A $q$-Robinson-Schensted-Knuth Algorithm and a $q$-polymer

In [Matveev-Petrov 2016](arXiv:1504.00666) a $q$-deformed Robinson-Schensted-Knuth algorithm ($q$RSK) was introduced. In this article we give reformulations of this algorithm in terms of the Noumi-Yamada description, growth diagrams and local moves. We show that the algorithm is symmetric, namely the output tableaux pairs are swapped in a sense of distribution when the input matrix is transposed. We also formulate a $q$-polymer model based on the $q$RSK, prove the corresponding Burke property, which we use to show a strong law of large numbers for the partition function given stationary boundary conditions and $q$-geometric weights. We use the $q$-local moves to define a generalisation of the $q$RSK taking a Young diagram-shape of array as the input. We write down the joint distribution of partition functions in the space-like direction of the $q$-polymer in $q$-geometric environment, formulate a $q$-version of the multilayer polynuclear growth model ($q$PNG) and write down the joint distribution of the $q$-polymer partition functions at a fixed time.


Introduction and main results
The RSK algorithm was introduced in [Knu70] as a generalisation of the Robinson-Schensted (RS) algorithm introduced in [Rob38,Sch61]. It transforms a matrix to a pair of semi-standard Young tableaux. For an introduction of the RS(K) algorithms and Young tableaux see e.g. [Ful97,Sag00].
The gRSK algorithm, as a geometric lifting of the RSK algorithm, that is replacing the maxplus algebra by the usual algebra in its definition, was introduced in [Kir01].
There are several equivalent formulations of the (g)RSK algorithms. The commonest definition of the RSK algorithm is based on inserting a row of the input matrix into a semi-standard Young tableau. However for the needs of defining gRSK, the insertion was reformulated as a map transforming a tableau row and an input row into a new tableau row and the input row to insert into the next tableau row. This was introduced in [NY04], and henceforce we call it the Noumi-Yamada description. It will be the first reformulation of the algorithms in this article, from which we derive all of our main results.
The symmetry properties state that the pair of output tableaux are swapped if the input matrix is transposed. One way to prove this is to reformulate the RSK algorithms as growth diagrams. The growth diagram was developed in [Fom86,Fom95], see also exposition in [Sag00, Section 5.2]. It is a rectangular lattice graph whose vertices record the growth of the shape of the output tableaux, and can be generated recursively by the local growth rule.
Much of the attention the (g)RSK algorithms receive these days come from the relation to the directed last passage percolation (DLPP) and the directed polymer (DP). The Greene's theorem [Gre74] (see for example [Sag00] for a modern exposition) characterises the shape of the output tableaux as lengths of longest non-decreasing subsequences. As an immmediate consequence, this can be viewed as the RSK algorithm transforming a matrix to a multilayer non-intersecting generalisation of the DLPP. Specifically the first row of the output tableaux corresponds to precisely the DLPP. When randomness is introduced into the input matrix, this connection yields exact formulas for the distribution of DLPP in geometric and exponential environments [Joh00]. The geometric lifting of the DLPP is the partition function of the directed polymer (DP) where the solvable environment is that of the log-Gamma weights [Sep12]. And unsurprisingly the gRSK is related to the DP the same way as RSK is related to the DLPP. This was used in [COSZ14] to obtain exact formulas for the distribution of the partition function of DP in a log-Gamma environment.
The DLPP and DP can be defined locally using a similar growth rule to the (g)RSK. And given the solvable environment there present reversibility results of this local growth rule of the partition function called the Burke property. It is used to show the cube root variance fluctuations of the partition functions [BCS06,Sep12].
Also in these solvable models, the distribution of the shape of the tableaux are related to certain special functions. In the RSK setting it is the Schur measure [O'C03], related to the Schur functions, and in gRSK setting it is the Whittaker measure, related to the gl +1 -Whittaker functions [COSZ14]. This kind of results can often be obtained using a combination of Doob's h-transform and the Markov function theorem [RP81].
In [OSZ14] a reformulation of the gRSK called the local moves was used to give a more direct treatment than the Markov function theorem to show the connection between the gRSK and the Whittaker functions.
The local moves can be generalised to take an array of Young diagram shape. This idea can be found in the proof of the Two Polytope Theorem in [Pak01]. 1 In [NZ16] this idea was used to yield the joint laws of the partition functions of the log-Gamma polymer in the space-like direction. Specifically it was used to formulate a geometric version of the multilayer polynuclear growth model (PNG) introduced in [Joh03], from which the joint law of the polymer partition functions at a fixed time could be written down.
One direction for generalisation of the (g)RSK algorithms is to interpolate using q-deformation. The Macdonald polynomials were introduced in [Mac88]. See [Mac98] for a detailed introduction. They are symmetric polynomials of two parameters q and t. We only consider t = 0, in which case they are also the q-Whittaker functions with some prefactors, as they are eigenfunctions of the q-deformed quantum Toda Hamiltonian [GLO10]. On the one hand the q-Whittaker functions interpolate between the Schur functions (q = 0) and the Whittaker functions (q → 1 with proper scalings [GLO12]). On the other hand the simiarlity among structures of the Macdonald polynomials, Schur polynomials and the Whittaker functions makes the Macdonald processes and measures [FR02,BC14] possible. This motivates the search for q-deformed RS(K) algorithms.
The qRS algorithms were introduced in [OP13] (column insertion version) and in [BP13, Dynamics 3, h = (1, 1, . . . , 1)] (row insertion version). In [MP16] several q-deformed RSK (qRSK) algorithms were introduced. In all these q-deformations the algorithms transform inputs into random pairs of tableaux, rather than just one pair of tableaux. These q-algorithms all have the desired property of transforming the input into various q-Whittaker processes.
In this article we work on the qRSK row insertion algorithm introduced in [MP16, Section 6.1 and 6.2]. It was shown in that paper that the qRSK algorithm transforms a matrix with q-geometric weights into the q-Whittaker process, and the push-forward measure of the shape of the output tableaux is the q-Whittaker measure.
We give the Noumi-Yamada description of this algorithm, from which we obtain a branching growth diagram construction similar to that in [Pei14], and show that the algorithm is symmetric: Theorem 1. Let φA(P, Q) = P(qRSK(A) = (P, Q)) be the probability of obtaining the tableau pair (P, Q) after performing qRSK on matrix A, then where A is the transpose of A.
We also formulate a q-polymer model which corresponds to the first row of the output tableaux of the qRSK. It interpolates between the DLPP (q = 0) and the DP (q → 1 with proper scaling). The Burke's property also carries over to the q-setting naturally, with which one immediately obtains some asymptotic results for the q-polymer with stationary boundary conditions. See Section 3 for more details. Also see Section 2.2 for definition of (x; q)∞ that appears in the theorem.
Theorem 2. Let Z be the partition function of the q-polymer. With stationary boundary conditions defined in Section 3.2, Finally we formulate a q-local move that agrees with the qRSK when taking a matrix input. Like in [Hop14,NZ16], we use the q-local moves to generalise the qRSK to take arrays on a Young diagram as the input, propose the corresponding qPNG model, and write down the joint distribution of the q-polymer partition functions in the space-like direction.
Like in [OSZ14,NZ16], the basic operation of the q-local moves, called ρ n,k , works on diagonal strips (i, j) i−j=n−k of the input.
In those two papers, when the gRSK is defined as a composition of the ρ n,k , they are defined in two different ways, row-by-row or antidiagonal-by-antidiagonal.
In [Hop14], ρ n,k (or more precisely the map b n,k in [OSZ14, (3.5)], see also (22)(23)) were referred to as "toggles". It was shown there the map called RSK can be of any composition of the toggles whenever they agree with a growth sequence of the underlying Young diagram of the input array.
In this article, we generalise this to the q-setting. By identifying the input pattern as an array on a Young diagram Λ, we show that the qRSK map TΛ can be of any composition of the ρ's whenever they agree with a growth sequence of Λ. For details of definitions of ρ n,k and TΛ see Section 4.
We fit the input (wi,j) (i,j)∈Λ into an infinite array A = (wi,j) i,j≥1 ∈ N N + ×N + and define TΛ such that when acting on an infinite array like A it only alters the topleft Λ part of the array.
Given a q-geometrically distributed input array, we can write down the explicit formula of the push-forward measure of TΛ.
In this article we let (αi) and (αj) be parameters such thatαiαj ∈ (0, 1) for all i, j. Also note for integer n, denote (n)q to be the q-Pochhammer (q; q)n (see Section 2.2).
Theorem 3. Given that the input pattern (wi,j) (i,j) have independent q-geometric weights We define an outer corner of a Young diagram to be any cell without neighbours to the right of below itself. More precisely, (n, m) is an outer corner of λ if λn = m and λn+1 < m.
The rest of the article is organised as follows. In Section 2 we review some preliminaries on (g)RSK, q-deformations and the q-Whittaker functions. Then we give the Noumi-Yamada description and growth diagram formulations of the qRSK algorithm, with which we prove the symmetry property Theorem 1. In Section 3 we formulate the q-polymer, define and discuss the Burke relations, prove the q-Burke property, with which we prove Theorem 2 In Section 4 we formulate the q-local moves, show their relation to the qRSK, prove Theorem 3, propose the qPNG, and discuss a measure on the matrix and its classical limit to a similar measure in [OSZ14].

Notations
We list some notations we use in this article.
• N is the set of the nonnegative integers, and N+ the set of the positive integers.
• IA is the indicator function on the set A. • w n,1:k is (wn,1, wn,2, . . . , w n,k ) • w1:n,1:m is a matrix (wi,j)n×m • := means (re)evaluation or definition. For example u := u + a means we obtain a new u which has the value of the old u adding a.  In this section we review the basics of the theory of Young tableaux and the Noumi-Yamada description of the (g)RSK. We also review some q-deformations and related probability distributions. Then we formulate the Noumi-Yamada description and growth diagram for the qRSK, and show how to use the latter to prove the symmetry property Theorem 1.
A Gelfand-Tsetlin (GT) pattern is a triangular array (λ k j ) 1≤j≤k≤m satisfying the interlacing constraints, that is The exact constraints of the GT pattern are thus We refer to the indices of the GT pattern coordinates in the following way. Given a coordinate λ k j , we call the superscript (k here) the level, the subscript (j here) the edge. Later when we consider the time evolution of the GT patterns, we use an argument in the bracket to denote time. Therefore λ k j ( ) is the coordinate at time , kth level and jth edge. We visualise a GT pattern, for example with 5 levels as follows, where we also annotate the interlacing relations: So the levels correspond to rows and edges corresponds to edges from the right in the picture.
Throughout this article we do not take powers of λj so the superscript on λ is always an index rather than a power. The same applies to notations a j k in Noumi-Yamada description of the qRSK, as well as the t in the proof of Theorem 5.
A semi-standard Young tableau, which we simply refer to as a tableau, is an Young diagramshaped array filled with positive integers that are non-descreasing along the rows and increasing along the columns. The underlying Young diagram is called the shape of the tableau.
Clearly the shape of a tableau (λ k j ) 1≤j≤k≤m is the bottom row in the visualisation of the GT pattern λ m = (λ m 1 , λ m 2 , . . . , λ m m ). The RSK algorithm takes in a matrix A = (wij)n×m as the input and gives a pair of tableaux (P, Q) as the output. We call the output P -tableau the insertion tableau, and the Q-tableau the recording tableau. When we mention the output tableau without specifying, it is by default the P -tableau, as most of the time we focus on this tableau. We usually denote the corresponding GT pattern for the P -and Q-tableaux as respectively (λ k j ) and (µ k j ). The RSK algorithm is defined by the insertion of a row (a 1 , a 2 , . . . , a m ) ∈ N m of nonnegative integers into a tableau (λ j k ) to obtain a new tableau (λ j k ). We call this insertion operation the RSK insertion and postpone its exact definition to Definition 1.
When applying the RSK algorithm to a matrix w1:n,1:m, we start with an empty tableau λ k j (0) ≡ 0, and insert w1,1:m into (λ k j (0)) to obtain (λ k j (1)), then insert w2,1:m into (λ k j (1)) to obtain (λ k j (2)) and so on and so forth. The output P -tableau is the GT pattern at time n: λ k j = λ k j (n), and the Q-tableau is the GT pattern at level m: µ j = λ m j ( ). For a traditional definition of the RSK insertion, see e.g. [Ful97]. The definition we give here is the Noumi-Yamada description.
Definition 1 (The Noumi-Yamada description of the RSK insertion). Suppose at time − 1 we have a tableau (λ j k ) = (λ j k ( − 1)) and want to RSK-insert row w ,1:m into it to obtain a new tableau (λ j k ) = (λ j k ( )). This is achieved by initialising a 1:m 1 = w ,1:m and recursive application (first along the edges, starting from 1 and incrementing, then along the levels, starting from the edge index and incrementing) of the followingλ The Noumi-Yamada description does not rely on wij being integers and hence extends the RSK algorithm to take real inputs. Similarly one can define the Noumi-Yamada description for the gRSK algorithm, which is simply a geometric lifting of the RSK algorithm. It also takes real inputs.
Definition 2 (The Noumi-Yamada description for the gRSK algorithm). Suppose at time − 1 we have a tableau (z j k ) = (z j k ( − 1)) and want to gRSK-insert a row w ,1:m into it to obtain a new tableau (z j k ) = (z j k ( )). This is done by initialising (a i 1 )i=1:m = (e w ,i )i=1:m and the recursive application (in the same way as in the RSK insertion) of the following Before defining the qRSK algorithm, let us review some q-deformations.

q-deformations
A good reference of the q-deformations is [GR04]. In this article we assume 0 ≤ q < 1. Define the q-Pochhammers and the q-binomial coefficients as We also define three q-deformed probability distributions.

q-geometric distribution
Definition 3. Given α ∈ (0, 1), a random variable X is said to be distributed according to the q-geometric distribution qGeom(α) if it has probability mass function (pmf) The first moment of the q-geometric distribution with parameter α is where Eq(α) = (α; q) −1 ∞ is a q-deformation of the exponential function (see for example (1.3.15) of [GR04]).

q-binomial distribution
There are several q-deformations of the binomial distribution. The one that is used in [MP16] to construct the qRSK is also called q-Hahn distribution. It appeared in [Pov13]. Apart from the dependency on q, it is has three parameters (ξ, η, n). For 0 ≤ η ≤ ξ < 1, and n ∈ N ∪ {∞}, the pmf is The fact that it is a probability distribution can be found in, for example [GR04, Exercise 1.3].

q-hypergeometric distribution
The q-hypergeometric distribution we consider here is defined as follows. For m1, m2, k ∈ N with The corresponding q-Vandermonde identity As with the usual hypergeometric distribution, the support of qHyp(m1, m2, k) is When m2 = ∞, the distribution is symmetric in m1 and k: This distribution appeared in [Blo52]. When k = 0 or m1 = 0, by (4) the distribution is supported on {0}: The fact that the qHyp is a probability distribution yields the following identities, where the second follows by taking m2 = ∞: 2.2.4 From the q-binomial distribution to the q-hypergeometric distribution The q-binomial distribution is related to the q-hypergeometric distribution in the following way: Proof. By the definition of the q-hypergeometric distribution it suffices to show n k q to the left hand side to turn the q −1 -Pochhammers into q-Pochhammers. The left hand side thus becomes Furthermore using (q −n ; q) k = (n)q (n−k)q (−1) k q ( k 2 )−nk the above formula becomes the right hand side of (8). Given a tableau (λ k j ) define its type ty((λ k j )) by

The (t = 0)-Macdonald polynomials and the q-Whittaker functions
Then the (t = 0)-Macdonald polynomials of rank N −1 indexed by λ and the q-Whittaker function ψx(λ) are defined as The q-Whittaker measure discussed in this article is the one induced by the Cauchy-Littlewood identity:

Classical limits
In this section we let q = e − for small > 0 and collect some results concerning the classical limits. Let is the dilogarithm function.
Item 1 can be found, for example as a special case of Theorem 3.2 in [BW16]. Item 2 was proved as Lemma 3.1 of [GLO12]. Item 3 can be derived as follows:

The Noumi-Yamada description of the qRSK
Now we can define a Noumi-Yamada description for the qRSK. Throughout this article we adopt the convention that for any Young diagram λ, the 0th edge are ∞: λ0 = ∞.
Theorem 4. The qRSK algorithm can be reformulated as the following Noumi-Yamada description.
Suppose at time − 1 we have a tableau (λ j k ) = (λ j k ( − 1)) and want to insert row w ,1:m into it to obtain a new tableau (λ j k ) = (λ j k ( )). We initialise a 1:m 1 = w ,1:m and recursively apply the following Proof. Let us recall the algorithm as described in [MP16, Section 6.1 and 6.2]. In natural language it works as follows. Suppose we want to insert row (a1:m) into the tableau (λ j k ) 1≤j≤k≤m . The top particle λ 1 1 receives a push a1 from the input row and finishes its movement. Recursively, when all the particles at level j − 1 finishes moving, the increment of the kth particle splits into two parts l j−1 k and r j−1 k , which contribute to the increment of the k + 1th and the kth particles at level j respectively. The right increment r j−1 k is a q-binomial distributed random variable. On top of that the rightmost particle λ j 1 of the GT pattern receives a push aj from the input row.
To be more precise we present a pseudocode description.
In this article we write a j k (n) in place of a j k when the insertion is performed at time n, namely to transform (λ j k (n − 1)) into (λ j k (n)). An alternative way of writing down the Noumi-Yamada description is as follows

Properties of the qRSK
As with the usual RSK, the qRSK preserves the interlacing constraints of the GT patterns along levels and time.
Lemma 4. Starting from the empty initial condition, Proof. We show this by induction. The empty initial condition is the initial condition for the induction. Assume for any n , j , k such that 0 ≤ n < k ≤ j , n ≤ n, j ≤ j, k ≤ k, (n , j , k ) = (n, j, k) we have λ j k (n ) = 0.
If j > k > 1, then by the induction assumption and (5). The other cases (j = k and k = 1) are similar and less complicated.
The following lemma can be viewed as the boundary case "dual" to (9). This duality will become clear when defining the q-local moves. When k = n, by Theorem 4 and Lemma 4 λ k n (n) = λ k n (n − 1) + a k n (n) = a k n (n) The next lemma shows that qRSK, like the usual RSK, is weight-preserving.
Lemma 6. Given empty initial condition, let (λ k j (n)) be the output of qRSK taking a matrix (wi,j), then almost surely Proof. We first show by induction that When n = 0 the empty initial condition shows that the above formula is true for all k ≥ 1. By recursively applying (9) and noting a 1 1 (n) = wn,1, we see the above formula is true for k = 1 and all n ≥ 0.
Assuming the above formula is true for (k , n ) ∈ {(k − 1, n), (k, n − 1), (k − 1, n − 1)}, summing over j = 1 : k in (10), and by noting a k 1 (n) = w n,k one has Then using Lemma 4 on (15) we arrive at the identity in the statement of the lemma.

The growth diagrams and the symmetry property
The growth diagram was developed in [Fom86,Fom95], see also for example [Sag00, Section 5.2] for an exposition. For the RSK algorithm it is an integer lattice [n] × [m], where each vertex is labelled by a Young diagram, and each cell labelled by a number. More specifically, the ( , j)-cell is labelled by w ,j , the ( , j)-th entry of the input matrix, whereas the label of vertex ( , j) is the Young diagram λ j 1:j ( ). The local growth rule is a function FRSK(λ, µ 1 , µ 2 , x) such that for all j and . The local growth rule generates the whole diagram. To see this, one may label the boundary vertices (0, 0 : m) and (0 : n, 0) with the empty Young diagrams, and apply FRSK recursively.
By the definition of the P -and Q-tableaux, the labels of the top row and the labels of the right most column of the growth diagram characterise the P -and Q-tableaux respectively. Therefore the symmetry property of the RSK algorithm is reduced to the symmetry property of the local rule: FRSK(λ, µ 1 , µ 2 , x) = FRSK(λ, µ 2 , µ 1 , x).
To see this, note that transposing the matrix amounts to transposing the lattice with the cell labels. By the symmetry property of the local rule, it is invariant under this transposition, therefore one can transpose the vertex labels as well. As a result the P -and Q-tableaux are swapped. This argument will be made more symbolic in the proof of Theorem 1.

The symmetry property for the qRSK
In the case where the algorithm itself is randomised, or weighted, the local rule branches, and the weights can be placed on the edges.
One example of this is both the column and the row qRS defined in [OP13, BP13] whose symmetry property was proved using this branching version of the growth diagram in [Pei14].
In this section we prove the symmetry property for the qRSK in the same way.
As remarked before, we use the convention that for any Young diagram λ, λ0 = ∞. By Theorem 4 we can write where x1 = x and x k+1 ∼ qHyp(µ 1 k − λ k , λ k−1 − µ 1 k , µ 2 k − λ k ) has pmf symmetric in µ 1 and µ 2 : Note that the local rule F does not "see" either the level or the time of the insertion. Therefore the Young diagrams have to be padded with infinitely many trailing 0s. This is why we let the edge index k range from 1 to ∞ in (16). It is consistent with the Noumi-Yamada description in the boundary case j = k and the "null" case j < k. When j = k, . Similarly when j < k all terms in the right hand side of (17) are 0, so thatλ j k can stay 0. The rest follows the same argument as in the proof of [Pei14, Theorem 3]. Here we produce a less visual and more symbolic argument.

From RSK algorithms to polymers
For the RSK algorithm, due to the Greene's Theorem [Gre74] the first edge of the output tableaux are the partition function of the directed last passage percolation (DLPP) of the input matrix: Let (λ j k ( )) be the output of the RSK algorithm taking matrix (wi,j)n×m, then Z0( , j) := λ j 1 ( ) = max π:(1,1)→( ,j) (i,j)∈π wi,j, where π : (1, 1) → ( , j) indicates π is an upright path from (1, 1) to ( , j). Locally, Z0 satisfies the following recursive relation, which is what happens at the first edge in the Noumi-Yamada description: Similarly for the gRSK, the first edge corresponds to the partition functions of the directed polymer (DP) of the matrix: Because of this, we define the q-polymer by focusing on the first edge Z(n, m) := λ m 1 (n). 3 Then by the Noumi-Yamada description of the qRSK the q-polymer can be defined locally by Zq(1, 1) = w1,1, Zq(n, 1) = Zq(n − 1, 1) + wn,1, n > 1 It is not known whether a more global interpretation of Zq for 0 < q < 1 exists, like the first definitions of Z0 and Z1 involving directed paths. More generally, the full Greene's theorem interpretes the sum of the first k edges of a fixed level of the (g)RSK-output triangular patterns as similar statistics of k directed non-intersecting paths, but the q version of this theorem is also unknown, so is a sensible definition of the q version of k-polymers.
But locally, the DLPP, DP and q-polymer models are very similar, as we shall see now.

q-Burke property
Fix and j, let Lemma 7. For 0 ≤ q ≤ 1 we have Proof. Immediate from the Noumi-Yamada descriptions at the first edge.
We call (B1.q) (B2.q) the Burke relations. When q = 0 or 1, the Burke relations define the RSK algorithms because the dynamics are the same along all edges of the GT patterns, whereas when q ∈ (0, 1) the qRSK dynamics in the non-first edges are different from the Burke relation.
Also for q = 0 or 1, when Uq, Vq and Xq are random with certain distributions, the Burke relations yield the Burke properties in the DLPP and DP cases.
Then in each of the above cases The Burke property with geometric weights can be found in e.g. [Sep09, Lemma 2.3], the one with the exponential weights in [BCS06], the one with log-gamma weights in [Sep12].
The q-Burke property is similar.
Proof. By the definitions of the q-geometric and the q-hypergeometric distributions, where the last identity is due to (7).
When q = 0 or 1 the converse of Proposition 1 is also true (see e.g. [Sep12] for the q = 1 case). That is, the Burke relation and the indentification in law of the triplets implies the specific distributions (geometric, exponential and loggamma) under reasonable assumptions thanks to the characterisation results in [Fer64,Fer65,Luk55]. The converse of the q-Burke property is open.
The q-Burke property allows one to tackle the q-polymer on the N 2 lattice (obtained by a simple shift of the model on the N 2 >0 lattice in previous considerations) with the following condition: We call such a configuration a q-polymer with stationary boundary conditions. Now we can show the strong law of large numbers of the partition functions.
Using (3) we obtain (1), and with the usual strong law of large numbers we obtain (2).
In [MP16, Section 6.3], the dynamics of the first edge of the tableaux was formulated as an interacting particle system, called the geometric q-pushTASEP. Therefore it has a natural correspondence with the q-polymer, where Z(n, m) + m = ξm(n) is the location of the mth particle at time n.
Here we describe the geometric q-pushTASEP whose initial condition corresponds to the qpolymer with stationary boundary condition.
Thus via the translation of (the arguments in the proof of) Theorem 2 we have Corollary 5. Let ξ0:∞ be the locations of the stationary geometric q-pushTASEP. Then we have the following 1. For any j ≥ 0, the jth particle performs a simple random walk with increments distributed according to qGeom(α).
2. At any time, the gap between neighbouring particles are independently distributed according to qGeom(β).

Classical limit of the Burke relations
It is natural to guess that under the classical limit of q-Burke relation (B1.q) and (B2.q) becomes the Burke relation (B1.1) and (B2.1) of the DP. Here we give a heuristic argument justifying this statement. The argument may be compared to that in the proof of [MP16,Lemma 8.17].
In the rest of this section, for convenience we omit the · when an integer is required. Given U, V we define and sample X ∼ qHyp(U , ∞, V ), then by Items 2 and 3 of Lemma 2 Using the reflection property of the dilogarithm function By taking derivatives of f we also have Hence f achieves unique maximum 0 at − log(e −U + e −V ). Now we can define X( ) by the relation X = m( ) + −1 X( ) and obtain and we have recovered (B2.1).

q-local moves
In this section we define the q-local moves and prove Theorem 3. In a sense, the local moves are very fundamental building blocks, as they unify the PNG and the RSK.
Let us define an object by adding to a 2 by 2 matrix a labelled edge connecting the 21-and 22-entries: The q-deformation of local moves consist of two maps: where a is a random variable with q-hypergeometric distribution qHyp (c − a, e, b − a).
On the boundary we define And l : Given an array A = (wij) i,j≥1 with labelled horizontal edges connecting neighbouring entries in the same rows, let lij and l ij be l and l acting on the (i, j)-th sub-2 by 2 matrix, namely li,j (resp. l i,j ) acts on A by acting on wi−1,j−1 wi−1,j wi,j−1 wi,j and keeping other entries unchanged. Similarly as in [OSZ14], define ρij by where for any integer n we denote (n) + := n ∨ 0 to be the positive part of n. The operator l ij are purely auxiliary, as they only serve to store the differences like ti,j − ti,j−1 = λ j−1 k−1 −λ j−1 k before ti,j is unrecoverably changed (see the proof of Theorem 5 for more details).
Given an input array (wij), we initialise by labelling all the horizontal edges between wi−1,j and wi,j with ei,j = ∞.
For two paritions λ and µ, denote by λ µ if λ ≺ µ and µ = λ + ei for some i. Let Λ be a Young diagram of size N , and ∅ = Λ(0) Λ(1) Λ(2) · · · Λ(N ) = Λ be a sequence of growing Young diagrams, which we call a growth sequence of Λ. For λ µ, denote µ/λ as the coordinate of the box added to λ to form µ. For example, if λ = (4, 2, 1) and µ = (4, 3, 1) then µ/λ = (2, 3). As aside, it is well known that a growth sequence Λ(0 : N ) of Λ corresponds to a standard Young tableau T of shape Λ, where T can be obtained by filling the box with coordinate Λ(i)/Λ(i − 1) by i. Now define to be an operator acting on integer arrays on N 2 >0 . It does not depend on the choice of the sequence as we shall see in the proof of Theorem 5, hence it is well defined.
Denote by S(Λ) the boundary of Λ: The set S(Λ) determines a coordinate system of all cells in Λ. To see this, for any (i , j ) ∈ Λ, there exists unique (i, j) ∈ S(Λ) and unique k ≥ 1 such that (i , j ) and (i, j) are at the same diagonal and their "distance" is k − 1: In this case we call (i, j, k) the Λ-coordinate of (i , j ).
In Theorem 5. Let (tij) = TΛA. For any (i , j ) ∈ Λ with Λ-coordinate (i, j, k) where (λ j k (i)) is the output of qRSK(A(I(Λ))). Note the above equality is an identity in joint distribution over all boxes (i , j ).
Here is an illustration, where we show the shape Λ, and t i j which corresponds to λ i k (j).
Proof. We prove it by induction. When Λ = (1), that is, it is a one-by-one matrix, applying the local move TΛ = ρ1,1 = l1,1 on A we obtain the correct result λ 1 1 (1) = w1,1. Let Θ Ξ be two Young diagrams such that Ξ/Θ = (n, k). We assume the theorem is true for Λ = Θ and want to show it holds for Λ = Ξ.
For all (i, j) with i − j = n − k, on the one hand, the Θ-coordinate and the Ξ-coordinate of (i, j) coincide, and on the other hand, as ρ n,k , by its definition, only alters the entries along the diagonal i − j = n − k. Therefore it suffices to show (ρ n,k TΘA)(n − + 1, k − + 1) = λ k (n), = 1 : n ∧ k.
It suffices to show that for = 1 : n ∧ k − 1 and for = n ∧ k, t n−i+1,k−i+1 = λ k i (n)∀i = 1 : n ∧ k. We consider the bulk case, that is when n, k > 1, as the boundary cases are similar and much easier.
For = 1, when l n,k acts on t 0 , by the Noumi-Yamada description it alters the submatrix (note that w n,k = a k 1 (n)) For 1 < < n ∧ k, given the induction assumption, l n− +1,k− +1 acts on t −1 by changing and that l n− ,k− +1 transforms the submatrix which stores the correct argument for a possible future operation ρ n,k+1 . For = n ∧ k, say n > k, then by the induction assumption and the definition of the local moves at the left boundary (20), l n−k+1,1 acts on t k−1 by changing . This is the boundary case (9) of the Noumi-Yamada description. Similarly when n = k and n < k, the upper boundary and upper-left boundary cases (19)(21) correspond to Lemma 5.

The push-forward measure of the q-local moves
In this section we prove Theorem 3. Before starting the proof, we show some illustrations to help with the readability.
Here is an illustration of the measure µq,Λ for Λ = (5, 5, 3, 3, 1). Some of the t-entries have been labelled. We focus on the products of q-Pochhammers: we use solid (resp. dashed) lines to indicate endpoints whose differences contribute to the q-Pochhammers in the denominator (resp. numerator). For example, the special solid line on the top left corner connecting 0 and t11 corresponds to (t11 − 0) −1 in the measure.
The proof is about transformation by ρ n,k from measure µΘ,q to µΛ,q where Λ/Θ = (n, k). Without loss of generality assume n > k. Intuitively speaking, after cancellations of q-Pochhammers that are not affected during this transformation, it suffices to show that ρ n,k has the following illustrated effect: Where the ai's, bi's, ci's and di's are aliases of t ,j 's on the tridiagonal area {( , j) : n − k − 1 ≤ − j ≤ n − k + 1} and the precise definition can be found in the proof. Now let us turn to the complete proof. It may be compared to that of [OSZ14, Theorem 3.2].
Once again we show this by an induction argument. When Λ = (1) has just one coordinate, the left hand side of (24) becomes is the right hand side of (24). Let Θ be a Young diagram such that Θ Λ. Let (n, k) = Λ/Θ. Since TΛ = ρ n,k • T θ , we can rewrite the left hand side of (24) as where the last identity comes from the induction assumption. So it suffices to show We assume n > k, as what follows can be adapted to the case n < k due to the symmetry. The proof when n = k is similar with very minor changes. For example, the right hand side of (26) will be the same except (a k−1 − b k )q and (d k − b k ) −1 q are replaced by (a k−1 )q and (d k ) −1 q respectively due to the involvement of (t 11 ) −1 q and (t11) −1 q .
By the structure of the products in MΘ and MΛ, we see that all the products outside of the diagonal strip near (i − j) = n − k are cancelled out in M k−1 (t )/M k (t). More precisely, when k > 1, by denoting It is time to calculate P(ρ n,k (t ) = t). This is the probability of mapping (d1, d2, . . . , d k ) to (a0, a1, . . . , a k−1 ).
By the definition of the q-local moves (also see (22) and (23)) we have By denoting d1 = X0, we can pin down the Xi's in terms of the other variables.
Clearly given that all droplets are sampled independently, the height functions are a Markov process because their values at time n only depend on their values at time n − 1 and the droplets at time n.
It is known that the PNG model observes the same dynamics as the RSK algorithm acting on a staircase tableau. More specifically, one may identify the top-level droplets for PNG with the input data for RSK: where the t n 's are the output of local moves taking the staircase tableau (wij) i+j≤m+1 , one may recover the dynamics of the PNG model (31) from the dynamics of the local moves.
Using the same correspondence, the gPNG model was defined using the gRSK dynamics, as per [NZ16].
It can be seen from this formula that, similar the usual PNG model, the height function hm(k) is a function of the heights at neighbouring positions at the previous time hm(k − 1), hm(k), hm(k + 1) and the droplet dm(k).
In [Joh03] the PNG model was used to show that asymptotically the partition functions of DLPP at the same time are the Airy process.
Here by applying the q-local moves on the staircase Young diagram and use Theorem 3 and Theorem 5, we obtain a q-version and the joint distribution of partition functions of the q-polymer at a fixed time in Corollary 2 in Section 1.
Our result on joint distributions of polymers, Corollary 1 and 2 are q-versions of Theorem 2.8 and 3.5 in [NZ16] respectively. To obtain anything more, such as the q-version of the two-point Laplace transform in Theorem 2.12 in that paper or the central limit of one point partition function in Theorem 1 in [BCR13], a natural question arises whether one can obtain a q-Whittaker version of Corollary 1.8 (writing one-point Laplace transform as a Fredholm determinant) in [BCR13], which is the main tool to show the two results.
This measure was introduced in [OSZ14] as the push-forward measure of local moves acting on a matrix with log-Gamma weights. The next proposition demonstrates the classical limit from µq to L. Proof. Quite straightforward by plugging in Items 1 and 2 in Lemma 2.