A note on limit shapes of minimal difference partitions
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n A note on limit shapes of minimal difference partitions
Alain Comtet , , Satya N. Majumdar , and Sanjib Sabhapandit Laboratoire de Physique Th´eorique et Mod`eles Statistiques, Universit´e deParis-Sud, CNRS UMR 8626, 91405 Orsay Cedex, France Institut Henri Poincar´e, 11 rue Pierre et Marie Curie, 75005 Paris, France
Abstract
We provide a variational derivation of the limit shape of minimal differ-ence partitions and discuss the link with exclusion statistics.
This paper is dedicated to Professor Leonid Pastur for his 70th anniversary .A partition of a natural integer E [1] is a decomposition of E as a sum ofa nonincreasing sequence of positive integers { h j } , i.e., E = P j h j such that h j ≥ h j +1 , for j = 1 , . . . . For example, 4 can be partitioned in 5 ways: 4, 3 + 1,2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1. Partitions can be graphically representedby Young diagrams (also called Ferrers diagrams) where h j corresponds to theheight of the j -th column. The { h j } ’s are called the parts or the summands of thepartition. One can put several constraints on such partitions. For example, onecan take the number of columns N to be fixed or put restrictions on the heights.In this paper we focus on a particular constrained partition problem called theminimal difference p partitions (MDP– p ). The MDP– p problem is defined byrestricting the height difference between two neighboring columns, h j − h j +1 ≥ p .For instance the only allowed partitions of 4 with p = 1 are 4 and 3 + 1. A typicalYoung diagram for MDP– p problem is shown in figure 1. Consider now the set ofall possible partitions of E satisfying E = P j h j and h j − h j +1 ≥ p . Since this isa finite set, one can put a uniform probability measure on it, which means thatall partitions are equiprobable. Then, a natural question is: what is the typicalshape of a Young diagram when E → ∞ ?In the physics literature this problem was first raised by Temperley, who wasinterested in determining the equilibrium profile of a simple cubic crystal grownfrom the corner of three walls at right angles. The two dimensional version ofthe problem —where walls (two) are along the horizontal and the vertical axesand E “bricks” (molecules) are packed into the first quadrant one by one suchthat each brick, when it is added, makes two contact along faces— corresponds1 p > ≥ p h j ≥ p E = X j h j ≥ p W h h j Figure 1: A typical Young diagram for MDP– p problem. The thick solid bordershows the height profile or the outer perimeter. W h is the width of the Youngdiagram at a height h , i.e., W h is the number of columns whose heights ≥ h .to the p = 0 partition problem. Temperley [2] computed the equilibrium profileof this two dimensional crystal. In the mathematics literature the investigationof the shape of random Young tableaux was started by Vershik and Kerov [3] andindependently by Logan and Shepp [4]. The case of uniform random partitionswas treated by Vershik and collaborators [5, 6, 7] who obtained the limit shapes forthe p = 0 and p = 1 cases and also the average deviations from the limit shapes [8].Some of these results were extended by Romik [9] to the MDP– p for p = 2. Theseproblems belong to the general framework of asymptotic combinatorics, a subjectwhich displays unexpected links with random matrix theory [10]. In this note wecompute the limit shapes of MDP– p for all p ≥ p = 0). Let P = ( i, h i ) and Q = ( j, h j ) be two points belonging to theouter perimeter of the Young diagram of a given partition. We evaluate the totalnumber of subdiagrams which connects these two points. These subdiagramsare lattice staircases with the only restriction that each step either goes right ordownward. The total number of horizontal steps is j − i , the total number ofvertical steps is h i − h j , and the total number of steps is j − i + h i − h j . Therefore,2he total number of configurations isΩ ( P, Q ) ≡ Ω ( i, h i ; j, h j ) = (cid:18) j − i + h i − h j j − i (cid:19) . (1)If P and Q are far apart (i.e., a = j − i ≫ , b = h i − h j ≫
1) we may use theStirling formula which givesln Ω ( P, Q ) = − a ln aa + b − b ln ba + b = p a + b φ ( −→ n ) , (2)where −→ n ≡ ( n , n ) = ( b, a ) / √ a + b is the unit vector orthogonal to −−→ P Q and φ ( −→ n ) = − n ln n n + n − n ln n n + n . (3)Heuristically one expects that in the limit E → ∞ , h → ∞ , W h → ∞ , theprofile of the Young diagram will be described by a smooth curve y = y ( x ) where y = h/ √ E and x = W h / √ E are the scaling variables. The normal vector can beparameterized as −→ n = − y ′ ( x ) p y ′ ( x ) , p y ′ ( x ) ! . Therefore φ ( −→ n ) = y ′ ( x ) p y ′ ( x ) ln (cid:20) − y ′ ( x )1 − y ′ ( x ) (cid:21) − p y ′ ( x ) ln (cid:20) − y ′ ( x ) (cid:21) . (4)In the lattice model, the points P and Q were taken to be far apart. Howeverin the new scale ( x, y ) one now assumes that they are close enough in order toensure that the interface is locally flat. The total number of Young diagrams Ωwith a given area E is obtained by adding all such local configuration, i.e.Ω = exp (cid:18) √ E Z ∞ d x p y ′ ( x ) φ ( −→ n ) (cid:19) , (5)with the area constraint Z ∞ y ( x )d x = 1 . (6)For large E , the most dominant contribution to Ω arises from the optimalcurve y = y ( x ) which maximizes the integral in (5) with the constraint (6). Thisoptimal curve describe the limit shape of the Young diagrams. Thus we are ledto the variational problem of extremizing L = Z ∞ d x (cid:20) y ′ ( x ) ln − y ′ ( x )1 − y ′ ( x ) − ln 11 − y ′ ( x ) (cid:21) − λ Z ∞ y ( x ) d x, (7)3here λ is a Lagrange multiplier. This leads to the Euler-Lagrange equation,which simplifies to dd x ln − y ′ ( x )1 − y ′ ( x ) = − λ. (8)We solve this equation with the boundary conditions y ( ∞ ) = 0 and y ( x → →∞ . The later condition follows from the fact that y ≡ h/ √ E ∼ ln E when x ≡ W h / √ E → E [14]. Therefore y (0) diverges in the limit E → ∞ .The solution gives the equation of the limiting shape as y ( x ) = − λ ln (cid:0) − e − λx (cid:1) with λ = π √ , (9)where λ is obtained by using the constraint (6).The goal of this paper is to extend this derivation to the MDP– p with p > p with p > p = 0).Let { h j } denote the set of non-zero heights in a given unrestricted partition( p = 0) E = P Nj =1 h j , where h j ≥ h j +1 for all j = 1 , , . . . , N −
1. Let usnow define a new set of heights h ′ j = h j + p ( N − j ) for j = 1 , , . . . , N . Thus h ′ j − h ′ j +1 = h j − h j +1 + p for all j = 1 , , . . . , N − h ′ N = h N >
0. Since h j − h j +1 ≥
0, the new heights thus satisfy the constraint h ′ j − h ′ j +1 ≥ p for all j = 1 , , . . . , N −
1. Since the mapping is one to one, the total number of localMDP– p configuration satisfiesΩ p ( i, h ′ i ; j, h ′ j ) = Ω ( i, h i ; j, h j ) . Moreover, h i − h j = h ′ i − h ′ j − p ( j − i ). Therefore using (1),Ω p = (cid:18) ( j − i )(1 − p ) + h ′ i − h ′ j j − i (cid:19) . The fact that the mapping does not preserve the total area does not spoil theargument since here we only deal with local MDP– p configurations. The areaconstraint is a global one which is implemented at the end of the calculationvia a Lagrange multiplier. Following the same steps as before we arrive at thevariational problem of extremizing L p = Z ∞ d x (cid:20)(cid:0) p + y ′ ( x ) (cid:1) ln − p − y ′ ( x )1 − p − y ′ ( x ) − ln 11 − p − y ′ ( x ) (cid:21) − λ Z ∞ y ( x ) d x. (10)Using the same Euler-Lagrange formalism, finally leads us to the equation of thelimit shape for p > y = − λ ln(1 − e − λx ) − px. (11)4 p = 3)( p = 2)( p = 1)( p = 0) λ ( p ) y λ ( p ) x Figure 2: Limit shapes for the minimal difference p partitions with p = 0 , ,
2, and3, where λ (0) = π/ √ λ (1) = π/ √ λ (2) = π/ √
15, and λ (3) = 0 . . . . .The Lagrange multiplier λ in (11) can be determined by using condition y ≥ R x m y ( x ) d x = 1, where x m is the solution of the equation y ( x m ) = 0. Writing exp( x m ) = y ∗ , it satisfies y ∗ − y ∗ − p = 1, and in terms of y ∗ one finds λ ≡ λ ( p ) = π − Li (1 /y ∗ ) − p y ∗ ) , (12)where Li ( z ) = P ∞ k =1 z k k − is the dilogarithm function. λ ( p ) is a constant whichdepends on the parameter p . For instance for p = 0 , λ (0) = π/ √ λ (1) = π/ √
12 and λ (2) = π/ √
15 in agreement with the earlier knownresults [5, 9]. Figure 2 shows the limit shapes for the MDP– p with p = 0 , , x ( y ) = λ − ln φ ( y ) satisfies φ ( y ) − e − λy φ ( y ) − p = 1 (13)Amazingly this equation appears in several apparently unrelated contexts.1. The generating function S ( t ) = 1 + P ∞ k =1 s k ( q ) t k for the number of con-nected clusters s k ( q ) of size k in a q-ary tree satisfies [15] S ( t ) − tS q ( t ) = 1 . (14)This establishes a formal link between two different combinatorial objects, on onehand the q-ary trees and on the other hand the MDP– p problem with p = 1 − q .5n graph theory [16], s k ( q ) is known as the generalized Catalan number, which isgiven by s k ( q ) = 1 k (cid:18) qkk − (cid:19) .
2. Consider the generating function of MDP– p problem Z ( x, z ) = X E X N ρ p ( E, N ) x E z N where ρ p ( E, N ) is the total number of MDP– p of E in N parts. In the limit E → ∞ the number of such partitions will be controlled by the singularities of Z ( x, z ) near x = 1. By setting x = e − β , one gets for β → Z ( x, z ) → Z ∞ ln y p (cid:16) z e − βǫ (cid:17) d ǫ, (15)where the function y p ( t ) is given by the solution of the equation y p ( t ) − t y − pp ( t ) = 1 . (16)3. In the physics literature (13) also arises in the context of exclusion statis-tics. Exclusion statistics [18, 19, 20, 21, 22]—a generalization of Bose and Fermistatistics—can be defined in the following thermodynamical sense. Let Z ( β, z )denote the grand partition function of a quantum gas of particles at inverse tem-perature β and fugacity z . Such a gas is said to obey exclusion statistics withparameter 0 ≤ p ≤
1, if Z ( β, z ) can be expressed as an integral representationln Z ( β, z ) = Z ∞ ˜ ρ ( ǫ ) ln y p (cid:16) z e − βǫ (cid:17) d ǫ, (17)where ˜ ρ ( ǫ ) denotes a single particle density of states and the function y p ( t ),which encodes fractional statistics is given by the solution of (13). Well knownmicroscopic quantum mechanical realizations of exclusion statistics are the Low-est Landau Level (LLL) anyon model [19] and the Calogero model [20], with ˜ ρ ( ǫ )being, respectively, the LLL density of states and the free one dimensional densityof states.The fact that the same equation appears in all three cases is obviously notfortuitous. The link between 2 and 3 follows from the fact that exclusion statis-tics has a combinatorial interpretation in terms of minimal difference partitionswhich generalizes the usual combinatorial interpretation of Bose statistics (respFermi) in terms of partitions without (with) restrictions. Let us briefly recall thiscorrespondence. Let n i be the number of columns of height h = i in a Young di-agram of a given partition of E , then E = P i n i ǫ i can be interpreted as the total6nergy of a non-interacting quantum gas of bosons where ǫ i = i for i = 1 , , . . . , ∞ represent equidistant single particle energy levels and n i = 0 , , , . . . , ∞ repre-sents the occupation number of the i -th level. If one now puts the restrictionthat h j > h j +1 (e.g. allowed partitions of 4 are: 4 and 3 + 1), then the restrictedpartition problem corresponds to a non-interacting quantum gas of fermions, forwhich n i = 0 ,
1. If in addition, one restricts the number of summands to be N ,then clearly N = P i n i represents the total number of particles. For example, if E = 4 and N = 2, the allowed partitions are 3 + 1 and 2 + 2 in the unrestrictedproblem, whereas the only allowed restricted partition is 3 + 1. The number ρ ( E, N ) of ways of partitioning E into N parts is simply the micro-canonicalpartition function of a gas of quantum particles with total energy E and totalnumber of particles N : ρ ( E, N ) = X { n i } δ E − ∞ X i =1 n i ǫ i ! δ N − ∞ X i =1 n i ! . (18)For both unrestricted and restricted partitions, one can readily check that thegrand partition function Z ( e − β , z ) = P N P E z N e − βE ρ ( E, N ), in the limit β →
0, is nothing but the one in (15), with p = 0 and p = 1 respectively.For a quantum gas obeying exclusion statistics with parameter p it is a priorinot obvious how to provide a combinatorial interpretation since the underlyingphysical models with exclusion statistics describe interacting models. However insome specific cases , such as the Calogero model one can show that the spectrumcan be parameterized as a free spectrum with some restrictions on the quantumnumbers which reflect the fact that the Pauli principle is replaced by a strongerexclusion principle [23, 24]. This exclusion is enforced at the level of the Youngdiagrams by the constraint h j − h j +1 ≥ p . The link between 1 and 3 expressesthis correspondence in terms of counting of states. Exclusion statistics can beimplemented by putting n particles in m sites on a one-dimensional lattice, underthe restriction that any two particles be at least p sites apart. For a periodiclattice, the number of ways of doing the above is [25] D m,n = m Γ (cid:0) m + (1 − p ) n (cid:1) Γ (cid:0) n + 1 (cid:1) Γ (cid:0) m + 1 − pn (cid:1) . One can check that D ,n = s n (1 − p ) which allows to interpret the generalizedCatalan numbers as quantum degeneracy factors.We acknowledge the support of the Indo-French Centre for the Promotion ofAdvanced Research (IFCPAR/CEFIPRA) under Project 3404-2.7 eferences [1] G.E. Andrews, The Theory of Partitions (Cambridge University Press, Cam-bridge, 1998).[2] H.N.Y. Temperley, Statistical mechanics and the partition of numbers, theform of crystal surfaces, Proc. Cambridge Philos. Soc. , 683 (1952).[3] A.M. Vershik and S.V. Kerov, Asymptotics of the Plancherel measure of thesymmetric group and the limiting shape of Young tableaux, Soviet Math.Dokl. , 527 (1977)[4] B. F . Logan and L. A. Shepp, A variational problem for random Youngtableaux, Adv. Math. , 206 (1977).[5] A.M. Vershik, Statistical mechanics of combinatorial partitions and theirlimit shapes, Funct. Anal. Appl. , 90 (1996).[6] G. Freiman, A.M. Vershik and Yu.V. Yakubovich, A local limit theorem forrandom strict partitions, Theory Probab. Appl , 453, (2000).[7] A.M. Vershik and Yu.V. Yakubovich, The limit shape and fluctuations of ran-dom partitions of naturals with fixed number of summands, Moscow Math.J. , 457 (2001).[8] A. Dembo, A.Vershik and O. Zeitouni, Large deviations for integer parti-tions, Markov Processes and Related Fields , 147 (2000).[9] D. Romik, Identities arising from limit shapes of constrained random parti-tions, preprint (2003).[10] L. Pastur, Lectures given at the Institut Henri Poincar´e trimester “Phenom-ena in large dimensions” (2006).[11] S. Shlosman, Geometric variational problems of statistical mechanics and ofcombinatorics, J. Math. Phys. , 1364 (2000).[12] P.L. Krapivsky, S. Redner and J. Tailleur, Dynamics of an unbounded inter-face between ordered phases, Phys. Rev. E , 026125 (2004).[13] R. Rajesh and D. Dhar, Convex lattice polygons of fixed area with perimeter-dependent weights, Phys. Rev. E , 016130 (2005).[14] P. Erd¨os and J. Lehner, The distribution of the number of summands in thepartitions of a positive integer, Duke Math. J. , 335 (1951).815] M.E. Fisher and J.W. Essam, Some cluster size and percolation problems,J. Math. Phys. , 609 (1961).[16] P. Hilton and J. Pedersen, Catalan numbers, their generalization and theiruses, Math. Intell. , 64 (1991).[17] A. Comtet, S.N. Majumdar and S. Ouvry, Integer partitions and exclusionstatistics, J. Phys. A: Math. Theor. (2007) 11255[18] F.D.M. Haldane, Fractional statistics in arbitrary dimensions: a generaliza-tion of the Pauli principle, Phys. Rev. Lett. , 937 (1991).[19] A. Dasni`eres de Veigy and S. Ouvry, Equation of state of an anyon gas ina strong magnetic field, Phys. Rev. Lett. , 600 (1994); One dimensionalstatistical mechanics for identical particles-the Calogero and anyon cases,Mod. Phys. Lett. B , 271 (1995).[20] S.B. Isakov, Fractional statistics in one dimension, Int. J. Mod. Phys. A, ,2563 (1994).[21] Y.S. Wu, Statistical distribution for generalized ideal gas of fractional statis-tics particles, Phys. Rev. Lett. , 922 (1994).[22] M.V.N. Murthy and R. Shankar, Thermodynamics of a one dimensional idealgas with fractional exclusion statistics, Phys. Rev. Lett. , 3331 (1994 );Haldane statistics and second virial coefficient ibid , 3629 (1994).[23] V. Pasquier, A lecture on the Calogero Sutherland models, in Lecture Notesin Physics (Integrable Models and Strings) , pp. 36-48 (Springer Berlin/Heidelberg 1994).[24] A.P. Polychronakos, Physics and mathematics of Calogero particles, J.Phys.A , 12793, (2006).[25] A.P. Polychronakos, Probabilities and path-integral realization of exclusionstatistics, Phys. Lett. B365