A Gaussian distribution for refined DT invariants and 3D partitions
AA GAUSSIAN DISTRIBUTION FOR REFINED DT INVARIANTSAND 3D PARTITIONS.
ANDREW MORRISON
Abstract.
We show that the refined Donaldson–Thomas invariants of C ,suitably normalized, have a Gaussian distribution as limit law. Combinatori-ally these numbers are given by weighted counts of 3D partitions. Our tech-nique is to use the Hardy–Littlewood circle method to analyze the bivariateasymptotics of a q -deformation of MacMahon’s function. The proof is basedon that of E.M. Wright who explored the single variable case. Introduction.
In [7] physicists suggested that the (IIA) string theory associated to a Calabi–Yauthreefold X should produce an algebra of BPS states: H BP S = (cid:77) γ ∈ Γ H γ where here the (charge) lattice Γ is identified with the even cohomology of X ,Γ = (cid:76) i =0 H ( X, Z ). Moreover each individual vector space H γ should have anadditional Z -grading coming from a symmetry in the little group Spin(3) [6].Mathematically, we consider the cohomological Hall algebra [12] as giving the al-gebra of BPS states on X = C . In this case Γ = H ( X, Z ) = Z and the γ = n thpiece is given by the critical cohomology of Hilb n ( C ) . Moreover each of thesevector spaces has a cohomological Z -grading. The Betti numbers of these gradedpieces are know as refined DT invariants. These numbers are dependent on thesingularities of Hilb n ( C ). For example they do not satisfy a Poincare duality.However, on a recent visit to EPFL, T. Hausel shared with me the output of acomputer experiment. He conjectured that the refined DT invariants, suitablynormalized, would have a Gaussian distribution as limit law, i.e. for large n plot-ting the Betti numbers against cohomological degree should give the bell curve ofa Gaussian distribution (cf. [18]). The goal of this paper is to prove that conjecture.In fact this proposal is entirely combinatorial. The Hilbert-Poincare series for thecohomological Hall algebra, computed in [2], equals M ( t, q / ) where t gives theΓ-grading, q gives the cohomological grading, and M δ ( t, q ) = (cid:89) m ≥ m − (cid:89) k =0 − q δ +2 k +1 − m t m . In addition we add a single D6-brane filing the C or mathematically a framing. a r X i v : . [ m a t h . AG ] M a r ANDREW MORRISON
Expanding this series gives an explicit formula [14] for the t n coefficent (cid:88) π (cid:96) n q δw ( π )+ w + ( π ) − w − ( π ) where the sum is over all plane partitions of n which we now explain.A plane partition is given by a two dimensional array of positive integers inthe first quadrant of Z that are weakly decreasing in both the x, y directions. Inthis way plane partitions are a generalization of ordinary (line) partitions [1]. Theanalogue of the Young diagram in this situation is a stack of three dimensionalboxes in Z ≥ . Such a collection gives a plane partition if and only if the stack isstable under the pull of gravity along the (1 , ,
1) axis. For example let π be theplane partition given alternatively as... ... ... ...0 0 0 02 2 1 0 0 · · · · · · · · · · · · Figure 1. integers
Figure 2. boxesHere the total sum of the integers/boxes is | π | = 35 and we say that the partitionhas size 35. The statistics appearing in the above formula for refined DT invariantsare, w + ( π ) = (cid:88) i
The distribution of the random vairable δ · X n + X + n − X − n for large n has the Gaussian distribution as limit law with µ = δζ (2) / (2 ζ (3)) / and σ = 1 / (2 ζ (3)) / . Setup.
First we split the problem into two parts, one of which has already been solved.Straight away we see that the covariance of X n and X + n − X − n is zero due tosymmetry E (( X n − µ X n )( X + n − X − n − µ X + n − X − n )) = E (( X n − µ X n )( X + n − X − n ))= E ( X n ( X + n − X − n )) − E ( µ X n ( X + n − X − n ))= E ( X n X + n − X n X − n ) − µ X n ( E ( X + n − X − n ))= 0 − . GAUSSIAN DISTRIBUTION FOR REFINED DT INVARIANTS AND 3D PARTITIONS. 3
The following is a result of E.P. Kamenov and L.R. Mutafchiev:
Theorem 2.1 ([11]) . Let a = ζ (2) / (2 ζ (3)) / and b = (cid:112) / / (2 ζ (3)) / . Then as n → ∞ we have X n ∼ N ( a, bn − ln n ) . Now the sum of two Gaussian random variables is again Gaussian with meanthe sum of the means and variance the sum of the variances plus a covariance term.Since the above covariance was zero, Theorem 1.1 will now follow from the resultjust mentioned together with:
Theorem 2.2.
Let c = 1 / (2 ζ (3)) / . Then as n → ∞ we have X + n − X − n ∼ N (0 , c ) . To prove this result we use the method of moments. That is we show that thelimiting distribution has the same moments as a Gaussian random variable withvariance 1 / (2 ζ (3)) / . Specifically we will show that in the limit E (cid:16)(cid:0) X + n − X − n (cid:1) k (cid:17) = (cid:26) k is odd,( k − ζ (3)) − k/ if k is even.Consider the generating series M ( t, q ) = (cid:89) m ≥ m − (cid:89) k =0 − q k +1 − m t m = (cid:88) π q w + ( π ) − w − ( π ) t | π | and let p n ( q ) be the coefficient of t n then we have E (cid:16)(cid:0) X + n − X − n (cid:1) k (cid:17) = n − k/ ∂ k p n ( q ) (cid:12)(cid:12) q =1 p n ( q ) | q =1 where ∂ = q ddq . Notice by symmetry this already implies that all the odd momentsvanish. The method of proof given in the next section follows the proof of E.Wright[20] who provided an asymptotic formula for the number of plane partitions of np n ( q ) | q =1 ∼ ζ (3) / / √ πn / e ( ζ (3)4 ) / ( n / ) / + ζ (cid:48) ( − . Wright’s proof in turn generalized the pioneering work of Hardy and Ramanujan [9]who first applied the Hardy-Littlewood circle method to get an asymptotic formulafor the number of ordinary partitions. Using this method in the next section wewill show that ∂ k p n ( q ) | q =1 ∼ n k/ · ( k − ζ (3)) − k/ · ζ (3) / / √ πn / e ( ζ (3)4 ) / ( n / ) / + ζ (cid:48) ( − when k is even. This gives the correct moments and shows that X + n − X − n has aGaussian limit law as promised. Typo warning: Wrights formula on page 179 is missing factor of √ ANDREW MORRISON Proof.
As explained in the previous section we are going to use the Hardy–Littlewoodcircle method to estimate the coefficients in the generating series M k ( t ) := ∂ k M ( t, q ) | q =1 . Given any function A ( t ) = (cid:80) n ≥ a n t n analytic on the interior of the unit disk wecan compute the n th coefficient in its MacLaurin series using the Cauchy formula a n = 12 πi (cid:90) C N t − n − A ( t ) dt where C N is the circle or radius e − /N . The idea of the circle method is that byunderstanding the singularities of A on the unit circle one can approximate thisintegral by an integral over a small subarc of the circle when N, n (cid:29) A ( t ) = M ( t ) = M ( t, q ) | q =1 is MacMahon’s function then letting t = e z Wright defines the major arc C (cid:48) N to be the points such that im( z ) < /N and the minor arc C (cid:48)(cid:48) N to be the remaining points on the circle C N . In some senseMacMahon’s function is most singular at z = 0 and so the integral over C N is wellapproximated by that over the small arc C (cid:48) N . The following two Lemmas makethis precise and will be very useful to us later: Lemma 3.1 (E.M.Wright [20] Lemma I) . There exists constants N , K so that forall N > N and t = e z ∈ C (cid:48) N along the major arc we have (cid:12)(cid:12)(cid:12) M ( t ) − e − ζ (cid:48) (1) z / e ζ (3) /z (cid:12)(cid:12)(cid:12) < KN − / − e ζ (3) N . Lemma 3.2 (E.M.Wright [20] Lemma II) . Given any (cid:15) > there exists an N such that for all N > N and t ∈ C (cid:48)(cid:48) N along the minor arc we have | M ( t ) | < e ( ζ (3) − / (cid:15) ) N . Using Lemma 3.2 one shows that the integral along the minor arc is relativelysmall. Then using Lemma 3.1 the integral along the major arc can be approxi-mated using the curve of steepest decent. This gives Wright’s asymptotic formulamentioned earlier [20] and illustrates the idea of the circle method.Recall, we are specifically interested in computing the coefficients of the series M k ( t ) = ∂ k M ( t, q ) | q =1 defined at the start of the section as a means to com-pute the moments of the random variables X + n − X − n . Let us write this series as M k ( t ) = F k ( t ) · M ( t ) then by Wright’s two lemmas above we have a good under-standing of the singularities of the factor M ( t ) it remains to analyze F k ( t ). Example 3.3.
Computing F ( t ) . Let us differentiate M ( t, q ) twice using ∂ = q ddq this gives (cid:88) m ≥ m − (cid:88) k =0 (1 − m + 2 k ) q k − m +2 t m (1 − q k +1 − m t m ) + (1 − m + 2 k ) q k − m +1 t m (1 − q k +1 − m t m ) · M ( t, q ) Wright’s first Lemma is more refined than this. We mearly extract his leading order approx-imation suitable for our purposes.
GAUSSIAN DISTRIBUTION FOR REFINED DT INVARIANTS AND 3D PARTITIONS. 5 + (cid:88) m ≥ m − (cid:88) k =0 (1 − m + 2 k ) q k − m +1 t m − q k − m +1 t m · M ( t, q ) . Then setting q = 1 we get (cid:88) m ≥ m − (cid:88) k =0 (1 + 2 k − m ) t m (1 − t m ) · M ( t ) = 13 · (cid:88) m ≥ m ( m − t m (1 − t m ) · M ( t ) deducing that F ( t ) = 13 (cid:88) m ≥ m ( m − t m (1 − t m ) . In the next two Lemmas we analyze the behavior of F k ( t ) along the major andminor arcs. Lemma 3.4.
Given k even, there exist constants N , K such that for all N > N and t = e z ∈ C (cid:48) N along the major arc we have (cid:12)(cid:12)(cid:12) F k ( t ) − ( k − ζ (3)) k/ z − k (cid:12)(cid:12)(cid:12) < KN k − . Proof.
First let us consider the case k = 2 used to compute the variance. By whatwe saw in Example 3.3 above F ( t ) = 13 (cid:88) m ≥ m ( m − e mz (1 − e mz ) . By using the Mellin transform e − τ = πi (cid:82) σ + i ∞ σ − i ∞ Γ( s ) τ − s ds we can express thisfunction as an integral, F ( t ) = 13 (cid:88) m ≥ m ( m − e mz (1 − e mz ) = 13 (cid:88) m ≥ (cid:88) i ≥ m ( m − ie imz = 16 iπ (cid:90) σ + i ∞ σ − i ∞ Γ( s ) (cid:88) m ≥ (cid:88) i ≥ m ( m − i ( imz ) s ds = 16 iπ (cid:90) σ + i ∞ σ − i ∞ Γ( s ) z − s ζ ( s − ζ ( s − − ζ ( s − ds. This integrand has a double pole at s = 2 and a simple pole at s = 4 coming fromthe Riemann zeta function while the gamma function contributes simple poles at s = 0 , − , − , . . . . Doing the residue calculus we see that F ( t ) = Γ(4) ζ (3)3 z − − γ Γ(2) ζ ( − z − + O (1)where γ is Euler’s constant coming from the Laurent expansion of the zeta functionabout s = 1.Computing the higher order derivatives is essentially an application of Wick’stheorem. Indeed if k = 2 r then we have F k =2 r ( t ) = ( k − F ( t )) r + G k ( t )where the combinatorial coefficient ( k − r differentials ∂ r . Using the product rule for differentiation we see the first ANDREW MORRISON term appearing. The remaining terms in G k ( t ) come from the other terms gener-ated in the product rule. Computing their Mellin transform shows the poles theycontribute are of order at least two less. (cid:3) Lemma 3.5.
Given k even, there exists positive constants N , C k , A k such that forall N > N and t = e z ∈ C (cid:48)(cid:48) N along the minor arc we have | F k ( t ) | < C k N A k . Proof.
Looking at the definition of F k ( t ) we see that ultimately it can be writtenas a finite sums and products of series like (cid:88) n ≥ n a t nc (1 − t n ) b where a, b, c ∈ N . Each of these sums can be bounded like (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) n ≥ n a t nc (1 − t n ) b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:88) n ≥ n a | t | n (1 − | t | n ) b using the Mellin transform as in the previous lemma gives bounds on this sum like C a,b N A a,b where | t | = e − /N and A a,b , C a,b are constants depending only on a, b .In total this gives the polynomial bounds claimed. (cid:3) Now we have got bounds on the series M k ( t ) along the major and minor arcs weare ready to estimate the Cauchy integral. Let us define the two quantities we areinterested in computing I (cid:48) k,n = 12 πi (cid:90) C (cid:48) N t − n − M k ( t ) dt,I (cid:48)(cid:48) k,n = 12 πi (cid:90) C (cid:48)(cid:48) N t − n − M k ( t ) dt. From now onwards we choose N = ( n/ ζ (3)) / so that n = 2 ζ (3) N . Now wecan get a bound on the integral along the minor arc that is sufficient to show thisintegral is negligible: Lemma 3.6.
Given k even then for all (cid:15) > there exists positive constants N , K, A such that for all N > N we have | I (cid:48)(cid:48) k,n | < K · N A · e ( − + (cid:15) ) N · e ζ (3) N . Proof.
Using Lemmas 3.2 and 3.5 we get | I (cid:48)(cid:48) k,n | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) πi (cid:90) C (cid:48)(cid:48) N F k ( t ) · M ( t ) · t − n − dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ π | πi | · sup C (cid:48)(cid:48) N ( | F k ( t ) | ) · sup C (cid:48)(cid:48) N ( | M ( t ) | ) · sup C (cid:48)(cid:48) N ( | t − n − | ) ≤ K · N A · e ( ζ (3) − / (cid:15) ) N · e ( − n − N − substituting n = 2 ζ (3) N gives the result. (cid:3) GAUSSIAN DISTRIBUTION FOR REFINED DT INVARIANTS AND 3D PARTITIONS. 7
From now all that remains is to estimate the integral I (cid:48) n,k . Combining Lemma3.1 and Lemma 3.4 we have I (cid:48) n,k = 12 πi (cid:90) C (cid:48) N F k ( t ) M ( t ) t − n − dt = e ζ (cid:48) ( − πi ( k − ζ (3)) k/ (cid:90) (1+ i ) /N (1 − i ) /N z − k +1 / e ζ (3) z +2 ζ (3) N z dz + O ( N − k − / − e ζ (3) N )= ( k − N k (2 ζ (3)) k/ · e ζ (cid:48) ( − N − / − πi (cid:90) i − i v − k +1 / e ζ (3) N (2 v + v − ) dv + O ( N − k − / − e ζ (3) N )where we set v = N z .Notice the prefactor in the above expression is essentially the term we are lookingfor. From now on we work with the integral P (cid:48) n,k := 12 πi (cid:90) i − i v − k +1 / e ζ (3) N (2 v + v − ) dv basically it will be enough to show that, for large N , this is independent of k . Thenin the limit we have I (cid:48) n,k = ( k − N k (2 ζ (3)) k/ · I (cid:48) n, . We are able to achieve this by localizingthe integral P (cid:48) n,k to an even smaller arc using the method of steepest descents.In the exponent of the integrand we have the function 2 v + v − . Roughly speakingthe integrand will be largest when this exponent is real. The curve of steepestdescent is defined to be the real curve given by im(2 v + v − ) = 0, specificallytaking v = X + iY we have ( X + Y ) = X. This is the closed curve C meeting the lines X = 0 at (0 , X = 1 at (1 , X = − Y at D = (2 − , − − ), and X = Y at E = (2 − , − ) as seen in Figure 3.As in [20] we consider an alternative integral along this curve C rather than alongthe straight line F G .Making a branch cut from 0 to −∞ along the real axis we consider the value of v / which is real and positive at v = 1 and take the contour for C parameterizedin the anti-clockwise direction. Following Wright we define ξ k ( v ) = v − k +1 / πi e ζ (3) N (2 v + v − ) and J (cid:48) k,n = (cid:90) C ξ k ( v ) dv. On the straight lines
EG, DF and along the arcs OE and OD we have goodbounds on ξ k ( v ). Since here Re( v − ) = ( X − Y ) / ( X + Y ) ≤ Y = rX gives (cid:12)(cid:12)(cid:12) v − k +1 / e ζ (3) N (2 v + v − ) (cid:12)(cid:12)(cid:12) ≤ X − k (1 + r ) − k e ζ (3) N (cid:16) − r r X − (cid:17) e ζ (3) N X . When r > X →
0, and when r = 1 along the lines EG, DF there are easy bounds. In summary, we have a bound | ξ k ( t ) | < Ke ζ (3) N X alongthese contours OE, OD, EC, and DF . Using contour integration to compare the ANDREW MORRISON
Figure 3.
Curve of steepest descent.original integral P (cid:48) n,k to the new integral J (cid:48) n,k along the curve C we see that (cid:12)(cid:12) P (cid:48) n,k − J (cid:48) n,k (cid:12)(cid:12) < (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) DF ξ k ( v ) dv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) D (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) GE (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < Ke ζ (3) N this allows us to integrate along the curve of steepest descent instead. To parame-terize this curve we choose t = − i (cid:18) v − v (cid:19) (2 v + 1) so that t = 3 − v − v − . Now the problem transforms to an integral over the realline J (cid:48) n,k = e ζ (3) N (cid:90) ∞−∞ χ k ( t ) e − ζ (3) N t dt, with χ k ( t ) = v − k +1 / πi dvdt . The most serious piece of this integral is located about t = 0 i.e. v = 1. To under-stand the behavior here we take a convergent power series χ k ( t ) = (cid:80) ∞ m =0 a m t m ina small neighborhood of t = 0. Next observe that for some constant K we have dvdt = iv (2 v + 1) v + v ⇒ (cid:12)(cid:12)(cid:12)(cid:12) v − k +1 / πi dvdt (cid:12)(cid:12)(cid:12)(cid:12) < K | v − k | on all of the real line. Moreover for some possibly larger K we have | v − k | = | − v − t | k ≤ Kt k on the compliment of the above radius of convergence about t = 0. All in all weget that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) χ k ( t ) − k +1 (cid:88) m =0 a m t m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < Kt k +2 GAUSSIAN DISTRIBUTION FOR REFINED DT INVARIANTS AND 3D PARTITIONS. 9 over the whole real line. Finally our integral is approximated by J (cid:48) n,k = k +1 (cid:88) m =0 a m (cid:90) ∞−∞ t m e − ζ (3) N t dt + M k where | M k | ≤ K (cid:90) ∞−∞ t k +2 e − ζ (3) N t dt < KN k +3 . By symmetry all the above odd integrals are zero. The even ones are given by (cid:90) ∞−∞ t m e ζ (3) N t dt = Γ( m + )( ζ (3) N ) m + . So to get the leading order asymptotics we need only the constant term a = π √ in the expansion of χ k ( t ). In particular to leading order there is no dependence on k as we claimed earlier. Substituting N = ( n/ ζ (3)) / gives the formula describedat the end of Section 2. 4. Final Remarks.
Some remarks about the asymptotics of DT invariants coming from this investi-gation and math/physics literature.
Asymmetry . Since the refined DT invariants are given by M ( t, q / ) their distri-bution is shifted by the trace of the plane partition coming from 3 X n . This shiftingis relevant to the refined topological vertex in physics [10]. This was discussed in[13] where a minor discrepancy was noticed with calculations in [5] for the case ofthe refined DT invariants of the resolved conifold singularity. Dimension of moduli space . In his lecture notes on quiver moduli M. Reinekedescribes a conjecture on M. Douglas on the asymptotic growth of Euler numbersof spaces of representations of Kronecker quivers [17]. Reineke proposes a general-ization of the conjecture would giveln ( χ ( M d ( Q ))) ∼ C Q (cid:112) dim( M d ( Q ))where M d ( Q )) is a suitable smooth model for the quiver moduli, d is a large di-mension vector, and C Q is an interesting constant to be determined.For sheaves on a Calabi-Yau threefold the moduli spaces will be singular andnot of the expected dimension. However in our case we do know that C − C (cid:112) dim(Hilb n ( C )) < ln (cid:0) | vir (Hilb n ( C )) | (cid:1) < C + C (cid:112) dim(Hilb n ( C ))for large n , where vir is the virtual Euler number or numerical DT count for thismoduli space, and C − C , C + C are constants. A proof of this follows from Wrightstheorem [20] for the virtual Euler number and Briancon and Iarrobino’s asymp-totics for the dimension of Hilb n ( C ) [3]. Geometrically this relationship betweenthe virtual Euler number and dimension of the moduli space seems like a strangecoincidence specific to C ? Orbifold: C × [ C / Z ]. In the case of numerical DT invariants, Panario, Rich-mond, and Young investigated the orbifold C × [ C / Z ], here the charge lattice istwo dimensional and one studies the bivariate asymptotics of colored partitions see[16]. BPS black holes . A major goal of string theory is to unify quantum mechanicswith Einstein’s general relativity. In the 90’s Strominger and Vafa showed thatindeed the topological string described the physics of some BPS black holes in acertain limit of the string coupling constant [19]. Later with Ooguri they conjec-tured that in a certain limit the entropy of BPS black holes should be determinedfrom the square of the topological string partition function [15].Recently in the case of D6 - D2/D0 states this conjecture has been developedby Denef and Moore [4]. If we let X be a Calabi-Yau threefold and I ( X, β, n ) bethe moduli space of ideal sheaves with Chern character (1 , , − β, n ), their paperspeculates that lim γ →∞ (cid:32) ln (cid:0) ln (cid:0) | vir ( I ( X, λ β, λ n )) | (cid:1)(cid:1) ln( λ ) (cid:33) = 2 . By Wright’s theorem this is true when β = 0. In [8] physicists checked this con-jecture for the quintic threefold. However as it is a hard problem to compute thehigher genus Gromov-Witten invariants in this case these checks are only partial. Acknowledgements.
Primarily I wish to thank T. Hausel for his hospitality in inviting me to EPFLand sharing the ideas that lead to this paper. Also thanks to J. Bryan, R. Pand-haripande, and B. Young for their support and comments, and to M. Marcolli whowas my mentor at MSRI where this paper was written.This paper was finished during a postdoc at MSRI in the spring 2013 programNon-Commutative Algebraic Geometry and Representation Theory. Normally I ama postdoc at ETH Z¨urich in the research group of R. Pandharipande sponsored bySwiss grant 200021143274.
References [1] G.E. Andrews,
The theory of partitions , Addison-Wesley Pub. Co., Advanced Book Program,1976[2] K. Behrend, J. Bryan, B. Szendr˝oi,
Degree zero motivic Donaldson-Thomas invariants , In-ventiones Mathematicae, June 2012.[3] J. Briancon and A. Iarrobino,
Dimension of the punctual Hilbert scheme , Journal of Algebra55, (1978)[4] F. Denef and G. Moore,
Split states, entropy enigmas, holes and halos , Journal of HighEnergy Physics, 2011.[5] T. Dimofte and S. Gukov
Refined, motivic, and quantum , Letters in Mathematical Physics,2010[6] D. S. Freed, Five Lectures on Supersymmetry, American Mathematical Soc., 1999[7] J. A. Harvey and G. Moore,
On the Algebras of BPS States , Communications in MathematicalPhysics, 1998, Volume 197, Issue 3, pp 489-519[8] M. x. Huang, A. Klemm, M. Marino and A. Tavanfar,
Black Holes and Large Order QuantumGeometry
Physical Review D, 2009[9] G. H. Hardy and S. Ramanujan.
Asymptotic formulae in combinatory analysis . Proc. LondonMath. Soc., 17:75 115, 1918.
GAUSSIAN DISTRIBUTION FOR REFINED DT INVARIANTS AND 3D PARTITIONS. 11 [10] A. Iqbal, C. Kozaz, C. Vafa,
The refined topological vertex , Journal of High Energy Physics,2009.[11] E. P. Kamenov and L. R. Mutafchiev,
The limiting distribution of the trace of a randomplane partition , Acta Mathematica Hungarica.[12] M. Kontsevich and Y. Soibelman,
Cohomological Hall algebra, exponential Hodge structuresand motivic DonaldsonThomas invariants , arXiv:1006.2706[13] A. Morrison, S. Mozgovoy, K. Nagao, B. Szendr˝oi,
Motivic DonaldsonThomas invariants ofthe conifold and the refined topological vertex , Advances in Mathematics, Volume 230, 2012[14] A. Okounkov and N. Reshetikhin, Random skew plane partitions and the Pearcey process.Comm. Math. Phys., 269(3):571609, 2007.[15] H. Ooguri, A. Strominger, C. Vafa,
Black hole attractors and the topological string , Phys.Rev. D 70 (2004)[16] D. Panario, B. Richmond, B.Young,
Bivariate asymptotics for striped plane partitions , SIAMpublications. 2009[17] M. Reineke
Moduli of representations of quivers . Proceedings of the ICRA XII conference,Torun, 2007.[18] M. Reineke.
Cohomology of non-commutative Hilbert schemes . Algebras and RepresentationTheory 8 (2005)[19] A. Strominger and C. Vafa,
Microscopic origin of the Bekenstein-Hawking entropy , PhysicsLetters B, Volume 379, Issues 14, 27 June 1996, Pages 99104[20] E. M. Wright.
Asymptotic partition formulae I. Plane partitions.