Packing dimers on $(2p + 1) \times (2q + 1) $ lattices

Abstract

We use computational method to investigate the number of ways to pack dimers on \emph{odd-by-odd} lattices. In this case, there is always a single vacancy in the lattices. We show that the dimer configuration numbers on $(2k+1) \times (2k+1)$ \emph{odd} square lattices have some remarkable number-theoretical properties in parallel to those of close-packed dimers on $2k \times 2k$ \emph{even} square lattices, for which exact solution exists. Furthermore, we demonstrate that there is an unambiguous logarithm term in the finite size correction of free energy of odd-by-odd lattice strips with any width $n \ge 1$. This logarithm term determines the distinct behavior of the free energy of odd square lattices. These findings reveal a deep and previously unexplored connection between statistical physics models and number theory, and indicate the possibility that the monomer-dimer problem might be solvable.

Figures (5)

• Figure 1: The configurational states of one lattice site. Suppose the strip with width $n$ is expanding in the vertical direction. The four states that each site can have are depicted here for the central site in the figure. (a) State $0$: the site is empty; (b) State $1$: the site is occupied by the first half of a vertical dimer; (c) State $2$: the site is occupied by the second half of a vertical dimer; (d) State $3$: the site is occupied by a horizontal dimer. Since the lattice strip is growing vertically, for horizontal dimers we do not need to distinguish first half or second half.
• Figure 2: (Color online) Free energy per lattice site ($\ln a_N/n^2$) of odd and even square lattices in unit of $-k_B T$ as a function of $n$, the size of the lattice. The values for odd lattices are in solid circle, those for even lattices in solid square. The value in the thermodynamic limit, $\lim_{n \rightarrow \infty} n^{-2} \ln a_N(n) = 0.291560904$, is also shown as the dotted horizontal line. The values for even lattices beyond $n=20$ are calculated from the exact results Kasteleyn1961Temperley1961. In the inset more data are shown for the even lattices to make it clearer the trend for $\ln a_N/n^2$ to approach the thermodynamic limit. Also shown in solid diamond are the values from lattices where there is a single vacancy restricted at certain specific sites on the boundary of the lattices Tzeng2003.
• Figure 3: (Color online) The original data of $\ln(a_N)/(mn)$ for $n=3$ and the fitted curves. In each panel the data points and curve in the upper part are for odd $m$, and those in the lower part are for even $m$. The dashed horizontal line is $a_0^e(3) = 0.219493$ from exact expression Eq. \ref{['E:a_0_e']}. The data $m \ge m_0 = 100$ are used in the fitting, and they are shown in the top panel. In the bottom panel the same curves are shown together with the original data for $1 \le m < m_0 = 100$, which are not used in the fitting.
• Figure 4: (Color online) The original data of $\ln(a_N)/(mn)$ for $n=15$ and the fitted curves. The dashed horizontal line is $a_0^e(15) = 0.280527$ from exact expression Eq. \ref{['E:a_0_e']}. See the legend of Figure \ref{['F:fit-3']}.
• Figure 5: (Color online) Fitting of $\ln a_N/n^2$ on odd square lattices to Eq. \ref{['E:fit-square']}. Data in the range of $9 \le n \le 19$ are used in fitting. The dashed horizontal line is the value in the thermodynamic limit: $0.291560904$.

Theorems & Definitions (2)

• Conjecture 1
• Conjecture 2