Volume Laws for Boxed Plane Partitions and Area Laws for Ferrers Diagrams
VVolume Laws for Boxed Plane Partitions and AreaLaws for Ferrers Diagrams
U. SchwerdtfegerFakult¨at f¨ur MathematikUniversit¨at BielefeldPostfach 10 01 31, 33501 Bielefeld, GermanyNovember 20, 2018
Abstract
We asymptotically analyse the volume-random variables of general, symmetricand cyclically symmetric plane partitions fitting inside a box. We consider the re-spective symmetry class equipped with the uniform distribution. We also prove arealimit laws for two ensembles of Ferrers diagrams. Most of the limit laws are Gaussian. A plane partition fitting inside an ( r, s, t ) -box is an r × s -array Π of non-negative integers p i,j ≤ t with weakly decreasing rows and columns. It can be visualised as a pile of unitcubes in the box B ( r, s, t ) := [0 , r ] × [0 , s ] × [0 , t ] “flushed into the corner”, see figure 1 and[2]. The figure also illustrates the connection to tilings of a hexagon of side lengths r, s and t by lozenges. Yet another interpretation is viewing such a pile of cubes as an order idealin the product of three finite chains (total orders) with the respective lengths r, s and t. In the following we mean by “plane partition” one which fits inside an ( r, s, t )-box. Thevolume of a plane partition is the sum of its parts or the number of unit cubes in the pileor the cardinality of the order ideal, respectively. If one side length, say t of the boundingbox is one, such a pile can be regarded as Ferrers diagram fitting inside an r × s -rectangleof area equal to the volume. Let PP ( r, s, t ) denote the set of all plane partitions fittinginside B ( r, s, t ) . A plane partition Π ∈ PP ( r, r, t ) is called symmetric if the correspondingpile of cubes is symmetric about x = y. To put it differently the cube ( i, j, k ) belongs tothe pile, if and only if ( j, i, k ) does. The respective tiling is symmetric w.r.t. a verticalaxis. A plane partition Π ∈ PP ( r, r, r ) is called cyclically symmetric if the correspondingtiling is invariant under a rotation about 2 π/ , i.e. the cube ( i, j, k ) belongs to the pile ifand only if ( k, i, j ) and ( j, k, i ) do. Call these subsets SPP ( r, t ) and CSPP ( r ) respectively.1 a r X i v : . [ m a t h . C O ] J a n r s33 2 2 211 10 0 0 0 0 000 0 0 00 Figure 1: A plane partition in (4 , , ? ] an “Arctic Circle Theorem” for tilings of a largehexagon is proved, which states that a typical tiling looks periodic near the corners of thehexagon and unordered inside the inscribed ellipse.It is natural to ask also for volume limit laws as the generating functions counting planepartitions by volume are available for the above classes. For the following formulae werefer to [2]. The volume generating function of PP ( r, s, t ) is r (cid:89) i =1 s (cid:89) j =1 t (cid:89) k =1 − q i + j + k − − q i + j + k − . (1.1)Symmetric plane partitions in B ( r, r, t ) have the volume generating function (cid:32) r (cid:89) i =1 t (cid:89) k =1 − q i + k − − q i + k − (cid:33) (cid:32) (cid:89) ≤ i 1) (1 − q α ijk ) α ijk (1 − q α ijk − ) . (1.4)In the following g ( x ) ( N ) denotes the N th derivative of g w.r.t. x. Moments of X of arbitraryorder N exist and can be computed via E (cid:0) X N (cid:1) = ( − I ) N (cid:0) P ( e Ix ) (cid:1) ( N ) (cid:12)(cid:12)(cid:12) x =0 , (1.5)where P ( e Ix ) ( I = − 1) is the characteristic function of X. Denote by µ, µ spp and µ cspp the mean value and by σ, σ spp and σ cspp the standard deviationof the volume variables on PP ( r, s, t ) , SPP ( r, t ) and CSPP ( r ) , respectively. Lemma 2.1. The volume distributions are symmetric about half the volume of the boundingbox, and hence the expected volumes are µ = rst/ , µ spp = r t/ and µ cspp = r / . Proof. Let v be the volume of the bounding box B . We show that X and v − X are equallydistributed. Let a plane partition of volume k be represented by a pile of k green cubesinside B . Now fill up the empty space in B with red cubes. The red cubes represent a planepartition of volume v − k. This construction describes a bijection between plane partitionsof volume k and v − k which respects symmetry and cyclic symmetry. Now the lemmafollows as E ( X ) = E ( v − X ) . In order to prove the Gaussian limits we consider the characteristic functions P ( e Ix )of the random variables X. More precisely, since sums are easier to handle than products,we will study the logarithms of the characteristic functions. The following lemma enablesus to compute explicit formulas for the variances of the random variables and to estimatethe Taylor coefficients of the functions in question. Lemma 2.2. For positive real numbers α, c with α > c ≥ , we have the expansion log (cid:18) ( α − c ) (1 − e αx ) α (1 − e ( α − c ) x ) (cid:19) = (cid:88) N ≥ H N,c ( α ) x N . (2.1)3 ere H N,c is a polynomial of degree N − in α. In particular we have H ,c ( α ) = and H ,c ( α ) = − c α + c . Furthermore there is a positive constant D, such that the inequality | H N,c ( α ) | ≤ D · α N − · (2 c ) N (2.2) holds for all N ∈ N . Proof. Define the function g ( t ) by g ( t ) = log (cid:18) e t − t (cid:19) . (2.3)Then the lhs of (2.1) is easily seen to be equal to g ( αx ) − g (( α − c ) x ) . We have thefollowing series expansion for g ( t ) : g ( t ) = log (cid:18) e t − t (cid:19) = log (cid:18) t + 13! t + . . . (cid:19) = (cid:88) N ≥ b N t N . (2.4)Observe that the singularities of g of smallest modulus are ± πI, so the numbers | b N | decay like (2 π ) − N and hence are bounded by some constant D. The N th coefficient in theTaylor expansion of g ( αx ) − g (( α − c ) x ) about x = 0 is in fact a polynomial of degree N − , namely H N,c ( α ) = b N · (cid:0) α N − ( α − c ) N (cid:1) = b N · N (cid:88) k =1 (cid:18) Nk (cid:19) ( − k +1 c k α N − k . (2.5)As α > c ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b N · N (cid:88) k =1 ( − k +1 (cid:18) Nk (cid:19) c k α N − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ D · α N − · c N · N (cid:88) k =1 (cid:18) Nk (cid:19) ≤ D · α N − · c N · N . (2.6)This finishes the proof of Lemma 2.2.Now we can easily compute the variances. The formula for σ already appeared in [9]. Lemma 2.3. We have1. σ = ( r st + rs t + rst ) = rst ( r + s + t ) , σ spp = tr + t r − t r + tr − tr and3. σ cspp = r − r . roof. Recall that the variance of a random variable Y can be obtained as the V ( Y ) = − log (cid:0) P ( e Ix ) (cid:1) (cid:48)(cid:48) (cid:12)(cid:12)(cid:12) x =0 , where P ( q ) is the probability generating function of Y. In the PP ( r, s, t )case we apply this to P ( q ) as in (1.4) and obtain − log (cid:0) P ( e Ix ) (cid:1) (cid:48)(cid:48) (cid:12)(cid:12)(cid:12) x =0 = − r (cid:88) i =1 s (cid:88) j =1 t (cid:88) k =1 log (cid:32) ( α ijk − (cid:0) − e α ijk Ix (cid:1) α ijk (cid:0) − e ( α ijk − Ix (cid:1) (cid:33) (cid:48)(cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 , (2.7)where α ijk = i + j + k − . According to Lemma 2.2, each summand on the rhs is equal to2! H , ( α ijk ) = − α ijk + 112 (2.8)and a straightforward calculation yields r (cid:88) i =1 s (cid:88) j =1 t (cid:88) k =1 (cid:18) 16 ( i + j + k − − (cid:19) = 112 (cid:0) r st + rs t + rst (cid:1) . (2.9)The other variances are calculated in an analogous way with suitable choices of α and c. Now we can investigate concentration properties of the families of volume randomvariables, i.e. we study the quotients of standard deviation and mean when the box getslarge. According to Lemma 2.1 and Lemma 2.3 these quotients tend to zero if at least twoside lengths of the bounding box tend to infinity. The latter is in particular satisfied when r → ∞ in the SPP ( r, t ) and CSPP ( r ) case. In the general case of PP ( r, s, t ) we have (cid:18) σµ (cid:19) = 13 (cid:18) st + 1 rt + 1 rs (cid:19) . (2.10)If say r and s are unbounded, the right hand side of (2.10) tends to zero for r, s → ∞ andthe family is concentrated to the mean, i.e. for every δ > P (cid:18) − δ ≤ X E ( X ) ≤ δ (cid:19) −→ r, s → ∞ )On the other hand, if two coordinates are fixed, say r, s, then the quotient is bounded awayfrom zero by rs for t → ∞ . These families are not concentrated. Similar considerationshold in the case of SPP ( r, t ) when r is fixed and t → ∞ . In the next section we investigatethe limit laws in these two cases. We first give a result for the concentrated families.5 roposition 3.1. If at least two of r, s, t tend to infinity, the family (cid:18) Y := X − µσ (cid:19) of normalised volume random variables on PP ( r, s, t ) converges in distribution to a stan-dard normal distributed random variable. The same statement holds for the volume vari-ables on SPP ( r, t ) if at least r → ∞ and on CSPP ( r ) if r → ∞ . Proof. For PP ( r, s, t ) . Let r, s → ∞ and t vary arbitrarily. The characteristic function of Y is φ ( x ) := e − µIx/σ P (cid:0) e Ix/σ (cid:1) , with P as in eq. (1.4). We prove that log ( φ ( x )) → − x for x ∈ R if r, s → ∞ . Then by Levy’s continuity theorem [4] the assertion follows.By Lemma 2.1 Y and − Y have the same distribution. So the characteristic function is realvalued and the coefficients for odd N vanish. By Lemma 2.2 we have the Taylor expansion φ ( x ) = − x (cid:88) N ≥ r (cid:88) i =1 s (cid:88) j =1 t (cid:88) k =1 H N, ( i + j + k − (cid:18) xIσ (cid:19) N (3.1)According to the estimate (2.2) we can bound the modulus of the 2 N th summand, N ≥ D r (cid:88) i =1 s (cid:88) j =1 t (cid:88) k =1 N ( i + j + k − N − (cid:18) | x | σ (cid:19) N ≤ N D | x | N rst ( r + s + t ) N − σ N . (3.2)Plugging the explicit expression for σ of Lemma 2.3 into the rhs of (3.2) we obtain theestimate for the 2 N th summand, N ≥ , in the expansion (3.1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r (cid:88) i =1 s (cid:88) j =1 t (cid:88) k =1 H N, ( i + j + k − (cid:18) xIσ (cid:19) N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ D | x | N · N (cid:18) rs + 1 rt + 1 st (cid:19) N − . (3.3)Summing in (3.3) over N ≥ (cid:12)(cid:12) log( φ ( x )) + x / (cid:12)(cid:12) ≤ D | x | | x | (cid:0) rs + rt + st (cid:1) − | x | (cid:0) rs + rt + st (cid:1) . (3.4)The rhs tends to zero for any fixed real x if r and s tend to infinity. This proves theassertion for the PP ( r, s, t ) case. The other cases are shown with similar estimates.Now we consider the non-concentrated case. If r = s = 1 we have a single columnof unit cubes of height t. The volume clearly is uniformly distributed. So, if rs > { , . . . , t } . Now the easiest guess is that for large t the dependence vanishes and thevolume of a single column is uniformly distributed. This guess is the right one as thefollowing proposition shows. 6 roposition 3.2. If r and s are fixed and t tends to infinity, the family (cid:18) Z t := X rst t (cid:19) of rescaled random variables converges in distribution to the ( rs ) -fold convolution of theuniform distribution on [0 , . In the SPP ( r, t ) case the so rescaled sequence convergesin distribution to the convolution of r factors of the uniform distribution on [0 , and r ( r − / factors of the uniform distribution in [0 , . Proof. For PP ( r, s, t ) . We show that the Fourier transform of Z t converges pointwise to (cid:16) e Ix − Ix (cid:17) rs , which is the Fourier transform of the rs -fold convolution of the uniform distri-bution on [0 , . The Fourier transform of Z t is P ( e Ix/t ) , with P as in (1.4). We expandthe product running from 1 to t in P ( e Ix/t ) . All but the last term in the numerator andthe first term in the denominator cancel out: P ( e Ix/t ) = r (cid:89) i =1 s (cid:89) j =1 ( i + j − − e Ix ( i + j + t − /t )( i + j + t − − e Ix ( i + j − /t )The single factors are easily seen to converge to e Ix − Ix . The SPP ( r, t ) case is worked out analogously. A Ferrers diagram is a convex lattice polygon which contains both upper corners and thelower left corner of its smallest bounding rectangle. When t = 1 a boxed plane partitioncan be viewed as a Ferrers diagram fitting inside an r × s rectangle. Then formula (1.1)reduces to the q -binomial coefficient [ r + ss ] q . The class of Ferrers diagrams with h rows and w columns has the area generating polynomial q h + w − (cid:2) h + w − h − (cid:3) q (4.1)since at least the h + w − m ≥ m + 2 is obtained by summing formula (4.1)over all pairs ( h, w ) ∈ N with h + w = m + 2 . This can be written as q m +1 m (cid:88) h =0 (cid:20) mh (cid:21) q . (4.2)7bserve that the index of summation in (4.2) can be interpreted as the height minus oneand that the total number of such Ferrers Diagrams is 2 m . So the following probabilitygenerating function (PGF) 2 − m q m +1 u m (cid:88) h =0 u h (cid:20) mh (cid:21) q (4.3)describes the ensemble of Ferrers diagrams of half-perimeter m + 2 counted by height andarea. The additional height parameter allows us to use the results for height- and width-ensembles dicussed above. More precisely, we condition the area variable in the perimeter-ensemble on the height variable. In what follows we will prove that the joint distribution ofthe (properly rescaled) height and area variables in this ensemble converge in distributionto the two-dimensional standard normal distribution. The following proposition from [6](see also [8]) is taylor made for this situation. Proposition 4.1. Let X m , Y m be real valued random variables and let Y m be supported in alattice L m := { α m + kδ m | k ∈ Z } , where δ m > and α m ∈ R , i.e. P ( Y m ∈ L m ) = 1 . Suppose Y m satisfies a local limit law µ with a density g ( y ) w.r.t. the Lebesgue measure on R (thisimplies δ m → ), i.e. for all y ∈ R and every sequence ( y m ) with y m ∈ L m and y m → y wehave P ( Y m = y m ) /δ m → g ( y ) . Suppose further that for µ -almost all y ∈ R the conditionaldistributions P ( X m ∈ ·| Y m = y m ) converge weakly to a measure ν ( · , y ) ( y m → y, y m ∈ L m ).Then the joint distribution of ( X m , Y m ) converges weakly to the measure ν defined by ν ( A × B ) := (cid:90) B ν ( A, y )d µ ( y ) for all Borel sets A, B ⊆ R . For q = 1 formula (4.3) is simply 2 − m u (1 + u ) m , the PGF of the height random variable H m . It satisfies the assumptions of [5, Theorem IX.14] and hence a Gaussian local limitlaw arises. The mean and standard deviation of the height are easily computed to beasymptotically equal to m/ √ m/ , respectively. So let Y m = 2( H m − m/ / √ m. The random variable Y m is supported in Z / √ m. Let y ∈ R and y m → y, where each y m ∈ Z / √ m. The area random variable A m , conditioned on the event { Y m = y m } , hasPGF q m +1 (cid:18) mm/ y m √ m/ (cid:19) − (cid:20) mm/ y m √ m/ (cid:21) q . (4.4)An application of Lemma 2.1 and Lemma 2.3 with r = m/ y m √ m/ , s = m − r and t = 1shows that the mean and the variance of the conditioned area variables are asymptoticallyequal to m / m / , respectively. So according to Proposition 3.1, the rescaled areavariable X m := ( A m − m / / (cid:112) m / 48 conditioned to { Y m = y m } converges weakly to aGaussian random variable with mean zero and variance one. Now Proposition 4.1 yields Proposition 4.2. Denote A m and H m the random variables of area and height of a Ferrersdiagram in the uniform fixed perimeter ensemble described by (4.3) . Let X m = A m − m / (cid:112) m / , Y m = H m − m/ √ m/ . hen, as m → ∞ , ( X m , Y m ) converges weakly to the two-dimensional standard normal-distribution. The last result can also be obtained by analysing a q -difference equation satisfied bythe generating function of Ferrers diagrams counted by height, area and half-perimeter. Acknowledgements The author thanks C. Richard for critical and helpful comments on the manuscript. Hewould also like to acknowledge financial support by the German Research Council (DFG)within the CRC 701. References [1] M. Bousquet-M´elou, A method for the enumeration of various classes of column-convexpolygons, Discrete Math. (1996), 1-25.[2] D. Bressoud, Proofs and Confirmations , Cambridge: Cambridge University Press(1999).[3] G.M. Constantine and T.H. Savits, A multivariate Fa`a di Bruno formula with appli-cations, Trans. Am. Math. Soc. (1996), 503–520.[4] W. Feller, Probability Theory and its Applications Vol.II, New York: Wiley (1971).[5] P. Flajolet and R. Sedgewick, Analytic Combinatorics , Cambridge: Cambridge Uni-versity Press (2008), to be published.[6] M.A. Fligner, A note on limit theorems for joint distributions with applications tolinear signed rank statistics, J. Roy. Statist. Soc. Ser. B (1981), 61–64.[7] T. Prellberg and A.L. Owczarek, Stacking models of vesicles and compact clusters, J.Stat. Phys. (1995), 755–779.[8] J. Sethuraman, Some limit theorems for joint distributions, Sankhya, Ser. A (1961),379–386.[9] D.B. Wilson, Diagonal sums of boxed plane partitions, Electron. J. Combin.8