PHASE TRANSITION IN THE ISING MODEL

Transcription

1 PHASE TRANSITION IN THE ISING MODEL 1. The Isng Model The Isng model s a crude but extremely mportant mathematcal model of a ferromagnetc metal ntroduced by Isng about 70 years ago. Its mportance stems from the fact that t s the one of the smplest mathematcal models to exhbt a phase transton: at hgh temperature, there s a unque equlbrum state for the system, but at temperatures below a certan crtcal temperature, there are several dstnct equlbrum states. Ths corresponds to the physcal phenomenon of spontaneous magnetzaton: If unmagnetzed ron s cooled to a very low temperature, t wll magnetze; and f a magnet s heated to a suffcently hgh temperature, t wll demagnetze. The latter may be verfed easly by experment, usng only a floppy dsk and a household stove Gbbs States. Let X be a fnte set and H : X R a functon, called the Hamltonan of the system. In physcal applcatons H(x) represents the energy of the system when t s n state x. The Gbbs state µ = µ β for β = 1/(kT ), where k = Boltzman s constant and T = temperature, s the probablty measure on X defned by (1.1) (1.2) µ β (x) = e βh(x) /Z(β), where Z(β) = x X e βh(x). The normalzng constant Z(β) s called the partton functon. The famly {µ β } β>0 s a oneparamter exponental famly of probablty measures on X, wth β playng the role of the natural parameter, H(x) the suffcent statstc, and log Z(β) the role of the ψ functon. Observe that, snce the sum n (1.2) s fnte, the partton functon s well-defned and (real-)analytc n the doman β > The Isng Hamltonan. In condensed-matter physcs, feld theory, and varous other parts of statstcal physcs, the state space X s often of the form (1.3) X = A V where V s a set of stes (whch we wll also call vertces) and A s a fnte set. Elements of V usually represent spatal locatons, and elements of A may represent atomc elements (n models of alloys), presence (+1) or absence (0) of partcles (n models of gases), spns (n models of magnetsm and n quantum feld theory), and so on. In the Isng model, A s the two-element set A = {±1}, and V s the set of vertces of a graph G; the most nterestng case, from the standpont of the physcst, s that where V s a subset of the d-dmensonal nteger lattce Z d. The Isng Hamltonan s defned as follows: for any 1

2 2 PHASE TRANSITION IN THE ISING MODEL confguraton x X := { 1, +1} V (1.4) H(x) = J,j V : j Here j means that vertces, j are nearest neghbors, that s, there s an edge of the graph G connectng and j; each edge counts only once n the sum. The constant J s called the couplng constant: f J < 0 the model s called ferromagnetc, and f J > 0 t s ant-ferromagnetc. Unless otherwse specfed, t s henceforth assumed that J < 0. Observe that n ths case, the system prefers confguratons n whch neghborng spns are algned, as these have lower energy. The degree to whch ths s true depends, of course, on the nverse temperature β for larger values of β the preference for low-energy states s stronger The Thermodynamc Lmt. The sort of magnet that you mght carry around n your pocket would have on the order of ron atoms. The exact number sn t mportant what s mportant s that the number s bg (even to a computer scentst). Thus, t makes sense to study the behavor of the Isng model when the vertex set V s large, and n fact to nqure about the lmtng behavor as V becomes nfnte. There are two ways to go about ths: (1) Look at the lmtng behavor of the Gbbs states µ β for fnte V as V becomes larger; or (2) Try to extend the defnton of Gbbs state to confguraton spaces on nfnte graphs. Program (2) s the theory of DLR states (for Dobrushn, Lanford, and Ruelle); t requres more mathematcal machnery than I wsh to nvest n now, and so I shall only dscuss program (1). Let G = (V, E) be a countably nfnte locally fnte graph (such as the nteger lattces Z d ; locally fnte means that for each vertex there are only fntely many edges ncdent to ). Let X = {±1} V be the space of confguratons on the vertex set V, and For any vertex V, let X : X {±1} be the th coordnate evaluaton map X (x) = x. The Borel σ algebra B on the space X s the smallest σ algebra that makes all of the random varables X measurable; equvalently, B s the σ algebra generated by the open sets of the product topology on X. For each fnte subset Λ V defne (1.5) H Λ (x) = J x x j ; Λ,j V j here the sum s over all edges of the graph G wth at least one endpont n Λ. For each z X and each fnte subset Λ V defne the Gbbs state µ z Λ = µz Λ,β on Λ wth external boundary condton z to be the dscrete Borel probablty measure on X determned by the rule (1.6) x x j µ z Λ(x) = exp{ βh Λ (x)}/z Λ,z (β) f x Λ c = z Λ c = 0 otherwse where Z Λ,z (β) s the approprate normalzng constant. Observe that µ z Λ s concentrated on a fnte set of confguratons, namely, those that agree wth z outsde of Λ. Also, f z, w X are two confguratons that agree outsde Λ thenµ z Λ = µw Λ.

3 PHASE TRANSITION IN THE ISING MODEL 3 The two most nterestng boundary condtons (at least for now) are z + 1 and z 1: we shall denote by µ + Λ and µ Λ the Gbbs states wth these external boundary condtons. It s mportant to note that these two Gbbs states are mrror mages, n the followng sense: f ρ : X X s the mappng that flps every spn, that s, ρ(x) = x, then for every fnte Λ V, (1.7) µ Λ = µ+ Λ ρ Phase Transton n Dmenson 2. The observable physcal phenomenon of spontaneous magnetzaton (and demagnetzaton) has a mathematcal analogue n the Isng model n dmensons two and hgher, a fact dscovered by R. Peerls n the 1930s, some years after Isng ntroduced hs model. 1 Let G = (Z 2, E) be the standard two-dmensonal lattce (the edges e E connect ponts of Z 2 that dffer by (1, 0) or (0, 1)), and let Λ n be the square of sde 2n + 1 centered at the orgn o. Denote by µ + n and µ n the Gbbs states wth external boundary condtons z + and z on the square Λ n. Theorem 1. There exsts β c satsfyng 0 < β c < such that the followng s true: For each vertex Z 2 (1.8) lm n µ+ n {X = +1} > 1/2 f β > β c (1.9) lm n µ+ n {X = +1} = 1/2 f β β c. The fact that β c < s, n essence, Peerls dscovery. I do not know who frst proved that β c > 0, but I consder ths just as mportant. Theorem 2. For each β, as n, the measures µ + n converge n dstrbuton to a probablty measure µ +, and the measures µ n converge n dstrbuton to a probablty measure µ. These lmtng measures are translaton nvarant, and µ s stochastcally domnated by µ +. It follows that the lmts n Theorem 1 equal µ + {X = +1}. Snce µ + s translatonnvarant, t also follows that µ + {X = +1} = µ + {X o = +1} for all vertces. By the symmetry relaton (1.7), µ {X o = 1} = µ {X o = +1}. Corollary 3. (1.10) (1.11) µ + µ f β > β c ; µ + = µ f β β c. Proof. The measure µ + stochastcally domnates the measure µ for any value of β. By the extenson of Strassen s monotone couplng theorem to nfnte confguraton spaces (HW set 2), there exst, on some probablty space, random confguratons Y +, Y wth dstrbutons µ +, µ, respectvely, such that Y Y + almost surely. But by Theorem 2 and relaton (1.9), f β β c then for every vertex, P {Y + = +1} = 1/2 = P {Y = +1}, 1 Isng s Ph. D. thess supervsor Lenz had suggested to Isng that a phase transton mght exst n the Isng model; Isng was able to prove that there s no phase transton n one dmenson, but proved nothng about the behavor n hgher dmensons.

4 4 PHASE TRANSITION IN THE ISING MODEL and so t must be that Y + = Y almost surely. In secton 4 below, we shall prove that the lmtng relaton (1.8) holds for suffcently large β (the low-temperature lmt), by Peerls orgnal and elegant argument. In secton 5 we shall prove that (1.9) holds for suffcently small β by relatng the constructon of random felds wth the dstrbutons µ ± Λ to ste percolaton. 2. The Markov Property for Gbbs States For any confguraton x X = {±1} V and any subset Λ V, denote by x Λ the restrcton of the confguraton x to the set Λ, that s, x Λ = (x ) Λ. Smlarly, denote by X Λ the vector (X ) Λ of coordnate evaluaton mappngs for stes n Λ. Proposton 4. Let Λ and Σ be dsjont, fnte subsets of V. For any confguraton x X the followng s true: (2.1) µ x Λ Σ(X Λ = x Λ X Σ = x Σ ) = µ x Λ(X Λ = x Λ ). Thus, µ x Λ s the condtonal dstrbuton under µx Λ Σ of X Λ gven that X Σ = x Σ. It s ths mutual consstency property that allows the possblty of extendng the defnton of a Gbbs state to nfnte confguraton spaces. For any subset Λ V, defne the outer boundary Λ of Λ to be the set of all vertces at dstance one from Λ. Proposton 5. For any confguraton x and any fnte subset Λ V, the probablty µ x Λ (X Λ = x Λ ) depends only on x Λ Λ. The proofs of Propostons 4 and 5 are left as exercses. Proposton 5 s qute easy. Proposton 4 s slghtly more subtle: You wll fnd t easest to begn by showng that t suffces to consder the case where Σ s a sngleton. Corollary 6. Let Σ Λ V be fnte subsets of V such that Σ Λ, and let x, y be confguratons such that x Λ = y Λ. Then (2.2) µ x Λ(X Σ = x Σ X Λ Σ = x Λ Σ ) = µ y Λ (X Σ = x Σ X Λ Σ = x Λ Σ ). Let s make t a hat trck ths one s also an exercse. 3. Stochastc Monotoncty Results Proposton 7. For any fnte subset Λ V and any two confguratons z, y X such that z y, the probablty measure µ z Λ s stochastcally domnated by µy Λ. Ths wll be proved by appeal to a theorem of Holley. Let Λ be a fnte set and X Λ = { 1, +1} Λ be the space of spn confguratons on Λ. For any ste Λ, defne the spn operators σ +, σ : X Λ X Λ by { (σ ± (x)) x j, f j j = ±1, f j =.

5 PHASE TRANSITION IN THE ISING MODEL 5 Theorem 8. Let Λ be a fnte set. Let µ and ν be probablty dstrbutons on { 1, +} Λ such that µ(x) > 0 and ν(x) > 0 for each x X. If for every ste and every par x, y of confguratons such that x y t s the case that (3.1) then µ s stochastcally domnated by ν. µ(σ + (x)) µ(σ (x)) ν(σ+ (y)) ν(σ (y)), Proof. The strategy s to buld a dscrete-tme Markov chan (X n, Y n ) n 0 on the space X Λ X Λ of confguraton pars n such a way that (a) X n Y n for all n; (b) (X n ) n 0 s an aperodc rreducble Markov chan wth statonary dstrbuton µ; and (c) (Y n ) n 0 s an aperodc rreducble Markov chan wth statonary dstrbuton ν. The reader should convnce hmself/herself that ths wll mply µ ν. To buld the (X n, Y n ) chan, t s enough to specfy the transton rules and then check that (a) (c) hold. The transton rule goes lke ths: Assume that the current state s (x, y), where x y. Choose I Λ at random, unformly on Λ; the confguratons x, y wll only be modfed at ste I, f at all. Gven that I =, update the spns (x, y ) at ste wth probabltes as follows: ( ) (++) ( ) ( ) ( +) (++) wth probablty ɛ wth probablty 1 ɛ wth probablty ɛ ( +) ( ) wth probablty ɛ ν(σ y) ν(σ + y) ( +) ( +) wth probablty 1 ɛ ɛ ν(σ y) ν(σ + y) (++) ( ) wth probablty ɛ ν(σ y) ν(σ + y) (++) ( +) wth probablty ɛ µ(σ x) µ(σ + x) ɛν(σ y) ν(σ + y) (++) (++) wth probablty 1 ɛ µ(σ x) µ(σ + x), where ɛ > 0 s chosen suffcently small that all of the probabltes are postve and less than 1. Note that the hypothess (3.1) guarantees that µ(σ x) µ(σ + x) ν(σ y) ν(σ + 0. y) It s clear that the transton probabltes specfed above are such that the (X n, Y n ) chan wll only vst states (x, y) such that x y. Therefore, to complete the proof, t suffces to show that the margnal processes X n and Y n are aperodc rreducble Markov chans wth

6 6 PHASE TRANSITION IN THE ISING MODEL statonary dstrbutons µ and ν, respectvely. Consder X n. Gven that X n = x, Y n = y, I n = and any specfcaton of the past, the condtonal dstrbuton of X n+1 satsfes X n+1 = σ + (x) wth probablty ɛ f x = σ (x), X n+1 = σ (x) wth probablty ɛµ(σ (x)) µ(σ + (x)) f x = σ + (x). Snce these condtonal probabltes don t depend on y or the past, t follows that the process (X n ) n 0 s Markov and aperodc rredcble. Moreover, the transton probabltes p(, ) for ths Markov chan satsfy the detaled balance equatons µ(σ + (x)) p(σ+ (x), σ (x)) = µ(σ+ (x)) ɛ µ(σ (x)) Λ µ(σ + (x))) = µ(σ (x)) ɛ Λ = µ(σ (x)) p(σ (x), σ+ (x)), where Λ denotes the cardnalty of Λ. Thus µ s the statonary dstrbuton of the Markov chan (X n ) n 0. A smlar calculaton shows that (Y n ) n 0 s Markov, and has statonary dstrbuton ν. Proof of Proposton 7. Under µ x Λ, the confguraton X V Λ = x V Λ almost surely. Thus, t suffces to show that f z y then the dstrbuton of X Λ under µ z Λ s stochastcally domnated by ts dstrbuton under µ y Λ. For ths, we use the suffcent condton provded by Theorem 8. Let z, ỹ be confguratons that concde wth z, y, respectvely, outsde Λ, and such that z ỹ. Then µ z Λ (σ+ ( z)) µ z = exp{ 2βJ z j } Λ (σ ( z)) j:j exp{ 2βJ ỹ j } = µy Λ (σ+ (ỹ)) µ y Λ (σ (ỹ)). j:j Consequently, by Theorem 8, µ z Λ s stochastcally domnated by µy Λ. Corollary 9. For each nteger n 1, (3.2) µ + n µ + n+1, (3.3) µ n µ n+1, and (3.4) µ n µ + n. Proof. Set Λ = Λ n, Λ = Λ n+1, and Σ = Λ n+1 Λ n. By Proposton 4, µ + n s the condtonal dstrbuton, under µ + n+1, of X Λ gven the event X Σ = z + Σ (recall that z+ = all pluses). By Proposton 7, ths domnates the condtonal dstrbuton of X Λ under any other event X Σ = x Σ. Therefore, µ + n stochastcally domnates the uncondtonal dstrbuton of X Λ

7 PHASE TRANSITION IN THE ISING MODEL 7 under µ + n+1 (Exercse: Explan why.) Ths proves that µ+ n µ + n+1. The second nequalty s smlar, and the thrd follows drectly from Proposton 7. Proof of Theorem 2. Snce µ + n µ + n+1, on some probablty space there exst X -valued random varables X (n) wth margnal dstrbutons µ + n such that X (n) X (n+1) for all n. (Ths s a consequence of Strassen s Monotone Couplng Theorem for measures on X.) Snce the components X (n) are ±, t follows that the confguratons X (n) converge coordnatewse, monotoncally, to a lmt X. The dstrbuton µ of X must be the weak lmt of µ + n. A smlar argument shows that the measures µ n converge to a lmt µ, and that µ = µ + ρ. That µ + µ follows from the fact that µ + n µ n. The transton nvarance of the measures µ +, µ can alos be proved by a stochastc comparson argument (the detals are omtted for now). 4. Peerls Contour Argument Peerls argument s based on the observaton that the Isng Hamltonan H Λ defned by (1.5) depends only on the number of +/ nearest neghbor pars n the confguraton: (4.1) (4.2) H Λ (x) = 2JL Λ (x) + C Λ L Λ (x) = δ(x, x j ), Λ,j V j where wth δ(, ) beng the Kronecker delta functon. Evaluaton of L Λ (x) can be accomplshed by parttonng the vertces of Λ Λ nto (maxmal) connected clusters of + spns and spns n x, as n Fgure ; L Λ (x) s the number of edges n Λ Λ connectng + clusters to clusters. For two-dmensonal graphs, L Λ (x) may be evaluated by drawng boundary contours around the connected clusters, as shown n the followng lemma. For the remander of ths secton, assume that G s the standard two-dmensonal nteger lattce Z 2. Lemma 10. For each vertex Λ Λ, let K = K (x) be the maxmal connected set of vertces j such that stes and j have the same spn n confguraton x. Then for any two vertces, j such that K K j there s a smple closed curve γ = γ,j, called a boundary contour (possbly empty) separatng K from K j. The curve γ s a fnte unon of horzontal and vertcal segments n the dual lattce. Each such segment bsects an edge connectng a vertex n K to a vertex n K j. Proof. The curve γ may be constructed usng a maze-walkng algorthm. Begn by choosng an edge e connectng K to K j (f there s one), and let the frst segment γ 1 of γ be a perpendcular bsector of e. Defne (orented) segments γ n, for n = 2, 3,..., nductvely, n such a way that f one traverses the segment γ n then a vertex of K s on the rght and a vertex of K j s on the left. Eventually the sequence γ n wll enter a cycle. Ths cycle must nclude all of the segments γ n because otherwse the rght/left rule would be volated somewhere. Therefore, the cycle determnes a closed curve. Ths closed curve must completely separate the regons K and K j, because otherwse one of them could not be connected. Consult your local topologst for further detals.

8 8 PHASE TRANSITION IN THE ISING MODEL Corollary 11. L Λ (x) =,j γ,j. Assume now that the regon Λ s a square. Fx a vertex Λ, and let x X be a confguraton such that x Λ c = z + Λ. If x c = 1, then t must be that the vertex s completely surrounded by a contour that separates t from Λ, as the vertces outsde Λ all have + spns. In partcular, the boundary of the connected cluster K = K (x) of spns to whch vertex belongs contans a unque contour γ := γ, that separates K from the exteror Λ c of the square Λ. (Note that ths contour may n general surround other connected components K j.) Defne C γ ot be the set of all vertces j that are surrounded by γ; defne confguraton x to be the confguraton obtaned from x by flppng all spns nsde γ, { x j f j C γ (4.3) ( x) j = +x j f j C γ. Lemma 12. Let x X be any confguraton such that x = 1, and let γ = γ, be the contour that separates K from Λ c. If x s the confguraton defned by (4.3), then (4.4) µ + Λ (x) µ + = exp{ 2βJ γ }. Λ ( x) Proof. For all nearest neghbor pars j, k, the spn products x j x k and x j x k are related as follows: Consequently, x j x k = x j x k f j, k are on opposte sdes of γ; = + x j x k otherwse. H Λ ( x) H Λ (x) = 2J γ. Lemma 13. The mappng x ( x, γ) s one-to-one. Proof. Gven ( x, γ), one can recover x by negatng n the regon C γ surrounded by γ. Proposton 14. For each β > 0 and each square Λ contanng vertex, (4.5) µ + Λ (X = 1) n3 n e 2βJn. Proof. On the event X = 1 the connected cluster K of spns contanng the vertex must be separated from the connected cluster K of + spns contanng the vertces on the boundary Λ. Let Γ be the boundary contour of K = K (X) that separates K from K. By Lemmas 13 and 12, the µ + Λ probablty that X = 1 and Γ = γ satsfes Consequently, n=4 µ + Λ (X = 1 and Γ = γ) exp{ 2βJ γ }. µ + Λ (X = 1) γ exp{ 2βJ γ },

9 PHASE TRANSITION IN THE ISING MODEL 9 where the sum s over all contours n the (dual) nteger lattce surroundng. To estmate the number of such surroundng contours of length k, observe that any such contour must ntersect the vertcal upward ray emanatng from vertex at some pont wthn dstance k of. Startng from ths ntersecton pont, the contour s formed by attachng successve lne segments, one at a tme; at each stage, there are at most 3 such segments to choose from. Hence, the number of surroundng contours of length k s at most k3 k. The estmate (4.5) now follows easly. Snce the sum on the rght sde of nequalty (4.5) s less than 1/2 for all suffcently large values of β Proposton 14, together wth Theorem 2, mples that (1.8) holds at low temperature. 5. The Hgh Temperature Lmt In ths secton we shall prove the followng proposton, whch mples that (1.9), and hence also (1.11), hold at hgh temperature. Proposton 15. (5.1) tanh( 4βJ) < 1/4 = lm n µ+ n {X = 1} = 1/2. The proof, unlke Peerls argument, does not really depend on planarty of the underlyng graph, and may be extended not only to the hgher-dmensonal nteger lattces but to arbtrary vertex-regular graphs (graphs wth the property that all vertces have the same number of ncdent edges). We shall only dscuss the case G = Z Bernoull-p Ste Percolaton. The upper bound of 1/4 n (5.1) for tanh( 4βJ) emerges from the world of ste percolaton. In ts smplest ncarnaton, ste percolaton has to do wth the connectvty propertes of the random graph obtaned from the twodmensonal nteger lattce by tossng a p con at every vertex, then erasng the vertex, and all edges ncdent to t, f the con toss results n a T. Percolaton s the event that the resultng subgraph of Z 2 has an nfnte connected cluster of vertces, equvalently, that Z 2 has an nfnte connected cluster of H vertces. Proposton 16. If p < 1/4 then percolaton occurs wth probablty 0. Proof. It s enough to show that for any vertex, the probablty that s part of an nfnte connected cluster of Hs s zero. Denote by K the (maxmal) connected cluster of vertces contanng at whch the con toss s H. Defne sets F 0, F 1, F 2,... nductvely as follows: Let F 0 = {}, and for each n 0 defne F n+1 to be the set of all vertces at whch the con toss s H that are nearest neghbors of vertces n F n and that have not been lsted n n j=0 F j. I clam that (5.2) E F n+1 4pE F n. To see ths, observe that, for each vertex j F n there are at most 4 vertces adjacent to j that can be ncluded n F n+1. For each of these, the condtonal probablty that t s ncluded n F n+1, gven the con tosses that have resulted n constructng F 0, F 1,..., F n, s at most p; consequently, the expected number that are ncluded s no more than 4p.

10 10 PHASE TRANSITION IN THE ISING MODEL The cluster K s the unon of the sets F 0, F 1,..., and so ts expected cardnalty s bounded by n E F n. By nequalty (5.2), f 4p < 1 then E K <, n whch case K s fnte wth probablty 1. Fx a ste V = Z 2, and denote by Λ n the square of sde 2n + 1 centered at the orgn. Say that percolates to Λ n f the connected cluster of Hs contanng ste extends to the boundary of Λ n, equvalently, f there s a path of H vertces from ot the boundary of Λ n. Denote ths event by A(, n). Corollary 17. If p < 1/4 then lm n P p (A(, n)) = 0 for each ste Monotone Couplng of Gbbs States. Proposton 18. Fx β > 0, and set p = tanh( 4βJ). On some probablty space (Ω, F, P ), there exst X valued random varables Z (n) Y (n) wth margnal dstrbutons µ n and µ + n, respectvely, such that (5.3) P (Z (n) Y (n) ) P p (A(, n)), where P p (A(, n)) s the probablty that ste percolates to Λ n n Bernoull-p ste percolaton. Observe that ths proposton and Corollary 17 mply Proposton 15, because Corollary 17 mples that the probablty that ste percolates to Λ n converges to zero f p = tanh( 4βJ) < 1/4. The proof of Proposton 18 wll use the followng lemma, whch explans the occurrence of the quantty tanh( 4βJ). Lemma 19. For any two confguratons z, y such that z y, and for any fnte regons Σ Λ and any ste Λ Σ, (5.4) µ + Λ (X = +1 X Σ = y Σ ) µ Λ (X = +1 X Σ = z Σ ) tanh( 4βJ). Proof. In vew of the Markov property (Proposton 4), t suffces to show that for any two confguratons x, y, (5.5) µ x Λ (X = +1) µ y Λ (X = +1) tanh( 4βJ). The two probabltes n (5.5) are easly calculated, as they depend only on the spns x j, y j at the four nearest neghbors of. The maxmum dscrepancy occurs when the x spns are all +1 and the y spns are all 1: t s tanh( 4βJ). Proof of Proposton 18. Fx n, and abbrevate Λ = Λ n, Z = Z (n), and Y = Y (n). There are N := (2n + 1) 2 stes n the square Λ: label these 1, 2,..., N n order, startng from the stes at dstance 1 from Λ, then proceedng through the stes at dstance 2 from Λ, and so on, but omttng ste untl the very end, so that t s lsted as ste N. We wll construct Z, Y one ste at a tme, proceedng through the stes 1, 2,..., N n order, usng ndependent unform-(0, 1) random varables U 1, U 2,..., U N. (The values Z = 1 and Y = +1 are determned by the requrement that the margnal dstrbutons of Z and Y are µ n and µ + n, respectvely.) To construct Z 1, Y 1, use the unform U 1 to choose ± spns from the Gbbs dstrbutons µ n (X 1 dx) and µ + n (X 1 dx). By Proposton 7, these dstrbutons are stochastcally

11 PHASE TRANSITION IN THE ISING MODEL 11 ordered, so the assgnment of spns may be done n such a way that Z 1 Y 1. Moreover,by Lemma 19, µ n (X 1 = +1) µ + n (X 1 = +1) p, so the probablty that Z 1 < Y 1 s no larger than p. Hence, the assgnment of spns may be dome n such a way that the event Z 1 < Y 1 s contaned n the event U 1 < p. Now suppose that Z j, Y j, for 1 j < m, are defned. Condtonal on the event Z j = z j, Y j = y j, wth z j y j, use the unform random varable U m to assgn the spns ± at Z m and Y m usng the condtonal dstrbutons Z m µ n (X m dx X j = z j 1 j < m) and Y m µ + n (X m dx X j = y j 1 j < m). Snce z j y j, these condtonal dstrbutons are agan stochastcally ordered, by Proposton 7 and Proposton 4; consequently, the assgnment of spns may be done n such a way that Z m Y m. Moreover, by Lemma 19, the condtonal probablty that Z m < Y m, gven the assgnments Y j = y j and Z j = z j for 1 j < m, s, once agan, no larger than p; consequently, the assgnments may be done n such a way that the event Z m < Y m s contaned n the event U m < p. It remans to show that the nequalty (5.3) holds. By constructon, Z j < Y j can only occur f U j < p. Moreover, by Corollary 6, f n the course of the constructon t develops that Z j = Y j for all stes j n a contour that surrounds ste, then t must be the case that Z = Y, as the condtonal dstrbutons of the spns Z k and Y k wll concde for all stes k after completon of the contour. Thus, Z Y can only occur f there s a connected path of stes j leadng from ste to Λ along whch U j < p. But ths s precsely the event that ste percolates to Λ n Bernoull-p percolaton.