Large Deviation Principle for One Dimensional Vector Spin Models with Kac Potentials
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Large deviation principle for one dimensional vector spin models with Kac potentials Paolo Butta Pierre Picco. N ˚ 3193 19 Juin 1997. THEME 4. ISSN 0249-6399. apport de recherche.
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(72) TVgi`tadXmfmNenvfcohVm^_el^dgih¿´ i6;´:$TbXvfX!enTVXvfX!^_myhbgmfk:gihQe}cohbXgi`Vm6W+coihVX.¶ ef^_©@c el^dgih¿´hVgVvfXc p g$h2go ÕenTbX!^_hLefXvlhc amlqEWYW+XenvqV`be6enTVX p Xrc@q¯go eã¦gk:go^_hQenm rgivlvfXadc el^dgih ¡`VhVref^_gihVm$^dm6XstkgohVXhQef^dc ac e6Tb^doT*enXWYkXvnc ef`VvfX!coh p gihbaqkgoaqEhVgiW3^dc a c eadg enXW+k:Xvfc en`bvfXo¤'S
(73) TVXXsE^_mlefXhVrX!go enTt^dmenvfcohVm^_el^dgih¦com$kVvfg¸iX p v^dogivfgi`bmlaq Qq b¤Üv iTba^_r}T4 -SN¤:]EkXhVrXv´: Ý Ú<;¤ ãhenTVvlXXgivjW+govfX p ^_W+XhVml^_gihVm6^_ej
(74) c m6kVvlg¸iX p Lq¯b¤8Üv oTba_^_r}Tµ´¥y¤:]E^_WYgihµ´ -SN¤ ]tkXhVrXv´
(75) Ýo;´&efTc eHefTVX3 ¡XvfvlgiW+coihVXef^_rYkba£c hcovHvfg e}c efgivcoh p radcomlml^_r@c a'6X^dmlXhL:Xvf W+g p Xadm X.sbTt^db^emfk:gihQe}cohbXgi`Vm W+coohVXel^d©c ef^_gih coh p mqtWYWYXefvlq VvlX@c Q^_hV=c e ml` 3¶ r^_XhQefaq°adg-efXWYkXvnc ef`VvfX ¤ , rgiWYkba_XenX p Xmlrv^dkbel^dgoh go cBa_a²X.stefvfXW cBaP´efvncohbmladc ef^_gih ^dhQ¸ covl^dcohQeNX³L`b^_a^_Vvl^_`VW mle}cBenXmj
(76) c m p gihbXQq2b¤:Üv oTba_^_r}T y¤uÕmlefXv´ ÝBZ"P´8$TVXvfX ^_e¦com6c admlgYkVvfg¸iX p efTc e$enTVXml`Vvl JcorXHenXhbml^_gih¸ coht^dmlTVXm¤ ÜvlgiW $TcBe p ^_mfr`VmfmlX p c g¸iXY^e!mlXXW+m³L`b^enX+hc en`bvnc aFefgme}cove!coh c hc aqtm^dmgo enTVX¢corN¸iXvfm^dgih¯go efTVXradcomfm^drc a 6z§givWYgivlXoXhVXvnc aa_qra£c mfml^_r@cBa " 'WYg p Xadm´ com p gihVX3 ¡giv6enTVX0ãml^_hV ªã¢cor3gihVX ¤0*givlXg¸iXv´coXhVXvnc aFkt^dren`VvlX+goh°efTVXYmlenc ef^_mlel^drc a W+Xr}Tcohb^_rm$go &ht^_efXHvncohVoX " WYg p Xa_m
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(80) a^dW3^e'g efTVX
(81) enTVXvfWYg p qEhcoW3^drjk:goenXhLel^£cBadmF ¡giv " W+g p Xadm´y¤ES
(82) TVgiWYkVmlgih ¤t]E^_a¸iXv´ ZoZ"P´LkVvlg¸iX p enTVX6¨)X:g·^en©.ªxuXhVvlgimfXjenTVXgivfXW ¡giv
(83) enTVXkVvlXmfml`VvfX ¤ ,6hcohcBadgiogi`VmvfXmf`baej ¡givenTVXr@c hVgihb^_r@c a& ¡vlXXXhVXvfoq ^_m$W3^dmlml^_hVV¤ S
(84) TVX$kVvfgEgo :go :mf`Vr}TcNvfXmf`bae²^dm²mlencoh p c v p ^_ gihVXTVcom²igtg p adcovfiX p X¸L^dc ef^_gih+Xmlel^dW+c efXm ¡givFenTVXjrgivlvfXmfk:gih p ^_hV!^_h p XkXh p XhLeH¬J^;¤ X ¤ÕhVgih+^dhQenXvncoref^_hV ® WYg p XaP¤²S
(85) Tt^dm²^dm rgohLenc ^චhVX p ^dh gi`Vv S
(86) TVXgivfXW Ýt¤ïÝ3¬¡ ¡giv # ® enTc e²'X p ^ p hVgoe²h p ^dhYenTbX$a^enXvfc en`bvfX6coh p ¦X:Xa^dX¸iX!^_hQenXvfXmef^_hV ^_h^_efmjg$hµ¤ S
(87) TbXyvfmemlefXk ^_henTVXymlqEmlefXW+c ef^_rmlef` p q go ¢corNWYg p Xa_m¦^dm'eng\rgihVml^ p Xv
(88) enTVXNgohVX p ^dWYXhVml^_gihc aVr@comlXo´o$TVXvfX
(89) 'Xr@cohXstkXreFefgNTc@¸iX$cyvfc enTbXvÕrgiWYkbadXenX p Xmfrvl^_kbef^_gihYgo eãqtkt^drc a)rgohbi`Vvfc ef^_gihVm¤ S
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(92) ^_efT*c h " 㪠mlqEWYW+Xenvq enTVX*vfXml`baenmgo 0à<P¤ ãh enTt^dm kcok:Xv 'XrgihVml^ p XvenTVX¯¢cor¸oXvlml^_gih go NenTbX* ¡Xvlvfg ¶ W c ihVXel^dr ra£c mfml^_r@cBa 6z coh p jX^_mfXhE:XvlWYg p Xa_m¬ # ® ¤ X kVvfg¸oX*c¯adcovliX p X¸L^dc ef^_gih kVv^dhVr^dktadXy ¡gov'enTbXHcomfmlgtr^£cBenX p ¬¡^dhthb^enXy¸ogoad`bW+X ®
(93) ¹ ^dVbm
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(99) TbX ckVv^dgiv^HkVvlgicob^a^_eãq p ^_mlefvl^_V`bel^dgih!º ¡givenTVX" m ^_m*c mfmf`bW+X p eng:X enTVX2hVgovfW+c a^d©X p mf`Vv JcorX°WYX@coml`VvfX°gih # %$'&.´jenTVX2`bhb^_emlkVTVXvfX°^dh( iÙ) +*,! ./ # %$'& //$'&0* /1 / -32 *,P¤ X p XhVgoefXyQ4q enTVX6mfkb^_h rgihti`Vvfc ef^_gi6h 5798';:< =Hcoh p ´ ¡giv¦c hLq+vlXo^_gih?>=go @N´AB ^dm enTVX6vfXmlenv^dref^_gih go enCg >6¤FÜ^_hc aa_qY'Xyrc a_Ea D6F´ D@BefTVX mfkcorXgo ÕcBa_a)efTVXmfkb^_h*rgihbo`Vvnc el^dgohVm$gih yF´ > vlXmfk:Xref^¸iXa_qo¤ S
(100) TbX+XhVXvloq°go enTVXYrgihbi`bvnc el^dgi"h ^dhchb^enXYvfXo^_giG h > g H c h p ^_efTM ¡vlXX gi`bh p covq rgih p ^_el^dgohVm^dm 2 IKJ*L B # - ( - X Y [Z\]P ¬ ® M W J*U ; ONUPRT QSP B V. $TVXvlX ^Z p XhVgoenXm
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(103) ^_h? _ o¤$WJ U ´;:<y´ 6: 5`2 2;´ ^dmc=¢c rMk:goefXhQef^dc a;Ù aJ*U 9# / / ^_efT cb ´db!e 5b´yhVgih hVXQcBef^¸iXcoh p rgihQef^_hL`Vgi`Vm!^dh + 5`22P´Õmenv^drefaq k:giml^ef^¸iX ^dh !5`2 coh p ^_efT mf`Vkbkgive^_h + 5`2 2;¤ X comfml`VWYX3c a_mfg efTc e7 TVcomy:gi`Vh p X pp Xvl^¸ c ef^¸iX^_h !5 `2 coh p enTcBe6^e6^_myhbgivfW+c a^d©X p mfgYenTc e f,g *]h / h / i # 2;j ¬PÝ ® S
(104) TVXeãqEkb^_r@cBa6r}Tbgo^drX^_mc b l # kYm n N &po ´ enTVXr}TcovfcorefXv^dmef^_r* ¡`VhVref^_gih=go 0+ 5 `2 2;¤ Ü8gov enXr}Tbhb^drc argihQ¸iXht^dXhVrX ´
(105) c emfgiWYXk:go^_hLe^dh efTVXkc kXv´Õ'X^_aa$comlmf`VWYX go enTb^_m kcovlel^dr`badcov
(106) ¡givfW¤ S
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(112) :gi`Vh p covlq!rgih p ^_el^dgohVm^_menTVX¦kVvfgiVcob^a_^eãq p ^_mlefvl^_V`bel^dgih3gohCD@B p XhVX p com 2 - xIKJ*U B 2 + *F ¬JZ ® B N J U *,B # B N J % '*) +6 uQSB $TVXvl X B N J ^_m6enTbX\kcovef^ef^_gih¯ ¡`VhVref^_gihµ´:^P¤ Xo¤jenTVX\hbgivfW+c a^d©c ef^_gih¯ Jc rengovjefTc eNW+c Xm enTVXv¤ Tµ¤ m¤²go ¦¬;Z ® c3kVvlgicot^_a^_eãqWYXcomf`VvlXo¤ ,6hc a_giigi`bmlaq'X p XVhVXenTVX ¹ ^_VVm\WYXcomf`VvlX^dh > ^_efTgo`Vh p covq rgih p ^ef^_gih Bsr$comenTbXkbvfgic b^_a^eãq 2 - ICJ L B / B r 2 U *,! j ¬P ® B N J U *,B / B r # BN
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(122) r J ^dmy'Xaa p XhVX p ´µ^_e p gtXmNhVgoe p Xk:Xh p gihefTVX\:gi`Vh p covlq*rgih p ^_el^dgoh BsrHcoh p ^_eyTcom engYXH^_hQenXvlkVvfXenX p comenTVX ¹ ^dbVm ¡vfXXXhVXvloqgo enTVXW+corvlgimfrgikb^_r!mqtmenXW´:mfXX!X ¤ b¤ Ýi P¤ S
(123) TbXH¨)X:g·^en©.ªxuXhVvlgimfXHenTbXgivlXW vfXadc efXm¦efTVX6a^dW3^e
(124) ¡vfXXNXhbXvf qcom 3 5eng!efTVX rgivlvfXmlkgoh p ^_hVgihVXkVvlX p ^_refX p LqenTVX
(125) ¸ coh p Xv cicBadm$¬JW+Xcoh3Xa p8® efTVXgivq3go :kVTc mfX envnc hVml^ef^_gihµ´rgiWYkVvfXTVXhVm^_¸oXgo enTbX *cBsE¦Xa_avf`ba_Xo¤ãhgi`VvrgihQenXsEeÙ Ç J* J ÑÊ!ÆBÎÌÉ Ø
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(128) TbXXhQenvfgokLq ¡`VhVrel^dgoh + J ' ¡giv
(129) gi`VvWYg p Xaµr@coh:XHrgiWYkV`befX p com ¡goaadg$m¤ X ^dhQenvlg p `VrXHenTbXWYgiWYXhLe6iXhVXvfc ef^_hVY ¡`VhVrel^dgoh f - + *2 & 35456 7 0:6 ¬l@Û ® / 1 0 ; coh p ¦X p XhVX g + J ; - Q 8) 5:0 Z J - / 1 01R = J :< j ¬lo ® 4 9 ¥'q2mlqEWYWYXefvlq / 01 6^_mc ¡`Vhbref^_gihMgo 7;/ 0 /8gihbaq°coh p ´Qq°`Vm^dhb*k:goadcovHrgEgiv p ^dhcBenXm´ gihVXX@com^_aq rgiWYkV`benXm ->=@? /A0 n / U$'(&6;/ 0 C/ B& D /;0 / ^^ ## bt´´ ¬lÝ ® / 1 0 # / < ;/ 0 / ^ J. .
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(133) TVXh 0 HZ J - / 10 ^ / J / !2i´ + J # = ¬lZ ® ^ / / I ! i 2 ´ J * ^_efT 0 # 0 " J # 1/ J / J ´ / J / 2i´Õ$TbXvfX # / J / H^dmefTVX2¬J`Vhb^_³E`bX ® kgim^_el^_¸oX!hL`VW!Xvj ¡gov
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(135) cohQq J : ^ $'&.´ + J Õ^_m¦hb^enX6V`be¦efTVX mf`VkVvlXW!`VW ^dm$hVg eyc r}Tb^dX¸iX p ^d4h i¤ S
(136) TbX6vlXmf`ta_e¦mle}cBenX p ^dhS
(137) TVXgovfXW ÝE¤_ p gEXm hVgoe p Xk:Xh p gihYenTVX$a£cBefef^_rX p ^_WYXhVm^dgih ¬J'X*Tc@¸oX¯rgohVml^ p XvlX p efTVXr@comlX * # 2engMml^_WYkba^_ Éq=hVgoenc ef^_gih ® coh p ^e X.stTb^dt^_efmc kVTcomlX\envnc hVml^ef^_gih°go ²WYX@coh2Xa p eãqEk:Xo¤ ãh2 JcoreN ¡givHc hL(q 23enTVXvfX\^_mHc kgoml^ef^¸iX mfgoa_`bef^_gih J go enTVXX³L`cBef^_gih ¬l@Ú ® J # +< J coh p cohQq\W+ coohVXel^d©c ef^_gih J gih\enTbXmfkVTVXvfXgo :vfc p ^_`Vm J W3^_hb^dW3^_©Xm²efTVX¦ ¡vlXXXhbXvf q ¡`VhVrel^dgoh J ¤'S
(138) TVXH^dhQ¸iXvfmlX ¬;WYX@c hXa p® rvl^ef^_r@c a&efXWYkXvnc ef`VvfXH^_mjefTL`Vm i # 2 ¬JenTt^dm¸ c a_`VXrgiWYXm$ ¡vfgoW gi`VvhbgivfW+c a^d©c ef^_gihr}TVgo^_rX ¬ Ý ® ¡gov
(139) enTVXH^_hLefXvfcorel^dgih ® ¤ S&gjgi`Vv LhVgadX p iX'efTVXvlXF^dmhVgjkVvfgEgo tgo S
(140) TVXgivfXW Ýt¤²^_h!efTVXFa_^enXvnc ef`VvfX ¤ÕS
(141) TVgoW+kE¶ mfgihXeHc a;´ ZiZ6P´:Qq*'giv Q^_hV^enT¯efTVXivfcoh p rcohVgiht^drc akVcovlel^_el^dgih* ¡`VhVrel^dgohµ´¿kVvlg¸iX enTVXycohc a_giigi`bm'me}c efXWYXhQe ¡giv²enTVXjenTVXvfWYg p qEhcoW3^drykVvfXmfmf`bvfX J* 01 ´ELq mfTbg·^_hV . . . . . . . . .
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