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Aizenman, Jadwin Hall, P.O.Box 708, Princeton, NY 08544- 0708. {\rtf1\defformat\mac\deff2 {\fonttbl{\f0\fswiss Chicago;}{\f2\froman New York;}{\f3\fswiss Geneva;}{\f4\fmodern Monaco;}{\f5\fscript Venice;}{\f13\fnil Zapf Dingbats;}{\f16\fnil Palatino;}{\f18\fnil Zapf Chancery;}{\f20\froman Times;}{\f21\fswiss Helvetica;} {\f22\fmodern Courier;}{\f23\ftech Symbol;}{\f129\fnil Script Math;}{\f133\fnil S%math%1;}{\f134\fnil S%math%2;}{\f139\fnil Machias;}{\f140\fnil Lovell;}{\f2515\fnil MT Extra;}}{\colortbl\red0\green0\blue0;\red0\green0\blue255;\red0\g reen255\blue255; \red0\green255\blue0;\red255\green0\blue255;\red255\green0\bl ue0;\red255\green255\blue0;\red255\green255\blue255;}{\stylesh eet{\s231\qj\li1440\ri720\sb240\sl360\tldot\tx8280\tqr\tx8640 \f20 \sbasedon0\snext0 toc 3;}{ \s232\qj\li720\ri720\sb240\sl360\tldot\tx8280\tqr\tx8640 \f20 \sbasedon0\snext0 toc 2;}{\s233\qj\ri720\sb240\sl360\tldot\tx8280\tqr\tx8640 \f20 \sbasedon0\snext0 toc 1;}{ \s234\qj\li2160\ri- 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\s247\qj\li720\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \i\f20\fs20 \sbasedon0\snext0 heading 9;}{ \s248\qj\li720\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \i\f20\fs20 \sbasedon0\snext0 heading 8;}{ \s249\qj\li720\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \i\f20\fs20 \sbasedon0\snext0 heading 7;}{ \s250\qj\li720\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \f20\fs20\ul \sbasedon0\snext0 heading 6;}{ \s251\qj\li720\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \b\f20\fs20 \sbasedon0\snext0 heading 5;}{ \s252\qj\li360\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \f20\ul \sbasedon0\snext0 heading 4;}{\s253\qj\ri- 90\sb240\sl360\tx720\tx5130\tx7020\tx7920\tqdec\tx8370\tx87 29 \i\f20 \sbasedon0\snext0 heading 3;}{\s254\qj\ri- 90\sb120\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \f20\ul \sbasedon0\snext0 heading 2;}{ \s255\qj\ri- 90\sb240\sl280\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \b\f20\ul \sbasedon0\snext0 heading 1;}{ \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \f20 \sbasedon222\snext0 Normal;}{ \s2\qj\ri-90\sb240\sl- 200\keep\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx3060\tx3 600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \f20 \sbasedon0\snext0 equation;}{\s4\qj\fi-980\li980\ri- 90\sb240\tx980\tqdec\tx8280\tqdec\tx8370 \f20 \sbasedon0\snext4 refs;}{ \s6\qj\ri- 100\sb240\sl280\tx900\tx2880\tx5130\tx7020\tqdec\tx8370 \f20 \sbasedon0\snext6 regular;}}\widowctrl\ftnbj {\ftnsep \pard\plain \qj\ri- 90\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx3060\tx3600\tx 5130\tx7020\tx7920\tqdec\tx8370\tx8729 \f20 \chftnsep \par \pard \qj\ri- 90\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx3060\tx3600\tx 5130\tx7020\tx7920\tqdec\tx8370\tx8729 \par }\sectd \pgnrestart\linemod0\linex0\cols1\endnhere\titlepg {\header \pard\plain \qj\ri-720\tx810\tx1440\tx4680\tqr\tx9360 \f20 {\f16\fs20 Slow decay of correlations }{\fs20 in the\tab }\chpgn \par \pard \qj\ri-720\tx810\tx1440\tx4500\tqr\tx9360 {\f16\fs20 topologically restrained O(2) model}{\fs20 \tab }{\f16 }\par \pard\plain \s244\qj\ri-90\sl360\tqc\tx4320\tqr\tx8640 \f20 {\fs20 \par }}{\footer \pard\plain \s243\qj\ri-90\tqc\tx4320\tqr\tx8640 \f20 \par }{\headerf \pard\plain \s244\qj\ri-90\tqc\tx4320\tqr\tx8460 \f20 \par }{\footerf \pard\plain \s243\qj\ri100\sl360\brdrt\brdrs \tqc\tx4320\tqr\tx8640 \f20 - Work supported in part by NSF Grant PHY-9214654.\par \pard \s243\qj\ri100\sl360\brdrt\brdrs \tqc\tx4320\tqr\tx8640 \par }\pard\plain \qc\ri- 360\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx3 060\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370 \f20 {\fs48 On the slow decay of O(2) correlations in the absence of topological excitations; remark on the Patrascioiu - Seiler model\par }\pard \qc\ri- 360\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx3 060\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370 {\fs36 \par }\pard \qc\ri-360\sb240\sl360\tx1260\tx5580 Michael Aizenman\par \pard \qc\ri-360\sb240\keep\tx360\tx5300 Departments of Physics and Mathematics\par \pard \qc\ri-360\keep\tx1340\tx4940 Princeton University\par \pard \qc\ri-360\keep\tx1340\tx5300 Jadwin Hall, P.O. Box 708\par \pard \qc\ri-440\keep\tx1340\tx5300 Princeton, New Jersey 08544-0708\par \pard\plain \s243\qj\ri-100\sl360\tqc\tx4320\tqr\tx9000 \f20 \tab \par \tab \par {\f18\fs28 \tab Dedicated to Oliver Penrose on the occasion of his sixty-fifth's birthday.}\par {\f18\fs28 \par }\pard\plain \qc\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \f20 {\b Abstract\par }\pard \qj\ri- 100\sb240\sl520\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx3 060\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 For spin models with O(2)-invariant ferromagnetic interactions, the Patrascioiu-Seiler constraint is: |arg({\ul S}(x))-arg({\ul S }(y))| \'b2\~{\f23 q}{\fs20\dn4 o }for all |x-y|=1. It is shown that in two dimensional systems of two-component spins the imposition of such constraints with {\f23 q}{\fs20\dn4 o} small enough indeed results in the suppression of exponential clustering. More explicitly, it is shown that in such systems on every scale the spin-spin correlation function is found to obey: <{\ul S}(x){\fs20\up6 .}{\ul S(}y)> \'b3 \|f(3,4 |x-y|{ \fs20\up6 2}) , at any temperature - including T={\f23\fs28 \'a5}. The derivation is along the lines proposed by A. Patrascioiu and E. Seiler [1], with the yet unproven conjectures invoked there replaced by another geometric argument. \par \pard \qj\ri- 100\sb240\sl480\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx3 060\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 {\ul Key words:}\tab Continuous symmetry, Kosterlitz-Thouless transition, decay of correlations, Fortuin-Kasteleyn representation, topological charges.\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx7020\tx8729 \tab \tab \tab \tab \tab \tab \tab \tab 5/12/94\par \pard\plain \s255\qj\ri- 90\sb240\sl280\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \b\f20\ul \page 1. Introduction{\plain \f20 \par }\pard\plain \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \f20 \tab Two dimensional classical spin models with co ntinuous symmetries exhibit interesting relations between the decay of equilibrium-averaged correlations and the topological features of 'typical' spin configurations. Such relations have been seen in numerous works, and at the intuitive level the can be related to elementary (free-)energy estimates. However our understanding of this mechanism is not complete. \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab For instance, in studying the O(2) invariant plane-rotor model on {\f129 Z}{\up10 2}, with {\ul S}{\fs20\dn4 x} two dimensional unit vectors and the interaction\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 {\up34 \tab \tab }{{\pict\macpict\picw137\pich27 057f00000000001b00891101a10064000eb77870720000000f00000 00c0000a0008c01000a00000000001b0089070000000022001b008 90000a000bea100c0000d63757272656e74706f696e740da000bf22 000000000000a000be0300140d000c2b010c0148290b0120030017 2906013d030014290a01200300172904012d 030014291901202904014a0d00072b060301780300172904012d03 0014290501790d000c28000c005201202905015307000100012200 0d00570400290601282904017829070129030017290501d7030014 2906015322000d007404002906012829050179290601290d000728 00170028017b290401782904012c2902017929 03017d0300170d001228000e002a01e5a100c003d32f4d547361766 5207361766520646566203330206469637420626567696e0d63757 272656e74706f696e742033202d3120726f6c6c20737562206e6567 2033203120726f6c6c207375620d34333834206469762038363420 33202d3120726f6c6c206578636820646976 207363616c650d63757272656e74706f696e74207472616e736c617 465203634203338207472616e736c6174650d2f746869636b20302 06465660d2f7468207b20647570207365746c696e6577696474682 02f746869636b206578636820646566207d206465660d313620746 8203237333920343037206d6f7665746f2031 3732203020726c696e65746f207374726f6b650d333635322034303 7206d6f7665746f20313732203020726c696e65746f207374726f6b 650d2f6673203020646566202f63662030206465660d2f7366207b6 578636820647570202f667320657863682064656620647570206e6 567206d6174726978207363616c65206d61 6b65666f6e7420736574666f6e747d206465660d2f6631207b66696 e64666f6e7420647570202f63662065786368206465662073667d20 6465660d2f6e73207b63662073667d206465660d2f7368207b6d6f7 665746f2073686f777d206465660d333834202f54696d65732d526f 6d616e2066310d284829202d3720333436 2073680d284a292031393139203334362073680d28532920323732 33203334362073680d2878292033303633203334362073680d2853 292033363336203334362073680d28792920333937372033343620 73680d323234206e730d2878292032303932203434322073680d28 79292032333737203434322073680d28782920 31333530203731392073680d287929203135343720373139207368 0d333834202f54696d65732d526f6d616e2066310d282029203332 36203334362073680d28202920383438203334362073680d282029 2031373632203334362073680d2820292032353739203334362073 680d285c28292032393233203334362073680d 285c29292033323635203334362073680d285c2829203338333620 3334362073680d285c29292034313738203334362073680d323234 206e730d287b292031323437203731392073680d282c2920313437 33203731392073680d287d292031363532203731392073680d3338 34202f53796d626f6c2066310d283d29203533 30203334362073680d282d2920393730203334362073680d285c33 3237292033343434203334362073680d323234206e730d282d2920 32323239203434322073680d353736206e730d285c333435292031 323938203433332073680d656e64204d547361766520726573746f 72650da000bfa10064008d4d41544800010081 18c702000003000a0112834802822002863d02822002862d031d00 000102822012834a030f01000b0112837802862d1283790011000a 028220031000000112835300000282281283780282290286d70298 0802980403100000011283530000028228128379028229000b0102 827b12837802822c12837902827d00110d0286 e50000007500a0008dff}}{\up30 , J}{\fs20\up24 x-y}{\up30 \'b3 0 ,\tab \tab (1.1)\par }\pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 one can associate to each spin configuration a configuration of charges \{q{\fs20\dn4 p}\} , located on the dual lattice, of plaquettes \{p\}, which may be viewed as representing "curl arg ({\ul S})" (with arg({\ul S}(x)) defined only mod. 2\'b9). For a given plaquette, q{\fs20\dn4 p} is defined as the sum, along the boundary of p, of the increments of the angles arg({\ul S}(x)), with the convention that the magnitude of the increase over each pair of neighboring sites does not exceed {\fs28 \'b9}. The values assumed by q{\fs20\dn4 p} are integer multiples of 2\'b9, and their definition makes sense regardless of H. There is evidence, present in both heuristic discussions and rigorous work on the Kosterlitz - Thouless transition [1,2], that the binding of such topological charges \{q {\fs20\dn4 p}\} into neutral clusters is related to slow decay of spin-spin correlations (power law instead of exponential). Nevertheless, there is no clear cut statement (at least to the author's knowledge) that in any sense one is either sufficient or necessary for the other. \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab Related to this question is a striking observation made by A. Patrascioiu and E. Seiler [1] in their discussion of the mechanisms associated with the decay of correlations in O(N) models. They present arguments which made it very plausible that in constrained versions o f such two dimensional O(2) models, with the spins restricted to satisfy\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 | arg( {\ul S}(x) ) - arg({\ul S}(y) ) | \'b2 {\f23 q}{\fs20\dn4 o} whenever |x-y| = 1,\tab \tab \tab (1.2)\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 if {\f23 q}{\fs20\dn4 o} small enough than at no temperature do the spin- spin correlations decay exponentially. T heir analysis is based on percolation theoretic and topological arguments, and is done under the assumption that some yet unproven conjectures are valid. \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab The main purpose of this note is to point out that the Patrascioiu - Seiler effect does indeed take place - regardless of the validity of the conjectures made in [1]. The result is summarized by the following statement, which provides a more explicit and assumption-free version of the one envisioned there. \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab We consider here models with O(2) - invariance, with interactions which are more general than (1.1), having the form \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab \tab \tab {{\pict\macpict\picw143\pich27 059b00000000001b008f1101a10064000eb77870720000000f00000 00c0000a0008c01000a00000000001b008f070000000022001b008f 0000a000bea100c0000d63757272656e74706f696e740da000bf220 00000000000a000be0300140d000c2b010c0148290b01200300172 906013d030014290a01200300172904012d 03001429190120290401670d00072b070301782904012c29020179 0d000c28000c004f012829060153070001000122000d0056040029 0701282904017829060129030017290601d7030014290601532200 0d007205002906012829040179290701290d00072800170028017b 290401782904012c290201792903017d030017 0d001228000e002a01e50300140d000c28000c00890129a100c003d e2f4d5473617665207361766520646566203330206469637420626 567696e0d63757272656e74706f696e742033202d3120726f6c6c20 737562206e65672033203120726f6c6c207375620d3435373620646 976203836342033202d3120726f6c6c2065 78636820646976207363616c650d63757272656e74706f696e7420 7472616e736c617465203634203338207472616e736c6174650d2f7 46869636b2030206465660d2f7468207b20647570207365746c696 e657769647468202f746869636b206578636820646566207d20646 5660d3136207468203236393620343037206d 6f7665746f20313732203020726c696e65746f207374726f6b650d3 336303920343037206d6f7665746f20313732203020726c696e6574 6f207374726f6b650d2f6673203020646566202f636620302064656 60d2f7366207b6578636820647570202f667320657863682064656 620647570206e6567206d61747269782073 63616c65206d616b65666f6e7420736574666f6e747d206465660d2 f6631207b66696e64666f6e7420647570202f636620657863682064 65662073667d206465660d2f6e73207b63662073667d206465660d 2f7368207b6d6f7665746f2073686f777d206465660d333834202f5 4696d65732d526f6d616e2066310d284829 202d37203334362073680d2867292031393132203334362073680d 2853292032363830203334362073680d2878292033303230203334 362073680d2853292033353933203334362073680d287929203339 3334203334362073680d323234206e730d28782920323132312034 34322073680d28792920323331382034343220 73680d2878292031333530203731392073680d2879292031353437 203731392073680d333834202f54696d65732d526f6d616e206631 0d28202920333236203334362073680d2820292038343820333436 2073680d2820292031373632203334362073680d285c2829203234 3638203334362073680d285c28292032383830 203334362073680d285c29292033323232203334362073680d285c 28292033373933203334362073680d285c29292034313335203334 362073680d285c29292034333435203334362073680d323234206e 730d282c292032323434203434322073680d287b29203132343720 3731392073680d282c29203134373320373139 2073680d287d292031363532203731392073680d333834202f5379 6d626f6c2066310d283d2920353330203334362073680d282d2920 393730203334362073680d285c3332372920333430312033343620 73680d353736206e730d285c333435292031323938203433332073 680d656e64204d547361766520726573746f72 650da000bfa1006400974d4154480001008b1ada02000003000a011 2834802822002863d02822002862d031d000001028220128367030 f01000b0112837802822c1283790011000a0282280298040310000 00112835300000282281283780282290286d702980802980403100 000011283530000028228128379028229000b 0102827b12837802822c12837902827d00110d0286e5000a029804 02822900006f72a0008dff}}{\up30 , \tab \tab \tab \tab (1.3)}{\up26 \par }\pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 with monotone increasing functions g{\fs20\dn4 x,y}(-) ({\f23 \'ad} ). The equilibrium averages for such system in a finite region {\f23 L}, with either the free or (when meaningful) the periodic boundary conditions, and subject to the constraints (1.2), take for form:\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 {\up50 \tab }{{\pict\macpict\picw252\pich56 0af100000000003800fc1101a10064000eb77870720000001900000 00c0000a0008c01000a00000000003800fc070000000022003800fc0 000a000bea100c0000d63757272656e74706f696e740da000bf2200 0000000000a000be0300170d000c2b011f013c030014290a0166290 501282904015329060128030017290401 d70300142902022929030017290b013e0d00072b0703014c030014 0d000c28001f003b01200300172907013d030014290a012028000f0 0590166290501282906015329070128030017290301d7030014290 3012929060129290601200300170d0007280006008e012d2904016 2030014290401480300170d00052b0602014c 0300140d000728000600a1012829030153290401290d000f280010 008701650d000c28000f00ad014b29090128290401532906012903 0017291401720300140d00072b0703016f28001a00c50178030017 290401ce2905014c0d001228001100c401d50300140d000c28000f0 0df012829050264530d00072b0d0301780d00 0c28000f00f501290300170d0012280013005201f20300140d000c2 b1a1701200300170d0007280022007d012d2904016203001429040 1480300170d00052b0602014c0300140d000728002200900128290 20153290501290d000f28002b007501650d000c28002a009a014b2 9090128290401532907012903001729130172 0300140d00072b0703016f28003600b20178030017290401ce2905 014c0d001228002d00b201d50300140d000c28002a00cd01282904 0264530d00072b0d0301780d000c28002a00e301290300170d0012 28002e006401f2070001000120001b0051001b00f9a100c006ea2f4 d547361766520736176652064656620333020 6469637420626567696e0d63757272656e74706f696e742033202d 3120726f6c6c20737562206e65672033203120726f6c6c207375620 d383036342064697620313739322033202d3120726f6c6c2065786 36820646976207363616c650d63757272656e74706f696e7420747 2616e736c617465203634202d323834362074 72616e736c6174650d2f746869636b2030206465660d2f7468207b2 0647570207365746c696e657769647468202f746869636b2065786 36820646566207d206465660d31362074682032353336203337333 9206d6f7665746f2035343132203020726c696e65746f207374726f 6b650d2f6673203020646566202f63662030 206465660d2f7366207b6578636820647570202f66732065786368 2064656620647570206e6567206d6174726978207363616c65206d 616b65666f6e7420736574666f6e747d206465660d2f6631207b666 96e64666f6e7420647570202f63662065786368206465662073667d 206465660d2f6e73207b63662073667d2064 65660d2f7368207b6d6f7665746f2073686f777d206465660d33383 4202f53796d626f6c2066310d283c29202d3920333833382073680d 285c333237292039303520333833382073680d283e292031333237 20333833382073680d283d29203230353820333833382073680d28 5c3332372920333438322033333239207368 0d323234206e730d282d29203434393020333036382073680d285c 33313629203633383620333639392073680d282d29203339333920 333934342073680d285c3331362920353738372034353735207368 0d353736206e730d285c3332352920363233332033343137207368 0d285c33363229203235363720333436332073 680d285c33323529203536333420343239332073680d285c333632 29203331363620343333392073680d333834202f54696d65732d52 6f6d616e2066310d2866292032393620333833382073680d285329 2035383520333833382073680d2866292032373931203333323920 73680d28532920333136322033333239207368 0d284b29203534383720333332392073680d285329203539303120 333332392073680d28645329203732343020333332392073680d28 4b29203438383820343230352073680d2853292035333032203432 30352073680d28645329203636343120343230352073680d323234 206e730d284829203437343820333036382073 680d285329203531383620333036382073680d286f292036393534 20333432352073680d287829203632353720333639392073680d28 7829203736343920333432352073680d2848292034313937203339 34342073680d285329203436333520333934342073680d286f2920 3633353520343330312073680d287829203536 353820343537352073680d28782920373035302034333031207368 0d333834202f54696d65732d526f6d616e2066310d285c282920343 53520333833382073680d285c28292037383520333833382073680 d285c295c29292039383820333833382073680d282029203138353 420333833382073680d282029203233373620 333833382073680d285c2829203239353020333332392073680d28 5c2829203333363220333332392073680d285c2929203335363520 333332392073680d285c2929203337373520333332392073680d28 2029203339343820333332392073680d285c282920353737312033 3332392073680d285c29292036313038203333 32392073680d285c2829203731303320333332392073680d285c29 29203738303720333332392073680d282029203333393720343230 352073680d285c2829203531373220343230352073680d285c2929 203535303920343230352073680d285c2829203635303420343230 352073680d285c292920373230382034323035 2073680d323234206e730d285c2829203530393720333036382073 680d285c2929203533313920333036382073680d285c2829203435 343620333934342073680d285c2929203437363820333934342073 680d323234202f53796d626f6c2066310d284c29203135353820333 933362073680d284c29203635333020333639 392073680d284c29203539333120343537352073680d313630206e 730d284c29203439333720333132362073680d284c292034333836 20343030322073680d323234202f53796d626f6c2066310d2862292 03436323620333036382073680d286229203430373520333934342 073680d333834206e730d2872292036373430 20333332392073680d287229203631343120343230352073680d34 3830202f54696d65732d526f6d616e2066310d2865292034323538 20333336342073680d286529203337303720343234302073680d65 6e64204d547361766520726573746f72650da000bfa1006401944d4 154480001018844ad02000003000a0102863c 1283660282281283530282280286d702822902822902863e030f01 000b0102854c0011000a02822002863d028220030e000001031500 00011283660282280298041283530282280286d702822902980402 8229028220032a0000110b110102862d028462128348030f01000c 0102854c0011000b028228128353028229000d 0102980409032002816500000a02980412834b0282281283530282 29031f000001028472030f01000b0112836f0011000001128378028 6ce02854c00110d0286d5000a028228128364128353030f01000b01 1283780011000a028229000b11110d0286f200000a010315000001 028220032a0000110b110102862d02846212 8348030f01000c0102854c0011000b028228128353028229000d01 02980409032002816500000a12834b028228128353028229031f00 0001028472030f01000b0112836f00110000011283780286ce02854 c00110d0286d5000a028228128364128353030f01000b011283780 011000a028229000b11110d0286f200000000 000286a0008dff}}{\up50 .\tab \tab \tab (1.4)\par }\pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 Here {\f23 r}{\fs20\dn4 o} (-) is the uniform measure on the circle, H is given by (1.3), and K(S) the indicator function which is 1 if the constraints (1.2) are satisfied.\par {\ul Theorem 1} In a two dimensional model of the type presented above, if the constraining angle meets the condition\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab \tab {{\pict\macpict\picw178\pich34 050e00000000002200b21101a10064000eb77870720000000e0000 000c0000a0008c01000a00000000002200b2070000000022002200 b20000a000bea100c0000d63757272656e74706f696e740da000bf2 2000000000000a000be2c000900170653796d626f6c0300170d000c 2e00040000ff002b011401712c0008001405 54696d65730300140d00072b0703016f0300170d000c280014000f0 1a328000a001f01700300142908012f29060134291103666f72290e 1920746865206375626963206c6174746963652020202020202003 001728001c001f01700300142908012f2906013229121a666f72207 4686520747269616e67756c6172206c6174 7469636503001728000b001902ec202a0902ed202a0902ee20a100c 002ef2f4d547361766520736176652064656620333020646963742 0626567696e0d63757272656e74706f696e742033202d3120726f6c 6c20737562206e65672033203120726f6c6c207375620d353639362 064697620313038382033202d3120726f6c 6c206578636820646976207363616c650d63757272656e74706f696 e74207472616e736c617465203634203439207472616e736c61746 50d2f6673203020646566202f63662030206465660d2f7366207b65 78636820647570202f667320657863682064656620647570206e65 67206d6174726978207363616c65206d616b 65666f6e7420736574666f6e747d206465660d2f6631207b66696e6 4666f6e7420647570202f63662065786368206465662073667d206 465660d2f6e73207b63662073667d206465660d2f7368207b6d6f76 65746f2073686f777d206465660d333834202f53796d626f6c20663 10d287129202d3136203539312073680d32 3234202f54696d65732d526f6d616e2066310d286f2920313936203 638372073680d333834206e730d28666f722920313934312032393 32073680d333834202f53796d626f6c2066310d285c323433292034 3334203539312073680d28702920393337203239332073680d2870 2920393430203836392073680d285c333534 2920373432203335352073680d285c333535292037343220363536 2073680d285c3335362920373432203935382073680d333834202f 54696d65732d526f6d616e2066310d282f29203132303820323933 2073680d282f292031323131203836392073680d333834202f5469 6d65732d526f6d616e2066310d283429203133 3833203239332073680d2832292031333739203836392073680d33 3834202f54696d65732d526f6d616e2066310d2820746865206375 626963206c61747469636520202020202020292032333837203239 332073680d28666f722074686520747269616e67756c6172206c617 474696365292031393532203836392073680d 656e64204d547361766520726573746f72650da000bfa1006400f74 d415448000100eb37fd02000003000a010284710b030f0100011283 6f0011000a0286a303020100010501010102020000010286700282 2f028834000112836612836f128372028120028174028168028165 028120028163028175028162028169028163 02812002816c028161028174028174028169028163028165028120 028120028120028120028120028120028120000102867002822f02 8832000102816602816f0281720281200281740281680281650281 2002817402817202816902816102816e02816702817502816c0281 6102817202812002816c028161028174028174 02816902816302816500000002967b0000003120a0008dff}}{\up30 \tab \tab \tab \tab (1.5)}\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 then the equilibrium correlation function does not decay faster than \|f(3,4|x- y|{\fs20\up6 2}) in the sense that: {\up34 \par }\pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 {\up34 \tab \tab }{\dn14 {\pict\macpict\picw164\pich41 092700000000002900a41101a10064000eb7787072000000180000 000c0000a0008c01000a00000000002900a4070000000022002900a 40000a000bea100c0000d63757272656e74706f696e740da000bf22 000000000000a000be0300140d000c2b1611036d61780d000a2800 1b0001014c030017290701a3030014290601 7c290201780300172905012d030014290701792905017c03001729 0201a3030014290d01320700010001220018002901000700010002 230102070001000122001a002c03f82305002908014c280025000b0 130030017290601a30300142905017c290201782906027c2c29040 17c290201792905027c2c030017290501a303 00142906014c0300170d000c2800110041013c030014290c0153220 012004d04000d000a2b060301780300170d000c280011005901d70 3001429060153220012006004000d000a2b070301790300170d000 c280011006f013e030014290a0120030017290601b3030014290a01 202800090095013328001b008f0134290601 4c0d000a280015009b013222000d008e1300a100c006982f4d54736 17665207361766520646566203330206469637420626567696e0d6 3757272656e74706f696e742033202d3120726f6c6c20737562206e 65672033203120726f6c6c207375620d3532343820646976203133 31322033202d3120726f6c6c206578636820 646976207363616c650d63757272656e74706f696e74207472616e7 36c617465203634203432207472616e736c6174650d2f746869636b 2030206465660d2f7468207b20647570207365746c696e65776964 7468202f746869636b206578636820646566207d206465660d2f73 7172207b0d3320696e64657820646976202f 746869636b2065786368206465660d67736176650d7472616e736c 6174650d64757020647570206e6567207363616c650d6475702034 202d3120726f6c6c2065786368206469762033203120726f6c6c206 469760d30207365746c696e6577696474680d6e6577706174680d3 02030206d6f7665746f0d647570202e333420 6d756c20302065786368206c696e65746f0d2e333735202e3231342 0726c696e65746f0d64757020746869636b2061646420647570202 e3337352065786368206c696e65746f0d3220696e64657820657863 68206c696e65746f0d64757020746869636b203220646976207375 6220647570203320696e6465782065786368 206c696e65746f0d2e362065786368206c696e65746f0d2e3337352 030206c696e65746f0d636c69700d746869636b207365746c696e65 77696474680d6e6577706174680d647570202e3334206d756c2030 2065786368206d6f7665746f0d2e3135202e30383520726c696e657 46f0d2e3337352030206c696e65746f0d74 6869636b2032206469762073756220647570202e36206578636820 6c696e65746f0d6c696e65746f0d7374726f6b650d67726573746f72 650d7d206465660d33393120333039203332302031323634203835 31203133207371720d3136207468203234313820353633206d6f76 65746f20313732203020726c696e65746f20 7374726f6b650d3330313620353633206d6f7665746f20313732203 020726c696e65746f207374726f6b650d3434383520343033206d6f 7665746f20363535203020726c696e65746f207374726f6b650d2f6 673203020646566202f63662030206465660d2f7366207b6578636 820647570202f6673206578636820646566 20647570206e6567206d6174726978207363616c65206d616b6566 6f6e7420736574666f6e747d206465660d2f6631207b66696e64666 f6e7420647570202f63662065786368206465662073667d2064656 60d2f6e73207b63662073667d206465660d2f7368207b6d6f766574 6f2073686f777d206465660d333834202f54 696d65732d526f6d616e2066310d286d6178292036343220353032 2073680d2820292033383135203530322073680d28202920343332 35203530322073680d333230206e730d287c292033383820383235 2073680d287c292031303031203832352073680d287c2920363635 20313134362073680d287c2c29203839382031 3134362073680d287c29203130333720313134362073680d287c2c 29203132373020313134362073680d333230202f54696d65732d52 6f6d616e2066310d284c29202d33203832352073680d2878292034 3537203832352073680d28792920383338203832352073680d284c 292031373438203832352073680d2878292037 333420313134362073680d28792920313130372031313436207368 0d284c29203136313820313134362073680d287829203236313720 3631352073680d2879292033323136203631352073680d33383420 6e730d2853292032343032203530322073680d2853292033303030 203530322073680d284c292034373230203833 322073680d333230202f53796d626f6c2066310d285c32343329203 23131203832352073680d282d2920363339203832352073680d285 c323433292031303634203832352073680d285c323433292034383 820313134362073680d285c3234332920313432312031313436207 3680d333834206e730d283c29203230343220 3530322073680d285c333237292032383038203530322073680d28 3e292033353033203530322073680d285c32363329203430313220 3530322073680d333230202f54696d65732d526f6d616e2066310d 2832292031343737203832352073680d2830292033313320313134 362073680d2832292034393234203636302073 680d333834206e730d2833292034373232203236312073680d2834 292034353133203833322073680d656e64204d5473617665207265 73746f72650da000bfa1006400ef4d415448000100e32a5b0200000 3000a01032701000112826d128261128278000b0401010112834c0 286a302827c12837802862d12837902827c02 86a3030d000001028832000c11000b02980412834c000102980502 88300286a302827c12837802827c02822c02827c12837902827c028 22c0286a312834c000011000a02863c029804029802031000000112 83530000030f01000b011283780011000a0298080286d702980802 980403100000011283530000030f01000b01 1283790011000a02863e0282200286b3028220030e000001028833 000102883412834c030f00000b11010288320000000000006620a0 008dff}}{\up34 \tab \tab \tab \tab (1.6)\par }\pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 at any temperature (including t = {\f23\fs28 \'a5}) and for any finite volume {\f23 L} containing the square [1,L]{\f23 \'b4} [1,L]. {\up34 \par }\pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab With few improvements - the essential one being the geometric argument given in Section 3, the proof of Theorem 1 is based on ideas of Patrascioiu and Seiler [1] . A key step is the following reduction of the spin-spin correlation to a connectivity function, which is enabled by an extension of the Fortuin - Kasteleyn construction to this model ([1]). \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 {\ul Lemma 1: } In a constrained system, with {\f23 q}{\fs20\dn4 o} satisfying the condition of Theorem 1, \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab \tab {\f129 E}( {\ul S}(x){\fs20\up6 .}{\ul S}(y) ) \'b3 \|f({\fs20 3,2}) Prob( x <\|o(=,{\fs20\up18 P})> y ) := \|f({\fs20 3,2)} {\f23 t}(x,y) \tab \tab (1.7)\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 where x <\|o(=,{\fs20\up18 P})> y denotes the event that, for the specified spin configuration {\ul S}({\fs20\up6 .} ), x and y are *-connected by a path along which the spins satisfy the 'broad polar cap condition':\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab \tab \tab \tab | {\ul S}(u) {\fs20\up6 . }{\ul i} | \'b3 1/\|r({\fs20 2}) , for {\ul i} = (1,0) .\tab \tab \tab (1.8) \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab A path is referred to as * connected if it is connected in the sense in which the external boundary of any nearest-neighbor connected set is connected. For the rectangular lattice that means that the path's consecutive sites are either nearest-neighbors or next-nearest-neighbors. On the triangular lattice the notion of connectedness is self dual (i.e., *connected = connected).\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab One may note that under the constraint (1.2), with {\f23 q}{\fs20\dn4 o} as in the Theorem, the admissible configurations are free of topological charges, i.e., q{\fs20\dn4 p} {\f23 \'ba} 0. This observation raises a question about an alternative approach to this topic, on which we comment in Section 4.\par \pard\plain \s255\qj\ri- 90\sb240\sl280\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \b\f20\ul \par 2. The clique representation\par \pard\plain \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \f20 \tab In this section we summarize the proof of Lemma 1, which involves mainly the Fortuin - Kasteleyn representation [4] which was extended to the systems considered here in ref. [1]. Its essential features are:\par \tab 1) Attention is focused on the discrete symmetry which consists of flipping just the first coordinate of the spin variables. Correspondingly, the two-component unit spin vectors S{\fs20\dn4 x} = (S{\f23\fs20\dn4 ||}(x) , S{\f23\fs20\dn4 ^} (x) ) are written as:\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab \tab S{\fs20\dn4 x} = {\fs28 ( }{\f23 s}(x) |S{\f23\fs20\dn4 ||}(x)| , S{\f23\fs20\dn4 ^}(x) {\fs28 ) } \tab \tab \tab \tab (2.1)\par with {\f23 s}{\fs20\dn4 x} = \'b1 1, and |S{\f23\fs20\dn4 ||}(x)| = \|r(1-S{\f23\fs20\dn4 ^}(x){\fs20\up6 2}) . The flip, R, consists of the reversal of {\f23 s}. \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab 2) For specified values of the orthogonal spin components, S{\f23\fs20\dn4 ^}({\fs20\up6 .} ), the conditional distribution of the Ising variables {\f23 s}({\fs20\up6 .}) in the constrained system is governed by a pair potential of the form\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab \tab {{\pict\macpict\picw194\pich27 063600000000001b00c21101a10064000eb77870720000000f0000 000c0000a0008c01000a00000000001b00c2070000000022001b00c 20000a000bea100c0000d63757272656e74706f696e740da000bf22 000000000000a000be0300140d000c2b010c0148290b0120030017 2906013d030014290a01200300172904012d 030014291d01200d00072800170028017b290401782904012c2902 01792903017d030017290301ce030014290501420300170d001228 000e002f01e50300140d000c28000c004101680d00072b070301782 903012c290301790d000c28000c005202282003001729090173030 014290a012829040178290702292c03001729 080173030014290a01282904017929060129290601202906017c29 050120290401530300170d00072b0703015e0300140d000c28000c 00ae0128030017290401d7030014290201292906022029a100c004 292f4d547361766520736176652064656620333020646963742062 6567696e0d63757272656e74706f696e742033 202d3120726f6c6c20737562206e65672033203120726f6c6c20737 5620d3632303820646976203836342033202d3120726f6c6c20657 8636820646976207363616c650d63757272656e74706f696e74207 472616e736c617465203634202d32313934207472616e736c61746 50d2f6673203020646566202f636620302064 65660d2f7366207b6578636820647570202f667320657863682064 656620647570206e6567206d6174726978207363616c65206d616b 65666f6e7420736574666f6e747d206465660d2f6631207b66696e6 4666f6e7420647570202f63662065786368206465662073667d206 465660d2f6e73207b63662073667d20646566 0d2f7368207b6d6f7665746f2073686f777d206465660d333834202 f54696d65732d526f6d616e2066310d284829202d3720323537382 073680d286829203230323720323537382073680d2878292033333 23620323537382073680d287929203432343520323537382073680 d285329203531303720323537382073680d32 3234206e730d287829203133353020323935312073680d28792920 3135343720323935312073680d2842292031383937203239353120 73680d287829203232343420323637342073680d28792920323434 3120323637342073680d333834202f54696d65732d526f6d616e20 66310d2820292033323620323537382073680d 2820292038343820323537382073680d2820292031393130203235 37382073680d285c282029203235393120323537382073680d285c 2829203331383620323537382073680d285c292c29203335323820 323537382073680d285c2829203431303420323537382073680d28 5c2929203434343620323537382073680d2820 29203436313920323537382073680d287c29203438303120323537 382073680d282029203439363320323537382073680d285c282920 3535313420323537382073680d285c292920353731372032353738 2073680d28205c2929203538393020323537382073680d32323420 6e730d287b2920313234372032393531207368 0d282c29203134373320323935312073680d287d29203136353220 323935312073680d282c29203233363720323637342073680d3338 34202f53796d626f6c2066310d283d2920353330203235373820736 80d282d292039373020323537382073680d285c333237292035363 33420323537382073680d323234206e730d28 5c33313629203137353620323935312073680d285e292035333234 20323637342073680d353736206e730d285c333435292031343436 20323636352073680d333834202f53796d626f6c2066310d2873292 03238363720323537382073680d287329203337383520323537382 073680d656e64204d54736176652072657374 6f72650da000bfa1006400b14d415448000100a5236302000003000 a0112834802822002863d02822002862d031d000001028220000b0 102827b12837802822c12837902827d0286ce12834200110d0286e 5000a1283680b030f01000112837802822c1283790011000a02822 8028220028473029804028228128378028229 02822c02847302980402822812837902822902822002827c028220 1283530b030f01000102865e0011000a0282280286d70282290282 2002822900007075a0008dff}}{\up30 \tab \tab \tab (2.2)}\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 where: i) h{\fs20\dn4 x,y} is allowed to assume the value +{\f23\fs28 \'a5} (but not -{\f23\fs28 \'a5} ) as a way of imposing the constraint (1.2), and ii) the sum is over the collection of bonds ({\f23 \'ba} pairs of sites) B which includes those for which J{\fs20\dn4 x,y} \'ad 0, and also those for which the constraint (1.2) is imposed. In its dependence on {\f23 s}({\fs20\up6 .}) the interaction is ferromagnetic in the sense that for each specified configuration of the S{\f23\fs20\dn4 ^} variables the two-body potentials satisfy:\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 h{\fs20\dn4 x,y} (+,+ | S{\f23\fs20\dn4 ^}({\fs20\up6 .}) ) = h{\fs20\dn4 x,y} (-,- | S{\f23\fs20\dn4 ^}({\fs20\up6 .}) ) \'b2 h{\fs20\dn4 x,y} (+, - | S{\f23\fs20\dn4 ^}({\fs20\up6 .}) ) = h{\fs20\dn4 x,y} (+,- | S{\f23\fs20\dn4 ^}({\fs20\up6 .}) )\tab \tab (2.3)\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab 3) The property (2.3) permits to invoke the Fortuin - Kasteleyn 'random cluster' representation of the conditional state of { \f23 s}({\fs20\up6 .}), conditioned on S{\f23\fs20\dn4 ^}({\fs20\up6 .}). The representation may be viewed as polarizing the ferromagnetic state into a superposition of states in which any two spins are either totally independent or rigidly aligned{ \fs18\up6 *{\footnote \pard\plain \s246\qj\ri- 90\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx3060\tx3600\tx 5130\tx7020\tx7920\tqdec\tx8370\tx8729 \f20\fs20 {\fs18\up6 *\tab }Clique repres entations have been found useful for numerical simulations (beating the critical slowing down [5]) and for rigorous results [4,6], and have also been encountered in quantum systems [7]. }} . More completely, the state is depicted as a superposition of ones in which the spins are organized into 'cliques'. For any specified partition the spins {\f23 s(}{\fs20\up6 .} ) are assuming uniform values on each clique, but the values for different cliques are independent and symmetrically distributed. \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab For this representation, the model is augmented by bond variables \{n(x,y)\}{\fs20\dn4 \{x,y\}}{\f23\fs20\dn4 \'ce}{\fs20\dn4 B} with values in \{0,1\}. For a specified bond configuration n({\fs20\up6 .}) the cliques are the clusters of sites connected by bonds with n(x,y)=1. The marginal distribution of the spins, obtained by a complete integration (or summation) over n({ \fs20\up6 .}) reproduces the measure seen in (1.4). \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab The joint distribution of \{{\f23 s}({\fs20\up6 .}), S{\f23\fs20\dn4 ^}({\fs20\up6 .}), n({\fs20\up6 .})\} can be written explicitly ([1]). We shall not repeat here the somewhat standard expression, but just note that it has has the following key properties.\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab i) For specified values of \{S{\f23\fs20\dn4 ^}({\fs20\up6 .}), n({\fs20\up6 .})\}, the conditional distribution of {\f23 s(}{ \fs20\up6 .}) has the clique structure described above.\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab ii) For specified values of \{{\f23 s}({\fs20\up6 .}), S{\f23\fs20\dn4 ^}({\fs20\up6 .})\}, the conditional distribution of n({ \fs20\up6 .}) is that of the independent bond percolation model, with the probability that a given bond is occupied given by\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab {{\pict\macpict\picw307\pich34 07d300000000002201331101a10064000eb77870720000000e0000 000c0000a0008c01000a0000000000220133070000000022002201 330000a000bea100c0000d63757272656e74706f696e740da000bf2 2000000000000a000be0300140d000c2b0114025072290d026f622 90c01282907016e2906012829040178290601 2c29050179290601290300172907013d030014290801312908017c 0300172904017303001429080128030017290401d7030014290202 292c290801530300170d00072b0703015e0300140d000c28001400 7a0128030017290301d70300142903022929290901200300172907 013d03001428000a00c601302937026966290a 0120030017290501440300142907016829080120030017290601a3 030014290a013028001c009e01310300172908012d030014290904 657870280300172916012d29070162290901440300142908016829 0601292905032020202915026966290a0120030017290501440300 1429070168290801200300172906013e030014 290a013003001728000b009a02ec202a0902ed202a0902ee20a100c 0053b2f4d547361766520736176652064656620333020646963742 0626567696e0d63757272656e74706f696e742033202d3120726f6c 6c20737562206e65672033203120726f6c6c207375620d393832342 064697620313038382033202d3120726f6c 6c206578636820646976207363616c650d63757272656e74706f696 e74207472616e736c617465203634202d31383038207472616e736 c6174650d2f6673203020646566202f63662030206465660d2f7366 207b6578636820647570202f667320657863682064656620647570 206e6567206d6174726978207363616c6520 6d616b65666f6e7420736574666f6e747d206465660d2f6631207b6 6696e64666f6e7420647570202f6366206578636820646566207366 7d206465660d2f6e73207b63662073667d206465660d2f7368207b 6d6f7665746f2073686f777d206465660d333834202f54696d65732 d526f6d616e2066310d28507229202d3420 323434382073680d285c28292037373620323434382073680d285c 2829203132303120323434382073680d282c292031353330203234 34382073680d285c2929203138363720323434382073680d287c29 203235393720323434382073680d285c2829203239383520323434 382073680d285c292c29203331383820323434 382073680d285c2829203338343720323434382073680d285c295c 2929203430353020323434382073680d2820292034333530203234 34382073680d282029203833373120323135312073680d28202920 3930313620323135312073680d286578705c282920353535392032 3732372073680d285c29292037323136203237 32372073680d2820202029203733383920323732372073680d2820 29203833373020323732372073680d282029203930313520323732 372073680d333834202f54696d65732d526f6d616e2066310d286f6 2292033393020323434382073680d286e292031303031203234343 82073680d2878292031333431203234343820 73680d287929203136363620323434382073680d28532920333434 3020323434382073680d2869662920383035342032313531207368 0d286829203837363420323135312073680d286829203730303920 323732372073680d28696629203830353320323732372073680d28 6829203837363320323732372073680d333834 202f53796d626f6c2066310d283d292032303834203234343820736 80d285c33323729203331303520323434382073680d285c3332372 9203339363720323434382073680d283d292034353534203234343 82073680d285c32343329203932313320323135312073680d282d2 9203532363620323732372073680d282d2920 3632353520323732372073680d283e292039323134203237323720 73680d285c33353429203438363720323230392073680d285c3335 3529203438363720323531332073680d285c333536292034383637 20323831382073680d323234206e730d285e292033363537203235 34342073680d333834202f54696d65732d526f 6d616e2066310d283129203233363020323434382073680d283029 203632373620323135312073680d28302920393531372032313531 2073680d283129203530323320323732372073680d283029203935 313720323732372073680d333834202f53796d626f6c2066310d287 329203237343820323434382073680d286229 203634363720323732372073680d333834202f53796d626f6c20663 10d284429203835323920323135312073680d28442920363737342 0323732372073680d284429203835323820323732372073680d656 e64204d547361766520726573746f72650da000bfa1006400f14d41 5448000100e533d702000003000a01128250 12827212836f12836202822802980412836e02822812837802822c 12837902822902863d02883102980402827c0284730282280286d7 02822902822c1283530b030f01000102865e0011000a0282280286 d702822902822902822002863d0302010001050101010202000001 02883000011283691283660282200285441283 680282200286a3028830000102883102862d128265128278128270 02822802862d028462029804028544128368028229028220028220 028220000112836912836602822002854412836802822002863e02 883000000002967b000000660da0008dff}}{\up28 \tab \tab \tab (2.4)}\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 with\tab \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab {{\pict\macpict\picw274\pich16 076400000000001001121101a10064000eb7787072000000060000 000c0000a0008c01000a0000000000100112070000000022001001 120000a000bea100c0000d63757272656e74706f696e740da000bf2 2000000000000a000be0300170d000c2b010a01440300142908016 8290801200300172906013d030014290a0220 20290801680d00072b070301782904012c290201790d000c28000a 003b01280300172904012d29070173030014290701282905017829 0602292c0300172908017303001429070128290501792906012929 06017c290501530300170d00072b0703015e0300140d000c28000a 008a0128030017290301d70300142903012929 0601290300172907012d030014290901680d00072b070301782904 012c290201790d000c28000a00b801280300172904017303001429 07012829050178290602292c030017290801730300142907012829 050179290601292906017c290501530300170d00072b0703015e03 00140d000c28000a00ff0128030017290401d7 0300142903012929060129a100c004f22f4d5473617665207361766 520646566203330206469637420626567696e0d63757272656e747 06f696e742033202d3120726f6c6c20737562206e65672033203120 726f6c6c207375620d3837363820646976203531322033202d3120 726f6c6c206578636820646976207363616c 650d63757272656e74706f696e74207472616e736c6174652036342 02d33303230207472616e736c6174650d2f6673203020646566202f 63662030206465660d2f7366207b6578636820647570202f667320 657863682064656620647570206e6567206d617472697820736361 6c65206d616b65666f6e7420736574666f6e 747d206465660d2f6631207b66696e64666f6e7420647570202f636 62065786368206465662073667d206465660d2f6e73207b6366207 3667d206465660d2f7368207b6d6f7665746f2073686f777d206465 660d333834202f53796d626f6c2066310d284429202d32203333343 02073680d333834202f54696d65732d526f 6d616e2066310d2868292032333320333334302073680d28682920 3132363020333334302073680d2878292032353637203333343020 73680d287929203334303420333334302073680d28532920333934 3620333334302073680d286829203532363520333334302073680d 287829203633343120333334302073680d2879 29203731373820333334302073680d285329203737323020333334 302073680d323234206e730d287829203134373720333433362073 680d287929203136373420333433362073680d2878292035343832 20333433362073680d287929203536373920333433362073680d33 3834202f54696d65732d526f6d616e2066310d 2820292034383520333334302073680d2820202920313030372033 3334302073680d285c2829203138323420333334302073680d285c 2829203234323720333334302073680d285c292c29203237363920 333334302073680d285c2829203332363320333334302073680d28 5c2929203336303520333334302073680d287c 29203338303020333334302073680d285c28292034333533203333 34302073680d285c2929203435353620333334302073680d285c29 29203437363620333334302073680d285c28292035383239203333 34302073680d285c2829203632303120333334302073680d285c29 2c29203635343320333334302073680d285c28 29203730333720333334302073680d285c29292037333739203333 34302073680d287c29203735373420333334302073680d285c2829 203831323720333334302073680d285c2929203833333020333334 302073680d285c2929203835343020333334302073680d32323420 6e730d282c2920313630302033343336207368 0d282c29203536303520333433362073680d333834202f53796d62 6f6c2066310d283d292036383920333334302073680d282d292031 39363620333334302073680d285c33323729203434373320333334 302073680d282d29203439363720333334302073680d285c333237 29203832343720333334302073680d32323420 6e730d285e29203431363320333433362073680d285e2920373933 3720333433362073680d333834202f53796d626f6c2066310d28732 9203231393020333334302073680d2873292033303236203333343 02073680d287329203539363420333334302073680d28732920363 8303020333334302073680d656e64204d5473 61766520726573746f72650da000bfa1006400d94d415448000100c d2c4302000003000a0102854412836802822002863d02822002822 01283680b030f01000112837802822c1283790011000a028228028 62d02847302822812837802822902822c028473028228128379028 22902827c1283530b030f01000102865e0011 000a0282280286d702822902980402822902862d1283680b030f01 000112837802822c1283790011000a028228028473028228128378 02822902822c02847302822812837902822902827c1283530b030f 01000102865e0011000a0282280286d70282290298040282290000 7368a0008dff}}{\up14 .\tab \tab \tab (2.5)\par }\pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab A key implication of the clique structure is that the spin - spin correlation function can be related to a connectivity probability. Averaging over {\f23 s}({\fs20\up6 .}) at given n({\fs20\up6 .}) and - for concreteness sake, at given S{\f23\fs20\dn4 ^ }({\fs20\up6 .}), one gets\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab \tab \tab {\f129 E}( {\f23 s}(x) {\f23 s}(y) | n({\fs20\up6 .}), S{\f23\fs20\dn4 ^}({\fs20\up6 .}) ) = I[ x <\|o(=,{ \fs20\up18 n})> y ] ,\tab \tab \tab (2.6)\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 where x <\|o(=,{\fs20\up18 n} )> y denotes the event that x and y belong to the same connected cluster, with respect to the bond configuration \{n\}, and I[ - ] is the corresponding indicator function. \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab Next, the average over n({\fs20\up6 .}), with its appropriate distribution conditioned on S{\f23\fs20\dn4 ^}({\fs20\up6 .} ) (whose details do not matter here) results in the equality\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab \tab \tab {\f129 E}( {\f23 s}(x) {\f23 s}(y) | S{\f23\fs20\dn4 ^}({\fs20\up6 .}) ) = Prob( x <\|o(=,{\fs20\up18 n})> y | S{ \f23\fs20\dn4 ^}({\fs20\up6 .}) ) ,\tab \tab \tab (2.7)\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 which can be employed for the following expression for the spin-spin correlation function \par {\f129 \tab E}( S(x){\fs20\up6 .}S(y) ) = 3 {\f129 E}{\fs28 ( } |S{\f23\fs20\dn4 ||}(x)| |S{\f23\fs20\dn4 ||}(y)| {\f23 s}(x) {\f23 s}(y) ) =\par \tab = 3 {\f129 E}{\fs36 ( } |S{\f23\fs20\dn4 ||}(x)| |S{\f23\fs20\dn4 ||}(y)| {\f129 E}{\fs28 ( }{\f23 s}(x) {\f23 s}(y) | S{\f23\fs20\dn4 ^}({\fs20\up6 .})) {\fs28 )} \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab = 3 {\f129 E}{\fs36 ( } |S{\f23\fs20\dn4 ||}(x)| |S{\f23\fs20\dn4 ||}(y)| Prob( x <\|o(=,{\fs20\up18 n})> y | S{ \f23\fs20\dn4 ^}({\fs20\up6 .}) ) {\fs28 )} .\tab \tab \tab (2.8)\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab The above equation yields the useful bound:\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab {\f129 E}( S(x){\fs20\up6 .}S(y) ) \'b3 \|f{\fs20 (3,2)} {\f129 E}{\fs28 ( }I[ |S{\f23\fs20\dn4 ||}(x)|, |S{\f23\fs20\dn4 || }(y)| \'b3 {\fs20 1/\|r(2) }] {\fs18\up6 . }Prob( x <\|o(=,{\fs20\up18 n})> y | S{\f23\fs20\dn4 ^}({\fs20\up6 .}) {\fs28 )} ,\tab (2.9)\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 which is not far from being an equality, in the sense that without the factor \|f({\fs20 1,2} ) I[ - ] the right side provides a bound in the opposite direction.\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab We note that eq. (2.9) does not depend on the value of the constraining angle {\f23 q}{\fs20\dn4 o} , and is just a reflection of the clique structure. In particular, it holds also for the unconstrained O(2) model. \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab A notable effect of the constraint (1.2), within the random cluster extension of the model, is that for any configuration { \ul S}({\fs20\up6 .}) in which the spin {\f23 s}(x) cannot be flipped without violating the relation (1.2) between {\ul S}(x) to {\ul S}(y), the conditional probability that n(x,y) = 1 is one. (In terms of (2.4): {\f23 D}h = +{\f23\fs28 \'a5} .) In effect, a new constraint is dynamically generated, that any bond with\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab | arg(S(x) ) - arg(S(y) ) | \'b2 {\f23 q}{\fs20\dn4 o} \'b2 | arg(RS(x) ) - arg(S(y) ) | \tab \tab (2.10) \par is (a.s.) occupied. [R{\ul S}(x) being the spin obtained by reflecting S{\fs20\dn4 ||}(x)]. Hence\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab \tab Prob( x <\|o(=,{\fs20\up18 n})> y | S{\f23\fs20\dn4 ^}({\fs20\up6 .}) ) \'b3 I[ x <\|o(=,{\fs20\up18 C})> y ] \tab \tab \tab (2.11)\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 where x <\|o(=,{\fs20\up18 C})> y denotes the event that, for the specified spin configuration S({\fs20\up6 .} ), x and y are connected by a path along which at each step the condition (1.9) is satisfied. \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab When {\f23 q}{\fs20\dn4 o} is small enough, as spelled in Theorem 1, x\~<\|o(=,{\fs20\up18 C})>\~ y is guaranteed to occur whenever {\ul S} is an allowed configuration in which x and y are *-connected by a path along which the spins take values only within the polar caps defined by (1.8). I.e., \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab \tab I[ x <\|o(=,{\fs20\up18 C})> y] \'b3\~ I[x <\|o(=,{\fs20\up18 P})> y] .\tab \tab \tab \tab (2.12)\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 The combination of the three statements (2.9), (2.11) and (2.12) proves Lemma 1.\par \par \pard\plain \s255\qj\ri- 90\sb240\sl280\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \b\f20\ul 3. The geometric argument\par \pard\plain \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \f20 \tab The lower bound for the correlation function stated in Theorem 1 is now a consequence of the following geometric considerations, stated initially for the cubic lattice. Let us focus on the spin configuration within the rectangle [1,L]{\f23 \'b4} [1,L]. The boundary of this rectangle consists of the four intervals: B{\fs20\dn6 1 }(top), ..., B{\fs20\dn6 4} (ordered consecutively). \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab In each configuration {\ul S}({\fs20\up6 .}), there either is a path connecting a pair of sites on the top and bottom faces, x{ \f23 \'ce}B{\fs20\dn6 1} and y{\f23 \'ce}B{\fs20\dn6 3}, in the sense denoted here by x<\|o(=,{\fs20\up18 P})>y, or else the <\|o(=,{\fs20\up18 P})> connected cluster of B{\fs20\dn6 1} is separated from B{\fs20\dn6 3} . In the latter case, the boundary of that cluster forms a connected path (in the regular sense) which reaches from a point on the left face, u{\f23 \'ce}B{\fs20\dn6 2}, to a point on the right face, v{\f23 \'ce}B{\fs20\dn6 4}{\f23 , } along which the spins fail (1.8), i.e., satisfy the complementary condition: \par \pard \qj\ri- 90\sb240\sl480\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab \tab \tab \tab | S{\f23\fs20\dn4 ^}(u) | \'b3 1/\|r({\fs20 2}) .\tab \tab \tab \tab (3.1)\par \pard \qj\ri- 90\sb240\sl480\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 Let this dual event be denoted u <\|o(=,{\fs20\up18 P}{\f23\fs18\up26 ^})> v. \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 {\up30 \tab }By rotation invariance, and the fact that the *connectedness required for u <\|o(=,{\fs20\up18 P}{\f23\fs18\up26 ^} )> v is less stringent than the connectedness needed for u <\|o(=,{\fs20\up18 P})> v, \par \pard \qj\ri- 90\sb240\sl480\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab Prob( u <\|o(=,{\fs20\up18 P}{\f23\fs18\up26 ^})> v) \'b3\~ Prob(u <\|o(=,{\fs20\up18 P})> v) = {\f23 t}(u,v) .\tab \tab \tab (3.2)\par \pard \qj\ri- 90\sb240\sl480\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 The dichotomy spelled above means that one of the events in the union always occurs, and thus \par \pard \qj\ri- 90\sb240\sl480\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab {{\pict\macpict\picw237\pich33 07f800000000002100ed1101a10064000eb77870720000000f00000 00c0000a0008c01000a00000000002100ed2c000b09d3084d542045 787472610309d30d000c2e00040000ff002a2102202007000000002 2002100ed0000a000bea100c0000d63757272656e74706f696e740d a000bf22000000000000a000be2c000800 140554696d65730300142800120001025072290d026f62290c0128 291f017b290501782c000900170653796d626f6c030017290b033c3 d3e0300140d0007280008005101500d000c2b0f0a01792905017d0 d000728001d001e0178030017290401ce030014290501420d00052 b040201310d000728001d002f012c29020179 030017290401ce030014290401420d00052b050201330309d30d00 12280014002a01550300140d000c280012006b0120292e017b2905 0175030017290b033c3d3e0300140d000728000800ae0150030017 0d000528000400b3015e0300140d000c2b0d0e01762904017d0d00 0728001d007d0175030017290401ce03001429 0401420d00052b060201320d000728001d008e012c290301790300 17290401ce030014290401420d00052b050201340309d30d001228 001400890155280014007001550300140d000c28001200cc01290d 000d290501200300172907013d030014290a012029040131a100c0 04cd2f4d547361766520736176652064656620 3330206469637420626567696e0d63757272656e74706f696e7420 33202d3120726f6c6c20737562206e65672033203120726f6c6c207 375620d373538342064697620313035362033202d3120726f6c6c2 06578636820646976207363616c650d63757272656e74706f696e7 4207472616e736c617465203634202d363130 39207472616e736c6174650d2f6673203020646566202f636620302 06465660d2f7366207b6578636820647570202f667320657863682 064656620647570206e6567206d6174726978207363616c65206d6 16b65666f6e7420736574666f6e747d206465660d2f6631207b6669 6e64666f6e7420647570202f636620657863 68206465662073667d206465660d2f6e73207b63662073667d2064 65660d2f7368207b6d6f7665746f2073686f777d206465660d33383 4202f54696d65732d526f6d616e2066310d28507229202d3420363 638352073680d285c28292037373620363638352073680d287b292 03137393120363638352073680d287d292033 31383120363638352073680d282029203333383720363638352073 680d287b29203438343420363638352073680d287d292036323336 20363638352073680d285c2929203634373920363638352073680d 323234206e730d282c29203134343820373035372073680d282c29 203435303820373035372073680d343136206e 730d282029203636353220363638352073680d2820292037313939 20363638352073680d333834202f54696d65732d526f6d616e2066 310d286f62292033393020363638352073680d2878292031393435 20363638352073680d287929203330323220363638352073680d28 7529203530303120363638352073680d287629 203630383020363638352073680d323234206e730d285029203235 343820363336382073680d2878292039313920373035372073680d 284229203131383920373035372073680d28792920313532322037 3035372073680d284229203137393120373035372073680d285029 203535323120363336382073680d2875292033 39353320373035372073680d284229203432323320373035372073 680d287929203435383220373035372073680d2842292034383531 20373035372073680d333834202f53796d626f6c2066310d283c3d3 e29203232393520363638352073680d283c3d3e292035333531203 63638352073680d323234206e730d285c3331 3629203130343820373035372073680d285c333136292031363530 20373035372073680d285c33313629203430383220373035372073 680d285c33313629203437313020373035372073680d313630206e 730d285e29203536373220363236382073680d343136206e730d28 3d29203638363420363638352073680d313630 202f54696d65732d526f6d616e2066310d28312920313333372037 3131332073680d283329203139343920373131332073680d283229 203433383420373131332073680d28342920353031352037313133 2073680d343136206e730d28312920373332312036363835207368 0d353736202f4d542d45787472612066310d28 5529203132393620363737322073680d2855292034333439203637 37322073680d285529203335333020363737322073680d656e6420 4d547361766520726573746f72650da000bfa1006401344d4154480 0010128352702000003000a0112825012827212836f12836202822 8032300000102980402827b12837802980202 9804032700000102863c02863d02863e000b110112835000000a02 980412837902827d000b011283780286ce128342030f01000c01028 8310011000b02822c1283790286ce128342030f01000c0102883300 1100000b110d028b55000a02822003230200010323000001029804 02827b128375029802029804032700000102 863c02863d02863e000b1101128350030f00000c110102865e00000 0000a02980412837602827d000b011283750286ce128342030f010 00c010288320011000b02822c1283790286ce128342030f01000c01 028834001100000b110d028b5500000b11110d028b55000a029804 0282290900a002822002863d028220028831 00006d6fa0008dff}}{\up30 .\tab \tab \tab (3.3)\par }\pard \qj\ri- 90\sb240\sl480\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab Since the number of events listed above is (2 L{\fs20\up6 2}), the probability of at least one of them exceeds \|f(1,2L{ \fs20\up6 2}) . For the pair of corresponding sites, say \{x,y\}, we have\par \pard \qj\ri- 90\sb240\sl480\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab \tab <{\ul S}(x){\fs20\up6 .}{\ul S}(y)> \'b3 \|f{\fs20 (3,2)} {\f23 t}(x,y) \'b3 \|f(3,4L{\fs20\up6 2}) .\tab \tab \tab (3.4)\par \pard \qj\ri- 90\sb240\sl480\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 Since x and y are on opposite faces of the boundary of [1,L]{\f23 \'b4} [1,L], (3.4) establishes the slow decay described in Theorem 1.\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab The above argument is robust, and for other lattices requires only a trivial adjustment: the sites x, y, u, and v need not lie on the boundary of [1,L]{\f23 \'b4}[1,L], but instead they will be within distance one from it. \par \pard\plain \s255\qj\ri- 90\sb240\sl280\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \b\f20\ul 4.\tab Discussion\par \pard\plain \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \f20 \tab It was noted by A. Polyakov, that the result of Theorem 1 fits well with the folk-lore that exponential decay requires the presence of unbound charges. It would be of interest to see a derivation of a bound along these lines. The following observation m ay be of relevance. For a bond b{\f23 \'ba}\{x,y\} let\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab \tab {\f23 Q}{\fs20\dn4 x,y} = arg({\ul S}(x)) - arg({\ul S}(y)) + k 2\'b9\tab \tab \tab \tab (4.1)\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 with k = 0, \'b11 chosen so that |{\f23 Q}{\fs20\dn4 x,y} | \'b2\~\'b9. Using Stokes theorem, in the absence of the topologica l charges (defined in the Introduction):\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab \tab {{\pict\macpict\picw197\pich30 06e400000000001e00c51101a10064000eb7787072000000120000 000c0000a0008c01000a00000000001e00c5070000000022001e00c 50000a000bea100c0000d63757272656e74706f696e740da000bf22 000000000000a000be0300140d000c2b020c0120292c07202020202 02020030017291701510300140d000a2b09 03016228001a00060162030017290601ce290601b6030014290401 5b290301312904012c2903014c2907015d030017290301b4030014 2905015b290301312905012c2902014c2907015d0300170d001228 000e002001e50300140d000c28000c005601200300172906013d03 0014292e052020202020291101710d000a2b07 03015028001a00660150030017290601ce0300142906015b290301 312904012c2903014c2907015d030017290301b40300142905015b 290301312905012c2902014c2907015d0300170d001228000e007c 01e50d000c28000c00ac013d030014290a02202029070130a100c00 4872f4d547361766520736176652064656620 3330206469637420626567696e0d63757272656e74706f696e7420 33202d3120726f6c6c20737562206e65672033203120726f6c6c207 375620d3633303420646976203936302033202d3120726f6c6c206 578636820646976207363616c650d63757272656e74706f696e742 07472616e736c617465202d33322033382074 72616e736c6174650d2f6673203020646566202f636620302064656 60d2f7366207b6578636820647570202f667320657863682064656 620647570206e6567206d6174726978207363616c65206d616b656 66f6e7420736574666f6e747d206465660d2f6631207b66696e6466 6f6e7420647570202f636620657863682064 65662073667d206465660d2f6e73207b63662073667d206465660d 2f7368207b6d6f7665746f2073686f777d206465660d333834202f5 4696d65732d526f6d616e2066310d2820292039362033343620736 80d2820202020202020292031353233203334362073680d2820292 032373935203334362073680d282020202020 292034343738203334362073680d28202029203538363420333436 2073680d333230206e730d285b2920373537203831352073680d28 2c2920393833203831352073680d285d2920313238372038313520 73680d285b292031353636203831352073680d282c292031373932 203831352073680d285d292032303936203831 352073680d285b292033373032203831352073680d282c29203339 3238203831352073680d285d292034323332203831352073680d28 5b292034353131203831352073680d282c29203437333720383135 2073680d285d292035303431203831352073680d333834202f5379 6d626f6c2066310d2851292032323434203334 362073680d333230202f54696d65732d526f6d616e2066310d2862 292032353438203434332073680d28622920323534203831352073 680d284c292031303736203831352073680d284c29203138383520 3831352073680d2850292035323333203434322073680d28502920 33333134203831352073680d284c2920343032 31203831352073680d284c292034383330203831352073680d3338 34206e730d2871292035303133203334362073680d333230202f53 796d626f6c2066310d285c3331362920343233203831352073680d 285c323634292031333932203831352073680d285c333136292033 353035203831352073680d285c323634292034 333337203831352073680d353736206e730d285c33343529203130 3539203433332073680d285c333435292034303134203433332073 680d333834206e730d283d292032393939203334362073680d283d 292035353436203334362073680d333230202f53796d626f6c20663 10d285c323636292036303820383135207368 0d333230202f54696d65732d526f6d616e2066310d283129203834 37203831352073680d2831292031363536203831352073680d2831 292033373932203831352073680d28312920343630312038313520 73680d333834206e730d2830292036313131203334362073680d65 6e64204d547361766520726573746f72650da0 00bfa1006400e54d415448000100d92e7502000003000a01028220 031d00000102822002822002822002822002822002822002822002 8551030f01000b0112836200110000011283620286ce0284b60282 5b02883102822c12834c02825d0286b402825b02883102822c1283 4c02825d02980400110d0286e5000a02822002 863d0900a0031d00000a0102822002822002822002822002822012 8371030f01000b0112835000110000011283500286ce0298020282 5b02883102822c12834c02825d0286b402825b02883102822c1283 4c02825d00110d0286e5000a02863d02822002822002883000006d 61a0008dff}} {\up30 .\tab \tab \tab (4.2)\par }\pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 That implies that the collection of the bond variables \{{\f23 Q}{\fs20\dn4 b}\} are strongly correlated. Can one extract from that a statement like Theorem 1? Further insight along these lines may also be useful in the discussion of other phenomena related to the Kosterlitz - Thouless transition.\par \tab It may also be of interest to point out that in the absence of the topological charges (i.e., when q{\fs20\dn8 P}{\f23 \'ba}0) there is a one to one correspondence between the angles {\f23 q}(x) = arg({\ul S}(x)) and real variables {\f23 j} (x) satisfying: {\f23 j}(0) = {\f23 q}(0) / {\f23 q}{\fs20\dn4 o} , and\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab \tab \tab \tab {\f23 j}(x) {\f23 -} {\f23 j}(y) = {\f23 Q}{\fs20\dn4 x,y} / {\f23 q}{\fs20\dn4 o} \tab \tab \tab \tab (4.3)\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 (without the corrections seen in (4.1) ). In terms of the variables {\f23 j}({\fs20\up6 .}), the constraint (1.2) on {\f23 q}({ \fs20\up6 .}) (at T={\f23\fs28 \'a5}) corresponds to the "hammock" potential:\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab {{\pict\macpict\picw239\pich34 075600000000002200ef1101a10064000eb77870720000000e0000 000c0000a0008c01000a00000000002200ef070000000022002200e f0000a000bea100c0000d63757272656e74706f696e740da000bf220 00000000000a000be0300140d000c2b02140120290401562909022 8200300172909016a030014290701282904 0178290701290300172906012d2909016a03001429070128290501 79290601292906022029290801200300172906013d030014290a01 2028000a0078013029160220202909017c0300172904016a030014 2908012829040178290601290300172907012d2909016a03001429 07012829040179290701292906017c29050120 030017290601a30300142909013103001728001c0073022ba50300 14291b0220202909017c0300172904016a03001429080128290401 78290601290300172907012d2909016a0300142907012829040179 290701292906017c290501200300172906013e0300142909013103 001728000b006d02ec202a0902ed202a0902ee 20a100c004f82f4d547361766520736176652064656620333020646 9637420626567696e0d63757272656e74706f696e742033202d312 0726f6c6c20737562206e65672033203120726f6c6c207375620d37 3634382064697620313038382033202d3120726f6c6c2065786368 20646976207363616c650d63757272656e74 706f696e74207472616e736c617465203634203437207472616e736 c6174650d2f6673203020646566202f63662030206465660d2f7366 207b6578636820647570202f667320657863682064656620647570 206e6567206d6174726978207363616c65206d616b65666f6e7420 736574666f6e747d206465660d2f6631207b 66696e64666f6e7420647570202f636620657863682064656620736 67d206465660d2f6e73207b63662073667d206465660d2f7368207 b6d6f7665746f2073686f777d206465660d333834202f54696d6573 2d526f6d616e2066310d2820292030203539332073680d285c2820 2920343334203539332073680d285c282920 393435203539332073680d285c2929203132383720353933207368 0d285c28292032303133203539332073680d285c29292032333535 203539332073680d28205c29292032353238203539332073680d28 20292032373937203539332073680d282029203333313920353933 2073680d282020292034343933203239342073 680d287c292034373731203239342073680d285c28292035313537 203239342073680d285c29292035343939203239342073680d285c 28292036323235203239342073680d285c29292036353637203239 342073680d287c292036373632203239342073680d282029203639 3234203239342073680d282020292034343933 203837302073680d287c292034373731203837302073680d285c28 292035313537203837302073680d285c2929203534393920383730 2073680d285c28292036323235203837302073680d285c29292036 353637203837302073680d287c292036373632203837302073680d 2820292036393234203837302073680d333834 202f54696d65732d526f6d616e2066310d28562920313533203539 332073680d2878292031303835203539332073680d287929203231 3534203539332073680d2878292035323937203239342073680d28 79292036333636203239342073680d287829203532393720383730 2073680d287929203633363620383730207368 0d333834202f53796d626f6c2066310d286a2920373038203539332 073680d286a292031373736203539332073680d286a29203439323 0203239342073680d286a292035393838203239342073680d286a2 92034393230203837302073680d286a29203539383820383730207 3680d333834202f53796d626f6c2066310d28 2d292031343838203539332073680d283d29203330303120353933 2073680d282d292035373030203239342073680d285c3234332920 37313231203239342073680d282b5c323435292033363335203837 302073680d282d292035373030203837302073680d283e29203731 3233203837302073680d285c33353429203334 3430203335352073680d285c333535292033343430203635382073 680d285c333536292033343430203936322073680d333834202f54 696d65732d526f6d616e2066310d28302920333737382032393420 73680d2831292037333932203239342073680d2831292037333933 203837302073680d656e64204d547361766520 726573746f72650da000bfa1006400cf4d415448000100c32b76020 00003000a0102822012835602822802822002846a0282281283780 2822902862d02846a0282281283790282290282200282290282200 2863d0282200302010001050101010202000001028830000102822 002822002827c02846a028228128378028229 02862d02846a02822812837902822902827c0282200286a3028831 000102862b0286a5000102822002822002827c02846a0282281283 7802822902862d02846a02822812837902822902827c0282200286 3e02883100000002967b0000006469a0008dff}}{\up36 }{\up32 .\tab \tab \tab (4.4)\par }\pard \qj\ri- 90\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx3060\tx3 600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab Under the correspondence {\f23 q}({\fs20\up6 .}){\f129 <>} {\f23 j}({\fs20\up6 .}) Theorem 1 acquires the following form.\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 {\ul Theorem 2:} Let V({\fs20\up6 .}) be a monotone non-decreasing function on [0,{\f23\fs28 \'a5}) which diverges for |z|>{\f23 1}. Then, in d=2 dimensional lattice systems of real-valued variables \{{\f23 j}(x)\} with the distribution \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab {{\pict\macpict\picw276\pich29 078e00000000001d01141101a10064000eb7787072000000040000 000c0000a0008c01000a00000000001d0114070000000022001d01 140000a000bea100c0000d63757272656e74706f696e740da000bf2 2000000000000a000be0300170d000c2b011901720300142906012 8290401640300172906016a03001429080129 0300172907013d0300140d000d290a01650300170d000a28000900 30012d290601620300142916012029050156290701282905017c03 00172902016a030014290701282904017829050129030017290401 2d2906016a0300142906012829040179290501292903017c290302 20290300170d0009280013003b013c03001429 0601782905012c290201790300172905013e0d000a280009004301 e50d000c2b51100164030014290601280300172904016a03001429 07012829050130290602292929090120030017290501500300140d 000a2b080301780d000c28001900cc01640300172906016a0300142 90701282904017829070129290503202f2029 0b044e6f726d291c012ea100c005182f4d547361766520736176652 0646566203330206469637420626567696e0d63757272656e74706 f696e742033202d3120726f6c6c20737562206e6567203320312072 6f6c6c207375620d3838333220646976203932382033202d312072 6f6c6c206578636820646976207363616c65 0d63757272656e74706f696e74207472616e736c617465203634202 d31313635207472616e736c6174650d2f6673203020646566202f63 662030206465660d2f7366207b6578636820647570202f66732065 7863682064656620647570206e6567206d6174726978207363616c 65206d616b65666f6e7420736574666f6e74 7d206465660d2f6631207b66696e64666f6e7420647570202f63662 065786368206465662073667d206465660d2f6e73207b636620736 67d206465660d2f7368207b6d6f7665746f2073686f777d20646566 0d333834202f53796d626f6c2066310d287229202d3139203139363 52073680d286a2920353130203139363520 73680d286429203436373320313936352073680d286a2920353030 3220313936352073680d286a29203636363120313936352073680d 333230206e730d286229203136363620313435372073680d286a29 203330303420313435372073680d286a2920333831332031343537 2073680d333834202f54696d65732d526f6d61 6e2066310d285c28292031383120313936352073680d285c292920 37353420313936352073680d285c28292034383639203139363520 73680d285c2829203532333920313936352073680d285c295c2929 203535373720313936352073680d28202920353837372031393635 2073680d285c28292036383938203139363520 73680d285c2929203732343020313936352073680d28202f202920 3734313320313936352073680d282e292038363634203139363520 73680d333230206e730d282029203233393220313435372073680d 285c2829203237363920313435372073680d287c29203239343120 313435372073680d285c282920333230362031 3435372073680d285c2929203335303220313435372073680d285c 2829203430313520313435372073680d285c292920343331312031 3435372073680d287c29203434313220313435372073680d28205c 2929203434383620313435372073680d323838206e730d282c2920 3231383020313830302073680d333834202f54 696d65732d526f6d616e2066310d28642920333138203139363520 73680d286429203634363920313936352073680d28782920373033 3820313936352073680d284e6f726d292037373731203139363520 73680d343136206e730d286529203132383020313936352073680d 333230206e730d285629203235333020313435 372073680d287829203333323820313435372073680d2879292034 31333820313435372073680d287829203632393320323036312073 680d323838206e730d287829203230333120313830302073680d28 7929203232363620313830302073680d333834202f53796d626f6c2 066310d283d29203937312031393635207368 0d285029203630323820313936352073680d333230206e730d282d 29203134383420313435372073680d282d29203336323120313435 372073680d285c33343529203231303620313437382073680d3238 38206e730d283c29203138353320313830302073680d283e292032 34323820313830302073680d333834202f5469 6d65732d526f6d616e2066310d2830292035333736203139363520 73680d656e64204d547361766520726573746f72650da000bfa1006 400de4d415448000100d22f1f02000003000a010284720282281283 6402846a02822902863d0900a01283650a030f00000b110102862d 028462031d00000102822012835602822802 980402827c02846a02822812837802822902862d02846a02822812 837902822902827c028220028229000c0102863c12837802822c128 37902863e00110b0286e50000000a02846402822802846a0282280 288300282290282290282200286500b030f0100011283780011000 a12836402846a028228128378028229028220 02822f02822012834e12836f12837212836d02822e00003031a0008 dff}}{\up8 \tab \tab \tab (4.5)\par }\pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 the following bound holds, at any {\f23 b}\'b30:\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab \|i(,, cos{\fs28 (}t [{\f23 j}(x) {\f23 -} {\f23 j}(0)]{\fs28 )} {\f23 r}(d{\f23 j})) " \'b3 " \|f(3,4 |x|{\fs20\up6 2} ) , for all | t | < {\f23 q}{\fs20\dn4 o} ,\tab \tab (4.6)\par \pard \qj\ri- 90\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx3060\tx3 600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 where "\'b3" is to be read in the sense made explicit in (1.6).\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab The intuitive explanation of (4.6) is that the long distance behavior of ({\f23 j}(x) {\f23 -} {\f23 j} (0)) is similar to that seen in Gaussian models with short short-range elastic forces, for which (in d=2 dimensions) <|{\f23 j}(x) {\f23 -} {\f23 j}(0)|{\fs20\up6 2}> grows, but at only a logarithmically slow rate. An interesting discussion, and a number of results on the rate of growth of |{\f23 j}(x) {\f23 -} {\f23 j}(0)| in two dimensional models (which, however, do not include the hammock potential) can be found in ref. [8]. \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 {\b\ul Acknowledgement} This article is dedicated to Oliver Penrose, whose cultured insights on the subject of statistical mechanics, and science in general, I have often had the privilege to enjoy.\par \pard \qj\ri- 90\sb240\sl360\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \tab I also thank A. Sokal, T. Spencer, and A. Polyakov for stimulating discussions of topics related to this work.\par \pard\plain \s255\qj\ri- 90\sb240\sl280\tx720\tx810\tx1440\tx1710\tx2160\tx2880\tx30 60\tx3600\tx5130\tx7020\tx7920\tqdec\tx8370\tx8729 \b\f20\ul REFERENCES\par \pard\plain \s4\qj\fi-980\li980\ri- 90\sb240\tx980\tqdec\tx8280\tqdec\tx8370 \f20 [1]\tab A. Patrascioiu and E. Seiler: Phase Structure of Two-Dimensional Spin Models and Percolation. J. Stat. Phys. {\b 69}, 573 (1992).\par \pard \s4\qj\fi-980\li980\ri- 90\sb240\tx980\tqdec\tx8280\tqdec\tx8370 [2]\tab J.M. Kosterlitz and D.J. Thouless, J. Phys. C {\b 6}, 1181 (1973).\par [3]\tab J. Frohlich and T. Spencer, Comm. Math. Phys. {\b 81}, 527 (1981).\par \pard \s4\qj\fi-980\li980\ri- 90\sb240\tx980\tqdec\tx8280\tqdec\tx8370 [4]\tab P.W. Kasteleyn and C.M. Fortuin, J. Phys. Soc. {\b 26} (Suppl.): 11 (1969), and C.M. Fortuin and P. W. Kasteleyn: Physica {\b 57}, 536 (1972).\par \pard \s4\qj\fi-980\li980\ri- 90\sb240\tx980\tqdec\tx8280\tqdec\tx8370 [ 5]\tab R. H. Swendsen and J. S. Wang.: Phys. Rev. Lett, {\b 56}, 86 (1987).\par \pard \s4\qj\fi-980\li980\ri- 90\sb240\tx980\tqdec\tx8280\tqdec\tx8370 [6] \tab M. Aizenman, J.T. Chayes, L. Chayes, and C.M. Newman, J. Stat. Phys. {\b 50}, 1 (1988).\par [7]\tab M. Aizenman and B. Nachtergaele: Geometric Aspects of Quantum Spin States, to appear in Commun. Math. Phys. (1994).\par \pard \s4\qj\fi-980\li980\ri- 90\sb240\tx980\tqdec\tx8280\tqdec\tx8370 [8]\tab H.J. Braskamp, E.H. Lieb, and J.L. Lebowitz: The statistical mechanics of anharmonic lattices. In {\i Bulletin of the International Statistical Institute} , vol. XLVI, book 1. (Proceedings of the 40th session, Warsaw 1975).\par }