Correlation in tree The (ferromagnetic) Ising model

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Brianna Chambers
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1 5/3/00 :\ jh\slf\nots.oc\7 Chaptr 7 Blf propagato corrlato tr Corrlato tr Th (frromagtc) Isg mol Th Isg mol s a graphcal mol or par ws raom Markov fl cosstg of a urct graph wth varabls assocat wth th vrtcs. Th graph, usually a lattc, has a valu of ± call sp assg to ach varabl at a vrtx. Th probablty (Gbbs xx j gs (, j ) x x j masur) of a gv cofgurato of sps s proportoal to whr x ± s th valu assocat wth vrtx. Thus (,,, ) whr s a ormalzato costat. xx j p x x x xx j Th valu of th summato s smply th ffrc th umbr of gs whos vrtcs hav th sam sp mus th umbr of gs whos vrtcs hav oppost sp. Th costat s vw as vrs tmpratur. Hgh tmpratur corrspos to a low valu of. At low tmpratur, hgh, ajact vrtcs hav tcal sps whr as at hgh tmpratur th sps of ajact vrtcs ar ucorrlat. O qusto of trst s gv th abov probablty strbuto what s th corrlato btw two varabls say x a x j. To aswr ths, w wat to trm Prob Prob x. If Prob( x ) pt of th th x as a fucto of ( j ) valu of Prob( x j ), w say th valus ar ucorrlat. Cosr th spcal cas whr th graph G s a tr. I ths cas a phas trasto occurs at 0 l + whr s th gr of th tr. For a suffctly tall tr a for > 0 th probablty that th root has valu + s bou away from ½ a ps o whthr th majorty of lavs hav valu + or -. For < 0 th probablty that th root has valu + s ½ pt of th valus at th lavs of th tr. Cosr a hght o tr of gr. If of th lavs hav sp + a - hav sp -, th th probablty of th root havg sp + s proportoal to ( ) ( ). If th probablty of a laf bg + s p, th th probablty of lavs bg + a - bg - s p ( p )

2 5/3/00 :\ jh\slf\nots.oc\7 Chaptr 7 Blf propagato corrlato tr Thus, th probablty of th root bg + s proportoal to ( ) A p ( p) ( p ) ( p) p + a th probablty of th root bg s proportoal to B p p p p p+ p ( ) ( ) ( ) ( ). Th probablty of th root bg + s A q A+ B p + p p p ( p) C D whrc p + D p + + p+ p. a Now th slop of th probablty of th root bg wth rspct to th probablty of a laf bg ths hght o tr s C q D C D At hgh tmpratur th probablty q of th root of ths hght o tr bg s / pt of p. At low tmpratur q gos from low probablty of blow p/ to hgh probablty of abov p/. How cosr a vry tall tr. If th p s th probablty that a root has valu +, w ca trat th formula for th hght o tr a w s that at low tmpratur th probablty of th root bg o covrgs to som valu. At hgh tmpratur th probablty of th root bg o s ½ pt of p. At th phas trasto, th slop of q at p/ s o.

3 5/3/00 :\ jh\slf\nots.oc\7 Chaptr 7 Blf propagato 3 corrlato tr hgh tmpratur Probablty q of th root bg as a fucto of p low tmpratur Probablty p of a laf bg at phas trasto slop of q quals at p/ Sc th slop of th fucto q(p) at p/ wh th phas trasto occurs s o w q ca solv for th valu of wh th phas trasto occurs. Frst w show that p 0 p. D p + p + p+ p p + + p+ p p p Th q D C p D D C C p p p ( ) p p + p + p + p + p Sttg

4 5/3/00 :\ jh\slf\nots.oc\7 Chaptr 7 Blf propagato 4 corrlato tr ( ) + yls + + l + Shap of q as a fucto of p. To complt th argumt w to show that q s a mootoc fucto of p. To s ths wrt. A s mootocally crasg fucto of p a B s mootocally q + B A crasg. From ths t follows that q s mootocally crasg. Not: Th jot probablty strbuto for th tr s of th form xx j gs (, j ) x x j. Suppos w ow that x has valu wth probablty p. Th w coul f a p for x fuctoφ, call vc, such that φ ( x) ( p ) x+ a p for x multply th jot probablty fucto byφ. Not howvr that th margal probablty of x s ot p. I factor t may b furthr from p aftr multplyg th cotoal probablty fucto by th fuctoφ. Not that t th trato gog from p to q w o ot gt th tru margal probablts at ach lvl sc w gor th ffct of th porto of th tr abov. Howvr, wh w gt to th root w o gt th tru margal for th root. To gt th tru margal s for th tror os w to s mssags ow from th root. Exrcss Exrcs : For a tr wth gr o, a cha of vrtcs, s thr a phas trasto? Work out mathmatcal what happs.

5 5/3/00 :\ jh\slf\nots.oc\7 Chaptr 7 Blf propagato 5 corrlato tr Soluto: Lt th vrtcs of th cha b x, x,, x. Th jot probablty strbuto xx s p( x, x,, x ) +. If w multply th jot probablty strbuto by th xx + vc fucto φ ( x ) ( p ) x + w gt p ( x, x,, x ) φ ( x ) summg ovr x w x x x x + x x x p + p s vc for x x p + p + xx + xx + gt (,,, ) φ ( ) p x x x x p p Th fucto probablty fucto w gt blow. a x. If w ormalz t so t s a.. For varous valus of w gt th fgur tmp0 0 tmp Exrcs : Cosr a by lattc. By xprmt trm th valu of at whch a phas trasto occurs for th corrlato of th sp of th org vrtx as a fucto of th sp o th bouary of th squar. Exrcs 3:

Math Tricks. Basic Probability. x k. (Combination - number of ways to group r of n objects, order not important) (a is constant, 0 < r < 1)

Math Tricks. Basic Probability. x k. (Combination - number of ways to group r of n objects, order not important) (a is constant, 0 < r < 1) Math Trcks r! Combato - umbr o was to group r o objcts, ordr ot mportat r! r! ar 0 a r a s costat, 0 < r < k k! k 0 EX E[XX-] + EX Basc Probablt 0 or d Pr[X > ] - Pr[X ] Pr[ X ] Pr[X ] - Pr[X ] Proprts