Power series expansions for the planar monomer-dimer problem

We compute the free energy of the planar monomer-dimer model. Unlike the classical planar dimer model, an exact solution is not known in this case. Even the computation of the low-density power series expansion requires heavy and nontrivial computations. Despite the exponential computational complexity, we compute almost three times more terms than were previously known. Such an expansion provides both lower and upper bounds for the free energy and makes it possible to obtain more accurate numerical values than previously possible. We expect that our methods can be applied to other similar problems.

Figure 1

$F_5^{\left(6\right)}$.

Figure 2

Tilings of $F_5^{\left(6\right)}$ to tilings of $F_{10}^{\left(7\right)}$ with property $A_0$.

Figure 3

Tilings of $F_8^{\left(7\right)}$ with property $A_3$ to tilings of $F_{10}^{\left(7\right)}$ with property $A_2$.

Figure 4

Tilings of $F_8^{\left(7\right)}$ with property $A_3$ to tilings of $F_{14}^{\left(7\right)}$ with property $A_2$.

Figure 5

“Large” and “thin” polyomino for $N=15$.

Figure 6

Plots of $L_{64}\left(p\right)$ and $U_{64}\left(p\right)$: (a) on $[0,1]$; (b) on $[0.9,1]$.