at the moment. Theorem 1.3. Assume q ∈ (0, 1), θ ∈ N and set t = q θ .
1. There exists a homeomorphism N : N → Ω q,t sending each ν ∈ N to the (q, t)-central probability measure M ν ∈ Ω q,t determined by the relations λ∈GTm M ν m (λ) In (1.2), we denoted by {M ν m } m≥1 the corresponding sequence of pushforwards of M ν under the projection maps Proj m : T → GT m . The left side in (1.2) is absolutely convergent on T m , T = {z ∈ C : |z| = 1}, and the functions Φ ν in the right side are defined in (5.2) and (5.8). The probability measure M ν is determined uniquely by the relations (1.2).

2.
For each k ∈ Z, the probability measures M ν and M A k ν are related by Another main result of this article is Theorem 7.9, where we characterize the Martin boundary of the (q, t)-Gelfand-Tseltin graph for t = q θ and θ ∈ N. In fact, we first prove that the Martin boundary is homeomorphic to N and then show that the minimal boundary Ω q,t coincides with the Martin boundary. See Sections 7.1 and 7.2 for the definition and characterization of the Martin boundary.

Comments on Theorem and connections to existing literature
Our first comment is that Theorem 1.3 is a generalization of the main theorem in the article of Vadim Gorin [21], which is the special case θ = 1 of our theorem, and characterizes the boundary of the q-Gelfand-Tsetlin graph. Some ideas in the proofs are the same, especially the overall scheme of using the ergodic method of Vershik-Kerov, see [47], but we need many new arguments as well. For example, [21] makes heavy use of the shifted Macdonald polynomials, in particular the binomial formula for shifted Macdonald polynomials at t = q, [34], while we do not use them at all. Moreover, in order to prove that the boundary of the q-Gelfand-Tsetlin graph is homeomorphic to N , [21] made use of the following closed formula for the shifted-Schur generating function of M ν,θ=1 N , in the case that ν 1 ≥ 0: In the formula above, s * λ (x 1 , . . . , x N ; q) is the shifted Macdonald polynomial at t = q. In addition to the usefulness of the closed formula (1.5) above, the multiplicative structure is surprising. It would be interesting to find a closed formula for the shifted Macdonald generating function of the measures M ν N , for general θ ∈ N, and find out if the multiplicative structure still holds in this generality.
It is shown in [21] that their main statement is equivalent to the characterization of certain Gibbs measures on lozenge tilings. A conjectural characterization of positive q-Toeplitz matrices is also given in that paper. Finally, it is mentioned that the asymptotics of q-Schur functions is related to quantum traces and the representation theory of U (gl ∞ ). It would be interesting to extend some of these statements to the Macdonald case, especially to connect the asymptotics of Macdonald characters to the representation theory of inductive limits of quantum groups.
Several other "boundary problems" have appeared in the literature in various contexts. For instance, in the limiting case t = q → 1, the problem of characterizing the boundary Ω q,t becomes equivalent to characterizing the space of extreme characters of the infinite-dimensional unitary group U(∞) = lim → U(N ). The answer also characterizes totally positive Toeplitz matrices [19,47,48]. Also in the degenerate case t = q 2 or t = q 1/2 and q → 1, the boundary problem becomes equivalent to characterizing the space of extreme spherical functions of the infinitedimensional Gelfand pairs (U(2∞), Sp(∞)) and (U(∞), O(∞)), respectively. This question, and in fact a more general one-parameter "Jack"-degeneration, was solved in [37]. A similar degenerate question in the setting of random matrix theory was studied in [42]. Some of the tools in this paper can be degenerated easily to these scenarios and they may provide an alternative approach to their proof as well; for example, the special case of our toolbox in the case t = q → 1 was used in [24] to study the corresponding boundary problem, and in [15] we also study refine the asymptotic result that is needed to solve the boundary problem of [37].
Another similar boundary problem in a somewhat different direction is the following. Assume we consider the Young graph instead of the Gelfand-Tsetlin graph, e.g., see [8]. Assume also that the cotransition probabilities coming from the branching rule of Macdonald polynomials are replaced by the cotransition probabilities coming from the Pieri-rule. In this setting, the boundary problem has not been solved yet, but it is expected that the answer is given by Kerov's conjecture, which characterizes Macdonald-positive specializations, see [4,Section 2].
Finally, it was brought to my attention, after I completed the results of this paper, that Grigori Olshanski has obtained a characterization of the extreme set of (q, t)-central measures in the extended Gelfand-Tsetlin graph for more general parameters q, t by different methods. His work follows the setting of the paper [23] of Gorin-Olshanski, which is some sort of analytic continuation to our proposed boundary problem. Interestingly, new features arise, e.g., two copies of the space N characterize the boundary in his context, one can define and work with suitable analogues of zw-measures, etc. Another related work in the t = q case is his recent article [41].

Organization of the paper
The present work is organized as follows. In Section 2, we briefly recall some important algebraic properties of Macdonald polynomials that will be used to obtain our main results. We prove integral representations for Macdonald characters of one variable in Section 3. Next, in Section 4, we obtain multiplicative formulas for Macdonald characters of a given number of variables m ∈ N in terms of those of one variable. By making use of our formulas, in Section 5 we obtain asymptotics of Macdonald characters as the signatures grow to infinity in a specific limit regime. In Sections 6 and 7, we define and characterize the boundary of the (q, t)-Gelfand-Tsetlin graph in the case that θ ∈ N and t = q θ . The asymptotic statements of Section 5 play the key role in the characterization of the boundary.
In Appendix A, we have bundled the necessary language and results of q-theory that are used throughout the paper. In Appendix B, we make some computations with expressions that appear in the multiplicative formulas for Macdonald polynomials.

Symmetric Laurent polynomials
A canonical reference for symmetric polynomials is [32]. We choose to give a brief overview of the tools that we need from [32], in order to fix terminology and to introduce lesser known objects, such as signatures and Macdonald Laurent polynomials.
Partitions can be graphically represented by their Young diagrams. The Young diagram of partition λ is the array of boxes with coordinates (i, j) with 1 ≤ j ≤ λ i , 1 ≤ i ≤ (λ), where the coordinates are in matrix notation (row labels increase from top to bottom and column labels increase from left to right), see Fig. 1.
A signature is a sequence of weakly decreasing integers λ = (λ 1 ≥ λ 2 ≥ · · · ≥ λ k ), λ i ∈ Z ∀ i. A positive signature is a signature whose elements are all nonnegative. The length of a signature, or positive signature, is the number k of elements of it. Positive signatures which differ by trailing zeroes are not identified, in contrast to partitions; for example, (4, 2, 2, 0, 0) and (4, 2, 2) are different positive signatures, the first of length 5 and the second of length 3. We shall denote GT N (resp. GT + N ) the set of signatures (resp. positive signatures) of length N . Evidently GT + N can be identified with the set of all partitions of length ≤ N . Under this identification, we are allowed s Figure 1. Young diagram for the partition λ = (5,4,4,2). Square s = (3, 3) has arm length, arm colength, leg length and leg colength given by a(s) = 1, a (s) = 2, l(s) = 0, l (s) = 2. to talk about the Young diagram of a positive signature λ ∈ GT + N , its size, the dominance order, and other attributes that are typically associated to partitions. Note, however, that length is defined differently for partitions and for positive signatures.
Let us now switch to notions pertaining to symmetric (Laurent) polynomials. Fix a positive integer N . Consider the field F = C(q, t) and recall the algebra Λ F [x 1 , . . . , x N ] of symmetric polynomials on the variables x 1 , . . . , x N with coefficients in F . For any m ∈ Z ≥0 , recall also the subalgebra Λ m F [x 1 , . . . , x N ] of symmetric polynomials on x 1 , . . . , x N that are homogeneous of degree m; then We also denote by Λ F [x ± 1 , . . . , x ± N ] the algebra of symmetric (with respect to the transpositions The connection between partitions/signatures and symmetric polynomials comes from the where S N · λ is the orbit of λ under the permutation action of S N , and the sum runs over distinct elements µ of that orbit. It is implied that
• Orthogonality relation: The Macdonald polynomials are orthogonal with respect to the inner product (·, ·) q,t on Note that P ∅ (q, t) = 1. If N < (λ), we set P λ (x 1 , . . . , x N ; q, t) def = 0 for convenience. When we are talking about Macdonald polynomials and some of their properties which hold regardless of the number N of variables, as long as N is large enough, we simply write P λ (q, t) instead of P λ (x 1 , . . . , x N ; q, t).
From the triangular decomposition of Macdonald polynomials, P λ (q, t) is a homogeneous polynomial of degree |λ|.
Finally, we have the following index stability property: (2.1) As pointed out before, the set of partitions of length ≤ N is in bijection with GT + N . Thus we can index the Macdonald polynomials by positive signatures rather than by partitions: for any λ ∈ GT + N , we let P λ (x 1 , . . . , x N ; q, t) be the Macdonald polynomial corresponding to the partition associated to λ. We can slightly extend the definition above and introduce Macdonald Laurent polynomials P λ (x 1 , . . . , x N ; q, t) for any λ ∈ GT N . Let λ ∈ GT N be arbitrary. If λ N ≥ 0, then λ ∈ GT + N and P λ (x 1 , . . . , x N ; q, t) is already defined. If λ N < 0, choose m ∈ N such that λ N + m ≥ 0 and so (λ 1 + m, . . . , λ N + m) ∈ GT + N . Then define By virtue of the index stability property, the Macdonald (Laurent) polynomial P λ (x 1 , . . . , x N ; q, t) is well-defined and does not depend on the value of m that we choose. For simplicity, we call P λ (x 1 , . . . , x N ; q, t) a Macdonald polynomial, whether λ ∈ GT + N or not. In a similar fashion, we can define monomial symmetric polynomials m λ , for any λ ∈ GT N .
Recall the definitions of the arm-length, arm-colength, leg-length, leg-colength a(s), a (s), l(s), l (s) of the square s = (i, j) of the Young diagram of λ, given by a(s) = λ i − j, a (s) = j − 1, l(s) = λ j − i, l (s) = i − 1; we note that λ j = |{i : λ i ≥ j}| is the length of the jth part of the conjugate partition λ , see Fig. 1.
From the second equality of Theorem 2.3 and the definition of dual Macdonald polynomials, we obtain Since the Macdonald polynomial P λ (x 1 , x 2 , . . . , x N ; q, t) is symmetric in x 1 , x 2 , . . . , x N , it is also a symmetric polynomial on x 2 , . . . , x N ; thus it is a linear combination of Macdonald polynomials P µ (x 2 , . . . , x N ; q, t) with coefficients in F [x 1 ]. More precisely, we have the so-called branching rule for Macdonald polynomials: where the branching coefficients are and the sum is over positive signatures µ ∈ GT + N −1 that satisfy the interlacing constraint which is written succinctly as µ ≺ λ.
We come to our final tool on Macdonald polynomials. It is the main theorem of [30], and is called the Jacobi-Trudi formula for Macdonald polynomials. For any n ∈ N, nonnegative integers τ 1 , . . . , τ n , variables u 1 , . . . , u n , define the rational functions C (q,t) τ 1 ,...,τn (u 1 , . . . , u n ) by where M (N ) is the set of strictly upper-triangular matrices with nonnegative entries, and for each 1 ≤ s ≤ N , the integers τ + s , τ − s , depend only on the indexing matrix τ and are defined by Remark 2.8. Observe that, even though M (N ) is an infinite set, the only nonvanishing terms in the sum above are those τ ∈ M (N ) such that λ s + τ + s − τ − s ≥ 0 ∀ s = 1, . . . , N . In other words, the sum is indexed by points of the discrete N (N −1) Let us introduce the last piece of terminology and main object of study in this paper.
and call P λ (x 1 , . . . , x m ; N, q, t) the Macdonald unitary character of rank N , number of variables m and parametrized by λ. For simplicity of terminology, we call P λ (x 1 , . . . , x m ; N, q, t) a Macdonald character rather than a Macdonald unitary character. Observe that if q, t ∈ C are such that |q|, |t| ∈ (0, 1), the evaluation identity for Macdonald polynomials, Theorem 2.3, shows that the denominator of (2.7) is nonzero.

Integral formulas for Macdonald characters of one variable
In this section, assume q is a real number in the interval (0, 1). There will also be a parameter θ, typically θ > 0, but we also consider cases when θ is a complex number with θ > 0. In either case, the parameter t = q θ satisfies |t| < 1.

Statements of the theorems
The simplest contour integral representation is the following, which works only when t = q θ , θ ∈ N, and involves a closed contour around finitely many singularities.
Theorem 3.2. Let θ > 0, t = q θ , N ∈ N, λ ∈ GT N and x ∈ C \ {0}, |x| ≤ 1. The integral below converges absolutely and the equality holds Contour C + is a positively oriented contour consisting of the segment , for some − π 2 ln q > r > 0 and λ N > M , see Fig. 2. Observe that C + encloses all real poles of the integrand (which accumulate at +∞) and no other poles.
The reader is referred to Appendix A for a reminder of the definition of the q-Gamma function, its zeroes and poles. Theorem 3.3. Let θ > 0, t = q θ , N ∈ N, λ ∈ GT N and x ∈ C, |x| ≥ 1. The integral below converges absolutely and the equality holds P λ x, t, t 2 , . . . , t N −1 ; q, t P λ 1, t, t 2 , . . . , t N −1 ; q, t = ln q q − 1  Remark 3.5. The formulas in Theorems 3.2 and 3.3 probably hold for more general contours C + , C − , but we do not need more generality for our purposes. Remark 3.7. The infinite contours in Theorems 3.2 and 3.3 are needed because there are infinitely many real poles in the integrand and the contour needs to enclose all of them. When θ ∈ N, the integrands have finitely many poles, and we can therefore close the contours, obtaining eventually Theorem 3.1. More generally, if θ > 0 is such that θN ∈ N, a similar remark applies. In fact, we can write the product of q-Gamma function ratios appearing in the integrand (3.2) as , (3.4) and since Γ q (t + 1) = 1−q t 1−q Γ q (t), we conclude that the product above is a rational function in q −z with finitely many real poles. Thus formula (3.2) is true if we replaced contour C + by a closed contour C 0 containing all finitely many real poles of the integrand. Similarly, we can replace C − by a closed contour C 0 in (3.3).

An example
Before carrying out the proofs of the theorems above in full generality, we prove some very special cases, by means of the residue theorem and the q-binomial formula. For simplicity, let |x| < 1 be a complex number, and consider the empty partition λ = ∅, or equivalently the N -signature λ = 0 N = (0, 0, . . . , 0). As remarked in Section 2.2, we have P (0 N ) (q, t) = 1, and therefore the left-hand sides of identities (3.1) and (3.2) are both equal to 1, when λ = (0 N ). Let us prove that the right-hand sides of (3.1) and (3.2) also equal 1, for λ = (0 N ).
Let us begin with the case θ / ∈ N, i.e., the right-hand side of (3.2). Since the contour C − encloses all real poles in the integrand in its interior, then the right-hand side of (3.2) equals From the definition of q-Gamma functions, see Appendix A, it is evident that, for any n ∈ Z ≥0 , we have The latter indeed equals 1 because of the q-binomial theorem, Theorem A.3, applied to z = xq θN = xt N , a = q 1−θN , and the equality q θN n (q 1−θN ; q) n = (−1) n q ( n+1 2 ) (q θN −n ; q) n ∀ n ≥ 0. Second, let us consider the case θ ∈ N, i.e., the right-hand side of (3.1). Observe that for λ = (0 N ), the integrand in (3.1) can be rewritten as (1 − q z−i ) −1 , whose set of poles enclosed in the interior of C 0 is {0, 1, 2, . . . , θN − 1}. Since Res z=n (1 − q z−n ) −1 = −(ln q) −1 = (ln (1/q)) −1 , similar considerations as above lead us to conclude that the right side of (3.1) is equal to the finite sum The latter equals 1, because of Corollary A.4 of the q-binomial formula, applied to z = xq −1 and M = θN − 1.
A simple argument involving the index stability for Macdonald polynomials, see (2.1), shows that if Theorems 3.1 and 3.2 hold for λ ∈ GT N , then they hold for (λ 1 + n ≥ λ 2 + n ≥ · · · ≥ λ N + n) ∈ GT N , and any n ∈ Z, cf.
Step 5 in Section 3.3 below. Thus the present example shows how to prove Theorems 3.1 and 3.2 for signatures of the form (n, n, . . . , n) ∈ GT N , only by use of the classical q-binomial theorem.
3.3 Integral formula when t = q θ , θ ∈ N: Proof of Theorem 3.1 Assume θ ∈ N and t = q θ . The proof of Theorem 3.1 is broken down into several steps. In the first four steps, we prove the statement for positive signatures λ ∈ GT + N (when all coordinates are nonnegative: λ 1 ≥ · · · ≥ λ N ≥ 0), and in step 5 we extend it for all signatures λ ∈ GT N (when some coordinates of λ could be negative).
Step 1. We derive a contour integral formula for the ratio P λ (q r t N −1 ,t N −2 ,...,t,1;q,t) P λ (t N −1 ,...,t,1;q,t) of Macdonald polynomials, and any r ∈ N. The index-argument symmetry, Theorem 2.2, applied to λ = (λ 1 , . . . , λ N ) and µ = (r) = (r, 0 N −1 ), gives The denominator P (r) (t N −1 , . . . , t, 1; q, t) has a simple expression due to the evaluation identity, Theorem 2.3; it is particularly simple for the row partition (r): Since we also have P (r) (q, t) = (q;q)r (t;q)r g r (q, t), see (2.4), identity (3.6) becomes The symmetric polynomials g r (q, t), in addition to being essentially one-row Macdonald polynomials, can be defined in terms of their generating function as follows, see [32,Chapter VI ]. If we fix nonzero values x 1 , . . . , x N ∈ C \ {0}, the identity above is an equality of real analytic functions in the domain {y ∈ C : max 1≤i≤N |x i y| < 1}. Thus we have the following contour integral representation where C is any circle around the origin and radius smaller than . . , N in the integral representation of g r (q, t), and replace it into the right-hand side of (3.7); then where C can be taken to be any circle around the origin of radius smaller than 1 (we need here . (3.10) Step 2. We obtain a new contour integral representation by modifying (3.10). The resulting contour integral representation involves an open contour C + , that looks like that of Fig. 2 (but has a slight difference from that in Theorem 3.2).
Observe that the absolute value of the integrand in (3.10) is of order o(R −r−1 ) = o(R −2 ), if |y| = R is large. An application of Cauchy's theorem yields that the value of the integral is unchanged if the closed contour C is deformed into the "keyhole" contour C shown in Fig. 4. Let us describe the contour C in words: it is a positively oriented contour, formed by two lines away from the origin, of arguments ±3π/4, and the portion of a semicircle of some radius 0 < δ < 1. Evidently, the straight lines are part of the level lines (ln(y)) = ±3π/4, while the portion of the semicircle is part of the level line (ln(y)) = δ (where ln is defined in C \ (−∞, 0]).
Next, we make the change of variables y = q −z , or z = − ln y ln q , where ln is defined on its principal branch. Based on the previous observations about the contour C being composed by level lines, it is clear that the resulting contour for z is a negatively oriented contour formed by one segment and two straight lines, but we can easily reverse the orientation of the contour at the cost of switching signs. Let us call the positively oriented contour C + , see Fig. 2; the integral formula becomes Note that points in the horizontal lines of contour C + in (3.11) have imaginary parts ± 3π 4 ln q , while points in the vertical segment of C + have real part − ln δ/ ln q < 0. Thus contour C + encloses exactly all the real poles of the integrand in (3.11) and no other poles (it also encloses the origin, though this not important).
Step 3. We make some final modifications to formula (3.11); the resulting contour integral representation will include a closed contour C 0 as in the statement of the theorem.
Observe that the integrand in (3.11) has finitely many real poles and they are all enclosed by contour C + ; also the integrand is exponentially small as |z| → ∞ along the contour C + . Therefore, as an application of Cauchy's theorem, we can replace C + by a closed contour C 0 that encloses all finitely many real poles of the integrand. Also note that for r > θN , we can write Thus for r > θN , equation (3.11) can be rewritten as (3.12) We let x = q r t N = q r+θN and replace all instances of q r in (3.12) above by x/t N ; then multiply both sides of the identity by t |λ| . We claim that the resulting equation is exactly equality (3.1).
In fact, the left-hand side of our equation is by homogeneity of the Macdonald polynomials. On the other hand, the right-hand side of our equation is which can be shown to be equal to the right-hand side of (3.1), by simple algebraic manipulations.
The conclusion is that we have proved identity (3.1) for all x = q m with m ∈ N large enough (to be precise, m is of the form r + θN and r > θN , so the statement was proved for all integers m > 2θN ).
Step 4. Still assuming λ ∈ GT + N , we prove Theorem 3.1 for all Observe that x = 0 is imposed to make sense of the term x z = exp(z ln x) and x / ∈ {q, q 2 , . . . , q θN −1 } is necessary so that the denominator of the right-hand side of (3.1) is nonzero.
We claim that both sides of (3.1) are rational functions of x. This would prove the desired result, given that we have shown in step 3 above that (3.1) holds for infinitely many points q m , where m is large enough. The left-hand side of (3.1) is obviously a polynomial on x. In the right-hand side of (3.1), we have the product We only need to check that the contour integral is also a rational function on x. In fact, this follows from the residue theorem and the fact that there are finitely many poles in the interior of C 0 , all of these being simple and integral. The fact that the poles considered above are simple and integral can be easily checked and is equivalent to fact that all the values Step 5. We extend equality (3.1) to all signatures λ ∈ GT N . Let λ ∈ GT N be arbitrary and we aim to prove (3.1). If λ ∈ GT + N , the result is already proved in the first four steps above. Otherwise, choose m ∈ N such that λ N + m ≥ 0, and so λ If we multiply the right-hand side of equality (3.1) by x m , and also make the change of variables z → z − m in the contour integral, we obtain dz.
Since we have proved equality (3.1) for the positive signature λ already, then (3.1) also holds for λ, since we have multiplied both sides by x m and obtained equal expressions.
3.4 Integral formula for general q, t I: Outline of proof of Theorems 3.2 and 3.3 Theorems 3.2 and 3.3 are analytic continuations of Theorem 3.1, with respect to the variable θ. We shall need the weak version of Carlson's lemma below, which is proved in [2, Theorem 2.8.1] (in this reference, the statement is given for functions that are analytic and bounded on {z ∈ C : z ≥ 0}, but a simple change of variables z → z + M , for some large positive integer M ∈ N leads us to the version below).
We outline the steps of the proof of Theorem 3.2 below, and we shall carry out the detailed proof of each step in the next section. Theorem 3.3 can be proved analogously and we leave the details to the reader.

•
Step 0. Prove that the right-hand side of (3.2) is well-defined, i.e., the contour integral is absolutely convergent. We prove the absolute convergence for all |x| ≤ 1, x = 0, and θ > 0.
• Step 1. Reduce the general statement to the case |x| < 1, x = 0, and λ ∈ GT + N . Assume the latter conditions are in place for the remaining steps in the proof.
• Step 2. Prove that the equation (3.2) of Theorem 3.2 holds when t = q θ and θ ∈ N.

Integral formula for general q, t II: Proof of Theorem 3.2
In this section, we prove Theorem 3.2; as we already mentioned, Theorem 3.3 can be proved in an analogous way, and we leave the details to the reader. From the last paragraph of the previous section, Theorem 3.2 will be proved if we verify steps 0-4 stated there.
From the definition of the q-Gamma function, F q (z; θ) is a holomorphic function in a neighborhood of the contour C + , thus C + F q (z; θ)dz is a well-posed integral. To prove its absolute convergence, it suffices to show that |F q (z; θ)| ≤ c 1 · q c 2 z , for some c 1 , c 2 > 0 and all z ∈ C + with z large enough.
Because |x| ≤ 1 and all z in the contour C + have bounded imaginary parts, we have |x z | = exp( z ln |x| − z arg(x)), z ∈ C + , is upper-bounded by a constant. Thus it will suffice to show for some c 1 , c 2 > 0 and all z ∈ C + with z large enough. Since each λ i + θ(N − i) is real and | z| is constant for z ∈ C + , when z is large enough, the statement reduces to showing for some c 1 , c 2 > 0 and all z ∈ C + with z large enough. Let π/2 > d > 0 be the value such that | z|, for any point z ∈ C + with z large enough, equals −d/ ln q (such d exists by our assumptions on the contour C + ). There exists a real number a > 0 large enough so that the following inequality holds q a 1 + q θ ≤ 2 cos d. (3.14) Now we consider only points z ∈ C + with z large enough so that | z| = −d/ ln q and z > θ + a From the definition of a in (3.14), the restriction on the values of z, and q ∈ (0, 1), one can easily obtain As a consequence of inequality (3.15) and the definition of the q-Gamma function, we have Since M = z − θ − a , the previous inequality almost shows (3.13). We are left to show that the absolute value of (q θ−z+M +1 ; q) ∞ /(q −z+M +1 ; q) ∞ is upper bounded by a constant independent of z ∈ C + as long as z is large enough. In fact, we have in particular continuous, the absolute value |(x; q) ∞ | attains a maximum c max ≥ 0 and a minimum value c min ≥ 0 on the compact subset {x ∈ C : |x| ≤ q −θ−a , | x| ≥ q 1−a sin d}, and moreover c min > 0 because all the roots of (x; q) ∞ = 0 are real. We have thus proved Proof of Step 1. Assume we proved identity (3.2) when x ∈ C \ {0}, |x| < 1. Then an easy application of the dominated convergence theorem (we know the integral converges absolutely when x = 1 because of step 1) shows the equation would also hold for all x ∈ C \ {0}, |x| ≤ 1.
Also, observe that step 5 of the proof of Theorem 3.1 can be repeated almost word-by-word to extend the theorem to all λ ∈ GT N , assuming that it was proved for all λ ∈ GT + N . Therefore we will assume for convenience in the next steps that |x| < 1, x = 0, λ ∈ GT + N , and prove the theorem only in that case, without loss of generality.
Proof of Step 2. In this step, we consider the case θ ∈ N. Since we are also assuming |x| < 1, x = 0, then clearly xt N / ∈ 0, q, q 2 , . . . , q θN −1 and therefore equality (3.1) holds if x was replaced with xt N . After multiplying both sides of the resulting equation by t −|λ| , we claim that we arrive at the desired (3.2) with C 0 in place of C + . In fact, the left-hand side of our equation is thanks to the homogeneity and symmetry of Macdonald polynomials. On the other hand, the right-hand side of our equation is which can be shown equal to the right-hand side of (3.2) (with C + replaced by C 0 ) after simple algebraic manipulations. Finally observe that contour C 0 can be replaced by C + by an application of Cauchy's theorem. Proof of Step 3. Let us begin by proving holomorphicity of the left-hand side of (3.2) with respect to the variable θ; observe that θ only appears inside the variable t = q θ . The Macdonald polynomials P λ (x 1 , . . . , x N ; q, t) are holomorphic functions of θ on {θ ∈ C : θ > 0} because all the branching coefficients ψ µ/ν (q, t) are holomorphic on this domain. Then P λ xt N −1 , t N −2 , . . . , t, 1; q, t and P λ t N −1 , t N −2 , . . . , t, 1; q, t are also holomorphic. It follows that the ratio of these two quantities is holomorphic if we proved that the denominator P λ t N −1 , t N −2 , . . . , t, 1; q, t never vanishes for θ > 0, or equivalently for |t| < 1; this is evident from the evaluation identity for Macdonald polynomials, Theorem 2.3.
Next we prove holomorphicity of the right-hand side of (3.2) as a function of θ in the domain {θ ∈ C : θ > 0}. Clearly xq θN ; q ∞ Γ q (θN ) is holomorphic in the given domain of θ, but it is less clear that the integral C + F q (z, θ)dz is holomorphic in the right half-plane, where .
First we claim that F q (z; θ) is holomorphic on U × {θ ∈ C : θ > 0}, for some neighborhood U of C + . Indeed, the factor x z is clearly entire on z, and does not depend on θ. We can write the product of ratios of q-Gamma functions in the definition of F q (z; θ), as we did in Remark 3.7, see (3.4).
which is clearly holomorphic on (z, θ) ∈ C 2 . Finally the remaining factor can be written as And also there is a neighborhood U of C + on which the function q λ N −z ; q −1 ∞ of z is holomorphic on U . Secondly, we claim that C + F q (z; θ)dz is absolutely convergent and moreover C + |F q (z; θ)|dz is uniformly bounded on compact subsets of {θ ∈ C : θ > 0}; this will be a consequence of the stronger statement Claim 3.9. Consider any compact subset K ⊂ {θ ∈ C : θ > 0}. There exists a constant M 1 > 0, depending on K, such that Let us first conclude the proof of step 3 from the claim above.
Proof of Claim 3.9. Since we have shown before that F q (z; θ) is holomorphic on U × {θ ∈ C : θ > 0}, then it suffices to prove inequality (3.16) for all z ∈ C + , z > M 2 and all θ ∈ K, where M 2 is an arbitrarily large positive constant. We can express the product of q-Gamma function ratios in the definition of F q (z; θ) as we did in (3.4). The last ratio is because we are assuming λ ∈ GT + N (and so λ 1 ≥ λ N ≥ 0). Plugging the equality above into (3.4), we obtain Thus we are left to deal with (3.18), which by definition of the q-Gamma function equals (1 − q) θN (q θN −z ;q)∞ (q −z ;q)∞ . For any θ with θ > 0, we have |(1 − q) θ | < 1. Thus we only need to prove the existence of M 1 , M 2 > 0 such that z ∈ C + , z > M 2 , implies There exists a > 0 large enough such that Note that such a exists because K ⊂ {θ ∈ C : θ > 0} is compact and so inf θ∈K θ > 0. Now consider only z ∈ C + with z > a + 1, and let M = M (z) it follows that Thus we only need to show that the right-hand side of (3.21) is bounded by a constant, for all z ∈ C + with z large enough.
and since | z| is a constant between 0 and − π 2 ln q for z ∈ C + with z large enough, then m 1 (z) = m 1 > 0 is a strictly positive constant independent of z ∈ C + as long as z is large enough. Since the function ( . Thus the right-hand side of (3.21) is upper bounded by the constant c max /c min < ∞.
Proof of Step 4. We prove a stronger statement than step 4. Let M > 0 be any positive number. We show that both sides of (3.2) are uniformly bounded on {θ ∈ C : θ ≥ M }. Let us begin with the left-hand side of (3.2). Observe that θ appears in the left side only within the variable t = q θ and |t| = q θ ≤ q M . Name = q M ∈ (0, 1); we have to prove that there exists a constant C > 0 such that Thanks to the branching rule for Macdonald polynomials, Theorem 2.5, and the assumptions |x| ≤ 1, Given λ ∈ GT + N , there are finitely many µ ∈ GT + N −1 with µ ≺ λ. Thus it suffices to prove that there exist constants C 1 , C 2 > 0 such that where C 1 , C 2 do not depend on t, though they may depend on µ.
The branching coefficient |ψ λ/µ (q, t)|, due to the expression in Theorem 2.5, is a finite product for all |t| ≤ and 2(1 − q c d ) −1 does not depend on t, so the boundedness of |ψ λ/µ (q, t)| follows. Due to the evaluation identity for Macdonald polynomials, Theorem 2.3, we have and similarly for n(µ). The last two terms above are products of a finite number of , and as we saw above it is implied that the absolute value of the last two terms above are upper bounded by a constant independent of t (as long as |t| ≤ ). Thus our only goal is to show there is an upper bound for t (N −1)(|λ|−|µ|)+n(µ)−n(λ) ; this fact follows if the exponent is nonnegative. In fact, we have Let us proceed to prove uniform boundedness of the right-hand side of (3.2) on {θ ∈ C : θ ≥ M }. First of all, the triangular inequality gives has an upper-bounded absolute value. We are left to deal with and prove its absolute value is uniformly bounded on {θ ∈ C : θ ≥ M }.
For any M 2 > 0, the contribution of the portion C + ∩ {z ∈ C : z ≤ M 2 } of the contour is bounded by a constant. In fact, θ ≥ M implies |t| = |q θ | = q θ ≤ q M and θ appears in the integrand of (3.2) only as part of the exponent of some q, thus the integrand can be written as a function of z and t (with q ∈ (0, 1) fixed). Thus for (z, t) in the compact subset and the contribution of the integral in the portion C + ∩ {z ∈ C : z ≤ M 2 } of the contour is upper bounded by L times the length of that finite portion.
Since |x| < 1, the term x z decreases exponentially as |z| → ∞, z ∈ C + . Thus to deal with the infinite portion of the integral uniformly over all z ∈ C + , z > M 2 , and θ ∈ C, θ ≥ M . We can bound the absolute value of (3.23) by If z is large enough, the product . Since λ 1 is real and z is constant for z ∈ C + , z large enough, it suffices to prove the following statement: there exist constants M 1 , M 2 > 0 such that z ∈ C + , z > M 2 , and θ ≥ M imply This statement was proved above in step 3, see (3.19). In that case, θ varied over a compact subset K ⊂ {θ ∈ C : θ > 0}, but in this case θ varies over a closed infinite domain of the form {θ ∈ C : θ ≥ M }. However, the expression (q θN −z ;q)∞ (q −z ;q)∞ depends on θ only by means of q θ , so the proof of (3.19) above can be repeated word-by-word, since we only used |t| ≤ q inf θ∈K θ < 1 in that proof.

Multiplicative formulas for Macdonald characters
We now come to the multiplicative formulas. All of our results require parameter θ to be a positive integer. In this section, q is typically a variable (but of course, we can specialize q to a complex number later).

Statement of the multiplicative theorem and some consequences
We need some non-standard terminology on q-difference operators. The q-shift operators {T q,x i : i = 1, . . . , m} are linear operators on C(q)[x 1 , . . . , x m ] that act as The q-difference operators that appear in the multiplicative formulas for Macdonald polynomials are finite sums of terms τ j,i ) be the sum of the entries of τ to the right of (i, i) (resp. sum of entries of τ above (i, i)), cf. (2.6). The main theorem of this section is the following.

2)
where for any τ ∈ M (m) θ , we denoted The proof of Theorem 4.1 is given in the next subsection. We derive here some conclusions, namely two special cases when the operator D (m) q,θ has a simple form. The first simple case is m = 2.
Corollary 4.2. In the same setting as Theorem 4.1 (for m = 2), we have Proof . For m = 2, we have M We let The statement of the theorem can be easily reduced to prove that the coefficient of T n q,x 1 T θ−n q,x 2 in the product n , i.e., we prove a (θ) n for all θ, n ∈ N, 0 ≤ n ≤ θ, and we do it by induction on θ. The case θ = 1 can be easily dealt with, using Lemma B.2. Now assume a (θ−1) n = b (θ−1) n for all 0 ≤ n ≤ θ − 1, and some θ ≥ 2. We prove a (θ) .
It is not difficult to conclude from these relations, and the inductive hypothesis a n for all 0 ≤ n ≤ θ, as desired.
When θ = 1 (equivalently t = q), the result has a compact form as well. Let us recall that when t = q, the Macdonald polynomials become the well known Schur polynomials s λ (x 1 , . . . , x N ) = P λ (x 1 , . . . , x N ; q, q). The Schur (Laurent) polynomials s λ (x 1 , . . . , x N ), λ ∈ GT N , can also be defined by the simple determinantal formula For any m ∈ N with 1 ≤ m ≤ N , we consider and call it a q-Schur character of rank N , number of variables m and parametrized by λ; it was defined before in [21]. We recover the following theorem.
Proof . Letting θ = 1 in Theorem 4.1, we see that the equation above holds if D  is the product of (−1) k T k q,x i T 1−k q,x j , where k ∈ {0, 1} ranges over the elements of τ strictly above the main diagonal. Thus the term corresponding to τ is We are left to show Both sides of (4.5) are of the form T p 1 q,x 1 · · · T pm q,xm , for some p 1 , . . . , p m ∈ Z ≥0 , so we simply need to check the equality between exponents p k of T q,x k in both sides, for an arbitrary 1 ≤ k ≤ m. In the left side, there are k − 1 factors of the form T τ k,i = τ + k to the exponent of T q,x k . Therefore the power of T q,x k in the left-hand side of (4.5) is T Evidently, the power of T q,x k in the right-hand side of (4.5) if also T , which finishes the proof.
Example 4.4. We discuss the first nontrivial example of the multiplicative formula for Macdonald polynomials (an example that is not dealt with in the Corollaries above): θ = 2, m = 3. The formula in this case is We can also write the q-difference operator in terms of the q-degree operators {D q,x i : i = 1, 2, 3} by using 1 q−1 T q,x i = 1 q−1 + D q,x i . Then we can replace the operator D q,2 with the sum of operators D and f i 1 ,i 2 ,i 3 (x 1 , x 2 , x 3 ; q, 2) are certain rational functions. With the help of Sage, we found There are nontrivial rational functions f i 1 ,i 2 ,i 3 (x 1 , x 2 , x 3 ; q, 2) as well, for some i 1 + i 2 + i 3 ≥ 3, e.g., . q,θ is, in general, homogeneous of degree θ m 2 as a functions of the operators {D q,x i : i = 1, 2, 3}. However, this example disproves it. Second, the terms f i 1 ,i 2 ,i 3 (x 1 , x 2 , x 3 ; q, 2) above make us suspect that We have checked this fact in the computer. In fact, we believe that the analogous statement for general m, θ ∈ N holds true, but the author could not prove it.

Proof of Theorem 4.1
Fix a positive signature λ ∈ GT + N and let us prove equation (4.1); we extend the result for all signatures λ ∈ GT N at the end.
Let us consider m positive integers n 1 > n 2 > · · · > n m > θ(N + m). By the index-argument symmetry, Theorem 2.2, applied to λ ∈ GT + N and µ = (n 1 ≥ · · · ≥ n m ≥ 0 ≥ · · · ≥ 0) ∈ GT + N , as well as the definition of the dual Macdonald polynomials Q µ (·; q, t), we obtain Apply the Jacobi-Trudi formula for Macdonald polynomials, Theorem 2.7, to the numerator of (4.8), then multiply and divide the term parametrized by τ by the product (In equation (4.9), we are setting g n (q, t) = 0 if n is a nonpositive integer.) Recall that t = q θ , θ ∈ N. In view of Lemma B.1, the only terms in the sum (4.9) with nonzero contributions are those parametrized by m × m matrices whose entries belong to the set {0, 1, . . . , θ}; define M (m) θ to be the set of such matrices. Notice that θ , because of our initial assumption on the values of n 1 , . . . , n m . By another application of the index-argument symmetry (and of the identity (2.4) above), we have Plugging (4.2) into (4.9), we obtain and rewrite some terms from (4.11) in these new variables. Clearly the left-hand side of equality (4.11) is the Macdonald character P λ (z 1 , . . . , z m ; N, q, q θ ). It is also evident that the variable u i in the term C (q,q θ ) τ 1,s+1 ,...,τ s,s+1 can be rewritten as for any p ∈ N, p > θN . By similar, but more complicated, computations we find , (4.14) for any partition (p 1 > p 2 > · · · > p m > 0) with p i > θ(N − i + 1) for all 1 ≤ i ≤ m. From (4.13) and (4.14), we obtain Observe that we used our assumption n 1 > n 2 > · · · > n m > θ(N + m) to guarantee that (4.13) and (4.14) are applicable. We have to rewrite both (4.16) and (4.17) in terms of z s . Let us begin with (4.16), which is a product of m terms; for 1 ≤ r ≤ m, the r th term is Under the change of indexing j → θ(N − r + 1) − j, the product in the denominator of (4.18) becomes whereas the numerator of (4.18) can be expressed as Define also which is the power of q coming from the terms (4.18), for 1 ≤ r ≤ m. Thus (4.16) equals On the other hand, (4.17) can be expressed in terms of the variables z i as follows: The denominator of (4.21) is 1≤i<j≤m 0≤k<θ (z i − q k z j ), and the numerator is . This is an easy exercise.
We have just proved that the statement of Theorem 4.1 holds for all x s = q ns+θ(N −s) , for all n 1 > n 2 > · · · > n m > θ(N + m). Since both sides of the equality (4.1) are evidently rational functions in x 1 , . . . , x m , an easy algebro-geometric argument shows the equality holds for all x 1 , . . . , x m ∈ C, as desired.
We still have to extend the theorem to all signatures. Let us prove (4.1) for an arbitrary λ ∈ GT N . If λ ∈ GT + N , then we are done. Otherwise, choose any p ∈ N large enough so that λ def = (λ 1 + p, λ 2 + p, . . . , λ N + p) ∈ GT + N . By homogeneity of Macdonald polynomials, we have We know that (4.1) holds for λ; from the expressions above, it holds also for λ provided that (the powers of q match) The latter equation is easy to check, and the proof of Theorem 4.1 is therefore finished.

Asymptotics of Macdonald characters
In the remaining of the paper, starting here, we denote the Macdonald polynomial P λ (x 1 , . . . , x N ; q, t) simply by P λ (x 1 , . . . , x N ). In this section, assume that θ ∈ N and let t = q θ . We study the asymptotics of (certain normalization of) Macdonald characters of a fixed number m of variables, as the rank N tends to infinity and the signatures λ(N ) stabilize in certain way that we define next.

Asymptotics of Macdonald characters of one variable
Theorem 5.2. Let t = q θ , θ ∈ N, and {λ(N )} N ≥1 , λ(N ) ∈ GT N , be a sequence of signatures that stabilizes to ν ∈ N . Then If we use instead Theorems 3.2 and 3.3, we could possibly extend the theorem for general q, t ∈ (0, 1). Since our main application does require θ ∈ N and t = q θ , we do not bother to pursue the more general case q, t ∈ (0, 1).
Proof . Let us first prove that the limit (5.1) holds uniformly for x in compact subsets of U. We use the integral representation for Macdonald characters of one variable, Theorem 3.1, for the signature λ(N ) ∈ GT N , and with x replaced by xt N , x ∈ U. Since P λ(N ) is a homogeneous polynomial of degree |λ(N )|, then the left-hand side is After simple algebraic manipulations, the right-hand side becomes Therefore we conclude (make change of variables i → N − i + 1 in the inner product) where C 0 is a finite contour encloses all real poles of the integrand and no other poles. We have the limit uniformly for x belonging to compact subsets of U.
Next modify the contour C 0 into an infinite contour C + as described in the statement of the theorem. This is possible to do because all the poles of the integrands of (5.3) belong to the interior of C + , as N grows. Moreover the resulting integral is well-posed for large enough N since the integrand is of order |x|q θN z and for any compact set K ⊂ C, there exists N 0 ∈ N such that sup We now look at the asymptotics of the integral in (5.3), with C 0 replaced by C + . The denominator in the integrand has the following limit The limit above can be justified properly by using the dominated convergence theorem and the estimate To prove the limit in the statement of the theorem, we are left to show We already proved the pointwise convergence; to make use of the dominated convergence theorem, we simply need estimates on the contribution of the tails of C + that are uniform in N and uniform for x belonging to compact subsets of C \ {0}. Parametrize the tails of C + as z = r + π √ −1 ln q or z = r − π √ −1 ln q ; for r ranging from some large R > 1 to +∞, we want to show that the contribution of each of these lines is small. We have We have proved so far that the limit in the theorem holds uniformly for x in compact subsets of U. Since the set {q −1 , q −2 , . . . } is discrete and has no accumulation points, Cauchy's integral formula allows us to deduce the uniform convergence in compact subsets of C \ {0} as soon as we show that Φ ν (x; q, t) admits an analytic continuation to C \ {0}.
By virtue of Riemann's theorem of removable singularities, it will suffice to show that Φ ν (x; q, t) is uniformly bounded in an open neighborhood of each pole q −k . Let R > 0, R / ∈ {q n : n ∈ Z}, be arbitrary. From what we have shown so far, it is clear that Thanks to the branching rule for Macdonald polynomials, Theorem 2.5, the fact that all the branching coefficients ψ µ/ν (q, t) are nonnegative when q, t ∈ (0, 1), and the fact that each From the pointwise limits we deduce that the sequences P λ(N ) R ±1 , t −1 , . . . , t 1−N /P λ(N ) 1, t −1 , . . . , t 1−N N ≥1 are uniformly bounded. As a result of the estimates above, Thus Φ ν (x; q, t) admits an analytic continuation to all the poles in z ∈ C : 1 R ≤ |z| ≤ R ∩ q −1 , q −2 , . . . . Since R > 0 was arbitrary (outside of a lattice), we conclude that Φ ν (x; q, t) admits an analytic continuation to C \ {0}.
If ν 1 ≥ 0, then lim N →∞ λ N (N ) = ν 1 shows that λ N (N ) ≥ 0 for all N > N 0 and N 0 ∈ N large enough. For all N > N 0 , the functions P λ(N ) x, t −1 , . . . , t 1−N /P λ(N ) 1, t −1 , . . . , t 1−N are polynomials in x and therefore holomorphic on C. Similar considerations as above allow us to analytically continue Φ ν (x; q, t) to C in this case, and also show that the limit (5.1) holds uniformly for x belonging to compact subsets of C.
We end this subsection with a Lemma that will be used in Section 7.
Lemma 5.4. If ν, ν ∈ N are such that Proof . In the integral representation of Φ ν (x; q, t), the set of poles of the integrand is {ν r + θ(r − 1) + s} and they are all enclosed by contour C + . We can therefore expand at least formally, as the sum of residues (5.5) The series above converges absolutely for all x ∈ C \ {0}, and uniformly for x in compact subsets of C \ {0}; in fact, we can argue as in the special case θ = 1 of [24, Section 6.2]. We have the bounds for any 1 ≤ m < r, Choose an arbitrary m ∈ N and fix it. For r > m, the general term in brackets at (5.5) has modulus upper bounded by where c(m, θ; q) > 0 is a constant depending on m, θ, as well as ν 1 , . . . , ν m , but not on r. It follows that the sum (5.5) converges absolutely for any x ∈ C \ {0} with |x| < q −mθ . Since m ∈ N was arbitrary, it follows that (5.5) converges absolutely for any x ∈ C \ {0}. The uniform convergence also follows from the bound above. In particular, (5.5) is the Fourier expansion of (xq;q)∞ (q;q)∞ Φ ν (x; q, t). A similar Fourier expansion can be given for (xq;q)∞ (q;q)∞ Φ ν (x; q, t). After multiplying equality (5.4) by (xq; q) ∞ /(q; q) ∞ , we have We can expand both sides of (5.6) as above, to get an equality of the form More precisely, (5.5) gives that the set {k ∈ Z : c k (ν) = 0} of indices which appear in the expansion of the left-hand side of (5.6) is The equality of these sets, and the inequalities ν 1 ≤ ν 2 ≤ · · · , ν 1 ≤ ν 2 ≤ · · · , imply that ν r = ν r for all r ≥ 1, i.e., ν = ν.

Asymptotics of Macdonald characters of a f ixed number m of variables
The following theorem is the main result for asymptotics of Macdonald characters of any given rank m ∈ N as N tends to infinity.
Theorem 5.5. Let θ ∈ N, t = q θ , and {λ(N )} N ≥1 , λ(N ) ∈ GT N , be a sequence of signatures that stabilizes to ν ∈ N . Also let m ∈ N be arbitrary. Then and admits an analytic continuation to the domain The convergence (5.7) is uniform on compact subsets of (C \ {0}) m and if ν 1 ≥ 0, then it is uniform on compact subsets of C m .
Remark 5.6. For θ = 1, our theorem has a different form than [24,Theorem 6.5]. It is not immediately clear that the two answers are the same.
Proof . This result is a consequence of Theorem 5.2 and the multiplicative formula for Macdonald polynomials, Theorem 4.1. Let us give more details. As before we prove first the uniform limit on compact subsets of U m . Begin by applying Theorem 4.1 for the signature λ(N ) ∈ GT N and t N −1 x i instead of x i , for i = 1, . . . , m. Since P λ(N ) is a homogeneous Laurent polynomial of degree |λ(N )|, the resulting left-hand side is As for the right side, the factor 1≤i<j≤m 0≤k<θ 1≤i<j≤m 0≤k<θ .
We can also obtain easily the polynomial equality Therefore for any k ∈ N, we have Observe also that the rational functions are invariant under the simultaneous transformations By combining (5.9), (5.10), (5.11) for k = θ(N − m + 1), θN , and (5.12), we obtain The following limits hold uniformly for (x 1 , . . . , x m ) belonging to compact subsets of U m We have moreover the following limit holds uniformly for (x 1 , . . . , x m ) belonging to compact subsets of (C \ {0}) m , because of Theorem 5.2, It is not difficult to observe that if U ⊂ C m is a domain preserved by the map of multiplication by q, and {f n } n≥1 , f are sequences of holomorphic functions on U for which lim uniformly for x belonging to compact subsets of U . As an implication, the order of the limit as N → ∞ and the q-difference operator D (m) q,θ can be interchanged. All the considerations above immediately imply the desired uniform limit for (x 1 , . . . , x m ) belonging to compact subsets of U m .
As in the proof of Theorem 5.2, the limit in the statement will hold also uniformly for compact subsets of (C \ {0}) m if we show that Φ ν (x 1 , . . . , x m ; q, t) admits an analytic continuation to this larger domain. The extension of Riemann's theorem for removable singularities for several complex variables, [45,Theorem 8], shows that Φ ν (x 1 , . . . , x m ; q, t) admits an analytic continuation to all ({z ∈ C \ {0} : |z| ≤ R}) m if we showed only that Φ ν (x 1 , . . . , x m ; q, t) is bounded on ({z ∈ C \ {0} : |z| ≤ R}) m ∩ U m . The latter can be proved by repeating the argument in the proof of Theorem 5.

Preliminaries on the (q, t)-Gelfand-Tsetlin graph
In this section, assume q, t ∈ (0, 1). We use the notation P − lim k→∞ M k = M to indicate that a sequence of probability measures {M k } k≥1 converges weakly to M .

The (q, t)-Gelfand-Tsetlin graph
The (q, t)-Gelfand-Tsetlin graph is an undirected, Z ≥0 -graded graph with countable vertices, together with a sequence of cotransition probabilities between the levels of the graph (considered as discrete spaces).
In general, the numbers Λ N +1 N (λ, µ) depend on the values q, t, but for simplicity we omit that dependence from the notation. By virtue of the evaluation identity, Theorem 2.3, and the assumption q, t ∈ (0, 1), we have or more explicitly By duality, the kernel Λ M N also determines a map M prob (GT M ) → M prob (GT N ) between the spaces of probability measures on GT M and GT N , that we denote by the same symbol Λ M N . For example, if λ ∈ GT M and δ λ is the delta mass at λ, then Λ M N δ λ is the probability measure on GT N given by , such that each M N is a probability measure on GT N , is called a (q, t)-coherent sequence if the following relations are satisfied The set of (infinite) (q, t)-coherent sequences {M N } N ≥0 is a convex set. Theorem 1.3 is, in different terms, a characterization of the extreme points of the set of (q, t)-coherent sequences.

The path-space T and (q, t)-central measures
The set of (q, t)-coherent sequences defined before is in bijection with a class of probability measures in the path-space of the GT graph T that we define next. Definition 6.3. The path-space T is the set of (infinite) paths in the GT graph that begin at ∅ ∈ GT 0 : For any finite path φ = (φ (0) ≺ φ (1) ≺ · · · ≺ φ (n) ), define the cylinder set S φ (or simply S(φ)) by The set T is equipped with the σ-algebra generated by the cylinder sets S φ , where φ varies over all finite paths in the GT graph. We always consider T as a measurable space.
An interesting class of probability measures on T consists of the ones that are coherent with the sequence of stochastic matrices Λ N +1 N N ≥0 . To clarify what such coherence is, define the natural projection maps It is a standard exercise to show that the σ-algebra of T is the smallest one for which all the maps Proj N are measurable. Consequently, for any probability measure M on T , we can associate to it the sequence of its pushforwards under the maps Proj N , namely the sequence Definition 6.4. A probability measure M on T is said to be a (q, t)-central measure if the following relations hold for all N ≥ 0, all finite paths φ = φ (0) ≺ · · · ≺ φ (N ) , and for some probability measures M N on GT N . The branching coefficients ψ µ/ν (q, t) are explicit in the statement of Theorem 2.5.
One can verify easily that, if the relations above hold, then the measure M N is the pushforward (Proj N ) * M , for all N ≥ 0. Moreover, {M N } N ≥0 is automatically a (q, t)-coherent sequence.
We denote by M prob (T ) the set of (q, t)-central (probability) measures. The set of (q, t)-central measures is a convex set. The set of extreme points of M prob (T ), equipped with its inherited topology, is called the boundary of the (q, t)-GT graph 4 and is denoted by Ω q,t .
The following proposition implies that the correspondence between the set of (q, t)-central measures and the set of (q, t)-coherent sequences is a bijection. Proof . Similar statements are known for other branching graphs, e.g., the case t = q of our proposition is given in [21,Propositions 4.4 and 4.9], and the case t = q → 1 is in [40,Proposition 10.3]. In the case t = q, this proposition is given in [21,Propositions 4.4 and 4.9]. The proof at our level of generality can be easily adapted from the proofs in [21]; details are left to the reader.

Macdonald generating functions
We introduce Macdonald generating functions, which will be very helpful in our study of (q, t)coherent sequences. Definition 6.6. Let M N be a probability measure on GT N , then its Macdonald generating function is the formal sum Note that P Mm (x 1 , . . . , x m ) depends on the values q, t, but we omit such dependence for simplicity.
The sum above is absolutely convergent on the torus (x 1 , . . . , x m ) ∈ T m . In fact, Theorem 2.5 and the fact that all the branching coefficients ψ µ/ν (q, t) are nonnegative imply P λ x 1 , x 2 t, . . . , x N t N −1 ≤ P λ |x 1 |, |x 2 t|, . . . , x N t N −1 = P λ 1, t, . . . , t N −1 . Thus not only is P M N (x 1 , . . . , x N ) well-defined as a function on T N , but also sup Therefore P M N ∈ L ∞ (T m ). If M N is supported on the set of positive signatures GT + N , then each P λ x 1 , x 2 t, . . . , x N t N −1 is a polynomial in x 1 , . . . , x N and therefore the sum defining P M N is absolutely convergent on the closed unit disk (x 1 , . . . , x m ) ∈ D m . Moreover P M N ∈ L ∞ (D m ) if M N is supported on GT + N . In general, P M N ∈ L ∞ (T m ) ⊂ L 2 (T m ). The Fourier series expansion of P M N can be obtained by using Corollary 2.6. In fact, where the interchange in the order of summation follows from the absolute convergence of all the sums involved. (For the absolute convergence, the nonnegativity of all coefficients c λ,µ is needed.) From the expansion above, we can extract the coefficient of x κ 1 1 · · · x κ N N in the Fourier series, for any κ = (κ 1 ≥ · · · ≥ κ N ) ∈ GT N . In fact, such term appears only in m κ x 1 , . . . , x N t N −1 with coefficient t n(κ) , where n(κ) = κ 2 + 2κ 3 + · · · + (N − 1)κ N . Thus the Fourier coefficient of If M N is supported on GT + N , then the sum defining f κ 1 ,...,κ N above is finite. Indeed the only signatures with a nonzero contribution are λ ∈ GT + N with |λ| = |κ|. But then |κ| = |λ| ≥ λ 1 , and there are finitely many signatures λ ∈ GT N with |κ| ≥ λ 1 ≥ · · · ≥ λ N ≥ 0. This observation will be put to use several times. Proof . Both P M N (x 1 , . . . , x N ) and P M N (x 1 , . . . , x N ) belong to L 2 T N . The equality of these functions implies that their Fourier coefficients agree. From (6.6), this means Observe that we have restricted the sum above to µ ∈ GT + N , because M N , M N are supported on positive signatures. Let n ∈ Z ≥0 be arbitrary. We show that M N (κ) = M N (κ) for all κ ∈ GT + N with |κ| = n.
Proof . Let us prove the first part. Let {M N } N ≥0 be a (q, t)-coherent sequence. By making use of the branching rule, Theorem 2.5, the fact that P µ is homogeneous of degree |µ|, and making a change in the order of summation, we obtain We can easily show that all sums above are absolutely convergent, so the change in the order of summations can be justified. Next we prove the converse statement. Assume that M N , M N +1 are probability measures on GT N , GT N +1 . Assume that they are supported on GT + N , GT + N +1 , respectively, and also that µ) is a stochastic matrix and M N +1 is a probability measure on GT N +1 , then M N is a probability measure on GT N . Moreover since M N +1 is supported on GT + N +1 , it follows that M N is supported on GT + N . In fact, if µ / ∈ GT N \ GT + N (or equivalently µ N < 0), then for any λ ∈ GT N +1 , either λ N +1 < 0 in which case M N +1 (λ) = 0, or λ N +1 ≥ 0 in which case Λ N +1  ∀ (x 1 , . . . , x N ) ∈ T N . (6.8) The convergence above is uniform on T N .
Proof . Let > 0 be a very small real number. Since M is a probability measure, there exists c > 0 large enough so that From the weak convergence P − lim is finite and has cardinality no greater than (2c + 1) N . Also, since GT We are ready to make the desired estimate. Use (6.9), (6.10), (6.11) and the triangle inequality to argue that for any m > max{N 1 , N 2 }, we have A partial converse to the previous proposition is Proposition 6.10 below. Proof . All functions {P M m } m≥1 , P M belong to L 2 (T m ). Therefore the limit lim m→∞ P M m (x 1 , . . . , x N ) = P M (x 1 , . . . , x N ) implies the convergence of the Fourier coefficients. Due to (6.6), this implies that, for any κ ∈ GT N , we have We show that lim m→∞ M m (λ) = M (λ) for any λ ∈ GT + N . In fact, let n ∈ Z ≥0 be arbitrary and let us prove lim m→∞ M m (λ) = M (λ) for any λ ∈ GT + N with |λ| = n. Consider the finite, square matrix C whose rows and columns are parametrized by λ ∈ GT + N , |λ| = n, and whose entries are C(κ, λ) = c λ,κ /P λ 1, t, . . . , t N −1 . Also let {M m } m≥0 , M be column vectors whose entries are parametrized by λ ∈ GT + N with |λ| = n, and whose entries, at λ ∈ GT + N , are {M m (λ)} m≥0 , M (λ).

Automorphisms A k
Recall the set N of nonincreasing integer sequences, given in Definition 5.1. Equip N with the topology of pointwise convergence. We denote a generic element of N by ν = (ν 1 ≤ ν 2 ≤ · · · ). For each k ∈ Z, we can define the continuous map A k : N → N by ν → A k ν = (ν 1 +k ≤ ν 2 +k ≤ · · · ). It is clear that A k has inverse A −k , so each A k is a homeomorphism.
Similar automorphisms can be constructed for GT and T . In detail, we can define the map A k : GT → GT by λ → A k λ = (λ 1 + k ≥ λ 2 + k ≥ · · · ), A k ∅ = ∅, whose inverse is A −k , and moreover it restricts to automorphisms GT N → GT N for each N ∈ Z ≥0 . It is clear that µ ≺ λ implies A k µ ≺ A k λ, so the automorphism A k of GT induces the automorphism of measurable spaces A k : The same notation A k is used to define automorphisms of the spaces N , GT and T , but there should be no confusion each time it is used in the future.
We have introduced the automorphisms A k because, in Lemma 6.12 below, we will relate the extreme central probability measures associated to ν and A k ν. The starting point is the following simple statement, which has nothing to do with probability. Lemma 6.11. Recall the functions Φ ν (x 1 , . . . , x m ; q, t), defined in the statement of Theorem 5.5. Let ν ∈ N and k ∈ Z be arbitrary. The following equality holds (6.13) Proof . As both sides of the identity (6.13) are analytic functions on (C \ {0}) m , we only need to prove the equality for (x 1 , . . . , x m ) ∈ U m , where the domain U m was defined in the statement of Theorem 5.5. We can now make use of formula (5.8) for Φ ν (x 1 , . . . , x m ; q, t). Observe that the only place where ν appears in the right-hand side is inside the univariate functions Φ ν (x i ; q, t).
The operator D (m) q,θ satisfies that for any Laurent polynomial f on variables x 1 , . . . , x m , the following identity holds We deduce that the lemma will be proved for all m ∈ N once we prove it for m = 1, that is, . The latter statement easily follows from the integral definition of Φ ν (x; q, t) in Theorem 5.2 (to be precise, our desired statement follows after a change of variables z → z + k in the integral).
Next Observe that if we let δ λ be the probability measure on GT m given by the delta mass at λ ∈ GT m , then A k δ λ = δ A k λ for any k ∈ Z.
Similarly if M is a probability measure on T , define A k M as the pushforward of M under the automorphism A k of T . This can be described concretely as follows. The automorphism A k of T induces automorphisms A k on the set of paths of length m in the GT graph, for any m ∈ N: It is therefore natural to define also an automorphism on the family of cylinder sets by The boundary of the (q, t)-Gelfand-Tsetlin graph In this section we prove Theorem 1.3, which characterizes the (minimal) boundary of the (q, t)-GT graph. Along the way, we also define and characterize the Martin boundary of the (q, t)-GT graph.
Assume throughout this section that q ∈ (0, 1), θ ∈ N and set t = q θ . Recall the notation P − lim k→∞ M k = M indicates that a sequence of probability measures {M k } k≥1 converges weakly to M .

The Martin boundary: def inition and preliminaries
For any λ ∈ GT N , let δ λ be the delta mass at λ. As remarked in Section 6.1, there exists a unique (q, t)-coherent sequence {M λ m } m=0,1,...,N such that each M λ m is a probability measure on GT m and M λ N = δ λ . Such a sequence is given explicitly by M λ N = δ λ and M λ m = Λ N m δ λ ∀ 0 ≤ m ≤ N − 1, where the probability measures Λ N m δ λ on GT m are given explicitly in (6.4). Moreover recall that for a (q, t)-central probability measure M on T , we can associate a (q, t)-coherent sequence {M m } m≥0 as in Definition 6.4. Proof . We use the Macdonald generating functions of Section 6.3 above. By Proposition 6.8, we have and then In the sum of the right-hand side above, n ranges from λ(N ) N to λ(N ) 1 because of the branching rule for Macdonald polynomials. We multiply the equality by z −λ(N ) N and then set z = 0, so in the right-hand side one clearly picks up the coefficient of z λ(N ) N , namely M λ(N ) 1 (λ(N ) N ). By the index stability of Macdonald polynomials, 2.2), the left-hand side is Thanks to Theorem 2.3, we can then obtain a lower bound for M λ(N ) 1 (λ(N ) N ) as follows: On the other hand, since (t; t) ∞ ∈ (0, 1) and P − lim We conclude that −N 1 ≤ λ(N ) N ≤ N 1 , for all N > N 2 . Therefore the sequence {λ(N ) N } N ≥1 is bounded.
One actually has the following more general statement.
Proof . The argument here is very similar to that of the previous proof for k = 1. As before, by making use of Macdonald generating functions, we can derive a more general equation than (7.1), which is In the sum of the right-hand side above, note that µ ranges over signatures in GT k such that This is a consequence of the branching rule for Macdonald polynomials. Another relevant observation is that for any µ ∈ GT k satisfying (7.3), any monomial c m 1 ,...,m k z m 1 and so on until, if m 2 = λ(N ) N −k+2 , . . . , m k ≥ λ(N ) N , then m 1 ≥ λ(N ) N −k+1 . This is a consequence of the triangularity property of the Macdonald polynomials, see Definition/Proposition 2.1.
In equation (7.2) above, multiply both sides by z −λ(N ) N k and then set z k = 0. From the branching rule for Macdonald polynomials and the fact that the branching coefficients satisfy the resulting left-hand side is (Note that the argument z k is no longer present in the numerator.) Similarly, from the property ψ µ/(µ 1 ,...,µ k−1 ) (q, t) = 1, for any µ ∈ GT k , the resulting right-hand side is After that, multiply both sides by (z k−1 t −1 ) −λ(N ) N −1 and then set z k−1 = 0; this gives Repeat the same procedure k times, until we have multiplied both sides by z 1 t 1−k −λ(N ) N −k+1 and set z 1 = 0. The end result is Therefore we conclude that −N 1 ≤ λ(N ) N −k+1 , . . . , λ(N ) N ≤ N 1 for all N > N 2 . We conclude that each sequence {λ(N ) N −i+1 } N ≥1 , i = 1, . . . , k, is uniformly bounded by a constant.
Next we show that Ω Martin q,t is in bijection with the set N . We do so by first constructing a map Ω Martin q,t → N that will later be shown to be bijective. Proposition 7.4. Let M be a (q, t)-central probability measure on T and {M m } m=0,1,2,... be its associated (q, t)-coherent system. If M belongs to Ω Martin q,t , then there exists a unique ν ∈ N such that The function Φ ν (z 1 , . . . , z m ; q, t) was defined in Theorem 5.2 for m = 1, and in Theorem 5.5 for general m.
Analogously, using that In a similar fashion, we can define subsequences N k 1 < N k 2 < · · · inductively. Now consider the subsequence N 1 = N 1 1 < N 2 = N 2 2 < N 3 = N 3 3 < · · · ⊂ N. By construction, the sequence {λ(N k )} k≥1 is such that the limits lim By combining (7.5), (7.6) and (7.7), we conclude , there exists a unique ν ∈ N such that equation (7.4) is satisfied for all m ∈ N. Thus there is a well-defined map of sets which is determined by setting ν = N(M ) be the unique element of N such that (7.4) is satisfied. We prove below that N is bijective, but first we show the convenient fact that N commutes with the automorphisms A k , k ∈ Z, of Section 6.4.
Take any m ∈ Z ≥0 , µ ∈ GT m , then where we have used Λ N m (A k ν, A k κ) = Λ N m (ν, κ), previously stated in the proof of Lemma 6.12. Let us move on to the second part of the lemma. Let N(M ) = ν; to show N(A k M ) = A k ν, we need For any m ∈ N, (x 1 , . . . , x m ) ∈ T m , we have where the last equality follows from Lemma 6.11.
Proposition 7.6. The Martin boundary Ω Martin q,t of the (q, t)-GT graph is bijective to N under the map N defined in (7.8) above.
Proof . Step 1. We prove N is surjective.
Let ν ∈ N be arbitrary; we want to show it belongs to the range of N. From Lemma 7.5, we may assume ν 1 = 0 without any loss of generality. Consider the sequence {λ(N ) = (ν N ≥ ν N −1 ≥ · · · ≥ ν 1 )} N ≥1 of signatures that stabilizes to ν. Let m ∈ N be arbitrary. The first claim is that the limit lim on T m . Correspondingly, the Fourier coefficients of the normalized Macdonald characters converge to those of Φ ν (x 1 t 1−m , . . . , x m−1 t −1 , x m ; q, t). Proposition refprop:coherentsequences and the expansion of Corollary 2.6 give Let κ ∈ GT m be arbitrary, and denote n(κ) def = κ 2 + 2κ 3 + · · · + (m − 1)κ m . Observe that x κ 1 1 · · · x κ m appears only in the monomial symmetric polynomial m κ (x 1 , x 2 t, . . . , x m t m−1 ) and the corresponding term is t n(κ) x κ 1 1 · · · x κm m , so (7.10) is essentially the Fourier expansion of the prelimit functions in (7.9). Then we have that, for any κ ∈ GT m , the following limit . . , t m−1 (7.11) exists and equals the Fourier coefficient of x κ 1 1 · · · x κm m in the function Φ ν (x 1 t m−1 , . . . , x m ; q, t). As mentioned before, M λ(N ) m (µ) = 0 unless µ m ≥ ν 1 ≥ 0. Thus we can restrict the sum in (7.11) to µ ∈ GT + m . From the same analysis as in Proposition 6.10, we can obtain that the limit lim Immediately from the definition (and Fatou's lemma) it follows that Next consider the following function . . , x m t m−1 P µ 1, t, . . . , t m−1 .
Clearly F defines a function on T m , as it is defined by an absolutely convergent series on the m-dimensional torus. Moreover its absolute value is upper bounded by 1, therefore F ∈ L ∞ (T m ) ⊂ L 2 (T m ). By the same argument preceding (6.6), one shows, for any κ ∈ GT m , that the Fourier coefficient of Observe that, if κ ∈ GT m is fixed, the sums in both (7.11) and (7.12) are finite because c µ,κ M m (µ) = 0 unless µ ≥ κ and µ ∈ GT + m . Therefore lim implies the equality between (7.11) and (7.12). In other words, the following convergence holds in the sense that all Fourier coefficients of the left side of (7.13) converge to the corresponding Fourier coefficients of F (x 1 , . . . , x m ), as N tends to infinity. But we already knew that the Fourier coefficients of the left side of (7.13) converge to the corresponding Fourier coefficients of Φ ν (x 1 t 1−m , . . . , x m ; q, t). Therefore all the Fourier coefficients of the difference F (x 1 , . . . , x m ) − Φ ν (x 1 t 1−m , . . . , x m ; q, t) of square-integrable functions on T m must be zero. It follows that F (x 1 , . . . , x m ) = Φ ν (x 1 t 1−m , . . . , x m ; q, t). In particular, the equality holds for x 1 = · · · = x m = 1, resulting in and shown that Φ ν (x 1 t 1−m , . . . , x m ; q, t) is the generating function of M m . Complete the sequence with M 0 = δ ∅ , the delta mass at ∅. We claim that {M m } m≥0 is a (q, t)-coherent sequence. In fact, since {M λ(N ) m } m=0,1,...,N is a (q, t)-coherent sequence, then for any 0 ≤ m < N , µ ∈ GT m . As N goes to infinity, the left side of (7.14) tends to M m (µ). By an argument similar to that in the proof of Proposition 6.9, one shows that the right side of (7.14) converges to µ). Indeed, the argument simply relies on the weak convergence M λ(N ) m+1 → M m+1 and the uniform (on λ) bound |Λ m+1 m (λ, µ)| = Λ m+1 m (λ, µ) ≤ 1. Therefore the limit of (7.14) as N → ∞ is for any m ∈ Z ≥0 , µ ∈ GT m . Thus {M m } m≥0 is a (q, t)-coherent sequence and has an associated probability measure M on T , as given by Proposition 6.5. By the definition of N, we conclude N(M ) = ν.
Step 2. Next we show that N is injective. Let M, M ∈ Ω Martin q,t have the same image ν under the map N. The goal is to prove M = M . From Lemma 7.5, we may assume ν 1 = 0 without any loss of generality. Furthermore, we can assume that M is the element of Ω Martin q,t such that N(M ) = ν and that was contructed in the first step.
Let {M m } m≥0 and {M m } m≥0 be the (q, t)-coherent sequences associated to M, M , then As it was mentioned in Section 6.3, Macdonald generating functions are uniformly bounded on the torus, in particular, P m ∈ L ∞ (T) ⊂ L 2 (T) for any probability measure m on GT 1 = Z. Write the first equality of (7.15) for x = e iθ as follows: Both sums above are expansions of a square integrable function on T in terms of the basis {e inθ } ⊂ L 2 (T). Thus the (Fourier) coefficients in both sums must agree, i.e., M 1 (n) = M 1 (n) ∀ n ∈ Z, and so M 1 = M 1 . We aim to apply a similar argument to show that M m = M m for any m ∈ N. We will be done once this is proved, as Proposition 6.5 would then show M = M . For general m, we make use of the fact that M ∈ Ω Martin q,t is the probability measure constructed in step 1: we use that each M m is supported on GT + m . As above, the definition of the map N implies The equality of the functions above implies the equality of corresponding Fourier coefficients. It follows that for any κ ∈ GT m we have, see (6.6), When κ / ∈ GT + m (or equivalently κ m < 0), we claim that the left side of (7.16) is zero. In fact, for any µ ∈ GT m , either c µ,κ = 0 when µ m ≥ 0 or M m (µ) = 0 when µ m < 0, because M m is supported on GT + m . Then also the right side of (7.16) is zero if κ / ∈ GT + m . By using also the properties of the coefficients c µ,κ stated in Corollary 2.

Characterization of the Martin boundary
Our next goal is to characterize the topological space Ω Martin q,t completely. Recall the map N : Ω Martin q,t → N , defined above in (7.8), by letting N(M ) = ν be the unique element of N such that Φ ν x 1 t 1−m , . . . , x m−1 t −1 , x m ; q, t = P Mm (x 1 , . . . , x m ) ∀ (x 1 , . . . , x m ) ∈ T m , ∀ m ≥ 1.
Proposition 7.6 shows that N is a bijection, so the inverse map N −1 is well-defined.
Definition 7.7. For any ν ∈ N , we denote N −1 (ν) by M ν and the corresponding (q, t)-coherent sequence by {M ν m } m≥0 . From step 1 of the proof of Proposition 7.6, and Lemma 7.5, we have that for the sequence of signatures {λ(N ) = (ν N ≥ · · · ≥ ν 1 )} N ≥1 which stabilizes to ν, the following weak convergence holds In fact, we note the same analysis as in step 1 of the proof of Proposition 7.6 shows that the weak convergence above holds for any sequence of signatures {λ(N )} N ≥1 stabilizing to ν.
We need the following lemma to prove that N is a homeomorphism.
Finally we move on to (3). To get started, we prove it for m = 1, so let ν, ν ∈ N be such that ν i ≥ ν i ∀ i ≥ 2 and ν 1 = ν 1 . Consider the following pair of sequences of signatures { λ(N ) = ( ν N ≥ · · · ≥ ν 2 ≥ ν 1 )} N ≥1 and {λ(N ) = (ν N ≥ · · · ≥ ν 2 ≥ ν 1 )} N ≥1 . Then we have the weak limits By the calculations in Lemma 7.2, we have Thus taking into account the limits (7.17) for m = 1, we deduce M ν 1 (ν 1 ) ≥ M ν 1 (ν 1 ). The proof of the third item for a general m ∈ N follows from similar calculations that are used to prove Lemma 7.3. We leave the details to the reader. Proof . The first step shows that N −1 is continuous and the second one shows that N is continuous.
Step 1. Let M ν (i) i≥1 ⊂ Ω Martin q,t and ν (i) i≥1 be the corresponding images under the map N. If lim i→∞ ν (i) = ν ∈ N pointwise, then we prove the weak limit P − lim i→∞ M ν (i) = M ν .
We are left with the task of proving (7.18). We show the uniform convergence in a neighborhood of the torus T m . But it suffices to prove the limit on compact subsets of U m . From the definition of Φ ν (x 1 , . . . , x m ; q, t) on U m , see (5.8), and observing that T m ⊂ U m , it follows that the result holds for any m ∈ N provided it holds for m = 1. We prove (7.18) for m = 1 on an open neighborhood of T.
Clearly a small enough open neighborhood of T is a subset of U. By the definition of Φ ν (x; q, t) on U, see (5.2), the desired uniform convergence will hold if we verify uniformly on compact subsets of x ∈ C \ {0}. Note that, since all ν (i) 1 i≥1 , ν 1 , are nonnegative, we can take the same contour C + for both of the integrals in (7.20).
The pointwise convergence of integrands in (7.20) is clear. We still need some uniform estimates for the contribution of the tails of the left side in (7.20). This is similar to the proof of Theorem 5.2.
Let K ⊂ C \ {0} be any compact set. Parametrize the tails of C + as z = r ± π √ −1 ln q ; for r ranging from some large R > 0 to +∞, we want to show that the contribution of each of these lines is small. We have sup x∈K x z · ∞ j=1 q −z+ν (i) j t j ; q ∞ q −z+ν (i) j t j−1 ; q ∞ ≤ const × |x| r × ∞ j=1 1 1 + q −r+ν (i) j +θ(j−1) · · · 1 + q −r+ν (i) for any z = r ± π √ −1 ln q , and any k ∈ N. The constant above depends on K but not on i. Choose k ∈ N large enough so that a def = sup x∈K |x| · q k ∈ (0, 1). Since lim q −z+ν (i) j t j ; q ∞ q −z+ν (i) j t j−1 ; q ∞ ≤ c · a r , if z = r ± π √ −1 ln q .
Since ∞ R a r dr = −e R ln a / ln a R→∞ − −−− → 0, we have just shown that the contribution of the tails of C + is uniformly small. We can then apply dominated convergence theorem to conclude (7.20), as desired.
Step 2. As in the step above, let M ν (i) i≥1 be a sequence in Ω Martin shows that any subsequence ν (ir) r≥1 ⊂ ν (i) i≥1 must have a subsubsequence ν (i r(s) ) s≥1 ⊂ ν (ir) r≥1 such that a pointwise limit lim s→∞ ν (i r(s) ) = ν exists. Then M = M ν = M ν , but since the map N is bijective, we have ν = ν . We therefore conclude that the sequence ν (i) i≥1 itself must converge to ν, finishing the proof.

Relation between the Martin boundary and the (minimal) boundary
The basic relation between the Martin and minimal boundary of the (q, t)-GT graph is the following statement, which actually holds in a much greater generality. In many examples, especially in the context of asymptotic representation theory, it is known that the Martin boundary of a branching graph is equal to its minimal boundary, e.g., [21,37]. In our case, we will also prove that this is the case by following the ideas in [21,41]. The following statement, which also holds in greater generality, will be useful. Other important identities we use in our paper are the q-binomial theorems. To state them, we need to define the q-Pochhammer symbols (x; q) n and (x; q) ∞ , for any x ∈ C and n ∈ Z ≥0 by Note that |q| < 1 implies that the product defining (x; q) ∞ is uniformly convergent for x ∈ C, and thus (x; q) ∞ is an entire function. The q-binomial formula is the following Proof . Let a = q −M in Theorem A.3, and use (q; q) n = (−1) n q ( n+1 2 ) (q −1 ; q −1 ) n . The statement is then proved for any |z| < 1; therefore it also holds for any z ∈ C because both sides are polynomials on z.
Another application of the q-binomial theorem is the following limit. Remark A.6. If b − a ∈ Z, the limit in Theorem A.5 holds uniformly on compact subsets of C \ {1}.