Metastable Behavior of Bootstrap Percolation on Galton-Watson Trees
MMetastable Behavior of Bootstrap Percolation on Galton-WatsonTrees
Assaf Shapira ∗ Abstract
Continuing the study of [6] on the critical probability of the bootstrap percolation on Galton-Watsontrees, we analyze the metastable states near criticality. We find that, depending on the exact choice ofthe offspring distribution, it is possible to have several distinct metastable states, with varying scaling oftheir duration while approaching criticality.
Bootstrap percolation is a deterministic dynamics in discrete time first introduced in [7] in order to modeldisordered magnetic systems, and broadly studied since in many different contexts. Fix a graph G and aparameter r ∈ N . Each vertex of the graph can be in one of two states – infected or healthy, which areinitially distributed independently with probabilities p and q = 1 − p . At each time step we update thesestates, such that the infected vertices remain infected, and a healthy vertex becomes infected if it has at least r infected neighbors. One may also consider more general infection conditions, such as the oriented bootstrappercolation – when the graph G is oriented, and we require at least r edges to point at infected vertices.Bootstrap percolation on various deterministic graphs has been the subject of extensive research. Onthe grid [ n ] d , the probability that all vertices are eventually infected, as a function of p (or equivalently q ),has been profoundly studied in [1, 11, 2]. For ( d + 1) -regular infinite trees, with ≤ r ≤ d , it is shownin [3] that a phase transition occurs. Defining q c to be the supremum over all q such that starting withprobability q to be healthy all vertices end up being infected with probability , an explicit expression for q c is found, and it is furthermore proven that q c lies in the open interval (0 , . In addition, it is determined,depending on d and on r , when the transition is continuous and when it is discontinuous. In [5] the details ofthe metastability properties are studied, describing the time evolution of the probability that the root stayshealthy near criticality.Random environments have also been of interest in this field, e.g., the bootstrap percolation on a pollutedgrid [10, 9], the random graph G n,p [13], the random regular graph [4, 12], and the Galton-Watson tree [6].In this paper, we will analyze the metastability of the bootstrap percolation on a directed Galton-Watsontree, i.e., the time behavior near criticality of the probability that the root is infected. In Section 3.1 we ∗ I acknowledge the support of the ERC Starting Grant 680275 MALIG a r X i v : . [ m a t h . P R ] A p r resent an interpretation of this probability as the almost sure prevalence – the limiting ratio of infectedvertices. In Section 3.2 we will study the zoology of the metastabilities for different offspring distributions,showing that this model introduces a vast variety of possible behaviors. Finally, in Section 5 we comment onother phase transitions that may occur. Fix an infection threshold r ≥ , and consider a Galton-Watson tree G whose offspring distribution issupported on r, r + 1 , . . . That is, defining ξ k to be the probability that a vertex has k children, we require ξ k = 0 for k < r .In the beginning, we decide for each vertex of G whether it is infected or healthy, independently withprobabilities p and q = 1 − p respectively. Then, at each time step t , a healthy vertex will get infected ifit has at least r infected children. Let us denote by φ Gt the (random) probability that the root is healthyat time t , so in particular φ G = q . Note also that φ Gt is decreasing in t . The expected value of φ Gt over allgraphs G , generated with offspring distribution ξ , will be denoted φ ξt .One particular case, that has been studied in [3, 5, 8, 7], is the case of a rooted ( d + 1) -regular tree, i.e., ξ k = k = d . Here, one can find φ dt recursively using the relation φ dt +1 = h d (cid:0) φ dt (cid:1) ; (2.1) h d ( x ) = q P [ Bin ( d, − x ) ≤ r − . (2.2)For the GW tree, such a recursion still holds for the expected value φ ξt : φ ξt +1 = h ξ (cid:16) φ ξt (cid:17) ; (2.3) h ξ ( x ) = ∞ (cid:88) k = r ξ k h k ( x ) . (2.4) φ t The relation in equation 2.3 allows us to find the expected value of φ Gt , but for a specific realization of G , φ Gt may differ from that value. For example, fixing t , there is a nonzero probability that a finite neighborhood ofthe root will have many vertices of high degree, which will result in a smaller φ Gt . However, we will see that φ ξt describes almost surely another observable – the prevalence, i.e., the limiting fraction of infected vertices.First, denote by B ( R ) the ball of radius R around the root. We can then define the R -prevalence at time t as ρ R ( t ) = |{ infected vertices in B ( R ) at time t }|| B ( R ) | . It is natural to expect ρ R ( t ) to be close to − φ ξt , and this is indeed the case, as shown in the following2roposition: Proposition 1.
Fix t . Then lim R →∞ ρ R ( t ) = 1 − φ ξt almost surely (in both the graph and the initial statemeasures). Following [3, 6], we define the critical probability q c = sup [0 , (cid:8) q : φ ξ ∞ = 0 (cid:9) . (3.1)In order to analyze this criticality, define g k ( x ) = h k ( x ) qx , (3.2) g ξ ( x ) = h ξ ( x ) qx . (3.3)In [6], the following fact is shown: Fact 1.
Fix ξ . Then:1. For a given q , φ ξ ∞ is the maximal solution in [0 , of g ξ ( x ) = q , and if no such solution exists.2. q c = [0 , g ξ ( x ) . We will consider here the behavior near criticality, at q slightly smaller than q c . Definition 1.
For < x < and some positive δ , the δ -entrance time of x is τ − x,δ ( q ) = min { t : φ ξt < x + δ } , and the δ -exit time is defined as τ + x,δ ( q ) = min { t : φ ξt < x − δ } . Definition 2.
Fix δ > . We say that the critical point is δ - ( ν , . . . , ν n ) -metastable at x > · · · > x n > if,for q (cid:37) q c , the following hold:1. τ − x ,δ = O (1) .2. log (cid:16) τ + xi,δ − τ − xi,δ (cid:17) log( q c − q ) q (cid:37) q c −−−→ − ν i for i = 1 , . . . , n .3. τ − x i +1 ,δ − τ + x i ,δ = O (1) for i = 1 , . . . , n and x n +1 = 0 .We say that the critical point is ( ν , . . . , ν n ) -metastable at x > · · · > x n if it is δ - ( ν , . . . , ν n ) -metastable at x > · · · > x n for small enough δ . See Figure 3.1.The following theorem gives a full classification of the metastability properties:3igure 3.1: A schematic picture of φ ξt as a function of t for a ( ν , . . . , ν n ) -metastable criticality at x > · · · >x n . Theorem 1.
Fix ξ . Then the metastable behavior is determined by one of the following cases:Case 1. g ξ attains its maximum at . In this case the critical probability is .Case 2. g ξ has a unique maximum at . In this case the phase transition is continuous. At the criticalpoint log( φ ξt )log t t →∞ −−−→ − ν , (3.4) where ν is determined by the asymptotic expansion g ξ ( x ) = q c − Cx ν + o ( x ν ) .Case 3. The maximum of g ξ is attained at the points x , . . . , x n for > x > · · · > x n > , and possiblyalso at . In this case the phase transition is discontinuous. For i = 1 , . . . , n we may write around x i g ξ ( x ) = 1 q c − C i ( x − x i ) ν i + o (cid:16) ( x − x i ) ν i (cid:17) , (3.5) with some C i > .Then the critical point is ( ν , . . . , ν n ) -metastable at x > · · · > x n .Remark . In the first case, where the critical probability is , it is not clear whether or not an asymptoticexpansion exists, since g ξ is not guaranteed to be analytic. When it does exist, one can recover a resultsimilar to Case 3.Finally, we show the main result – that the different metastability behaviors described above are possible: Theorem 2.
1. Let ν ∈ N . Then there exists ξ such that the phase transition is continuous, and satisfies equation 3.4at criticality. 4. Let ( ν , . . . , ν n ) ∈ N n . Then there exist ξ and x > · · · > x n such that the critical point is ( ν , . . . , ν n ) -metastable at x > · · · > x n . Proof of Proposition 1.
The idea of the proof is to notice that the main contribution to the prevalence comesfrom the sites close to the boundary, and then use their independence. Thus, we fix a width w , and consider ρ R,w ( t ) = |{ infected vertices in B ( R ) \ B ( R − w ) at time t }|| B ( R ) \ B ( R − w ) | . First, we claim that ρ R ( t ) is approximated by ρ R,w ( t ) for large w . More accurately, we have | B ( R − w ) | ≤ − w | B ( R ) | , which also implies that the number of infected vertices in B ( R ) \ B ( R − w ) is the same as thenumber of infected vertices in B ( R ) , up to a correction of order − w | B ( R ) | . Then ρ R ( t ) = ρ R,w ( t ) + O (cid:0) − w (cid:1) . (4.1)We would now like to bound the distance between ρ R,w ( t ) and − φ ξt . Let ε > , and, by equation 4.1,take w big enough such that | ρ R ( t ) − ρ R,w ( t ) | < ε uniformly in R . Note that ρ R,w ( t ) is a weighted averageof the w random variables ρ R, ( t ) , ρ R − , ( t ) , . . . , ρ R − w +1 , ( t ) , and consider one of these variables, ρ r, ( t ) .This variable is the average of the random variables v is infected for all vertices v of distance r from the root,and since these are independent Bernoulli random variables with mean − φ ξt , and since there are at least R − w +1 such variables, we can use a large deviation estimate, yielding P (cid:104)(cid:12)(cid:12)(cid:12) ρ r, ( t ) − (cid:16) − φ ξt (cid:17)(cid:12)(cid:12)(cid:12) > ε (cid:105) ≤ e − c R − w +1 for a positive c that only depends on ε and on φ ξt . Since for − φ ξt to be far from ρ R,w ( t ) it must be far fromat least one of the variables ρ R, ( t ) , ρ R − , ( t ) , . . . , ρ R − w +1 , ( t ) , we have P (cid:104)(cid:12)(cid:12)(cid:12) ρ R,w ( t ) − (cid:16) − φ ξt (cid:17)(cid:12)(cid:12)(cid:12) > ε (cid:105) ≤ we − c R − w +1 . (4.2)Hence, ρ R ( t ) is ε -close to − φ ξt with probability larger than − we − c R − w +1 , which concludes the proofby the Borel-Cantelli lemma.Before proving Theorems 1 and 2, we will need a couple of small results. Claim . g k is a polynomial of degree k − , whose lowest degree monomial is of degree k − r . Proof.
By equations 3.2 and 2.2 g k ( x ) = P [ Bin ( k, − x ) ≤ r − x = r − (cid:88) i =0 (cid:18) ki (cid:19) (1 − x ) i x k − i − ; k − r and k − . The coefficient of x k − r is (cid:0) kr − (cid:1) (cid:54) = 0 , and thecoefficient of x k − is (cid:80) r − i =0 (cid:0) ki (cid:1) ( − i , which is also nonzero since < r − < k . This concludes the proof. Claim . g r ( x ) , . . . , g m ( x ) , x m − r +1 , . . . , x m − is a basis of the linear space of polynomials of degree smalleror equal to m − . Proof.
Denote v ( x ) = g r ( x ) , . . . , v m − r +1 ( x ) = g m ( x ) , v m − r +2 ( x ) = x m − r +1 , v m ( x ) = x m − . By Claim1, all v ’s are of degree smaller or equal to m − . Moreover, the matrix whose ( i, j ) entry is the coefficientof x j in the polynomial v i is upper triangular, with nonzero diagonal. This shows that { v i } mi =1 is indeed abasis.We will also use the following result from [6]: Claim . For ξ k = r − k ( k − , g ξ ( x ) = 1 .We are now ready to prove Theorems 1 and 2. Proof of Theorem 1.
First, we note that g k (1) = 1 for all k , so in particular the series (cid:80) ∞ k = r ξ k g k ( x ) convergesat x = 1 . By Claim 1, the monomials of degree up to n of the partial sum (cid:80) Nk = r ξ k g k ( x ) are fixed once N > n + r . From these two facts we conclude that g ξ ( x ) is analytic in ( − , and continuous at . Thus,cases 1, 2 and 3 exhaust all possibilities.The result will then follow from general arguments of dynamical systems near a bifurcation point. Sincethe exact calculations are a bit tedious, we only give here a short sketch of the argument, referring to theappendix for the complete proof.For case 2, the expression φ t +1 = φ t − Cq c φ ν +1 t + o ( φ ν +1 t ) could be estimated by comparing to the differential equationd φ d t = − Cq c φ ν +1 t . This equation could be solved explicitly, yielding the asymptotics of equation 3.4.For case 3, the approximate differential equation will bed φ d t = − x i q c ( q c − q ) − C i q c x i ( φ − x i ) ν i . The solution of this equation has a plateau around x i , whose length diverges as ( q c − q ) − νi . Proof of Theorem 2.
For the first part, it will be enough to show that there exist an offspring distribution ξ and a polynomial Q ( x ) = b + · · · + b r − x r − such that1. g ξ ( x ) = Const − x ν Q ( x ) .2. Q ( x ) > for all x ∈ [0 , . 6his ξ , according to Theorem 1 and the fact that b > , will indeed satisfy equation 3.4. Ratherthan ξ , it will be easier to find a sequence { χ k } ∞ k = r with a finite sum together with a polynomial P ( x ) = a + · · · + a r − x r − , such that1. g χ ( x ) = (cid:80) k χ k g k ( x ) = 1 − x ν P ( x ) .2. χ k ≥ .3. P ( x ) > for all x ∈ [0 , .Taking ξ = (cid:80) χ k χ k will then conclude the proof. Let χ k = r − k ( k − r ≤ k ≤ ν + r − k ≥ ν + r . (4.3)Using Claim 3, we may write g χ ( x ) = 1 − ∞ (cid:88) k = ν + r r − k ( k − g k ( x ) . By Claim 1 g χ is a polynomial of degree ν + r − , therefore (cid:80) ∞ k = ν + r r − k ( k − g k ( x ) equals a polynomial ofdegree ν + r − . Using again Claim 1, we can define the polynomial P ( x ) = ∞ (cid:88) k = ν + r r − k ( k − g k ( x ) x ν . It is left to show that P ( x ) > for all x ∈ [0 , . By equations 3.2 and 2.2, P ( x ) is non-negative andcould only vanish at x = 0 . But by Claim 1, P (0) = r − ν + r )( ν + r − (cid:16) g ν + r ( x ) x ν (cid:17) x =0 (cid:54) = 0 . This concludes the firstpart. Remark . Note that, by Claim 2, we can define the projection Pr from the space of polynomials of degree atmost r + ν − to its subspace spanned by x ν , . . . , x ν + r − with kernel spanned by g r ( x ) , . . . , g ν + r − ( x ) . Definealso M to be the map from the space of polynomials of degree at most r − to the space of polynomials ofdegree at most r + ν − given by the multiplication by x ν . Then the first of the conditions above can bewritten as Pr M P = Pr . Since Pr ◦ M is bijective, this equation has a unique solution; and what we have shown in the proof isthat this solution satisfies the necessary positivity conditions.We will now prove the second part of the theorem. In analogy with the first one, we will find ξ , Q ( x ) = b + · · · + b r − x r − and x > · · · > x n such that:1. g ξ ( x ) = Const − ( x − x ) ν . . . ( x − x n ) ν n Q ( x ) .2. Q ( x ) > for all x ∈ [0 , .Similarly to the previous part, we will look for { χ k } ν + r − k = r and P ( x ) = a + · · · + a r − x r − satisfying:7. g χ ( x ) = (cid:80) k χ k g k ( x ) = 1 − ( x − x ) ν . . . ( x − x n ) ν n P ( x ) .2. χ k > .3. P ( x ) > for all x ∈ [0 , .Note that choosing ν = 2 ν + · · · +2 ν n , χ k (defined in equation 4.3) is strictly positive for r ≤ k ≤ ν + r − .Since P was required to be strictly positive, we may hope that also after adding a small perturbation ( x , . . . , x n ) around there still exists a positive solution P . More precisely, let us denote by M x ,...,x n themultiplication by ( x − x ) ν . . . ( x − x n ) ν n , acting on the polynomials of degree at most r − . In particular,for x , . . . , x n = 0 this is M defined in Remark 2. Then, we want to show that the solution ofPr M x ,...,x n P = Pr satisfies the positivity conditions 2 and 3. By continuity of the determinant, when ( x , . . . , x n ) is in a smallneighborhood of the operator Pr M x ,...,x n is invertible. Moreover, in an even smaller neighborhood of thepolynomial ( Pr M x ,...,x n ) − Pr will satisfy the positivity condition 3 – matrix inversion is continuous, andthe set of polynomials satisfying this condition is open and contains ( Pr M ) − Pr by the first part of theproof. Finally, since coordinate projections of − ( x − x ) ν . . . ( x − x n ) ν n ( Pr M x ,...,x n ) − Pr with respectto the basis defined in Claim 2 are continuous in ( x , . . . , x n ) , and since for ( x , . . . , x n ) = 0 condition 2 issatisfied, by taking ( x , . . . , x n ) in a further smaller neighborhood of we are guaranteed to find a polynomial P satisfying the required conditions. φ t Consider, for example, r = 2 and ξ k = k =2 + k =5 . The function g ξ ( x ) is maximal at g ξ (0) = , thenit has a local minimum, followed by a local maximum (see Figure 5.1). In this case, recalling Fact 1, φ ξt willhave a discontinuity at this local maximum, that is, a second phase transition occurs. We may then expectthat one can find ξ giving rise to as many (decreasing) local maxima of g ξ as we wish: Conjecture 1.
Let ν (1)1 , . . . , ν (1) n , ν (2)1 , . . . , ν (2) n , . . . , ν ( m ) n m . Then there exists g ξ , { q i } mi =1 , (cid:110) x ( i ) j (cid:111) ≤ i ≤ m, ≤ j ≤ n i such that q i is a critical point which is (cid:16) ν ( i )1 , . . . , ν ( i ) n i (cid:17) -metastable at x ( i )1 , . . . , x ( i ) n i . Another possible phase transition, studied in [8] for the case of regular trees, is when infinite infected clustersstart to appear, but the prevalence is still smaller than . Following the proof of Proposition 3.9 in [8], onesees that it applies also for the bootstrap percolation on GW trees, showing that the critical probability q ( ∞ ) c above which infinite clusters no longer appear is strictly bigger than q c defined in equation 3.1, unless ξ k = r . 8igure 5.1: g ξ for r = 2 and ξ k = k =2 + k =5 . We show three lines q for three parameters q , intersecting g ξ at φ ξ ∞ . One sees here the discontinuity when q equals the value of g ξ at the local maximum. The problem of bootstrap percolation in disordered systems raises many questions. Related to the workpresented here, one may be interested in the metastable regime for other systems, such as G n,p or therandom regular graph. Another natural problem is the analysis of the bootstrap percolation on the randomgraph with a given degree sequence, that has a GW local structure, with analogy to the regular tree structureof the random regular graph. Acknowledgments
I would like to thank Cristina Toninelli for the introduction of the subject and for useful discussion, and toLucas Benigni for his help throughout the writing of this paper.
Appendix
This paper concerns with the analysis of a phase transition originating in the appearance of a new fixed pointfor a certain recurrence relation, i.e., a bifurcation. In this appendix, we will try to understand in a moregeneral context the time scaling in systems of that type. Let us then consider a sequence of reals { x n } ∞ n =0 ,defined by the value x and a recursion formula for n > : x n = f ( x n − ) . (A1)9e will also fix now some positive δ < , that will be used throughout this appendix as the windowaround the new fixed point in which we are interested.First, we will study the time scaling at the bifurcation point, when the new fixed point is first created.In this case, we may expect f to be tangent to the identity function at the fixed point, so we will start ourdiscussion with the following assumptions: Assumption A1. f has a fixed point y , such that for y ∈ ( y , y + δ ) : y − c ( y − y ) α ≤ f ( y ) ≤ y − c ( y − y ) α , for some α > , < c ≤ c < δ − ( α − . Assumption A2. x ∈ ( y , y + δ ) . We first mention the following fact:
Fact A1.
The sequence is decreasing and bounded from below by y .Proof. By Assumption A1, x n +1 < x n whenever x n ∈ ( y , y + δ ) . Moreover, x n +1 − y ≥ x n − y − c ( x n − y ) α = ( x n − y ) (cid:16) − c ( x n − y ) α − (cid:17) ≥ ( x n − y ) (cid:0) − cδ α − (cid:1) > . Therefore, since x ∈ ( y , y + δ ) by assumption A2, the entire sequence is in the interval ( y , y + δ ) , andit is decreasing.The following theorem will describe the asymptotic of the sequence: Theorem A1.
Let { x n } ∞ n =0 be the sequence defined in equation A1, satisfying Assumptions A1 and A2.Then y + a ( n + n ) − α − ≤ x n ≤ y + an − α − , where a = (cid:104) ( α −
1) (1 − δ ) − α c (cid:105) − α − , a = [( α − c ] − α − , and n = ( x − y ) − α ( α − − δ ) − α c are all positive constants.Proof. Let us first define a sequence t n = ( x n − y ) − α , and note that t n is positive for all n . Then usingFact A1 and Assumption A1, fixing c (cid:48) = ( α −
1) (1 − δ ) − α c and c (cid:48) = ( α − c , we can estimate: t n = ( f ( x n − ) − y ) − α t n = ( f ( x n − ) − y ) − α ≤ ( x n − − c ( x n − − y ) α − y ) − α ≥ ( x n − − c ( x n − − y ) α − y ) − α = (cid:18) t − α n − − ct α − α n − (cid:19) − α = (cid:18) t − α n − − ct α − α n − (cid:19) = t n − (cid:0) − ct − n − (cid:1) − α = t n − (cid:0) − ct − n − (cid:1) − α ≤ t n − (cid:0) c (cid:48) t − n − (cid:1) ≥ t n − (cid:0) c (cid:48) t − n − (cid:1) = t n − + c (cid:48) ; = t n − + c (cid:48) .10e have used here the fact that, for any < z < δ < , we can approximate (1 − z ) − α using its derivativesat and at δ : − (1 − α ) ≤ (1 − z ) − α − z ≤ − (1 − α )(1 − δ ) − α . We then also use ct − n − = ( x n − y ) α − < δ α − < δ .Finally, x n = y + t − α − n x n ≥ y + (cid:16) ( x − y ) − α + c (cid:48) n (cid:17) − α − ≤ y + (cid:16) ( x − y ) − α + c (cid:48) n (cid:17) − α − = y + (cid:16) c (cid:48) (cid:16) n + ( x − y ) − α c (cid:48) (cid:17)(cid:17) − α − ≤ y + an − α − ; = y + a ( n + n ) − α − . Next, we will be interested in the behavior near the bifurcation point, just before the new fixed pointappears. For this purpose we will consider a family { x εn } ∞ n =0 of sequences, each defined by the value x ε anda recursion formula for n > : x εn = f ε (cid:0) x εn − (cid:1) , (A2)and assume: Assumption A3.
There is a point y such that for | y − y | < δ and ε < ε y − c ( y − y ) α − ε ≤ f ε ( y ) ≤ y − c ( y − y ) α − ε, for an integer α > and positive constants c and c . Assumption A4. < x − y < δ . In order to study the asymptotic behavior of x εn for small values of ε , we will need the following definition: Definition A1.
The exit time N δ ( ε ) is the minimal n such that x εn < y − δ .Replacing Fact A1 will be the following: Fact A2.
For all ε < ε , N δ ( ε ) is finite, and for n < N δ ( ε ) the sequence x εn is decreasing.Proof. By Assumption A3, for n < N δ ( ε ) , if x εn < y + δ then x εn +1 < x εn < y + δ . Hence, the sequenceremains in the interval ( y − δ, y + δ ) an long as n < N δ ( ε ) . Since in this interval the sequence is decreasing,the result follows by Assumption A4.For our analysis, we will compare this sequence to the solution of the following differential equations, thatwill approximate x εn − y : d ζ d s = − c ζ α − ε, d ζ d s = − c ζ α − ε,ζ (0) = z ε = x ε − y ; ζ (0) = z ε = x ε − y . ζ is strictly decreasing, and in particular one can define its inverse t : [ −∞ , z ε ] → [0 , ∞ ] , and τ n = t ( x εn − y ) . t and τ n will be defined in the same manner. Note that these all depend on ε , even thoughthis dependence is omitted from the notation. The next lemma will show that the continuous crossing times τ n and τ n are close to the discrete one, namely n . Lemma A2.
For all n ≤ N δ ( ε ) , (1 − κ c,δ,ε ) n ≤ τ n ≤ τ n ≤ (cid:0) κ c,δ,ε (cid:1) n, where for all c > , κ c,δ,ε = max( C ε α − , αδ α − ) . C is a positive constant depending on δ, c and ε givenexplicitly in the proof, and bounded when δ and ε are not too big. For example, if ε < and cδ α − < , C < (3 + 4 α c ) α .Proof. Let z n = x n − y . Then τ n = t ( f ε ( x n − ) − y ) ≤ t (cid:0) z n − − cz αn − − ε (cid:1) = z n − − cz αn − − ε ˆ z d z − cz α − ε = t ( z n − ) − z n − − cz αn − − ε ˆ z n − d zcz αn − + ε − z n − − cz αn − − ε ˆ z n − (cid:18) d zcz α + ε − d zcz αn − + ε (cid:19) = τ n − + 1 − z n − − cz αn − − ε ˆ z n − (cid:18) d zcz α + ε − d zcz αn − + ε (cid:19) . In order to study the error term, we will use the following estimation:
Claim . Fix w ∈ ( − δ, δ ) , and c > . Let I = w ˆ w − cw α − ε (cid:18) cw α + ε − cw α + ε (cid:19) d w. Then | I | ≤ κ c,δ,ε . Proof.
We will first consider the case in which the integration interval passes through , that is < w 1+ 12 α , rather thanjust bounds on its limsup and liminf. Such a direct application of the theorem, however, forces us to choosean initial condition x ε that converges to y as ε goes to . To overcome this issue, we can use the estimation14bove with a fixed δ until x n reaches δ ε , which happens at n of order δ ε ´ z d z − cz α − ε (cid:28) ε − α . Then restartthe dynamics using the estimation with δ ε until reaching − δ ε , which takes an order ε − α of steps, andthen using again the estimation for our fixed δ show that the number of steps required to reach − δ is muchsmaller than ε − α . This would yield lim ε → N δ ( ε ) ε − α = ∞ ˆ −∞ d ucu α + 1 . References [1] Michael Aizenman and Joel L Lebowitz. Metastability effects in bootstrap percolation. Journal ofPhysics A: Mathematical and General , 21(19):3801, 1988.[2] József Balogh, Béla Bollobás, Hugo Duminil-Copin, and Robert Morris. The sharp threshold for boot-strap percolation in all dimensions. Transactions of the American Mathematical Society , 364(5):2667–2701, 2012.[3] József Balogh, Yuval Peres, and Gábor Pete. Bootstrap percolation on infinite trees and non-amenablegroups. Combinatorics, Probability and Computing , 15(5):715–730, 2006.[4] József Balogh and Boris G Pittel. Bootstrap percolation on the random regular graph. Random Structures& Algorithms , 30(1-2):257–286, 2007.[5] Marek Biskup and Roberto H Schonmann. Metastable behavior for bootstrap percolation on regulartrees. Journal of Statistical Physics , 136(4):667–676, 2009.[6] Béla Bollobás, Karen Gunderson, Cecilia Holmgren, Svante Janson, and Michał Przykucki. Bootstrappercolation on Galton-Watson trees. Electronic Journal of Probability , 19, 2014.[7] John Chalupa, Paul L Leath, and Gary R Reich. Bootstrap percolation on a Bethe lattice. Journal ofPhysics C: Solid State Physics , 12(1):L31, 1979.[8] Luiz RG Fontes and Roberto H Schonmann. Bootstrap percolation on homogeneous trees has 2 phasetransitions. Journal of Statistical Physics , 132(5):839–861, 2008.[9] Janko Gravner and Alexander E Holroyd. Polluted bootstrap percolation with threshold two in alldimensions. arXiv preprint arXiv:1705.01652 , 2017.[10] Janko Gravner and Elaine McDonald. Bootstrap percolation in a polluted environment. Journal ofStatistical Physics , 87(3):915–927, 1997.[11] Alexander E Holroyd. Sharp metastability threshold for two-dimensional bootstrap percolation. Proba-bility Theory and Related Fields , 125(2):195–224, 2003.1512] Svante Janson. On percolation in random graphs with given vertex degrees.