Quantum Mechanics of Plancherel Growth
Arghya Chattopadhyay, Suvankar Dutta, Debangshu Mukherjee, Neetu
QQuantum Mechanics of Plancherel Growth
Arghya Chattopadhyay a,b , Suvankar Dutta b , Debangshu Mukherjee b,c , Neetu b a Institute of Mathematical Sciences, Homi Bhaba National Institute (HBNI),IV Cross Road, Taramani, Chennai 600113, India b Department of Physics Indian Institute of Science Education and Research Bhopal,Bhopal bypass, Bhopal 462066, India c Chennai Mathematical Institute, SIPCOT IT Park,Siruseri 603103, India.
Abstract
Growth of Young diagrams, equipped with Plancherel measure, follows the automodel equation ofKerov. Using the technology of unitary matrix model we show that such growth process is exactlysame as the growth of gap-less phase of Gross-Witten and Wadia (GWW) model. Our analysisalso o ff ers an alternate proof of limit shape theorem of Vershik-Kerov and Logan-Shepp. We alsostudy fluctuations of random Young diagrams in this paper. We map Young diagrams in automodelclass to di ff erent shapes of two dimensional phase space droplets of underlying non-interactingfermions. Fluctuation of these Young diagrams correspond to small ripples on the boundaries ofsuch droplets. We quantise this classical system using Hamiltonian dynamics and show that thedi ff erent modes of these fluctuations satisfy U (1) Kac-Moody algebra. We further construct theHilbert space of this algebra and find a correspondence between the states in Hilbert space andautomodel diagrams. In particular the Kac-Moody primary corresponds to null Young diagram(no box) whereas automodel diagrams are mapped to descendants of Kac-Moody primary. Keywords:
Plancherel growth of Young diagrams, unitary matrix model, Kac-Moodyalgebra.
Contents1 Introduction 22 Plancherel Growth of Young diagrams 5
Email addresses: [email protected] (Arghya Chattopadhyay), [email protected] (Suvankar Dutta), [email protected] (Debangshu Mukherjee), [email protected] (Neetu) a r X i v : . [ h e p - t h ] N ov The Partition Function 8 k analysis of partition function 10 ff erent branches of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.1.1 Rectangular Young diagram - symmetric solution . . . . . . . . . . . . . 124.1.2 Asymmetric solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 A connection with automodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Young diagrams (or sometimes simply diagrams in this paper) play an important role in mathemat-ics as well as in physics. It provides a convenient diagrammatic way to describe the representationsof symmetric group and general linear groups. Various properties of these representations can beunderstood very easily using this diagrammatic technique. There are di ff erent notations in litera-ture to depict a Young diagram. In this paper we follow the “English notation”. In this notation,boxes are arranged in horizontal rows with the condition that number of boxes in a row is alwaysless than or equal to that in the row above. In general as one goes to higher and higher dimen-sional representations the number of boxes in Young diagram increases. It has been observed thatirrespective of underlying group or representation the growth of Young diagrams behaves veryinterestingly.Young diagrams can be classified in terms of total number of boxes. Let us consider Y k to bethe set of all Young diagrams with k boxes. Then all the diagrams in Y k + can be obtained byadding one box to each diagram in Y k in all possible allowed ways. See figure 1. Such process iscalled growth process of Young diagrams. One can construct all possible Young diagrams at anyarbitrary level starting from null diagram ( ∅ ), means no box .For a restricted growth process, one can assign a probability to every transition. Denoting a par-ticular Young diagram at level k by λ k we associate a transition probability P transition ( λ k , λ k + ) for atransition from λ k to λ k + P transition ( λ k , λ k + ) = k + λ k + dim λ k (1.1)if λ k + is obtained from λ k by adding one box, otherwise P transition ( λ k , λ k + ) =
0. A growth process,following the above probability measure, is called
Plancherel growth process (see [1, 2] for a2 igure 1:
Growth of Young diagrams. comprehensive review). Note that the probability to get a diagram at level k + k does not depend on the history of transition from level k − k . Thus, the growth processis Markovian.It was shown by Vershik and Kerov [3] and independently by Logan and Shepp [4] that Youngdiagrams following Plancherel growth process converge to a universal diagram in the large k limitwhen normalised (scaled) appropriately such that the area of the diagram is unity. The boundaryof such normalised diagram becomes smooth under scaling. A universal Young diagram meansthe boundary curve takes a particular form, which is called limit shape . The limit shape followsthe famous arcsin law [3, 5].In the continuum (large k ) limit Kerov introduced a di ff erential model to capture the growth ofYoung diagrams [1, 5]. He associated a ‘time’ parameter with continuous diagrams to study theevolution of those diagrams with respect to that. It turns out that Young diagrams equipped withPlancherel measure follow a first order partial di ff erential equation. The model was named as automodel [1, 5]. The class of diagrams satisfying such growth or evolution equation is calledautomodel class. The limit shape is a unique solution of the automodel equation in far future with3 as initial condition in far past.The first goal of this paper is to write down a matrix model which captures the growth of Youngdiagrams in automodel class. In order to achieve that we define a Young lattice Y = ∞ (cid:91) k = Y k . (1.2)All the members of Y k have same number of boxes but di ff erent shapes. Y k can be thought of as anensemble of Young diagrams with the same macroscopic variable k . Therefore the Young latticecan be thought of as a grand canonical ensemble and one can write down a partition function forthe entire lattice Q Y = ∞ (cid:88) k = z k Z Y k (1.3)where z is called fugacity ( z >
0) and Z Y k is the canonical partition function for Y k , given by Z Y k = (cid:88) λ k P ( λ k ) δ ( k − | λ k | ) . (1.4) P ( λ k ) is any measure associated with the Young diagram λ k . To capture the Plancherel growthprocess it is natural to take P ( λ k ) to be Plancherel measure (2.4). Further discussions on Plancherelmeasure is deferred till section 2.It turns out that the above partition function is solvable in large k limit. The large k limit is similarto classical limit in physics. In this limit the partition function is dominated by a single Youngdiagram. Surprisingly, we find that this dominant diagram has the boundary, exactly same as thatof automodel diagram obtained in [3–5]. We also observe that the dominant Young diagramsobtained from (1.4) satisfy similar automodel equation, obtained by Kerov, with time parameterrelated to an inbuilt parameter of the matrix model. Therefore, the matrix model (1.3) captures thegrowth of Young diagrams in automodel class for P ( λ k ) equal to Plancherel measure.The most fascinating observation in this paper is the dynamics of large k fluctuations of Youngdiagrams in automodel class. Large k fluctuations of limit shape diagrams is an interesting sub-ject to study in mathematics [6–17]. However, in this paper we study such fluctuations from aphase space point of view. First, we map di ff erent automodel diagrams to di ff erent 2 dimensionaldroplets of an underlying free fermi system. The mapping between automodel diagrams and freefermi droplets follows from the equivalence between the partition functions of automodel growth(1.3) and Gross-Witten and Wadia (GWW) model, which is a unitary matrix model. GWW modelhas two phases in the large N limit (where N being the rank of unitary matrices) and the auto-model class corresponds to gap-less phase of GWW. Thus automodel diagrams can also have anequivalent description in terms of eigenvalue distribution on a unit circle. Since the eigenvaluesof unitary matrices in GWW model (or in any generic unitary matrix model) behave like positionof free fermions [18], it was shown in [19] and other follow up papers [20, 21] that automodelclass can be described in terms of phase space droplets of underlying free fermi theory. These4roplets are similar to Thomas-Fermi droplets at zero temperature. Di ff erent automodel diagramsare mapped to di ff erent shapes of free fermi droplets. Large k fluctuations of automodel diagramscorrespond to small ripples at the boundary of the droplets. To study the dynamics of such fluctu-ations we construct a single particle Hamiltonian from the shape of the droplets [21]. We employHamiltonian dynamics to study the evolution of classical boundary of a droplet and then quantisethe system. It turns out that di ff erent modes of fluctuations satisfy an abelian Kac-Moody algebra.We also construct the Hilbert space of this algebra and find a one to one correspondence betweenthe states in Hilbert space and automodel diagrams. In particular the Kac-Moody primary corre-sponds to null Young diagram (no box) whereas automodel diagrams are mapped to descendantsof Kac-Moody primary. We also map the Gaussian fluctuations of Young diagrams to descendantstates in the Hilbert space.
2. Plancherel Growth of Young diagrams
To make the growth process meaningful it is customary to assign a probability for each diagram atlevel k in the Young lattice Y . There is a natural way to assign probability to di ff erent diagrams.We count the total number of inequivalent paths one can follow to come to a particular diagram atlevel k starting from ∅ . See figure 1. It turns out that the Plancherel measure is proportional to thesquare of that number. The proportionality constant is fixed by the normalization condition. Tocalculate the number of paths heading to a diagram λ k we look at growth of Young tableaux ratherthan Young diagrams. Starting from ∅ we keep on adding one box at each level with increasingnumber. Therefore the readers can easily convince themselves that at each level k we have di ff erentYoung tableaux and a particular tableau can be reached from ∅ by a unique path only. Thus thenumber of paths available to reach a particular Young diagram λ k is equal to the number of standardYoung tableaux f λ k of that given shape. It is well known that[22, 23], (cid:88) λ k ∈Y k ( f λ k ) = k ! . (2.1)Hence we get the Plancherel measure P ( λ k ) for a diagram λ k P ( λ k ) = f λ k k ! . (2.2)We use this probability to write the partition function for the growth process. The number f λ k isequal to the dimension of the representation λ k , i.e. f λ k = dim λ k . (2.3)and thus we have, P ( λ k ) = (dim λ k ) k ! . (2.4)When a Young lattice is equipped with Plancherel measure (2.4), the transition probability betweentwo diagrams λ k and λ k + in that lattice is also fixed and is given by (1.1) [5, 24, 25]. Therefore5ither of the probabilities (1.1 or 2.4) can be used to study the growth process. It was observed in[3, 4] that a Young lattice equipped with Plancherel measure terminates to a universal diagram inthe limit k → ∞ when the diagrams are scaled properly. Although we are using the “English” notation for Young diagrams, but the limit shape of Youngdiagrams takes a simple form in rotated French notation. A typical shape of Young diagram inFrench notation is given in fig. 2. The centres of boxes are marked with ( X , Y ) coordinates. Thefunction X ( Y ) specifies a particular shape of Young diagram in this notation. However, it is more X Y
Figure 2:
Typical structure of a Young diagram in French notation. convenient to rotate this diagram anti-clock wise by π/ u =
12 ( Y − X ) v =
12 ( Y + X ) . (2.5)A Young diagram in this notation is depicted in figure 3. For finite number of boxes the function v ( u ) is rough and zig-zag i.e. v (cid:48) ( u ) = ±
1. As the number of boxes goes very large we define arescaled function ˆ v k ( u ) = v ( u √ k ) √ k (2.7) There is another advantage to draw the Young diagrams in rotated French notation. A transition from λ k to λ k + occurs when one keeps a box at any of the minima of rotated diagram. Putting a box at di ff erent minima correspondsto di ff erent diagrams at k + µ a where a is the position ofa minimum. To find µ a we define two polynomials P ( x ) = (cid:81) na = ( x − x a ) and Q ( x ) = (cid:81) n − a = ( x − y a ), where x , · · · , x n are positions of consecutive minima and y , · · · , y n − are consecutive maxima. The transition probability from λ k to λ k + by adding a box at a th minima is given by decomposing the quotient into partial fraction n (cid:88) a = µ a x − x a = Q ( x ) P ( x ) . (2.6) Y Figure 3:
Typical structure of a 45 ◦ anti-clockwise rotated Young diagram. such that the area under the curve is finite and the boundary curve becomes smooth. It was ob-served by [3, 4] that when the growth process follows Plancherel transition probability (1.1) theasymptotic shape of rescaled Young diagrams converges uniformly to a unique curve given bylim k →∞ ˆ v k ( u ) ≡ Ω ( u ) = π ( u sin − u + √ − u ) if | u | ≤ | u | if | u | > . (2.8)In the continuum limit Kerov introduced [5] charge of a diagram, denoted by σ ( u ) and is given by(we are using the notation that ˆ v ( u ) = ˆ v k ( u ) in the large k limit) σ ( u ) =
12 (ˆ v ( u ) − | u | ) . (2.9)Therefore, σ (cid:48) ( u ) = (cid:26) + ˆ v (cid:48) ( u )2 for u < − + ˆ v (cid:48) ( u )2 for u > . (2.10)One can define moments of a diagram p n = − n (cid:90) u n − d σ ( u ) (2.11)such that area of a diagram (area covered under the curve ˆ v ( h )) is given by A = ( p − p ) /
2. It isconvenient to consider a moment generating function S ( x ) = ∞ (cid:88) n = p n n x − n = (cid:90) d σ ( u ) u − x . (2.12)The moment generating function as well as the sequence of moments determine the charge andhence the diagram (ˆ v ( u )) completely. The moment generating function plays an important role inour large k analysis of partition function. 7n [5] Kerov introduced a dynamical model for the growth of Young diagrams. For every contin-uous Young diagram characterized by the function ˆ v ( u ), one can define the function v ( u , t ), calledthe automodel tableaux which depends on two variables u and t asˆ v ( u , t ) = √ t ˆ v ( u / √ t ) for t > . (2.13)Kerov showed that the Young diagrams, following Plancherel growth, belong to automodel classand satisfy the equation ∂ t ˆ v ( u , t ) = t (ˆ v ( u , t ) − u ∂ u ˆ v ( u , t )) . (2.14)In terms of charges the automodel equation is given by, ∂ t σ (cid:48) ( u , t ) + u t σ (cid:48)(cid:48) ( u , t ) = . (2.15)With this preliminary discussion on Plancherel growth process of Young diagrams and automodelclass we are in a position to write down a partition function for the Young lattice.
3. The Partition Function
The grand canonical partition function for Y is given by Q Y = ∞ (cid:88) k = z k (cid:88) λ k (dim λ k ) k ! δ ( k − | λ k | ) , z > . (3.1)This partition function is related to Poissonised Plancherel measure [26]. The above ensemblesometimes is known as
Meixner ensemble in literature [27, 28].Our goal is to solve this matrix model in the large k limit. In that limit the partition function isdominated by a particular Young diagram and it turns out that the shape of this dominant large k Young diagram falls into the automodel class of Kerov [5] and for a particular value of parameter itbecomes limit shape [3–5]. Before we present the calculation to obtain the universal diagram, weshow that the partition function (3.1) is remarkably equivalent to the partition function of Gross-Witten-Wadia model and its cousins. This was also observed in [19].
The Gross-Witten-Wadia model is a well studied unitary matrix model in physics. The partitionfunction for this model is defined over an ensemble of N × N unitary matrices with a real potentialTr U + Tr U † , where the trace has taken over fundamental representation. The partition function ofGWW model is given by Z GWW = (cid:90) [ dU ] e N λ (Tr U + Tr U † ) , λ ≥ . (3.2)8ross and Witten [29] (and independently by Wadia [30]) studied this matrix model in the contextof lattice QCD and observed that the system undergoes a third order phase transition at λ = ff erent phases of this model are characterised by the topology of distribution of eigenvalues ofunitary matrix U on a unit circle. The strong coupling phase ( λ >
2) corresponds to a gap-lessdistribution of eigenvalues whereas weak coupling phase ( λ <
2) shows a finite gap in eigenvaluedistribution.A close cousin of GWW model [31, 32] is Z c = (cid:90) [ dU ] e a Tr U Tr U † . (3.3)The phase structure and eigenvalue distributions of this model are similar to those of GWW upto a redefinition of parameters : a (cid:104) Tr U (cid:105) = N /λ [20]. Expanding the exponential in (3.3) we get Z c = (cid:90) [ dU ] ∞ (cid:88) k = a k k ! (Tr U ) k (Tr U † ) k . (3.4)Using Frobenius formula for the characters of symmetric group we can write(Tr U ) k = (cid:88) R χ R (1 k )Tr R U , and (Tr U † ) k = (cid:88) R χ R (1 k )Tr R U † (3.5)where (cid:80) R is sum over representations of U ( N ) (or S U ( N )) and χ R (1 k ) is the character of conjugacyclass (1 k ) of symmetric group S k in representation R . Finally using the normalization conditionfor the characters of unitary group (cid:90) [ dU ]Tr R U Tr R (cid:48) U † = δ RR (cid:48) (3.6)we arrive at the final expression for Z c written in terms of sum over representations of U ( N )[19] Z c = ∞ (cid:88) k = a k (cid:88) R ( χ R (1 k )) k ! . (3.7)It is well know that character of conjugacy class (1 k ) of symmetric group S k in representation R isequal to the dimension of the representations [22] χ R (1 k ) = dim R . (3.8)Representations of U ( N ) can be expressed in terms of Young diagrams. Since χ R (1 k ) is non-zeroonly when total number of boxes in the Young diagram is k we have Z c = ∞ (cid:88) k = a k (cid:88) λ k (dim λ k ) k ! δ ( k − | λ k | ) . (3.9)9hus we see that the partition function for (cousin of) GWW model is same as that of Young latticewith a playing the role of fugacity (see [19] for details). However, there is a small di ff erencebetween these two partition functions. The sum in (3.9) runs over representations of unitary groupwhere as in (3.1) the sum runs over representations of symmetric group. This di ff erence makesthese two systems behave di ff erently for certain range of parameter (fugacity). We shall get backto this issue at appropriate place.
4. Large k analysis of partition function The large k analysis of partition function (3.1) was explicitly done in [19]. We briefly review theprocedure for the readers, not familiar with matrix model techniques (for a more comprehensivetreatment of matrix models, see [33–35]). To analyse the partition function (3.1) we denote a validYoung diagram of symmetric group S k by a set of numbers { n i } Li = where n i denotes the numberof boxes in i th row. L is an arbitrary positive integer greater than or equal to the height of the first Figure 4:
A generic Young diagram in English notation. Here L is an arbitrary positive integer. The numberof boxes in the first column is less than or equal to L . In general, ∃ a number 0 < M ≤ L such that n i = i = M + , · · · , L . column. See figure 4. The dimension of a representation λ k of S k is given by [22]dim λ k = k ! h ! h ! · · · h L ! L (cid:89) i = i < j ( h i − h j ) (4.1)where, h i = n i + L − i (4.2)is the hook lengths of the first box in i th row. 10e consider the large L limit of the partition function (3.1). In this limit the hook numbers h i ∼ L (4.2). Therefore we define the following continuous functions to describe Young diagrams at large L n ( x ) = n i L , h ( x ) = h i L , where x = iL , x ∈ [0 , . (4.3)Functions n ( x ) or h ( x ) captures the distribution of boxes in a large k Young diagram. The relationbetween n ( x ) and h ( x ) follows from equation (4.2) and is given by h ( x ) = n ( x ) + − x . (4.4)The number of boxes in a Young diagram in the large L limit is given by k = L (cid:88) i = n i −→ L (cid:34)(cid:90) dx ( h ( x ) + − x ) (cid:35) = L (cid:34)(cid:90) dxh ( x ) − (cid:35) = L k (cid:48) (4.5)where k (cid:48) = (cid:90) dxh ( x ) −
12 (4.6)is the renormalised box number and is a O (1) quantity. Thus we see that the number of boxes in aYoung diagram in the large L limit goes as ∼ O ( L ) and hence L ∼ O ( √ k ). The partition function(3.1) in L → ∞ limit is given by, Q Y = (cid:90) [ Dh ( x )] e − L S e ff [ h ( x )] (4.7)where − S e ff [ h ( x )] = (cid:90) dx − (cid:90) dy ln | h ( x ) − h ( y ) | − (cid:90) dxh ( x ) ln h ( x ) + k (cid:48) ln( zk (cid:48) ) + k (cid:48) + . (4.8)In the large L limit the dominant contribution to the partition function comes from the extrema of S e ff [ h ( x )]. Varying S e ff [ h ( x )] with respect to h ( x ) we get the saddle point equation − (cid:90) u ( h (cid:48) ) dh (cid:48) h − h (cid:48) = ln (cid:32) h ξ (cid:33) , where ξ = zk (cid:48) (4.9)where, Young diagram density u ( h ) is given by u ( h ) = − ∂ x ∂ h . (4.10)Monotonicity of h ( x ) implies 0 ≤ u ( h ) ≤ u ( h ) also satisfies two conditions (cid:90) dhu ( h ) = , and (cid:90) hu ( h ) dh = k (cid:48) + . (4.11)Our goal is to solve this saddle point equation (4.9) to find Young diagram density such that itsatisfies the constraints (4.11). 11 .1. Di ff erent branches of solutions All possible large L solutions of (4.9) were thoroughly discussed in [19] and it was observed that(4.9) admits two possible solutions. However, here we look at the problem more carefully keepingthe symmetry of the growth process in mind. From the Plancherel measure (2.4) we see that at anylevel k , two Young diagrams related to each other by transposition, have same probability P ( λ k ).Therefore the large L solution of (4.9) must be invariant under transposition. Young diagrams,symmetric under transposition, are called rectangular diagrams [2, 5]. Following [19], we can take the following ansatz for u ( h ) to get a rectangular Young diagram u ( h ) = (cid:26) h ∈ [0 , p )˜ u ( h ) h ∈ ( p , q ] . (4.12)To solve the saddle point equation we define a resolvent H ( h ) = (cid:90) h U h L u ( h (cid:48) ) dh (cid:48) h − h (cid:48) . (4.13)After a little algebra, we find that the resolvent H ( h ) is given by [19] H ( z ) = ln h (cid:16) h − − (cid:112) ( h − − ξ (cid:17) ξ . (4.14)The resolvent is same as the moment generating function for the rectangular diagrams defined in(2.12) [5]. The resolvent has a branch cut in the complex h plane. Young diagram density is givenby the discontinuity of H ( h ) about the branch cut˜ u ( h ) = π cos − (cid:34) h − ξ (cid:35) , for p ≤ h ≤ q . (4.15)The supports p and q are given by, p = − ξ, q = + ξ. (4.16)This particular class of solution exists subject to the following condition k (cid:48) = ξ . (4.17)Since p ≥
0, this solution is valid for 0 ≤ ξ ≤ /
2. From the definition of ξ ( ξ = zk (cid:48) ) we also seethat this solution exists for either ξ = k (cid:48) = z = . (4.18)12 igure 5: A Young diagram in English notation for automodel class (0 < ξ < / The case ξ = k (cid:48) = z = ξ which is similar to the length of the firstrow. Also the function ˜ u ( h ) is symmetric about body diagonal. A typical Young diagram for thisclass has been depicted in figure 5. In this phase the renormalised number of boxes (i.e. k (cid:48) ) ina Young diagram grows from k (cid:48) = k (cid:48) = / ξ changes from 0 to 1 /
2. For any value of ξ between 0 and 1 / ξ = / u ( h ) = π cos − ( h − . (4.19)This terminal distribution is same as the universal curve or the limit shape obtained by [3, 4].Hence we see that the limit shape Young diagram corresponds to GWW transition point in matrixmodel side.We calculate Plancherel measure (2.4) for this dominant diagram. Following [19], the Plancherelmeasure in large k limit is given by1 L ln P λ k = (cid:90) q dhu ( h ) − (cid:90) q dh (cid:48) u ( h (cid:48) ) ln | h − h (cid:48) | − (cid:90) q u ( h ) h ln h dh + k (cid:48) + + k (cid:48) ln k (cid:48) . (4.20)Evaluating the right hand side for symmetric solutions (4.12) and (4.15) we get P λ k = + O (cid:32) L (cid:33) . (4.21)Thus we see that in the large k (or large L ) limit the symmetric solution (4.12, 4.15) is the maximumprobable solution. Probability of having other diagrams is suppressed by powers of L . This givesan alternate proof of limit shape theorem of Vershik-Kerov and Logan-Shepp result [3, 4].13 .1.2. Asymmetric solutions In large k limit the matrix model (3.3) renders another class of solution [19]. This solution is givenby u ( h ) = π cos − h + ξ − / (cid:112) ξ h , for p ≤ h ≤ p = √ p = (cid:112) ξ − √ , and √ q = (cid:112) ξ + √ z (or a ) is given by z = ξ ξ − . (4.24)The solution is valid for ξ > /
2. The Young diagrams for this distribution is not symmetric undertransposition.This is a valid solution in the context of GWW model. In case of GWW model the sum in equation(3.9) was over the representations of unitary group U ( N ) for which the maximum number ofboxes in the first column of a Young diagram is N . Therefore, the symmetric representation failsto be a valid solution of GWW when the first column saturates this bound. As a result, GWWmodel undergoes a third order phase transition at ξ = /
2, known as Gross-Witten-Wadia phasetransition.However we are dealing with partition function (1.3) where the sum is running over the represen-tations of symmetric group. In this case there is no such restriction on L (number of boxes in thefirst column). Thus we do not see any such phase transition here. To make a precise connection between automodel class and our solution, we need to set up adictionary between the variables defined in ( u , v ) plane and ( h , x ) plane. The relation between French notation and
English notation is Y = n and X = x . We use the following transformationbetween ( n , x ) and ( u , v ) so that u = v = n = , x = u = n − xv = n + x . (4.25)Using this mapping one can show that the Young diagram distribution function (4.10) is related to v (cid:48) ( u ) in the following way u ( h ) = − v (cid:48) ( u ) with u = h − . (4.26)14ne can also check that the terminal diagram (4.19) is exactly same as the limit shape defined in(2.8). Thus we see that the Young diagram density u ( h ) is related to charge σ ( u ) defined in (2.9)by u ( h ) = − σ (cid:48) ( u ). The resolvent (4.14) for this symmetric solution same as the moment generatingfunction for charges (2.12).We also observe that the symmetric distributions (4.12) for 0 < ξ < / ∂ ξ u ( h , ξ ) + h − ξ ∂ h u ( h , ξ ) = . (4.27)Since for this branch we have ξ = k (cid:48) , the above equation can be written as, ∂ k (cid:48) u ( h , k (cid:48) ) + h − k (cid:48) ∂ h u ( h , k (cid:48) ) = . (4.28)Thus we see that the Young diagram density satisfies the automodel equation (2.15) with k (cid:48) playingthe role of automodel time t . This is natural to expect that the renormalised box number k (cid:48) playingthe role of growth parameter t in Kerov’s paper [5]. Hence we conclude that the partition function(3.1), in the limit of large box number, is dominated by Young diagram belonging to the automodelclass of Kerov.
5. Fluctuations of automodel diagrams, Kac-Moody algebra and state-diagram correspon-dence
Gapless phase of GWW model is a classical solution of the model. Study of large N fluctuationsor quantum fluctuations of the classical solution is always interesting on its own. Since automodeldiagrams are mapped to gapless phase of GWW model, large N fluctuations of classical solution,therefore, correspond to large k fluctuations of automodel diagrams. Such fluctuation of Youngdiagrams have been under investigation primarily in the mathematics literature [6–17].Kerov studied the Gaussian fluctuations around the limit shape of Young diagrams (denoted by Ω , as defined in (2.8)) endowed with Plancherel measure in [6]. In [8], Ivanov and Olshankireconstructed a proof of Kerov’s result on fluctuations around the limit shape from his unpublishedwork notes, 1999. A rescaled Young diagram defined in (2.7) in the k → ∞ limit takes the form oflimit shape Ω . However there can be large k corrections to this result and we call such correctionsas fluctuations of limit shape diagram. The central result pertains to large k corrections to the limitshape which can be stated as lim k →∞ ˆ ν k ( u ) ∼ Ω ( u ) + √ k ∆ ( u ) (5.1)The sub-leading piece ∆ ( u ) is a Gaussian process defined for | u | ≤
2. More precisely, ∆ ( u ) is arandom trigonometric series given by ∆ ( u ) = ∆ (2 cos θ ) = π ∞ (cid:88) n = α n √ n sin( n θ ) ; u = θ (5.2)15here α n are independent Gaussian random variables with mean 0 and variance 1. Further inves-tigations has been done towards understanding the central limit theorem for Gaussian fluctuationsaround the limit shape [10, 11]. Fluctuations of random Gaussian and Wishart matrices have beenrelated to the notion of free probability and free cumulants in earlier works [15–17].Here we take a di ff erent route to study the dynamics of such fluctuations. Our approach is quitegeneric and has importance and implications more on matrix model and gauge theory side. Uni-tary matrix models in large N ( N being the rank of matrices) limit renders di ff erent solutions orphases [29, 30, 36–39]. Such phases are also corroborated by numerical studies of lattice gaugetheories in the limit of large number of colours [40–42]. These classical solutions (large N phasesof unitary matrix model) can be described in terms of phase space droplets in two dimensions[19]. These droplets are similar to Thomas-Fermi distributions at zero temperature. Therefore,quantum fluctuations ( O (1 / N )) of these classical solutions can be thought of as small ripples onthe boundary of these droplets. In this section we study the dynamics of such ripples and showthat di ff erent modes of fluctuations satisfy an abelian Kac-Moody algebra. Automodel diagramsare captured by unitary matrix model (GWW, in particular) and hence can be represented as freefermi droplets. As a result, large N (large √ k ) fluctuations of automodel diagrams satisfy the samealgebra. We further construct the Hilbert space of this algebra and find a one to one correspon-dence between the states in Hilbert space and automodel diagrams. In particular the Kac-Moodyprimary corresponds to null Young diagram (no box) whereas automodel diagrams are mapped todescendants of Kac-Moody primary.Droplet description for classical phases is based on the fact that the partition function of unitarymatrix model can also be equivalently written in terms of representations of unitary group. Forexample GWW model partition function has two descriptions : one in eigenvalue basis (3.2) or(3.3) and the second one in Young diagram basis (3.9). Hence di ff erent large N phases can bedescribed either in terms of eigenvalue distributions or Young diagram distributions. There is aone-to-one correspondence between these two descriptions. It is well known that eigenvalues ofunitary matrices in a unitary matrix model behave like position of free fermions [18]. On theother hand hook lengths in Young diagram representation are like momenta of these fermions [19,43]. A relation between these two pictures o ff ers droplet or phase space description for di ff erentclassical phases [20, 21]. The phase space is two dimensional and spanned by hook numbers and eigenvalues - ( h , θ ).The gapless phase of GWW model is characterised by the eigenvalue density [19, 31, 32], ρ gapless ( θ ) = π (1 + ξ cos θ ) for 0 ≤ ξ < / . (5.4) The gapped phase occurs for ξ > /
2. The eigenvalue density for gapped phase is given by ρ gapped ( θ ) = ξ (cid:115) ξ − sin θ θ θ ≤ ξ . (5.3)The asymmetric solution (4.22) is mapped to one-gap phase (5.3). ff erent large N solutions. We define a phase space distri-bution function ω ( h , θ ) in ( h , θ ) plane ω ( h , θ ) = Θ (cid:32) ( h − h − ( θ ))( h + ( θ ) − h )2 (cid:33) (5.5)such that ω ( h , θ ) = h − ( θ ) < h < h + ( θ ) and zero otherwise. The eigenvalue and Young diagramdistributions are obtained from ω ( h , θ ) by integrating over h and θ respectively ρ ( θ ) = π (cid:90) ω ( h , θ ) dh and u ( h ) = π (cid:90) ω ( h , θ ) d θ. (5.6)Thus, the two dimensional distribution function ω ( h , θ ) captures information about both the distri-butions. Since eigenvalue and Young diagram densities are normalised, the distribution function ω ( h , θ ) also satisfies the normalisation condition12 π (cid:90) ω ( h , θ ) d θ dh = . (5.7)We also define a quantity S ( θ ), called momentum densityS ( θ ) = πρ ( θ ) (cid:90) h ω ( h , θ ) dh = h + ( θ ) + h − ( θ )2 . (5.8)Using this definition and eigenvalue distribution defined in (5.6) we have h ± ( θ ) = S ( θ ) ± πρ ( θ ) . (5.9)Since ω ( h , θ ) = ω ( h , θ ), it is actually the shape (i.e. boundary) of this distribution functionwhich captures information about di ff erent large N phases of the theory. To find the shape of thedistribution we need to find h ± ( θ ) for di ff erent phase of the theory. For a generic class of matrixmodel, it was observed in [20, 21] that h ± ( θ ) is given by h ± ( θ ) = W ( θ ) ± πρ ( θ ) (5.10)where the function W ( θ ) depends on the matrix model under consideration. For GWW matrixmodel, which is our current interest, W ( θ ) is given by [21] W ( θ ) = + ξ cos θ. (5.11)Hence for gap-less phase we have, h + ( θ ) = + ξ cos θ, h − ( θ ) = . (5.12)17 a) ξ = ∅ . (b) < ξ < / (c) ξ = / Figure 6:
Droplets for automodel diagrams with h as the radial coordinate and θ being the angular one. The distributions for di ff erent values of ξ are plotted in figure 6. ξ = ξ the shape of the distribution starts deforming. ξ = / π andindependent of ξ . Therefore, evolution (with respect to ξ ) of automodel diagrams (4.27) maps todeformation of these droplets keeping the area constant. One can think of these distributions asincompressible fluid droplets. One important thing to notice here is that the origin ( h =
0) remainsinside the droplet for 0 ≤ ξ < /
2, i.e. the distribution is single valued . Similar droplet pictureexists for one-gap phase also. The quantum fluctuations ( N corrections) of classical solution ( √ k fluctuations of automodel Youngdiagrams) corresponds to small ripples on the boundary of classical droplets like in figure 7. Tostudy the dynamics of these fluctuations we need to know how the boundary of these droplets For one-gap phase ( ξ > /
2) the distribution is determined by [19] h ± ( θ ) = + ξ cos θ ± πρ gapped ( θ ) . (5.13)It was shown that for this phase the origin remains outside the droplet. Thus if we take the origin out from the 2 d plane, ξ < / ξ > / ff erent. However, this is not the focus of this current work. igure 7: Large k fluctuations about classical geometry for 0 < ξ < / evolve with time. To incorporate dynamics into the picture, we first obtain the single particleHamiltonian for the underlying fermi system. The distribution functions ω ( h , θ ) for di ff erent clas-sical phases are similar to Thomas-Fermi (TF) distribution at zero temperature. Thomas-Fermidistribution at zero temperature is given by ∆ ( p , q ) = Θ ( µ − h ( p , q )) (5.14)where µ is chemical potential and h ( p , q ) is single particle Hamiltonian density. Comparingour phase space distribution (5.5) with TF distribution we find the Hamiltonian density is givenby h ( h , θ ) = h − S ( θ ) h + g ( θ )2 + µ, where g ( θ ) = h + ( θ ) h − ( θ ) . (5.15)Total Hamiltonian can be obtained by integrating h ( h , θ ) over the phase space H h = π (cid:126) (cid:90) d θ (cid:90) dh ω ( h , θ ) h ( h , θ ) . (5.17)We have taken into account the fact that one state occupies a phase space area of 2 π (cid:126) in semi-classical approximation. We also need to modify the normalisation of phase space density π (cid:126) (cid:90) dhd θω ( h , θ ) = N , with (cid:126) N = N is total number of states available inside a droplet ( N ∼ O ( √ k )). The classical limitcorresponds to (cid:126) → , L → ∞ with (cid:126) N = One can show that [21] integrating over h , the total Hamiltonian (without the (cid:126) factor) is same as the collectivefield theory Hamiltonian of Jevicki and Sakita [44] H h = (cid:90) d θ (cid:32) S ρ + π ρ + V e f f ( θ ) ρ (cid:33) + µ (5.16)with an e ff ective potential. For automodel solution number of boxes in the first column is always less than or equal to N , hence we take L = N . h = S (cid:48) ( θ ) h − g (cid:48) ( θ )2 , ˙ θ = h − S ( θ ) . (5.19)These are the set of equations for a particle moving on circle under the influence of an e ff ectivepotential. The above set of equations o ff ers the following solutions for gapless phase ( g ( θ ) = θ ( t ) = − (2 ξ +
1) tan (cid:16) (cid:112) − ξ t (cid:17)(cid:112) − ξ , h ( t ) = − ξ − ξ cos (cid:16) (cid:112) − ξ t (cid:17) . (5.20)The solutions are plotted in figure 8. For ξ = θ ( t )and h ( t ) are constant. As we increase ξ the particle starts spending more time at θ ( t ) = ± π :momentum is minimum when the particle reaches at ± π . At ξ = /
2, we get an instanton likesolution. Eliminating t from the above solutions one can find that the phase space trajectory for (a) ξ = (b) < ξ < / (c) ξ = / Figure 8:
Single particle trajectories for automodel class for di ff erent values of ξ . the particle. The trajectory is given by h ( t ) = + ξ cos θ ( t ) (5.21)20hich is the boundary of the classical droplet. Thus we see that shape of large N droplets can bemapped to phase space trajectories of a classical particle moving on a circle under the influence ofan e ff ective potential. We use the set of Hamilton’s equations (5.19) to study the evolution of theboundary of classical droplets [45, 46].Since gapless distributions are single valued ( h − ( θ ) = h = h + ( θ )and the boundary evolution is governed by ˙ h ( θ ) = h ( θ ) h (cid:48) ( θ )2 . (5.22)We would like to introduce a Poisson bracket between h ( θ ) and h ( θ (cid:48) ) such that the boundary evo-lution equation can be written as, ˙ h ( θ ) = { h ( θ ) , H h } (5.23)where H h is the total Hamiltonian (5.17). Integrating over h , H h can be written as, H h = − π (cid:126) (cid:90) d θ (cid:48) (cid:32) h ( θ (cid:48) )12 (cid:33) . (5.24)Defining the following Poisson bracket { h ( θ ) , h ( θ (cid:48) ) } = π (cid:126) δ (cid:48) ( θ − θ (cid:48) ) , (5.25)one can check that the equation (5.23) boils down to (5.22).To quantise the above classical system we promote the Poisson bracket (5.25) to commutationrelation [ h ( θ ) , h ( θ (cid:48) )] = π i (cid:126) δ (cid:48) ( θ − θ (cid:48) ) . (5.26)We decompose the fluctuations of h ( θ ) in Fourier modes h ( θ ) = ∞ (cid:88) n = −∞ f n e in θ . (5.27)The reality of h ( θ ) leads to the condition f † n = f − n . The commutation relation (5.26) in h ( θ ) impliesthat the di ff erent modes of fluctuations satisfy the following commutation relations[ f m , f n ] = − m (cid:126) δ m + n , . (5.28)This is a U (1) Kac-Moody algebra . Demanding the fluctuations to be area preserving leads to theconstraint f = . (5.29)Non-zero modes do not cost any change in area of the droplets. Redefining f n f n = (cid:126) √ n a † n for n ≥ , (5.30) Since h − =
0, we denote boundary of a droplet by h ( θ ), instead of h + ( θ ).
21t follows from (5.28) that [ a m , a † n ] = δ m , n for m , n ≥ Heisenberg algebra .The representation of (5.31) defines the Hilbert space of the system. We define a ground state | (cid:11) as a n | (cid:11) = , ∀ n ≥ . (5.32)A generic excited state can be written as, | (cid:126) q (cid:11) = (cid:89) n ( a † n ) q n | (cid:11) ∀ q n ∈ Z ≥ . (5.33) There exists a one to one correspondence between states in the Hilbert space and Young diagramsfor the representations of permutation groups via phase space distribution . In order to find thesame we first define a bilinear of h ( θ ), called “Sugawara stress tensor” T ( θ ) = h ( θ )2 (cid:126) . (5.34)The commutation relation (5.26) implies that the stress tensor satisfies the following commutationrelation [ T ( θ ) , T ( θ (cid:48) )] = π i (cid:0) T ( θ ) + T ( θ (cid:48) ) (cid:1) δ (cid:48) ( θ − θ (cid:48) ) (5.35)Decomposing T ( θ ) in Fourier modes T ( θ ) = n =+ ∞ (cid:88) n = −∞ L n e in θ (5.36)we find that the Virasoro generators L m s satisfy the Witt algebra [ L m , L n ] = ( n − m ) L m + n . (5.37)The Virasoro generators can be written in terms of modes of h ( θ ) L m = (cid:126) (cid:88) n = , Z f n f m − n . (5.38)The zero mode, in particular, is given by L = (cid:126) + (cid:88) n > na † n a n . (5.39) A mapping between Young diagrams and operators / states was considered in[47, 48]. We map states to dropletsand hence to Young diagrams. There is a constant part in L which is equal to ζ ( −
22 generic state | (cid:126) q (cid:11) is an eigenstate of L . We denote L eigenvalue of | (cid:126) q (cid:11) by h (cid:126) q . From (4.11), it iseasy to check that the L operator is related to total box number operator and hence L eigenvalue h (cid:126) q measures the total number of boxes in the corresponding Young diagram associated with a state | (cid:126) q (cid:11) h (cid:126) q = k + N . (5.40)For example, ground state | (cid:11) has L eigenvalue N / L | (cid:11) = (cid:126) | (cid:11) (5.41)and hence, the corresponding diagram has zero box ( ∅ ). Ground state | (cid:11) a primary state of Kac-Moody as well as Virasoro.To complete the mapping between the states and the Young diagrams we also need to specifythe shape of the Young diagram with k boxes corresponding to a state in Hilbert space. Since,a particular Young diagram corresponds to a droplet in phase space, we define an operator ˆ S ( θ )which captures the shape of the droplet in phase space associated with a state in Hilbert space.From (5.27) we see that an excitation of the ground state by n th mode corresponds to a cos n θ deformation of the boundary, therefore we define the “shape” operatorˆ S ( θ ) = + (cid:126) √ k (cid:88) n > √ n cos n θ a † n a n (5.42)where k is the total number of boxes. The eigenvalue of ˆ S ( θ ) define the shape function h ( θ ) inphase space and hence the distribution of boxes in the corresponding Young diagram. For example | (cid:11) state has ˆ S ( θ ) eigenvalue one and therefore gives a circular droplet, which corresponds to ∅ diagram.An automodel diagram corresponds to a particular descendant state of U (1) Kac-Moody alge-bra | , ξ (cid:11) = (cid:16) a † (cid:17) q | (cid:11) , with q = N ξ . (5.43)This state corresponds to a Young diagram with total number of boxes N ξ and the shape of thedroplet is given by h = + ξ cos θ .We consider generic fluctuations of automodel diagrams. The corresponding states in Hilbertspace are given by | F (cid:11) = (cid:89) n (cid:16) a † n (cid:17) α n | , ξ (cid:11) , α n ∈ Z ≥ . (5.44)The number of boxes in Young diagram for state | F (cid:11) is given by N ξ + N ξ (cid:88) n n α n (5.45)23nd the corresponding shape is given by1 + ξ cos θ + (cid:126) √ k (cid:88) n √ n α n cos n θ. (5.46)This shape corresponds to a random fluctuations of automodel Young diagrams as α n ’s are ran-dom. One can consider these α n ’s to be random Gaussian integers with mean zero. Thus thesefluctuations are similar to fluctuations given by equation (5.2) except the fact that here α n s areintegers .
6. Conclusion
In this paper we show that the growth of Young diagrams equipped with Plancherel measure canbe studied through a simple matrix model. We write down a partition function for such growthprocess and solve the model in the continuum limit. In [5], Kerov introduced a di ff erential modelfor the growth of Young diagrams, known as the automodel . We find that in continuum limitour one parameter solution falls in the automodel class of Kerov with renormalised box numberplaying the role of time . At the limiting value of the parameter the dominant solution matcheswith the limit shape of [3, 4]. Our analysis also o ff ers an alternate proof of limit shape theorem ofVershik-Kerov and Logan-Shepp.We observe that the evolution of Young diagrams in automodel class can be mapped to di ff erentshapes of incompressible fluid droplets in two dimensions. Automodel evolution corresponds toarea preserving deformation of these fluid droplets. Such identification was possible due to theequivalence between GWW model and automodel partition function [19]. In particular we seethat GWW transition point maps to limit shape of [3, 4]. Since eigenvalues of unitary matricesbehave like position of free fermions, two dimensional fluid droplets are identified with classicalphase space of these free fermions [19, 21]. From this distribution we construct the one particleHamiltonian. The Hamiltonian describes dynamics of a particle moving on a circle of unit radiusunder the influence of an e ff ective potential. The classical phase space trajectories of such particlecaptures the information about the automodel Young diagrams.In view of the above connection between automodel diagrams and two dimensional phase spacedroplets, any fluctuations of automodel diagrams correspond to small ripples on the boundary ofthese droplets. Using the Hamiltonian equations we quantise the dynamics of such ripples and findthat di ff erent modes of these fluctuations satisfy abelian Kac-Moody algebra. The fluctuationsof classical droplets studied in this paper seems quite universal in nature. Edge excitations offractional quantum Hall fluid also satisfy similar algebra [49]. Fluctuations of large N fermionsin harmonic oscillator potential also belong to the similar class [45, 46, 50]. Since automodelclass is equivalent to gapless phase of GWW model, quantum fluctuations of the classical weak (5.2) considers fluctuations of ˆ v k ( u ), where as our Young diagram density is related to derivative of ˆ v k ( u ). Hencethe generic fluctuation (5.46) will match with (5.2) if we take a derivative of the same. U (1) currents and express the Virasoro modes in terms of Kac-Moody modes. It turnsout that the zero mode of the Virasoro is proportional to box number operator. We also constructthe Hilbert space for the Kac-Moody algebra. It turns out that there is a correspondence betweendi ff erent states in the Hilbert space and Young diagrams. The Kac-Moody primary (which isa Virasoro primary as well) corresponds to ∅ (null) diagram. Automodel diagrams correspondto descendants of Kac-Moody algebra. Fluctuations about automodel diagrams corresponds tofluctuations about automodel states. The shape of those fluctuations matches with the fluctuationsof limit shape studied by [8]. It would be interesting to study these Gaussian fluctuations of limitshape diagrams in the context of states in Hilbert space. Acknowledgments:
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