Random plane partitions and corner distributions

We explore some probabilistic applications arising in connections with $K$-theoretic symmetric functions. For instance, we determine certain corner distributions of random lozenge tilings and plane partitions. We also introduce some distributions that are naturally related to the corner growth model. Our main tools are dual symmetric Grothendieck polynomials and normalized Schur functions.


Introduction
Combinatorics arising in connections with K-theoretic Schubert calculus is quite rich. Accompanied by certain families of symmetric functions, it usually presents some inhomogeneous deformations of objects behind classical Schur (or Schubert) case. While the subject is intensively studied from combinatorial, algebraic and geometric aspects, see [Len00,Buc02,Buc05,Vak06,LP07,IN13,Yel17] and many references therein, much less is known about probabilistic connections (unlike interactions between probability and representation theory). Some work in this direction was done in [TY11] and related problems were addressed in [Yel19a].
In this paper we give several probabilistic applications obtained with tools from combinatorial K-theory. We mostly focus on one deformation of Schur functions, the dual Grothendieck polynomials, whose associated combinatorics is fairly neat. In particular, we show that these functions are naturally related to the corner growth model [Joh00,Joh01,Sep09,Rom15] (which can also be viewed as a totally asymmetric simple exclusion process or a directed last passage percolation).
We are now going to discuss our results across few related models.
1.1. N-matrices with bounded last passage time. A lattice path Π with vertices indexed by N 2 is called a monotone path if it uses only steps of the form (i, j) → (i + 1, j), (i, j + 1). An N-matrix is a matrix of nonnegative integers with only finitely many nonzero entries. Let a, b, c be positive integers. Given an N-matrix D = (d ij ) with b rows and c columns, the last passage time G(b, c) is defined as where the maximum is taken over monotone paths Π from (1, 1) to (b, c). Let BM(a, b, c) be the set of b × c N-matrices whose last passage time is bounded by a, i.e. Consider the uniform probability measure on BM(a, b, c). For a random matrix D ∈ BM(a, b, c), define the column marginals i.e. as the sum of entries in ℓ-th column of D where ℓ ∈ [1, c]. It is natural to ask what are distributions of C ℓ . We determine these distributions. Let us mention some results that we obtain (more general results about C ℓ are presented later via lozenge tilings).
The first property is the following symmetry.
Theorem 1.1 (Equidistribution of marginals). For all i, j ∈ [1, c], the random variables C i and C j have the same distribution, i.e. P(C i = n) = P(C j = n) for all n.
Next we obtain limiting distributions when a, b, c grow proportionally.
Theorem 1.2 (Asymptotic normality of marginals). Assume that a = ⌊αn⌋, b = n, and c = ⌊γn⌋ for α, γ > 0. As n → ∞ we have convergence in distributioñ Moreover, the joint distribution of any collection ofC ℓ converges to the distribution of independent standard Gaussian random variables.
By symmetry, we also get similar results for row marginals. In fact, we prove a more general statement with different asymptotic regimes for the parameters a, b, c. Note that in some regimes the variables C ℓ become asymptotically independent. We also determine joint distributions for collections of C ℓ .
By default we ignore zeros of π. We denote by sh(π) := {(i, j) : π ij > 0} the shape of π. Let PP(a, b, c) be the set of boxed plane partitions that fit inside the box a × b × c, i.e. the first row of the shape is at most a, the first column is at most b, and the first entry is at most c. We show that |PP(a, b, c)| = |BM(a, b, c)| (Theorem 2.2).
Consider the uniform probability measure on the set PP(a, b, c). Recall that classical MacMahon's theorem on boxed plane partitions gives the explicit product formula Note that we also have where s λ is the Schur polynomial and 1 n := (1, . . . , 1) repeated n times. (See e.g. [Sta99,Ch. 7

])
For a uniformly random plane partition π ∈ PP(a, b, c), let X ℓ be the number of columns of π that contain entry ℓ ∈ [1, c]. In fact, the variables X ℓ become image of the marginals C ℓ described previously under the bijection which we describe in Sec. 2. Similarly, X i and X j have the same distribution for all i, j ∈ [1, c]. We now consider limiting distributions in another asymptotic regime.
Moreover, the joint distribution of any collection of X ℓ converges to the distribution of independent Poisson random variables with rate t.
By symmetry, we also get similar results for the random variables Y ℓ , the number of rows containing ℓ in a random plane partition. Hence the theorem also implies some bounds (deviation) for the area containing ℓ.
1.3. Lozenge tilings of a hexagon. A lozenge tiling is a tiling of a planar domain with three types of lozenges which we refer to as left, top and right tiles. Let LT(a, b, c) be the set of lozenge tilings of a hexagon with sides (a, b, c, a, b, c) as in Fig. 1. For a lozenge tiling T ∈ LT(a, b, c), define: • left corners (or simply corners) of T , that are local tile configurations of the form • for each corner α, its height h(α) ∈ [1, c] is the z-coordinate of the top tile of α, when T is viewed in R 3 as a pile of cubes (boxed plane partition), see Fig. 1.
Consider the uniform probability measure on the set LT(a, b, c). We obviously have |L(a, b, c)| = |PP(a, b, c)|, where the bijection is given by writing the height of each top tile. The random variables C ℓ and X ℓ defined for previous models translate into the following random variables for lozenge tilings. We similarly have the symmetry: for all i, j ∈ [1, c], Γ i and Γ j have the same probability distribution (Corollary 5.4).
We find limits of corner distributions Γ in several asymptotic regimes.
Theorem 1.5. We have convergence in distributions in the following regimes: (i) Poisson: for a, b, c → ∞ with ab/c → t > 0 we have (ii) Negative binomial: for b fixed and a, c → ∞ with a/(a + c) → q ∈ (0, 1) we have Moreover, the joint distribution of any collection of random variables Γ ℓ weakly converges to the distribution of independent random variables given in corresponding regimes (i), (ii) or (iii).
Consider the Gaussian regime, say for a = b = c = n → ∞. It is known that random lozenge tilings have the arctic circle phenomenon: there is a circle inscribed in the hexagon, outside of which there are frozen regions and inside there is a liquid region [CLP98, CKP01, Ken09], see Fig. 2. Note that left corners on the topmost level at height c form a profile for the upper frozen region. Then the number of such corners is about n/2 with fluctuations of order √ n, which is compatible with known edge behavior. However what may seem surprising is the behavior across height levels. There is a kind of invariance there, the number of corners still has the same distribution on each level (even though some are inside the liquid region). Moreover, the theorem also tells that there is asymptotic independence between height levels which is different from the situation between slices parallel to the sides of the hexagon. As it will be clear later, the arctic phenomenon also passes to the first object of N-matrices with bounded last passage time, so that upper frozen boundary is recorded in the last column of a large matrix and the matrix itself has frozen regions in first and last quadrants, see Fig. 2.
There is a general CLT for linear statistics on determinantal point processes with Hermitian symmetric correlation kernels, see [Sos00]. Even though overall distributions of lozenges form a determinantal process, its kernel has a non-Hermitian form, see [Pet14], which means that the general result cannot be used to every lozenge statistics. For a certain height-type statistics, asymptotic normality with logarithmic fluctuations was proved in [BF14].
A well studied statistics on lozenge tilings is distributions of a particular lozenge type across corresponding slices parallel to the sides of the hexagon, see [BP14,Pet14,GP15] and references therein. In our case we consider right lozenges (which complement left corners). We also determine joint distributions of Γ with that statistics, see Sec. 6.
We prove our results about distributions of Γ by showing that their probability generating function can be expressed via normalized Schur polynomials which in turn follows from a result on dual Grothendieck polynomials defined below.
Theorem 1.6 (cf. Theorem 5.1). The probability generating function of Γ has the following explicit formula Our results crucially rely on properties of these symmetric functions. The dual symmetric Grothendieck polynomials g λ can be defined via the following combinatorial presentation where the sum runs over plane partitions T of shape λ and c i (T ) is the number of columns of T containing i. These polynomials were explicitly introduced and studied in [LP07] (and earlier implicitly in [Len00,Buc02]) and they are related to K-homology of Grassmannians. More properties of these functions can be found in [Yel17,Yel19]. The polynomials g λ can be considered as an inhomogeneous K-theoretic deformation of Schur polynomials. By definition, it is easy to see that the top degree homogeneous component of g λ is the Schur polynomial s λ , i.e. g λ = s λ + lower degree terms.
1.5. g-measure on plane partitions. Let (q 1 , . . . , q c ) ∈ (0, 1) c . For a plane partition π define the descent set Des(π) := {(i, j) : π ij > π i+1 j } and let des(π) = |Des(π)|. Consider the probability distribution P g,b,c , which we call g-measure or g-distribution, on the (infinite) set PP(∞, b, c) of plane partitions with at most b rows and maximal entry at most c, defined as follows: where the associated partition function (normalization) can be computed as (see Corollary 7.2) Proposition 1.7. Let π ∈ PP(∞, b, c) and λ = (λ 1 ≥ · · · ≥ λ b ≥ 0). We have By this formula it is convenient to view P g,b,c as a distribution on integer partitions λ with at most b parts. We write P g,b,c (λ) to mean this distribution.
1.6. Corner growth model. Let W = (w ij ) i,j≥1 be a random matrix with iid entries w ij that have geometric distribution with parameter q ∈ (0, 1), i.e.
Consider the last passage times for all m, n ≥ 1 where the maximum is over monotone paths Π from (1, 1) to (m, n). The corner growth model can be viewed as evolution of a random Young diagram Y (t) given at time t by the region Bar01], see also [Joh01,Sep09,Rom15] and references therein for more on the topic. In Sec. 8 we review and use some known results.
The main relationship between this stochastic process and the g-distribution defined above is the following.
Theorem 1.9. Let λ = (λ 1 , . . . , λ b ) be a partition with at most b parts. We have This formula is useful for both objects that it relates, i.e. as a formula for row distributions of the performance table as well as a source for new properties of dual Grothendieck polynomials.

A bijection between plane partitions and N-matrices
Given a plane partition π, define the descent level sets i.e. D iℓ is the set of column indices of the entry ℓ in ith row of π that are strictly larger than the entry below.
Before proceeding, let us mention some important properties of these sets: • The elements in D iℓ form a consecutive segment.
• For (i, ℓ) = (i 1 , ℓ 1 ) such that i ≥ i 1 , ℓ ≥ ℓ 1 , the sets D iℓ and D i 1 ℓ 1 are disjoint (more precisely, every element in D i 1 ℓ 1 is larger than any element of D iℓ ). Proof. Suppose π ∈ PP(a, b, c). Let us show that D = Φ(π) ∈ BM(a, b, c). By construction, it is clear that D has at most b rows and c columns. Take any monotone path (1, 1) = (i 1 , ℓ 1 ) → · · · → (i k , ℓ k ) ≤ (b, c) in D. Then the descent level sets D isℓs for s = 1, . . . , k are pairwise disjoint by the property mentioned above. Hence Let us now describe the inverse map Φ −1 . Given an N-matrix D ∈ BM(a, b, c), we show how to uniquely reconstruct a plane partition π ∈ PP(a, b, c) such that Φ(π) = D. We will build π sequentially by scanning the columns of D starting from the last one.
Let us show how to add an element ℓ in i-th row of some plane partition π ′ . To do so, find the first available column of π ′ whose length is less than i, then add the elements ℓ in that column so that its length becomes i. For example, suppose π ′ already had the following shape and we want to add ℓ in the 3rd row: add ℓ in row 3 → ℓ ℓ Let us initially set π = ∅. To add the ℓ-th column (d 1ℓ , . . . , d bℓ ) T of D for ℓ = c, c − 1, . . . , 1 we do the following. For each i = b, b − 1, . . . , 1 add ℓ in i-th row of π exactly d iℓ times.
Let us check that after this procedure we have π ∈ PP(a, b, c). By construction, it is clear that π has at most b rows and the largest entry at most c. We know that in D the maximal weight of a monotone path from (1, 1) to (b, c) is at most a. Suppose to the contrary that the length of the first row of π is larger than a. Consider the elements of the first row of π, say (ℓ 1 ≥ · · · ≥ ℓ k ) for k > a and suppose they were added in rows i 1 , . . . , i k during the procedure. Then we must have i 1 ≥ · · · ≥ i k and hence there is a monotone path in D that passes through all points (i k , ℓ k ) → · · · → (i 1 , ℓ 1 ) (here if we have the same point (i j , ℓ j ) repeated several times we may use it just once as the its multiplicity is recorded in d i j ℓ j ). The weight of any such monotone path is at least j d i j ℓ j ≥ k > a which is a contradiction. Finally, it is easy to see that π has the desired property Φ(π) = D. Note that by letting a, b, c → ∞ one can view Φ as a bijection between plane partitions and N-matrices.
We discuss few more properties of the bijection Φ that will be needed later.
(i) Let c ℓ (π) be the number of columns of π containing entry ℓ, and c ℓ (D) = i d iℓ be column sums of D. Then we have c ℓ (π) = c ℓ (D) for all ℓ ≥ 1.
(ii) Let λ = sh(π) be the shape of π. We have for all k ≥ 1 where the maximum taken over monotone paths Π from (k, 1) to (b, c), if D has b rows and c columns.
Proof. The part (i) is clear by construction. Indeed, by definition of Φ we have (ii) Take any monotone path Π from (k, 1) to (b, c). Then the descent level sets D iℓ for (i, ℓ) ∈ Π are pairwisely disjoint. Using this and since i ≥ k for all (i, ℓ) ∈ Π, we obtain On the other hand, suppose the k-th row of π has entries (ℓ 1 ≥ · · · ≥ ℓ m ) where m = λ k . Assume the entries ℓ 1 , . . . , ℓ m end in rows i 1 ≥ · · · ≥ i m of π. Then there is a monotone path Π from (k, 1) to (b, c) that contains all points (i m , ℓ m ), . . . , (i 1 , ℓ 1 ). The weight of any such path is at least j d i j ℓ j ≥ m = λ k . Combining this with the inequality (4) we obtain (3).
Remark 2.6. The Robinshon-Schensted-Knuth (RSK) correspondence can also be considered as a bijection between plane partitions and N-matrices, see e.g. [Sta99,Ch. 7]. The bijection Φ has another description and different yet related properties. We explore more properties and applications of this bijection in [Yel19b].
x c i i and c i is the number of columns of T containing i.
The following properties hold: g λ is an inhomogeneous symmetric polynomial whose top degree component is the Schur polynomial, i.e. g λ = s λ + lower degree elements.
The following properties will be important for our results.
Lemma 3.4 (Coincidence lemma). Let ρ := (a b ). We have Proof. We show that the Jacobi-Trudi formulas for both polynomials coincide under these conditions. Note firstly that by the classical Jacobi-Trudi formula we have For the polynomials g, the Jacobi-Trudi identity (5) for λ = ρ gives (6).
Corollary 3.5. We have the following formula Proof. Let ρ = (a b ). By combinatorial definition of g, we have the branching formula On the other hand, g ρ (x, 1) = s ρ (1 b , x) by Lemma 3.4.
Lemma 3.6. The following identity holds Proof. This identity implies from the Cauchy identity for the pair G λ , g λ of Grothendieck polynomials and the following property of the symmetric Grothendieck polynomials G λ (1 b ) = 1 if ℓ(λ) ≤ b and 0 otherwise, see [Yel19a]. In fact, one can also derive it from the bijection Φ, we address this in [Yel19b].

Normalized Schur polynomials
The normalized Schur polynomial S λ (x; N ) is defined for N ≥ max(k, ℓ(λ)) as follows We are interested in a special case when λ = (a b ) has rectangular shape. We present some known asymptotics for normalized Schur polynomials.
Proposition 4.1 (see [BP14]). Let N = b + c, ρ = (a b ) and x i ∈ C. We have: For b fixed and a, c → ∞ with a/(a + c) → q ∈ (0, 1) Consider now the following Gaussian asymptotic regime. Let u > 0, q ∈ (0, 1) be real parameters. Suppose a = ⌊uN ⌋, b = ⌊qN ⌋, b + c = N and let N → ∞. Then the partitions ρ = (a b ) (when each row is rescaled by N ) have the following simple limit shape f : [0, 1] → R given by Define two more parameters Proposition 4.2 ([GP15]). Let a, b, c be in the Gaussian regime as above. As N → ∞, we have uniformly on compact subsets of (R \ {0}) k .
Remark 4.3. In fact, this is a special case of a more general result proved in [GP15]. Proof. By definition of g λ and the bijection between lozenge tilings and plane partitions so that . . . , x c ). (11) Using (7) and the fact that Z abc = s ρ (1 b+c ) we obtain as needed.
In the first Poisson regime a, b → ∞ with ab/c → t by Proposition 4.1 we have Hence by (12), Γ weakly converges to k iid Poisson random variables with parameter t.
In the second negative binomial regime for fixed b and a, c → ∞ with a/(a + c) → q ∈ (0, 1) again by Proposition 4.1 we have Hence by (12), Γ weakly converges to k iid negative binomial random variables with parameters b, q.

5.2.
Distributions of X ℓ in plane partitions and marginals C ℓ in N-matrices. The results on Γ ℓ easily translate to results on X ℓ on random boxed plane partitions and C ℓ on N-matrices with bounded last passage time.
Take a uniformly random plane lozenge tilings T ∈ LT(a, b, c) and let Ψ be the map that records heights of all top tiles, the result is clearly a plane partition π = Ψ(T ) ∈ PP(a, b, c) and Ψ is a bijection. We have Γ ℓ (D) = X ℓ (π). In particular, Theorem 1.3 follows from Theorem 1.5 in the Poisson regime. Now take a uniformly random matrix D ∈ BM(a, b, c), then by Lemma 2.5 (i), we have C ℓ (D) = X ℓ (π) for π = Φ −1 (D). Then Theroem 1.2 follows from Theorem 1.5 in the Gaussian regime for u = α/(1 + γ) and q = 1/(1 + γ). Theorem 1.1 then follows from Corollary 5.4.

Joint distributions of left corners and right lozenges
Definition 6.1 (Right lozenge distributions). Let Y (k) be the vector of positions i.e. ycoordinates of the right tiles along k-th slice parallel to the side a of the hexagon, see Fig. 4. For k ≤ min(b, c), these positions have the form µ 1 ≥ · · · ≥ µ k ≥ 0.
Let P n,k := {µ : µ ⊢ n, ℓ(µ) ≤ k} be the set of partitions of n with length at most k. We have the following Schur expansion (see [BP14]) of normalized Schur polynomials where ρ = (a b ), N = b + c, and k ≤ min(b, c). The distribution of Y has the formula which is a case of more general orthogonal polynomial ensembles. Alternatively, there is also another formula where µ c = (a − µ b , . . . , a − µ 1 ) is a complement to µ in ρ.
It is proved in [JN06,GP15] that for every k, as N → ∞ we have convergence in distribution where GUE k is the distribution of the spectrum of a random k × k Hermitian matrix from the Gaussian unitary ensemble (GUE) (see Proposition 4.2).
Now we would like to relate the two distributions Γ and Y .
(ii) The distribution of Γ (k) can be computed as follows where the conditional probability is determined as Proof. By Corollary 5.3 and the Schur polynomial expansion (14) we have Therefore, To find the joint distribution we have where we used the formula (15).

g-measure
Let (q 1 , . . . , q c ) ∈ (0, 1) c . Recall that we define the probability distribution, called g-measure or g-distribution, on the (infinite) set PP(∞, b, c) of plane partitions with at most b rows and maximal entry at most c where Z b,c is the normalization constant.
Corollary 7.2. We have Proof. Follows from the proposition and the identity (8).
Theorem 7.3. For π ∈ PP(∞, b, c), let λ = (λ 1 ≥ · · · ≥ λ b ≥ 0) be the shape of π. The distribution of the first part λ 1 can be expressed as Proof. For the distribution of λ 1 we now have where we used Lemma 3.4 in the last identity.
Corollary 7.4. The distribution of λ 1 can be expressed via the Toeplitz determinant (1 + q i z −1 ).
Proof. Using the Jacobi-Trudi identity for Schur polynomials we obtain Note that Remark 7.5. From the Toeplitz determinant we can also obtain Fredholm determinant formula of the form det[I − K] for some kernel K using the Borodin-Okounkov formula [BO00].

Corner growth model
Consider a random matrix W = (w ij ) with iid entries w ij that are geometrically distributed with parameter q, i.e.
For m, n ≥ 1, recall that the last passage times G(m, n) are defined as where the maximum is taken over monotone paths Π from (1,1) to (m, n). Note that we have the recurrence relation − 1, j), G(i, j − 1)) + w ij , i, j ≥ 1 (one needs to set the boundary conditions G(i, 0) = G(0, i) = 0). The corner growth model can be viewed as evolution of a random Young diagram Y (t) given at time t by the region One can see that Y (t + 1) adds boxes to some corners of Y (t). We refer to [Sep09,Rom15] for more on the topic. The main result of this section is the following. where q i = q for all i ∈ [1, c] in the g-measure.
Lemma 8.2. Let D = (D ij ) b,c i,j=1 be a random b × c matrix, where D ij are iid geometrically distributed random variables with parameter q. Then P(Φ −1 (D) = π) = P g,b,c (π), where q i = q for all i ∈ [1, c] in the g-measure.
Proof of Theorem 8.1. Implies now from Lemma 2.5 (ii), Lemma 8.2 and Proposition 7.1. In particular, λ b ∼ G(1, c) has binomial distribution with parameters c, q. For λ 1 ∼ G(b, c) combining with Theorem 7.3 we obtain. More generally using the Jacobi-Trudi-type determinantal formula (5) we get Let us now discuss several results about the corner growth model that imply results on the shape λ from the g-distribution.
It was proved in [Joh00] that for x > 0 we have almost sure convergence to deterministic limit shape lim b→∞ 1 b G(⌊xb⌋, b) → ψ(x) = (1 − q) −1 (qx + q + 2 √ qx) and fluctuations around this limit shape are of order b 1/3 given by where σ(x) = (1 − q) −1 (qx) 1/6 ( √ x + √ q) 2/3 (1 + √ qx) 2/3 and F TW (t) is the Tracy-Widom distribution, the limiting distribution of the properly scaled largest eigenvalue in GUE [TW94]. Therefore, combining these with Corollary 8.3, λ from the g-distribution has limit shape where x ∈ (0, 1], c = b; and similar fluctuations. It was proved in [Bar01] that for fixed b, as c → ∞, the variables jointly converge in distribution to largest eigenvalues of b principal minors in b × b GUE matrix, which is known as GUE corners process (or GUE minors process [JN06]). Therefore, for λ from the g-distribution we have λ b−k+1 − q 1−q c √ q 1−q √ c , k = 1, . . . , b jointly converge in distribution to largest eigenvalues in b × b GUE corners process. It is also known [GW91] that as c → ∞ the random variables (18) converge in distribution to the process where B i are independent standard Brownian motions.