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\centerline{Dimer $\la_3 = .453 \pm .001$ and Some Other Very
Intelligent Guesses}
\vspace{1in}
\centerline{Paul Federbush}
\centerline{Department of Mathematics}
\centerline{University of Michigan}
\centerline{Ann Arbor, MI 48109-1043}
\centerline{(pfed@umich.edu)}
\vspace{1in}
\centerline{\underline{Abstract}}
Working with a presumed asymptotic series for $\la_d$ developed in
previous work, we make some intelligent guesses for $\la_d$ with
$d=3, 4, 5$; and estimates for the corresponding errors. We present
arguments in favor of these guesses, we earnestly believe they will
turn out to be correct. Such approximate values may help stimulate
people working on rigorous bounds. In addition to suggesting bounds
to prove, there will be the strong motivation to prove me wrong.
\vfill\eject
In a previous paper, [1], we developed an asymptotic expansion for $
\la_d$ of the dimer problem in powers of $1/d$. In [2] computer
calculations were performed to obtain some terms in this expansion,
and also related quantities arising in the theory. Using results in
[2] we herein will argue for the following estimates.
\begin{eqnarray}
\la_2 &=& .296 \pm .007 \\
\la_3 &=& .453 \pm .001 \\
\la_4 &=& .5748 \pm .0006\\
\la_5 &=& .6785 \pm .0001
\end{eqnarray}
This entire note is to argue for (2), (3), and (4). The resut for $
\la_2$ is exactly known, consistent with (1), which is included as
the bellwether example of our algorithm to obtain (1) - (4).
In [2] we obtained in dimensions 2 and 3 a series $B_0, ..., B_5$,
eq. (41)-(52) of [2]. Using eq. (22)-(27) of [2] and the theory
developed in [1] one can obtain such a series $B_0,...,B_5$ in any
dimension. Our algorithm to obtain (1)-(4) above, in given dimension
$d$, is to seek the two successive $B's$, $B_g$ and $B_{g+1}$, with
minimum value of $|B_g - B_{g+1}|$. Then with
\be a = \frac 1 2 \big( B_g + B_{g+1} \big) \ee
and
\be b = |B_g - B_{g+1}| \ee
our estimate is
\be \la_d = a \pm b . \ee
The following table encapsulates the $B$ series for $d=2,3,4,5$.
\[
\begin{tabular}{c|c|c|c|c|}
& $d=2$ & $d=3$ & $d=4$ & $d=5$ \\ \hline
& & & & \\
$B_0$ & .1931 & .3959 & .5397 & .6513 \\
& & & & \\
$B_1$ & .2556 & .4375 & .5710 & .6763 \\
& & & & \\
$B_2$ & .2921 & .4538 & .5801 & .6821 \\
& & & & \\
$B_3$ & .2993 & .4524 & .5781 & .6803 \\
& & & \\
$B_4$ & .2906 & .4468 & .5751 & .6786 \\
& & & & \\
$B_5$ & .2814 & .4445 & .5745 & .6785
\end{tabular} \]
It is generally believed that it is hard to compute the $\la_d, \ d >
2$, with great precision. We note that the elements $B_5$ in our
table required weeks of computer time to evaluate, and we question
whether the $B_6$ terms will ever be computed. We are only claiming $
\la_3$, say, with a certain accuracy; we do not know how to get $\la_3
$ with greater accuracy. Because we do not have the $B_6$ terms, it
might be better to double the error bounds of (3) and (4) above.
We give the following arguments in favor of our algorithm, (5)-(7)
above.
1) The $B_i$ term in the series of $B's$ is the sum of the first $i$
terms in a power series in $x$ (set equal to 1) that is presumed
asymptotic. Equations (5)-(7) encode the general rule of thumb
wisdom for extracting a `best value' of the sum of an asymptotic
expansion in a simple way.
2) This algorithm gives a correct estimate for $\la_2$, eq. (1),
which is known exactly. We take this as rather compelling.
3) $B_g$ as a function of $d$ is an expansion in powers $1/d$ up to
power $1/d^g$. The asymptotic expansion of [1] for $\la_d$ in powers
of $1/d$ yields a `best approximation' in high dimensions that is a
power series in $1/d$ to a high power. For the approximation of our
algorithm to match this approximation, $g$ {\it must increase with
dimension}.
We have found this last argument hard to explain here, and it would
also be hard to write down a precise statement corresponding to the
discussion (though it could be done). But for us the fact that $g=2$
for $d=2$ and $d=3$, and $g=4$ for $d=4$ and $d=5$, was as important
as arguments 1) and 2) above. It gives the algorithm a final ring of
truth.
\vspace{1in}
\centerline{\underline{References}}
\begin{itemize}
\item[[1]] Paul Federbush, Hidden Structure in Tilings, Conjectured
Asymptotic Expansion for $\la_d$ in Multidimensional Dimer Problem, \
arXiv : 0711.1092V9 [math-ph].
\item[[2]] Paul Federbush, Dimer $\la_d$ Expansion Computer
Computations, \ arXiv : 0804.4220V1 [math-ph].
\end{itemize}
\end{document}