The devil's staircase for chip-firing on random graphs and on graphons
aa r X i v : . [ m a t h . C O ] A p r The devil’s staircase for chip-firing on random graphsand on graphons
Viktor Kiss a,1 , Lionel Levine b,2 , Lilla T´othm´er´esz c,3 a Alfr´ed R´enyi Institute of Mathematics, Re´altanoda u. 13–15, H-1053 Budapest, Hungary b Cornell University, Ithaca, New York 14853-4201, USA c MTA-ELTE Egerv´ary Research Group, P´azm´any P´eter s´et´any 1/C, Budapest, Hungary
Abstract
We study the behavior of the activity of the parallel chip-firing upon increasingthe number of chips on an Erd˝os–R´enyi random graph. We show that in varioussituations the resulting activity diagrams converge to a devil’s staircase as weincrease the number of vertices. Our method is to generalize the parallel chip-firing to graphons, and to prove a continuity result for the activity. We alsoshow that the activity of a chip configuration on a graphon does not necessarilyexist, but it does exist for every chip configuration on a large class of graphons.
Keywords:
Abelian sandpile model, parallel chip-firing, devil’s staircase,random graph, graphon
1. Introduction
In this paper, we study the behavior of the activity of the parallel chip-firingupon increasing the number of chips in the system. Numerical experiments ofBagnoli, Cecconi, Flammini, and Vespignani [2] suggested that for planar grids,upon increasing the number of chips in the system, the activity asymptoticallyincreases as a Devil’s staircase. Later, Levine [8] proved a similar statementfor complete graphs, i.e., if we take a sequence of complete graphs whose sizetends to infinity, and a sequence of chip configurations on them that convergein a certain sense, then the activity diagrams tend to a Devil’s staircase. Inthis paper, we prove analogous statements in various situations for sequences ofErd˝os–R´enyi random graphs. Our method is to generalize the parallel chip-firing
Email addresses: [email protected] (Viktor Kiss), [email protected] (Lionel Levine), [email protected] (Lilla T´othm´er´esz) Partially supported by NSF grant DMS-1455272, and by the National Research, Devel-opment and Innovation Office – NKFIH, grants no. 104178, 124749, 129211, and 128273. Partially supported by NSF grant DMS-1455272. Partially supported by NSF grant DMS-1455272, and by the National Research, Devel-opment and Innovation Office – NKFIH, grants no. 128673, and 132488.
Preprint submitted to Elsevier o graphons, and then to prove a continuity theorem for the activity. Levine’sresults can be interpreted as a Devil’s staircase result for the constant graphon.Using our continuity theorem, we can handle the case of sequences of graphonsconverging to a constant graphon.
Let G be a graph with vertex set V ( G ) and edge set E ( G ). We will denotethe number of edges connecting vertex u and v by e G ( u, v ), and the degree of avertex v by deg G ( v ). We will often consider Erd˝os–R´enyi random graphs. By G ( n, p ) we denote the random graph on n vertices, where each edge is presentindependently with probability p .For a graph G , a chip configuration assigns to each vertex a non-negativeamount of chips. Hence a chip configuration is a function σ : V ( G ) → R ≥ ,where R ≥ = { x ∈ R : x ≥ } . In the literature, a chip-configuration isusually considered to be integer-valued, but since we will be interested in thechange of dynamics as we gradually increase the amount of chips, we choose toallow nonintegrality. For two chip configurations σ and σ ′ , σ ≥ σ ′ means that σ ( v ) ≥ σ ′ ( v ) for each vertex v .Firing a node v means that the fired vertex passes a chip along each edgeincident to it, i.e., the chip configuration σ gets modified to σ ( u ) + e G ( u, v ) if u = v,σ ( u ) − deg G ( v ) if u = v. During a step of the parallel chip-firing, each vertex v of G fires f ( v ) = j σ ( v )deg G ( v ) k times, where we call f = f ( G, σ ) the firing vector of σ . The resultingconfiguration is U σ ( v ) = σ ( v ) − deg G ( v ) f ( v ) + X u ∈ V ( G ) f ( u ) e G ( u, v ) . We will denote by U n σ for n ∈ N the chip configuration after n steps of theparallel chip-firing.We denote by u n ( v ) = u n ( G, σ )( v ) the number of times v fired duringthe first n turns, and call this function the odometer , that is, u n ( G, σ )( v ) = P n − k =0 f ( G, U k σ )( v ).It is easy to see that a parallel chip-firing started from the configuration σ on a graph G eventually enters a periodic state, and if G is connected then eachvertex fires the same number of times in a period. Hence, lim n →∞ u n ( v ) n existsand is the same for each v ∈ V ( G ). We call this quantity the activity of σ anddenote it by a ( G, σ ).We will be interested in the way the activity changes when we add a smallamount of chips to each node. The activity diagram of G and σ is s ( y ) = s ( G, σ )( y ) = a ( G, σ + y · deg G ). Numerical experiments of [2] suggested that forplanar grids of growing size, the activity diagrams tend to a Devil’s staircase.We will call a function c : [ a, b ] → [0 ,
1] a
Devil’s staircase , if c ( a ) = 0, c ( b ) =2, it is continuous, nondecreasing, but locally constant on an open dense set.Levine [8] proved such a phenomenon for complete graphs, i.e. that if we takea sequence of complete graphs whose size tends to infinity, and a sequence ofchip configurations on them that converge in a certain sense, then (with a mildassumption on the limiting chip configuration) the activity diagrams tend to aDevil’s staircase. We take this analysis further, and are able to handle the caseof sequences of Erd˝os–R´enyi random graphs. To analyze the activity diagram on a (dense) graph, we use the theory ofgraphons. In order to do so, we introduce parallel chip-firing on graphons. Ona graphon, the parallel chip-firing is not necessarily eventually periodic, and theactivity of a chip configuration might not exist (see Proposition 4.1). However,we show that if there is a lower bound on the degrees in the graphon then theactivity exists for any chip configuration (see Theorem 4.11).We show a continuity theorem for the activity. This theorem says that fora graphon with a lower bound on the degrees, and a chip configuration that iscompatible with the graphon in a mild sense, if another graphon with a lowerbound on the degrees is close in cut distance, and the chip configurations areclose to each other in the L distance, then the activities are also close to eachother. For a precise statement, see Theorem 5.2. Using the method of [8], weshow in Theorem 6.6 that with some mild assumptions on the chip configuration σ , for the constant p graphon C p , the activity diagram s ( C p , σ ) is a Devil’sstaircase. Combining this result with the continuity theorem, we are able toprove Theorem 6.8 that gives a condition on a sequence of Erd˝os–R´enyi randomgraphs and chip configurations that the activity diagrams converge to a Devil’sstaircase. Finally, we give a concrete example for a one-parameter family ofrandom chip configurations, where the activities tend to a Devil’s staircase, seeTheorem 6.9.
2. Parallel chip-firing on graphons
We now review the notion of graphons defined by Lov´asz and Szegedy [10]as limits of sequences of dense graphs, then introduce parallel chip-firing onthem. We follow [9] in introducing graphons. See [9] for more informationabout graphons.A graphon is a Lebesgue measurable function W : [0 , → [0 ,
1] which is symmetric , that is, W ( x, y ) = W ( y, x ) for all x, y ∈ [0 , W is the unit interval [0 , x and y are connected in W ornot, we have a real number W ( x, y ) = W ( y, x ) describing how well they areconnected.Here, and everywhere else where we do not specify the measure, we meanthe Lebesgue measure, that we denote by λ .Note that for every (labeled) graph G with vertices { v , v , . . . , v n } one canconstruct a corresponding graphon W G the following way: partition [0 ,
1] into3 measurable sets A , . . . , A n with λ ( A ) = λ ( A ) = · · · = λ ( A n ). Then set W G ( x, y ) = 1 if x ∈ A i and y ∈ A j with ( v i , v j ) ∈ E ( G ), and W G ( x, y ) = 0otherwise.The degree of a vertex x of W is deg W ( x ) = R W ( x, y ) dy . Note that thedegree is well-defined for almost all x ∈ [0 , W ) = inf { ε : λ ( { x : deg W ( x ) ≥ ε } ) > } . We use the notation deg W ( x, A ) = R A W ( x, y ) dy To define the convergence of graphon sequences, a notion of distance ofgraphons is needed. It turns out that the right notion is the cut distance ofgraphons.
Definition 2.1.
The (labeled) cut distance of the graphons U and W is definedby d (cid:3) ( U, W ) = sup
S,T ⊆ [0 , (cid:12)(cid:12)(cid:12)(cid:12)Z S Z T U ( x, y ) − W ( x, y ) dy dx (cid:12)(cid:12)(cid:12)(cid:12) . The labeled cut distance corresponds to comparing the similarity of twographons when identifying vertices of the same label. The unlabeled cut distance corresponds to the case where we want to find the best identification of the twovertex sets: δ (cid:3) ( U, W ) = inf ϕ d (cid:3) ( U, W ϕ ) where ϕ runs over the invertible mea-sure preserving transformations of [0 ,
1] to itself, and W ϕ ( x, y ) = W ( ϕ ( x ) , ϕ ( y ))(see [9, Subsection 8.2.2]). For a (labeled) graph G and a graphon W , we usethe notation d (cid:3) ( G, W ) = d (cid:3) ( W G , W ), and similarly for δ (cid:3) . δ (cid:3) is a pseudo-metric on the space of graphons, and by factorizing with the graphons at zerounlabeled cut distance, one obtains a compact metric space.In our applications, we consider sequences of Erd˝os–R´enyi graphs G ( n, p ), n = 1 , , . . . . Such a sequence is known to converge to the constant p graphon C p with probability 1 in the distance δ (cid:3) . Since C ϕp = C p for any invertible,measure preserving transformation ϕ , δ (cid:3) ( W, C p ) = d (cid:3) ( W, C p ) for any graphon W . This fact, and the simpler formalization are the reasons that in this paper,we use the labeled cut distance as a metric on graphons.The definition of the parallel chip-firing on graphons is analogous to thaton finite graphs. A chip configuration on a graphon is an (almost everywhere)non-negative function σ ∈ L ([0 , W by Chip( W ), and use k σ k to denote the L norm of a chipconfiguration σ . For a given chip configuration σ on a graphon W , the parallelupdate rule is defined similarly as in the case of finite graphs. Let the firingvector of σ be f ( x ) = f ( W, σ )( x ) = ( j σ ( x )deg W ( x ) k if deg W ( x ) >
00 if deg W ( x ) = 0 , then we can define the update rule by U σ ( x ) = σ ( x ) − deg W ( x ) f ( x ) + Z f ( y ) W ( x, y ) dy. f ( W, σ ) is defined almost everywhere, and it followsfrom the following claim that
U σ ( x ) is finite almost everywhere. Claim 2.2.
For a arbitrary graphon W and σ ∈ Chip( W ) , U σ ∈ Chip( W ) and k U σ k = k σ k .Proof. From the definition of the firing vector, σ ( x ) − deg W ( x ) f ( x ) ≥
0, hence
U σ is non-negative, and k U σ k = Z | σ ( x ) − deg W ( x ) f ( x ) | dx + Z Z f ( y ) W ( x, y ) dy dx = Z σ ( x ) dx − Z deg W ( x ) f ( x ) dx + Z Z f ( y ) W ( x, y ) dx dy = Z σ ( x ) dx − Z deg W ( x ) f ( x ) dx + Z f ( y ) deg W ( y ) dy = k σ k , where we used Fubini’s theorem for non-negative functions to interchange theintegrals.As in the case of finite graphs, the odometer u n ( x ) = u n ( W, σ )( x ) denotesthe number of times x fired during the first n turns, i.e., u n ( x ) = u n ( W, σ )( x ) = n − X i =0 f ( W, U i σ )( x ) . To talk about any notion of activity, we need to assume that the graphon W is connected , that is, there is no measurable partition [0 ,
1] = A ∪ B with λ ( A ), λ ( B ) > W ( x, y ) = 0 for almost all ( x, y ) ∈ A × B . As we will see inSection 4.1, connectedness itself is not enough: there is a connected graphon W with a reasonably nice chip configuration σ such that lim n →∞ u n ( W,σ )( x ) n doesnot exists for any x ∈ [0 , W and chip configuration σ there is a real number a = a ( W, σ ) such that lim n →∞ u n ( W,σ )( x ) n exists and isequal to a for almost all x ∈ [0 ,
1] then we say that the activity exists and isequal to a . As we will see in Theorem 4.11, the activity of any chip configurationexists on a graphon with a lower bound on the degrees.We can also introduce the activity diagram of a chip configuration σ on agraphon W as a straightforward generalization of the graph case: s ( W, σ )( y ) = a ( W, σ + y · deg W ). The following claim tells us that activity diagrams aremonotone increasing. Lemma 2.3. If σ ′ ≥ σ almost everywhere, then u n ( W, σ ′ )( x ) ≥ u n ( W, σ )( x ) for any graphon W , n ∈ N and almost all x ∈ [0 , .Proof. We proceed by induction on n . The statement is clear for n = 0, since u ( W, σ ′ )( x ) = u ( W, σ )( x ) = 0 for all x ∈ [0 , u n ( W, σ ′ )( x ) ≥ u n ( W, σ )( x ) for almost all x ∈ [0 , x ∈ [0 ,
1] has the properties that σ ′ ( x ) ≥ σ ( x ) and u n ( W, σ ′ )( x ) ≥ u n ( W, σ )( x ). Fix such an x ∈ [0 ,
1] towards showing that u n +1 ( W, σ ′ )( x ) ≥ n +1 ( W, σ )( x ). By induction hypothesis, u n ( W, σ ′ )( x ) = u n ( W, σ )( x ) + k forsome k ≥
0, moreover, U n σ ( x ) = σ ( x ) − u n ( W, σ )( x ) deg W ( x ) + Z u n ( W, σ )( y ) W ( x, y ) dy ≤ σ ′ ( x ) − ( u n ( W, σ ′ )( x ) − k ) deg W ( x ) + Z u n ( W, σ ′ )( y ) W ( x, y ) dy = U n σ ′ ( x ) + k deg W ( x ) . It follows that u n +1 ( W, σ )( x ) = u n ( W, σ )( x ) + (cid:22) U n σ ( x )deg W ( x ) (cid:23) ≤ u n ( W, σ )( x ) + (cid:22) U n σ ′ ( x )deg W ( x ) (cid:23) + k = u n ( W, σ ′ )( x ) + (cid:22) U n σ ′ ( x )deg W ( x ) (cid:23) = u n +1 ( W, σ ′ )( x ) . This finishes the proof.
3. The finite diameter condition
In this section we formulate a notion for graphons that is an analogue of thediameter of finite graphs. We will be able to give a sufficient condition for theexistence of the activity of a chip configuration using this notion.For a measurable set A ⊆ [0 , A ) the neighborhood of A in W , i.e. Γ( A ) = { x ∈ [0 ,
1] : ∃ y ∈ A such that W ( y, x ) > } . For ε >
0, we denote by Γ ε ( A ) the set of those neighbors of A that receiveat least ε chips by firing the set A once, that is,Γ ε ( A ) = (cid:26) x ∈ [0 ,
1] : Z A W ( y, x ) dy ≥ ε (cid:27) . We denote Γ k ( A ) = Γ ◦ · · · ◦ Γ | {z } k ( A ) and similarly for Γ kε ( A ).The following definition plays a key role in our results. Definition 3.1 (Finite diameter condition) . A graphon W : [0 , → [0 ,
1] issaid to have finite diameter , if there is an N ∈ N such that for all measurablesubset A ⊆ [0 ,
1] with λ ( A ) > ε > λ ( A ∪ Γ ε ( A ) ∪ Γ ε ( A ) ∪· · · ∪ Γ Nε ( A )) = 1.It is reasonable to call the smallest such N the diameter of W , but wewill not use this notion. There are many equivalent ways to define the finitediameter property. Above we tried to give the most natural definition. Thefollowing theorem gives two more equivalent formulations that will play a rolein this paper. We also note that we could use Γ ′ ε ( A ) = A ∪ Γ ε ( A ) and get thesame property. 6 heorem 3.2. The following statements are equivalent for a graphon W :(i) W has a finite diameter;(ii) there exist N ∈ N and ε > such that for all measurable A ⊆ [0 , with λ ( A ) ≥ , λ ( A ∪ Γ ε ( A ) ∪ Γ ε ( A ) ∪ · · · ∪ Γ Nε ( A )) = 1 ;(iii) W is connected and there exists δ > such that deg W ( x ) ≥ δ for almostall x . Notice that (ii) is different from (i) in that we require the existence of an ε that is suitable for every “large” measurable set. Proof.
First we prove that (i) and (ii) both imply (iii).If W is not connected and A ∪ B = [0 ,
1] is a partition witnessing this,then by supposing λ ( A ) ≥ /
2, we see that for each N ∈ N and each ε > λ ( A ∪ Γ ε ( A ) ∪ Γ ε ( A ) ∪ · · · ∪ Γ Nε ( A )) = λ ( A ) <
1, contradicting the assumptionsof both (i) and (ii).If the degrees of W are not bounded from below (so for every ε > λ ( { x :deg W ( x ) < ε } ) >
0) then the degrees are not bounded from below on [0 , )(i.e. for every ε > λ ( { x ∈ [0 , ) : deg W ( x ) < ε } ) >
0) or on [ , , ) and let A = [ , ε >
0, the set B ε = { x ∈ [0 , ) : deg W ( x ) < ε } is of positive measure, so λ ( A ∪ Γ ε ( A ) ∪ Γ ε ( A ) ∪ · · · ∪ Γ Nε ( A )) ≤ λ ([0 , \ B ε ) < N ∈ N . Thuswe have proved the directions (i) ⇒ (iii) and (ii) ⇒ (iii).To show (iii) ⇒ (i) and (iii) ⇒ (ii), we first prove the following. Claim 3.3. If W is connected then for each interval [ a, b ] ⊂ (0 , there exists ε > so that R A R A c W ( x, y ) dx dy ≥ ε for all measurable subset A ⊆ [0 , with λ ( A ) ∈ [ a, b ] .Proof. Suppose towards a contradiction that for some [ a, b ] there is a sequenceof subsets ( A n ) n ∈ N such that λ ( A n ) ∈ [ a, b ] but R A n R A cn W ( x, y ) dx dy → W by coming up with a subset A ⊆ [0 ,
1] with λ ( A ) ∈ [ a, b ] and R A R A c W ( x, y ) dx dy = 0.Since { A n : n ∈ N } is a bounded subset of L ∞ ([0 , L ∞ is the dualof L , there is a weak ∗ convergent subsequence of ( A n ) n ∈ N tending to f ∈ L ∞ by the Banach–Alaoglu theorem. We can suppose that the subsequence is theoriginal one, hence Z g ( x ) · A n ( x ) dx → Z g ( x ) · f ( x ) dx (3.1)for all g ∈ L ([0 , f ( x ) ∈ [0 ,
1] for almost all x ∈ [0 , R f ( x ) dx ∈ [ a, b ] by using g ( x ) = { x : f ( x ) < } , g ( x ) = { x : f ( x ) > } and g ( x ) ≡ A n that Z Z W ( x, y ) A n ( x ) A cn ( y ) dx dy = Z Z W ( x, y ) A n ( x )(1 − A n ( y )) dx dy → . Now we show that R R W ( x, y ) f ( x )(1 − f ( y )) dx dy = 0. The function x W ( x, y ) is in L for almost all y ∈ [0 , Z W ( x, y ) A n ( x ) dx → Z W ( x, y ) f ( x ) dx for almost all y . For a fixed ε >
0, let n be large enough so that for B = (cid:26) y : ∀ n ≥ n (cid:12)(cid:12)(cid:12)(cid:12)Z W ( x, y ) A n ( x ) dx − Z W ( x, y ) f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε (cid:27) ,λ ( B ) ≥ − ε . Then for each n ≥ n , (cid:12)(cid:12)(cid:12)(cid:12) Z Z W ( x, y ) A n ( x )(1 − A n ( y )) dx dy − Z Z W ( x, y ) f ( x )(1 − A n ( y )) dx dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z B (cid:12)(cid:12)(cid:12)(cid:12)Z W ( x, y ) A n ( x ) dx − Z W ( x, y ) f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) (1 − A n ( y )) dy + Z B c (cid:12)(cid:12)(cid:12)(cid:12)Z W ( x, y ) A n ( x ) dx − Z W ( x, y ) f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) (1 − A n ( y )) dy ≤ ε, using that every function here has values in [0 ,
1] and that λ ( B c ) ≤ ε . Thefunction y R W ( x, y ) f ( x ) dx is in L , hence Z Z W ( x, y ) f ( x )(1 − A n ( y )) dx dy → Z Z W ( x, y ) f ( x )(1 − f ( y )) dx dy. Thus, lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12) Z Z W ( x, y ) A n ( x )(1 − A n ( y )) dx dy − Z Z W ( x, y ) f ( x )(1 − f ( y )) dx dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε for every ε >
0, meaning that Z Z W ( x, y ) A n ( x )(1 − A n ( y )) dx dy → Z Z W ( x, y ) f ( x )(1 − f ( y )) dx dy, R R W ( x, y ) f ( x )(1 − f ( y )) dx dy = 0.Now we extract a subset from f . Since R f ( x ) dx ∈ [ a, b ] and f ( x ) ∈ [0 ,
1] foralmost all x , λ ( { x : f ( x ) = 1 } ) ≤ b and λ ( { x : f ( x ) > } ) ≥ a . Hence, there isa measurable set A with { x : f ( x ) = 1 } ⊆ A ⊆ { x : f ( x ) > } and λ ( A ) ∈ [ a, b ].It is easy to check that if for some ( x, y ) ∈ [0 , , f ( x )(1 − f ( y )) = 0, then A ( x )(1 − A ( y )) = 0. Therefore R R W ( x, y ) f ( x )(1 − f ( y )) dx dy = 0 impliesthat R R W ( x, y ) A ( x )(1 − A ( y )) dx dy = 0, hence A witnesses that W is notconnected, a contradiction.Now we prove (iii) ⇒ (i) and (iii) ⇒ (ii). It is enough to prove the cor-responding statements for the operator Γ ′ ε ( A ) = A ∪ Γ ε ( A ) in place of Γ ε ( A ),since one can easily see by induction on k that (Γ ′ ε ) k ( A ) = A ∪ · · · ∪ (Γ ′ ε ) k ( A ) ⊆ A ∪ Γ ε/N ( A ) ∪ · · · ∪ Γ kε/N ( A ) for each k ≤ N .The following claim proves (ii) from (iii) and will also be used to prove (i). Claim 3.4. If W is a connected graphon, and deg W ( x ) ≥ δ for almost every x for some δ > , then for every a ∈ (0 , there exist N ∈ N and ε > such that λ ((Γ ′ ε ) N ( A )) = 1 for every measurable set A with λ ( A ) ∈ [ a, .Proof. Using the lower bound on the degree, λ (Γ ′ ε ( A )) = 1 for every measurable A with λ ( A ) ≥ − δ ε ≤ δ ε ′ > a, b ] = [ a, − δ ], then for everymeasurable subset A with λ ( A ) ∈ [ a, − δ ], R R W ( x, y ) A ( x ) A c ( y ) dx dy ≥ ε ′ . Since R W ( x, y ) A ( x ) dx ≤ y ∈ A c , λ (cid:18)(cid:26) y ∈ A c : Z W ( x, y ) A ( x ) dx ≥ ε ′ (cid:27)(cid:19) ≥ ε ′ . In other words, λ (Γ ′ ε ′ / ( A ) \ A ) ≥ ε ′ for every measurable subset A with λ ( A ) ∈ [ a, − δ ]. Then, λ ((Γ ′ ε ′ / ) ⌈ /ε ′ ⌉ ( A )) ∈ [1 − δ ,
1] for every such A . Therefore,also using (3.2), N = ⌈ ε ′ ⌉ + 1 and ε = min { ε ′ , δ } satisfy the claim.The proof of (iii) ⇒ (ii) is complete using the claim, so now we move onto show (iii) ⇒ (i). The extra difficulty comes from sets of small measure.If A is a measurable subset with 0 < λ ( A ) ≤ δ then for almost all x ∈ A , R W ( x, y ) A c ( y ) dy ≥ δ , hence R R W ( x, y ) A ( x ) A c ( y ) dx dy ≥ δλ ( A )3 . Let B = (cid:26) y ∈ A c : Z W ( x, y ) A ( x ) dx ≥ δλ ( A )3 (cid:27) . Since R W ( x, y ) A ( x ) dx ≤ λ ( A ) for almost all y ∈ A c ,2 δλ ( A )3 ≤ Z Z W ( x, y ) A ( x ) A c ( y ) dx dy ≤ λ ( B ) · λ ( A ) + δλ ( A )3 , λ ( B ) ≥ δ , and thus λ (Γ ′ δλ ( A )3 ( A )) ≥ δ .Now we apply Claim 3.4 with a = δ to get N ′ and ε ′ such that λ ((Γ ′ ε ′ ) N ′ ( A )) =1 for every A with λ ( A ) ≥ δ . Then for any subset A of positive measure, N = N ′ + 1 (that is independent of A ) and ε = min { ε ′ , δλ ( A )3 } work, λ ((Γ ′ ε ) N ( A )) =1. Let us point out a nice property of graphons with finite diameter. In many re-spect, unbounded chip configurations are inconvenient. However, for a graphonwith finite diameter, any chip configuration becomes bounded after one step ofthe parallel chip-firing. Lemma 3.5.
If a graphon W has mindeg( W ) = d > , then for any chipconfiguration σ on W , n ≥ and almost all x ∈ [0 , , we have U n σ ( x ) ≤ deg W ( x ) + k σ k d ≤ k σ k d . Proof.
Since by Claim 2.2, k U σ k = k σ k , it is enough to prove the statementfor n = 1. U σ ( x ) = σ ( x ) − deg W ( x ) f ( x ) + R f ( y ) W ( x, y ) dy where f ( y ) = ( j σ ( y )deg W ( y ) k if deg W ( y ) >
00 if deg W ( y ) = 0 . For an x where deg W ( x ) >
0, we have σ ( x ) − deg W ( x ) f ( x ) ≤ deg W ( x ). Also,as W ( x, y ) ≤ x, y ∈ [0 , R f ( y ) W ( x, y ) dy ≤ R f ( y ) dy ≤ R σ ( y ) d dy = d k σ k as σ is almost everywhere nonnegative.
4. Existence of the activity
In this section we investigate the existence of the activity of a chip configura-tion on a graphon. Recall that by definition the activity of a chip configuration σ on a graphon W exists and is equal to a ∈ R if lim n →∞ u n ( x ) n exists and isequal to a for almost all x ∈ [0 , W is con-nected and σ ( x )deg W ( x ) is bounded then lim inf u n ( x ) n is the same for almost every x ∈ [0 , u n ( x ) n . In the main result of this section,Theorem 4.11, we show that the finite diameter condition implies the existenceof the activity for any chip configuration. Proposition 4.1.
There exist a connected graphon W and a bounded chip con-figuration σ on W such that the activity of σ does not exist. roof. In our construction we will have lim inf u n ( x ) n = and lim sup u n ( x ) n = 1for each x ∈ [0 , S m ∈ Z A m be a measurable partition of [0 ,
1] with λ ( A m ) > m ∈ Z , and let us denote by m : [0 , → Z the unique function with x ∈ A m ( x ) for every x ∈ [0 , W ( x, y ) = 1 if and only if | m ( x ) − m ( y ) | = 1 andlet W ( x, y ) = 0 otherwise. It is easy to see that W is connected.We say that a set A m is of type 1 with respect to a chip configuration ρ if ρ ( x ) = λ ( A m − ) for each x ∈ A m . We say that A m is of type 2 if ρ ( x ) = λ ( A m − ) + λ ( A m +1 ) for each x ∈ A m , and it is of type 3 if ρ ( x ) =2 · λ ( A m − ) + λ ( A m +1 ) for each x ∈ A m . In our example, we will choose thestarting configuration σ so that for each n , each set A m is of type i for some i with respect to U n σ . It is clear that every such chip configuration is bounded.Now let Z ∪ Z ∪ Z = Z be a partition of the integers with the propertythat m ∈ Z ⇔ m + 1 ∈ Z . (4.1)We define the chip configuration σ in the following way: σ ( x ) = λ ( A m ( x ) − ) if m ( x ) ∈ Z ,λ ( A m ( x ) − ) + λ ( A m ( x )+1 ) if m ( x ) ∈ Z , · λ ( A m ( x ) − ) + λ ( A m ( x )+1 ) if m ( x ) ∈ Z , that is, the type of A m is i with respect to σ if and only if m ∈ Z i .Using (4.1), we have that A m is of type 1 if and only if A m +1 is of type 3 with respect to σ . (4.2) Claim 4.2.
Suppose that a configuration ρ satisfies (4.2) and also that each A m is of some type with respect to ρ . Then for each m ∈ Z , A m is of type i withrespect to U ρ if and only if A m +1 is of type i with respect to ρ .Proof. We distinguish multiple cases according to the type of A m with respectto ρ .If A m is of type 1 with respect to ρ then using (4.2), A m +1 is of type 3 and A m − is of type 2 or 3. Hence, starting from ρ , the points of A m − ∪ A m +1 can fire once, but the points of A m cannot fire at all. Therefore, U ρ ( x ) =2 · λ ( A m − ) + λ ( A m +1 ) for each x ∈ A m , thus A m is indeed of type 3 withrespect to U ρ , as is A m +1 with respect to ρ .If A m is of type 2 with respect to ρ then again using (4.2), A m − is of type2 or type 3, and A m +1 is of type 1 or type 2. Since in this case the points of A m − ∪ A m can fire once, for each x ∈ A m , U ρ ( x ) = λ ( A m − ) if A m +1 is oftype 1, and U ρ ( x ) = λ ( A m − ) + λ ( A m +1 ) if A m +1 is of type 2. Thus the proofis also complete in this case.If A m is of type 3 with respect to ρ then by (4.2), A m − is of type 1, and A m +1 is of type 1 or type 2. Hence, for each x ∈ A m , U ρ ( x ) = λ ( A m − ) if A m +1 is of type 1, and U ρ ( x ) = λ ( A m − ) + λ ( A m +1 ) if A m +1 is of type 2. Thusthe proof of our claim is complete. 11t is easy to prove by induction on n using Claim 4.2, that for each n , everyset A m is of some type with respect to U n σ and also that U n σ satisfies (4.2). Itis also clear that u n ( x ) equals the cardinality of the set ( Z ∪ Z ) ∩{ m ( x ) , m ( x )+1 , . . . , m ( x ) + n − } . The only thing that remains to finish the construction,is to choose the partition Z = Z ∪ Z ∪ Z such that (4.1) is satisfied and alsofor every x the liminf of | ( Z ∪ Z ) ∩ { m ( x ) , m ( x ) + 1 , . . . , m ( x ) + n − }| as n tends to infinity is 1 / Z = { n ∈ Z : n ≤ } ∪ ∞ [ k =1 { n ∈ Z : (2 k + 1)! ≤ n < (2 k + 2)! } , where k ! denotes the factorial of k . We can add the remaining integers alter-natingly to Z and Z satisfying (4.1). It is straightforward to check that thisconstruction satisfies the above requirements. lim inf and lim supIn this section we prove the following theorem. Theorem 4.3. If W is a connected graphon and σ is a chip configurationon W such that σ ( x )deg W ( x ) < K almost everywhere, then there are real numbers u , u ∈ [0 , such that lim inf u n ( x ) n = u and lim sup u n ( x ) n = u for almost all x ∈ [0 , . We conjecture that this statement holds more generally, for any chip config-uration.
Conjecture 4.4. If W is a connected graphon and σ is a chip configurationon W , then there are real numbers u , u ∈ [0 ,
1] such that lim inf u n ( x ) n = u andlim sup u n ( x ) n = u for almost all x ∈ [0 , y W ( x,y )deg W ( x ) can beinterpreted as a density describing the neighborhood of x . Even though thesedensities can be quite different for different points, if we start a Markov chainat each point, the transition probabilities of this Markov chain will be close toeach other after a sufficiently large number of steps. We can approximate theamount of chips received by the points using these probabilities, showing thatthe lim inf and lim sup of u n ( x ) n do not depend on x .We start with collecting the notions regarding Markov chains that we willneed. Let ( X, A ) be a measurable space and P : X × A → [0 ,
1] denote thetransition probabilities of a Markov chain, that is, P ( x, · ) is a probability dis-tribution for each x ∈ X and P ( · , A ) is measurable for each A ∈ A . Thehigher-order transition probabilities are defined by P ( x, A ) = P ( x, A ) and P n +1 ( x, A ) = Z X P n ( x, dy ) P ( y, A ) . P is said to be irreducible if there exists anon-zero σ -finite measure φ on X such that for every A ∈ A with φ ( A ) > x ∈ X there exists n ∈ N with P n ( x, A ) > π is called stationary distribution if π ( A ) = Z X P ( x, A ) dπ ( x )for every A ∈ A . A Markov chain with a stationary distribution π is aperiodic if there do not exist disjoint, measurable subsets A , . . . , A d − ⊆ X with d ≥ ≤ i < d and all x ∈ A i , P ( x, A ( i +1) (mod d ) ) = 1, and π ( A ) > total variation norm , thatis, k ν k = sup A ∈A | ν ( A ) | . Theorem 4.5.
If a Markov chain on a state space with countably generated σ -algebra is irreducible and aperiodic, and has a stationary distribution π , thenfor π -a.e. x ∈ X , lim n →∞ k P n ( x, · ) − π ( · ) k = 0 . Now we are ready to prove the main result in this subsection.
Proof of Theorem 4.3.
We first claim that we can suppose that deg W ( x ) > x ∈ [0 , A = { x ∈ [0 ,
1] : deg W ( x ) = 0 } ,and define W ′ by W ′ ( x, y ) = 1 if x ∈ A or y ∈ A , and W ′ ( x, y ) = W ( x, y )otherwise. It is easy to check that W ′ is a graphon, and W ( x, y ) = W ′ ( x, y )for almost all ( x, y ) ∈ [0 , , since the connectedness of W implies λ ( A ) = 0.Hence, W ′ is also connected and deg W ′ ( x ) > x . One can also showby induction on n , that if deg W ( x ) > u n ( W, σ )( x ) = u n ( W ′ , σ )( x ), sincein this case W ′ ( x, y ) = W ( x, y ) for almost all y ∈ [0 , W ′ then we are also done for W . Thus wecan indeed suppose that deg W ( x ) > x ∈ [0 , W is Borel measurable; we can do so, since for any (Lebesgue measurable)graphon W there is a Borel measurable one W ′ such that W = W ′ almost ev-erywhere, hence, using that deg W ′ ( x ) = deg W ( x ) almost everywhere, it followsthat u n ( W ′ , σ )( x ) = u n ( W, σ )( x ) almost everywhere.Define a Markov chain using W by the transition probabilities P ( x, A ) = R A W ( x, y ) dy deg W ( x ) . Here, and everywhere else where we do not specify the measure, we integratewith respect to the Lebesgue measure.To check that P : [0 , × B ([0 , → [0 , B ([0 , σ -algebra, indeed defines a Markov chain, one can use e.g. [7, Exercise 17.36] toshow first that x deg W ( x ) is Borel, hence ( x, y ) W ( x,y )deg W ( x ) is also Borel, and13hen use the same exercise again to show that for each A ∈ B ([0 , x P ( x, A )is also Borel.Notice that each distribution P ( x, · ) is absolutely continuous with respectto the Lebesgue measure, since p ( x, y ) = W ( x, y )deg W ( x )is a density function. One can also show by induction on n that P n ( x, · ) has adensity function p n ( x, · ), which can be defined inductively by p ( x, y ) = p ( x, y )and p n +1 ( x, y ) = Z p n ( x, z ) p ( z, y ) dz = Z p n ( x, z ) W ( z, y )deg W ( z ) dz. We will also use that for k ≤ n , p n ( x, y ) = Z p k ( x, z ) p n − k ( z, y ) dz. (4.3)Let us define a probability distribution on [0 ,
1] by π ( A ) = R A deg W ( x ) dx R deg W ( x ) dx withdensity function d π ( x ) = deg W ( x ) R deg W ( y ) dy . Claim 4.6.
The Markov chain determined by P is irreducible, and has π as astationary distribution.Proof. The following calculation, using Fubini’s theorem for non-negative func-tions shows that π is indeed a stationary distribution: Z P ( x, A ) dπ ( x ) = Z P ( x, A ) d π ( x ) dx = Z Z A p ( x, y ) dy · d π ( x ) dx = Z Z A W ( x, y )deg W ( x ) dy · deg W ( x ) R deg W ( z ) dz dx = Z Z A W ( x, y ) dy dx · R deg W ( z ) dz = R A deg W ( y ) dy R deg W ( z ) dz = π ( A ) . To check irreducibility, we use φ = λ , the Lebesgue measure. Let x ∈ [0 , A n = { y ∈ [0 ,
1] : p n ( x, y ) > } . We claim that it is enoughto show that λ ( S n A n ) = 1. Indeed, if this is the case and A ∈ B ([0 , λ ( A ) >
0, then λ ( A ∩ A n ) > n , hence P n ( x, A ) = R A p n ( x, y ) dy ≥ R A ∩ A n p n ( x, y ) dy >
0, since p n ( x, · ) is positive on A ∩ A n and λ ( A ∩ A n ) > λ ( S n A n ) < B =[0 , \ S n A n . Then λ ( B ) >
0, and also λ ( B ) <
1, since for example 1 = R p n ( x, y ) dy = R A n p n ( x, y ) dy , showing that each A n is of positive measure.14sing the fact that W is connected, R B c R B W ( x, y ) dy dx >
0, thus there existssome n such that R A n R B W ( x, y ) dy dx >
0. It follows that for A ′ n = { x ∈ A n : R B W ( x, y ) dy > } , λ ( A ′ n ) >
0. Now we show that P n +1 ( x, B ) > P n +1 ( x, B ) = Z B p n +1 ( x, y ) dy = Z B Z p n ( x, z ) p ( z, y ) dz dy ≥ Z B Z A ′ n p n ( x, z ) p ( z, y ) dz dy = Z A ′ n p n ( x, z ) Z B W ( z, y )deg W ( z ) dy dz > , since p n ( x, z ) > z ∈ A ′ n ⊆ A n , R B W ( z,y )deg W ( z ) dy ≥ R B W ( z, y ) dy > z ∈ A ′ n , and λ ( A ′ n ) > W is bipartite , that is,there is a measurable partition [0 ,
1] = A ∪ B such that λ ( A ), λ ( B ) > W ( x, x ′ ) = 0 for almost all ( x, x ′ ) ∈ A and also W ( y, y ′ ) = 0 for almost all( y, y ′ ) ∈ B . For our purposes, another formulation of bipartiteness will beuseful. For a bipartite graphon W we call the decomposition [0 , ⊇ F = X ∪ Y into disjoint subsets a canonical decomposition , if λ ( X ), λ ( Y ) > λ ( F ) = 1,for all x ∈ X and for almost all x ′ ∈ X , W ( x, x ′ ) = 0, and also for all y ∈ Y and for almost all y ′ ∈ Y , W ( y, y ′ ) = 0. Note that every bipartite graphon hasa canonical decomposition.For such a decomposition we denote by π X the distribution on X definedby π X ( A ) = R A deg W ( x ) dx R X deg W ( x ) dx , and similarly π Y is a distribution on Y defined by π Y ( B ) = R B deg W ( y ) dy R Y deg W ( y ) dy . We denote the corresponding density functions by d π X and d π Y . Claim 4.7. If W is bipartite with canonical decomposition X ∪ Y , then P ( x, · ) for x ∈ X are transition probabilities for an irreducible Markov chain on X withstationary distribution π X . The analogous statement holds for Y as well.Proof. Using the fact that P ( x, · ) is absolutely continuous with respect to theLebesgue measure, and the properties of the canonical decomposition, one caneasily show by induction on n that P n ( x, X ) = 1 and P n +1 ( x, Y ) = 1 for every x ∈ X . (4.4)This implies that P ( x, · ) is a probability distribution on X for each x ∈ X .The measurability of P ( · , A ) can be shown as before for P ( · , A ), showing that P indeed defines a Markov chain.Claim 4.6 and (4.4) shows that P is irreducible on X , and a similar compu-tation as in the proof of Claim 4.6 shows that π X is a stationary distribution. Claim 4.8. If W is not bipartite then k p n ( x, · ) − d π k → for almost every x ∈ [0 , . If W is bipartite with canonical decomposition X ∪ Y , then k p n ( x, · ) − d π X k → for almost all x ∈ X and k P n ( y, · ) − d π Y k → for almost all y ∈ Y . roof. It is enough to prove that if W is not bipartite then k P n ( x, · ) − π k → x ∈ [0 , ν is a signed measure with density function d ν , then k d ν k = R | d ν ( x ) | dx = (cid:12)(cid:12) R d ν > d ν ( x ) dx (cid:12)(cid:12) + (cid:12)(cid:12) R d ν < d ν ( x ) dx (cid:12)(cid:12) ≤ · k ν k .It is also clear that π and λ are mutually absolutely continuous, since π hasa density function which is everywhere positive. Hence, a statement holds π -a.e.if and only if it holds λ -a.e. In the following discussion, where we can choosebetween the two, we always use the Lebesgue measure, as in the statement ofthis claim.To prove the first assertion we want to apply Theorem 4.5 for P . Using Claim4.6, it remains to show that P is aperiodic. Suppose towards a contradictionthat the Markov chain is not aperiodic, hence there exist measurable subsets A , . . . , A d − ⊆ [0 ,
1] with d ≥ i < d and all x ∈ A i , P ( x, A ( i +1) (mod d ) ) = 1, and λ ( A ) >
0. It follows that λ ( A i ) > i , since if we suppose this for some fixed i ≤ d − P ( x, A i +1 ) = R A i +1 p ( x, y ) dy for all x ∈ A i , showing λ ( A i +1 ) >
0. Let A = S i For a connected graphon W and a set A ⊆ [0 , with λ ( A ) > , P ( x, A ) > for almost all x ∈ A .Proof. Let B = { x ∈ A : P ( x, A ) = 0 } . Then, of course P ( x, B ) = 0 for all x ∈ B . Suppose towards a contradiction that λ ( B ) > 0. Then0 = Z B P ( x, B ) dx = Z B Z B p ( x, z ) dz dx = Z B Z B Z p ( x, y ) p ( y, z ) dy dz dx = Z Z B p ( x, y ) dx Z B p ( y, z ) dz dy. Since p ( x, y ) = 0 if and only if p ( y, x ) = 0, we have R B p ( x, y ) dx = 0 if andonly if R B p ( y, z ) dz = 0. Therefore p ( x, y ) = 0 for almost all ( x, y ) ∈ B × [0 , W is connected.We can use the lemma for A = A to get that P ( x, A ) > x ∈ A , hence it is not possible to have d ≥ P ( x, A ) = 1 for all x ∈ A and A ∩ A = ∅ . Therefore d ≤ 2, and d = 2 implies that W is bipartite. Itfollows that if W is not bipartite then P is aperiodic. The first assertion thenfollows from Theorem 4.5.Now suppose that W is bipartite with canonical decomposition X ∪ Y . Touse Theorem 4.5 to finish the proof, after applying Claim 4.7, it remains to showthat P is aperiodic on X . If this was not the case, there would exist disjointsets of positive measure A , A ⊆ X such that P ( x, A ) = 1 for all x ∈ A .Using Lemma 4.9 we obtain that P ( x, A ) > x ∈ A , whichcontradicts the existence of such sets. 16e are now ready to finish the proof of Theorem 4.3. We calculate lowerand upper estimates for u n ( x ) for an arbitrary x ∈ [0 , Claim 4.10. If W is a connected graphon and σ is a chip configuration on W such that σ ( x )deg W ( x ) < K almost everywhere, then u n ( x ) ≤ ( K − n for almostall x ∈ [0 , .Proof. We show by induction on n that U n σ ( x )deg W ( x ) < K . From this statement, theclaim easily follows. The statement for n = 0 is an assumption of the theorem.Suppose now that it holds for some n ∈ N towards showing it for n + 1. Clearly, U n +1 σ ( x ) < deg W ( x ) + Z ( K − W ( x, y ) dy < K deg W ( x ) , finishing the proof.Since R W ( x, y ) u n − ( y ) dy is the amount of mass received by x during thefirst n − u n ( x ) ≥ R W ( x,y ) u n − ( y ) dy deg W ( x ) − 1. Now let x ∈ [0 , 1] be arbitraryand k ≤ n , then u n ( x ) ≥ R W ( x , x ) u n − ( x ) dx deg W ( x ) − Z p ( x , x ) u n − ( x ) dx − ≥ Z p ( x , x ) R W ( x , x ) u n − ( x ) dx deg W ( x ) − ! dx − Z Z p ( x , x ) p ( x , x ) u n − ( x ) dx dx − Z p ( x , x ) dx − Z p ( x , x ) u n − ( x ) dx − ≥ · · ·≥ Z p k ( x , x k ) u n − k ( x k ) dx k − k. For the upper estimate, we use that u n ( x ) ≤ σ ( x )+ R W ( x,y ) u n − ( y ) dy deg W ( x ) for any x , hence u n ( x ) ≤ σ ( x ) + R W ( x , x ) u n − ( x ) dx deg W ( x )= σ ( x )deg W ( x ) + Z p ( x , x ) u n − ( x ) dx ≤ σ ( x )deg W ( x ) + Z p ( x , x ) σ ( x ) + R W ( x , x ) u n − ( x ) dx deg W ( x ) dx σ ( x )deg W ( x ) + Z p ( x , x ) σ ( x )deg W ( x ) dx + Z p ( x , x ) R W ( x , x ) u n − ( x ) dx deg W ( x ) dx = σ ( x )deg W ( x ) + Z p ( x , x ) σ ( x )deg W ( x ) dx + Z p ( x , x ) u n − ( x ) dx ≤ · · ·≤ σ ( x )deg W ( x ) + k − X i =1 (cid:18)Z p i ( x , x i ) σ ( x i )deg W ( x i ) dx i (cid:19) + Z p k ( x , x k ) u n − k ( x k ) dx k Now first suppose that W is not bipartite and let M ⊆ [0 , 1] be the set ofpoints x such that k p k ( x, · ) − d π k → 0. Using Claim 4.8, λ ( M ) = 1. We nowshow that the conclusion of the theorem holds for points in M , that is, for any x, x ′ ∈ M , lim inf u n ( x ) n = lim inf u n ( x ′ ) n and lim sup u n ( x ) n = lim sup u n ( x ′ ) n .Let x, x ′ ∈ M be arbitrary, and for a fixed ε > k ∈ N so that k p k ( x, · ) − d π k ≤ ε and k p k ( x ′ , · ) − d π k ≤ ε . Then, using Claim 4.10 and that σ ( x )deg W ( x ) < K for almost all x ∈ [0 , n ≥ k , u n ( x ) − u n ( x ′ ) ≤ σ ( x )deg W ( x ) + k − X i =1 (cid:18)Z p i ( x, y ) σ ( y )deg W ( y ) dy (cid:19) + Z (cid:0) p k ( x, y ) − p k ( x ′ , y ) (cid:1) u n − k ( y ) dy + k ≤ σ ( x )deg W ( x ) + k − X i =1 (cid:18)Z p i ( x, y ) K dy (cid:19) + Z n ( K − (cid:12)(cid:12) p k ( x, y ) − p k ( x ′ , y ) (cid:12)(cid:12) dy + k ≤ σ ( x )deg W ( x ) + kK + 2 n ( K − ε + k. One can similarly calculate a lower estimate for u n ( x ) − u n ( x ′ ), hence as n tendsto infinity, we get that lim sup (cid:12)(cid:12) u n ( x ) n − u n ( x ′ ) n (cid:12)(cid:12) ≤ K − ε for all ε > 0, hencelim (cid:12)(cid:12) u n ( x ) n − u n ( x ′ ) n (cid:12)(cid:12) = 0, thus lim inf u n ( x ) n = lim inf u n ( x ′ ) n and lim sup u n ( x ) n =lim sup u n ( x ′ ) n for all x, x ′ ∈ M . Thus the proof of the theorem is complete incase W is not bipartite.Now suppose that W is bipartite with canonical decomposition X ∪ Y . Let M be the union of the set of points x ∈ X with k p n ( x, · ) − d π X k → y ∈ Y with k p n ( y, · ) − d π Y k → 0. Using Claim 4.8, λ ( M ) =1. A similar argument to the above one shows that if x, x ′ ∈ X ∩ M thenlim (cid:12)(cid:12) u n ( x ) n − u n ( x ′ ) n (cid:12)(cid:12) = 0 and the same conclusion holds for points y, y ′ ∈ Y ∩ M .It remains to show the same for a pair ( x, y ) with x ∈ X ∩ M and y ∈ Y ∩ M .In the following, the density functions d π X and d π Y of π X and π Y are un-derstood to be defined on [0 , d π X vanishing outside X and d π Y van-ishing outside Y . We claim that for a fixed ε we can choose k ∈ N so that k p k ( x, · ) − d π X k ≤ ε and k p k +1 ( y, · ) − d π X k ≤ ε . To show this, first note thatfor all k ∈ N , (cid:13)(cid:13) p k +1 ( y, · ) − d π X (cid:13)(cid:13) = Z (cid:12)(cid:12)(cid:12)(cid:12)Z p ( y, u ) p k ( u, z ) du − d π X ( z ) (cid:12)(cid:12)(cid:12)(cid:12) dz = Z (cid:12)(cid:12)(cid:12)(cid:12)Z p ( y, u ) (cid:0) p k ( u, z ) − d π X ( z ) (cid:1) du (cid:12)(cid:12)(cid:12)(cid:12) dz = Z p ( y, u ) (cid:13)(cid:13) p k ( u, . ) − d π X (cid:13)(cid:13) du. Let k be large enough so that k p k ( x, · ) − d π X k ≤ ε and for that set H = { u ∈ X : k p k ( u, · ) − d π X k ≥ ε } , λ ( H ) ≤ ε deg W ( y )4 . Then, using that p ( y, u ) = 0for almost all u ∈ Y , and that p ( y, u ) ≤ W ( y ) and k p k ( u, · ) − d π X k ≤ u ∈ [0 , (cid:13)(cid:13) p k +1 ( y, · ) − d π X (cid:13)(cid:13) = Z p ( y, u ) (cid:13)(cid:13) p k ( u, . ) − d π X (cid:13)(cid:13) du ≤ Z H W ( y ) du + Z X \ H p ( y, u ) ε du ≤ ε ε ε, showing our claim.Then using Claim 4.10 again, for n ≥ k , u n ( x ) − u n +1 ( y ) ≤ σ ( x )deg W ( x ) + k − X i =1 (cid:18)Z p i ( x, z ) σ ( z )deg W ( z ) dz (cid:19) + Z (cid:0) p k ( x, z ) − p k +1 ( y, z ) (cid:1) u n − k ( z ) dz + ( k + 1) ≤ σ ( x )deg W ( x ) + kK + 2 n ( K − ε + ( k + 1) , with a similar calculation showing the opposite direction, proving together thatlim (cid:12)(cid:12) u n ( x ) n − u n +1 ( y ) n (cid:12)(cid:12) = 0. This implies that lim inf u n ( z ) n and lim sup u n ( z ) n is thesame for almost all z ∈ [0 , 1] even if W is bipartite. Therefore the proof of thetheorem is complete. In this section we give a sufficient condition for the existence of the activity.19 heorem 4.11. If the finite diameter condition holds for a graphon W , thenthe activity exists for any chip configuration. We believe that having finite diameter is not necessary for the existence ofthe activity of every chip configuration. Problem 4.12. Give a necessary and sufficient condition for a graphon W suchthat the activity a ( W, σ ) exists for each σ .We start the proof of Theorem 4.11 by investigating the properties of thefollowing two quantities: for a graphon W , a chip configuration σ , and each n ∈ N let m n = m n ( W, σ ) = inf { k : λ ( { x : u n ( x ) = k } ) > } ,M n = M n ( W, σ ) = sup { k : λ ( { x : u n ( x ) = k } ) > } . It is easy to see that m n ( W, σ ) is finite, however, M n ( W, σ ) could be infinite. Lemma 4.13. m n is superadditive, that is, m n + k ≥ m n + m k .Proof. We claim that it is enough to prove thatfor almost all x ∈ [0 , , u n + k ( x ) ≥ m n + u k ( x ) . (4.5)Indeed, suppose that (4.5) holds and let A = { x ∈ [0 , 1] : u n + k ( x ) = m n + k .Note that λ ( A ) > m n + k . For almost all x ∈ A , we have m n + k = u n + k ( x ) ≥ m n + u k ( x ). Since λ ( A ) > 0, and u k ( x ) < m k can onlyhold on a set of measure zero, we conclude that there exists x ∈ A such that u k ( x ) ≥ m k and m n + k ≥ m n + u k ( x ) both hold. Hence m n + k ≥ m n + m k .To prove (4.5), we proceed by induction on k . For k = 0, the statement istrivial. Suppose that the statement holds for k , i.e. u n + k ( y ) ≥ m n + u k ( y ) foralmost all y ∈ [0 , k + 1.Fix an arbitrary x ∈ [0 , 1] with deg W ( x ) > 0, and suppose that u n + k ( x ) = m n + u k ( x ) + a for some nonnegative integer a . If starting from σ we fire eachvertex y exactly m n + u k ( y ) times, we get to U k σ as firing each vertex m n timesdoes not change the chip configuration. Now U n + k σ ( x ) = σ ( x ) − u n + k ( x ) deg W ( x ) + Z u n + k ( y ) W ( x, y ) dy ≥ σ ( x ) − ( m n + u k ( x ) + a ) deg W ( x ) + Z ( m n + u k ( y )) W ( x, y ) dy = U k σ ( x ) − a · deg W ( x ) . Hence u n + k +1 ( x ) = u n + k ( x ) + (cid:22) U n + k σ ( x )deg W ( x ) (cid:23) ≥ m n + u k ( x ) + a + (cid:22) U k σ ( x ) − a · deg W ( x )deg W ( x ) (cid:23) = m n + u k +1 ( x ) . emma 4.14. M n is subadditive, that is, M n + k ≤ M n + M k .Proof. If M n is infinite, then we are ready. If M n is finite, then the statementcan be proved analogously to Lemma 4.13.We recall Fekete’s lemma [1], that states that for a superadditive sequence a n , lim n →∞ a n n exists and equals to sup n a n n , and for a subadditive sequence b n ,lim n →∞ b n n exists and equals to inf n b n n . Hence we have the following. Proposition 4.15. The limit lim n →∞ m n n exists and is equal to sup n m n n . Thelimit lim n →∞ M n n exists and is equal to inf n M n n . In particular, if the activity of ( W, σ ) exists then m k k ≤ lim n →∞ m n n ≤ a ( W, σ ) ≤ lim n →∞ M n n ≤ M k k for every k ≥ . Proposition 4.16. If the graphon W has finite diameter and σ is a chip con-figuration on W with U n σ ( x ) ≤ K for each n ∈ N and x ∈ [0 , , then thereexists k ∈ N such that M n − m n ≤ k for each n .Proof. Let us fix ε > N ∈ N according to the equivalent definition (ii) ofTheorem 3.2. Let A = { x ∈ [0 , 1] : u n ( x ) ≥ M n + m n } . Then either the measureof A or the measure of the complement of A is at least , so we are able to usethe condition (ii) for one these. Case 1: λ ( A ) ≥ , hence λ ( A ∪ Γ ε ( A ) ∪ Γ ε ( A ) ∪ · · · ∪ Γ Nε ( A )) = 1 by our choiceof ε and N .Let us fire each vertex m n times. Then only a measure-zero set of verticesare fired more times than they should be after the first n steps. After firingeach vertex m n times, the chip configuration is the same as originally. Thenwe additionally fire each vertex the necessary number of times to fire almost allvertex x exactly u n ( x ) times (except for the elements of the measure-zero set { x : u n ( x ) < m n } ). In this way, we get to a chip configuration that is equal to U n σ almost everywhere, hence it is essentially bounded by K .The vertices in A have to be fired at least M n − m n times additionally. Bythe definition of Γ ε ( A ), the vertices in Γ ε ( A ) receive at least ε chips if A isfired. Hence after the at least M n − m n additional firings, the vertices in Γ ε ( A )receive at least ε · M n − m n chips. Using the bound on the configuration, andthat the degree of a vertex is at most 1, each vertex of Γ ε ( A ) has to do at least (cid:6) ε · M n − m n − K (cid:7) ≥ ε · M n − m n − K additional firings.Continuing like this, we get that any vertex in A ∪ Γ ε ( A ) ∪ Γ ε ( A ) ∪· · ·∪ Γ Nε ( A )has to do at least ε N · M n − m n − K ( ε N − + · · · + ε + 1) additional firings. As thevertices that fire m n times do not need any additional firing and the measureof these vertices is positive by the definition of m n , we have that ε N · M n − m n − K ( ε N − + · · · + ε + 1) ≤ , M n − m n ≤ ( ε ) N · K ( ε N − + · · · + ε + 1), which does not depend on n . Case 2: λ ( A ) < / 2. Let B = [0 , \ A . Then λ ( B ) > / 2, hence λ ( B ∪ Γ ε ( B ) ∪ Γ ε ( B ) ∪ · · · ∪ Γ Nε ( B )) = 1Starting again from σ , let us fire each vertex M n times. Then the chipconfiguration remains the same. Now we “inverse fire” each vertex the necessarynumber of times so that in the end, the number of firings made by a vertex x is u n ( x ) for all x , except those that are in the measure zero set { x : u n ( x ) > M n } .Then we reached a chip configuration ρ , with ρ ( x ) = U n σ ( x ) almost everywhere.By definition, the vertices of B have to be inverse fired at least M n − m n times.By inverse firing B , the vertices in Γ ε ( B ) all lose at least ε chips, hence afterat least M n − m n inverse firings, they lose at least ε · M n − m n chips. As originallythey had at most K chips and at the end they have at least 0 chips, they haveto gain at least ε · M n − m n − K chips. By an inverse firing they can gain at mostone chip, hence they need to perform at least ε · M n − m n − K inverse firings.Continuing this, we get that each vertex in B ∪ Γ ε ( B ) ∪ Γ ε ( B ) ∪ · · · ∪ Γ Nε ( B )needs at least ε N · M n − m n − K ( ε N − + · · · + ε + 1) inverse firings. As the verticesthat fire M n times do not need any inverse firing and the measure of thesevertices is positive by the definition of M n , ε N · M n − m n − K ( ε N − + · · · + ε +1) ≤ M n − m n ≤ ( ε ) N · K ( ε N − + · · · + ε + 1), which also does not dependon n . We conclude that k = ( ε ) N · K ( ε N − + · · · + ε + 1) suffices. Proof of Theorem 4.11. Let σ be an arbitrary chip configuration on W . Sincethe activity of σ exists if and only if the activity of U σ exists, we can take a stepin the parallel chip-firing, and deal with σ ′ = U σ instead of σ . By Lemma 3.5,there exists a bound K ∈ R such that U n σ ′ ( x ) ≤ K for each n ∈ N and almostall x ∈ [0 , n →∞ m n ( W,σ ′ ) n and lim n →∞ M n ( W,σ ′ ) n bothexist, and by Proposition 4.16 they are the same.Since m n ( W,σ ′ ) n ≤ u n ( W,σ ′ )( x ) n ≤ M n ( W,σ ′ ) n for almost all x and every n , it alsofollows that the limit lim n →∞ u n ( W,σ ′ )( x ) n exists for almost all x , and equals tothe value lim n →∞ m n ( W,σ ′ ) n . 5. The continuity of the activity In this section we show a “continuity” theorem for the activity on graphonsof finite diameter. Definition 5.1 (Smooth pair) . A pair ( W, σ ), where W is a graphon and σ isa chip configuration is called smooth , iffor any n ∈ N , λ ( { x ∈ [0 , 1] : ∃ k ∈ N U n σ ( x ) = k · deg W ( x ) } ) = 0. (5.1)We also say that σ is a smooth chip configuration on W .The main goal of the current section is to prove the following theorem con-cerning the continuity of the activity. 22 heorem 5.2. Let ( W, σ ) be a smooth pair and d > where W is a connectedgraphon with mindeg( W ) ≥ d and σ is a chip configuration. Then for any ε > there exists δ > such that if W ′ is a connected graphon with mindeg( W ′ ) ≥ d , d (cid:3) ( W, W ′ ) < δ and σ ′ is a chip configuration with k σ − σ ′ k < δ then | a ( W, σ ) − a ( W ′ , σ ′ ) | < ε . Remark 5.3. We note that one cannot leave out the condition of ( W, σ ) beinga smooth pair. Take for example the graphon W ≡ σ ≡ 1. Then a ( W, σ ) = 1 as each vertex will fire once in each step. However,if we modify σ by decreasing its values by some ε > k σ − σ ′ k = ε , on the other hand, the chip configuration becomes stable, hence a ( W, σ ′ ) = 0. Question 5.4. Would Theorem 5.2 remain true if we only required finite diam-eter for W and W ′ , without asking for a common lower bound on the degrees?That is, is the following, stronger form of Theorem 5.2 true? Let ( W, σ ) be a smooth pair where W is a graphon of finite diameter and σ is a chip configuration. Then for any ε > there exists δ > such that if W ′ isa graphon of finite diameter with d (cid:3) ( W, W ′ ) < δ and σ ′ is a chip configurationwith k σ − σ ′ k < δ then | a ( W, σ ) − a ( W ′ , σ ′ ) | < ε . We prove Theorem 5.2 through a series of lemmas and propositions. First, inLemma 5.5 and 5.6 we show that if k u n ( W, σ ) − u n ( W ′ , σ ′ ) k is sufficiently smallthen the quantities m n and M n , as defined in Section 4.3, are also close for ( W, σ )and ( W ′ , σ ′ ). Next, in Proposition 5.8 and 5.10 we show that if d (cid:3) ( W, W ′ ) and k σ − σ ′ k are small then k U W σ − U W ′ σ ′ k and k f ( W, σ ) − f ( W ′ , σ ′ ) k are alsosmall. Finally, in the proof of Theorem 5.2, we put these ingredients togetherin an inductive argument. Lemma 5.5. Suppose σ is a chip configuration on the graphon W and σ ′ is a configuration on the graphon W ′ . Let d > and n ∈ N be given. If mindeg( W ′ ) ≥ d , k σ ′ k ≤ k σ k and k u n ( W, σ ) − u n ( W ′ , σ ′ ) k < d , then m n ( W ′ , σ ′ ) ≥ m n ( W, σ ) − (cid:16) d + k σ k d (cid:17) .Proof. Suppose that m n ( W ′ , σ ′ ) < m n ( W, σ ), otherwise we have nothing toprove. Since k u n ( W, σ ) − u n ( W ′ , σ ′ ) k ≤ d and the odometer is integer-valued,for the set A = { x ∈ [0 , 1] : u n ( W, σ )( x ) = u n ( W ′ , σ ′ )( x ) } we have λ ( A ) ≤ d .Let B = { x ∈ [0 , 1] : u n ( W ′ , σ ′ )( x ) = m n ( W ′ , σ ′ ) } . By definition of m n ( W ′ , σ ′ ), λ ( B ) > 0. Note that B ⊆ A except for a measure zero set. Foralmost all x ∈ B , deg W ′ ( x, [0 , \ A ) ≥ d , since λ ( A ) ≤ d . In the parallelchip-firing started from σ ′ on W ′ , almost all vertex of [0 , \ A fired at least m n ( W, σ ) − m n ( W ′ , σ ′ ) times more than almost all vertex in B , hence for almostall vertex x ∈ B , U nW ′ σ ′ ( x ) ≥ σ ′ ( x ) + ( m n ( W, σ ) − m n ( W ′ , σ ′ )) d . Using Lemma 3.5, U nW ′ σ ′ ( x ) ≤ k σ ′ k d , and thus U nW ′ σ ′ ( x ) ≤ k σ k d foralmost all x ∈ [0 , m n ( W, σ ) − m n ( W ′ , σ ′ ) ≤ d + k σ k d .23or the analogous claim about M n we need the chip configuration σ ′ to bebounded. Lemma 5.6. Suppose σ is a chip configuration on the graphon W and σ ′ is aconfiguration on the graphon W ′ . Let d > , K ∈ N and n ∈ N be given. If mindeg( W ′ ) ≥ d , k u n ( W, σ ) − u n ( W ′ , σ ′ ) k < d and σ ′ ( x ) < K for almost all x , then M n ( W ′ , σ ′ ) ≤ M n ( W, σ ) + Kd .Proof. The proof of the lemma is similar to the previous one. Using the bound-edness of σ ′ , M n ( W ′ , σ ′ ) < ∞ , hence we can suppose that M n ( W, σ ) < ∞ as well. We can also suppose that M n ( W ′ , σ ′ ) > M n ( W, σ ), otherwise wehave nothing to prove. Let A be defined as in the proof of Lemma 5.5, andlet B = { x ∈ [0 , 1] : u n ( W ′ , σ ′ )( x ) = M n ( W ′ , σ ′ ) } . Then again, λ ( A ) ≤ d , λ ( B ) > B ⊆ A and for almost all x ∈ B , deg W ′ ( x, [0 , \ A ) ≥ d .In the parallel chip-firing started from σ ′ on W ′ , almost all vertex of [0 , \ A fired at least M n ( W ′ , σ ′ ) − M n ( W, σ ) times less than almost all vertex in B , hencefor almost all vertex x ∈ B ,0 ≤ U nW ′ σ ′ ( x ) ≤ σ ′ ( x ) − ( M n ( W ′ , σ ′ ) − M n ( W, σ )) d . Using σ ′ ( x ) < K , it follows that M n ( W ′ , σ ′ ) − M n ( W, σ ) ≤ Kd .The following technical lemma is used many times in the proof of the nextproposition. Lemma 5.7. For two arbitrary graphons W and W ′ , k deg W − deg W ′ k ≤ (cid:3) ( W, W ′ ) . Therefore, for any η > , λ ( { x ∈ [0 , 1] : | deg W ( x ) − deg W ′ ( x ) | ≥ η } ) ≤ (cid:3) ( W,W ′ ) η .Proof. Let A = { x ∈ [0 , 1] : deg W ( x ) > deg W ′ ( x ) } . Thend (cid:3) ( W, W ′ ) ≥ Z A Z W ( x, y ) − W ′ ( x, y ) dy dx = Z A deg W ( x ) − deg W ′ ( x ) dx = Z A | deg W ( x ) − deg W ′ ( x ) | dx. Similarly, one can show that R A c | deg W ( x ) − deg W ′ ( x ) | dx ≤ d (cid:3) ( W, W ′ ), hencethe first assertion of the proposition follows.We state the next proposition in a more general form than what is neededto prove Theorem 5.2. As a consequence, the proof requires an extra technicalstep. However, the statement is simpler this way, and we believe it might beinteresting on its own. Proposition 5.8. Let σ be a chip configuration on a connected graphon W suchthat ( W, σ ) is a smooth pair. Then for any ε > there exists a δ > such thatif σ ′ is a chip configuration on a connected graphon W ′ with d (cid:3) ( W, W ′ ) < δ and k σ − σ ′ k < δ , then k U W σ − U W ′ σ ′ k < ε . roof. Throughout the proof we use the notation f ( x ) = f ( W, σ )( x ) for thegiven pair ( W, σ ), and similarly f ′ ( x ) = f ( W ′ , σ ′ )( x ) for a graphon W ′ and achip configuration σ ′ satisfying the conditions of the proposition for a δ > η > N = { x ∈ [0 , 1] : f ( x ) = f ′ ( x ) } ,U η = (cid:26) x ∈ [0 , 1] : σ ( x ) ≥ η (cid:27) , U ′ η = (cid:26) x ∈ [0 , 1] : σ ′ ( x ) ≥ η (cid:27) ,L η = { x ∈ [0 , 1] : deg W ( x ) ≤ η } , L ′ η = { x ∈ [0 , 1] : deg W ′ ( x ) ≤ η } ,A η = N ∪ U η ∪ U ′ η ∪ L η ∪ L ′ η . Then we have the following. Lemma 5.9. For any ε ′ > , there exist η > and δ > such that if W ′ is agraphon and σ ′ is a chip configuration with d (cid:3) ( W, W ′ ) < δ and k σ − σ ′ k < δ then R A η σ + σ ′ < ε ′ .Proof. We first claim that for a given ε ′′ > η > δ > λ ( A η ) < ε ′′ . Indeed, since σ is in L , for any ε ′ there exists ε ′′ such that λ ( A η ) < ε ′′ implies R A η σ < ε ′ . Letus fix such an ε ′′ . Since k σ − σ ′ k < δ , R A η σ ′ < δ + R A η σ . Therefore, if η and δ are small enough so that δ < ε ′ and λ ( A η ) < ε ′′ then R A η σ + σ ′ < ε ′ .Let us fix an ε ′′ > U η and U ′ η . Since σ ∈ L , there exists η > η ≤ η then λ ( U η ) ≤ ε ′′ . Then λ ( U ′ η \ U η ) < ηδ , since k σ − σ ′ k < δ . Hence, as U η ⊆ U η , λ ( U η ∪ U ′ η ) < ε ′′ + 2 ηδ . Clearly we can choose δ small enough sothat for any η ≤ η , λ ( U η ∪ U ′ η ) < ε ′′ .Next, we bound the measure of L η ∪ L ′ η . Since W is connected, λ ( { x :deg W ( x ) = 0 } ) = 0. It follows that there exists η > η ≤ η then λ ( L η ) ≤ ε ′′ . It is easy to check that L ′ η ⊆ L η ∪{ x : | deg W ( x ) − deg W ′ ( x ) | ≥ η } .Therefore, using Lemma 5.7 and the fact that d (cid:3) ( W, W ′ ) < δ , λ ( L ′ η \ L η ) ≤ d (cid:3) ( W,W ′ ) η < δη . Then, using L η ⊆ L η , for any fixed η ≤ η we can choose δ > λ ( L η ∪ L ′ η ) < ε ′′ .Finally, we deal with N . For ζ > B ζ = { x : ∃ t ∈ N ( t deg W ( x ) + ζ ( t + 1) < σ ( x ) < ( t + 1) deg W ( x ) − ζ ( t + 1)) } . Since ( W, σ ) is smooth and W is connected, the set B = { x : deg W ( x ) =0 or ∃ t ∈ N ( σ ( x ) = t deg W ( x )) } is of measure 0. Since S ζ> B ζ = B c and ζ < ζ ′ implies B ζ ⊇ B ζ ′ , there exists ζ > λ ( B ζ ) ≥ − ε ′′ . Let usfix such a ζ . 25et N ζ = { x ∈ B ζ : f ( x ) > f ′ ( x ) } ,N ζ = { x ∈ B ζ : f ( x ) < f ′ ( x ) } , and note that N ⊆ B cζ ∪ N ζ ∪ N ζ , hence it is enough to bound the measure ofthese two sets.If x ∈ N ζ then for some t ∈ N , σ ( x ) > t deg W ( x ) + ζ ( t + 1), and σ ′ ( x ) < t deg W ′ ( x ) . It follows that either σ ( x ) ≥ σ ′ ( x ) + ζ or deg W ′ ( x ) ≥ deg W ( x ) + ζ , hence N ζ ⊆ { x : σ ( x ) ≥ σ ′ ( x ) + ζ } ∪ { x : deg W ′ ( x ) ≥ deg W ( x ) + ζ } . Since k σ − σ ′ k < δ , λ ( { x : σ ( x ) ≥ σ ′ ( x )+ ζ } ) < δζ . Using Lemma 5.7, λ ( { x :deg W ′ ( x ) ≥ deg W ( x ) + ζ } ) ≤ (cid:3) ( W,W ′ ) ζ < δζ . It follows that λ ( N ζ ) < δζ .A similar argument yields that λ ( N ζ ) < δζ , hence, using N ⊆ B cζ ∪ N ζ ∪ N ζ , λ ( N ) < ε ′′ + δζ . It is clear, that by choosing a small enough δ , λ ( N ) < ε ′′ .Therefore we can complete the proof by first fixing any positive η ≤ min { η , η } and then choosing a small enough δ , so that λ ( U η ∪ U ′ η ) , λ ( L η ∪ L ′ η ) , λ ( N ) < ε ′′ . We apply Lemma 5.9 with ε ′ = ε to obtain η > δ , hence d (cid:3) ( W, W ′ ) <δ , k σ − σ ′ k < δ imply R A η σ + σ ′ < ε . The final value for δ will be chosen tobe less than δ .Now we investigate the effect of firing the vertices in [0 , \ A η and in A η separately. So let f = f + f be the unique decomposition with f ( x ) = 0 if x ∈ A η and f ( x ) = 0 if x ∈ [0 , \ A η . Let σ ( x ) = σ ( x ) − f ( x ) deg W ( x ) + Z f ( y ) W ( x, y ) dy,σ ( x ) = σ ( x ) − f ( x ) deg W ( x ) + Z f ( y ) W ( x, y ) dy. We define f ′ , f ′ , σ ′ and σ ′ analogously for W ′ and σ ′ . It is straightforward tocheck that σ = U W σ and σ ′ = U W ′ σ ′ , so it is enough to prove that k σ − σ ′ k <ε . Since k σ − σ ′ k ≤ k σ − σ k + k σ − σ ′ k + k σ ′ − σ ′ k , it is enough to bound these quantities.26hen k σ − σ k = Z (cid:12)(cid:12)(cid:12)(cid:12) − f ( x ) deg W ( x ) + Z f ( y ) W ( x, y ) dy (cid:12)(cid:12)(cid:12)(cid:12) dx ≤ Z f ( x ) deg W ( x ) dx + Z Z f ( y ) W ( x, y ) dy dx ≤ Z A η σ ( x ) dx + Z Z f ( y ) W ( x, y ) dx dy ≤ Z A η σ ( x ) dx + Z f ( y ) deg W ( y ) dy ≤ Z A η σ ( x ) dx + Z A η σ ( y ) dy ≤ ε, where we used Fubini’s theorem for non-negative functions to interchange theorder of integration. A similar calculation shows that k σ ′ − σ ′ k ≤ ε .It remains to show that k σ − σ ′ k < ε , which is the tricky part of the proof. k σ − σ ′ k ≤ k σ − σ ′ k + Z | f ( x ) deg W ( x ) − f ′ ( x ) deg W ′ ( x ) | dx + Z (cid:12)(cid:12)(cid:12)(cid:12)Z f ( y ) W ( x, y ) dy − Z f ′ ( y ) W ′ ( x, y ) dy (cid:12)(cid:12)(cid:12)(cid:12) dx ≤ δ + Z A cη f ( x ) | deg W ( x ) − deg W ′ ( x ) | dx + Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z A cη f ( y )( W ( x, y ) − W ′ ( x, y )) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx. Using the fact that f ( x ) ≤ σ ( x )deg W ( x ) ≤ η η = η for every x ∈ A cη and Lemma5.7, Z A cη f ( x ) | deg W ( x ) − deg W ′ ( x ) | dx ≤ Z A cη η | deg W ( x ) − deg W ′ ( x ) | dx ≤ (cid:3) ( W, W ′ ) η < δη . It remains to bound the integral R (cid:12)(cid:12)(cid:12)R A cη f ( y )( W ( x, y ) − W ′ ( x, y )) dy (cid:12)(cid:12)(cid:12) dx .This part is the most technical one.Let K be the largest integer with K ≤ η , and for 0 ≤ j ≤ K , let E j = { y ∈ [0 , 1] : f ( y ) = j } . For each fixed x ∈ [0 , R E j W ( x, y ) − W ′ ( x, y ) dy can be either negative and non-negative for each j ≤ K . These give us 2 K +1 many possibilities for the sign of the integrals, thus partitioning [0 , s ∈ { , } K +1 , where 0 corresponds tonon-negative integrals and 1 corresponds to negative ones, so let I s = n x ∈ [0 , 1] : ∀ j ≤ K (cid:16) s ( j ) = 0 ⇔ Z E j W ( x, y ) − W ′ ( x, y ) dy ≥ (cid:17)o . Z (cid:12)(cid:12)(cid:12) Z A cη f ( y )( W ( x, y ) − W ′ ( x, y )) dy (cid:12)(cid:12)(cid:12) dx ≤ Z X j ≤ K (cid:12)(cid:12)(cid:12) Z E j j ( W ( x, y ) − W ′ ( x, y )) dy (cid:12)(cid:12)(cid:12) dx = X s ∈{ , } K +1 Z I s X j ≤ K ( − s ( j ) Z E j j ( W ( x, y ) − W ′ ( x, y )) dy dx ≤ K X s ∈{ , } K +1 X j ≤ K Z I s Z E j ( − s ( j ) ( W ( x, y ) − W ′ ( x, y )) dy dx ≤ K ( K + 1)2 K +1 d (cid:3) ( W, W ′ ) < η (cid:18) η + 1 (cid:19) η +1 δ, where we used the definition of d (cid:3) and Fubini’s theorem for the integrablefunction ( x, y ) ( − s ( j ) ( W ( x, y ) − W ′ ( x, y )). Hence, k σ − σ ′ k ≤ δ + δη + η (cid:16) η + 1 (cid:17) η +1 δ . Therefore, by choosing δ small enough, we can make surethat k σ − σ ′ k < ε , completing the proof of the Proposition 5.8. Proposition 5.10. Suppose ( W, σ ) is a smooth pair, where W is a graphonwith finite diameter and σ is a chip configuration. Then for any ε > and d > there exists a δ > such that if σ ′ is a chip configuration on a graphon W ′ with mindeg( W ′ ) ≥ d , d (cid:3) ( W, W ′ ) < δ and k σ − σ ′ k < δ , then k f ( W, σ ) − f ( W ′ , σ ′ ) k < ε .Proof. Let θ = min { mindeg( W ) , d } , and apply Lemma 5.9 with ε ′ = εθ toobtain η > δ > 0. Then, as f ( W, σ )( x ) ≤ σ ( x )deg W ( x ) ≤ σ ( x ) θ for almost all x ∈ [0 , R A η f ( W, σ ) ≤ θ R A η σ < ε . Similarly, R A η f ( W ′ , σ ′ ) < ε . Therefore, k f ( W, σ ) − f ( W ′ , σ ′ ) k < ε for every W ′ and σ ′ with d (cid:3) ( W, W ′ ) < δ and k σ − σ ′ k < δ .We are now ready to prove the main theorem of this section. Proof of Theorem 5.2. Let K = 1 + 2 k σ k d . We first claim that it is enough toprove the theorem for chip configurations that satisfy σ ( x ) ≤ K , σ ′ ( x ) ≤ K for almost all x ∈ [0 , σ ( x ) ≤ K, σ ′ ( x ) ≤ K for almost all x ∈ [0 , σ and σ ′ be arbitrary.Then a ( W, U W σ ) = a ( W, σ ) and a ( W ′ , U W ′ σ ′ ) = a ( W ′ , σ ′ ), and using Lemma3.5, U W σ ( x ) ≤ k σ k d and U W ′ σ ′ ( x ) ≤ k σ ′ k d for almost all x ∈ [0 , δ to be less than k σ k , then K is an essential upper bound for U W σ and U W ′ σ ′ , hence the weak version allows us to find δ ′ small enough so thatthe conclusion of the theorem holds for U W σ and U W ′ σ ′ instead of σ and σ ′ provided that k U W σ − U W ′ σ ′ k < δ ′ . Then one can use Proposition 5.8 with ε = δ ′ to find δ ≤ δ ′ so that k σ − σ ′ k < δ implies k U W σ − U W ′ σ ′ k < δ ′ , andwe are done. 28o now let W , σ , d and ε be fixed with σ ( x ) ≤ K for almost all x ∈ [0 , K = 1 + 2 k σ k d . We need to show that one can find δ > δ ≤ k σ k suchthat for every ( W ′ , σ ′ ) with mindeg( W ′ ) ≥ d , d (cid:3) ( W, W ′ ) < δ , k σ − σ ′ k < δ and σ ′ ( x ) ≤ K for almost all x ∈ [0 , 1] we have | a ( W, σ ) − a ( W ′ , σ ′ ) | < ε .Using Propositions 4.16 and 4.15,lim n →∞ m n ( W, σ ) n = lim n →∞ M n ( W, σ ) n and m n ( W, σ ) n ≤ a ( W, σ ) ≤ M n ( W, σ ) n for each n .Therefore we can choose n large enough so that M n ( W, σ ) n − m n ( W, σ ) n < ε , d + 4 k σ k nd < ε Kdn < ε , where the latter two quantities come from Lemma 5.5 and Lemma 5.6. Notethat the choice of n only depends on ε, d, σ and W .We claim that it is enough to prove that if δ is small enough and W ′ , σ ′ satisfythe conditions of the theorem, moreover, σ ′ ( x ) ≤ K for almost all x ∈ [0 , m n ( W ′ , σ ′ ) ≥ m n ( W, σ ) − (cid:18) d + 4 k σ k d (cid:19) and M n ( W ′ , σ ′ ) ≤ M n ( W, σ ) + 2 Kd . (5.2)Indeed in this case m n ( W ′ ,σ ′ ) n ≥ m n ( W,σ ) n − ε ≥ a ( W, σ ) − ε , and M n ( W,σ ) n ≤ M n ( W,σ ) n + ε ≤ a ( W, σ ) + ε . Since W ′ has finite diameter, a ( W ′ , σ ′ ) = lim k →∞ m k ( W ′ , σ ′ ) k = sup k m k ( W ′ , σ ′ ) k ≥ m n ( W ′ , σ ′ ) n ≥ a ( W, σ ) − ε. Similarly, a ( W ′ , σ ′ ) = lim k →∞ M k ( W ′ , σ ′ ) k = inf k M k ( W ′ , σ ′ ) k ≤ M n ( W ′ , σ ′ ) n ≤ a ( W, σ ) + ε. By Lemmas 5.5 and 5.6, for (5.2) to hold, it is enough to chose δ smallenough so that if d (cid:3) ( W, W ′ ) < δ and k σ − σ ′ k < δ , then k σ ′ k ≤ k σ k and k u n ( W, σ ) − u n ( W ′ , σ ′ ) k < d . The first condition is satisfied since δ ≤ k σ k .To satisfy the second one, we apply Propositions 5.8 and 5.10 repeatedly. Itis clearly enough to choose δ small enough so that for every k ≤ n , k ≥ k ( u k ( W, σ ) − u k − ( W, σ )) − ( u k ( W ′ , σ ′ ) − u k − ( W ′ , σ ′ )) k = k f ( W, U k − W σ ) − f ( W ′ , U k − W ′ σ ′ ) k < d n .We first apply Proposition 5.10 to ( W, U n − W σ ) and ε = d n to get δ n > k U n − W σ − U n − W ′ σ ′ k < δ n and d (cid:3) ( W, W ′ ) < δ n k f ( W, U n − W σ ) − f ( W ′ , U n − W ′ σ ′ ) k < d n . Now let ε n − = min (cid:8) δ n , d n (cid:9) and apply both Proposition 5.8 and Proposition5.10 with ε = ε n − to get δ = δ n − > δ n − ≤ δ n so that k U n − W σ − U n − W ′ σ ′ k < δ n − and d (cid:3) ( W, W ′ ) < δ n − imply k U n − W σ − U n − W ′ σ ′ k < ε n − ≤ δ n and k f ( W, U n − W σ ) − f ( W ′ , U n − W ′ σ ′ ) k < ε n − ≤ d n . By continuing downwards in a similar fashion, we can arrive at δ = δ > k σ − σ ′ k < δ and d (cid:3) ( W, W ′ ) < δ imply k U k − W σ − U k − W ′ σ ′ k ≤ δ k and thus k f ( W, U k − W σ ) − f ( W ′ , U k − W ′ σ ′ ) k < d n for each k ≤ n , k ≥ k u n ( W, σ ) − u n ( W ′ , σ ′ ) k < d , and the proof of the theorem is finallycomplete.The next proposition shows that ( W, σ ) being a smooth pair is not a verystrong condition. Proposition 5.11. For any chip configuration σ : [0 , → R and any connectedgraphon W , the set { µ ∈ [0 , 1] : ( W, σ + µ · deg W ) is not a smooth pair } iscountable.Proof. Fix the chip configuration σ and for any µ ∈ [0 , σ µ = σ + µ · deg W . For a µ ∈ [0 , 1] and n, ℓ, k ∈ N , let bad ( µ, n, ℓ, k ) = { x ∈ [0 , 1] : u n ( W, σ µ )( x ) = ℓ, U n σ µ ( x ) = k · deg W ( x ) } . It is clear from the definition that bad ( µ, n, ℓ, k ) is measurable for each µ ∈ [0 , 1] and n, ℓ, k ∈ N . Let us fix n, ℓ, k ∈ N , we now show that if µ ′ = µ then λ ( bad ( µ, n, ℓ, k ) ∩ bad ( µ ′ , n, ℓ, k )) = 0. Suppose that µ ′ > µ , then byLemma 2.3, u n ( W, σ µ ′ )( x ) ≥ u n ( W, σ µ )( x ) for almost all x ∈ [0 , x ∈ bad ( µ, n, ℓ, k ) ∩ bad ( µ ′ , n, ℓ, k ) with deg W ( x ) > U n σ µ ′ ( x )= σ µ ′ ( x ) − u n ( W, σ µ ′ )( x ) · deg W ( x ) + Z u n ( W, σ µ ′ )( y ) W ( x, y ) dy = σ µ ′ ( x ) − u n ( W, σ µ )( x ) · deg W ( x ) + Z u n ( W, σ µ ′ )( y ) W ( x, y ) dy> σ µ ( x ) − u n ( W, σ µ )( x ) · deg W ( x ) + Z u n ( W, σ µ )( y ) W ( x, y ) dy = U n σ µ ( x ) , U n σ µ ′ ( x ) = U n σ µ ( x ) = k · deg W ( x ). Hence indeed,almost all x ∈ [0 , 1] cannot be in both bad ( µ, n, ℓ, k ) and bad ( µ ′ , n, ℓ, k ).If for fixed n, ℓ, k ∈ N uncountably many µ exists with λ ( bad ( µ, n, ℓ, k )) > λ ( bad ( µ, n, ℓ, k )) > ε for some ε > 0. Bytaking at least ε + 1 sets of those, two will intersect in a set of positive measure,a contradiction. We conclude that for fixed n, ℓ, k ∈ N , only countably many µ exists with the property that bad ( µ, n, ℓ, k ) is of positive measure. Thereforeall, but countably many µ has the property that λ ( bad ( µ, n, ℓ, k )) = 0 for every n, ℓ, k ∈ N , hence for all, but countably many µ ∈ [0 , W, σ µ ) is a smoothpair. 6. The Devil’s staircase phenomenon In this section we use our previous results to prove the Devil’s staircase phe-nomenon in some situations. First, we prove that under mild conditions, theactivity diagram of a chip configuration on an Erd˝os–R´enyi random graph isclose to a Devil’s staircase with high probability. Then we show a one-parameterfamily of random chip configurations on Erd˝os–R´enyi random graphs that ex-hibit the Devil’s staircase phenomenon with high probability. Let C p denotethe graphon with C p ( x, y ) = p for all x, y ∈ [0 , C p Here we give a sufficient condition for the activity diagram of a chip config-uration on C p to be a Devil’s staircase. We deduce the sufficient condition fromthe analogous theorem of Levine [8], which concerns C . Let us first note therelationship of the activity on C and on C p . Proposition 6.1. For any σ and < p ≤ , a ( C p , σ ) = a ( C , σp ) .Proof. It is enough to show that for each n , p U nC p ( σ ) = U nC ( σp ). This impliesthat ⌊ p U nC p ( σ )( x ) ⌋ = ⌊ U nC ( σp )( x ) ⌋ for each n , hence the odometers are the same.Proving p U nC p ( σ ) = U nC ( σp ) is straightforward by induction on n .Now we can use the results of [8] that gives a sufficient condition for theactivity diagram of a chip configuration on the graphon C to be a Devil’sstaircase. (We note that [8] uses a different terminology, in particular, it doesnot refer to graphons.)For [8], a generalized chip configuration is a measurable function σ : [0 , → [0 , ∞ ) (hence every chip configuration on a graphon as defined in the currentpaper is a generalized chip configuration as defined in [8]). The update operator U defined in equation (8) of [8] coincides with the parallel update rule for thegraphon C if σ ( x ) < x ∈ [0 , C is defined as lim n →∞ β n ( σ ) n (if it exists), where β n ( σ ) =31 n − i =0 λ ( { x : U i σ ( x ) ≥ } ). One can make the obvious generalization and foran arbitrary graphon W and chip configuration σ . Set β n ( W, σ ) = n − X i =0 λ ( { x : U i σ ( x ) ≥ deg W ( x ) } ) . Proposition 6.2. If W has finite diameter and σ ( x ) < W ( x ) for almostall x ∈ [0 , , then the two definitions of the activity coincide, that is, a ( W, σ ) =lim n →∞ β n ( W,σ ) n .Proof. By Theorem 4.11, if W has finite diameter, then there exist a ( W, σ )such that lim n →∞ u n ( x ) n = a ( W, σ ) for almost all x ∈ [0 , σ ( x ) < W ( x ) for almost all x ∈ [0 , 1] implies U i σ ( x ) < W ( x ) for all i ∈ N andalmost all x ∈ [0 , u n ( x ) = |{ i ∈ N : 0 ≤ i < n, U i σ ( x ) ≥ deg W ( x ) }| for almost all x ∈ [0 , β n ( W, σ ) = Z u n ( x ) dx. As lim n →∞ u n ( x ) n = a ( W, σ ) for almost all x ∈ [0 , ε , we can choose n such that for any n ≥ n , for A n = { x ∈ [0 , 1] : | u n ( x ) n − a ( W, σ ) | ≤ ε } , wehave λ ( A n ) ≥ − ε .Then for n ≥ n , β n ( W, σ ) n = Z u n ( x ) n dx ≤ Z A n ( a ( W, σ ) + ε ) dx + Z [0 , \ A n dx ≤ a ( W, σ ) + 2 ε. Hence lim n →∞ β n ( W,σ ) n ≤ a ( W, σ ).Similarly, for n ≥ n , β n ( W, σ ) n = Z u n ( x ) n dx ≥ Z A n ( a ( W, σ ) − ε ) dx ≥ (1 − ε )( a ( W, σ ) − ε ) . Hence lim n →∞ β n ( W,σ ) n ≥ a ( W, σ ), and the proof is complete.Now we collect the results from [8] that we need. The statements and argu-ments that follow are all present in [8], but not everything is in a form convenientfor us, so we repeat some of the arguments of that paper. We call a chip con-figuration σ on C preconfined if σ ( x ) < x ∈ [0 , σ , let us define the function f σ : R → R the following way. For x ∈ [0 , f σ ( x ) = λ ( { v : σ ( v ) ≥ } ) + λ ( { v : σ ( v ) ∈ [1 − x, ∪ [2 − x, } ) . (6.1)It is easy to check that f is an increasing function with f σ (1) = f σ (0) + 1.Hence there is a unique extension of f σ to R as an increasing function, which32e also denote by f σ , that satisfies f σ ( x + 1) = f σ ( x ) + 1. If f σ is continuousthen it has a well-defined Poincar´e rotation number ρ ( f σ ) = lim n →∞ f nσ ( x ) n , which is independent of x , see [8]. Lemma 6.3 ([8, Lemma 6]) . If σ is preconfined and f σ is continuous then a ( C , σ ) = ρ ( f σ ) . It is easy to check (and the computation can also be found in [8]) that if y ∈ R is given such that σ + y is also a preconfined chip configuration on C (thatis, 0 ≤ σ + y < f σ + y ( x − y ) = f σ ( x ). As stated also in [8], conjugatingby the homeomorphism R y : R → R defined by R y ( x ) = x + y does not changethe rotation number. Then, for any y ∈ R , ρ ( f σ + y ) = ρ ( R y ( f σ + y ( R − y ))). Thefunction inside is x R y ( f σ + y ( x − y )) = R y ( f σ ( x )), hence, using also theprevious lemma, we have the following. Lemma 6.4. If σ is a chip configuration on C , y ∈ R such that σ and σ + y arepreconfined, and both f σ and f σ + y are continuous, then a ( C , σ + y ) = ρ ( R y ◦ f σ ) . Let σ be a stable chip configuration on C , that is, σ ( v ) < v ∈ [0 , y ∈ [0 , σ and σ + y are both preconfined. Now defineΦ σ,y : R → R byΦ σ,y ( x ) = ⌈ x ⌉ − λ ( { v ∈ [0 , 1] : σ ( v ) < ⌈ x ⌉ − x } ) + y. It is easy to check that Φ σ,y is continuous if λ ( { v : σ ( v ) = x } ) = 0 for all x ∈ [0 , σ,y ( x ) = R y ( f σ ( x )), hence s ( C , σ )( y ) = a ( C , σ + y ) = ρ (Φ σ,y ) , where s is the activity diagram as defined in Section 2. Since Φ σ,y ( x + 1) =Φ σ,y ( x ) + 1 for every x ∈ R , it makes sense to denote by Φ σ,y the correspondingmap from R / Z = S to S . Theorem 6.5 ([8, Proposition 10]) . If σ ( v ) < for almost all v ∈ [0 , , λ ( { v : σ ( v ) = c } ) = 0 for each c ∈ R and Φ qσ,y = Id for any q ∈ N \ { } , then s ( C , σ ) is a Devil’s staircase. Moreover, if α is irrational, then s ( C , σ ) − ( α ) isa point, and if y is rational then s ( C , σ ) − ( y ) is an interval of positive length. Applying Proposition 6.1, we get the following corollary for C p . Theorem 6.6. For some < p ≤ , if σ ( v ) < p for almost all v ∈ [0 , , λ ( { v : σ ( v ) = c } ) = 0 for each c ∈ R and Φ q p σ,y = Id for any q ∈ N \ { } , then s ( C p , σ ) is a Devil’s staircase. Moreover, if α is irrational, then s ( C p , σ ) − ( α ) isa point, and if y is rational then s ( C p , σ ) − ( y ) is an interval of positive length. .2. Random graphs We give a sufficient condition for the activity diagrams of random graphsconverging to a Devil’s staircase. We collected the necessary background onrandom graphs in Appendix Appendix A.First, we need the following result. Theorem 6.7. [9, Theorem 11.32], [10, Corollary 2.6] If G n = G ( n, p ) is asequence of Erd˝os–R´enyi graphs then G n → C p with probability . We note here, that the referenced papers use the unlabeled cut distanceto prove the above theorem. However, as noted in Section 2, δ (cid:3) ( G n , C p ) =d (cid:3) ( G n , C p ) for each graph G n , hence the theorem remains true if the con-vergence is understood using the labeled cut distance. We also note that bydefinition, d (cid:3) ( G n , C p ) = d (cid:3) ( W G n , C p ), hence the convergence also holds forthe graphon version of the graphs.To deal with the convergence of chip configurations on graphs, we do thefollowing: for a graph G with vertices labeled v , . . . , v n , we define the graphonversion ˜ σ : [0 , → R of a chip configuration σ by ˜ σ ( x ) = n σ ( v i ) if i − n ≤ x < in .For a sequence of chip configurations ( σ n ) n such that σ n lives on the graph G n ,we say that they are convergent if the sequence of graphon versions (˜ σ n ) n isconvergent in the k . k norm. Theorem 6.8. Suppose that ( G n ) n is a sequence of (labeled) Erd˝os–R´enyirandom graphs, where G n = G ( n, p ) , < p ≤ , σ n is a chip configura-tion on G n for each n , and k ˜ σ n − σ k → for some chip configuration σ on C p such that σ ( x ) < p for almost all x ∈ [0 , . Moreover, suppose that λ ( { x ∈ [0 , 1] : σ ( x ) = c } ) = 0 for each c ∈ R and Φ q (1 /p ) σ,y = Id for any y ∈ [0 , and q ∈ N . Then with probability , the sequence of activity diagrams ( s ( G n , σ n )) n converges uniformly to the Devil’s staircase s ( C p , σ ) .Proof. Since the activity diagram s ( C p , σ ) is a Devil’s staircase by Theorem 6.6,it is enough to prove that with probability 1, s ( G n , σ n ) → s ( C p , σ ) uniformlyas n → ∞ .We would like to apply Theorem 5.2 for the graphon C p and the chip con-figuration σ y = σ + yp [0 , .As noted above, with probability 1, d (cid:3) ( G n , C p ) → 0. By our assumption,the graphon versions ˜ σ , ˜ σ , . . . converge to σ in k . k .We need to show that C p has finite diameter, but this is trivial, since forany set A with λ ( A ) > ε = pλ ( A ) works to show that Γ ε ( A ) = [0 , C p , σ y ) is a smooth pair for any value of y ∈ [0 , σ y ( x ) < p for each y ∈ [0 , 1] and x . This implies U i σ y ( x ) < p foreach i by induction. (Indeed, in any step, any vertex fires at most once. Henceany vertex can gain at most p chips in a step. But if a vertex already had atleast p chips, then it also fires, hence its number of chips does not increase.) Wenow claim that λ ( { x : U n σ y ( x ) = c } ) = 0 for any n ∈ N , y ∈ [0 , 1] and c ∈ R .Let us fix y ∈ [0 , 1] and define c i = λ ( { x : U i σ y ( x ) ≥ p } ). Our claim for n = 034s a condition of the theorem, and for n > { x : U n σ y ( x ) = c } = { x : U n − σ y ( x ) < p and U n − σ y ( x ) = c − c n − }∪{ x : U n − σ y ( x ) ≥ p and U n − σ y ( x ) = c − c n − + p } . One can easily show by induction on n , using the above equality, that { x : U n σ y ( x ) = c } is indeed a set of measure 0 for each n , y and c .Fix d < p and ε > 0. We show thatwith probability 1 there exists n such that if n ≥ n and y ∈ [0 , | a ( C p , σ y ) − a ( G n , σ n + y deg G n ) | < ε. (6.2)For any y ∈ [0 , C p , σ y ) with ε and d to get δ . Let U n = W G n be the graphon corresponding to G n . Noticethat the graphon version of σ n + y deg G n is ˜ σ n + y deg U n . It is also easy to seethat a ( G n , σ n + y deg G n ) = a ( U n , ˜ σ n + y deg U n ), since the chip-firings on G n and on U n correspond to each other. Therefore to get (6.2) using Theorem 5.2and Theorem 3.2, we need to show that with probability 1, there exists n suchthat for n ≥ n , U n is connected, has minimal degree at least d , d (cid:3) ( C p , U n ) < δ ,and for any y ∈ [0 , k σ y − (˜ σ n + y deg U n ) k < δ . U n has degree at least d for each point if and only if G n has degree at least dn . Hence by PropositionAppendix A.1 with probability 1 there exists an index n such that for each n ≥ n , the graphon U n has mindeg( U n ) ≥ d . If G n is connected, then U n is connected, hence by Proposition Appendix A.4, there exists n such thatfor n ≥ n , U n is connected. With probability 1, d (cid:3) ( C p , U n ) tends to 0 by theremark after Theorem 6.7, hence there exists an index n such that d (cid:3) ( C p , U n ) <δ for n ≥ n . Now k σ y − (˜ σ n + y deg U n ) k ≤ k σ − ˜ σ n k + k yp [0 , − y deg U n k . Here k σ − ˜ σ n k tends to 0 by the assumptions of the theorem, hence it is below δ/ n ≥ n for some index n . For a fixed x ∈ [0 , 1] let v be the vertex of G n such that x belongs to the part of [0 , 1] corresponding to v . Then, using that y ≤ P (cid:20) ∃ y ∈ [0 , (cid:18) | yp − y deg U n ( x ) | > δ (cid:19)(cid:21) ≤ P (cid:20) | p − deg U n ( x ) | > δ (cid:21) = P (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) p − deg G n ( v ) n (cid:12)(cid:12)(cid:12)(cid:12) > δ (cid:21) = P (cid:20) | pn − deg G n ( v ) | > δn (cid:21) ≤ e − δ n by Claim Appendix A.3. As G n has n vertices, P [ ∃ x ∈ [0 , 1] : | p − deg U n ( x ) | >δ/ ≤ ne − δ n . Since P n ≥ ne − δ n < ∞ , by the Borel–Cantelli lemma, withprobability 1 there exists n such that | p − deg U n ( x ) | ≤ δ/ n ≥ n andall x ∈ [0 , n ≥ n we have | yp − y deg U n ( x ) | ≤ δ/ x ∈ [0 , y ∈ [0 , k yp [0 , − y deg U n k = Z | yp − y deg U n ( x ) | dx ≤ δ/ . n = max { n , n , n , n , n } exists and theconditions of Theorem 5.2 are satisfied with W ′ = U n and σ ′ = ˜ σ n + y deg U n for n ≥ n and y ∈ [0 , ε > 0, with probability 1,there exists an index n with | a ( C p , σ y ) − a ( G n , σ n + y deg G n ) | < ε for n ≥ n and y ∈ [0 , ε values tending to 0, we conclude thatwith probability 1, s ( G n , σ n ) tends to s ( C p , σ ) uniformly, therefore the proof iscomplete. We show a concrete example where the activities of a one parameter familyof chip configurations on a random graph give a Devil’s staircase with highprobability. We will again take an Erd˝os–R´enyi random graph, but this time weput a random number of chips on the vertices independently following geometricdistribution, and look at how the activity changes if we increase the mean ofthe geometric distribution.Let G n = G ( n, p ) for some 0 < p ≤ 1. Suppose that for v ∈ V ( G n ), σ µn ( v ) ∼ Geometric ( µn ) independently for some µ > 0. Here we mean thegeometric distribution as P ( σ µn ( v ) = k ) = ( µn ) k / (1 + µn ) k +1 for k ≥ 0. Notethat this way, the expected value E σ µn ( v ) = µn . Let us relabel the vertices suchthat σ µn ( v ) ≤ σ µn ( v ) ≤ . . . , and let us denote by ˜ σ µn the corresponding chipconfiguration on the graphon W G n . Let us take these random chip configurationsindependently for each n ∈ N . For different values of µ , we couple the randomchip configurations in the following way. For each vertex v , we independentlygenerate countably many independent uniform random variables between 0 and1. For some value µ , we put k chips on v if the first k of its random variablesare between µn and 1, and the ( k + 1) th is between 0 and µn . This way weobtain independent Geometric ( µn ) random variables for each vertex.We show the following. Theorem 6.9. Suppose that G n = G ( n, p ) for < p ≤ , and σ µn is a chipconfiguration where the number of chips on each vertex is an independent Geo-metric random variable with mean µn , coupled for different values of µ as above.Then with probability one, the sequence of functions µ a ( G n , σ µn ) convergespointwise to a Devil’s staircase on the interval µ ∈ [0 , p log 2 ] . To prove this theorem, we need to find out the limit of the chip configura-tions. Let σ µ ( v ) = − µ log(1 − v ) be a chip configuration on C p , where by logwe mean the natural logarithm. We will show the following: Lemma 6.10. For any fixed µ > , k ˜ σ µn − σ µ k → with probability as n → ∞ . We will prove the lemma later. As G n → C p with probability 1, one needsto examine the behaviour of the activity of σ µ on C p . Unfortunately we cannotdirectly apply Theorem 6.6 here, as we do not talk about activity diagrams, buta different one-parameter family of chip configurations. However, we can stillshow the following. 36 emma 6.11. The map µ a ( C p , σ µ ) is a Devil’s staircase on [0 , p log 2 ] .Proof. The chip configuration σ µ is unbounded, but U σ µ is bounded, and since a ( σ µ ) = a ( U σ µ ), it is enough to deal with the latter. To calculate U σ µ , let usdenote by { x } p the p -fractional part of x ∈ R , that is, the unique number in[0 , p ) with the property that x + kp = { x } p for some k ∈ Z . Then U σ µ ( v ) = {− µ log(1 − v ) } p + p X n ≥ λ ( { u : − µ log(1 − u ) ≥ np } ) . (6.3)Now, as it is easier to handle monotone increasing chip configurations and U σ µ is not increasing (as a function v ( U σ µ )( v )), we try to rearrange it to anincreasing chip configuration σ µ with the property that λ ( { v : U σ µ ( v ) < x } ) = λ ( { v : σ µ ( v ) < x } ) for every x ∈ R . (6.4)Clearly, if (6.4) holds, the analogous statement will hold for U k ( U σ µ ( v )) and U k ( σ µ ), hence, by Proposition 6.2, a ( σ µ ) = a ( U σ µ ) = a ( σ µ ).To define an increasing σ µ satisfying (6.4) , our only option is that σ µ ( v ) = x if and only if λ ( { u : U σ µ ( u ) < x } ) = v . Since λ ( { u : − µ log(1 − u ) ≥ y } ) = e − yµ , (6.5) p X n ≥ λ ( { u : − µ log(1 − u ) ≥ np } ) = pe − pµ − e − pµ =: y ( µ ) . (6.6)From (6.5) we also have for x ∈ [0 , p ) that λ ( { v : {− µ log(1 − v ) } p < x } ) = λ (cid:18)(cid:26) v : − µ log(1 − v ) ∈ [ n ≥ [ np, np + x ) (cid:27)(cid:19) = X n ≥ e − npµ − e − np + xµ = 1 − e − xµ − e − pµ . Hence, using also (6.3) and (6.6), λ ( { u : U σ µ ( u ) < x + y ( µ ) } ) = 1 − e − xµ − e − pµ = v ⇔ x = − µ log(1 − v + ve − pµ ) , thus σ µ ( v ) = − µ log(1 − v + ve − pµ ) + y ( µ ) = − µ log(1 − v + ve − pµ ) + pe − pµ − e − pµ . It is easy to check that it satisfies (6.4), and also that σ µ ( v ) ≤ p if v ∈ [0 , µ ∈ [0 , p log(2) ]. Since − µ log(1 − v + ve − pµ ) ≥ v ∈ [0 , σ = σ µ − y ( µ ) and y = y ( µ ) to get that a ( σ µ ) = ρ (cid:0) R y ( µ ) p (cid:0) f σµ − y ( µ ) p (cid:1)(cid:1) , f σµ − y ( µ ) p is defined as in (6.1).With the notation f µ = R y ( µ ) p (cid:0) f σµ − y ( µ ) p (cid:1) , our task is to show that µ ρ ( f µ )is a Devil’s staircase. For x ∈ [0 , f µ ( x ) = R y ( µ ) p (cid:0) f σµ − y ( µ ) p (cid:1) ( x )= y ( µ ) p + λ (cid:18)(cid:26) v : − µp log(1 − v + ve − pµ ) ≥ − x (cid:27)(cid:19) = y ( µ ) p + e − p (1 − x ) µ − e − pµ − e − pµ = e − p (1 − x ) µ − e − pµ . To show that µ ρ ( f µ ) is a Devil’s staircase, as in [8], we need to showthat µ f µ is increasing, continuous with respect to the supremum norm, andthat ( f µ ) n = id R + k for each n ≥ k ∈ Z . (Note that the last condition saysthat if f µ : S → S is the circle map corresponding to f µ then ( f µ ) n is not theidentity.)To show that µ f µ is increasing, we need to show for x ∈ [0 , µ < µ ′ that f µ ( x ) ≤ f µ ′ ( x ). This inequality easily follows from e − p (1 − x ) µ < e − p (1 − x ) µ and 1 − e − pµ > − e − pµ ′ .Now we show that µ f µ is continuous with respect to the supremumnorm. For µ < µ ′ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − p (1 − x ) µ ′ − e − pµ ′ − e − p (1 − x ) µ − e − pµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − p (1 − x ) µ ′ − e − pµ ′ − e − p (1 − x ) µ ′ − e − pµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − p (1 − x ) µ ′ − e − pµ − e − p (1 − x ) µ − e − pµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) − e − pµ ′ − − e − pµ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − p (1 − x ) µ ′ − e − pµ − e − p (1 − x ) µ − e − pµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where, using the fact that for x ≥ 0, 1 − e − x ≤ x , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − p (1 − x ) µ ′ − e − pµ − e − p (1 − x ) µ − e − pµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ − e − pµ e − p (1 − x ) µ ′ (cid:18) − e − (cid:16) p (1 − x ) µ − p (1 − x ) µ ′ (cid:17) (cid:19) ≤ − e − pµ (cid:18) p (1 − x ) µ − p (1 − x ) µ ′ (cid:19) = 11 − e − pµ ( µ − µ ′ ) p (1 − x ) µµ ′ ≤ − e − pµ ( µ − µ ′ ) pµµ ′ . Thus, k f µ − f µ ′ k ≤ (cid:12)(cid:12)(cid:12)(cid:12) − e − pµ ′ − − e − pµ (cid:12)(cid:12)(cid:12)(cid:12) + 11 − e − pµ ( µ − µ ′ ) pµµ ′ , showing that µ f µ is continuous.It remains to show that ( f µ ) n = id R + k for any n ≥ k ∈ Z . Let us fix n ≥ 1, and choose ε > f µ ) k ((0 , ε )) does not contain an38nteger point for any k ≤ n . To finish the proof of the proposition, we now showthat ( f µ ) n is strictly convex on the interval (0 , ε ).For x ∈ (0 , f µ exists at x , and ispositive, since( f µ ) ′ ( x ) = pµ · e − p (1 − x ) µ − e − pµ ( f µ ) ′′ ( x ) = (cid:18) pµ (cid:19) e − p (1 − x ) µ − e − pµ , and using the property f µ ( x + 1) = f µ ( x ) + 1,( f µ ) ′ ( x ) > f µ ) ′′ ( x ) > x ∈ R \ Z . (6.7)Since the composition of twice differentiable functions is twice differentiable,( f µ ) k is twice differentiable on (0 , ε ). It is enough to show that (( f µ ) k ) ′′ ( x ) > x ∈ (0 , ε ) and k ≤ n , which we prove by induction on k together withthe statement (( f µ ) k ) ′ ( x ) > k = 1 the statements follows from (6.7). Now suppose that the state-ments are true for k < n , we wish to prove it for k + 1. By the choiceof ε , ( f µ ) k ((0 , ε )) ⊆ ( n, n + 1) for some n ∈ Z , hence f µ is twice differen-tiable on ( f µ ) k ((0 , ε )) with a positive derivative and second derivative. Hence,(( f µ ) k +1 ) ′ = ( f µ ◦ ( f µ ) k ) ′ = (( f µ ) ′ ◦ ( f µ ) k ) · (( f µ ) k ) ′ > , ε ) by the in-duction hypothesis and (6.7). Similarly, (( f µ ) k +1 ) ′′ = ( f µ ◦ ( f µ ) k ) ′′ = (( f µ ) ′′ ◦ ( f µ ) k ) · ((( f µ ) k ) ′ ) + (( f µ ) ′ ◦ ( f µ ) k ) · (( f µ ) k ) ′′ > 0, again using the inductionhypothesis and (6.7). Thus the proof of the proposition is complete. Proof of Lemma 6.10. Let X n , . . . X nn be independent Geometric random vari-ables with mean µn , i.e., X ni ∼ Geometric ( µn ) for all i ≤ n so that σ µn ( v i )is the i th smallest among { X n , . . . , X nn } . Let F n : [0 , ∞ ] → [0 , 1] be the ap-propriately normalized empirical distribution function, which in our case is F n ( t ) = n P nk =1 I { X nk ≤ tn } , where we normalize by n to match the graphoncase. Let E : [0 , ∞ ] → [0 , E ( t ) = 1 − e − tµ which is the inverse of σ µ takenas a function from [0 , 1] to R + . Notice that the graph of σ µ is the mirror imageof the graph of E . Moreover, if we connect the points ( x, lim y → x − ˜ σ µn ( x )) and( x, lim y → x + ˜ σ µn ( x )) for all jumping points in the graph of ˜ σ µn , and similarly forthe graph of F n , then the two obtained broken lines are once again mirror im-ages of each other. Hence k ˜ σ µn − σ µ k = k F n − E k . Thus, it is enough to provethat k F n − E k → n → ∞ .Let I nk ( t ) = I { X nk ≤ tn } . Then F n ( t ) = n P nk =1 I nk ( t ). Let F : [0 , ∞ ] → [0 , 1] be defined as F ( t ) = 1 n n X k =1 E I nk ( t ) = 1 − (cid:18) − 11 + µn (cid:19) ⌊ tn ⌋ . k F n − E k ≤ k F n − F k + k F − E k . We first bound the term k F − E k . k F − E k = Z ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:18) − 11 + µn (cid:19) ⌊ tn ⌋ − (cid:16) − e − tµ (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = Z ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − tµ − (cid:18) − 11 + µn (cid:19) ⌊ tn ⌋ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ≤ Z t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − tµ − (cid:18) − 11 + µn (cid:19) ⌊ tn ⌋ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt + Z ∞ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − tµ − (cid:18) − 11 + µn (cid:19) ⌊ tn ⌋ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt. Since R ∞ e − tµ dt < ∞ , for any fixed ε , for large enough t , R ∞ t | e − tµ | dt < ε .Using that (cid:16) − µn (cid:17) µn ≤ e and that ⌊ tn ⌋ µn > t µ for n ≥ 1, it is clearthat for large enough t , Z ∞ t (cid:18) − 11 + µn (cid:19) ⌊ tn ⌋ dt = Z ∞ t (cid:18) − 11 + µn (cid:19) µn ! ⌊ tn ⌋ µn dt < Z ∞ t e t µ dt < ε for any n ≥ 1. Let us fix a t large enough so that both conditions are satisfied,then R ∞ t (cid:12)(cid:12)(cid:12)(cid:12) e − tµ − (cid:16) − µn (cid:17) ⌊ tn ⌋ (cid:12)(cid:12)(cid:12)(cid:12) dt ≤ ε .For this fixed t , Z t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − tµ − (cid:18) − 11 + µn (cid:19) ⌊ tn ⌋ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = Z t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − tµ − (cid:18) − 11 + µn (cid:19) µn ! ⌊ tn ⌋ µn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ≤ Z t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − tµ − (cid:18) − 11 + µn (cid:19) µn ! tµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt + Z t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18) − 11 + µn (cid:19) µn ! tµ − (cid:18) − 11 + µn (cid:19) µn ! ⌊ tn ⌋ µn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt. In the first term, | e − tµ − (cid:0) (1 − µn ) µn (cid:1) tµ | is a continuous function in t , andas n increases, it monotonically tends to 0 pointwise. Hence by the theorem ofDini, | e − tµ − (cid:0) (1 − µn ) µn (cid:1) tµ | uniformly tends to the constant zero function as n → ∞ . Thus, for a large enough n , the first term is smaller than ε .40or the second term, Z t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18) − 11 + µn (cid:19) µn ! tµ − (cid:18) − 11 + µn (cid:19) µn ! ⌊ tn ⌋ µn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = Z t (cid:18) − 11 + µn (cid:19) µn ! tµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:18) − 11 + µn (cid:19) µn ! µ + ⌊ tn ⌋ µ − t − tnµµ (1+ µn ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ≤ Z t (cid:18) − 11 + µn (cid:19) µn ! tµ (cid:18) − 11 + µn (cid:19) µn ! − t − µµ (1+ µn ) − dt, where the inequality comes from the fact that (1 − µn ) µn is always less than1, so if the exponent, µ + ⌊ tn ⌋ µ − t − tnµµ (1+ µn ) is positive then multiplying the exponentby − ε for large enough n .This means that for an arbitrary ε , if n is large enough, then k F − E k ≤ ε .To bound the term k F n − F k , we copy the standard proof of the Glivenko–Cantelli theorem. Note that | I nk ( t ) − E I nk ( t ) | ≤ 1. Hence we can apply Azuma’sinequality to get P ( | F n ( t ) − F ( t ) | > s ) = P ( | P nk =1 ( I nk ( t ) − E I nk ( t )) | ≥ ns ) ≤ e − ns regardless of the value of t .Now take t = 0 , t , . . . , t m − , t m = ∞ such that F ( t i ) = im . This can bedone since F is continuous, it is zero in 0 and tends to one in infinity. Now P (cid:18) max i =1 ,...,m − {| F n ( t i ) − F ( t i ) |} > s (cid:19) ≤ m · e − ns . Take again an arbitrary t ≥ 0. There exists some i such that t i ≤ t < t i +1 .As F n and F are both monotone increasing, F n ( t i ) ≤ F n ( t ) ≤ F n ( t i +1 ) and F ( t i ) ≤ F ( t ) ≤ F ( t i +1 ) = F ( t i ) + m . Hence F n ( t ) − F ( t ) ≤ F n ( t i +1 ) − F ( t i ) = F n ( t i +1 ) − F ( t i +1 )+ m and F ( t ) − F n ( t ) ≤ F ( t i +1 ) − F n ( t i ) = F ( t i ) − F n ( t i )+ m .Thus, for any m , and any t ,sup t ∈ R | F n ( t ) − F ( t ) | ≤ max i =0 ,...m | F n ( t i ) − F ( t i ) | + 1 m . By choosing s = ε n / and m = n / ε , we get P (cid:18) sup t ∈ R {| F n ( t ) − F ( t ) |} > εn / (cid:19) ≤ P (cid:18) max i =0 ,...,m {| F n ( t i ) − F ( t i ) |} > ε n / (cid:19) ≤ n / ε · e − n / ε . ε , P ∞ n =1 P (sup t ∈ R {| F n ( t ) − F ( t ) |} > εn / ) < ∞ ,hence by the Borel–Cantelli lemma, with probability one, there exists n ∈ N such that for n ≥ n , sup t ∈ R {| F n ( t ) − F ( t ) |} ≤ εn / . Repeating this argumentfor a series ε , ε , . . . tending to zero, we get that with probability one, for each ε > 0, there exists n ∈ N such that for n ≥ n , sup t ∈ R {| F n ( t ) − F ( t ) |} ≤ εn / .We have proved that F n and F are uniformly close to each other for large n with high probability. Now we show that the integral of their difference issmall for large values of t . Fix ε > e − ε > 2. Then P ( X nk ≥ n / √ ε ) = (1 − µn ) l n / √ ε m , hence P (cid:18) max k X nk ≥ n / √ ε (cid:19) ≤ n (cid:18) − 11 + µn (cid:19) n / √ ε = n (cid:18)(cid:18) − 11 + µn (cid:19) µn (cid:19) n / µ √ ε . For large enough n , n ≤ n / µ √ ε and also (1 − µn ) µn ≤ e − ε . Hence forlarge enough n , P (cid:18) max k X nk ≥ n / √ ε (cid:19) ≤ (cid:18) e − ε (cid:19) n / µ √ ε . Since e − ε < 1, this means that ∞ X n =1 P (cid:18) max k X nk ≥ n / √ ε (cid:19) < ∞ . Once again using the Borel–Cantelli lemma, with probability one, there exists n ∈ N such that for n ≥ n , max k X nk ≤ n / √ ε . For ε > 0, if ε < ε then n / √ ε < n / √ ε , hence the above bound holds for each such ε .Notice that sup { t : F n ( tn ) < } = max { X nk : k = 1 , . . . , n } . Hence withprobability one, for each ε > ε < ε there exists n ∈ N such that for n ≥ n , F n (cid:16) n / √ ε (cid:17) = 1.Hence with probability one, for each ε > ε < ε , there exists n ∈ N suchthat for n ≥ n , k F n − F k = Z n / √ ε | F n ( x ) − F ( x ) | dx + Z ∞ n / √ ε | − F ( x ) | dx ≤ εn / · n / √ ε + Z ∞ n / √ ε e − xµ dx = √ ε + ( − µe − xµ ) | x = ∞ x = n / √ ε = √ ε + µe − n / µ √ ε . This proves that k F n − F k → n → ∞ .Altogether, we obtain that k F n − E k → n → ∞ .42 roof of Theorem 6.9. Fix an arbitrary µ ∈ [0 , C p and σ µ .We claim that ( C p , σ µ ) is s smooth pair. This can be proved analogously tothe corresponding statement in the proof of Theorem 6.8. Also, C p has finitediameter, as noted in the proof of Theorem 6.8.By Theorem 6.7, Lemma 6.10 and Proposition Appendix A.1, with proba-bility one we can apply Theorem 5.2 to ( C p , σ µ ) to get that for any ε > 0, if n is large enough, then | a ( C p , σ µ ) − a ( G n , σ µn ) | ≤ ε .Applying the above argument to a dense countable subset of µ values and asequence of ε values tending to zero, we get that with probability one, a ( G n , σ µn )tends to a ( C p , σ µ ) for a dense set of µ values. Because of the way we coupled therandom chip configuration σ µn , if we increase the value of µ , then the number ofchips monotonically increases on each vertex in each outcome. Hence on eachoutcome, a ( G n , σ µn ) monotonically increases if we increase µ , using Lemma 2.3. σ µ also increases pointwise in µ , hence a ( C p , σ µ ) also increases monotonically.As µ a ( C p , σ µ ) is continuous, if a ( G n , σ µn ) tends to a ( C p , σ µ ) for a dense setof µ values, then a ( G n , σ µn ) tends to a ( C p , σ µ ) for each µ ∈ [0 , a ( G n , σ µn ) tends to a ( C p , σ µ ) pointwise. As by Lemma6.11, the map µ a ( C p , σ µ ) is a Devil’s staircase, we obtained the statementof the Theorem. Appendix A. Basic properties of random graphs Here we collect some well-known basic properties of random graphs. Through-out the section, G ( n, p ) again means the random graph with n vertices, whereeach edge is present independently with probability p . Proposition Appendix A.1. Let d < p be a fixed constant. If ( G n ) n ∈ N is asequence of random graphs where G n = G ( n, p ) , then with probability one, thereexists an index n such that for each n ≥ n , mindeg ( G n ) ≥ dn . We will use the following form of Azuma’s inequality. Theorem Appendix A.2 (Azuma’s inequality) . Suppose that X , . . . , X n areindependent random variables, E [ X i ] = 0 for each i ∈ N , and for each i thereexist c i > such that, | X i | ≤ c i almost surely. Then P " n X i =1 X i > t ≤ e − t P ni =1 c i . Claim Appendix A.3. For a vertex v ∈ V ( G n ) , P [ | deg G n ( v ) − np | > ηn ] ≤ e − nη .Proof. We use Azuma’s inequality with X u = { uv is an edge } − p . Then { X u } u ∈ V \{ v } is a set of independent random variables, E [ X u ] = 0 and | X u | ≤ max { p, − p } ≤ u ∈ V \ { v } . Azuma’s inequality applied for { X u } u ∈ V \{ v } and for {− X u } u ∈ V \{ v } gives us the above bound.43 roof of Proposition Appendix A.1. 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