On the Kertész line: Some rigorous bounds
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b On the Kert´esz line: Some rigorous bounds ∗† Jean RUIZ ‡ Centre de Physique Th´eorique, CNRS Luminy case 907,F–13288 Marseille Cedex 9, France.Marc WOUTS § Modal’X, Universit´e Paris Ouest - Nanterre La D´efense. Bˆat. G.200 avenue de la R´epublique, 92001 Nanterre Cedex.October 24, 2018
Abstract
We study the Kert´esz line of the q –state Potts model at (inverse)temperature β , in presence of an external magnetic field h . This lineseparates two regions of the phase diagram according to the existence ornot of an infinite cluster in the Fortuin-Kasteleyn representation of themodel. It is known that the Kert´esz line h K ( β ) coincides with the line offirst order phase transition for small fields when q is large enough. Here weprove that the first order phase transition implies a jump in the densityof the infinite cluster, hence the Kert´esz line remains below the line offirst order phase transition. We also analyze the region of large fieldsand prove, using techniques of stochastic comparisons, that h K ( β ) equalslog( q − − log( β − β p ) to the leading order, as β goes to β p = − log(1 − p c )where p c is the threshold for bond percolation. One important feature of the Fortuin–Kasteleyn representation of Ising andPotts models [1] (the random cluster model), is that the geometrical transition,i.e. the apparition of an infinite cluster, corresponds precisely to the phasetransition leading to a spontaneous magnetization in the absence of an externalfield [2]. In [3], Kert´esz pointed out that this property is lost in the Ising modelwhen an external field h is introduced: while thermodynamic quantities areanalytic for any h >
0, a geometric transition appears in the correspondingrandom cluster model and there is a whole percolation transition line extendingfrom the Curie point ( h = 0) to infinite fields. As Kert´esz explained, theanalyticity of thermodynamic quantities and the existence of the percolationtransition are not contradictory because the free energy remains analytic. ∗ This document has been produced using TEX macs (see ) † Keywords:
Ising model, Potts model, percolation, random cluster model, random media,phase transition. ‡ Email: [email protected] § Email: [email protected] q , the first order transition extends to small, positive fields h and it is animportant issue to understand whether or not the Kert´esz line coincides withthe line of phase transition. Such a property was established in [4] for small h (and q large enough) and hence extends the relevance of the random cluster rep-resentation for the analysis of the phase transition in the corresponding region.Here we address some of the remaining issues: we prove the existence of theline, show that the first order phase transition results in a discontinuity of thepercolation density, and provide bounds on the Kert´esz line that are particularlyprecise in the region of large fields.In the Potts model, the spin variables σ i associated with lattice sites takevalues in the discrete set { , . . . ., q } . Considering a spin configuration in a finitebox Λ ⊂ Z d ( d > β , subject to anexternal ordering field h , is defined by the Gibbs measure µ PottsΛ ( σ ) = 1 Z PottsΛ Y h i,j i e β ( δ σi,σj − Y i e hδ σi, (1)Here the first product is over nearest neighbor pairs of Λ, the second runs oversites of Λ, Z PottsΛ denotes the partition function (normalizing factor) and δ isthe Kronecker symbol.To study the behavior of clusters, in the sense of FK clusters, we turn to thecorresponding Edwards–Sokal formulation [5], given by the joint measure µ ESΛ ( σ , η ) = 1 Z ESΛ Y h i,j i (cid:0) e − β δ η ij , + (1 − e − β ) δ η ij , δ σ i ,σ j (cid:1) Y i e hδ σi, . (2)This model can be thought as follows. Given a spin configuration, betweentwo neighboring sites with σ i = σ j , one put an edge ( η ij = 1) with probability1 − e − β and no edge w.p. e − β ; for σ i = σ j , no edge (or bond) is present. Whenthe field is infinite all spins take the value one and we are left with the classicalbond percolation problem. At finite fields, the spins are not uniformly equal toone yet we will see that percolation in the edge variable η still occurs at somefinite temperature.We call µ RCΛ , f the marginal law of η under µ ESΛ . This measure can be con-sidered as well for non-integer q > Theorem 1
Let β, h > and q > .i. The infinite volume limit µ RC f = lim Λ ր Z d µ RCΛ , f (3) exists. i. The probability θ = µ RC f ( the origin belongs to an infinite cluster of η ) (4) increases with β and h , and decreases with q .iii. Hence the Kert´esz line h K ( β ) = inf { h > θ > } (5) exists, and h K ( β ) decreases with β . Note that h K ( β ) = 0 if β > β c , where β c is the critical inverse temperaturefor the phase transition with no field, while h K ( β ) = + ∞ if β β p , where β p = − log(1 − p c ) (6)is the critical inverse temperature for percolation at infinite fields and p c thethreshold for bond percolation on Z d .Then we examine the consequences of the first order transition on the density θ of the infinite cluster: Theorem 2
A discontinuity in the parameter β in the mean energy e f = 11 − e − β µ RC f ( η ij ) (7) where i, j are neighboring sites, or in the magnetization, implies a discontinuityin the density θ of percolation. This means that θ has a jump on the line of first order phase transition.Consequently, the Kert´esz line cannot be found above the line of first orderphase transition. It is known [4] that both lines coincide at small fields when q is large, hence the question remains whether they coincide up to the otherextremity of the line of first order phase transition. In the corresponding meanfield analysis [6] we proved the existence of a cusp as soon as q >
2, that is,whenever appears a line of first order phase transition. However, in the twodimensional Potts model no bifurcation was noted numerically [4].We conclude our exposition of the results with upper and lower bounds onthe Kert´esz line, which are particularly efficient when β is taken slightly above β p = − log(1 − p c ), corresponding to the regime of large fields. The idea thatled to the next theorem is that the model can be understood as independentbond percolation over a random media : the spins not equal to 1 are consideredas defects , which become rare when h → + ∞ . Our proofs are reminiscent of[7] in which similar methods were employed to provide necessary and sufficientconditions for the phase transition in the dilute Ising model, see also [8] forbeautiful results on mixed percolation.3 heorem 3 For any d > , q > and β > β p , one has h K ( β ) − log q e β − e βp − − q − − log ( β − β p ) + log (2 p c ( q − O β → β + p ( β − β p ) (9) while h K ( β ) > − log e − β p − e − β p c ( q − − βd (10)= − log ( β − β p ) + log ( p c ( q − − (2 d − β p + O β → β + p ( β − β p ) . (11)Thus, to the leading order, h K ( β ) ≃ − log( β − β p )+log( q −
1) when β → β + p .The upper and lower asymptotes differ from the constant log(2) − (2 d −
1) log(1 − p c ) that does not depend on q .These upper and lower bounds are presented in Fig. 1 together with thenumerical results of [4]. h β Figure 1: A comparison between upper and lower bounds with the numericalresults of [4] for d = 2 and q = 10.To summarize, we have shown that for the lattice Potts model subject toan external field, the Kert´esz line is well defined. We have presented upperand lower bounds on this line. These bounds are very precise at high fieldsand complement the previous study [4] in which a precise approximation at lowfield was given. In addition, we have shown that a jump of the mean energy orof the magnetization implies a jump in the percolation density of the clustersassociated to the corresponding FK representation of the model. This last resultdoes not exclude the presence of an intermediate regime of the field where whendecreasing the temperature, one first encounters the percolation transition andthen for a lower temperature the percolation density would exhibits a jump. Acknowledgments.
It is a pleasure to thank Daniel Gandolfo for valuablediscussions. One of us (M. W.) is grateful to CPT and LATP for their kindhospitality. 4
Appendix
A.1 A random cluster representation
For any β, h > q > η have η ij ∈ { , } for all < i, j > nearest neighbor pairs in the domain. For Λ a finite subset of Z d and π a boundary condition on Λ, that is an edge configuration on Z d which restrictionto Λ has no open edge, we consider µ RCΛ , π ( η ) = 1 Z RCΛ , π Y h i,j i (cid:0) e − β δ η ij , + (1 − e − β ) δ η ij , (cid:1) Y C ∈C π Λ ( η ) w ( S ( C )) (12)where w ( S ) = 1 + ( q − e − hS . The first product runs over h i, j i nearest neighbor pairs in Λ. The second oneis over all connected components (clusters) C ∈ C π Λ ( η ), where C π Λ ( η ) is the setof clusters of Z d under the wiring π ∨ η (the edge configuration defined by( π ∨ η ) ij = max( π ij , η ij )), that own some site of Λ. For any such cluster, S ( C )stands for its number of sites.The variable η under the joint measure µ ESΛ defined at (2) follows the law µ RCΛ , f , where f stands for the free boundary condition , that is the edge config-uration with no edge open. Conditionally on η , for integer q ∈ { , , . . . } thedistribution of σ under the joint measure µ ESΛ is as follows: the spin is con-stant on each cluster of η , and a cluster with S sites obtains the color 1 withprobability e hS / ( e hS + q − q − / ( e hS + q − µ CESΛ ( σ , n ) = 1 Z CESΛ Y h i,j i [ e − β δ n ij , + (1 − e − β ) δ n ij , χ ( σ i = σ j = 1)+(1 − e − β ) δ η ij , χ ( σ i = σ j = 1)] Y i e hδ σi, (13)with the edges variables n ij taking values in the set { , , } . Let us consider,for a while, thermodynamics limits (the existence of thermodynamics limits willbe proven at the next section). We want to emphasize that the question ofpercolation for η under µ RC f and for the color 1 in n under µ CES are equivalent .Indeed, any infinite cluster for η under µ ES will be given the color σ = 1 withprobability one as soon as h > /q if h = 0). Hence, relabelling η into n according to the spin of clusters we obtain in fact an infinite clusterfor the color 1 in n , w. p. 1 (w. p. 1 /q if h = 0) and this shows that the5robability of percolation from the origin θ under µ RC f and θ for the color 1 in n under µ CES satisfy θ = (cid:26) θ/q if h = 0 θ if h > . A.2 Conditional probabilities and infinite volume limit
Here we give the proof of Theorem 1. Like the usual random cluster represen-tation, the measures µ RCΛ , π satisfy the DLR equations, which means that, givenany Λ ′ ⊂ Λ, the restriction of η to Λ ′ under the measure µ RCΛ , π conditioned on η = ω outside of Λ ′ has law µ RCΛ ′ , ω ∨ π . Consequently, the measures µ RCΛ , π arecharacterized by the law of η on a single edge ij given the boundary condition π : µ RC { ij } , π ( η ij = 1) = p π ij (14)where p π ij = p def . = 1 − e − β if π connects i and j , otherwise p π ij = pp + (1 − p ) w ( S π i ) w ( S π j ) w ( S π i + S π j ) (15)where S π i (resp. S π j ) is the number of sites of the cluster containing i (resp. j )under the connections π .It is easily verified that p π ij is an increasing function of β , h and π , decreasingwith q >
1. Thanks to the DLR equations, the hypothesis of Holley’s Lemma(see for instance Theorem 4.8 in [9] or Theorems 2.1 and 2.6 in [10]) are ver-ified and this implies that µ RCΛ , π stochastically increases with β, h and π , andstochastically decreases with q . Using again the DLR equations, we see thatthe measure µ RCΛ , f stochastically increases as Λ ր Z d , proving the existence ofthe weak limit µ RC f at (3). Point ii of the theorem follows from the variationsof µ RC f with β, h and q which are the same than those of µ RCΛ , π while point iii isan immediate consequence of ii . A.3 First order transitions
Theorem 2 is essentially a consequence of the uniqueness of infinite volumemeasures under the condition that the infinite cluster has the same density underboth infinite volume limits for free and wired boundary conditions (Theorem 4below). We adapt here the classical argument at h = 0 to our setting h > µ RC w of µ RCΛ , π under the wired boundary condition w , that has alledges open. As in [11, 12, 13] it happens that: Lemma 1
Given h > , q > , the set of β at which µ RC f = µ RC w is at mostcountable. roof Let y π Λ = 1 | Λ | log Y h i,j i (cid:0) e β − (cid:1) η ij Y C ∈C π Λ ( η ) w ( S ( C )) . (16)When π = f , the quantity y π Λ is sub-additive – when one cluster of size S iscut into two clusters of size S , S with S + S = S , then w ( S ) w ( S ) w ( S ).Hence y f Λ converges to some y ( β, h ) as Λ → Z d . The influence of the boundarycondition π on y π Λ is of order | ∂ Λ | / | Λ | : for any configuration η the product Q C ∈C π Λ ( η ) w ( S ( C )) decreases with π and conversely, Y C ∈C f Λ ( η ) w ( S ( C )) ( w (1)) | ∂ Λ | Y C ∈C w Λ ( η ) w ( S ( C )) (17)because C f Λ ( η ) contains at most | ∂ Λ | clusters not present in C w Λ ( η ), each of themhaving size S >
1. Hence for any sequence π Λ , any sequence of cubes Λ → Z d ,we have y π Λ → y . Now we show that y π Λ is a convex function of λ = log( e β − ∂y π Λ ∂λ = µ RCΛ , π | Λ | X η ij is an increasing function of β , hence of λ , and the convexity holds for both y π Λ and its limit y . Therefore y is derivable at all β / ∈ D h where D h (that dependson h ) is finite or countable. When this occurs, by convexity of the y π Λ we havelim Λ ∂y f Λ ∂λ = ∂y Λ ∂λ = lim Λ ∂y w Λ ∂λ which implies that the probability of opening a given edge is the same underboth free and wired boundary conditions: µ RC f ( η ij ) = µ RC w ( η ij ). Because of thestochastic domination µ RC f stoch µ RC w , the conclusion µ RC f = µ RC w follows. (cid:3) On the other hand we introduce the magnetization m w = µ RC w (cid:18)
11 + ( q − e − hS η i (cid:19) − q (18)under the infinite volume measure µ RC w with wired boundary condition, for any h >
0, where S η i is the number of sites of the cluster of η that contains i . Welet m f the same quantity under µ RC f . We consider also e f , the mean energyas in (7) and θ f = θ (see (4)) the density of the percolating cluster under themeasure µ RC f , and call e w and θ w the corresponding quantities under µ RC w . Wecan write e f and m f as increasing limits and e w , m w and θ w as decreasinglimits of continuous, increasing functions of β . For instance, e w = 1 p lim Λ ր Z d µ RCΛ , w ( η ij )7nd θ w = lim ∆ ր Z d lim Λ ր Z d µ RCΛ , w (0 η ↔ ∂ ∆)are decreasing limits while the functions β µ RCΛ , w ( η ij ) and β µ RCΛ , w (0 η ↔ ∂ ∆)are continuous, increasing. Hence: Lemma 2
For any h > and q > , e f and m f are left-continuous functionsof β , while e w , m w and θ w are right-continuous. As a consequence of Lemma 1 the equalities e f = e w , m f = m w and θ f = θ w hold true for all but countably many β . In view of Lemma 2, the energy (resp.the magnetization) is continuous at some β if and only if it has the same valueunder both µ RC f and µ RC w . Hence, at a point of discontinuity it is the case that µ RC f = µ RC w . But at such points we cannot have θ f = θ w in view of Theorem 4below and Theorem 2 follows. Theorem 4
The equality θ f = θ w implies the uniqueness of random clustermeasures – in other words, µ RC f = µ RC w when θ f = θ w . Theorem 4 was proven in [12] in the case of h = 0 (Theorem 5.2 in [12] ;see also Theorem 5.16 in [10] for the complete construction). The proof givenin [10] applies verbatim in the present setting.The reader might be interested as well in a simpler proof of the fact that θ w = 0 implies the uniqueness of random cluster measures (Theorem A.2 in [14])which shows as well that the Kert´esz line remains below the line of discontinuousphase transition. A.4 An upper bound on the Kert´esz line
Our upper bound is based directly on the conditional probabilities (14) and(15). Sinceinf π µ RC { ij } , π ( η ij = 1) = µ RC { ij } , f ( η ij = 1) = ˜ p def . = pp + (1 − p ) w (1) /w (2) , the measure µ RC f stochastically dominates independent bond percolation of pa-rameter ˜ p , and ˜ p > p c ensures that percolation occurs, i.e. that θ >
0. We recallthe notation β p = − ln(1 − p c ), which yields θ > ⇐ ˜ p > p c ⇔ (cid:0) q − e − h (cid:1) q − e − h < e β − e β p − ⇐ (cid:0) q − e − h (cid:1) < e β − e β p − .5 A lower bound on the Kert´esz line The former method yields here the only information that h K ( β ) = + ∞ for all β β p . Hence we consider another point of view : we use a joint measureanalogous to µ ESΛ (2) and compare the spins which are not of color 1 to randomdefects , which have a vanishing density in the limit h → ∞ .As we aim at a lower bound that holds for non-integer q >
1, we considera modified (monochrome) version of µ ESΛ that gives only two colors to spinconfigurations s . The color 1 plays effectively the role of a color in the Pottsmodel, and undergoes the external field. The color 0 condensates all q − q ). Let µ M Λ ( s , η ) = 1 Z M Λ ω ( s , η ) (20)where ω ( s , η ) = Y h i,j i (cid:0) e − β δ η ij , + (1 − e − β ) δ η ij , δ s i ,s j (cid:1) × Y i e hδ si, × ( q − N ( s , η ) , (21)and N ( s , η ) is the number of clusters of η that have spin s = 0. The marginallaw of η equals µ RCΛ , f , while the conditional law of η knowing s is the following: η has all edges closed between regions of s of different colors, while its restrictionto the regions with s = 1 follows a bond percolation process of parameter p = 1 − e − β , and its restriction to the regions with s = 0 follows the usualrandom cluster measure of parameters p = 1 − e − β and q ′ = q − s a spin configuration with ¯ s i = 1, and call ˜ s the modified configuration with ˜ s i = 0. For any η such that η ij = 0 for all j adjacent to i , one has ω (˜ s , η ) = ω (¯ s , η )( q − e − h . (22)Therefore, µ M Λ (˜ s ) µ M Λ (¯ s ) > ( q − e − h P η : η ij =0 , ∀ j ∼ i ω (¯ s , η ) P η ω (¯ s , η )= ( q − e − h × µ M Λ ( η ij = 0, for all j adjacent to i | s = ¯ s ) . (23)But the latter probability is at least e − βd and (23) implies thatsup ¯ s µ ( s i = 1 | s j = ¯ s j , ∀ i = j ) ρ def . = 11 + ( q − e − βd e − h . (24)Hence we have a lower bound on the density of defects : the process of goodsites ( s i = 1) is stochastically dominated by site percolation of parameter ρ ,and percolation cannot occur (i.e. θ = 0) if the mixed percolation process [8] of9ite density ρ and edge density p = 1 − e − β does not percolate, that is, if thereis no infinite cluster after the removal of closed sites and closed bonds.The order in which sites and bonds are close does not modify the properties ofthe mixed percolation process. Here we shall consider that the edge percolationat density p is done first, giving the diluted graph G made of the open edges andtheir vertices, and that the site percolation of parameter ρ is realized afterwards.It has been known for a long time that bond percolation of parameter ρ on G is more likely to succeed than site percolation (see [15, 16] for inductive proofsand [17], proof of Lemma 5 for a dynamical coupling ). But the process of bondpercolation with intensity ρ on the diluted graph G boils down to the classicalbond percolation on Z d with parameter p × ρ and we have shown that θ = 0 ⇐ p × ρ < p c ⇔ e − β p − e − β < p c ( q − e − βd e − h (25)which leads to the lower bound (10). References [1] C. M. Fortuin and P. W. Kasteleyn. On the random cluster model I:Introduction and relation to other models.
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