Percolation in a dependent random environment

Random Struct. Algorithms Pub Date : 2000-07-01 DOI:10.1002/1098-2418(200007)16:4%3C333::AID-RSA3%3E3.0.CO;2-C

J. Jonasson, Elchanan Mossel, Y. Peres

{"title":"Percolation in a dependent random environment","authors":"J. Jonasson, Elchanan Mossel, Y. Peres","doi":"10.1002/1098-2418(200007)16:4%3C333::AID-RSA3%3E3.0.CO;2-C","DOIUrl":null,"url":null,"abstract":"Draw planes in R3 that are orthogonal to the x axis, and intersect the x axis at the points of a Poisson process with intensity λ; similarly, draw planes orthogonal to the y and z axes using independent Poisson processes (with the same intensity). Taken together, these planes naturally define a randomly stretched rectangular lattice. Consider bond percolation on this lattice where each edge of length is open with probability e−, and these events are independent given the edge lengths. We show that this model exhibits a phase transition: for large enough λ there is an infinite open cluster a.s., and for small λ all open clusters are finite a.s. We prove this result using the method of paths with exponential intersection tails, which is not applicable in two dimensions. The question whether the analogous process in the plane exhibits a phase transition is open. Disciplines Statistics and Probability This journal article is available at ScholarlyCommons: https://repository.upenn.edu/statistics_papers/434 ! \" # $ $ &% '( )* %+ , -/.1032/45-/.1432/676 .14 8 9;:<0=2/432/4?>?.1676 @A9 B,CEDF2/9HG#@AI @A6 J 032/9;KL@MI76ON 4=PQD1@AI 6 PQR ST./U#V#@A:<0=4=.19W.1X/S(2F4=Y5Z*@M[=I7@A\\]N*4=PWD/@MI76 PWR S^./U -/@AI C=6 2/9;@AK _3@M[=I7C32/I`S(aEbdc/e/e1e f*gEh7i`j<kmlni o#prqtsvuxwyqnzm{7| }~zO \u0085xq qnp\u0085{HApmnAnzxqnw\u0085 m{ q<}|`MqzxO}~zM\u0085{7p|\u0085{7 d\u0085x{ q}y| q< \u0085m{&u/n}yzM|dqO=n}||Az ump { |\u0085|sH}~\u0085}yzM\u0085{7zx|}~ 1|}y}~wqpwynFmpqts5umwqzm{ |Hnp\u0085\u0085mAnAzxqw \u0085&m{qzx ¡q{7| ¢x|}yzm }~zxm{`u/{`zx{7zMH=A}y||\u0085nzump {7||\u0085{7|£QsH}¤*m{|\u0085qn¡{¥}yzA{`zx|\u0085}~¦ § ̈ qn©n{`z(\u0085nA{ m{`p H\u0085x{7|\u0085{aumwqzx{7| zxq<¢mpqnw~wyT{ «Fzm{ q prqzxAw~T|\u0085p{ rx{7Tp{7 qzxn¢mwqp wyq\u0085} {A§ ¬ nzF|}{`p FAzx®u/{`pr Awyq\u0085}ynz ̄nz®m}|,wq<\u0085\u0085} {*sHx{`p{O{7qAr®{7A{O wy{`zmn\u0085®°}| nu/{`z¡sH}¤¡uxp\u0085Axqm}ywy}¤,±A2F3`Aqnzx \u0085m{ |{{` ́A{`zM|3qnp\u0085{}yzx{7uF{7zx{`zMdA}~ ́A{`z \u0085m{H{ n{wy{`zmn\u0085x|7§ μa{¥|m<svxq< \u0085m}|d{`w/{ m}ym}~| q&umFqn|\u0085{H\u0085prqzx|\u0085}~\u0085}ynz ¶· ̧np wyqnp\u0085A{H{`zmA¢mn*O\u0085m{7p\u0085{#}y| qz }~z«Fzm}¤{,AuF{7z wy¢x|{`p¥qm§ |`§yFqnzx$ ̧np|\u0085qwywEaqwyw·AuF{7z wy¢x|{`pr| qp{«xzm}~\u0085{¡qm§ |`§μ {,ump< ́n{ \u0085m}| p{7|\u0085¢mw¤&¢F|}yzm \u0085x{O{ \u0085x1 ̄n ox»<1⁄4 ̧1⁄2M3⁄4O¿3ÀÁ1⁄4 ̧1⁄2 Â\u0085Ã oxÄ<Å/Â`Åx1⁄4;ÀQ»<Æ ÀÁÅx1⁄4ÇÂ Èr3⁄4`ÂrÉ 1⁄4;ÀQÄ<Å 1⁄4Ê»ÀÁÆË3⁄4 =sHm}yr ̄}| zmqumumwy}`qxw~{ }~zÌs $}~{7zx|}ynzF|`§ ̈m{¡ÍA¢x{7|\u0085}ynzsHm{`\u0085m{7p¥\u0085m{qzxqnw~AnA¢x|Hump {7||#}~z \u0085m{ umwqzx{¥{`m}~x}¤r| q¡umxqn|\u0085{¥\u0085prqzx|\u0085}~\u0085}ynz$}y|AuF{7z § Î Ï ÐÑEÒ3ÓOÔÕ Ö Ñ3× Ó Ð ØdÙ<ÚÜÛÞÝàßáHâ ã äå=Ùvæèçé`æ1ê ë3ì&æí îðï ñòê ê=ómï Ù®Ú ë æ1Ú Ú ëòÙvÙîòçÙï ̄æ1é Ù^ô ̧æ1å=ÙôÁÙîöõ,÷ ̧Ú ë\"ß;ê=ómï ï ÷Áå ôÁø îòÙ<ê=Ùí îòÙíFÚ7äOé`æí îòómùûú1æ1é`÷ ̧æ1å·ôÁÙïÌüýþnÿnþ #ìÚ æ /÷ ̧í çvú1æô ̧ñòÙï$÷ ̧í 1â TÙ <æô ̧ô¥Ú ëòÙ tómô ̧ôÁÙ tÚ ÷ ̧ómí üý þ ÿ þ Ú ëòÙ ! \" # $mìæí î% tómí ï ÷ ̧îòÙ<é ÷ ̧í îòÙ<ê=Ùí îòÙíFÚ êEÙ<é& tómô ̧æ1Ú ÷Áómí3ì$ü ' þ ÿ þ( ìO÷Qí Ú ëòÙ é`æí î ómù Ùíxú1÷Áé ómí·ùÙíxÚüý þ ÿ) +* óé ÙOê é`Ù <÷ ̧ï ÙôÁøaõ#Ù*î óÚ ëòÙ-,WómôQôÁónõ,÷Qíòç/. 021 æ Ù*æÌï æùê·ôÁÙó3, Ú ëòÙOú1æ1é`÷ ̧æ1å ôÁÙï¡üý þ ÿ æ3 4 tóé7î ÷ ̧íòçÚ óaÚ ëòÙ÷Áé!5 óm÷ ̧íxÚ&î·÷ ̧ï Ú é`÷Áå ñòÚ ÷Áómí6 087 Ù<Ú Ùæ3 `ë ÙîòçÙóê=Ùíèß:9<;>=?;A@3ï Ù<ÚB' þ ÝC nä¥õ,÷ÁÚ ë ̄ê·é óå·æ1å·÷ ̧ôQ÷ÁÚøý þ æí î <ôÁómï Ùîèß:9D;D= ;Qì/' þ Ý xä óÚ ëòÙ<é õ,÷ ̧ï Ùì ÷ ̧í îòÙ<ê=Ùí îòÙíFÚ ôÁøE,;óé,æô ̧ô=ÙîòçÙï FHG # I # JLK<J M =4N¥Û ÝPORQTSVU WYX Z%[/;]\\ ^ =VW =_=(`39 a N>aEbdcLe f_g(=4Wih·ßDX1äkjl ma&eLnV^]N>^#b3No9 Spb3c =VcIq?9>WrU cIfs=4cIN#üý þ ÿ þ( t#u aVb)ND9va>wx= a y z|{ ï ñ ê þ E} ý ~6 þ j8h·ßDX1ä(~6 y{\u0085 U Wb cLsU W 9<=4cIN= - b3N>^s9 c O Q b3c _b cI_n(U >A=(n ND9<U c]US8 9 a ND9>c n?N= #= a ü(1ÿ U cm+@ N>^ =pn(U W W = a /U c 9>cdq b W 9Dbg4A=VaOüý þD ÿ b W =9 c =/=4c =4cIN ; N>^ =4cb);aV;@6N>^/=VW == `)9vaVN>ab cE9>c wcI9>N=oU3W&9 =VcIN=(x/=4Wrn&U Ab3ND9DU3c n e#a N=4W ¡^9Dn4^9vaxNDWrb cLa&9 =VcIN3S4U W+a&9>f A= Wb c )U f¢ +b £ ; ß 1 ëòÙÌóé`÷ÁÙíFÚ Ùî <ô ̧ñ ï Ú Ù<é*ó3, æ®úÙ<é Ú ÙV¤ ¥^÷ ̧ïÚ ë ÙÌñ í ÷Áómí(ó3,æôQôHóê=ÙíTóé`÷ÁÙíFÚ Ùî(ê·æ1Ú ë·ïOÙùæí æ1Ú ÷Qíòç ,;é ómù¦¥I ~ä L JLK 9 q3=Vc j 2j @-n&U c/a&9<=4Wmb]Wb c )U f =4cLq?9>WU cLf_=VcINka4b3ND9va>S& 9>c N>^ = 9 c =:/=Vc =Vc n =n&U c 9DND9<U c dUrSpN>^ =_N>^ = U3W=4f@a&eLnV^\u0085N>^#b3N _ ý þ Ý S4U W 1â Yb c ã;! 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引用次数: 12

Abstract

Draw planes in R3 that are orthogonal to the x axis, and intersect the x axis at the points of a Poisson process with intensity λ; similarly, draw planes orthogonal to the y and z axes using independent Poisson processes (with the same intensity). Taken together, these planes naturally define a randomly stretched rectangular lattice. Consider bond percolation on this lattice where each edge of length is open with probability e−, and these events are independent given the edge lengths. We show that this model exhibits a phase transition: for large enough λ there is an infinite open cluster a.s., and for small λ all open clusters are finite a.s. We prove this result using the method of paths with exponential intersection tails, which is not applicable in two dimensions. The question whether the analogous process in the plane exhibits a phase transition is open. Disciplines Statistics and Probability This journal article is available at ScholarlyCommons: https://repository.upenn.edu/statistics_papers/434 ! " # $ $ &% '( )* %+ , -/.1032/45-/.1432/676 .14 8 9;:<0=2/432/4?>?.1676 @A9 B,CEDF2/9HG#@AI @A6 J 032/9;KL@MI76ON 4=PQD1@AI 6 PQR ST./U#V#@A:<0=4=.19W.1X/S(2F4=Y5Z*@M[=I7@A\]N*4=PWD/@MI76 PWR S^./U -/@AI C=6 2/9;@AK _3@M[=I7C32/I`S(aEbdc/e/e1e f*gEh7i`j=?;A@3ï Ù<ÚB' þ ÝC nä¥õ,÷ÁÚ ë ̄ê·é óå·æ1å·÷ ̧ôQ÷ÁÚøý þ æí î <ôÁómï Ùîèß:9D;D= ;Qì/' þ Ý xä óÚ ëòÙ<é õ,÷ ̧ï Ùì ÷ ̧í îòÙ<ê=Ùí îòÙíFÚ ôÁøE,;óé,æô ̧ô=ÙîòçÙï FHG # I # JLKaEbdcLe f_g(=4Wih·ßDX1äkjl ma&eLnV^]N>^#b3No9 Spb3c =VcIq?9>WrU cIfs=4cIN#üý þ ÿ þ( t#u aVb)ND9va>wx= a y z|{ ï ñ ê þ E} ý ~6 þ j8h·ßDX1ä(~6 y{ U Wb cLsU W 9<=4cIN= - b3N>^s9 c O Q b3c _b cI_n(U >A=(n ND9c n?N= #= a ü(1ÿ U cm+@ N>^ =pn(U W W = a /U c 9>cdq b W 9Dbg4A=VaOüý þD ÿ b W =9 c =/=4c =4cIN ; N>^ =4cb);aV;@6N>^/=VW == `)9vaVN>ab cE9>c wcI9>N=oU3W&9 =VcIN=(x/=4Wrn&U Ab3ND9DU3c n e#a N=4W ¡^9Dn4^9vaxNDWrb cLa&9 =VcIN3S4U W+a&9>f A= Wb c )U f¢ +b £ ; ß 1 ëòÙÌóé`÷ÁÙíFÚ Ùî <ô ̧ñ ï Ú Ù<é*ó3, æ®úÙ<é Ú ÙV¤ ¥^÷ ̧ïÚ ë ÙÌñ í ÷Áómí(ó3,æôQôHóê=ÙíTóé`÷ÁÙíFÚ Ùî(ê·æ1Ú ë·ïOÙùæí æ1Ú ÷Qíòç ,;é ómù¦¥I ~ä L JLK 9 q3=Vc j 2j @-n&U c/a&9<=4Wmb]Wb c )U f =4cLq?9>WU cLf_=VcINka4b3ND9va>S& 9>c N>^ = 9 c =:/=Vc =Vc n =n&U c 9DND9^ =_N>^ = U3W=4f@a&eLnV^N>^#b3N _ ý þ Ý S4U W 1â Yb c ã;! S_X Z [ b c #" ßr %$ h·ßDX1ä ä(~64@EN>^/=Vc%b);aV;N>^ =VW = =(`39 a N>a]b NDWb c/a 9<=4cIN_U W 9<=4cIN= /=4Wrn&U b)ND9DU c n4 e#a N=VW4; T 6 '&)( } ý ~6 þ Ý * +-, ~6. , ~6 X , Ý/ 01ß1 2$ nä âï ó tómí î·÷ÁÚ ÷Áómí ß 3äë ómô ̧î ïOê·é ónú1÷ ̧îòÙîèÚ ë æ1Ú 2" ßr 4$ h ßDX/ä ä(~64 5 6 ómí·ï ÷ ̧îòÙ<é!X÷ ̧í îòÙ<ê=Ùí îòÙíFÚ87 óm÷ ̧ï ï ómí$ê=óm÷ ̧íFÚdê é ó tÙï ï Ùï õ,÷ ̧Ú ëOê·æ1é`æùÙ<Ú Ù<é 1 ëòÙX:9 ,WómôQî 6 æ1é`Ú Ù;9 ï ÷ ̧æíÜê é ó1î ñL tÚ ó3,Ú ëòÙï ÙOê=óm÷ ̧íFÚê é ó tÙï ï Ùï<ì·õ,÷ÁÚ ë®Ú ëòÙ ï Ú æí î æ1é`îÜê é ó1î ñL tÚ æî 5 æ3 tÙíL tø®é Ùô ̧æ1Ú ÷Áómídì ÷ ̧ï æ^çé`æ1ê·ë ÷ ̧ï ómùóé ê·ë ÷v Ú ó O Q Ú ë æ1Úõ#Ù ̄é Ù4,;Ù<éÚ ó(æïæ=< a NDW =4N n4^ = "Ab3N 7 Ù<Úæí ÙîòçÙ| ̄ó3, ôÁÙíòçÚ ëA@ þ óê=Ùíõ,÷ ̧Ú ëê é óå æ1å·÷ ̧ô ̧÷ÁÚø*ý þ ÝðÙV¤/ê ßB$C@ þ ä7ì/÷ ̧í·îòÙ<ê=Ùí îòÙíFÚ ôÁø,;óé#æô ̧ô·ÙîòçÙï#çm÷ÁúÙíÌÚ ë Ù÷Áé ôÁÙíòçÚ ë·ï4 7 ÷QíL tÙD _ ýþ E ÝF _ @#þ ZûôÁóç ßr G0 ä Ý 1ì 6 óé ómôQô ̧æ1é ø IH5æ1ê ê·ô ̧÷ÁÙïÌÚ ó?Ú ë·÷ ̧ï ê é ó tÙï ï4 KJrí 7 Ù tÚ ÷Áómí [Ìõ¥Ù$ê é ónúÙL7é óê=ómï ÷ÁÚ ÷Áómí][# 1ì3õ,ë ÷ 7ëLÚ óçÙ<Ú ëòÙ<é*õ,÷ÁÚ ë 6 óé ómô ̧ô ̧æ1é`ø IH/ì ÙïM9 Ú æ1å·ô ̧÷Qï ëòÙïÚ ë æ1Ú*Ú ë ÷ ̧ïê=Ù<é( tómô ̧æ1Ú ÷ ̧ómíTê é ó tÙï ïOë æï æ®ê·ë æï Ù;9Ú é`æí ï ÷ÁÚ ÷ ̧ómíT÷ ̧íN OJÚ*÷ ̧ïOíòóÚ /íòóAõ,í(÷A, ï ñL 7ëèæLê·ë æï ÙÚ é7æí ï ÷ÁÚ ÷Áómí(ó# 4 <ñòé7ïT,;óéHX^ÝPHèßÊï Ù<Ù 6 ómí 5 Ù tÚ ñòé Ùm[# IHmä& 7 Ù<ÙRQ·çmñ é Ù| å=ÙôÁóAõl,;óé æ^ê·÷ tÚ ñ é Ùaó3, Ú ë ÷ ̧ï ê é`ó# tÙï ï ÷ ̧íSH^î ÷QùÙí ï ÷Áómí·ï4 1 ë ÷ ̧ï$ê·÷ tÚ ñ é Ùaõæï ê é ó1î ñL tÙî ñ ï`÷ ̧íòç^ï ó3,WÚõ¥æ1é`Ù õ&é`÷ ̧Ú Ú Ùí®åFøT3 7 tómô ̧í ÷v tónú U WV X J (Y i [Z) W L$ > 8$ G \ ^])_! D " # a`? > a ' b`4$? $GZ G c_= /$ $ Z) Dd $ G ÝeHgfIh 1 ë ÙÜé Ùï Úó3,Ú ëòÙ ê·æ1ê=Ù<é÷ ̧ïóé çmæí·÷ji<Ùî æï,WómôQôÁónõ,ï 1 ëòÙ<óé Ùù L÷ ̧ïê é ónúÙîð÷ ̧í 7 Ù tÚ ÷ ̧ómí H?ú/÷QæèÚ ëòÙTùÙ<Ú ë ó/îðó3,*ê æ1Ú ë ï õ,÷ÁÚ ëek` /U c =Vc ND9Db l &c N=VW&a4= n?ND9DU3c \Ib39 a ß m J 1 ä& 1 ë ÷ ̧ï ùÙ<Ú ëòó1î õæï ÷QíxÚ é ó1î ñL tÙî ÷ ̧í 6 ó ¤ æí·îonñòé é`Ù<Ú Ú ph æï ̄æèÚ ó/ómôi,Wóé å=ómñ í î ÷ ̧í ç té`÷ÁÚ ÷ <æô,ê é óå·æ1å·÷Qô ̧÷ÁÚ ÷ÁÙï<ì æí î(î Ù<úÙôÁóê=Ùî,;ñòé`Ú ëòÙ<éåFø\q¥Ùí 5 æùÌ÷ ̧í ÷Çìr7 ÙùæíFÚ ôÁÙÌæí îs7 Ù<é Ùï ÷ ̧í2 [ ìdõ,ëòó®ñ ï ÙîT÷ÁÚÚ ó®ê é óAúÙ Ú é`æí ï`÷ÁÙíL tÙó3,óé`÷ÁÙíFÚ Ùî <ô ̧ñ ï Ú Ù<é`ïo,;óé,÷< ÷< î6 1ê=Ù<é( tómô ̧æ1Ú ÷ ̧ómí6 Jrí 7 Ù tÚ ÷Áómís[¡õ¥Ù&æí æôÁøgi<Ù Ú ëòÙR< ï Ú é Ù<Ú 7ëòÙîô ̧æ1Ú Ú ÷v tÙt>OùÙíFÚ ÷ÁómíòÙîæ1å=ónúÙì/æí î ÷ ̧í 7 Ù tÚ ÷Áómívuõ¥Ù î ÷ ̧ï( <ñ ï ïæíòóÚ ëòÙ<é&æ1ê ê ô ̧÷ <æ1Ú ÷Áómí ó3, Ú ë ÙÚ ëòÙ<óé Ùù Ú óê é óAúÙÚ ëòÙÙV¤ò÷ ̧ï Ú ÙíL tÙó3,æ ê ë æï ÙÚ é`æí ï ÷ ̧Ú ÷Áómí6

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依赖随机环境中的渗透

在R3中绘制与x轴正交的平面，并在泊松过程的点处与x轴相交，强度为λ;同样，使用独立的泊松过程(具有相同的强度)绘制与y轴和z轴正交的平面。这些平面合在一起，自然地定义了一个随机拉伸的矩形晶格。考虑这种晶格上的键渗透，其中每条长度的边都以e−的概率打开，并且这些事件在给定的边长度下是独立的。我们证明了该模型表现出一个相变:当λ足够大时，存在一个无限开簇，而当λ足够小时，所有开簇都是有限开簇。我们使用具有指数相交尾的路径方法证明了这一结果，该方法不适用于二维。平面上的类似过程是否表现出相变的问题是开放的。这篇期刊文章可以在ScholarlyCommons上找到:https://repository.upenn.edu/statistics_papers/434 !# $ $ &% '()* %+， -/.1032/45-/。1432/676 .14 8 9;;1676 @A9 B,CEDF2/9HG#@AI @A6 J 032/9;KL@MI76ON 4=PQD1@AI 6 PQR ST./ u# V#@A:=?;A@3ï ÙaEbdcLe f_g(=4Wih·ßDX1äkjl ma&eLnV^]N>^#b3No9 Spb3c =VcIq?9 > WrU cIfs = 4 cin # uþyþ(t # u真空断路)ND9va > wx = y z | {i n eþe} y ~ 6þj8h·ßDX1a(~ 6y{u Wb cL苏W ^ 9 s9 c O问b3c_b cI_n”(u>= (n ND9c吗?N=# = u(1 yu cm+ @ N > ^ = pn (u W W =/ u c9 > cdq b W 9 dbg4a = VaOuþDyb W =9 c=/ = 4 c= 4 cin;N > ^ = 4 cb);aV;@6N > ^ /大众= == ')9 vavn > acE9 > c wcI9 > N= oU3W&9 = VcIN=(x/ = 4 wrn&uAb3ND9DU3c Ne # N= 4 w¡^9 dn4 ^9 vaxndwrb cLa&9 = VcIN3S4U w + 9 f >= wb c)U¢f + b£;ß1 eoUIoe”÷盟iFU UWU cLf_ = VcINka4b3ND9va > &9 > cN > ^ = 9 c=:/ = Vc= Vc N =N U c9 dnd9 ^ = _N”> ^ = U3W= 4 f@a&eLnV ^N > ^ # b3N _ yþy S4U W 1 c Yb;!S_X Z [b c#”ßr % h·美元ßDX1a一(4 ~ 6@en > ^ / = Vc % b);aV;N > ^ = = =(大众' 39 a N >] NDWb c / a 9 7 U 8 G \ ^]美元)_ !D“# a”?> a ' b ' 4$?广州G c_ = /美元Z) DdG YeHgfIh 1美元e UUe中U我o3,U eoU e·æ1 e = UOuUiFU÷AomioU我æ1 = onuUi /æ我÷̧我7 U你÷Aomivuo¥U我÷̧我(< n我æiooU eoU < eæ1 e e o̧÷<æ1 U÷奥米o3 U e UeoU < oe UU oe e非统UeoU紫外线¤o÷̧U UiL你o3,æe eæ我UU eæ我÷÷̧U Aomi6

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Random Struct. Algorithms

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