On the critical value function in the divide and color model
OOn the critical value function in the divide and color model
Andr´as B´alint ∗ Vincent Beffara † Vincent Tassion ‡ May 17, 2018
Abstract
The divide and color model on a graph G arises by first deleting each edge of G withprobability 1 − p independently of each other, then coloring the resulting connected compo-nents ( i.e. , every vertex in the component) black or white with respective probabilities r and1 − r , independently for different components. Viewing it as a (dependent) site percolationmodel, one can define the critical point r Gc ( p ).In this paper, we mainly study the continuity properties of the function r Gc , which is aninstance of the question of locality for percolation. Our main result is the fact that in thecase G = Z , r Gc is continuous on the interval [0 , / p = 0 forthe more general class of graphs with bounded degree. We then investigate the sharpnessof the bounded degree condition and the monotonicity of r Gc ( p ) as a function of p . Keywords:
Percolation, divide and color model, critical value, locality, stochastic domination.
AMS 2000 Subject Classification:
Introduction
The divide and color (DaC) model is a natural dependent site percolation model introduced byH¨aggstr¨om in [6]. It has been studied directly in [6, 4, 3, 2], and as a member of a more generalfamily of models in [7, 3, 1, 5]. This model is defined on a multigraph G = ( V , E ), where E is amultiset ( i.e. , it may contain an element more than once), thus allowing parallel edges betweenpairs of vertices. For simplicity, we will imprecisely call G a graph and E the edge set , evenif G contains self-loops or multiple edges. The DaC model with parameters p, r ∈ [0 , G with vertex set V and edge set E , is defined by the followingtwo-step procedure: • First step: Bernoulli bond percolation. We independently declare each edge in E to beopen with probability p , and closed with probability 1 − p . We can identify a bondpercolation configuration with an element η ∈ { , } E : for each e ∈ E , we define η ( e ) = 1if e is open, and η ( e ) = 0 if e is closed. • Second step: Bernoulli site percolation on the resulting cluster set. Given η ∈ { , } E , wecall p -clusters or bond clusters the connected components in the graph with vertex set V and edge set { e ∈ E : η ( e ) = 1 } . The set of p -clusters of η gives a partition of V . For each p -cluster C , we assign the same color to all the vertices in C . The chosen color is black ∗ Chalmers University of Technology, e-mail: [email protected] † UMPA-ENS Lyon, e-mail: [email protected] ‡ ENS Lyon, e-mail: [email protected] a r X i v : . [ m a t h . P R ] J u l ith probability r and white with probability 1 − r , and this choice is independent fordifferent p -clusters.These two steps yield a site percolation configuration ξ ∈ { , } V by defining, for each v ∈ V , ξ ( v ) = 1 if v is black, and ξ ( v ) = 0 if v is white. The connected components (via the edge set E )in ξ of the same color are called (black or white) r -clusters . The resulting measure on { , } V is denoted by µ Gp,r .Let E b ∞ ⊂ { , } V denote the event that there exists an infinite black r -cluster. By standardarguments (see Proposition 2.5 in [6]), for each p ∈ [0 , critical coloring value r Gc ( p ) ∈ [0 ,
1] such that µ Gp,r ( E b ∞ ) (cid:40) = 0 if r < r Gc ( p ) ,> r > r Gc ( p ) . The critical edge parameter p Gc ∈ [0 ,
1] is defined as follows: the probability that there existsan infinite bond cluster is 0 for all p < p Gc , and positive for all p > p Gc . The latter probabilityis in fact 1 for all p > p Gc , whence r Gc ( p ) = 0 for all such p . Kolmogorov’s 0 − µ Gp,r ( E b ∞ ) ∈ { , } ; nevertheless it is possiblethat µ Gp,r ( E b ∞ ) ∈ (0 ,
1) for some r > r Gc ( p ) ( e.g. on the square lattice, as soon as p > p c = 1 / µ Gp,r ( E b ∞ ) = r ). Statement of the results
Our main goal in this paper is to understand how the critical coloring parameter r Gc dependson the edge parameter p . Since the addition or removal of self-loops obviously does not affectthe value of r Gc ( p ), we will assume that all the graphs G that we consider are without self-loops.On the other hand, G is allowed to contain multiple edges.Our first result, based on a stochastic domination argument, gives bounds on r Gc ( p ) in termsof r Gc (0), which is simply the critical value for Bernoulli site percolation on G . By the degree ofa vertex v , we mean the number of edges incident on v (counted with multiplicity). Proposition 1.
For any graph G with maximal degree ∆ , for all p ∈ [0 , , − − r Gc (0)(1 − p ) ∆ ≤ r Gc ( p ) ≤ r Gc (0)(1 − p ) ∆ . As a direct consequence, we get continuity at p = 0 of the critical value function: Proposition 2.
For any graph G with bounded degree, r Gc ( p ) is continuous in p at . One could think of an alternative approach to the question, as follows: the DaC model canbe seen as Bernoulli site percolation of the random graph G p = ( V p , E p ) where V p is the setof bond clusters and two bond clusters are connected by a bond of E p if and only if they areadjacent in the original graph. The study of how r Gc ( p ) depends on p is then a particular caseof a more general question known as the locality problem : is it true in general that the criticalpoints of site percolation on a graph and a small perturbation of it are always close? Here, forsmall p , the graphs G and G p are somehow very similar, and their critical points are indeedclose.Dropping the bounded-degree assumption allows for the easy construction of graphs forwhich continuity does not hold at p = 0: Proposition 3.
There exists a graph G with p Gc > such that r Gc is discontinuous at .
2n general, when p >
0, the graph G p does not have bounded degree, even if G does;this simple remark can be exploited to construct bounded degree graphs for which r Gc hasdiscontinuities below the critical point of bond percolation (though of course not at 0): Theorem 4.
There exists a graph G of bounded degree satisfying p Gc > / and such that r Gc ( p ) is discontinuous at / . Remark 5.
The value / in the statement above is not special: in fact, for every p ∈ (0 , , itis possible to generalize our argument to construct a graph with a critical bond parameter above p and for which the discontinuity of r c occurs at p . Our main results concerns the case G = Z , for which the above does not occur: Theorem 6.
The critical coloring value r Z c ( p ) is a continuous function of p on the wholeinterval [0 , / . The other, perhaps more anecdotal question we investigate here is whether r Gc is monotonicbelow p c . This is the case on the triangular lattice (because it is constant equal to 1 / Z in simulations (see the companion paper [2]).In the general case, the question seems to be rather delicate. Intuitively the presence ofopen edges would seem to make percolation easier, leading to the intuition that the function p (cid:55)→ r c ( p ) should be nonincreasing. Theorem 2.9 in [6] gives a counterexample to this intuition.It is even possible to construct quasi-transitive graphs on which any monotonicity fails: Proposition 7.
There exists a quasi-transitive graph G such that r Gc is not monotone on theinterval [0 , p Gc ) . A brief outline of the paper is as follows. We set the notation and collect a few resultsfrom the literature in Section 1. In Section 2, we stochastically compare µ Gp,r with Bernoullisite percolation (Theorem 9), and show how this result implies Proposition 1. We then turn tothe proof of Theorem 6 in Section 3, based on a finite-size argument and the continuity of theprobability of cylindrical events.In Section 4, we determine the critical value function for a class of tree-like graphs, and inthe following section we apply this to construct most of the examples of graphs we mentionedabove.
We start by explicitly constructing the model, in a way which will be more technically convenientthan the intuitive one given in the introduction.Let G be a connected graph ( V , E ) where the set of vertices V = { v , v , v , . . . } is countable.We define a total order “ < ” on V by saying that v i < v j if and only if i < j . In this way, forany subset V ⊂ V , we can uniquely define min( V ) ∈ V as the minimal vertex in V with respectto the relation “ < ”. For a set S , we denote { , } S by Ω S . We call the elements of Ω E bondconfigurations , and the elements of Ω V site configurations . As defined in the Introduction, ina bond configuration η , an edge e ∈ E is called open if η ( e ) = 1, and closed otherwise; in asite configuration ξ , a vertex v ∈ V is called black if ξ ( e ) = 1, and white otherwise. Finally,for η ∈ Ω E and v ∈ V , we define the bond cluster C v ( η ) of v as the maximal connected inducedsubgraph containing v of the graph with vertex set V and edge set { e ∈ E : η ( e ) = 1 } , anddenote the vertex set of C v ( η ) by C v ( η ). 3or a ∈ [0 ,
1] and a set S , we define ν Sa as the probability measure on Ω S that assigns toeach s ∈ S value 1 with probability a and 0 with probability 1 − a , independently for differentelements of S . We define a functionΦ : Ω E × Ω V → Ω E × Ω V , ( η, κ ) (cid:55)→ ( η, ξ ) , where ξ ( v ) = κ (min( C v ( η ))). For p, r ∈ [0 , P Gp,r to be the image measure of ν E p ⊗ ν V r by the function Φ, and denote by µ Gp,r the marginal of P Gp,r on Ω V . Note that this definition of µ Gp,r is consistent with the one in the Introduction.Finally, we give a few definitions and results that are necessary for the analysis of the DaCmodel on the square lattice, that is the graph with vertex set Z and edge set E = {(cid:104) v, w (cid:105) : v =( v , v ) , w = ( w , w ) ∈ Z , | v − w | + | v − w | = 1 } . The matching graph Z ∗ of the squarelattice is the graph with vertex set Z and edge set E ∗ = {(cid:104) v, w (cid:105) : v = ( v , v ) , w = ( w , w ) ∈ Z , max( | v − w | , | v − w | ) = 1 } . In the same manner as in the Introduction, we define, for acolor configuration ξ ∈ { , } Z , (black or white) ∗ -clusters as connected components (via theedge set E ∗ ) in ξ of the same color. We denote by Θ ∗ ( p, r ) the P Z p,r -probability that the originis contained in an infinite black ∗ -cluster, and define r ∗ c ( p ) = sup { r : Θ ∗ ( p, r ) = 0 } for all p ∈ [0 ,
1] — note that this value may differ from r Z ∗ c ( p ). The main result in [3] is thatfor all p ∈ [0 , / r Z c ( p ) and r ∗ c ( p ) satisfy the duality relation r Z c ( p ) + r ∗ c ( p ) = 1 . (1)We will also use exponential decay result for subcritical Bernoulli bond percolation on Z .Let denote the origin in Z , and for each n ∈ N = { , , . . . } , let us define S n = { v ∈ Z : dist ( v, ) = n } (where dist denotes graph distance), and the event M n = { η ∈ Ω E : there is apath of open edges in η from to S n } . Then we have the following result: Theorem 8 ([8]) . For p < / , there exists ψ ( p ) > such that for all n ∈ N , we have that ν E p ( M n ) < e − nψ ( p ) . p = 0 In this section, we prove Proposition 1 via a stochastic comparison between the DaC measureand Bernoulli site percolation. Before stating the corresponding result, however, let us recallthe concept of stochastic domination.We define a natural partial order on Ω V by saying that ξ (cid:48) ≥ ξ for ξ, ξ (cid:48) ∈ Ω V if, for all v ∈ V , ξ (cid:48) ( v ) ≥ ξ ( v ). A random variable f : Ω V → R is called increasing if ξ (cid:48) ≥ ξ implies that f ( ξ (cid:48) ) ≥ f ( ξ ), and an event E ⊂ Ω V is increasing if its indicator random variable is increasing. Forprobability measures µ, µ (cid:48) on Ω V , we say that µ (cid:48) is stochastically larger than µ (or, equivalently,that µ is stochastically smaller than µ (cid:48) , denoted by µ ≤ st µ (cid:48) ) if, for all bounded increasingrandom variables f : Ω V → R , we have that (cid:90) Ω V f ( ξ ) dµ (cid:48) ( ξ ) ≥ (cid:90) Ω V f ( ξ ) dµ ( ξ ) . By Strassen’s theorem [11], this is equivalent to the existence of an appropriate coupling ofthe measures µ (cid:48) and µ ; that is, the existence of a probability measure Q on Ω V × Ω V suchthat the marginals of Q on the first and second coordinates are µ (cid:48) and µ respectively, and Q ( { ( ξ (cid:48) , ξ ) ∈ Ω V × Ω V : ξ (cid:48) ≥ ξ } ) = 1. 4 heorem 9. For any graph G = ( V , E ) whose maximal degree is ∆ , at arbitrary values of theparameters p, r ∈ [0 , , ν V r (1 − p ) ∆ ≤ st µ Gp,r ≤ st ν V − (1 − r )(1 − p ) ∆ . Before turning to the proof, we show how Theorem 9 implies Proposition 1.
Proof of Proposition 1.
It follows from Theorem 9 and the definition of stochastic dom-ination that for the increasing event E b ∞ (which was defined in the Introduction), we have µ Gp,r ( E b ∞ ) > r (1 − p ) ∆ > r Gc (0), which implies that r Gc ( p ) ≤ r Gc (0) / (1 − p ) ∆ . Thederivation of the lower bound for r Gc ( p ) is analogous.Now we give the proof of Theorem 9, which bears some resemblance with the proof ofTheorem 2.3 in [6]. Proof of Theorem 9.
Fix G = ( V , E ) with maximal degree ∆, and parameter values p, r ∈ [0 , < ” and the minimum of a vertex set with respect to this relationas defined in Section 1. In what follows, we will define several random variables; we will denotethe joint distribution of all these variables by P .First, we define a collection ( η ex,y : x, y ∈ V , e = (cid:104) x, y (cid:105) ∈ E ) of i.i.d. Bernoulli( p ) randomvariables ( i.e. , they take value 1 with probability p , and 0 otherwise); one may imagine havingeach edge e ∈ E replaced by two directed edges, and the random variables represent which ofthese edges are open. We define also a set ( κ x : x ∈ V ) of Bernoulli( r ) random variables. Givena realization of ( η ex,y : x, y ∈ V , e = (cid:104) x, y (cid:105) ∈ E ) and ( κ x : x ∈ V ), we will define an Ω V × Ω E -valuedrandom configuration ( η, ξ ) with distribution P Gp,r , by the following algorithm.1. Let v = min { x ∈ V : no ξ -value has been assigned yet to x by this algorithm } . (Notethat v and V, v i , H i ( i ∈ N ), defined below, are running variables, i.e. , their values will beredefined in the course of the algorithm.)2. We explore the “directed open cluster” V of v iteratively, as follows. Define v = v .Given v , v , . . . , v i for some integer i ≥
0, set η ( e ) = η ev i ,w for every edge e = (cid:104) v i , w (cid:105) ∈ E incident to v i such that no η -value has been assigned yet to e by the algorithm, and write H i +1 = { w ∈ V \ { v , v , . . . , v i } : w can be reached from any of v , v , . . . , v i by usingonly those edges e ∈ E such that η ( e ) = 1 has been assigned to e by this algorithm } . If H i +1 (cid:54) = ∅ , then we define v i +1 = min( H i +1 ), and continue exploring the directed opencluster of v ; otherwise, we define V = { v , v , . . . , v i } , and move to step 3.3. Define ξ ( w ) = κ v for all w ∈ V , and return to step 1.It is immediately clear that the above algorithm eventually assigns a ξ -value to each vertex.Note also that a vertex v can receive a ξ -value only after all edges incident to v have alreadybeen assigned an η -value, which shows that the algorithm eventually determines the full edgeconfiguration as well. It is easy to convince oneself that ( η, ξ ) obtained this way indeed has thedesired distribution.Now, for each v ∈ V , we define Z ( v ) = 1 if κ v = 1 and η ew,v = 0 for all edges e = (cid:104) v, w (cid:105) ∈ E incident on v ( i.e. , all directed edges towards v are closed), and Z ( v ) = 0 otherwise. Notethat every vertex with Z ( v ) = 1 has ξ ( v ) = 1 as well, whence the distribution of ξ ( i.e. , µ Gp,r )stochastically dominates the distribution of Z (as witnessed by the coupling P ).Notice that Z ( v ) depends only on the states of the edges pointing to v and on the value of κ v ; in particular the distribution of Z is a product measure on Ω V with parameter r (1 − p ) d ( v ) at5 , where d ( v ) ≤ ∆ is the degree of v , whence µ Gp,r stochastically dominates the product measureon Ω V with parameter r (1 − p ) ∆ , which gives the desired stochastic lower bound. The upperbound can be proved analogously; alternatively, it follows from the lower bound by exchangingthe roles of black and white. r Z c ( p ) on the interval [0 , / In this section, we will prove Theorem 6. Our first task is to prove a technical result valid onmore general graphs stating that the probability of any event A whose occurrence depends ona finite set of ξ -variables is a continuous function of p for p < p Gc . The proof relies on the factthat although the color of a vertex v may be influenced by edges arbitrarily far away, if p < p Gc ,the corresponding influence decreases to 0 in the limit as we move away from v . Therefore, theoccurrence of the event A depends essentially on a finite number of η - and κ -variables, whenceits probability can be approximated up to an arbitrarily small error by a polynomial in p and r . Once we have proved Proposition 10 below, which is valid on general graphs, we will applyit on Z to certain “box-crossing events,” and appeal to results in [3] to deduce the continuityof r Z c ( p ). Proposition 10.
For every site percolation event A ⊂ { , } V depending on the color of finitelymany vertices, µ Gp,r ( A ) is a continuous function of ( p, r ) on the set [0 , p Gc ) × [0 , . Proof.
In this proof, when µ is a measure on a set S , X is a random variable with law µ and F : S −→ R is a bounded measurable function, we write abusively µ [ F ( X )] for the expectationof F ( X ). We show a slightly more general result: for any k ≥ x = ( x , . . . , x k ) ∈ V k and f : { , } k → R bounded and measurable, µ Gp,r [ f ( ξ ( x ) , . . . , ξ ( x k ))] is continuous in ( p, r )on the product [0 , p Gc ) × [0 , { x , . . . , x k } such that the states of the x i suffices to determine whether A occurs, and take f to be the indicator function of A .To show the previous affirmation, we condition on the vector m x ( η ) = (min C x ( η ) , . . . , min C x k ( η ))which takes values in the finite set V = (cid:8) ( v , . . . , v k ) ∈ V k : ∀ i v i ≤ max { x , . . . , x k } (cid:9) , and weuse the definition of P Gp,r as an image measure. By definition, µ Gp,r [ f ( ξ ( x ) , . . . , ξ ( x k ))]= (cid:88) v ∈ V P Gp,r [ f ( ξ ( x ) , . . . , ξ ( x k )) |{ m x = v } ] P Gp,r [ { m x = v } ]= (cid:88) v ∈ V ν E p ⊗ ν V r [ f ( κ ( v ) , . . . , κ ( v k )) |{ m x = v } ] ν E p [ { m x = v } ]= (cid:88) v ∈ V ν V r [ f ( κ ( v ) , . . . , κ ( v k ))] ν E p [ { m x = v } ] . Note that ν V r [ f ( κ ( v ) , . . . , κ ( v k ))] is a polynomial in r , so to conclude the proof we only need toprove that for any fixed x and v , ν E p ( { m ( x ) = v } ) depends continuously on p on the interval[0 , p Gc ).For n ≥
1, write F n = {| C x | ≤ n, . . . , | C x k | ≤ n } . It is easy to verify that the event { m x = v } ∩ F n depends on the state of finitely many edges. Hence, ν E p [ { m x = v } ∩ F n ] isa polynomial function of p . 6ix p < p Gc . For all p ≤ p ,0 ≤ ν E p [ { m ( x ) = v } ] − ν E p [ { m x = v } ∩ F n ] ≤ ν E p [ F cn ] ≤ ν E p [ F cn ]where lim n →∞ ν E p [ F cn ] = 0, since p < p Gc . So, ν E p [ m ( x ) = v ] is a uniform limit of polynomials onany interval [0 , p ] , p < p Gc , which implies the desired continuity. Remark 11.
In the proof we can see that, for fixed p < p Gc , µ Gp,r ( A ) is a polynomial in r . Remark 12. If G is a graph with uniqueness of the infinite bond cluster in the supercriticalregime, then it is possible to verify that ν E p [ { m ( x ) = v } ] is continuous in p on the whole interval [0 , . In this case, the continuity given by the Proposition 10 can be extended to the whole square [0 , . Proof of Theorem 6.
In order to simplify our notations, we write P p,r , ν p , r c ( p ), for P Z p,r , ν E p and r Z c ( p ) respectively. Fix p ∈ (0 , /
2) and ε > δ = δ ( p , ε ) > p ∈ ( p − δ, p + δ ), r c ( p ) ≥ r c ( p ) − ε, (2)and r c ( p ) ≤ r c ( p ) + ε. (3)Note that by equation (1), for all small enough choices of δ > ≤ p ± δ < / r ∗ c ( p ) ≤ r ∗ c ( p ) + ε. (4)Below we will show how to find δ > p ∈ ( p − δ , p + δ ). Onemay then completely analogously find δ > p ∈ ( p − δ , p + δ ),and take δ = min( δ , δ ).Fix r = r c ( p ) + ε , and define the event V n = { ( ξ, η ) ∈ Ω Z × Ω E : there exists a verticalcrossing of [0 , n ] × [0 , n ] that is black in ξ } . By “vertical crossing,” we mean a self-avoidingpath of vertices in [0 , n ] × [0 , n ] with one endpoint in [0 , n ] × { } , and one in [0 , n ] × { n } .Recall also the definition of M n in Theorem 8. By Lemma 2.10 in [3], there exists a constant γ > p, a ∈ [0 ,
1] and L ∈ N :(3 L + 1)( L + 1) ν a ( M (cid:98) L/ (cid:99) ) ≤ γ, and P p,a ( V L ) ≥ − γ (cid:27) ⇒ a ≥ r c ( p ) . As usual, (cid:98) x (cid:99) for x > m such that m ≤ x . Fix such a γ .By Theorem 8, there exists N ∈ N such that(3 n + 1)( n + 1) ν p ( M (cid:98) n/ (cid:99) ) < γ for all n ≥ N . On the other hand, since r > r c ( p ), it follows from Lemma 2.11 in [3] that thereexists L ≥ N such that P p ,r ( V L ) > − γ. Note that both (3 L + 1)( L + 1) ν p ( M (cid:98) L/ (cid:99) ) and P p,r ( V L ) are continuous in p at p . Indeed, theformer is simply a polynomial in p , while the continuity of the latter follows from Proposition 10.Therefore, there exists δ > p ∈ ( p − δ , p + δ ),(3 L + 1)( L + 1) ν p ( M (cid:98) L/ (cid:99) ) ≤ γ, and P p,r ( V L ) ≥ − γ.
7y the choice of γ , this implies that r ≥ r c ( p ) for all such p , which is precisely what we wantedto prove.Finding δ > p ∈ ( p − δ , p + δ ) is analogous: one only needsto substitute r c ( p ) by r ∗ c ( p ) and “crossing” by “ ∗ -crossing,” and the exact same argument asabove works. It follows that δ = min( δ , δ ) > p ∈ ( p − δ, p + δ ), completing the proof of continuity on (0 , / Remark 13.
It follows from Theorem 6 and equation (1) that r ∗ c ( p ) is also continuous in p on [0 , / . In this section, we will study the critical value functions of graphs that are constructed byreplacing edges of an infinite tree by a sequence of finite graphs. We will then use several suchconstructions in the proofs of our main results in Section 5.Let us fix an arbitrary sequence D n = ( V n , E n ) of finite connected graphs and, for every n ∈ N , two distinct vertices a n , b n ∈ V n . Let T = ( V , E ) denote the (infinite) regular treeof degree 3, and fix an arbitrary vertex ρ ∈ V . Then, for each edge e ∈ E , we denote theend-vertex of e which is closer to ρ by f ( e ), and the other end-vertex by s ( e ). Let Γ D = ( ˜ V , ˜ E )be the graph obtained by replacing every edge e of Γ between levels n − n ( i.e. , suchthat dist ( s ( e ) , ρ ) = n ) by a copy D e of D n , with a n and b n replacing respectively f ( e ) and s ( e ).Each vertex v ∈ V is replaced by a new vertex in ˜ V , which we denote by ˜ v . It is well knownthat p Γ c = r Γ c (0) = 1 /
2. Using this fact and the tree-like structure of Γ D , we will be able todetermine bounds for p Γ D c and r Γ D c ( p ).First, we define h D n ( p ) = ν E n p ( a n and b n are in the same bond cluster), and prove the fol-lowing, intuitively clear, lemma. Lemma 14.
For any p ∈ [0 , , the following implications hold:a) if lim sup n →∞ h D n ( p ) < / , then p ≤ p Γ D c ;b) if lim inf n →∞ h D n ( p ) > / , then p ≥ p Γ D c . Proof.
We couple Bernoulli bond percolation with parameter p on Γ D with inhomogeneousBernoulli bond percolation with parameters h D n ( p ) on T , as follows. Let η be a random variablewith law ν ˜ Ep , and define, for each edge e ∈ E , W ( e ) = 1 if ˜ f ( e ) and ˜ s ( e ) are connected by apath consisting of edges that are open in η , and W ( e ) = 0 otherwise. The tree-like structureof Γ D implies that W ( e ) depends only on the state of the edges in D e , and it is clear that if dist ( s ( e ) , ρ ) = n , then W ( e ) = 1 with probability h D n ( p ).It is easy to verify that there exists an infinite open self-avoiding path on Γ D from ˜ ρ in theconfiguration η if and only if there exists an infinite open self-avoiding path on T from ρ in theconfiguration W . Now, if we assume lim sup n →∞ h D n ( p ) < /
2, then there exists t < / N ∈ N such that for all n ≥ N , h D n ( p ) ≤ t . Therefore, the distribution of the restriction of W on L = { e ∈ E : dist ( s ( e ) , ρ ) ≥ N } is stochastically dominated by the projection of ν E t on L .This implies that, a.s., there exists no infinite self-avoiding path in W , whence p ≤ p Γ D c by theobservation at the beginning of this paragraph. The proof of b) is analogous.We now turn to the DaC model on Γ D . Recall that for a vertex v , C v denotes the vertexset of the bond cluster of v . Let E a n ,b n ⊂ Ω E n × Ω V n denote the event that a n and b n are inthe same bond cluster, or a n and b n lie in two different bond clusters, but there exists a vertex8 at distance 1 from C a n which is connected to b n by a black path (which also includes that ξ ( v ) = ξ ( b n ) = 1). This is the same as saying that C a n is pivotal for the event that there isa black path between a n and b n , i.e. , that such a path exists if and only if C a n is black. It isimportant to note that E a n ,b n is independent of the color of a n . Define f D n ( p, r ) = P D n p,r ( E a n ,b n ),and note also that, for r > f D n ( p, r ) = P D n p,r (there is a black path from a n to b n | ξ ( a n ) = 1). Lemma 15.
For any p, r ∈ [0 , , we have the following:a) if lim sup n →∞ f D n ( p, r ) < / , then r ≤ r Γ D c ( p ) ;b) if lim inf n →∞ f D n ( p, r ) > / , then r ≥ r Γ D c ( p ) . Proof.
We couple here the DaC model on Γ D with inhomogeneous Bernoulli site percolationon T . For each v ∈ V \ { ρ } , there is a unique edge e ∈ E such that v = s ( e ). Here we denote D e ( i.e. , the subgraph of Γ D replacing the edge e ) by D ˜ v , and the analogous event of E a n ,b n forthe graph D ˜ v by E ˜ v . Let ( η, ξ ) with values in Ω ˜ E × Ω ˜ V be a random variable with law P Γ D p,r .We define a random variable X with values in Ω V , as follows: X ( v ) = ξ ( ˜ ρ ) if v = ρ, E ˜ v is realized by the restriction of ( η, ξ ) to D ˜ v ,0 otherwise.As noted after the proof of Lemma 14, if u = f ( (cid:104) u, v (cid:105) ), the event E ˜ v is independent of thecolor of ˜ u , whence ( E ˜ v ) v ∈ V \{ ρ } are independent. Therefore, as X ( ρ ) = 1 with probability r ,and X ( v ) = 1 is realized with probability f D n ( p, r ) for v ∈ V with dist ( v, ρ ) = n for some n ∈ N , X is inhomogeneous Bernoulli site percolation on T .Our reason for defining X is the following property: it holds for all v ∈ V \ { ρ } that˜ ρ ξ ↔ ˜ v if and only if ρ X ↔ v, (5)where x Z ↔ y denotes that x and y are in the same black cluster in the configuration Z . Indeed,assuming ˜ ρ ξ ↔ ˜ v , there exists a path ρ = x , x , · · · , x k = v in Γ such that, for all 0 ≤ i < k ,˜ x i ξ ↔ ˜ x i +1 holds. This implies that ξ ( ˜ ρ ) = 1 and that all the events ( E ˜ x i ) /
2, then there exists t > / N ∈ N such that for all n ≥ N , f D n ( p, r ) ≥ t . In this case, the distribution of the restriction of X on K = { v ∈ V : dist ( v, ρ ) ≥ N } is stochastically larger than the projection of ν E t on K . Let us further assume that r > X ( ρ ) = 1 with positive probability, and f D n ( p, r ) > n ∈ N . Therefore,under the assumptions lim inf n →∞ f D n ( p, r ) > / r > ρ is in an infinite black cluster in X (and, hence, ˜ ρ is in an infinite black cluster in ξ ) with positive probability, which can onlyhappen if r ≥ r Γ D c ( p ). On the other hand, if lim inf n →∞ f D n ( p, > /
2, then it is clear thatlim inf n →∞ f D n ( p, r ) > / r ≥ r Γ D c ( p )) for all r >
0, which implies that r Γ D c ( p ) = 0.The proof of part a) is similar. 9 Counterexamples
In this section, we study two particular graph families and obtain examples of non-monotonicityand non-continuity of the critical value function.
The results in Section 4 enable us to prove that (a small modification of) the constructionconsidered by H¨aggstr¨om in the proof of Theorem 2.9 in [6] is a graph whose critical coloringvalue is non-monotone in the subcritical phase.
Proof of Proposition 7.
Define for k ∈ N , D k to be the complete bipartite graph with thevertex set partitioned into { z , z } and { a, b, v , v , . . . , v k } (see Figure 1). We call e , e (cid:48) and e , e (cid:48) the edges incident to a and b respectively, and for i = 1 , . . . , k , f i , f (cid:48) i the edges incident to v i . Consider Γ k the quasi-transitive graph obtained by replacing each edge of the tree T by acopy of D k . Γ k can be seen as the tree-like graph resulting from the construction described atbeginning of the section, when we start with the constant sequence ( D n , a n , b n ) = ( D k , a, b ). z v k v e f f k f k e e z e . . .f ba Figure 1: The graph D k .We will show below that it holds for all k ∈ N that p Γ k c > / , (6) r Γ k c (0) < / , and (7) r Γ k c (1 / < / . (8)Furthermore, there exists k ∈ N and p ∈ (0 , /
3) such that r Γ k c ( p ) > / . (9)Proving (6)–(9) will finish the proof of Proposition 7 since these inequalities imply that thequasi-transitive graph Γ k has a non-monotone critical value function in the subcritical regime.Throughout this proof, we will omit superscripts in the notation when no confusion ispossible. For the proof of (6), recall that h D k is strictly increasing in p , and h D k ( p D k ) = 1 / − h D k ( p ) is the ν p -probability of a and b being in two different bond clusters, we havethat 1 − h D k (1 / ≥ ν / ( { e and e (cid:48) are closed } ∪ { e and e (cid:48) are closed } ) . h D k (1 / ≤ /
81, which proves (6).To get (7), we need to remember that for fixed p < p D k , f D k ( p, r ) is strictly increasing in r ,and f D k ( p, r D k ( p )) = 1 /
2. One then easily computes that f (0 , /
3) = 16 / > /
2, whence (7)follows from Lemma 15.Now, define A to be the event that at least one edge out of e , e (cid:48) , e and e (cid:48) is open. Then f D k (1 / , / ≥ P / , / ( E a,b | A ) P / , / ( A ) ≥ P / , / ( C b black | A ) · / , which gives that f D k (1 / , / ≥ / > /
2, and implies (8) by 15.To prove (9), we consider B k to be the event that e , e (cid:48) , e and e (cid:48) are all closed and thatthere exists i such that f i and f (cid:48) i are both open. One can easily compute that P p,r ( B k ) = (1 − p ) (cid:16) − (1 − p ) k (cid:17) , which implies that we can choose p ∈ (0 , /
3) (small) and k ∈ N (large) such that P p ,r ( B k ) > /
18. Then, f D k ( p , /
3) = P p ,r ( E a,b | B k ) P p ,r ( B k ) + P p ,r ( E a,b | B ck )(1 − P p ,r ( B k )) < (2 / · · / / , whence inequality (9) follows with these choices from Lemma 15, completing the proof. Proof of Proposition 3.
For n ∈ N , let D n be the graph depicted in Figure 2, and let G beΓ D constructed with this sequence of graphs as described at the beginning of Section 4. v ... bna Figure 2: The graph D n .It is elementary that lim n →∞ h D n ( p ) = p , whence p Gc = 1 / p = 0 is subcritical. Since lim n →∞ f D n (0 , r ) = r , Lemma 15 gives that r Gc (0) = 1 / √
2. On theother hand, lim n →∞ f D n ( p, r ) = p + (1 − p ) r for all p >
0, which implies by Lemma 15 that for p ≤ / r Gc ( p ) = 1 / − p − p → / p →
0, so r Gc is indeed discontinuous at 0 < p Gc .In the rest of this section, for vertices v and w , we will write v ↔ w to denote that thereexists a path of open edges between v and w . Our proof of Theorem 4 will be based on theLemma 2.1 in [10], that we rewrite here: Lemma 16.
There exists a sequence G n = ( V n , E n ) of graphs and x n , y n ∈ V n of vertices( n ∈ N ) such that1. ν E n / ( x n ↔ y n ) > for all n ; . lim n →∞ ν E n p ( x n ↔ y n ) = 0 for all p < / , and3. there exists ∆ < ∞ such that, for all n , G n has degree at most ∆ . Lemma 16 provides a sequence of bounded degree graphs that exhibit sharp threshold-typebehavior at 1 /
2. We will use such a sequence as a building block to obtain discontinuity at 1 / Proof of Theorem 4.
We first prove the theorem in the case p = 1 /
2. Consider the graph G n = ( V n , E n ) , x n , y n ( n ∈ N ) as in Lemma 16. We construct D n from G n by adding to it oneextra vertex a n and one edge { a n , x n } . More precisely D n has vertex set V n ∪ { a n } and edge set E n ∪ { a n , x n } . Set b n = y n and let G be the graph Γ D defined with the sequence ( D n , a n , b n )as in Section 4.We will show below that there exists r > r such that the graph G verify the followingthree properties:(i) 1 / < p Gc (ii) r Gc ( p ) ≥ r for all p < / r Gc (1 / ≤ r .It implies a discontinuity of r Gc at 1 / < p Gc , finishing the proof.One can easily compute h D n ( p ) = pν E n p ( x n ↔ y n ). Since the graph G n has degree at most ∆and the two vertices x n , y n are disjoint, the probability ν E n p ( x n ↔ y n ) cannot exceed 1 − (1 − p ) ∆ .This bound guarantees the existence of p > / n such that h D n ( p ) < / n , whence Lemma 14 implies that 1 / < p ≤ p Gc .For all p ∈ [0 , f D n ( p, r ) ≤ ( p + r (1 − p )) (cid:0) ν E n p ( x n ↔ y n ) + r (1 − ν E n p ( x n ↔ y n )) (cid:1) . which gives that lim n →∞ f D n ( p, r ) < (cid:0) r +12 (cid:1) · r . Writing r the positive solution of r (1 + r ) = 1, weget that lim n →∞ f D n ( p, r ) < / p < /
2, which implies by Lemma 15 that r Gc ( p ) ≥ r .On the other hand, f D n (1 / , r ) ≥ ν E n p ( x n ↔ y n ) (cid:0) r (cid:1) , which gives by Lemma 16 thatlim n →∞ f D n (1 / , r ) > · r . Writing r such that (1 + r ) = 1, it is elementary to checkthat r < r and that lim n →∞ f D n (1 / , r ) > /
2. Then, using Lemma 15, we conclude that r c (1 / ≤ r . Acknowledgments.
We thank Jeff Steif for suggesting (a variant of) the graph that appearsin the proof of Theorem 4. V.B. and V.T. were supported by ANR grant 2010-BLAN-0123-01.
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