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\title{\LARGE $B%@%$%^!<LO7?$H$=$N<~JU(B}
\author{$B9b:j6b5W(B (Kanehisa Takasaki)\\
$B5~ETBg3XBg3X1!?M4V!&4D6-3X8&5f2J(B\\
Graduate School of Human and Environmental Studies\\
Kyoto University}
\date{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

\maketitle
\bigskip

\section{$B$O$8$a$K(B}

$B%@%$%^!<!J(Bdimer$B!$#2NLBN!K$O%b%N%^!<$H8F$P$l$k(B
$BJ,;R$,#2$D=E9g$7$?7A(B \rowdimer $B$NJ,;R$r0UL#$9$k(B
$B!J$A$J$_$K!$%b%N%^!<$,B??t=E9g$7$?J,;R$,%]%j%^!<$G$"$k!K!%(B
$B%@%$%^!<LO7?$O%@%$%^!<$N$5$^$6$^$JG[CV(B $M$ $B$K4X$9$k(B
$B>uBVOB!JJ,G[H!?t!K(B
\beqnn
  Z = \sum_{M} e^{-E(M)}
\eeqnn
$B$GDj5A$5$l$kE}7WNO3XE*LO7?$G$"$k!%(B
$B=E$_!J(BBoltzmann weight$B!K(B$e^{-E(M)}$ $B$,(B $1$ 
$B$N>l9g$K$O%@%$%^!<G[CV$N?t$(>e$2$NLdBj$K$J$k!%(B

$B%@%$%^!<LO7?$r?t3XE*$KDj<02=$9$k$K$O%0%i%UM}O@$N(B
$B8@MU$rMQ$$$k!%%0%i%UM}O@$N8@MU$G$O!$%@%$%^!<$NG[CV$O(B
$B#2It%0%i%U$N40A4%^%C%A%s%0$KB>$J$i$J$$!%(B
$B$3$l$OC1$J$kDj<02=$NJX59$K$H$I$^$i$J$$0UL#$r$b$D!%(B
$BN%;6?t3X$NJ,Ln$G$O!V%^%H%m%$%I9=B$!W$N0l<o$H$7$F(B
$B$h$/CN$i$l$F$$$k$3$H$@$,!$40A4%^%C%A%s%0F1;N$O(B
$B!V2sE>!W$H$$$&A`:n$G8_$$$K0\$j9g$&!%%0%i%UM}O@$G$O(B
$B$3$N$3$H$,40A4%^%C%A%s%0$K4X$9$k=tLdBj(B
$B!JFC$K$"$k<o$N:GE,2=LdBj!K$r2r$/80$H$J$k!%(B
$B$8$D$O!$%@%$%^!<LO7?$NJ,G[H!?t$r07$&:]$K$b(B
$B$3$N!V2sE>!W$H$$$&8+J}$,=EMW$J<j$,$+$j$rM?$($k!%(B
$B$3$N$3$H$O(B1960$BG/Be$N(BKasteleyn$B$N8&5f$K$h$C$F(B
$BL@$i$+$K$J$C$?!%(B

Kasteleyn$B$OJ?J}3J;R%0%i%U$N>e$N%@%$%^!<LO7?$KBP$7$F(B
$BJ,G[H!?t$N$"$kI=<0!J6=L#?<$$$3$H$K!$#2<!85%$%8%s%0(B
$B!J(BIsing$B!KLO7?$NJ,G[H!?t$H$[$H$s$IF1$87A$r$7$F$$$k!K$rF@$?(B 
\cite{Kasteleyn61}$B!%$3$NI=<0$+$iLO7?$NG.NO3XE*6K8B(B
$B$K$*$1$k<+M3%(%M%k%.!<$,87L)$J7A$G5a$a$i$l$?!%(B
$B$5$i$K(BKasteleyn$B$OF1$89M$(J}$,0lHL$NJ?LLE*#2It%0%i%U$N(B
$B>l9g$K$bDLMQ$9$k$3$H$r<($7$?(B \cite{Kasteleyn67}$B!%(B
$B$3$l$i$N8&5f$K$*$$$F(BKasteleyn$B$,MQ$$$?$N$O!$(B
$B%0%i%U$NNY@\9TNs$KE,Ev$J=E$_$HId9f$r>h$8$FF@$i$l$k(B
$BH?BP>N9TNs!J:#F|$G$O!V%+%9%F%l%$%s!J(BKasteleyn$B!K9TNs!W(B
$B$H8F$P$l$k!K$G$"$k!%(BKasteleyn$B$O%@%$%^!<LO7?$NJ,G[H!?t$,(B
$B$3$N9TNs$N%Q%U<0!J(BPfaffian$B!K$H$7$FI=$;$k$3$H$r<($7!$(B
$BFC$KJ?J}3J;R%0%i%U$N>l9g$K$3$N%Q%U<0!J<B:]$K$O(B
$B$=$l$HF1Ey$J9TNs<0!K$r6qBNE*$K7W;;$7$F!$%$%8%s%0LO7?$N(B
$BJ,G[H!?t$NI=<0$K;w$?I=<0$rF3$$$?$N$G$"$k!%(B

Kasteleyin$B$,(B1960$BG/Be$K9T$C$?8&5f$O(B2000$BG/A08e$+$i(B
$B:F$S4X?4$r=8$a$k$h$&$K$J$j!$%@%$%^!<LO7?<+BN$KBP$9$k(B
$BM}2r$,?<$^$C$?!J(BKenyon$B$N2r@b(B \cite{Kenyon03} $B;2>H!K(B
$B$@$1$G$J$/!$?tO@$d4v2?3X$J$I$H$N;W$$$,$1$J$$4XO"$b(B
$B8+=P$5$l$?!J?"ED$N2r@b5-;v(B \cite{Ueda05} $B;2>H!K!%(B
$B$=$N8e8=:_$^$GB3$/8&5f$NCf$G$bFCI.$9$Y$-$J?JE8$H$7$F!$(B
Kenyon, Okounkov, Sheffield $B$N@.2L(B \cite{KOS03} 
$B$,7G$2$i$l$k!%H`$i$O<~4|E*6-3&>r7o$N$b$H$G$N%@%$%^!<LO7?(B
$B!J%H!<%i%9>e$N#2It%0%i%U$GDj<02=$5$l$k!K$r0lHLE*$J@_Dj$G(B
$B9M;!$7!%G.NO3XE*6K8B$K$*$1$k<+M3%(%M%k%.!<$r7hDj$7$?!%(B
$B$=$7$F!$$=$3$K$"$k<o$NJ?LLBe?t6J@~!J%9%Z%/%H%k6J@~!K(B
$B$H$=$l$KIU?o$9$k!V%"%a!<%P!W$H8F$P$l$k4v2?3XE*9=B$$H$N(B
$B;W$$$,$1$J$$4XO"$r8+=P$7$?!%(B

$B%"%a!<%P$O%m!<%i%sB?9`<0$N%K%e!<%H%sB?LLBN$K(B
$B4v2?3XE*2r<a$rM?$($k$3$H$rF05!$H$7$F@8$^$l$?35G0$G$"$j(B
\cite{GKZ94}$B!$K\Mh$OB?JQ?tD64v2?H!?t$N8&5f$H(B
$BL)@\$K4XO"$9$k$b$N$@$C$?$,!$$=$N8e!V%H%m%T%+%k4v2?3X!W(B
$B$J$I$N4QE@$G$b8&5f$5$l$F$$$k!%$5$i$K!$(B
$B%m!<%i%sB?9`<0$+$i$O!V%m%s%-%s!J(BRonkin$B!KH!?t!W(B
$B$H8F$P$l$kH!?t$,Dj5A$5$l$k$,!$$=$l$O%K%e!<%H%sB?LLBN$d(B
$B%a!<%P$N4v2?3XE*FCD'$HL)@\$K4XO"$9$kCx$7$$@-<A$r$b$D(B \cite{PR04}$B!%(B
$B!J0J>e$NOCBj$K$D$$$F$O(B Mikhalkin \cite{Mikhalkin04} 
$B$d(BKirillov$B$HA0Ln(B \cite{KM06} $B$N2r@b$,$"$k!%!K(B

Kenyon$B$?$A(B\cite{KOS03}$B$O<~4|E*%@%$%^!<LO7?$N(B
$B!V%9%Z%/%H%k6J@~!W$,FCJL$J@-<A$r$b$DJ?LLBe?t6J@~(B
$B$G$"$k$3$H$KCmL\$7!$G.NO3X6K8B$N<+M3%(%M%k%.!<$,(B
$B$3$NBe?t6J@~$KH<$&%m%s%-%sH!?t$KB>$J$i$J$$$3$H$rL@$i$+$K$7$?!%(B
$B$3$NJ}LL$N8&5f$O$=$N8e$bB3$$$F$$$k(B \cite{KO03,KO05}$B!%(B
$BB>J}$G$O!$D689M}O@$d%2!<%8M}O@$N8&5f$H$N;W$$$,$1$J$$4XO"$b(B
$B8+=P$5$l$F$$$k(B \cite{ORV06,HK05,FHKVW05,FHKV05,MN06}$B!%(B

$B0J2<$G$O$3$N%@%$%^!<LO7?$K$D$$$F4pACE*$JItJ,$r2r@b$9$k!%(B
$BFC$K!$J?LLE*#2It%0%i%U$N>l9g$KJ,G[H!?t$,%Q%U<0$d(B
$B9TNs<0$H$7$FI=$;$k;EAH$_$rCzG+$K2r@b$9$k!%(B
$B<~4|E*%@%$%^!<LO7?$KIU?o$7$F8=$l$k%"%a!<%P$d(B
$B%m%s%-%sH!?t$K$D$$$F$b:G8e$K4JC1$K?($l$k!%(B
$B$5$i$K>\$7$$$3$H$K$D$$$F$O!$:G8e$K<($9(B
$B;29MJ88%$r;2>H$5$l$?$$!%(B



\section{$B%0%i%UM}O@$r>/$7$@$1(B}


\subsection{$B%0%i%U$H$=$NI=8=J}K!(B}

$B%0%i%U$O!VE@$H@~!W$N9=B$$G$"$j!$$$$/$D$+$N(B
$B!J>l9g$K$h$C$F$OL58B8D$N!KD:E@$H$=$N4V$r7k$VJU$+$i$J$k!%(B
$B%0%i%U$r?^7AE*$KIA$$$F9M$($k$3$H$bB?$$$,!$(B
$BCj>]E*$J0UL#$G$N%0%i%U$OD:E@$N=89g(B $V$ $B$H(B
$BJU$N=89g(B $E$ $B$NAH(B $G = (V,E)$ $B$K2a$.$J$$!%(B
$BG$0U$N#2D:E@4V$KJU$,9b!9#1$D$7$+$J$$%0%i%U$G$O!$(B
$BJU$NN>C<(B $v,v'$ $B$r;XDj$9$l$PJU(B $e = (v,v')$ 
$B$,7h$^$k!%0J2<$G$OFC$KCG$i$J$$8B$j$=$N$h$&$J%0%i%U(B
$B$N$_$r9M$($k!J#2D:E@4V$KJ#?t$NJU$,$"$k%0%i%U$O(B
$B!VB?=E%0%i%U!W!$$=$N$h$&$JJU$O!VB?=EJU!W8F$P$l$k(B
\footnote{$B<~4|E*%@%$%^!<LO7?$NDj<02=$K$*$$$F$O!$(B
$BB?=EJU$r$b$D%0%i%U$r9M$($k$3$H$,$"$k!%(B}
$B!K!%N>C<$,F1$8$G$"$k$h$&$JJU(B $(v,v)$ $B$O%k!<%W$H8F$P$l$k!%(B

$BJU$K8~$-$,;XDj$5$l$F$$$k$+H]$+$K$h$C$F%0%i%U$O(B
$B!VM-8~%0%i%U!W$H!VL58~%0%i%U!W$KJ,$+$l$k!%(B
$BL58~%0%i%U$G$OJU$K8~$-$,$J$$$N$G!$(B$(v,v')$ 
$B$H(B $(v',v)$ $B$OF1$8JU$rI=$9$b$N$H2r<a$5$l$k!%(B
$B$3$l$i$r6hJL$9$l$PM-8~%0%i%U$H$J$k!%(B
$B0J2<$G$OL58~%0%i%U$N$_$r9M$($k$N$G!$(B
$B!VL58~!W$H$$$&8@MU$r>J$$$FC1$K%0%i%U$H$$$&$3$H$K$9$k!%(B

$B%0%i%U$rM?$($k$3$H$OD:E@$N=89g(B $V$ $B$N>e$K(B
$B!VNY@\4X78!W$rDj$a$k$3$H$HF1Ey$G$"$k!%(B
$BJU$G7k$P$l$?#2D:E@(B $v,v'$ $B$O!VNY@\$7$F$$$k!W(B
$B$H$$$&!%$3$l$OD:E@$N=89g(B $V$ $B$N>e$K#29`4X78!$(B
$B8@$$49$($l$P%V!<%kH!?t(B
\beqnn
  a:\ V \times V \to \{0,1\}, \quad 
  a(v,v') 
  = \begin{cases}
    1 & \text{$B!J(Bv,v'$B$,NY@\$9$k$H$-!K(B}\\
    0 & \text{$B!J$=$&$G$J$$$H$-!K(B}
    \end{cases}
\eeqnn
$B$rDj$a$k!%$3$NNY@\4X78$r(B $V \times V$ $B$N9TNs(B
\beqnn
  A = (a(v,v'))_{v,v'\in V} 
\eeqnn
$B$GI=$9$3$H$b$G$-$k!%$3$l$r!VNY@\9TNs!W$H$$$&!%(B

$BF1MM$N0UL#$G!$$3$lD:E@$N=89g(B $V$ $B$HJU$N=89g(B $E$ 
$B$N4V$K!V@\B34X78!W$,Dj$^$k!%$9$J$o$A!$D:E@(B $v$ $B$,(B
$BJU(B $e$ $B$NC<E@$G$"$k$H$-!$D:E@(B $e$ $B$HJU(B $v$ $B$O(B
$B!V@\B3$7$F$$$k!W$H$$$&!JM-8~%0%i%U$N>l9g$K$O!$(B
$B;OE@$G$"$k$+=*E@$G$"$k$+$K$h$C$F#2<oN`$N@\B34X78(B
$B$,9M$($kI,MW$,$"$k!K!%$3$l$i$r(B $V \times E$ $B9TNs$G(B
$BI=8=$7$?$b$N$r!V@\B39TNs!W$H$$$&!%0J2<$G$O@\B39TNs$O(B
$B@Q6KE*$KMQ$$$J$$$N$G!$FC$K5-9f$ODj$a$J$$!%(B

\paragraph*{$BNc(B}
$n$ $B8D$ND:E@$N4V$r$9$Y$FJU$G7k$s$@%0%i%U$r(B $n$ $B0L$N(B
$B40A4%0%i%U$H8F$S!$(B$K_n$ $B$H$$$&5-9f$GI=$9!%(B
$B$?$H$($P(B $K_5$ $B$O?^(B\ref{K_n-fig}$B$K<($7$?$h$&$J(B
$B%0%i%U$K$J$k!%NY@\9TNs$O!JD:E@$r$I$N=g=x$K(B
$B%i%Y%kIU$1$7$F$b!K<!$N$h$&$K$J$k!'(B
\beqnn
  A = \left(\begin{array}{ccccc}
      0 & 1 & 1 & 1 & 1 \\
      1 & 0 & 1 & 1 & 1 \\
      1 & 1 & 0 & 1 & 1 \\
      1 & 1 & 1 & 0 & 1 \\
      1 & 1 & 1 & 1 & 0 
      \end{array}\right) 
\eeqnn
\bigskip

\begin{figure}[t]
\begin{center}
\begin{picture}(160,100)
\put(80,90){\line(-1,-3){30}}
\put(50,0){\line(1,0){60}}
\put(110,0){\line(-1,3){30}}
\put(110,0){\line(-4,3){80}}
\put(30,60){\line(1,-3){20}}
\put(30,60){\line(5,3){50}}
\put(80,90){\line(5,-3){50}}
\put(130,60){\line(1,0){0}}
\put(130,60){\line(-1,-3){20}}
\put(130,60){\line(-4,-3){80}}
\put(30,60){\line(1,0){100}}
\put(80,90){\circle*{6}}
\put(130,60){\circle*{6}}
\put(30,60){\circle*{6}}
\put(110,0){\circle*{6}}
\put(50,0){\circle*{6}}
\end{picture}
\caption{$B40A4%0%i%U(B $K_5$}
\label{K_n-fig}
\end{center}
\end{figure}

$BJ?LL>e$GJU$,8r:5$7$J$$$h$&$KIA$1$k%0%i%U$O(B
$BJ?LLE*$G$"$k$H$$$&!%$?$H$($P!$#40L$^$G$N(B
$B40A4%0%i%U(B $K_2,K_3,K_4$ $B$OJ?LLE*$@$,!$(B$K_5$ $B$O(B
$BJ?LLE*$G$J$$!%#60L0J>e$N40A4%0%i%U$O(B $K_5$ $B$r4^$`(B
$B$N$GJ?LLE*$G$O$J$$!%(B


\subsection{$B#2It%0%i%U$H40A4%^%C%A%s%0(B}

$BD:E@=89g(B $V$ $B$,6&DLItJ,$r$b$?$J$$#2$D$N(B
$BItJ,=89g(B $V_1,V_2$ $B$KJ,$+$l$F(B
$B!J(B$V = V_0 \cup V_1$, $V_0 \cap V_1 
= \emptyset$$B!K!$JU$,(B $V_1$ $B$H(B $V_2$ $B$N4V$K$N$_(B
$BB8:_$9$k$h$&$J%0%i%U$r#2It%0%i%U$H8F$S!$(B
$G = (V_1,V_2,E)$$B!J(B$E$ $B$OJU$N=89g!K(B
$B$H$$$&$h$&$K#3$DAH$H$7$FDj<02=$9$k!%(B

$B#2It%0%i%U$O#2?'$K:L?'$5$l$?%0%i%U$H9M$($k(B
$B$3$H$b$G$-$k!%0lHL$K!$%0%i%U$N:L?'$H$O!$(B
$B$$$/$D$+$N?'$rMQ0U$7$F!$3FD:E@$K$=$N$&$A$N0l$D$r(B
$B;XDj$9$k$3$H$G$"$k!%$?$@$7!$NY@\$9$k#2D:E@$N?'$O(B
$B0lCW$7$J$$$H$$$&>r7o$rK~$?$9$b$N$H$9$k!%(B
$B#2It%0%i%U$O#2?'$G:L?'$G$-$k%0%i%U$KB>$J$i$J$$!$(B
$B0J2<$G$O!$(B$V_1$ $B$ND:E@$OGr!J(Bwhite$B!K$K!$(B
$V_2$ $B$ND:E@$K$O9u!J(Bblack$B!K$K:L?'$5$l$F$$$k$H8+$J$9!%(B

\paragraph*{$BNc(B}
$BJ?J}3J;R$d#63Q3J;R$O#2?'$K:L?'2DG=$G$"$j!$(B
$B#2It%0%i%U$H8+$J$9$3$H$,$G$-$k!J?^(B\ref{bgraph-fig}$B!K!%(B
$B$3$l$i$OJ?LLE*$G$b$"$k!%(B
\bigskip

\begin{figure}[t]
\begin{center}
\begin{picture}(120,100)
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\multiput(0,33)(30,0){4}{\line(0,1){24}}
\multiput(3,30)(30,0){3}{\line(1,0){24}}
\multiput(30,30)(60,0){2}{\circle*{6}}
\multiput(0,30)(60,0){2}{\circle{6}}
\multiput(0,3)(30,0){4}{\line(0,1){24}}
\multiput(3,0)(30,0){3}{\line(1,0){24}}
\multiput(30,0)(60,0){2}{\circle{6}}
\multiput(0,0)(60,0){2}{\circle*{6}}
\end{picture}
\begin{picture}(160,130)
\multiput(20,120)(40,0){4}{\circle{6}}
\multiput(23,118)(40,0){4}{\line(4,-3){14}}
\multiput(3,107)(40,0){4}{\line(4,3){14}}
\multiput(0,105)(40,0){5}{\circle*{6}}
\multiput(0,88)(40,0){5}{\line(0,1){14}}
\multiput(0,85)(40,0){5}{\circle{6}}
\multiput(3,83)(40,0){4}{\line(4,-3){14}}
\multiput(23,72)(40,0){4}{\line(4,3){14}}
\multiput(20,70)(40,0){4}{\circle*{6}}
\multiput(20,53)(40,0){4}{\line(0,1){14}}
\multiput(20,50)(40,0){4}{\circle{6}}
\multiput(23,48)(40,0){4}{\line(4,-3){14}}
\multiput(3,37)(40,0){4}{\line(4,3){14}}
\multiput(0,35)(40,0){5}{\circle*{6}}
\multiput(0,18)(40,0){5}{\line(0,1){14}}
\multiput(0,15)(40,0){5}{\circle{6}}
\multiput(3,13)(40,0){4}{\line(4,-3){14}}
\multiput(23,2)(40,0){4}{\line(4,3){14}}
\multiput(20,0)(40,0){4}{\circle*{6}}
\end{picture}
\caption{$B#2It%0%i%U$H$7$F$NJ?J}3J;R!&#63Q3J;R(B}
\label{bgraph-fig}
\end{center}
\end{figure}


$B#2It%0%i%U(B $G = (V_1,V_2,E)$ $B$K$*$$$F(B
$BC<E@$r6&M-$7$J$$$$$/$D$+$NJU$+$i$J$k=89g(B
\beqnn
  M = \{(w_1,b_1),\,(w_2,b_2),\,\cdots,\,(w_n,b_n)\}, 
  \quad 
  w_i \not= w_j,\; b_i \not= b_j \; (i \not= j)
\eeqnn
$B$r%^%C%A%s%0$H$$$&!%%^%C%A%s%0$,(B
\beqnn
  \{w_1,w_2,\cdots,w_n\} = V_1, \quad 
  \{b_1,b_2,\cdots,b_n\} = V_2 
\eeqnn
$B$H$$$&$h$&$K(B $G$ $B$N$9$Y$F$ND:E@$rJ$$&$H$-!$(B
$B40A4%^%C%A%s%0$H$$$&!J?^(B\ref{matching-fig}$B!K!%(B

\begin{figure}[t]
\begin{center}
\begin{picture}(140,100)
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\multiput(30,90)(60,0){2}{\circle*{6}}
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\put(63,63){\line(1,0){24}}
\put(3,93){\line(1,0){24}}
\put(63,93){\line(1,0){24}}
\put(100,0){\makebox{$M$}}
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\begin{picture}(140,100)
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\multiput(3,60)(30,0){3}{\line(1,0){24}}
\multiput(30,60)(60,0){2}{\circle{6}}
\multiput(0,60)(60,0){2}{\circle*{6}}
\multiput(0,33)(30,0){4}{\line(0,1){24}}
\multiput(3,30)(30,0){3}{\line(1,0){24}}
\multiput(30,30)(60,0){2}{\circle*{6}}
\multiput(0,30)(60,0){2}{\circle{6}}
\multiput(0,3)(30,0){4}{\line(0,1){24}}
\multiput(3,0)(30,0){3}{\line(1,0){24}}
\multiput(30,0)(60,0){2}{\circle{6}}
\multiput(0,0)(60,0){2}{\circle*{6}}
\put(3,3){\line(1,0){24}}
\put(63,3){\line(0,1){24}}
\put(3,33){\line(1,0){24}}
\put(93,3){\line(0,1){24}}
\put(3,63){\line(0,1){24}}
\put(63,63){\line(1,0){24}}
\put(33,63){\line(0,1){24}}
\put(63,93){\line(1,0){24}}
\put(100,0){\makebox{$M'$}}
\end{picture}
\caption{$BJ?J}3J;R%0%i%U$N40A4%^%C%A%s%0$NNc(B
$B!J#2=E@~$O%^%C%A%s%0$KB0$9$kJU$rI=$9!K(B}
\label{matching-fig}
\end{center}
\end{figure}

$BM?$($i$l$?#2It%0%i%U$K$*$1$k40A4%^%C%A%s%0$O(B
$B2sE>A`:n$rDL$8$F8_$$$K0\$j9g$&!%(B
$B$3$N$3$H$OFs$D$N40A4%^%C%A%s%0$N0c$$$r(B
$B!V8r8_JDO)!W$H8F$P$l$k$b$N$GI=8=$9$k$3$H$K$h$C$F(B
$B@bL@$5$l$k!%(B

$M$ $B$H(B $M'$ $B$r#2$D$N40A4%^%C%A%s%0$H$9$k!%(B
$B$3$l$i$r(B $G$ $B>e$K=E$M$FIA$$$?$b$N$O8_$$$K8r:5$7$J$$(B
$B$$$/$D$+$NJU$HC1=cJDO)$KJ,$+$l$k!J?^(B\ref{overlap-fig}$B!K!%(B
$B$3$N$&$AJDO)0J30$NJU$O(B $M$ $B$H(B $M'$ $B$K6&DL$N$b$N$G$"$j!$(B
$B$=$l$,=E$J$j$"$C$F$$$k$N$G0l$D$NJU$K8+$($F$$$k!%(B
$BB>J}!$JDO)$OFs$D$N40A4%^%C%A%s%0$,?)$$0c$C$F$$$k(B
$B$H$3$m$KAjEv$9$k!%$3$NJDO)$O(B $M$ $B$KB0$9$kJU$H(B $M'$ 
$B$KB0$9$kJU$,8r8_$K8=$l$k$H$$$&FCD'$r$b$D!%(B
$B$3$N$h$&$JJDO)$r8r8_JDO)$H$$$&!%(B$M$ $B$H(B $M'$ 
$B$N0c$$$O$3$l$i$N8r8_JDO)$N=89g(B $\mathcal{C} 
= \{C_1,\cdots,C_k\}$ $B$K$h$C$FI=8=$5$l$k!%(B

\begin{figure}[t]
\begin{center}
\begin{picture}(140,100)
\multiput(3,90)(30,0){3}{\line(1,0){24}}
\multiput(30,90)(60,0){2}{\circle*{6}}
\multiput(0,90)(60,0){2}{\circle{6}}
\multiput(0,63)(30,0){4}{\line(0,1){24}}
\multiput(3,60)(30,0){3}{\line(1,0){24}}
\multiput(30,60)(60,0){2}{\circle{6}}
\multiput(0,60)(60,0){2}{\circle*{6}}
\multiput(0,33)(30,0){4}{\line(0,1){24}}
\multiput(3,30)(30,0){3}{\line(1,0){24}}
\multiput(30,30)(60,0){2}{\circle*{6}}
\multiput(0,30)(60,0){2}{\circle{6}}
\multiput(0,3)(30,0){4}{\line(0,1){24}}
\multiput(3,0)(30,0){3}{\line(1,0){24}}
\multiput(30,0)(60,0){2}{\circle{6}}
\multiput(0,0)(60,0){2}{\circle*{6}}
\put(3,-3){\line(1,0){24}}
\put(63,-3){\line(1,0){24}}
\put(3,27){\line(1,0){24}}
\put(63,27){\line(1,0){24}}
\put(3,57){\line(1,0){24}}
\put(63,57){\line(1,0){24}}
\put(3,87){\line(1,0){24}}
\put(63,87){\line(1,0){24}}
\put(3,3){\line(1,0){24}}
\put(63,3){\line(0,1){24}}
\put(3,33){\line(1,0){24}}
\put(93,3){\line(0,1){24}}
\put(3,63){\line(0,1){24}}
\put(63,63){\line(1,0){24}}
\put(33,63){\line(0,1){24}}
\put(63,93){\line(1,0){24}}
\put(100,0){\makebox{$M \cup M'$}}
\end{picture}
\caption{$B?^(B\ref{matching-fig}$B$N#2$D$N(B
$B40A4%^%C%A%s%0$r9gJ;$7$?$b$N!J#6$D$NJU$H(B
$B#2$D$NC1=cJDO)$KJ,$+$l$k!K(B}
\label{overlap-fig}
\end{center}
\end{figure}

$B8+J}$rJQ$($l$P!$8r8_JDO)$N=89g(B $\mathcal{C} 
= \{C_1,\cdots,C_k\}$ $B$O#2$D$N(B
$B40A4%^%C%A%s%0(B $M,M'$ $B$N0lJ}$+$iB>J}$X>h$j49$($k(B
$B$?$a$N%G!<%?$H8+$J$;$k!%(B$M$ $B$+$i(B $M'$ $B$X(B
$B>h$j49$($k$K$O!$(B$C_1,\cdots,C_k$ $B$K$*$$$F(B $M$ 
$B$KB0$9$kJU$HB0$5$J$$JU!J8r8_$KJB$s$G$$$k!K$r(B
$BF~$lBX$($l$P$h$$!%(B$M'$ $B$+$i(B $M$ $B$X>h$j49$($k(B
$B>l9g$bF1MM$G$"$k!%$3$N>h$j49$($NA`:n$,(B
$B%^%C%A%s%0$NJU$rJDO)$K1h$C$F2sE>$9$k$h$&$K8+$($k$N$G!$(B
$B!V2sE>!W$H8F$P$l$k$N$G$"$k!J?^(B\ref{rotation-fig}$B!K!%(B

\begin{figure}[t]
\begin{center}
\begin{picture}(220,70)
\put(0,0){\line(1,0){30}}
\put(0,2){\line(1,0){30}}
\put(30,0){\line(1,0){30}}
\put(60,0){\line(1,0){30}}
\put(60,2){\line(1,0){30}}
\put(90,0){\line(0,1){30}}
\put(90,30){\line(-1,0){30}}
\put(90,32){\line(-1,0){30}}
\put(60,30){\line(0,1){30}}
\put(60,60){\line(-1,0){30}}
\put(60,62){\line(-1,0){30}}
\put(30,60){\line(0,-1){30}}
\put(30,30){\line(-1,0){30}}
\put(30,32){\line(-1,0){30}}
\put(0,30){\line(0,-1){30}}
\put(120,30){\vector(1,0){20}}
\put(160,0){\line(1,0){30}}
\put(190,0){\line(1,0){30}}
\put(190,2){\line(1,0){30}}
\put(220,0){\line(1,0){30}}
\put(250,0){\line(0,1){30}}
\put(252,0){\line(0,1){30}}
\put(250,30){\line(-1,0){30}}
\put(220,30){\line(0,1){30}}
\put(222,30){\line(0,1){30}}
\put(220,60){\line(-1,0){30}}
\put(190,60){\line(0,-1){30}}
\put(192,60){\line(0,-1){30}}
\put(190,30){\line(-1,0){30}}
\put(160,30){\line(0,-1){30}}
\put(162,30){\line(0,-1){30}}
\end{picture}
\caption{$B8r8_JDO)$K1h$C$?%^%C%A%s%0$N2sE>(B}
\label{rotation-fig}
\end{center}
\end{figure}



\section{$B%@%$%^!<LO7?$H$O2?$+(B}


\subsection{$B%@%$%^!<LO7?$NDj<02=(B}

$B#2It%0%i%U$N8@MU$rMQ$$$l$P!$%@%$%^!<LO7?$O(B
$B0J2<$N$h$&$KDj<02=$5$l$k!%(B
\begin{itemize}
\item[1)] $B#2It%0%i%U(B $G = (V_1,V_2,E)$ $B$r;XDj$9$k!%(B
\item[2)] $B3FJU(B $e = (w,b)$$B!J(B$w \in V_1$, $b \in V_1$$B!K(B
$B$KBP$7$F@5?t(B $W(e)$ $B$r;XDj$9$k!%(B$W(e)$ $B$O#18D$N(B
$B%@%$%^!<(B \rowdimer $B$r(B $e$ $B$KCV$/$H$-$NE}7WNO3XE*=E$_(B
$B!J(BBoltzmann weight$B!K$rI=$9!%(B
\item[3)] $G$ $B$N40A4%^%C%A%s%0(B $M$ $B$KBP$7$F(B
$B$=$N=E$_(B $W(M)$ $B$r(B
\beqnn
  W(M) = \prod_{e\in M} W(e) 
\eeqnn
$B$HDj$a$k!%$3$l$,(B $M$ $B$N3FJU$K%@%$%^!<(B \rowdimer 
$B$rG[CV$7$?>uBV$NJ,G[H!?t$X$N4sM?$H$J$k!%(B
\item[4)] $BJ,G[H!?t(B $Z = Z(G)$ $B$r(B
\beqnn
  Z = \sum_{M\in \mathcal{M}} W(M) 
\eeqnn
$B$HDj$a$k!%$3$3$G(B $\mathcal{M}$ $B$O(B $G$ $B$N(B
$B40A4%^%C%A%s%0A4BN$N=89g$rI=$9!%(B
\end{itemize}


$B8+$+$1$O0[$J$k$,$3$N$h$&$J%@%$%^!<LO7?$KK]Lu$G$-$k(B
$BLdBj$O$$$m$$$m$"$k!%$=$NNc$r$$$/$D$+>R2p$9$k!%(B


\subsection{$B%I%_%N$K$h$k%?%$%kD%$j(B}



$B@5J}7A$rJU$r6-$K$7$FJB$Y$F$G$-$k?^7A$K(B
$B#28D$N@5J}7A$r$D$J$$$@!V%I%_%N!W(B
\rowdomino, \coldomino $B$r(B
$B7d4V$J$/JB$Y$k$3$H$O$$$o$f$k!V%?%$%kD%$j!W(B
$B$N0l<o$G$"$k!%$3$l$O%@%$%^!<LO7?$KK]Lu$G$-$k!%(B
$B<B:]!$@5J}7A$N=E?4$r7k$s$GF@$i$l$k%0%i%U(B
$B!JJ?J}3J;R$rJ?LLE*%0%i%U$H8+$J$9$H$-$N!VAPBP%0%i%U!W(B
$B$KB>$J$i$J$$!K$O#2It%0%i%U$G$"$j!$%I%_%N%?%$%kD%$j$O(B
$B$3$N#2It%0%i%U$N40A4%^%C%A%s%0$K$A$g$&$IBP1~$9$k(B
$B$+$i$G$"$k!%(B

$B%I%_%N%?%$%kD%$j$N?t$(>e$2$N4JC1$JNc$H$7$F!$(B
$2 \times n$ $B$N@5J}7A$rJB$Y$?>l9g(B
\begin{center}
\begin{picture}(180,30)
\put(0,30){\line(1,0){150}}
\put(0,15){\line(1,0){150}}
\put(0,0){\line(1,0){150}}
\multiput(0,0)(15,0){6}{\line(0,1){30}}
\multiput(90,23)(7,0){3}{\circle*{2}}
\multiput(90,7)(7,0){3}{\circle*{2}}
\multiput(120,0)(15,0){3}{\line(0,1){30}}
\end{picture}
\end{center}
$B$r@bL@$7$h$&(B \cite{Knuth-etal}$B!%(B
$B%?%$%kD%$j$NAm?t$r(B $f_n$ $B$HI=$7$F!$$=$NA22=<0$r(B
$BF3$/$3$H$r9M$($k!%1&C<$N#2Ns$KCmL\$9$k$H!$(B
$B%I%_%N$NG[CV$O<!$N#2DL$j$KJ,$+$l$k!'(B
\begin{itemize}
\item[1)] $B1&C<$N#1Ns$K(B \coldomino $B$r#18DCV$$$?$b$N!%(B
\item[2)] $B1&C<$N#2Ns$K(B \rowdomino $B$r>e2<#28DCV$$$?$b$N!%(B
\end{itemize}
1) $B$N$h$&$J%?%$%kD%$j$NAm?t$O(B $2 \times(n-1)$ $B$N(B
$B>l9g$N%?%$%kD%$j$NAm?t(B $f_{n-1}$ $B$KEy$7$$!%(B
$B$^$?(B 2) $B$N$h$&$J%?%$%kD%$j$NAm?t$O(B $2 \times(n-2)$ $B$N(B
$B>l9g$N%?%$%kD%$j$NAm?t(B $f_{n-2}$ $B$KEy$7$$!%$7$?$,$C$F(B
\beqnn
  f_n = f_{n-1} + f_{n-2} 
\eeqnn
$B$H$$$&A22=<0$,F@$i$l$k!%B>J}!$(B$n = 1,2$ $B$N>l9g$K$O(B
\beqnn
  f_1 = 1,\quad f_2 = 2 
\eeqnn
$B$H$J$k!%$3$&$7$F(B $f_n$ $B$,(B
$B%U%#%\%J%C%A?t$KB>$J$i$J$$$3$H$,$o$+$C$?!%(B

$m \times n$ $B$N>l9g$N%I%_%N%?%$%kD%$j$NAm?t$O(B
$B$3$N$h$&$K4JC1$J9M;!$G$O$o$+$i$J$$$,!$(B
Kasteleyn $B$H(B Fisher, Temperley $B$O(B
$B9*L/$JJ}K!$K$h$C$F$=$l$r5a$a$?!%(B
Kastelyn$B$NJ}K!$K$D$$$F$O$"$H$G@bL@$9$k!%(B


\subsection{$BI)7A$K$h$k%?%$%kD%$j(B}

$B@5#33Q7A$rJU$r6-$K$7$FJB$Y$F$G$-$k?^7A(B
$B!J#33Q3J;R$N0lIt$r$J$9!K$KBP$7$F$O!$(B
$B@5#33Q7A$r#2$D$D$J$$$@$b$N!JI)7A$K$J$k!K$rJB$Y$k(B
$B%?%$%kD%$j$,9M$($i$l$k!J?^(B\ref{Lozenge-fig}$B!K!%(B

\begin{figure}[t]
\begin{center}
\begin{picture}(240,130)
\multiput(25,120)(50,0){4}{\line(5,-3){25}}
\multiput(0,105)(50,0){4}{\line(5,3){25}}
\multiput(0,105)(50,0){4}{\line(5,-3){25}}
\multiput(25,90)(50,0){4}{\line(5,3){25}}
%
\multiput(25,90)(50,0){4}{\line(5,-3){25}}
\multiput(0,75)(50,0){4}{\line(5,3){25}}
\multiput(0,75)(50,0){4}{\line(5,-3){25}}
\multiput(25,60)(50,0){4}{\line(5,3){25}}
%
\multiput(25,60)(50,0){4}{\line(5,-3){25}}
\multiput(0,45)(50,0){4}{\line(5,3){25}}
\multiput(0,45)(50,0){4}{\line(5,-3){25}}
\multiput(25,30)(50,0){4}{\line(5,3){25}}
%
\multiput(25,30)(50,0){4}{\line(5,-3){25}}
\multiput(0,15)(50,0){4}{\line(5,3){25}}
\multiput(0,15)(50,0){4}{\line(5,-3){25}}
\multiput(25,0)(50,0){4}{\line(5,3){25}}
%
\multiput(0,0)(25,0){9}{\line(0,1){120}}
\end{picture}
\begin{picture}(100,90)
\put(10,55){\thicklines\line(5,3){25}}
\put(10,55){\thicklines\line(5,-3){25}}
\put(35,40){\thicklines\line(5,3){25}}
\put(35,70){\thicklines\line(5,-3){25}}
\put(35,40){\line(0,1){30}}
%
\put(45,0){\thicklines\line(5,3){25}}
\put(45,0){\thicklines\line(0,1){30}}
\put(70,15){\thicklines\line(0,1){30}}
\put(45,30){\thicklines\line(5,3){25}}
\put(45,30){\line(5,-3){25}}
%
\put(0,15){\thicklines\line(0,1){30}}
\put(0,45){\thicklines\line(5,-3){25}}
\put(25,0){\thicklines\line(0,1){30}}
\put(0,15){\thicklines\line(5,-3){25}}
\put(0,15){\line(5,3){25}}
\end{picture}
\end{center}
\caption{$BI)7A!J1&?^$N$h$&$K#3DL$j$NCV$-J}$,$"$k!K(B
$B$K$h$k#33Q3J;R?^7A!J:8?^!K$N%?%$%kD%$j(B}
\label{Lozenge-fig}
\end{figure}

$B$3$N%?%$%kD%$j$O#3<!856u4VFb$G:BI8<4$KJ?9T$J(B
$BN)J}BN$r@Q$_>e$2$?>uBV$H$7$F2r<a$9$k$3$H$b$G$-$k!%(B
$B$3$N$h$&$JN)J}BN$O#3<!856u4V$N(B $(1,1,1)$ $B$N(B
$BJ}8~$+$iD/$a$l$P(B
\begin{center}
\begin{picture}(50,60)
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\end{picture}
\end{center}
$B$H$$$&$h$&$K8+$($k!%$3$N?^$O#3<!856u4V$+$i(B
$BJ?LL(B $x + y + z = 0$ $B$X$N<M1F$rIA$$$?$b$N$G$"$j!$(B
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$B!J?^(B\ref{cube-Lozenge-fig}$B!K!%(B

\begin{figure}[t]
\begin{center}
\begin{picture}(120,120)
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\put(50,120){\line(5,-3){50}}
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\put(100,30){\line(0,1){60}}
\put(0,30){\line(5,-3){50}}
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%
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\put(75,75){\line(5,3){25}}
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\put(25,15){\line(0,1){60}}
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\put(75,15){\line(0,1){60}}
%
\put(25,75){\line(5,-3){25}}
\put(50,60){\line(5,3){25}}
\put(50,30){\line(0,1){30}}
\end{picture}
\qquad
\begin{picture}(120,120)
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\put(50,120){\line(5,-3){50}}
\put(0,30){\line(0,1){60}}
\put(100,30){\line(0,1){60}}
\put(0,30){\line(5,-3){50}}
\put(50,0){\line(5,3){50}}
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\put(25,105){\line(5,-3){50}}
\put(25,75){\line(5,3){50}}
\put(0,90){\line(5,-3){25}}
\put(75,75){\line(5,3){25}}
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\put(25,15){\line(0,1){60}}
\put(50,0){\line(0,1){30}}
\put(75,15){\line(0,1){60}}
%
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\put(50,60){\line(5,-3){25}}
\put(50,60){\line(0,1){30}}
\end{picture}
\end{center}
\caption{$BN)J}BN$N@Q$_>e$2$rI)7A%?%$%kD%$j$H8+$J$9(B}
\label{cube-Lozenge-fig}
\end{figure}

$B#33Q3J;R$N=E?4$rNY@\$9$k#33Q7A$N4V$G7k$Y$P(B
$B#63Q3J;R%0%i%U!J$b$H$N#33Q3J;R%0%i%U$N(B
$BAPBP%0%i%U!K$,$G$-$k!%$3$l$O#2It%0%i%U$G$"$j!$(B
$BI)7A$K$h$k%?%$%kD%$j$O$3$N#2It%0%i%U$KBP$9$k(B
$B%@%$%^!<$NG[CV$HBP1~$9$k!J?^(B\ref{Lozenge-dimer-fig}$B!K!%(B
$B$3$&$7$F!$$3$N%?%$%kD%$j$b%@%$%^!<LO7?$KK]Lu$G$-$k!%(B

\begin{figure}[t]
\begin{center}
\begin{picture}(120,120)
\put(0,90){\line(5,3){50}}
\put(50,120){\line(5,-3){50}}
\put(0,30){\line(0,1){60}}
\put(100,30){\line(0,1){60}}
\put(0,30){\line(5,-3){50}}
\put(50,0){\line(5,3){50}}
%
\put(25,105){\line(5,-3){50}}
\put(25,75){\line(5,3){50}}
\put(0,90){\line(5,-3){25}}
\put(75,75){\line(5,3){25}}
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%
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%
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%
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\put(65,92){\line(1,0){20}}
%
\put(37,60){\line(1,-3){5}}
\put(62,45){\line(1,3){5}}
\put(40,77){\line(1,0){20}}
\end{picture}
\qquad
\begin{picture}(120,120)
\put(0,90){\line(5,3){50}}
\put(50,120){\line(5,-3){50}}
\put(0,30){\line(0,1){60}}
\put(100,30){\line(0,1){60}}
\put(0,30){\line(5,-3){50}}
\put(50,0){\line(5,3){50}}
%
\put(25,105){\line(5,-3){50}}
\put(25,75){\line(5,3){50}}
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%
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\put(50,60){\line(5,-3){25}}
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%
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\multiput(40,15)(0,30){4}{\line(1,0){20}}
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\multiput(10,45)(25,15){4}{\line(1,-3){5}}
\multiput(10,75)(25,15){3}{\line(1,-3){5}}
\multiput(10,45)(25,-15){3}{\line(1,3){5}}
\multiput(10,75)(25,-15){4}{\line(1,3){5}}
\multiput(35,90)(25,-15){3}{\line(1,3){5}}
%
\put(37,30){\line(1,-3){5}}
\put(12,45){\line(1,-3){5}}
\put(12,75){\line(1,-3){5}}
\put(62,15){\line(1,3){5}}
\put(87,30){\line(1,3){5}}
\put(87,60){\line(1,3){5}}
\put(15,92){\line(1,0){20}}
\put(40,107){\line(1,0){20}}
\put(65,92){\line(1,0){20}}
%
\put(37,60){\line(1,3){5}}
\put(62,75){\line(1,-3){5}}
\put(40,47){\line(1,0){20}}
\end{picture}
\end{center}
\caption{$B?^(B\ref{cube-Lozenge-fig}$B$KBP1~$9$k(B
$B%@%$%^!<$NG[CV!J#2=E@~$O%@%$%^!<$rI=$9!K!%(B
$BFs$D$N%@%$%^!<G[CV$,Cf1{$N#63Q7A$K$*$1$k(B
$B%^%C%A%s%0$N2sE>$K$h$C$F0\$j9g$&$3$H$KCm0U!%(B}
\label{Lozenge-dimer-fig}
\end{figure}

$B>e$N$h$&$JN)J}BN$N@Q$_>e$2>uBV$OJ?LLJ,3d(B
$B!J(Bplane partition$B!K$K$h$C$F$bI=8=$G$-$k!%(B
$BDL>o$NJ,3d!J(Bpartition$B!K$OM?$($i$l$?@5@0?t(B $n$ $B$r(B
\beqnn
  \lambda_1 + \lambda_2 + \cdots + \lambda_d = n 
\eeqnn
$B$H$$$&$h$&$K@5@0?t(B $\lambda_1,\lambda_2,\cdots,
\lambda_d$ $B$NOB$H$7$FI=$9$3$H$r0UL#$9$k!%(B
$B=g=x$rF~$lBX$($?$b$N$OF1$8J,3d$H8+$J$5$l$k$N$G!$(B
$\lambda_1 \ge \lambda_2 \ge \cdots 
\ge \lambda_d$ $B$H$$$&>r7o$rCV$/!%$3$N$h$&$J(B
$B@5@0?t$NJ,3d$O(B $n$ $B8D$N@5J}7A$rJB$Y$??<$5$,(B $d$ $B$N(B
$B%d%s%0?^7A$KBP1~$9$k!%$5$i$K!$(B
$d,n$ $B$K$D$$$F$N>r7o$r$O$:$7$F!$0lHL$K(B
\beqnn
  \lambda = (\lambda_1,\lambda_2,\cdots), \quad 
  \lambda_1 \ge \lambda_2 \ge \cdots
\eeqnn
$B$H$$$&@0?tNs(B $\lambda$$B!J$?$@$7!$$"$kHV9f$+$i@h$O(B $0$ 
$B$,JB$V$H$9$k!K$rJ,3d$H8F$V$3$H$b$"$k!%J?LLJ,3d$O(B
$B$=$N#2<!852=(B
\beqnn
  \pi = (\pi_{ij})_{i,j\ge 1}, \quad 
  \pi_{ij} \ge \pi_{i+1,j} \ge 0, \quad 
  \pi_{ij} \ge \pi_{i,j+1} \ge 0
\eeqnn
$B$G$"$k!%J?LLJ,3d$O#3<!85%d%s%0?^7A$KBP1~$9$k!%(B
$B$9$J$o$A!$J?LL$NBh#1>]8B$K@5J}7A$rI_$-5M$a$F!$(B
$(i,j)$ $BHVL\$N@5J}7A$N>e$K(B $\pi_{ij}$ $B8D$N(B
$BN)J}BN$r@Q$a$P!$#3<!85%d%s%0?^7A$,$G$-$k!%(B

$B$3$N$h$&$K!$#33Q3J;R?^7A$N%?%$%kD%$j!J8@$$49$($l$P(B
$B#63Q%0%i%U>e$N%@%$%^!<G[CV!K$OJ?LLJ,3d!&(B
$B#3<!85%d%s%0?^7A$HBP1~$7$F$$$k!%$3$N8+J}$,(B
$BD689M}O@$N$"$k<o$NLdBj$r9M$($k:]$K$bM-MQ$G$"$k(B
$B$3$H$,:G6a;XE&$5$l$F$$$k(B \cite{ORV06}$B!%(B



\section{$BJ,G[H!?t$N%Q%U<0!&9TNs<0I=<((B}


\subsection{$B%+%9%F%l%$%s9TNs(B}

$B0J2<$G$OJ?LLE*#2It%0%i%U(B $G = (V_1,V_2,E)$ 
$B$N>e$G%@%$%^!<LO7?$r9M$($k!%(BKasteleyn 
\cite{Kasteleyn61,Kasteleyn67} $B$O(B
$B$3$N>l9g$NJ,G[H!?t$,(B
\beq
  Z = |\Pf\tilde{K}| = |\det K| 
\label{Z=detK}
\eeq
$B$HI=<($G$-$k$3$H$r<($7$?!J(B$\Pf\tilde{K}$ 
$B$O(B $\tilde{K}$ $B$N%Q%U<0$rI=$9!K!%(B
$B$3$3$K8=$l$?9TNs(B
\beqnn
  \tilde{K} 
  = \left(\begin{array}{cc}
    0 & K \\
    - \tp{K} & 0 
    \end{array}\right) 
  \; \text{$B!J(B$V \times V$ $B9TNs!K(B}, 
  \quad 
  K = (K_{wb})_{w\in V_1,\, b\in V_2}
  \; \text{$B!J(B$V_1 \times V_2$ $B9TNs!K(B}
\eeqnn
$B$,%+%9%F%l%$%s9TNs$G$"$k!%(B$\tilde{K}$ $B$O(B $G$ 
$B$NNY@\9TNs(B $\tilde{A}$ $B$K=E$_$HId9f$rIU$1$?(B
$B$b$N$G$"$k!%$9$J$o$A!$(B$K$ $B$N9TNsMWAG$O(B
\beqnn
  K_{wb} 
  = \begin{cases}
    \pm W(w,b) & ((w,b) \in E) \\ 
    0          & ((w,b) \not\in E) 
    \end{cases}
\eeqnn
$B$H$$$&7A$r$7$F$$$k!%0J2<$G$O!$$3$NId9f(B $\pm$ $B$N(B
$BDj$aJ}$d!$$=$l$K$h$C$F(BKasteleyn$B$N8x<0$,@.N)$9$k(B
$B;EAH$_$r2r@b$9$k!%(B

\paragraph*{$BCm0U(B}

1) $BNY@\9TNs$,BP>N9TNs$G$"$k$N$KBP$7$F(B
$B%+%9%F%l%$%s9TNs$OH?BP>N9TNs$G$"$k!%(B
$B$@$+$i$3$=%Q%U<0(B $\Pf\tilde{K}$ $B$,0UL#$r$b$D!%(B
$BD:E@$r(B $1,2,\cdots,2n$ $B$G%i%Y%kIU$1$7$F(B $\tilde{K}$ 
$B$N9TNsMWAG$r(B $\tilde{K}_{ij}$$B!J(B$i,j = 1,\cdots,2n$$B!K(B
$B$HI=$;$P!$$3$N%Q%U<0$O(B
\beq
  \Pf\tilde{K} 
  = \sum_{\mu \in \mathcal{M}_{2n}} \sgn(\mu) 
    \tilde{K}_{\mu(1)\mu(2)}\cdots
    \tilde{K}_{\mu(2n-1)\mu(2n)}
\label{PfKtilde=terms}
\eeq
$B$H$$$&E83+$r$b$D!%$3$3$G!$(B$\mathcal{M}_{2n}$ 
$B$O(B $1,2,\cdots,2n$ $B$NCV49(B $\mu$ $B$G(B
\beqnn
\begin{aligned}
& \mu(1)<\mu(2),\; \mu(3)<\mu(4),\; 
  \cdots,\;\mu(2n-1) < \mu(2n), \\
& \mu(1) < \mu(3) < \cdots < \mu(2n) 
\end{aligned}
\eeqnn
$B$H$$$&>r7o$rK~$?$9$b$NA4BN$N=89g$G$"$k!%(B
$B$3$NE83+<0$+$i40A4%^%C%A%s%0$H$N4XO"$,(B
$B8+$($k$3$H$KCm0U$5$l$?$$!%<B:]!$(B
$B9TNs$NE:;z(B $1,2,\ldots,2n$ $B$r(B
$B40A4%0%i%U(B $K_{2n}$ $B$ND:E@!$(B
$(\mu(2i-1),\mu(2i))$ $B$r$=$NJU$H8+$J$;$P!$(B
$\mu \in \mathcal{M}_{2n}$ $B$O(B $K_{2n}$ 
$B$N40A4%^%C%A%s%0(B
\footnote{$K_{2n}$ $B$O#2It%0%i%U$G$O$J$$$,!$(B
$B%0%i%UM}O@$G$O$3$N$h$&$J>l9g$K$b40A4%^%C%A%s%0(B
$B$H$$$&8@MU$rMQ$$$k!%#2It%0%i%U$N40A4%^%C%A%s%0$O(B
$B$=$NFCJL$J>l9g$G$"$k!%(B}
$B$H8+$J$;$k!%(B
\par
2) $G$ $B$O#2It%0%i%U$G$"$k$+$i!$NY@\9TNs$O(B
\beqnn
  \tilde{A} 
  = \left(\begin{array}{cc}
    0 & A \\
    A & 0 
    \end{array}\right), \quad 
  A = (A_{wb})_{w\in V_1,b\in V_2}
\eeqnn
$B$H$$$&%V%m%C%/9=B$$r$b$D!%(B$\tilde{K}$ $B$N(B
$B%V%m%C%/9=B$(B
\beqnn
  \tilde{K} 
  = \left(\begin{array}{cc}
    0 & K \\
    - \tp{K} & 0 
    \end{array}\right), \quad 
  K = (K_{wb})_{w\in V_1,b\in V_2}
\eeqnn
$B$O$3$N$3$H$KM3Mh$9$k!%$3$N%V%m%C%/9=B$$+$i(B
\beq
  \Pf\tilde{K} = (-1)^{n(n-1)/2}\det K
\eeq
$B$H$$$&Ey<0$,=>$&!%(B
$B%@%$%^!<LO7?$NJ,G[H!?t$O$=$l$i$N@dBPCM$G(B
$BI=$5$l$k$N$G!$(B$\Pf\tilde{K}$ $B$H(B $\det K$ $B$N(B
$B$$$:$l$r9M$($F$b$h$$!%0J2<$G$O$*$b$K(B $\det K$ 
$B$r9M$($k!%(B
\par
3) $\tilde{K}$ $B$N%V%m%C%/9=B$$O%G%#%i%C%/:nMQAG$r(B
$BO"A[$5$;$k!%L5<ANL>l$N%G%#%i%C%/:nMQAG$O>l$N@.J,$N(B
$B%+%$%i%j%F%#$K$h$C$F%V%m%C%/J,$1!J%+%$%i%kI=<(!K$5$l$k!%(B
$B$=$3$G$O(B $K$ $B$KAjEv$9$k$b$N$H$7$F%o%$%k:nMQAG$,8=$l$k(B
$B!J$A$J$_$K!$4v2?3X$G$O%o%$%k:nMQAG$r%G%#%i%C%/:nMQAG$H(B
$B8F$V$3$H$,B?$$!K!%%o%$%k:nMQAG$N9TNs<0$O(B
$B4v2?3XE*$K=EMW$JBP>]$G$"$k!%%@%$%^!<LO7?$G$O(B
$B$=$l$KAjEv$9$k$b$N$,J,G[H!?t$rM?$($k$3$H$K$J$k!%(B
$B$3$N;v<B$OBgJQ<(:6E*$G$"$j!$%@%$%^!<LO7?$NGX8e$K(B
$B%U%'%k%_%*%s$,1#$l$F$$$k$3$H$r<(:6$7$F$$$k!%(B
$B$3$N$3$H$O%$%8%s%0LO7?!J%U%'%k%_%*%s$H$N4X78$O(B
$B$h$/CN$i$l$F$$$k!K$H$N4X78$+$i8+$F$b$b$C$H$b$i$7$$!%(B
$B<B:]!$(BKenyon$B$O$3$N$h$&$J4QE@$+$i!$%+%9%F%l%$%s9TNs$r(B
$B%o%$%k:nMQAG!J#2<!85;~6u$N>l9g$K$O%3!<%7!<!&%j!<%^%s:nMQAG(B
$B$K$J$k!K$NN%;6E*N`;w$H$7$FB*$($F!$N%;6E*%0%j!<%sH!?t$d(B
$BN%;6E*@5B'H!?t$J$I$rO@$8$F$$$k(B
$B!J2r@b5-;v(B \cite{Kenyon03} $B$H$=$N0zMQJ88%$r(B
$B;2>H$5$l$?$$!K!%(B


\subsection{$BM=HwE*9M;!(B}

$B#2It%0%i%U$NNY@\9TNs$N3F9TNsMWAG$K%@%$%^!<$N(B
$B=E$_(B $W(w,b)$ $B$r>h$8$?9TNs!J=E$_IU$-NY@\9TNs!K(B
$B$r2~$a$F(B $\tilde{A}$ $B$H$$$&5-9f$GI=$9!%$9$J$o$A(B
\beqnn
  \tilde{A} 
  = \left(\begin{array}{cc}
    0 & A \\
    A & 0 
    \end{array}\right), 
  \quad 
  A_{wb} 
  = \begin{cases}
    W(w,b) & ((w,b) \in E) \\
    0      & ((w,b) \not\in E)
    \end{cases}
\eeqnn
$B0J2<$G$O!$(B$V_1,V_2$ $B$KB0$9$kD:E@$r(B
$B6&DL$N(B $1,2,\cdots,n$ $B$G%i%Y%kIU$1$9$k!%(B
$B$3$l$KH<$C$F(B $A$ $B$N9TNs@.J,$r(B $A_{ij}$
$B!J(B$i,j = 1,2,\cdots,n$$B!K$HI=$9!%(B
$V_1$ $B$H(B $V_2$ $B$ND:E@$r6hJL$9$kI,MW$,$"$k(B
$B>l9g$K$O(B
\beqnn
  V_1 = \{w_1,w_2\cdots,w_n\}, \quad 
  V_2 = \{b_1,b_2,\cdots,b_n\} 
\eeqnn
$B$H$$$&%i%Y%kIU$1$rMQ$$$k$3$H$K$9$k!%(B

$B=E$_IU$-NY@\9TNs$N%V%m%C%/(B $A$ $B$KBP$7$F(B
$B$=$N%Q!<%^%M%s%H!J(Bpermanent$B!K!$$9$J$o$A(B
$B9TNs<0$NDj5A$+$iId9f0x;R$r<h$j=|$$$?$b$N(B
\beq
  \perm A
  = \sum_{\sigma \in S_n}
    A_{1\sigma(1)}\cdots A_{n\sigma(n)}
\label{permA=terms}
\eeq
$B$r9M$($F$_$h$&!%(B
$B1&JU$N3F9`(B $A_{1\sigma(1)}\cdots A_{n\sigma(n)}$ $B$K(B
\beqnn
  M = \{(1,\sigma(1)),\,\cdots,\,(n,\sigma(n))\} 
\eeqnn
$B$H$$$&JU$N=89g$rBP1~$5$;$l$P(B
\begin{itemize}
\item[(i)] $A_{1\sigma(1)}\cdots A_{n\sigma(n)} \not= 0$ 
\item[(ii)] $M$ $B$O#2It%0%i%U(B $G$ $B$N40A4%^%C%A%s%0$G$"$k(B
\end{itemize}
$B$H$$$&#2>r7o$OF1CM$J$N$G!$(B$0$ $B$G$J$$9`$O(B $G$ $B$N(B
$B40A4%^%C%A%s%0(B $M$ $B$K#1BP#1BP1~$7!$(B
$B$=$NCM$O(B $W(M)$ $B$KEy$7$$!%$3$&$7$F(B
$B%@%$%^!<LO7?$NJ,G[H!?t$KBP$9$k%Q!<%^%M%s%HI=<((B
\beq
  Z = \perm A 
\label{Z=permA}
\eeq
$B$,F@$i$l$k!%(B

$B$3$N%Q!<%^%M%s%HI=<((B
\footnote{$B%U%'%k%_%*%s$H$N4X78$G8@$($P!$(B
$B$3$l$O$$$o$P%\%>%sE*$JI=<($G$"$k!%(B}
$B$O;DG0$J$,$i7W;;$NLr$K$ON)$?$J$$!%(B
$B%Q!<%^%M%s%H$O9TNs<0$N$b$DNI$$@-<A$r(B
$B$b$?$J$$$+$i$G$"$k!%9TNs<0I=<($,$G$-$l$P!$(B
$B8GM-CMLdBj$dBP3Q2=$rMxMQ$7$F7W;;$r9T$&F;$b3+$1$k!%(B

$B$=$3$G!$=E$_IU$-NY@\9TNs(B $A = (A_{ij})$ $B$N(B
$B9TNsMWAG$KE,Ev$JId9f0x;R(B $\kappa_{ij} = \pm 1$ 
$B$r>h$8$?$b$N(B
\beq
  K_{ij} = \kappa_{ij} A_{ij}
\eeq
$B$rF3F~$7!$(B$K = (K_{ij})$ $B$N9TNs<0(B
\beq
  \det K 
  = \sum_{\sigma\in S_n}
    \sgn(\sigma)K_{1\sigma(1)}\cdots K_{n\sigma(n)} 
\label{detK=terms}
\eeq
$B$N1&JU$N$9$Y$F$N9`$,(B{\bf $BF1Id9f(B}$B$K$J$k$h$&$K(B
$BId9f0x;R(B $\kappa_{ij}$ $B$rA*$V$3$H$r9M$($k!%(B
$B$=$N$h$&$JId9fIU$1$,2DG=$G$"$l$P(B
\beqnn
  \sgn(\sigma)K_{1\sigma(1)}\cdots K_{n\sigma(n)} 
  = \pm A_{1\sigma(1)}\cdots A_{n\sigma(n)}
  \quad \text{$B!J(B$\pm$ $B$O6&DL!K(B}
\eeqnn
$B$H$$$&Ey<0$,@.N)$9$k$N$G!$J,G[H!?t$,(B $K$ $B$N(B
$B9TNs<0$H$7$FI=$;$k$3$H$K$J$k!%(B



\subsection{$BId9fIU$1$K4X$9$k9M;!$HDjM}(B}

$\det K$ $B$NE83+(B (\ref{detK=terms}) $B$N(B $0$ $B$G$J$$9`$O(B
$B#2It%0%i%U(B $G$ $B$N$"$k40A4%^%C%A%s%0(B $M$ $B$KBP1~$7$F$$$k!%(B
$B$9$G$K@bL@$7$?$h$&$K!$G$0U$N#2$D$N40A4%^%C%A%s%0(B $M,M'$ $B$O(B
$B$$$/$D$+$NJDO)$K$*$1$k2sE>A`:n$K$h$C$F8_$$$K0\$j9g$&!%(B
$B$7$?$,$C$F!$(B(\ref{detK=terms}) $B$N$9$Y$F$N9`$,(B
$BF10lId9f$r$b$D$3$H$r<($9$K$O!$0l$D$NJDO)$K$*$1$k(B
$B2sE>$K$h$C$F9`$NId9f$,JQ2=$7$J$$$3$H$r3N$+$a$l$P$h$$!%(B

$G$ $B$ND:E@$NNs(B $w_{i_1},\,b_{j_1},\,w_{i_2},\,b_{j_2},\,
\cdots,\,w_{i_p},\,b_{j_p}$$B!J(B$V_1,V_2$ $B$ND:E@$r(B $w_i,b_i$ 
$B$HI=$95,Ls$K=>$C$F$$$k!K$r$?$I$C$F(B $w_{i_1}$ $B$KLa$k(B
$BD9$5(B $2p$ $B$NJDO)(B
\beqnn
  C = (w_{i_1},\,b_{j_1},\,w_{i_2},\,b_{j_2},\,
      \cdots,\,w_{i_p},\,b_{j_p},w_{i_1}) 
\eeqnn
$B$r9M$($k!%(B$M$ $B$O(B $C$ $B$NJU$N$&$A(B $(w_{i_1},b_{j_1}),\,
(w_{i_2},b_{j_2}),\, \cdots,(w_{i_p},b_{j_p})$ $B$r4^$_!$(B
$B$=$l$i$N2sE>(B
\beqnn
  (w_{i_1},b_{j_1}) \to (w_{i_2},b_{j_1}),\; 
  (w_{i_2},b_{j_2}) \to (w_{i_3},b_{j_2}),\;
  \cdots,\, 
  (w_{i_p},b_{j_p}) \to (w_{i_1},b_{j_p}) 
\eeqnn
$B$K$h$C$F(B $M'$ $B$K0\$k$H$9$k!%(B(\ref{detK=terms}) 
$B$K$*$$$F(B $M$ $B$KBP1~$9$k9`$O(B
\beqnn
  \sgn\left(\begin{array}{ccccc}
    i_1 & i_2 & \cdots & i_p & \cdots \\
    j_1 & j_2 & \cdots & j_p & \cdots 
  \end{array}\right) 
  K_{i_1j_1}K_{i_2j_2}\cdots K_{i_pj_p}\cdots 
\eeqnn
$B$H$$$&7A$r$7$F$$$k!%$3$3$G!$9`$N9=B$$r8+$d$9$/$9$k$?$a!$(B
$\sgn(\sigma)$ $B$H(B $K_{1\sigma(1)}\cdots K_{n\sigma(n)}$ 
$B$N$=$l$>$l$K$*$$$F(B $i_1,\cdots,i_p,j_1,\cdots,j_p$ 
$B$KBP1~$9$kItJ,$rA0$NJ}$K4s$;$FI=$7$F$$$k!%(B
$BF1MM$N0UL#$G!$(B$M'$ $B$KBP1~$9$k9`$O(B
\beqnn
  \sgn\left(\begin{array}{ccccc}
    i_1 & i_2 & \cdots & i_p & \cdots \\
    j_p & j_1 & \cdots & j_{p-1} & \cdots 
  \end{array}\right) 
  K_{i_1j_p}K_{i_2j_1}\cdots K_{i_pj_{p-1}}\cdots 
\eeqnn
$B$H$J$k!%$3$3$K8=$l$?CV49$O(B
$B=d2sCV49(B $(j_1\;j_2\;\cdots\;j_p)$ $B$K$h$C$F(B
\beqnn
  \left(\begin{array}{ccccc}
    i_1 & i_2 & \cdots & i_p & \cdots \\
    j_1 & j_2 & \cdots & j_p & \cdots 
  \end{array}\right) 
 = (j_1\; j_2\;\cdots\;j_p) 
   \left(\begin{array}{ccccc}
     i_1 & i_2 & \cdots & i_p & \cdots \\
     j_p & j_1 & \cdots & j_{p-1} & \cdots 
   \end{array}\right) \\
\eeqnn
$B$H$$$&4X78$G7k$P$l$F$$$k$N$G!$Id9f$K4X$7$F(B
\beqnn
  \sgn\left(\begin{array}{ccccc}
    i_1 & i_2 & \cdots & i_p & \cdots \\
    j_1 & j_2 & \cdots & j_p & \cdots 
  \end{array}\right) 
= (-1)^{p-1}  
  \sgn\left(\begin{array}{ccccc}
    i_1 & i_2 & \cdots & i_p & \cdots \\
    j_p & j_1 & \cdots & j_{p-1} & \cdots 
  \end{array}\right)  
\eeqnn
$B$H$$$&Ey<0$,@.N)$9$k!%$7$?$,$C$F!$(B$A_{ij}$ $B$K>h$8$?(B
$BId9f0x;R(B $\kappa_{ij}$ $B$KBP$7$F(B

\beq
  \kappa_{i_1j_1}\kappa_{i_2j_2}\cdots\kappa_{i_pj_p} 
  = (-1)^{p-1}\kappa_{i_1j_p}\kappa_{i_2j_1}
    \cdots\kappa_{i_pj_{p-1}}
\eeq
$B$H$$$&Ey<0$,@.N)$9$l$P!$>e$N#29`$OF1Id9f$K$J$k!%(B

$\det K$ $B$NE83+(B (\ref{detK=terms}) $B$N$9$Y$F$N9`$,(B
$BF1Id9f$G$"$k$?$a$K$O!$:G8e$K<($7$?Ey<0$,(B $G$ $B$N(B
$B$9$Y$F$NJDO)(B $C$ $B$KBP$7$F@.N)$9$l$P$h$$!%(B
$B$3$NEy<0$r$b$&>/$78+$d$9$$7A$K=q$-D>$;$P(B
\beq
  \prod_{e\in C}\kappa_e = (-1)^{|C|/2-1} 
\label{sign-cond}
\eeq
$B$H$J$k!%$3$3$G(B $\kappa_e$ $B$OJU(B $e = (w_i,b_j)$ 
$B$KBP$9$kId9f0x;R(B $\kappa_{ij}$ $B$G$"$j!$(B$|C|$ $B$O(B
$BJDO)(B $C$ $B$ND9$5!J6v?t$K$J$k!K$rI=$9!%(B

$G$ $B$,J?LLE*#2It%0%i%U$G$"$k>l9g$K$O!$LL$N8D?t$K4X$9$k(B
$B5"G<K!$K$h$C$F!$(B(\ref{sign-cond}) $B$rK~$?$9Id9fIU$1$,(B
$B>o$K2DG=$G$"$k$3$H$r<($;$k(B \cite{Kasteleyn67}$B!%(B
$B$=$N>ZL@$r8+$l$P!$%0%i%U$N3FLL$N6-3&$KBP$7$F(B
(\ref{sign-cond}) $B$,@.N)$9$l$P==J,$G$"$k$3$H$b$o$+$k!%(B
$B$J$*!$$3$N$h$&$JId9fIU$1$O0l0UE*$G$O$J$$$N$G!$(B
$B>u67$K1~$8$FET9g$N$h$$$b$N$rA*$Y$P$h$$!%(B

$B0J>e$N7k2L$r$^$H$a$k$H<!$N$h$&$K$J$k!%(B

\paragraph*{$BDjM}!J(BKasteleyn$B!K(B}
$G$ $B$,J?LLE*#2It%0%i%U$G$"$l$P!$(B(\ref{sign-cond}) 
$B$rK~$?$9$h$&$JId9fIU$1(B $\kappa_e$$B!J(B$\kappa \in E)$ 
$B$,B8:_$9$k!%$3$N>r7o$,K~$?$5$l$l$P(B
\beqnn
  Z = |\det K| = |\Pf\tilde{K}| 
\eeqnn
$B$H$$$&Ey<0$,@.N)$9$k!%(B

\paragraph*{$BCm0U(B}
1) Kasteleyn\cite{Kasteleyn67} $B$O(B $K$ $B$N(B
$B$+$o$j$K(B $\tilde{K}$ $B$r9M$($F!$(B$\Pf\tilde{K}$ $B$N(B
$BE83+(B (\ref{PfKtilde=terms}) $B$N$9$Y$F$N9`$,(B
$BF1Id9f$K$J$k$h$&$J>r7o$H$7$F(B (\ref{sign-cond}) 
$B$rF3$$$?!%K\@a$G$O(BKasteleyn$B$,(B $\Pf\tilde{K}$ 
$B$KBP$7$FMQ$$$?O@K!$r(B $\det K$ $B$N>l9g$K>F$-D>$7$F>R2p$7$?!%(B
$B%Q%U<0$K$h$kDj<02=$O#2It%0%i%U0J30$K$b3HD%$G$-$k$H$$$&(B
$BMxE@$,$"$k$,!$$3$3$G$O@bL@$N$7$d$9$5$rM%@h$7$F(B
$B9TNs<0$K$h$kDj<02=$r:NMQ$7$?!%(B
\par
2) $BJ?J}3J;R%0%i%U$N>l9g$K$O!$3FLL!J#43Q7A$r$J$9!K(B
$B$N6-3&$K1h$&Id9f0x;R$N@Q$,(B $-1$ $B$G$"$l$P(B
(\ref{sign-cond}) $B$N>r7o$,K~$?$5$l$k!%(B
Kasteleyn$B$O?^(B\ref{sign-sqlattice-fig}
$B$K<($9$h$&$JId9fIU$1$r9T$C$?$,!$(B
Kenyon\cite{Kenyon03}$B$OId9f0x;R$NBe$o$j$K(B
$B5u?tC10L(B $i = \sqrt{-1}$ $B$G=E$_IU$-NY@\9TNs$r=$@5$7$F!$(B
$BJ,G[H!?t$N9TNs<0I=<($rM?$($?!%(BKenyon$B$NMQ$$$?9TNs$O(B
$BJ?J}3J;R%0%i%U$KFC2=$7$?$b$N$@$,!$9TNs<0$N7W;;$,4JC1$K$J$k(B
$B$H$$$&MxE@$r$b$D!J?"ED$N2r@b(B\cite{Ueda05}$B$b;2>H$5$l$?$$!K!%(B

\begin{figure}[t]
\begin{center}
\begin{picture}(120,100)
\multiput(3,90)(30,0){5}{\line(1,0){24}}
\multiput(30,90)(60,0){3}{\circle*{6}}
\multiput(0,90)(60,0){3}{\circle{6}}
\multiput(0,63)(30,0){6}{\line(0,1){24}}
\multiput(3,60)(30,0){5}{\line(1,0){24}}
\multiput(30,60)(60,0){3}{\circle{6}}
\multiput(0,60)(60,0){3}{\circle*{6}}
\multiput(0,33)(30,0){6}{\line(0,1){24}}
\multiput(3,30)(30,0){5}{\line(1,0){24}}
\multiput(30,30)(60,0){3}{\circle*{6}}
\multiput(0,30)(60,0){3}{\circle{6}}
\multiput(0,3)(30,0){6}{\line(0,1){24}}
\multiput(3,0)(30,0){5}{\line(1,0){24}}
\multiput(30,0)(60,0){3}{\circle{6}}
\multiput(0,0)(60,0){3}{\circle*{6}}
%
\multiput(3,73)(30,0){6}{\makebox{$\scriptstyle +$}}
\multiput(3,43)(30,0){6}{\makebox{$\scriptstyle +$}}
\multiput(3,13)(30,0){6}{\makebox{$\scriptstyle +$}}
\multiput(15,93)(30,0){5}{\makebox{$\scriptstyle -$}}
\multiput(15,63)(30,0){5}{\makebox{$\scriptstyle +$}}
\multiput(15,33)(30,0){5}{\makebox{$\scriptstyle -$}}
\multiput(15,3)(30,0){5}{\makebox{$\scriptstyle +$}}
\end{picture}
\end{center}
\caption{$BJ?J}3J;R%0%i%U$N>l9g$NId9fIU$1$NNc(B}
\label{sign-sqlattice-fig}
\end{figure}


\subsection{$BJ?J}3J;R%0%i%U$N>l9g(B}

Kasteleyn\cite{Kasteleyn61}$B$O>e$G>R2p$7$?(B
$B%Q%U<0!&9TNs<0I=<($rMQ$$$FG$0U$NJ?J}3J;R%0%i%U$KBP$9$k(B
$BJ,G[H!?t$r6qBNE*$K5a$a$?!%3J;R$N%5%$%:$r(B $2m \times 2n$$B!$(B
$B?eJ?J}8~$H?bD>J}8~$N%@%$%^!<$N=E$_$r(B
\beqnn
  W(\rowdimer) = a, \quad 
  W(\coldimer) = b 
\eeqnn
$B$H$9$k!%?bD>J}8~$N$9$Y$F$NJU$K(B $+$ $B$r!$(B
$B?eJ?J}8~$K$D$$$F$ONs$4$H$K8r8_$K(B $\pm$ $B$r(B
$B3d$jEv$F$l$P!J?^(B\ref{sign-sqlattice-fig})$B!$(B
$BId9fIU$1$N>r7o(B (\ref{sign-cond}) $B$OK~$?$5$l$k!%(B
Kasteleyn$B$O$3$NId9fIU$1$N$b$H$G(B $\tilde{K}$ 
$B$rDj5A$7!$$=$l$rBP3Q2=$7$F(B
\beq
  Z = \prod_{j=1}^m\prod_{k=1}^n 
      \Bigl(4a^2\cos^2\frac{\pi j}{2m+1} 
          + 4b^2\cos^2\frac{\pi k}{2n+1} \Bigr)
\label{Z-sqlattice}
\eeq
$B$H$$$&J,G[H!?t$NI=<($rF@$?!%$3$l$O%$%8%s%0LO7?$N(B
$BJ,G[H!?t$H$[$H$s$IF1$87A$r$7$F$*$j!$#2$D$NLO7?$N(B
$BL)@\$J4XO"$r<(:6$7$F$$$k!J%$%8%s%0LO7?$H$N4X78$K$D$$$F$O!$(B
Kasteleyn$B$N2r@b(B \cite{Kasteleyn67} $B$r;2>H$5$l$?$$!K!%(B
$B$A$J$_$K!$(BKasteleyn$B$O(B $\tilde{K}$ $B$rBP3Q2=$9$k(B
$B:]$K(B $\tilde{K}$ $B$KFb:_$9$k%F%s%=%k@Q9=B$$r(B
$BMxMQ$7$?$N$@$,!$BP3Q2=$N7W;;$OB?>/$N5;9*$rMW$9$k!%(B
Kenyon\cite{Kenyon03} $B$NMQ$$$?9TNs$O(B
$B$=$l<+BN$,%F%s%=%k@Q$N7A$r$7$F$$$k$N$G!$(B
$BBP3Q2=$N7W;;$,Cx$7$/4JC1$K$J$k!%(B

$BHy@QJ,3X$G$*$J$8$_$N6hJ,5a@QK!$N8x<0$rMQ$$$l$P!$(B
$B$3$&$7$FF@$i$l$?J,G[H!?t$NI=<($+$iG.NO3X6K8B$K$*$1$k(B
$B%@%$%^!<$"$?$j$N<+M3%(%M%k%.!<(B
\beqnn
  F = - \lim_{m,n\to\infty}\frac{\log Z}{4mn}
\eeqnn
$B$,$?$@$A$K5a$a$i$l$k!%$=$N7k2L$O(B
\beq
\begin{aligned}
  F &= - \frac{1}{4\pi^2}\int_0^\pi d\theta \int_0^\pi d\tau 
     \log(4a^2\cos^2\theta + 4b^2\cos^2\tau) \\
  &= - \frac{1}{4\pi^2}\int_0^\pi d\theta \int_0^\pi d\tau 
     \log\bigl(a^2(e^{i\theta} + e^{-i\theta})^2 
             + b^2(e^{i\tau} + e^{- i\tau})^2\bigr) 
\end{aligned}
\label{F-sqlattice}
\eeq
$B$H$$$&7A$N@QJ,$K$J$k!%$3$l$r(B $z = e^{2i\theta}$, 
$w = e^{2i\tau}$ $B$K4X$9$kJ#AG<~2s@QJ,$K=q$-D>$;$P!$(B
$B$"$H$G>R2p$9$k!V%m%s%-%sH!?t!W$NFC<lCM$H8+$J$;$k!%(B
$B$3$N@QJ,$O?tO@!&AP6J4v2?3X$J$I$H$b4XO"$7$F$$$k(B
$B!J>\$7$/$O(BKenyon$B$d?"ED$N2r@b(B \cite{Kenyon03,Ueda05} 
$B$H$=$N0zMQJ88%$r;2>H$5$l$?$$!K!%(B



\section{$B<~4|E*%@%$%^!<LO7?$H%"%a!<%P!$%m%s%-%sH!?t(B}


\subsection{$B%H!<%i%9>e$N#2It%0%i%U$KBP$9$k%@%$%^!<LO7?(B}

Kasteleyn$B$O#2=E<~4|E*6-3&>r7o$N$b$H$G$N(B
$BJ?J}3J;R%0%i%U!J8@$$49$($l$P%H!<%i%9>e$NJ?J}3J;R%0%i%U!K(B
$B$N%@%$%^!<LO7?$NJ,G[H!?t$b7W;;$7$?(B \cite{Kasteleyn61}$B!%(B
$B$=$N7W;;K!$OJ?J}3J;R%0%i%U$ND>@Q9=B$$rMxMQ$9$k$b$N$@$C$?!%(B
Kenyon, Okounkov, Sheffield \cite{KOS03} $B$O(B
$BD>@Q9=B$$r;}$?$J$$#2=E<~4|E*#2It%0%i%U!JE57?E*$K$O(B
$B#63Q3J;R$HJ?J}#83Q3J;R!K$K$bDLMQ$9$k?7$?$JJ}K!$r(B
$B9M0F$7$?!%(B

Kenyon$B$?$A$b0l<o$N%+%9%F%l%$%s9TNs$rMQ$$$k$N$@$,!$(B
$B$=$l$O#28D$NJ#AGJQ?t(B $z,w$ $B$K0MB8$9$k(B $V_1 \times V_2$ 
$B9TNs(B $K(z,w)$ $B$G$"$k!%$=$NDj5A$OJ?LL%0%i%U$N>l9g$H(B
$BF1MM$G$"$k$,!$$9$G$K@bL@$7$?Id9f0x;R$K2C$($F!$(B
$BJU(B $e = (w,b)$ $B$,%H!<%i%9$N4pK\NN0h$N6-3&$H(B
$B8r:5$9$k$H$-$K(B $z^{\pm 1}$ $B$d(B $w^{\pm 1}$ $B$,(B
$B0x;R$H$7$F8=$l$k!J(B$\pm$ $B$N$$$:$l$rA*$V$+$O(B $e$ $B$H(B
$B6-3&$N8r:5$N;EJ}$G7h$^$k!K!%(B$z,w$ $B$O<~4|E*6-3&>r7o2<$N(B
$B8GM-CMLdBj$G$O$*$J$8$_$N%V%m%C%[>h?t$KB>$J$i$J$$!%(B
$B:#$N>l9g$K$O%H!<%i%9$N#2$D$N<~4|$K$D$$$F$N(B
$B<~4|E*6-3&>r7o$KH<$C$F#28D$N%V%m%C%[>h?t$,8=$l$k!%(B

$K(z,w)$ $B$N9TNs<0$H$7$FFC@-B?9`<0(B
\beqnn
  P(z,w) = \det K(z,w) 
\eeqnn
$B$rDj5A$9$k!%$3$l$O(B $z,w$ $B$N%m!<%i%sB?9`<0$H$J$k!%(B
$B$5$i$K!$%9%Z%/%H%k6J@~$r(B
\beqnn
  P(z,w) = 0 
\eeqnn
$B$H$$$&J}Dx<0$K$h$C$FDj5A$9$k!%@53N$K8@$($P!$(B
$BJ#AGBe?t6J@~$H$7$F$N%9%Z%/%H%k6J@~(B
\beq
  \Gamma = \{(z,w) \in \CC^2 \mid P(z,w) = 0\} 
\eeq
$B$H$=$N<BIt(B
\beq
  \Gamma_{\RR} = \{(z,w) \in \RR^2 \mid P(z,w) = 0\} 
\eeq
$B$r6hJL$9$kI,MW$,$"$k!%(B

Kenyon$B$?$A$O$3$l$i$rMQ$$$F<!$N$h$&$J7k2L$r<($7$?!%(B
\begin{itemize}
\item[1)] $B<~4|E*%@%$%^!<LO7?$NJ,G[H!?t$O(B
\beq
  Z = \frac{1}{2}\bigl(- P(1,1) 
        + P(1,-1) + P(-1,1) + P(-1,-1)\bigr) 
\label{Z=spin-sum}
\eeq
$B$HI=$;$k!%(B
\item[2)] $B%9%Z%/%H%k6J@~$N<BIt(B  $\Gamma_{\RR}$ $B$O(B
$B!V%O%k%J%C%/!J(BHarnack$B!K@-!W$"$k$$$O!V6KBg@-!W(B
$B$H8F$P$l$kNI$$@-<A$r$b$D<BBe?t6J@~$G$"$k!%(B
$B$"$H$G@bL@$9$k%9%Z%/%H%k6J@~(B $\Gamma$ $B$+$i(B
$B%"%a!<%P$X$NBP?t<LA|$b$3$N@-<A$rH?1G$7$?FCD'$r$b$D!%(B
\item[3)] $BG.NO3XE*6K8B!J4pK\NN0h>e$N%0%i%U$N(B $N \times N$ 
$B8D$N%3%T!<$r=D2#(B $N$ $BG\$N<~4|$N%H!<%i%9>e$KJB$Y$F!$(B
$N \to \infty$ $B$N6K8B$r9M$($k!K(B
$B$,0UL#$r$b$D!%(B
\item[4)] $BG.NO3X6K8B$K$*$1$k4pK\NN0h$"$?$j$N<+M3%(%M%k%.!<$O(B
\beq
  F = - \oint_{|z|=1}\frac{dz}{2\pi iz} 
        \oint_{|w|=1}\frac{dw}{2\pi iw}\log|P(z,w)| 
\label{F=oint}
\eeq
$B$H$$$&@QJ,$GM?$($i$l$k!%(B
\item[4)] $BF1MM$N$3$H$,0lMM<'>l(B $(B_x,B_y)$ $B$NB8:_$9$k(B
$B>l9g$K$b8@$($k!%$3$N>l9g$N4pK\NN0h$"$?$j$N<+M3%(%M%k%.!<(B
$B!JI=LLD%NO!K$O(B $P(z,w)$ $B$N%m%s%-%sH!?t(B $R(B_x,B_y)$ $B$+$i(B
$B%k%8%c%s%I%kJQ49$r7PM3$7$F7h$^$k!%(B
\end{itemize}

$BMW$9$k$K!$$3$l$i$N7k2L$O<~4|E*%@%$%^!<LO7?$,(B
$B0l<o$N2D2rLO7?$G$"$k$3$H$r<($7$F$$$k!%$A$J$_$K!$(B
(\ref{Z=spin-sum})$B$O(BKasteleyn\cite{Kasteleyn61}$B$,(B
$BJ?J}3J;R%0%i%U$N>l9g$K8+=P$7$F$$$?;v<B$G$"$k!%(B
$B1&JU$N#4$D$N9`$O%H!<%i%9$N#4$D$N%9%T%s9=B$$KBP1~$9$k!%(B
$B$^$?!$(B(\ref{F=oint})$B$OHs<~4|E*J?J}3J;R%0%i%U$N(B
$BG.NO3XE*6K8B$+$iF@$i$l$?7k2L(B(\ref{F-sqlattice})
$B$r0lHL2=$7$?$b$N$H8+$J$;$k!%%"%a!<%P$H%m%s%-%sH!?t$K$D$$$F$O(B
$B@a$r2~$a$F@bL@$9$k$,!$(BKenyon$B$?$A$O$3$N4v2?3XE*!&H!?tO@E*35G0$+$i(B
$B%@%$%^!<LO7?$NG.NO3XE*6K8B$NAj9=B$$dAj4XH!?t$N@-<A$J$I$K4X$7$F$b(B
$B$5$^$6$^$J7kO@$r0z$-=P$7$F$$$k!J>\$7$/$O86O@J8(B \cite{KOS03}$B$d(B
$B$=$N8e$N8&5f(B \cite{KO03,KO05}$B$r;2>H$5$l$?$$!K!%(B



\subsection{$B%"%a!<%P$H%m%s%-%sH!?t(B}

$B%m%s%-%sH!?t$O(B (\ref{F=oint}) $B$N@QJ,$r(B
$BFC<lCM(B $R(0,0)$ $B$H$7$FM?$($kH!?t$G$"$j!$(B
\beq
  R(x,y) = \oint_{|z|=1}\frac{dz}{2\pi iz} 
           \oint_{|w|=1}\frac{dw}{2\pi iw}\log|P(e^xz,e^yw)|
\eeq
$B$H$$$&@QJ,$K$h$C$FDj5A$5$l$F!$0J2<$N$h$&$J@-<A$r$b$D!%(B
\begin{itemize}
\item[1)] $R(x,y)$ $B$O(B $\RR^2$ $B>e$NFLH!?t$G$"$k!%(B
\item[2)] $R(x,y)$ $B$O(B $P(z,w)$ $B$N%"%a!<%P(B
\beq
  A = \{(\log|z|,\log|w|) \mid (z,w) \in \Gamma \} 
\eeq
$B$N>e$G695AFL$G$"$k!%(B
\item[3)] $B%"%a!<%P$NJd=89g(B $\RR^2 \setminus A$ $B$O(B
$B$$$/$D$+$NO"7k@.J,$KJ,$+$l$k!%(B$R(x,y)$ $B$N(B
$B8{G[(B $\bsnabla R(x,y)$ $B$O3FO"7k@.J,$N>e$G0lDj$G$"$k!%(B
$B8@$$49$($l$P!$(B$R(x,y)$ $B$O%"%a!<%P$NJd=89g>e$G(B
$B6hJ,E*$K@~7A$G$"$k!%(B
\item[4)] $R(x,y)$ $B$O%"%a!<%P$N>e$G(B
$B%b%s%8%e!&%"%s%Z!<%kJ}Dx<0(B
\beq
    \frac{\rd^2R}{\rd x^2}\frac{\rd^2R}{\rd y^2} 
    - \Bigl(\frac{\rd^2R}{\rd x\rd y}\Bigr)^2 
    = \frac{1}{\pi^2} 
\label{Monge-Ampere}
\eeq
$B$rK~$?$9!%(B
\item[5)] $B0lMM<'>l(B $(B_x,B_y)$ $B$N2<$G$N4pK\NN0h$"$?$j$N(B
$B<+M3%(%M%k%.!<$O(B $R(B_x,B_y)$ $B$N%k%8%c%s%I%kJQ49(B $\sigma(s,t)$
$B$GM?$($i$l$k!%(B$\sigma(s,t)$ $B$O(B $P(z,w)$ $B$N%K%e!<%H%sB?3Q7A>e$N(B
$BH!?t$G!$$d$O$j%b%s%8%e!&%"%s%Z!<%kJ}Dx<0$rK~$?$9!%(B
\end{itemize}


\paragraph*{$BNc(B}
$B#63Q3J;R>e$N<~4|E*%@%$%^!<LO7?$G$O(B
$BGrD:E@$H9uD:E@$r$A$g$&$I#18D$:$D4^$`$h$&$J(B
$B4pK\NN0h$rA*$V$3$H$,$G$-$k!J?^(B\ref{p-honeycome-fig}$B!K!%(B
$B$3$&$7$FF@$i$l$k%H!<%i%9>e$N#2It%0%i%U$O(B
$BGrD:E@$H9uD:E@$r#18D$:$D;}$A!$(B
$B$=$N4V$r#3=EJU$G7k$s$@$b$N$K$J$k!%(B
\begin{figure}[t]
\begin{center}
\begin{picture}(200,130)
\multiput(30,120)(60,0){3}{\circle{6}}
\multiput(33,118)(60,0){2}{\line(3,-2){24}}
\multiput(3,102)(60,0){3}{\line(3,2){24}}
\multiput(0,100)(60,0){3}{\circle*{6}}
\multiput(0,72)(60,0){3}{\line(0,1){24}}
\multiput(0,70)(60,0){3}{\circle{6}}
\multiput(3,68)(60,0){3}{\line(3,-2){24}}
\multiput(33,52)(60,0){2}{\line(3,2){24}}
\multiput(30,50)(60,0){3}{\circle*{6}}
\multiput(30,23)(60,0){3}{\line(0,1){24}}
\multiput(30,20)(60,0){3}{\circle{6}}
\multiput(33,18)(60,0){2}{\line(3,-2){24}}
\multiput(3,2)(60,0){3}{\line(3,2){24}}
\multiput(0,0)(60,0){3}{\circle*{6}}
%
\multiput(30,85)(30,-50){2}{\line(1,0){60}}
\multiput(30,85)(60,0){2}{\line(3,-5){30}}
\end{picture}
\end{center}
\caption{$B<~4|E*#63Q3J;R$N4pK\NN0h!JJ?9T;MJU7A$G0O$^$l$?ItJ,!K(B}
\label{p-honeycome-fig}
\end{figure}
$BBP1~$9$k%+%9%F%l%$%s9TNs(B $K(x,y)$ $B$O(B $1 \times 1$$B!$(B
$B$9$J$o$A%9%+%i!<$G$"$j!$$=$l<+BN$,FC@-B?9`<0$KEy$7$$!%(B
$B6qBNE*$K$O(B
\beqnn
  P(z,w) = 1 + z + w
\eeqnn
$B$H$J$k!J%@%$%^!<<+BN$N=E$_$O(B $1$ $B$K$7$F$$$k!K!%(B
$B1&JU$N#3$D$N9`$OGrD:E@$H9uD:E@$r7k$V#3$D$NJU$KBP1~$9$k!%(B
$P(z,w) = 0$ $B$+$i(B $w = - z - 1$ $B$H$J$k$N$G!$(B
$B$3$NB?9`<0$N%"%a!<%P$O(B
\beqnn
  A = \{(\log|z|,\log|z + 1|) \mid z \in \CC\} 
\eeqnn
$B$HI=$;$k!%$9$J$o$A!$J?LL$NE@(B$(x,y)$ $B$,(B $A$ $B$K(B
$BB0$9$k$N$O$"$kJ#AG?t(B $z$ $B$KBP$7$F(B
\beqnn
  e^x = |z|, \quad e^y = |z + 1|
\eeqnn
$B$H$$$&Ey<0$,@.N)$9$k$H$-$G$"$k!%4v2?3XE*$K8+$l$P!$(B
$B$3$l$ODlJU$ND9$5$,(B $1$$B!$B>$N#2JU$ND9$5$,(B $e^x,e^y$ 
$B$N#33Q7A$,B8:_$9$k!$$H$$$&$3$H$G$"$k$+$i!$(B
$A$ $B$O#3$D$N#33QITEy<0(B
\beqnn
  e^x + 1 \ge e^y, \quad 
  e^y + 1 \ge e^x, \quad 
  e^x + e^y \ge 1 
\eeqnn
$B$,@.N)$9$k$h$&$J(B $(x,y)$ $B$N=89g$G$"$k!%(B
$B$3$l$O(B $x$ $B<4$NIi$NItJ,!$(B$y$ $B<4$NIi$NItJ,!$(B
$BBh#1>]8BFb$NH>D>@~(B $x = y$ $B$r4^$`(B
$BC1O"7kNN0h$K$J$k!JA4BN$H$7$F8+$l$P(B ``T''
$B$NJ8;z$rL58B$K0z$-1d$P$7$?$h$&$J7A$r$7$F$$$k!K!%(B
$B$=$NJd=89g$O(B
\beqnn
\text{(i)} \quad e^x + 1 < e^y \qquad
\text{(ii)} \quad e^y + 1 < e^x \qquad 
\text{(iii)} \quad e^x + e^y < 1 
\eeqnn
$B$H$$$&ITEy<0$G7h$^$k#3$D$NO"7k@.J,$+$i$J$k!%(B
(i), (ii), (iii) $B$K$*$1$k(B $R(x,y)$ 
$B$N8{G[$O(B $(0,1)$, $(1,0)$, $(0,0)$ $B$G$"$k!%(B
$B$3$N8{G[$NCM$O(B $P(z,w)$ $B$N%K%e!<%H%sB?3Q7A(B
$B!J#33Q7A$r$J$9!K$ND:E@$KBP1~$7$F$$$k!%(B


\subsection{$B%b%s%8%e!&%"%s%Z!<%kJ}Dx<0$K$D$$$F(B}

$B%m%s%-%sH!?t$O0lHL$N%m!<%i%sB?9`<0$KBP$7$F$b(B
$BDj5A$5$l$F(B 1), 2), 3) $B$N@-<A$r$b$D$,!$(B
$B$=$N>l9g$K$O%b%s%8%e!&%"%s%Z!<%kJ}Dx<0(B 
(\ref{Monge-Ampere}) $B$rK~$?$9$H$O8B$i$J$$(B
$B!JBe$o$j$K0l<o$NITEy<0$,@.N)$9$k(B \cite{PR04})$B!%(B
$B%b%s%8%e!&%"%s%Z!<%kJ}Dx<0$O%9%Z%/%H%k6J@~$N(B
$B<BIt(B $\Gamma_{\RR}$ $B$N6KBg@-$H4X78$7$F$$$k!%(B
$B$=$N0UL#$G!$$3$l$OAH$_9g$o$;E*$J9M;!$dBe?tE*$J7W;;$+$i(B
$B=P$;$k$3$H$G$O$J$/!$$$$o$PH!?tO@E*$J7k2L$G$"$k!%(B
$B$=$NMWE@(B \cite{KOS03} $B$r$4$/<jC;$+$K@bL@$7$F$*$3$&!%(B

$B80$O%9%Z%/%H%k6J@~$+$i%"%a!<%P$X$NBP?t<LA|(B
\beqnn
  \Log :\; \Gamma \to A, \quad 
  \Log(z,w) = (\log|z|,\log|z|) 
\eeqnn
$B$N@-<A$K$"$k!%$9$G$K=R$Y$?%9%Z%/%H%k6J@~$N(B
$B6KBg@-$+$i!$$3$NBP?t<LA|$,%"%a!<%P$N(B
$BFbIt(B $A^{\mathrm{int}}$ $B$G$O!J@53N$K$O(B
$B$=$NM-8B8D$NE@$r=|$$$F!K#2BP#1$G$"$k!$(B
$B$H$$$&$3$H$,$o$+$k!%$9$J$o$A!$!JM-8B8D$r=|$/!K(B
$BG$0U$N(B $(x,y) \in A$ $B$KBP$7$F(B
\beqnn
  P(z,w) = 0, \quad |z| = e^x, \quad |w| = e^y 
\eeqnn
$B$H$$$&>r7o$rK~$?$9(B $(z,w) \in \CC^2$ $B$O(B
$B#28D!JJ#AG6&Lr$JBP$r$J$9!K$7$+$J$$!%$3$N$3$H$r(B
Passare$B$H(BRullg{\aa}rd \cite{PR04}$B$,9T$C$F$$$k(B
$B9M;!$HAH$_9g$o$;$l$P!$%b%s%8%e!&%"%s%Z!<%kJ}Dx<0(B 
(\ref{Monge-Ampere}) $B$,=>$&$N$G$"$k!%(B

$B$3$N$h$&$J7A$G%b%s%8%e!&%"%s%Z!<%kJ}Dx<0$,8=$l$k(B
$B$3$H$OBgJQ6=L#?<$$!%2D@QJ,7O$N4QE@$+$i$=$N0U5A$r(B
$B>/$7O@$8$F$_$h$&!%(B

$B%b%s%8%e!&%"%s%Z!<%kJ}Dx<0$O%=%j%H%sJ}Dx<0$H$O(B
$B0[<A$NB8:_$G$"$j!$$`$7$mL5J,;62D@QJ,7O(B
\cite{Krichever94,TT-review}$B$d(B
$BN.BNNO3XE*2D@QJ,7O(B\cite{DN89,Tsarev91}$B$K6a$$!%(B
$BG.NO3X6K8B$X0\9T$9$kA0$NLO7?$O%U%'%k%_%*%s$H(B
$B4X78$,$"$j$=$&$J$N$G!$G.NO3XE*6K8B$r%U%'%k%_%*%s$N(B
$B=`8EE56K8B!JL5J,;68MED3,AX$dL5J,;6(BKP$B3,AX$O(B
$B$=$N$h$&$J$b$N$H$7$FM}2r$G$-$k!K$H$7$F07$&$3$H$,(B
$B$G$-$k$J$i$PL5J,;62D@QJ,7O$H$D$J$,$k2DG=@-$,$"$k!%(B
$B$5$i$K!$(B(\ref{Monge-Ampere}) $B$rJ#AG2=$7$?$b$N(B
$B!JJ#AG%b%s%8%e!&%"%s%Z!<%kJ}Dx<0!K(B
\beqnn
  \frac{\rd^2\Omega}{\rd x\rd\bar{x}}
  \frac{\rd^2\Omega}{\rd y\rd\bar{y}}
  - \frac{\rd^2\Omega}{\rd x\rd\bar{y}}
    \frac{\rd^2\Omega}{\rd\bar{x}\rd y} 
  = 1 
\eeqnn
$B$,%D%$%9%?!<M}O@$G07$($k$3$H(B\cite{Takasaki05}
$B$b%R%s%H$H8@$($k$+$b$7$l$J$$(B
$B!JJ#AG%b%s%8%e!&%"%s%Z!<%kJ}Dx<0$N<!854JLs$H$7$F(B
$BL5J,;68MEDJ}Dx<0$dL5J,;6(BKP$BJ}Dx<0$rF3$/$3$H$,$G$-$k!K!%(B
$B$3$N$h$&$K>u67>Z5r$O$$$/$D$+$"$k$,!$B>J}$G$O!$(B
$B%b%s%8%e!&%"%s%Z!<%kJ}Dx<0<+BN$rL5J,;62D@QJ,7O$+$i(B
$BF3$/$3$H$OFq$7$$!$$H$$$&;v<B$b$"$k!%(B
$B%@%$%^!<LO7?$N%m%s%-%sH!?t$r07$&OHAH$_$H$7$F$O(B
$BL5J,;62D@QJ,7O$h$j$bN.BNNO3XE*2D@QJ,7O$NM}O@$NJ}$,(B
$BE,$7$F$$$k$+$b$7$l$J$$!%(B

Kenyon$B$H(BOkounkov $B$,N)J}BN@Q$_>e$27?$NLO7?$N3&LL$N(B
$B!V6K8B7A!J(Blimit shape$B!K!W$K4X$7$F9T$C$?8&5f(B
\cite{KO05} $B$O$3$NLdBj$H$b4XO"$7$F$$$k$h$&$K(B
$B;W$o$l$k!%$=$3$G$O3&LL$N6K8B7A$K4X$9$kHyJ,J}Dx<0$H$7$F(B
$BJ#AG%P!<%,!<%9J}Dx<0$H$=$N0lHL2=$,EP>l$9$k!%(B
$B$5$i$K$3$N8&5f$OBe?t6J@~$N%H%m%T%+%k6K8B$r(B
$B07$C$F$$$kE@$K$*$$$F$bCmL\$5$l$k!%(B

$B%@%$%^!<LO7?$H$O0[$J$k@_Dj$G$O$"$k$,!$CfDE$HA0ED$,(B
$BD6BP>N%2!<%8M}O@$N%$%s%9%?%s%H%s7W;;$K4X$7$F9T$C$?(B
$B8&5f(B \cite{MN06} $B$b$3$NLdBj$HL54X78$G$O$J$$$h$&$K(B
$B;W$o$l$k!%$=$3$KEP>l$9$kBe?t6J@~$O%@%$%^!<LO7?$N(B
$B%9%Z%/%H%k6J@~$H$O0[$J$k$b$N$@$,!J$8$D$O(B
$B<~4|E*8MED3J;R$N%9%Z%/%H%k6J@~$NN`;wJ*$G$"$k!K!$(B
$B6KBg@-$=$NB>$N@-<A$O6&DL$7$F$$$k!%CfDE$HA0ED$O(B
$B$=$N%H%m%T%+%k6K8B$K$D$$$F$bD689M}O@$N4QE@$+$i(B
$B>\$7$/O@$8$F$$$k!%(B




\begin{thebibliography}{99}

\bibitem{Kasteleyn61}
P. Kasteleyn, The physics of dimers on a lattice, 
Physica {\bf 27}  (1961), 1209--1225. 

\bibitem{Kasteleyn67}
P. Kasteleyin, Graph theory and crystal physics, 
in ``Graph Theory and Theoretical Physics'' 
(Academics Press, 1967), pp. 43--110. 

\bibitem{Kenyon03}
R. Kenyon, An introduction to dimer models, 
arXiv:math.CO/0310326. 

\bibitem{Ueda05}
$B?"ED0l@P!$%I%_%N$K$h$k%?%$%kD%$j!$(B
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\end{document}
