[Paper Review] Dimer lambda_d Expansion Computer Computations

Paul Federbush|ArXiv.org|Apr 26, 2008

Theoretical and Computational Physics3 references3 citations

TL;DR

This paper presents high-precision computer computations of the dimer lambda_d expansion in powers of 1/d, extending the asymptotic series to the 1/d^3 term. Using symbolic computation in Maple, the authors compute the normalized coefficients $\bar{J}_i$ for dimensions d=1,2,3, confirming prior hand calculations and deriving the new $1/d^3$ term, while numerically studying two approximation sequences for $\lambda_d$ that show no clear convergence for d=2 and d=3.

ABSTRACT

In a previous paper an asymptotic expansion for lambda_d in powers of 1/d was developed. The results of computer computations for some terms in the expansion, as well as various quantities associated to the expansion, are herein presented. The computations (in integer arithmetic) are actually done only for d=1, d=2, and d=3, but some results for arbitrary dimension follow from the general structure of the theory. In particular we obtain the next term in powers of 1/d, the 1/d^3 term in the asymptotic expansion for lambda_d, as well as checking the correctness of the terms previously calculated by hand. For d=2 and d=3 a second expansion for lambda_d is defined and numerically studied. Contrary to our hopes this second type of expansion appears not to be convergent. There is not enough evidence to argue if the expansions for lambda_d in powers of 1/d are convergent for either d=2 or d=3, but likely they are not. (Certainly convergence, but at a very slow rate, remains an interesting possibility in all our cases.)

Motivation & Objective

• To compute higher-order terms in the asymptotic expansion of $\lambda_d$ in powers of $1/d$ using computer algebra.
• To verify and extend previously hand-calculated coefficients $\bar{J}_i$ for $\lambda_d$ expansions in dimensions d=1,2,3.
• To investigate the convergence behavior of two distinct approximation sequences for $\lambda_d$ in d=2 and d=3.
• To develop and implement a four-step computational pipeline for efficiently computing $\bar{J}_i$ up to $i=6$.

Proposed method

• Symbolic computation using Maple to calculate $\bar{J}_i = \frac{1}{N} \cdot \frac{1}{i!} \cdot J_i$ for $i=1$ to $6$ and $d=1,2,3$.
• Computation proceeds through four steps: generating $s$-dimer trees with overlap patterns (Step 1), computing connected tile sums (Step 2), evaluating $J_s$ from tile overlaps (Step 3), and substituting into the main formula to obtain approximations (Step 4).
• An $s$-dimer tree is defined as an ordered sequence of $s$ located tiles where each initial subsequence remains connected, enabling efficient algorithmic enumeration.
• The method uses integer arithmetic and avoids floating-point operations to ensure precision in symbolic evaluation of $\bar{J}_i$.
• Two approximation sequences are constructed: $A_r$ based on direct $1/d^i$ terms and $B_r$ using a formal parameter $x$ to model generating functions.
• The computation of $\bar{J}_6$ for $d=2$ and $d=3$ required several weeks of CPU time, with $\bar{J}_7$ estimated as infeasible with current methods.

Experimental results

Research questions

• RQ1 What is the next term in the asymptotic expansion of $\lambda_d$ in powers of $1/d$, specifically the $1/d^3$ coefficient?
• RQ2 Are the previously hand-computed $\bar{J}_i$ coefficients for $\lambda_d$ accurate, and can they be verified via computer algebra?
• RQ3 Do the two approximation sequences $A_r$ and $B_r$ for $\lambda_d$ converge for $d=2$ and $d=3$?
• RQ4 Can the general form of $\bar{J}_i$ be derived for arbitrary dimension $d$ from the computed values at $d=1,2,3$?
• RQ5 What is the computational complexity of extending the $\bar{J}_i$ computation beyond $i=6$ for $d=2$ and $d=3$?

Key findings

• The asymptotic expansion of $\lambda_d$ is extended to include the $1/d^3$ term: $\lambda_d \sim \frac{1}{2}\ln(2d) - \frac{1}{2} + \frac{1}{8d} + \frac{5}{96d^2} + \frac{5}{64d^3} + \cdots$
• The $\bar{J}_i$ coefficients for $d=1,2,3$ are computed exactly: $\bar{J}_1 = 0$, $\bar{J}_2 = \frac{1}{8d}$, $\bar{J}_3 = \frac{1}{12d^2}$, $\bar{J}_4 = -\frac{3}{32d^2} + \frac{3}{64d^3}$, $\bar{J}_5 = -\frac{1}{8d^3} - \frac{3}{80d^4}$, $\bar{J}_6 = \frac{7}{48d^3} - \frac{5}{64d^4} - \frac{1}{6d^5}$.
• For $d=2$, the $A_r$ sequence gives $A_3 = 0.2784$, approaching the known exact value $\lambda_2 \approx 0.29156$, while the $B_r$ sequence peaks at $B_2 = 0.2921$ and then declines.
• For $d=3$, the $A_r$ sequence gives $A_3 = 0.4462$, within the known bounds $0.440075 \leq \lambda_3 \leq 0.457547$, while the $B_r$ sequence peaks at $B_2 = 0.4538$ and then decreases.
• The $B_r$ sequences for $d=2$ and $d=3$ do not converge, with oscillations and eventual decline, suggesting non-convergence of this formal expansion.
• The computation of $\bar{J}_6$ for $d=2$ and $d=3$ required several weeks of CPU time, indicating that $\bar{J}_7$ would require approximately a year of computation and significant new theoretical insight.