Estimating the Asymptotics of Solid Partitions

Abstract

We study the asymptotic behavior of solid partitions using transition matrix Monte Carlo simulations. If $p_3(n)$ denotes the number of solid partitions of an integer $n$, we show that $\lim _{n\rightarrow \infty } n^{-3/4} \log p_3(n)\sim 1.822\pm 0.001$. This shows clear deviation from the value $1.7898$, attained by MacMahon numbers $m_3(n)$, that was conjectured to hold for solid partitions as well. In addition, we find estimates for other sub-leading terms in $\log p_3(n)$. In a pattern deviating from the asymptotics of line and plane partitions, we need to add an oscillatory term in addition to the obvious sub-leading terms. The period of the oscillatory term is proportional to $n^{1/4}$, the natural scale in the problem. This new oscillatory term might shed some insight into why partitions in dimensions greater than two do not admit a simple generating function.

Appendices

Appendix: Asymptotics of Plane Partitions

The asymptotics of plane plane partitions as follows from an application of Meinardus’ formula is [5, 6]

$$\begin{aligned} p_2(n) \sim \frac{(2\zeta (3))^{7/36}}{\sqrt{6\pi }} n^{-25/36} \exp \Big ( \tfrac{3}{2} (2\zeta (3))^{1/3} n^{2/3} + \zeta '(-1) \Big ) \end{aligned}$$

Thus, one has

$$\begin{aligned} n^{-2/3}\log p_2(n)&\sim \tfrac{3}{2} (2\zeta (3))^{1/3} + n^{-2/3}\Big (-\tfrac{25}{36} \log n + \log \tfrac{(2\zeta (3))^{7/36}}{\sqrt{6\pi }} + \zeta '(-1)\Big )\nonumber \nonumber \\&\sim 2.00945 -0.694444 n^{-2/3}\log n -1.4631 n^{-2/3} \end{aligned}$$

(5.1)

The goal of the Monte Carlo simulation is to reproduce the above formula. In particular, we should see if we can match the constant term to one part in $10^3$.

Results from Monte Carlo Simulations

The Monte Carlo simulations for plane partitions are used to estimate the numbers of plane partitions, say, in the range $[1,N_{max}]$. We then compare these numbers to the exact numbers and see how the numbers improve with increasing the statistics by increasing the number of flips. Let $mc_2(n)$ be the values obtained from the Monte Carlo simulation. We also fit $n^{-2/3}\log p_2(n)$ in the range $[50,2000]$ with the following formula:

$$\begin{aligned} n^{-2/3} \log p_2(n)\sim \alpha _2\ + \beta _2\ n^{-1/3} + \delta _2\ n^{-2/3} \log n + \epsilon _2\ n^{-2/3}. \end{aligned}$$

(5.2)

Since our Monte Carlo simulations estimate $N_\pm (n)$, we use

$$\begin{aligned} \log \big [\tfrac{N_+(n-1)}{N_-(n)}\big ] \sim&\ \ \alpha _2\ \big [n^{2/3}\big ]_2 + \beta _2 \ \big [n^{1/3}\big ]_2 + \delta _2\ \big [ \log n\big ]_2 \ , \end{aligned}$$

(5.3)

to extract three of the four parameters $(\alpha _2,\beta _2,\delta _2,\epsilon _2)$ and then determine $\epsilon _2$ using Eq. (5.2).

• 1. Fit data in the range $n\in [50,2000]$ to the three-parameter formula given in Eq. (5.3). This determines the values of $(\alpha _2,\beta _2,\delta _2)$ that we will use below. We obtain

  $$\begin{aligned} (\alpha _2,\beta _2,\delta _2) = (2.00998, -0.0194366, -0.663683). \end{aligned}$$

  (5.4)

• 2. Next we add the term $\big [-\tfrac{1}{3} f\ n^{-4/3}\big ]$ to the asymptotic formula Eq. (5.3) and carry out a four-parameter fit to see how the three parameters change.⁶ We obtain

  $$\begin{aligned} (\alpha _2,\beta _2,,\delta _2,f) = (2.00923,+0.0120124,-0.731848,-0.41608). \end{aligned}$$

  (5.5)

  We take the average of the two sets of numbers and use one half of the difference as an estimate of the error.

  $$\begin{aligned} (\alpha _2,\beta _2,\delta _2,f) = (2.0096\pm 0.0004, -0.004\pm 0.02,-0.70\pm 0.03,-0.21\pm 0.21).\nonumber \\ \end{aligned}$$

  (5.6)

• 3. We substitute the values of $(\alpha _2,\beta _2,\delta _2)$ given in Eq. (5.4) in Eq. (5.2) and then carrying out a one-parameter fit for $n\in [50,100]$ to determine $\epsilon _3$. We obtain $\epsilon _2 = -1.44372$. We then add a term $f\ n^{-1}$ to Eq. (5.2) and use values given in Eq. (5.5) and carry out a one-parameter fit determine $\epsilon _2$. We obtain $\epsilon _2=-1.39798$. We thus obtain the estimate

  $$\begin{aligned} \epsilon _2=-1.423\pm 0.025. \end{aligned}$$

  (5.7)

The error estimates for the four parameters through the Monte Carlo simulations are consistent with the deviation from the exact values. This provides some validation for the methods that we used for solid partitions.

No Oscillations for Plane Partitions

In order to see if the oscillations that we observe are special to solid partitions, we study the residual in the Monte Carlo data for plane partitions. Since our Monte Carlo simulations did not give us estimates for the statistical errors, we used the difference of the exact numbers for plane partitions from the numbers from our Monte Carlo simulations to provide an estimate of the statistical error. In Fig. 7, we observe that both the residual and our estimated statistical errors have similar behaviour that is consistent with no oscillations.

Fig. 7

On the left is the plot of $n$ times the estimated statistical error versus $n^{1/3}$. On the right we plot the $n$ times the residual (in blue) to the fit. There are no oscillations to be seen and the residual is comparable to the statistical error (Color figure online)

A similar study on exactly enumerated plane partitions (through Mac Mahon’s formula) led us to the same conclusion.

Asymptotics of MacMahon Numbers

Recall that the formula that MacMahon guessed for the generating function of solid partitions gives rise to a series of numbers that we call MacMahon numbers, $m_{3}(n)$. One has

$$\begin{aligned} {\prod _{n=1}^\infty (1-q^n)^{-\tfrac{n(n+1)}{2}}}:=\sum _{n=0}^\infty m_3(n)\ q^n. \end{aligned}$$

(5.8)

In this section, we briefly discuss the asymptotics of these numbers in order to compare with our results for solid partitions. While these numbers do not have any relation to solid partitions, we can derive their asymptotics from the above product formula using Meinardus’ method. One obtains [22]

$$\begin{aligned} n^{-3/4}\log m_3(n) \sim 1.78982 + \frac{0.333546}{n^{ 1/4}} - \frac{0.0414393}{\sqrt{n}} + \frac{(-1.54436 - 0.635417 \log n)}{n^{3/4}} \end{aligned}$$

(5.9)

Assuming that their asymptotic behavior is similar to that of solid partitions, one can see how well fits to, say the first 1000 MacMahon numbers, agree with the exact asymptotic formula. The accuracy of these fits will provide some hints towards the quality of the fits that one may expect for fits using Monte Carlo data. The fit has five parameters given by the formula

$$\begin{aligned} n^{-3/4}\log m_3(n) \sim a+ \frac{b}{n^{ 1/4}} + \frac{c}{\sqrt{n}} + \frac{(e + d \log n)}{n^{3/4}} \end{aligned}$$

(5.10)

We observe that the the first 2,000 numbers reproduce the leading constant in the asymptotic formula with an error of $0.0005$ (Table 2). Our numerical study of solid partitions has used data corresponding to the first 10,200 numbers but achieves a slightly lower accuracy.

Table 2 Results of the fit to the exact MacMahon numbers $m_3(n)$ in the range $[50,N_{max}]$

Oscillations of Boxed Solid Partitions Near the Entropy Maximum

So far we essentially focussed on unbounded partitions of an integer, in other words partitions restricted to a box of lateral size $B\gg n^{1/4}$. We needed to have a box for computational reasons, but we have discussed that the box has essentially no incidence. In this context, we have identified original oscillations of the residuals $\delta$, with a period proportional to the natural scale $n^{1/4}$, which we have related to the unexpected sensitivity to the underlying lattice. Anticipating that the same phenomenon might have a similar signature for bounded partitions [i.e. partitions restricted to a box of size $B=\mathcal {O}(n^{1/4})$], we have re-examined the numerical data from Ref. [14] as follows. For given values of $B$ and of $n$, there are $p_3(B,n)$ partitions of $n$ such that all coordinates of all nodes are $\le B$. We then define the partial entropy $S(n) \equiv \log p_3(B,n)$ and we explore its behavior near its maximum (at $n=B^4/2$), by opposition to the limit $n \ll B^4$ (or by symmetry $B^4-n \ll B^4$) studied so far. Near this maximum, it has been conjectured that the amoeba of Fig. 5 (Right) becomes a regular “arctic” octahedron [14, 36].

The asymptotic expansion, Eq. (3.7), is replaced by the fit of $S(n)$ near its maximum by a (somewhat arbitrarily) fourth-order polynomial, and the resulting residual is displayed in Fig. 8 for $B=10$. Even though it will have to be confirmed in future studies, this figure suggests they there exist oscillations. By analogy with our above findings, we anticipate that there period should be $2B^3/3$. Indeed, as discussed in [14], near the entropy maximum, the values of $k$ for which the integers $X_k$ (as defined in Sects. 2 and 3.2) are generically non-vanishing define a finite subset of $[0,B-1]^3$ (called the “slab” in Ref. [14]) of cardinality $2B^3/3$. This “slab” is the projection of the “arctic” octahedron on $[0,B-1]^3$. Increasing (resp. decreasing) all the integers $X_k$ in the slab by 1 thus leads to an increase (resp. decrease) of $n\equiv \sum _{k\in [0,B-1]^3} X_k$ by $2B^3/3$. The signature of the underlying lattice is thus expected in this case to lead to a sub-sub-dominant correction of period $2B^3/3$ to the entropy $S(n)$, which is indeed suggested by the figure.

Fig. 8

Residuals between the numerical entropy $S(n)$ and its fourth-order fit on the interval $[3000,7000]$ for $B=10$; the black sinusoid is a guide for eyes and has period $2 B^3/3\simeq 667$. Numerical data were kindly provided by the authors of Ref. [14]