%% Use plain TeX %% This paper contains 2 figures, whose PostScript form is appended %% to this file. %% JNL.TEX %% %% This is JNL.TEX Version 0.3 as of 6/12/85, %% modified by John Cardy %% \font\twelverm=cmr10 scaled 1200 \font\twelvei=cmmi10 scaled 1200 \font\twelvesy=cmsy10 scaled 1200 \font\twelveex=cmex10 scaled 1200 \font\twelvebf=cmbx10 scaled 1200 \font\twelvesl=cmsl10 scaled 1200 \font\twelvett=cmtt10 scaled 1200 \font\twelveit=cmti10 scaled 1200 \skewchar\twelvei='177 \skewchar\twelvesy='60 % Define \...point macros to change fonts and spacings consistently \def\twelvepoint{\normalbaselineskip=12.4pt \abovedisplayskip 12.4pt plus 3pt minus 9pt \belowdisplayskip 12.4pt plus 3pt minus 9pt \abovedisplayshortskip 0pt plus 3pt \belowdisplayshortskip 7.2pt plus 3pt minus 4pt \smallskipamount=3.6pt plus1.2pt minus1.2pt \medskipamount=7.2pt plus2.4pt minus2.4pt \bigskipamount=14.4pt plus4.8pt minus4.8pt \def\rm{\fam0\twelverm} \def\it{\fam\itfam\twelveit}% \def\sl{\fam\slfam\twelvesl} \def\bf{\fam\bffam\twelvebf}% \def\mit{\fam 1} \def\cal{\fam 2}% \def\tt{\twelvett} \textfont0=\twelverm \scriptfont0=\tenrm \scriptscriptfont0=\sevenrm \textfont1=\twelvei \scriptfont1=\teni \scriptscriptfont1=\seveni \textfont2=\twelvesy \scriptfont2=\tensy \scriptscriptfont2=\sevensy \textfont3=\twelveex \scriptfont3=\twelveex \scriptscriptfont3=\twelveex \textfont\itfam=\twelveit \textfont\slfam=\twelvesl \textfont\bffam=\twelvebf \scriptfont\bffam=\tenbf \scriptscriptfont\bffam=\sevenbf \normalbaselines\rm} % tenpoint \def\tenpoint{\normalbaselineskip=12pt \abovedisplayskip 12pt plus 3pt minus 9pt \belowdisplayskip 12pt plus 3pt minus 9pt \abovedisplayshortskip 0pt plus 3pt \belowdisplayshortskip 7pt plus 3pt minus 4pt \smallskipamount=3pt plus1pt minus1pt \medskipamount=6pt plus2pt minus2pt \bigskipamount=12pt plus4pt minus4pt \def\rm{\fam0\tenrm} \def\it{\fam\itfam\tenit}% \def\sl{\fam\slfam\tensl} \def\bf{\fam\bffam\tenbf}% \def\smc{\tensmc} \def\mit{\fam 1}% \def\cal{\fam 2}% \textfont0=\tenrm \scriptfont0=\sevenrm \scriptscriptfont0=\fiverm \textfont1=\teni \scriptfont1=\seveni \scriptscriptfont1=\fivei \textfont2=\tensy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \textfont\itfam=\tenit \textfont\slfam=\tensl \textfont\bffam=\tenbf \scriptfont\bffam=\sevenbf \scriptscriptfont\bffam=\fivebf \normalbaselines\rm} %% %% Various internal macros %% \def\beginlinemode{\endmode \begingroup\parskip=0pt \obeylines\def\\{\par}\def\endmode{\par\endgroup}} \def\beginparmode{\endmode \begingroup \def\endmode{\par\endgroup}} \let\endmode=\par {\obeylines\gdef\ {}} \def\singlespace{\baselineskip=\normalbaselineskip} \def\oneandathirdspace{\baselineskip=\normalbaselineskip \multiply\baselineskip by 4 \divide\baselineskip by 3} \def\oneandahalfspace{\baselineskip=\normalbaselineskip \multiply\baselineskip by 3 \divide\baselineskip by 2} \def\doublespace{\baselineskip=\normalbaselineskip \multiply\baselineskip by 2} \def\triplespace{\baselineskip=\normalbaselineskip \multiply\baselineskip by 3} \newcount\firstpageno \firstpageno=2 \footline={\ifnum\pageno<\firstpageno{\hfil}\else{\hfil\twelverm\folio\hfil}\fi} %\footline={\hfil\twelverm\folio\hfil} \let\rawfootnote=\footnote % We must set the footnote style \def\footnote#1#2{{\rm\singlespace\parindent=0pt\rawfootnote{#1}{#2}}} \def\raggedcenter{\leftskip=4em plus 12em \rightskip=\leftskip \parindent=0pt \parfillskip=0pt \spaceskip=.3333em \xspaceskip=.5em \pretolerance=9999 \tolerance=9999 \hyphenpenalty=9999 \exhyphenpenalty=9999 } \def\dateline{\rightline{\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\year}} \def\received{\vskip 3pt plus 0.2fill \centerline{\sl (Received\space\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \qquad, \number\year)}} %% %% Page layout, margins, font and spacing (feel free to change) %% \hsize=6.5truein %\hoffset=1truein \vsize=8.9truein %\voffset=1truein \parskip=\medskipamount \twelvepoint % selects twelvepoint fonts (cf. \tenpoint) \doublespace % selects double spacing for main part of paper (cf. % \singlespace, \oneandahalfspace) \overfullrule=0pt % delete the nasty little black boxes for overfull box %% %% The user definitions for major parts of a paper (feel free to change) %% \def\preprintno#1{ \rightline{\rm #1}} % Preprint number at upper right of title page \def\title % Title on title page {\null\vskip 3pt plus 0.2fill \beginlinemode \doublespace \raggedcenter \bf} \def\author % Author(s) name(s) on title page {\vskip 4pt plus 0.2fill \beginlinemode \singlespace \raggedcenter} \def\affil % Affiliations (can intermix with \author) {\vskip 3pt plus 0.1fill \beginlinemode \oneandahalfspace \raggedcenter \sl} \def\abstract % Begin abstract {\vskip 3pt plus 0.3fill \beginparmode \doublespace \narrower ABSTRACT: } \def\endtitlepage % End title page, begin body of paper {\endpage % This subsumes \body \body} \def\body % Begin text body; can be used to end {\beginparmode} % \title, \author, \affil, \abstract, % \reference, or \figurecaption modes \def\head#1{ % Head; NOTE enclose the text in {} \filbreak\vskip 0.5truein % e.g., \head{I. Introduction} {\immediate\write16{#1} \line{\bf{#1}\hfil}\par} \nobreak\vskip 0.25truein\nobreak} \def\subhead#1{ % Subhead; NOTE enclose the text in {} \vskip 0.25truein % e.g., \subhead{A. History of the Problem} {\line{\sl{#1}\hfil} \par} \nobreak\vskip 0.05truein\nobreak\noindent} \def\refto#1{$^{#1}$} % For references in text as superscript \def\references % Begin references -- basic format is Phys Rev {\head{References} % I.e., volume, page, year (space after commas). \beginparmode \frenchspacing \parindent=0pt \leftskip=0truecm \parskip=8pt plus 3pt \everypar{\hangindent=\parindent}} \gdef\refis#1{\indent\hbox to 0pt{\hss#1.~}} % Ref list numbers. \gdef\journal#1, #2, #3, 1#4#5#6{ % Journal reference. Comma sets {\sl #1~}{\bf #2}, #3, (1#4#5#6)} % off: name, vol, page, year \def\refstylenp{ % Nucl Phys(or Phys Lett) ref style: V, Y, P \gdef\refto##1{ [##1]} % Reference in text [] \gdef\refis##1{\indent\hbox to 0pt{\hss##1)~}} % Ref list numbers) \gdef\journal##1, ##2, ##3, ##4 { % Journal reference {\sl ##1~}{\bf ##2~}(##3) ##4 }} \def\refstyleprnp{ % Input like pr, output like np!! \gdef\refto##1{ [##1]} % Reference in text [] \gdef\refis##1{\indent\hbox to 0pt{\hss##1)~}} % Ref list numbers) \gdef\journal##1, ##2, ##3, 1##4##5##6{ % Journal reference {\sl ##1~}{\bf ##2~}(1##4##5##6) ##3}} \def\refstylejp{ % no numbers in text; in refs, Y, J, V, P \gdef\journal##1, ##2, ##3{ {\sl ##1~}{\bf ##2} ##3}} \def\refstylephysica{ % J, V, Y, P \gdef\journal##1, ##2, ##3, ##4{ ##1~{\bf ##2} (##3) ##4}} \def\pr{\journal Phys. Rev., } \def\pra{\journal Phys. Rev. A, } \def\prb{\journal Phys. Rev. B, } \def\prc{\journal Phys. Rev. C, } \def\prd{\journal Phys. Rev. D, } \def\prl{\journal Phys. Rev. Lett., } \def\jmp{\journal J. Math. Phys., } \def\rmp{\journal Rev. Mod. Phys., } \def\cmp{\journal Comm. Math. Phys., } \def\np{\journal Nucl. Phys., } \def\pl{\journal Phys. Lett., } \def\jph{\journal J. Phys., } \def\jp{\journal J. Phys., } \def\jpa{\journal J. Phys. A: Math. Gen., } \def\jsp{\journal J. Stat. Phys., } \def\endreferences{\body} \def\figurecaptions % Begin figure captions { \beginparmode \head{Figure Captions} % \parskip=24pt plus 3pt \everypar={\hangindent=4em} } \def\endfigurecaptions{\body} \def\endpage % Eject a page {\vfill\eject} \def\endpaper % Ways to say goodbye {\endmode\vfill\supereject} \def\endjnl {\endpaper} \def\endit {\endpaper\end} %% %% Various little user definitions %% \def\ref#1{Ref.\thinspace#1} % for inline references \def\refs#1{Refs.\thinspace#1} \def\Ref#1{Ref.\thinspace#1} % ditto \def\fig#1{Fig.\thinspace#1} \def\figs#1{Figs.\thinspace#1} \def\Equation#1{Equation (#1)} % For citation of equation numbers \def\Equations#1{Equations (#1)} % ditto \def\Eq#1{Eq.\thinspace(#1)} % ditto \def\eq#1{Eq.\thinspace(#1)} % ditto \def\Eqs#1{Eqs.\thinspace(#1)} % ditto \def\eqs#1{Eqs.\thinspace(#1)} % ditto \def\sec#1{Sec.\thinspace(#1)} \def\Sec#1{Sec.\thinspace(#1)} \def\frac#1#2{{#1 \over #2}} \def\ffrac#1#2{\textstyle{#1\over#2}\displaystyle} %for smaller fractions in \def\half{\textstyle{ 1\over 2}} %displayed equations \def\eg{{\it e.g.,\ }} \def\Eg{{\it E.g.,\ }} \def\ie{{\it i.e.,\ }} \def\Ie{{\it I.e.,\ }} \def\etal{{\it et al.}} \def\etc{{\it etc.}} \def\via{{\it via}} \def\sla{\raise.15ex\hbox{$/$}\kern-.57em} \def\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill} \def\twiddle{\lower.9ex\rlap{$\kern-.1em\scriptstyle\sim$}} \def\bigtwiddle{\lower1.ex\rlap{$\sim$}} \def\gtwid{\mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}}} \def\ltwid{\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}}} \def\square{\kern1pt\vbox{\hrule height 1.2pt\hbox{\vrule width 1.2pt\hskip 3pt \vbox{\vskip 6pt}\hskip 3pt\vrule width 0.6pt}\hrule height 0.6pt}\kern1pt} \def\ucsb{Department of Physics\\University of California\\ Santa Barbara CA 93106} \def\ucsd{Department of Physics\\University of California\\ La Jolla, CA 92093} \def\begintable{\offinterlineskip\hrule} \def\endtable{\hrule} \def\equalign{\eqalign} \def\equaligno{\eqaligno} %Figures in text. Use \midinsert\vskip xin\figure{number}{caption}\endinsert \def\figure#1#2{ \parindent=1in\hsize=5.5in\singlespace\item {\bf Figure {#1}} {#2}} \def\(#1){Eq.\thinspace(\call{#1})} \def\call#1{{#1}} \abovedisplayskip=18pt plus 3pt minus 9pt \belowdisplayskip=18pt plus 3pt minus 9pt % DEFS.TEX % This file contains several useful macros which should be input before %typing a paper. \def\eqro#1{\eqno{\rm (#1)}} \def\gap{$$\eqalign{&\cr&\cr&\cr&\cr&\cr&\cr&\cr}$$} \def\biggap{$$\eqalign{&\cr&\cr&\cr&\cr&\cr&\cr&\cr&\cr&\cr&\cr&\cr&\cr&\cr}$$} \def\lhs{left hand side} \def\rhs{right hand side} \def\Re{{\rm Re}} \def\Im{{\rm Im}} \mathchardef\duall="0414 \def\dual{{\!\check{\phantom{x}}}} \def\const{{\rm const.}} \def\inv#1{{#1}^{-1}} \def\mod#1{|#1|} \def\no#1{:\!#1\!:} %%normal ordering \def\ev#1{\langle #1\rangle} \def\ov#1{\overline{#1}} \def\bra#1{\langle #1\vert} \def\ket#1{\vert #1\rangle} \def\bkt#1#2{\langle #1 \vert #2 \rangle} \def\mel#1#2#3{\bra{#1}{#2}\ket{#3}} \def\tr{{\rm Tr}\,} \def\vac{\ket0} \def\sn{\mathop{\rm sn}\nolimits} \def\cn{\mathop{\rm cn}\nolimits} \def\dn{\mathop{\rm dn}\nolimits} \def\snh{\mathop{\rm snh}\nolimits} \def\dim{\mathop{\rm dim}\nolimits} \def\ch{\mathop{\rm ch}\nolimits} \def\sh{\mathop{\rm sh}\nolimits} \def\deg{\mathop{\rm deg}\nolimits} \def\Ln{\mathop{\rm Ln}\nolimits} \def\ab{\bar\a} \def\pb{\bar p} \def\qb{\bar q} \def\zb{\bar z} \def\hb{\bar h} \def\kb{\bar k} \def\nb{\bar n} \def\rb{\bar r} \def\tb{\bar\tau} \def\qb{\bar q} \def\wb{\bar w} \def\sb{\bar s} \def\xb{\bar x} \def\betabar{\bar\beta} \def\sib{\bar\sigma} \def\Tb{\overline T} \def\Lb{\overline L} \def\Wb{\overline W} \def\psib{\overline\psi} \def\partialb{\overline\partial} \def\xt{\tilde x} \def\qt{\tilde q} \def\phit{\widetilde\phi} \def\Pt{\widetilde P} \def\Z#1{{\rm Z}_#1} %%for Z_n groups \def\Z2{${\rm Z}_2$} \def\one{{\bf 1}} %Don't try to use this except between $ signs! \def\H{{\cal H}} \def\O{{\cal O}} \def\A{\hat A} \def\ope{operator product expansion} \def\rg{renormalization group} \def\fp{fixed point} \def\fps{fixed points} \def\sco{scaling operator} \def\scos{scaling operators} \def\bc{boundary condition} \def\bcs{\bc s} \def\cf{correlation function} \def\cfs{\cf s} \def\sm{statistical mechanics} \def\qft{quantum field theory} \def\qfts{quantum field theories} %\def\dim{scaling dimension} \def\dims{scaling dimensions} \def\repn{representation} \def\repns{\repn s} \def\IR{irreducible \repn} \def\ir{\IR} \def\irrep{\IR} \def\IRS{\IR s} \def\irs{\IRS} \def\hw{highest weight} \def\hwr{highest weight \repn} \def\hwrs{\hwr s} \def\cft{conformal field theory} \def\cfts{conformal field theories} \def\dd#1#2{{\partial{#1}\over\partial{#2}}} \def\dbd#1{{\partial\over\partial{#1}}} \def\etc{{\it etc.}} \def\eg{{\it e.g.}} \def\ie{{\it i.e.}} \def\viz{{\it viz.}} \def\ibid{{\it ibid.}} \def\uhp{upper half plane} \def\lhp{lower half plane} \def\ode{ordinary differential equation} \def\odes{\ode s} \def\pde{partial differential equation} \def\pdes{\pde s} \def\a{\alpha} \def\b{\beta} \def\g{\gamma} \def\d{\delta} %\def\d{{\rm d}} \def\e{\epsilon} \def\z{\zeta} \def\th{\theta} \def\k{\kappa} \def\l{\lambda} \def\m{\mu} \def\n{\nu} \def\x{\xi} \def\r{\rho} \def\s{\sigma} \def\t{\tau} \def\ph{\phi} \def\ps{\psi} \def\om{\omega} \def\w{\omega} \def\G{\Gamma} \def\D{\Delta} \def\Th{{\mit\Theta}} \def\L{\Lambda} \def\X{\Xi} \def\S{\Sigma} \def\U{\Upsilon} \def\Ph{\Phi} \def\Ps{\Psi} \def\Om{\Omega} \def\E{\varepsilon} \def\vth{{\rm sign}} \def\dT{\delta T} \def\dg{\delta g} \def\i{{\rm i}} \def\per{\quad.} \refstylenp %% REFORDER.TEX 6/7/85 Doug E. %% (mods: 3/25/87 R.G.Palmer) %% %% This macro package is intended for use with JNL. %% It will automatically order and sort the references in a paper %% by order of first citation.(!!) To use, say \input reforder %% after \input jnl (and after all definitions of \refto etc., %% in particular after any use of the \refstyleXX macros), %% but before any use of \refto etc. Use \refto{} (or \ref{} and %% \Ref{}) to cite references in the text. Use \refis{} to supply %% the references, SKIP A LINE after each reference. Open the %% reference listing with \references and close it with \endreferences. %% %% REFORDER depends on the %% JNL macros \refto{}, \ref{}, \Ref{} to identify citation of references. %% REFORDER also contains a macro \cite{} which can be used to cite %% references; e.g., ``Reference \cite{19} blah...'' will produce %% output ``Reference 19 blah''. Multiple citations can be separated %% by commas. E.g., \refto{24,26,27} and \cite{3,7} %% are legal. Also legal is \refto{3-7}, which expands to mean the same %% as \refto{3,4,5,6,7}. Reference ``numbers'' can in general be any %% alphanumeric string; e.g. BjorkenAndDrell is perfectly OK used in %% the form\ref{BjorkenAndDrell}; such strings should contain no blanks. %% %% If you have your own pet macros to cite references such as, e.g., %% \def\referpet#1{$^(#1)$)}, you can bring it to the attention of %% REFORDER so all \referpet's will be properly cited simply by %% declaring ``\citeall\referpet'' once near the beginning, after %% \referpet is defined and %% after \input reforder. This has the effect of redefining the macro %% as e.g., \def\referpet#1{$^(\cite{#1})$}. (Such \citeall'ed macros %% must have exactly one argument #1, as in \referpet.) See e.g., %% the end of this file where \refto, \ref and \Ref are \citeall'ed. %% %% REFORDER depends on the macro \refis{} to supply each reference. %% \refis{} can be used to supply a reference anywhere in the paper %% after its first citation. The macro \endreferences actually triggers %% sorting and listing of references. Skip a line after a reference %% listing (or, alternatively, end each listing in \par). %% %% Use \ignoreuncited after \input reforder if you wish to ignore %% references that are supplied but not cited. This is particularly %% useful if you maintain a master file of references (each supplied %% with \refis{}) but only use a subset of these in a given paper. %% Include your reference file (with \input) between \references %% and \endreferences. %% %% Use \referencefile after \input reforder if you want an ordered %% source listing of the references in file .ref %% %% The \reftorange macro can be used to produce a %% reference range, like $^{10-15}$. (The \refto macro always %% lists the references one by one, even for e.g. \refto{10-15}). %% Use e.g. \reftorange{10}{11-14}{15} -- the references in the %% middle group are cited but only 10-15 appears in the text. %% Note that \reftorange does NOT check for increasing order. \catcode`@=11 \newcount\r@fcount \r@fcount=0 \newcount\r@fcurr \immediate\newwrite\reffile \newif\ifr@ffile\r@ffilefalse \def\w@rnwrite#1{\ifr@ffile\immediate\write\reffile{#1}\fi\message{#1}} \def\writer@f#1>>{} \def\referencefile{% Stuff to write .REF file \r@ffiletrue\immediate\openout\reffile=\jobname.ref% \def\writer@f##1>>{\ifr@ffile\immediate\write\reffile% {\noexpand\refis{##1} = \csname r@fnum##1\endcsname = % \expandafter\expandafter\expandafter\strip@t\expandafter% \meaning\csname r@ftext\csname r@fnum##1\endcsname\endcsname}\fi}% \def\strip@t##1>>{}} \let\referencelist=\referencefile \def\citeall#1{\xdef#1##1{#1{\noexpand\cite{##1}}}} \def\cite#1{\each@rg\citer@nge{#1}} % Variable No. of args, separated by "," \def\each@rg#1#2{{\let\thecsname=#1\expandafter\first@rg#2,\end,}} \def\first@rg#1,{\thecsname{#1}\apply@rg} % each@ag is a general purpose \def\apply@rg#1,{\ifx\end#1\let\next=\relax% variable no. of arg. macro. \else,\thecsname{#1}\let\next=\apply@rg\fi\next}% args separated by commas \def\citer@nge#1{\citedor@nge#1-\end-} % Check for M-N range (M and N numbers) \def\citer@ngeat#1\end-{#1} \def\citedor@nge#1-#2-{\ifx\end#2\r@featspace#1 % Single argument \else\citel@@p{#1}{#2}\citer@ngeat\fi} % M-N range of arguments \def\citel@@p#1#2{\ifnum#1>#2{\errmessage{Reference range #1-#2\space is bad.}% \errhelp{If you cite a series of references by the notation M-N, then M and N must be integers, and N must be greater than or equal to M.}}\else% {\count0=#1\count1=#2\advance\count1 by1\relax\expandafter\r@fcite\the\count0,% \loop\advance\count0 by1\relax% Loop from M to N \ifnum\count0<\count1,\expandafter\r@fcite\the\count0,% \repeat}\fi} \def\r@featspace#1#2 {\r@fcite#1#2,} % Eat spaces at beginning or end of arg \def\r@fcite#1,{\ifuncit@d{#1}% Cite individual reference \newr@f{#1}% \expandafter\gdef\csname r@ftext\number\r@fcount\endcsname% {\message{Reference #1 to be supplied.}% \writer@f#1>>#1 to be supplied.\par}% \fi% \csname r@fnum#1\endcsname} \def\ifuncit@d#1{\expandafter\ifx\csname r@fnum#1\endcsname\relax}% \def\newr@f#1{\global\advance\r@fcount by1% \expandafter\xdef\csname r@fnum#1\endcsname{\number\r@fcount}} \let\r@fis=\refis % Save old \refis, redefine \def\refis#1#2#3\par{\ifuncit@d{#1}% Use two params #2 #3 to strip blank \newr@f{#1}% \w@rnwrite{Reference #1=\number\r@fcount\space is not cited up to now.}\fi% \expandafter\gdef\csname r@ftext\csname r@fnum#1\endcsname\endcsname% {\writer@f#1>>#2#3\par}} \def\ignoreuncited{% redefine \refis if ignoring uncited references \def\refis##1##2##3\par{\ifuncit@d{##1}% \else\expandafter\gdef\csname r@ftext\csname r@fnum##1\endcsname\endcsname% {\writer@f##1>>##2##3\par}\fi}} \def\r@ferr{\endreferences\errmessage{I was expecting to see \noexpand\endreferences before now; I have inserted it here.}} \let\r@ferences=\references \def\references{\r@ferences\def\endmode{\r@ferr\par\endgroup}} \let\endr@ferences=\endreferences \def\endreferences{\r@fcurr=0% Save old \endreferences, redefine {\loop\ifnum\r@fcurr<\r@fcount% Loop over refnum and produce text \advance\r@fcurr by 1\relax\expandafter\r@fis\expandafter{\number\r@fcurr}% \csname r@ftext\number\r@fcurr\endcsname% \repeat}\gdef\r@ferr{}\endr@ferences} % Save old \endpaper, redefine it to write parting message. \let\r@fend=\endpaper\gdef\endpaper{\ifr@ffile \immediate\write16{Cross References written on []\jobname.REF.}\fi\r@fend} \catcode`@=12 \citeall\refto % These macros will generate citations \citeall\ref % \citeall\Ref % %% EQNORDER.TEX 11/05/85 Doug E. %% %% This macro package is intended for use with JNL. %% It will automatically order and sort the equations in a paper %% by order of appearance. To use, say \input eqnorder %% after \input jnl (and after all definitions of \eqno etc., %% but before any use of \eqno etc. Use \() to cite equations %% in the text. Use \eqno() or \tag to put the numbers on displayed %% equations; or use &() with \eqalignno{} as explained in the TeXBOOK. %% %% EQNORDER depends on the macro \() to refer to equations in the %% text; use it as Equation \() or Eq. \() or Eqs. \(), etc. %% EQNORDER also contains a macro \call{} which can be used to refer %% equations; e.g., ``Equation \call{19} blah...'' will produce %% output ``Equation 19 blah''. Multiple citations must be separated %% by commas. E.g., \(24,26,27) and \call{3,7} are legal. A sequence %% of equation numbers can be referred to by, e.g., \(3-7) which means %% the same as (3,4,5,6,7). %% %% Equation ``numbers'' can actually be any alphanumeric string; %% e.g., equation \tag Schroedinger $$ can be referred to by %% \(Schroedinger). In fact, if you expect to renumber the equations, %% it is actually easier and less confusing to tag them with names %% rather than numbers. %% %% There is one big rule: You cannot refer to an equation before you %% display it. There is a limited loophole: You can refer to the %% first, second or third equation number just below where you are %% as \(+1), \(+2), or \(+3). In the same way, the equation number %% just above can be called \(0), and the three preceding numbers %% \(-1), \(-2), \(-3). %% %% TeX keeps the equation number as a count \tagnumber. \tagnumber %% is initially 0, and it is incremented by 1 just BEFORE it is used %% to tag a displayed equation. You are free to reset \tagnumber, %% which you can do just by writing e.g. \tagnumber=23. %% %% If you label a displayed equation with a null number, \tag $$ %% or \eqno() or &(), an incremented \tagnumber will be generated for %% the equation, but the only way to refer to that equation is %% via the \(+n) or \(0) or \(-n) notation. %% %% In long papers, the author often numbers the equations anew %% in each section in the style \tag 6.1 $$, \tag 6.2 $$, and so %% forth; the equations are then referred to by \(6.1) etc. %% One reason for doing this is to minimize chaos when equations %% have to be renumbered --- but this is what EQNORDER already does! %% If you still want to use such a style, just declare \taghead{6.} %% for example at the beginning of Section 6. 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Thus constructions like \tag 25 ' $$ %% and \tag 25 '' $$ are legal. %% %% If you have your own pet macros to call equations such as, e.g., %% \def\eqnpet#1{($#1$))}, you can bring it to the attention of %% REFORDER so all \eqnpet's will be properly calld simply by %% declaring ``\callall\eqnpet'' after \eqnpet is defined and %% after \input eqnorder. This has the effect of redefining the macro %% as e.g., \def\eqnpet#1{($\call{#1}$)}. (Such \callall'ed macros %% must have exactly one argument #1, as in \eqnpet.) \catcode`@=11 \newcount\tagnumber\tagnumber=0 \immediate\newwrite\eqnfile \newif\if@qnfile\@qnfilefalse \def\write@qn#1{} \def\writenew@qn#1{} \def\w@rnwrite#1{\write@qn{#1}\message{#1}} \def\@rrwrite#1{\write@qn{#1}\errmessage{#1}} \def\taghead#1{\gdef\t@ghead{#1}\global\tagnumber=0} \def\t@ghead{} \expandafter\def\csname @qnnum-3\endcsname {{\t@ghead\advance\tagnumber by -3\relax\number\tagnumber}} \expandafter\def\csname @qnnum-2\endcsname {{\t@ghead\advance\tagnumber by -2\relax\number\tagnumber}} \expandafter\def\csname @qnnum-1\endcsname {{\t@ghead\advance\tagnumber by -1\relax\number\tagnumber}} \expandafter\def\csname @qnnum0\endcsname {\t@ghead\number\tagnumber} \expandafter\def\csname @qnnum+1\endcsname {{\t@ghead\advance\tagnumber by 1\relax\number\tagnumber}} \expandafter\def\csname @qnnum+2\endcsname {{\t@ghead\advance\tagnumber by 2\relax\number\tagnumber}} \expandafter\def\csname @qnnum+3\endcsname {{\t@ghead\advance\tagnumber by 3\relax\number\tagnumber}} \def\equationfile{% \@qnfiletrue\immediate\openout\eqnfile=\jobname.eqn% \def\write@qn##1{\if@qnfile\immediate\write\eqnfile{##1}\fi} \def\writenew@qn##1{\if@qnfile\immediate\write\eqnfile {\noexpand\tag{##1} = (\t@ghead\number\tagnumber)}\fi} } \def\callall#1{\xdef#1##1{#1{\noexpand\call{##1}}}} \def\call#1{\each@rg\callr@nge{#1}} \def\each@rg#1#2{{\let\thecsname=#1\expandafter\first@rg#2,\end,}} \def\first@rg#1,{\thecsname{#1}\apply@rg} \def\apply@rg#1,{\ifx\end#1\let\next=\relax% \else,\thecsname{#1}\let\next=\apply@rg\fi\next} \def\callr@nge#1{\calldor@nge#1-\end-} \def\callr@ngeat#1\end-{#1} \def\calldor@nge#1-#2-{\ifx\end#2\@qneatspace#1 % \else\calll@@p{#1}{#2}\callr@ngeat\fi} \def\calll@@p#1#2{\ifnum#1>#2{\@rrwrite{Equation range #1-#2\space is bad.} \errhelp{If you call a series of equations by the notation M-N, then M and N must be integers, and N must be greater than or equal to M.}}\else% {\count0=#1\count1=#2\advance\count1 by1\relax\expandafter\@qncall\the\count0,% \loop\advance\count0 by1\relax% \ifnum\count0<\count1,\expandafter\@qncall\the\count0,% \repeat}\fi} \def\@qneatspace#1#2 {\@qncall#1#2,} \def\@qncall#1,{\ifunc@lled{#1}{\def\next{#1}\ifx\next\empty\else \w@rnwrite{Equation number \noexpand\(>>#1<<) has not been defined yet.} >>#1<<\fi}\else\csname @qnnum#1\endcsname\fi} \let\eqnono=\eqno \def\eqno(#1){\tag#1} \def\tag#1$${\eqnono(\displayt@g#1 )$$} \def\aligntag#1$${\gdef\tag##1\\{&(\displayt@g##1 )\cr}\eqalignno{#1\\}$$ \gdef\tag##1$${\eqnono(\displayt@g##1 )$$}} \let\eqalignnono=\eqalignno \def\eqalignno#1{\displ@y \tabskip\centering \halign to\displaywidth{\hfil$\displaystyle{##}$\tabskip\z@skip &$\displaystyle{{}##}$\hfil\tabskip\centering &\llap{$\displayt@gpar##$}\tabskip\z@skip\crcr #1\crcr}} \def\displayt@gpar(#1){(\displayt@g#1 )} \def\displayt@g#1 {\rm\ifunc@lled{#1}\global\advance\tagnumber by1 {\def\next{#1}\ifx\next\empty\else\expandafter \xdef\csname @qnnum#1\endcsname{\t@ghead\number\tagnumber}\fi}% \writenew@qn{#1}\t@ghead\number\tagnumber\else {\edef\next{\t@ghead\number\tagnumber}% \expandafter\ifx\csname @qnnum#1\endcsname\next\else \w@rnwrite{Equation \noexpand\tag{#1} is a duplicate number.}\fi}% \csname @qnnum#1\endcsname\fi} \def\ifunc@lled#1{\expandafter\ifx\csname @qnnum#1\endcsname\relax} \let\@qnend=\end\gdef\end{\if@qnfile \immediate\write16{Equation numbers written on []\jobname.EQN.}\fi\@qnend} \catcode`@=12 %% DEBUG %%\def\see#1 {\expandafter\show\csname#1\endcsname} \preprintno{UCSBTH--91--56} \title Critical Percolation in Finite Geometries \author John L. Cardy \affil\ucsb \abstract The methods of conformal field theory are used to compute the crossing probabilities between segments of the boundary of a compact two-dimensional region at the percolation threshold. These probabilities are shown to be invariant not only under changes of scale, but also under mappings of the region which are conformal in the interior and continuous on the boundary. This is a larger invariance than that expected for generic critical systems. Specific predictions are presented for the crossing probability between opposite sides of a rectangle, and are compared with recent numerical work. The agreement is excellent. \vfill \line{PACS Nos: 64.60.Ak, 05.50.+k\hfil} \eject Conformal field theory has been very successful in determining universal quantities associated with two-dimensional isotropic systems at their critical points\refto{CARDYDL,CARDYFSS}. The range of predictions which can be made appears to be bounded by the enthusiasm and industriousness of the theorist rather than by any intrinsic limitations of the theory. However, the underlying assumptions of conformal field theory, and their appropriateness for describing the scaling limit of critical lattice systems, are not rigorously founded, and it remains important to perform precise numerical tests of the theory whenever possible. Recently\refto{LANG}, extensive numerical work has been carried out to estimate crossing probabilities in rectangular geometries for critical percolation in very large but finite lattices, with the principal aim of establishing their universality between different models. Percolation provides an important test of the ideas of conformal field theory because large-scale numerical simulations are more readily performed. In this letter we consider the general problem of crossing probabilities in the language of conformal field theory, and derive exact expressions which may be compared with the numerical work. The most familiar way to think about percolation as a critical phenomenon is through the $q\to1$ limit of the $q$-state Potts model\refto{KF}. In that model, spins $s(r)$ at the sites of the lattice are allowed to be in one of $q$ possible states $(\a,\b,\ldots)$, and the partition function is the trace of a product over links of the form $$ Z=\prod_{(r,r')}\left(1+x\d_{s(r),s(r')}\right) \eqno() $$ The terms in the expansion in powers of $Z$ in powers of $x$ are in 1-1 correspondence with configurations of bonds appearing in the bond percolation problem, and in the limit $q\to1$ they are weighted appropriately if $x=p/(1-p)$. Two sites in the same cluster are necessarily in the same state of the Potts model. Consider now two disjoint segments $S_1$ and $S_2$ of the piecewise differentiable boundary of a simply connected compact region. Let $Z_{\a\b}$ be the partition function of the $q$-state Potts model with the constraint that all spins at lattice sites on $S_1$ are fixed in the state $\a$, and all the spins on $S_2$ are fixed in the state $\b$. The rest of the boundary spins are unrestricted. Then the crossing probability between $S_1$ and $S_2$ is given by $$ \pi(S_1,S_2)=\lim_{q\to1}\left(Z_{\a\a}-Z_{\a\b}\right) \eqno() $$ where, in the second term $\a\not=\b$. In fact, in the limit when $q=1$, the first term is unity. The interior of the compact region may be mapped conformally to the \uhp, so that the boundary is mapped onto the real axis. If there are corners on the boundary, the map will be singular but continuous at these points. Conformal field theory relates the partition functions in the two geometries, in a manner to be described later. Thus, if the images of $S_1$ and $S_2$ are the intervals $(x_1,x_2)$ and $(x_3,x_4)$ respectively (where we may assume that the $x_i$ are placed in increasing order), the problem reduces to that of finding the respective partition functions $Z_{\a\a}$ and $Z_{\a\b}$ in this geometry. The study of boundary conditions in conformal field theory\refto{CARDYSURF,CARDYBC} shows that, for a particular theory, there is a given set of boundary conditions consistent with the conformal symmetry of the theory. In general they correspond to the possible fixed points of the \rg\ in the semi-infinite system: thus a generic boundary condition becomes equivalent in the continuum limit to one of those allowed by conformal symmetry. In addition, points on the boundary at which the boundary condition changes may be identified\refto{CARDYBC} with points of insertion of boundary operators, that is, scaling operators of the theory corresponding to highest weight states of the Virasoro algebra\refto{BPZ,CARDYSURF}. Situations where more than one change of boundary condition occurs then correspond to correlation functions of these boundary operators. In the case in question, let us denote the boundary condition where the spins are free by $(f)$, and those where they are fixed in a given state by $(\a)$. Denote the boundary operator corresponding to a switch from boundary condition $(i)$ to $(j)$ at the point $x$ by $\phi_{(i|j)}(x)$. Then the partition functions we need are given in terms of correlators by $$ \eqalign{ Z_{\a\a}&=Z_f\,\ev{\phi_{(f|\a)}(x_1)\,\phi_{(\a|f)}(x_2)\,\phi_{(f|\a)}(x_3) \,\phi_{(\a|f)}(x_4)}\cr Z_{\a\b}&=Z_f\,\ev{\phi_{(f|\a)}(x_1)\,\phi_{(\a|f)}(x_2)\,\phi_{(f|\b)}(x_3) \,\phi_{(\b|f)}(x_4)}\cr} \eqno(ZZ) $$ where $Z_f$ is partition function with free boundary conditions all along the real axis. Note that, in the \uhp, all three partition functions in general diverge in the infinite volume limit, and \(ZZ) strictly should be interpreted as being valid only for a large but finite lattice. However, when $q=1$, $Z_f=1$ identically, and this problem does not arise. In order to compute the above \cfs\ using the methods of \cft, we need to understand to which representations of the Virasoro algebra the boundary operators belong. It has been known for some time\refto{DOTFAT,SACLAY} that the critical $q$-state Potts model corresponds, in the continuum limit, to a \cft\ with conformal anomaly number\refto{BPZ} $c=1-6/m(m+1)$, where $q=4\cos^2(\pi/m+1)$, with $m\geq1$. Thus percolation has $c=0$. This is consistent with the fact that $c$ is related to the finite-size corrections to the free energy\refto{CARDYPESCH} in certain geometries, and the free energy vanishes identically when $q=1$. However, the problem of boundary operator assignment has not been addressed so far, except for the cases $q=2$ and $q=3$\refto{CARDYOPC,CARDYBC,SB}. However, it is not difficult to determine the assignment for the operators $\phi_{(f|\a)}$. For minimal \cfts, all the scaling operators have the property that their corresponding representations contain null states\refto{BPZ}. This has the consequence that the allowed values of their scaling dimensions are given by the Kac formula $$ h=h_{r,s}={\big(r(m+1)-sm\big)^2-1\over 4m(m+1)} \eqno(KAC) $$ where $r$ and $s$ are positive integer are positive integers. In addition, the correlators involving these operators obey differential equations of order at most $rs$. For unitary models, for example those with positive definite Boltzmann weights, all allowed operators must be of this type. Although this condition is not applicable to the $q$-state Potts model for general $q$, the fact that it does apply for $q=2$ and $q=3$ suggests that those operators whose position $(r,s)$ in the Kac table does not appear to change as a function of $c(q)$ do correspond to representations with null states even in the non-unitary case. Indeed, it was conjectured in \ref{CARDYSURF} that the spin operator of the Potts model, when inserted at a boundary with free boundary conditions, corresponds to $(r,s)=(1,3)$. This agrees with known results for $q=2,3$\refto{CARDYOPC,CARDYBC,SB}, and is also consistent with the known assignment of operators in the bulk. It gives a prediction for the case $q=1$ which agrees with numerical estimates to within their accuracy\refto{DEBELL}. There are $(q-1)$ independent such spin operators. The continuum limit of duality symmetry\refto{POTTS} for the critical $q$-state Potts model maps the free boundary condition $(f)$ onto a fixed boundary condition $(\a)$. Exactly which state $\a$ is chosen is arbitrary, since just one spin on the boundary has to be assigned a given value in order to make the duality mapping 1-1. An insertion of the spin operator at the point $x$ on the free boundary is mapped into an insertion of the disorder operator $\phi_{(\a|\b)}(x)$ where $\b\not=\a$. There are just $(q-1)$ such operators, for a fixed $\a$. This duality symmetry implies that that the correlators of $\phi{(\a|\b)}$ are simply related to those of the boundary spin operator with free boundary conditions, and hence that it also corresponds to $(r,s)=(1,3)$ in the Kac table. However, we are interested in the operators $\phi_{(\a|f)}$. Consider the insertion of two such operators $\phi_{(\a|f)}(x)\phi_{(f|\b)}(x')$ as the points $x$, $x'$ approach each other. This is given by the \ope, which symbolically must have the structure $$ \phi_{(\a|f)}\cdot\phi_{(f|\b)}\sim\d_{\a\b}\,\one+\phi_{(\a|\b)}+\cdots \eqno(OPE) $$ where $\one$ is the identity operator (no change in boundary condition). According to the fusion rules of \cft\refto{BPZ}, there is one such operator in the Kac table which has such a simple \ope\ with itself, namely $(r,s)=(1,2)$. We therefore conjecture that this is the correct assignment for the operators $\phi_{(f|\a)}$, for general $q$. This agrees with the known results for $q=2$ and $q=3$\refto{CARDYOPC,CARDYBC,SB}. It implies that the correlators involving these operators satisfy second order differential equations. From the Kac formula \(KAC), we see that, in the limit $q\to1$, their scaling dimensions are given by $$ h=h_{1,2}(0)=0\quad. \eqno() $$ This vanishing of the scaling dimension will turn out to have remarkable consequences when the result is transformed back into the original geometry. In fact, it has a natural explanation. Consider a compact region on whose boundary there is a single segment $S_1$ on which the Potts spins are fixed into the state $\a$. On the remainder of the boundary, the spins are free. In the limit $q\to1$, the partition function in this geometry is equal to unity, and equal to $Z_f$, since in either case any spin can only be in a single state. But the ratio of these partition functions is equal to the correlation function $\ev{\phi_{(f|\a)}\phi_{(\a|f)}}$, which in general will scale like distance to the power $2h$. This is only consistent if $h=0$. However, the form of the four-point functions, although simplified by this result, is nontrivial. Consider the half plane geometry, when the points lie along the real axis. Conformal invariance implies\refto{BPZ} that they are of the form $F(\eta)$, depending only on the invariant cross-ratio $\eta=(x_4-x_3)(x_2-x_1)/(x_3-x_1)(x_4-x_2)$. The absence of other prefactors multiplying $F$ is a consequence of $h=0$. The fact that the correlators satisfy second order differential equations implies that $F(\eta)$ satisfies a Riemann equation, whose general solution is\refto{BPZ,CARDYSURF} $$ F=P\left\{\matrix{0&\infty&1&{}\cr 0&-4h_{1,2}&0&\eta\cr h_{1,3}&-4h_{1,2}+h_{1,3}&h_{1,3}&{}\cr}\right\} \eqno() $$ Which solution is chosen depends on whether we calculate $Z_{\a\a}$ or $Z_{\a\b}$. Although is straightforward to solve this problem for arbitrary $q$, we restrict ourselves to $q=1$ for simplicity. In that limit, one of the solutions of the Riemann equation reduces to a constant, and the second solution is proportional to $\eta^{1/3}{}_2F_1(\ffrac13,\ffrac23,\ffrac43;\eta)$. The combination corresponding to $Z_{\a\b}$ is determined by the requirement that as $(x_3-x_2)\to0$, that is $\eta\to1$, the \ope\ \(OPE) requires that the solution vanish like $(1-\eta)^{1/3}$. In addition, in the opposite limit $\eta\to0$, we expect that $Z_{\a\b}\to Z_{\a\a}=1$. Using simple identities on hypergeometric functions, we then find for the crossing probability $$ \eqalign{ \pi\big((x_1,x_2),(x_3,x_4)\big)&= {3\G(\ffrac23)\over\G(\ffrac13)^2} \eta^{1/3}{}_2F_1(\ffrac13,\ffrac23,\ffrac43;\eta)\cr &=1- {3\G(\ffrac23)\over\G(\ffrac13)^2} (1-\eta)^{1/3}{}_2F_1(\ffrac13,\ffrac23,\ffrac43;1-\eta)\cr}\quad. \eqno(PI) $$ Now consider the transformation of the \uhp\ onto the interior of a simply connected compact region by a conformal mapping $z\to w$. If the boundary of the region is a differentiable curve, this mapping may be taken to be conformal also on the boundary. In that case, correlation functions of operators on the boundary transform in the standard manner summarized by the formula $$ \ev{\phi_1(w_1)\phi_2(w_2)\ldots}=\prod_i|w'(z_i)|^{-h_i} \ev{\phi_1(z_1)\phi_2(z_2)\ldots} \eqno(TRANS) $$ where the correlation functions on the left and \rhs s refer to the new and the old geometry respectively, and $h_i$ is the scaling dimension of $\phi_i$. In our case, however, since $h=0$, no such prefactors arise, and the correlation function is truly invariant. For a general critical system, the partition function for a compact region (without any operator insertions) is not itself scale invariant, but picks up a factor $(L/L_0)^{ac}$ where $L$ has the dimensions of length and gives the overall size of the region, $L_0$ is some non-universal microscopic scale (\eg\ the lattice spacing), and $a$ is geometry dependent\refto{CARDYPESCH}. However, for the case of percolation, $c=0$ and $Z=1$, so such effects are absent. In general, there is a further complication when the boundary of the compact region is only piecewise differentiable, and boundary operators happen to sit at the corners. In this case \(TRANS) does not apply. Instead there appear additional non-scale invariant factors of the form $(L/L_0)^{-(\pi/\g)h}$, where $\g$ is the interior angle at the corner. Such factors have been treated explicitly for the Ising model with various boundary conditions\refto{KLEBAN}. However, once again, since $h=0$ for the problem at hand, such factors are absent. We conclude that crossing probabilities are indeed invariant under mappings which are conformal in the interior and are piecewise conformal on the boundary, but that this is not generic for all critical systems, for example when $q\not=1$. As an example, consider the case treated in \ref{LANG} of the crossing probability between opposite sides of a rectangle of aspect ratio $r$. This is the image of the \uhp\ under a Schwartz-Christoffel transformation. Taking the points $x_j$ to be at $(-k^{-1},-1,1,k^{-1})$, the aspect ratio of the rectangle is given by $r=K(1-k^2)/2K(k^2)$, where $K(u)$ is the complete elliptic integral of the first kind. The prediction is then that the crossing probability is given by \(PI), with $\eta=((1-k)/(1+k))^2$. The results of this are illustrated in \figs{1,2}, and compared with the numerical data obtained in \ref{LANG} for bond percolation on square lattices with approximately $4\cdot10^4$ sites. It is seen from \fig{2} that the deviations between the numerical experiment and the theory are consistent with the internal scatter of the data, although there appears also to be a systematic difference which may be due to finite-size effects. It is possible to generalize the above methods to treat the case of correlations between different crossing events. As long as the segments involved are not adjacent, such probabilities may always be related to correlation functions of $\phi_{(f|\a)}$ operators, and they should have the same invariance properties as the simple crossing probabilities considered here. The fact that they enjoy these properties, which are not expected to hold for analogous quantities in generic critical systems, suggests that some of the ideas of conformal invariance might usefully be reformulated for the percolation problem without invoking the mapping to the Potts model. The author is grateful to R. P. Langlands for providing a copy of \ref{LANG} before publication, and the numerical data shown in \figs{1,2}. He also thanks T. Spencer and M. Aizenman for stimulating his interest in this problem, and P. Kleban for communicating the results of \ref{KLEBAN}. This work was supported by NSF Grant PHY 86-14185. \head{Figure Captions} \item{1)} Theory {\it vs.} the numerical data of \ref{LANG} for the horizontal crossing probabilities $\pi_h(r)$ for rectangles of aspect ratio $r$. In the figure, $\Ln\big((1-\pi_h)/\pi_h\big)$ is plotted against $\Ln r$. The numerical data is represented by by points, and the solid curve is the theoretical prediction. \item{2)} Deviation between numerical estimates and theoretical predictions of crossing probabilities $\pi_h(r)$ and $1-\pi_v(r)$. \references \refis{CARDYDL} Cardy J L in {\sl Phase Transitions and Critical Phenomena}, v.11, C. Domb and J. L. Lebowitz eds. (Academic, 1987) \refis{CARDYFSS} See articles in {\sl Finite-Size Scaling, Current Physics -- Sources and Comments}, v.2, J. L. Cardy, ed. (North-Holland, 1988) \refis{LANG} Langlands R P, Pichet C, Pouliot Ph and Saint-Aubin Y, {\sl On the Universality of Crossing Probabilities in Two-Dimensional Percolation}, preprint CRM-1785, October 1991. \refis{KF} Kasteleyn P W and Fortuin C M 1989 {\sl J. Phys. Soc. Japan} {\bf 26} {\sl (Supp.)} 11 \refis{CARDYSURF} Cardy J L 1984 {\sl Nucl. Phys.} {\bf B240} 514 \refis{CARDYBC} Cardy J L 1989 {\sl Nucl. Phys.} {\bf B324} 581 \refis{BPZ} Belavin A A, Polyakov A M and Zamolodchikov A B 1984 {\sl Nucl. Phys.} {\bf B241} 333 \refis{DOTFAT} Dotsenko Vl S and Fateev V A 1984 {\sl Nucl. Phys.} {\bf B240} 312 \refis{SACLAY} Di Francesco P, Saleur H and Zuber J-B 1987 {\sl J. Stat. Phys.} {\bf 49} 57 \refis{CARDYPESCH} Cardy J L and Peschel I 1988 {\sl Nucl. Phys.} {\bf B300} 377 \refis{CARDYOPC} Cardy J L 1986 {\sl Nucl. Phys.} {\bf B275} 200 \refis{SB} Saleur H and Bauer M 1989 {\sl Nucl. Phys.} {\bf B320} 591 \refis{DEBELL} De'Bell K and Essam J W 1980 {\sl J. Phys. C} {\bf 13} 4811 \refis{POTTS} Potts R B 1952 {\sl Proc. Camb. Phil.} {\bf 48} 106 \refis{KLEBAN} Kleban P and Vassileva I, in preparation. \endreferences \endit %% This is the PostScript file for Figure 1 %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart % Scaling calculations 0.02397 0.46679 0.01494 0.07644 [ [(Fig. 1)] 0.5 0.62428 0 -1 Msboxa [(0.5)] 0.25737 0.00244 0 1 Msboxa [(1)] 0.49077 0.00244 0 1 Msboxa [(1.5)] 0.72416 0.00244 0 1 Msboxa [(2)] 0.95756 0.00244 0 1 Msboxa [(Ln[r])] 1.00625 0.01494 -1 0 Msboxa [(2)] 0.01147 0.16781 1 0 Msboxa [(4)] 0.01147 0.32069 1 0 Msboxa [(6)] 0.01147 0.47357 1 0 Msboxa [(Ln[\(1-pi\)/pi])] 0.02397 0.62428 0 -1 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray 0 setgray [(Fig. 1)] 0.5 0.62428 0 -1 Mshowa gsave gsave 0.002 setlinewidth 0 0.01494 moveto 1 0.01494 lineto stroke 0.25737 0.00869 moveto 0.25737 0.02119 lineto stroke 0 setgray [(0.5)] 0.25737 0.00244 0 1 Mshowa 0.49077 0.00869 moveto 0.49077 0.02119 lineto stroke 0 setgray [(1)] 0.49077 0.00244 0 1 Mshowa 0.72416 0.00869 moveto 0.72416 0.02119 lineto stroke 0 setgray [(1.5)] 0.72416 0.00244 0 1 Mshowa 0.95756 0.00869 moveto 0.95756 0.02119 lineto stroke 0 setgray [(2)] 0.95756 0.00244 0 1 Mshowa 0 setgray [(Ln[r])] 1.00625 0.01494 -1 0 Mshowa 0.02397 0 moveto 0.02397 0.61803 lineto stroke 0.01772 0.16781 moveto 0.03022 0.16781 lineto stroke 0 setgray [(2)] 0.01147 0.16781 1 0 Mshowa 0.01772 0.32069 moveto 0.03022 0.32069 lineto stroke 0 setgray [(4)] 0.01147 0.32069 1 0 Mshowa 0.01772 0.47357 moveto 0.03022 0.47357 lineto stroke 0 setgray [(6)] 0.01147 0.47357 1 0 Mshowa 0 setgray [(Ln[\(1-pi\)/pi])] 0.02397 0.62428 0 -1 Mshowa grestore grestore 0 0 moveto 1 0 lineto 1 0.618034 lineto 0 0.618034 lineto closepath clip newpath 0 setgray gsave gsave gsave gsave 0.004 setlinewidth 0.02381 0.01488 moveto 0.2052 0.07805 lineto 0.30538 0.116 lineto 0.37884 0.1464 lineto 0.43839 0.17321 lineto 0.48921 0.19799 lineto 0.53396 0.22152 lineto 0.57417 0.24423 lineto 0.61082 0.2664 lineto 0.64458 0.2882 lineto 0.67593 0.30972 lineto 0.70523 0.33106 lineto 0.73274 0.35227 lineto 0.75869 0.37338 lineto 0.78325 0.39441 lineto 0.80657 0.4154 lineto 0.82877 0.43634 lineto 0.84996 0.45726 lineto 0.87022 0.47816 lineto 0.88964 0.49904 lineto 0.90828 0.51991 lineto 0.92621 0.54077 lineto 0.94347 0.56162 lineto 0.96012 0.58247 lineto 0.97619 0.60332 lineto stroke grestore grestore grestore gsave gsave 0.008 setlinewidth 0.02397 0.01472 Mdot 0.04719 0.02259 Mdot 0.07058 0.0305 Mdot 0.09366 0.03914 Mdot 0.11718 0.04658 Mdot 0.14284 0.05538 Mdot 0.16371 0.06251 Mdot 0.18667 0.07115 Mdot 0.20949 0.07906 Mdot 0.23215 0.0877 Mdot 0.25688 0.09666 Mdot 0.27983 0.10527 Mdot 0.30453 0.11519 Mdot 0.32604 0.12334 Mdot 0.35079 0.13348 Mdot 0.37561 0.14394 Mdot 0.39708 0.1531 Mdot 0.42002 0.16381 Mdot 0.44283 0.17382 Mdot 0.46946 0.18614 Mdot 0.49226 0.19789 Mdot 0.51514 0.21025 Mdot 0.53943 0.22289 Mdot 0.55838 0.23255 Mdot 0.58256 0.24686 Mdot 0.60809 0.26171 Mdot 0.63254 0.27829 Mdot 0.65361 0.29154 Mdot 0.6793 0.30795 Mdot 0.70026 0.32368 Mdot 0.72606 0.34257 Mdot 0.74805 0.3606 Mdot 0.76999 0.37746 Mdot 0.79193 0.39891 Mdot 0.82024 0.42463 Mdot 0.84316 0.44671 Mdot 0.86607 0.46598 Mdot 0.88902 0.49375 Mdot 0.91284 0.51605 Mdot 0.93675 0.54443 Mdot 0.95515 0.56288 Mdot grestore grestore grestore % End of Graphics MathPictureEnd %% This is the PostScript file for Figure 2 %! 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