Existence of graphs with sub exponential transitions probability decay and applications
aa r X i v : . [ m a t h . P R ] A p r EXISTENCE OF GRAPHS WITH SUB EXPONENTIALTRANSITIONS PROBABILITY DECAY AND APPLICATIONS .
CL´EMENT RAU
Abstract.
In this paper, we present a complete proof of the construction ofgraphs with bounded valency such that the simple random walk has a returnprobability at time n at the origin of order exp ( − n α ) , for fixed α ∈ [0 ,
1[ andwith Folner function exp ( n α − α ). We begin by giving a more detailled proofof this result contained in (see [4]).In the second part, we give an application of the existence of such graphs. Weobtain bounds of the correct order for some functional of the local time of asimple random walk on an infinite cluster on the percolation model. Contents
1. Introduction and results 11.1. Example of application of proposition 1.2. 22. Proof of proposition 1.2 32.1. Wreath products and explanation of our method 32.2. case ≤ α < ≤ α ≤ Z d Introduction and results
A graph G is a couple ( V ( G ) , E ( G )), where V ( G ) stands for the set of verticesof G et E ( G ) stands for the set of edges of G . All graphs G which are consideredhere are infinite and have bounded geometry and we denote by ν ( g ) the number ofneighbors of g in G .We study the following random walk X on G defined by: (cid:26) X = g, P ( X n +1 = b | X n = a ) = ν ( a )+1 (1 { ( a,b ) ∈ E ( G ) } + 1 { a = b } )(1)The random walk X jumps uniformly on the set of points formed by the pointwhere the walker is and his neighbors. Thus X admits reversible measures whichare proportionnal to m ( x ) = ν ( x ) + 1. In this context, the transition probabilties are linked by the isoperimetric profile.For a graph G and for a subset A of G , we introduce the boundary of A relativelyto graph G defined by ∂ G A = { ( x, y ) ∈ E ( G ); x ∈ A et y ∈ V ( G ) − A } . Actually, we will rather work with Følner function to deal with isoperimetry. Let G be a graph, we note F ol G the Følner function of G defined by: F ol G ( k ) = min {| U | ; U ⊂ V ( G ) et | ∂ G U || U | ≤ k } . If G ′ ⊂ G is a subgraph of G , we will use the Folner function of G ′ relatively to G defined by: F ol GG ′ ( k ) = min {| U | ; U ⊂ V ( G ) et | ∂ G U || U | ≤ k } . We have the following proposition (see coulhon [1])
Proposition 1.1.
Let m = inf V ( U ) m > and X be the random walk definedby (1). Assume that F ol ( n ) ≥ F ( n ) with F a non negative and non decreasingfunction, then sup x,y P ( X n = y | X = x ) (cid:22) v ( n ) , where v satisfies : ( v ′ ( t ) = − v ( t )8( F − (4 /v ( t ))) ,v (0) = 1 /m . (We recall that a n (cid:22) b n if there exists constants c and c such that for all n ≥ , a n ≤ c b c n and a n ≈ b n if a n (cid:22) b n and a n (cid:23) b n .)For example, we retrieve that in Z d , the random walk X defined above has transi-tions decay at time n less than n − d/ and in F the Cayley graph of the free groupwith two elements, the transition decay of the random walk are less than e − n . Anatural question is to know if there exists graphs with intermediate transitions de-cay. Some others motivations can be found in section 3.From Z , one can perhaps adjust some weigths on edges to get the expecteddecay but we look after a graph with no weigths.Indeed, there are combinatoricsarguments in section 3 that will not work if any weigths are present.Our main result is : Proposition 1.2.
Let α ∈ [0; 1[ , F := e x α − α and σ ( n ) := e − n α . There exists agraph D F = ( V ( D F ) , E ( D F )) with bounded valency such that :(i) F ol D F ≈ F ,(ii) there exists a point d ∈ V ( D F ) such that, for all n, p D F n ( d , d ) ≈ σ ( n ) ,where p D F n ( , ) stands for the transitions probability of the random walk X definedabove when G = D F . Example of application of proposition 1.2.
With the help of these graphsand with some good wreath products, we will be able to find upper bound offunctional of type: E ( e − λ P F ( L x,n ,x ) ) where L x,n = { k ∈ [0; n ]; X k = x } on the graph C g getafter a surcritical percolation on edges of Z d , where edges are kept or removed withrespect Bernouilli independant variables. The points of C g are the point of the infinite connected component C which contains the origin, we will give more detailsin section 3. In particular, we will prove the following property: Theorem 1.3.
Consider a simple random walk X on the infinite cluster of Z d thatcontains the origin Q a.s on the set |C| = + ∞ , and for large enough n we have: ∀ α ∈ [0 , E ω ( e − λ P z ; Lz ; n> L αz ; n { X n =0 } ) ≈ e − n η , (2) ∀ α > / E ω ( Y z ; L z ; n > L − αz ; n { X n =0 } ) ≈ e − n dd +2 ln ( n ) d +2 , (3) where η = d + α (2 − d )2+ d (1 − α ) . The constants present in the relation ≈ do not depend on the cluster ω. Remark 1.4.
If we take α = 0 in equation (2), we retrieve the Laplace transformof the number of visited points N n (see [6] ), E ω ( e − λN n ) ≈ e − n d/d +2 . In the whole article,
C, c are constants which value can evolve from lines to lines.2.
Proof of proposition 1.2
In this section, we first recall the definition of the wreath product of two graphsand we explain our strategy aimed at the construction of our expected graphs. Thisleads naturally towards two cases corresponding to the two last subsections.2.1.
Wreath products and explanation of our method.
Let A a graph and( B z ) z ∈ A a family of graphs. Definition 2.1.
The wreath product of A and ( B z ) z ∈ A is the graph noted by A ≀ B z such that: V ( A ≀ B z ) = { ( a, f ); a ∈ A and f : A → ∪ z B z with supp ( f ) < ∞ and ∀ z ∈ A, f ( z ) ∈ B z } and E ( A ≀ B ) = { (cid:16) ( a, f )( b, g ) (cid:17) ; ( f = g and ( a, b ) ∈ E ( A )) or ( a = b and ∀ x = a f ( x ) = g ( x ) and (cid:0) f ( a ) , g ( a ) (cid:1) ∈ E ( B a )) } This graph can be interpreted as follow: imagine there is a lamp in each point a of A such that each point of B a defined a different intensity of the lamp. Thedifferent intensity of each lamp can be represented by a configuration f : A → ∪ a B a which encodes the intensity of the lamp at point a by the value f ( a ). A point inthe wreath product is the couple formed by the position of a walker in graph A andthe state of each lamp. A particular case is when the graph B a (called the fiber) isthe same for all a ∈ A .Let us now explain the way we construct graph D F of proposition 1.2. Considerthe wreath product of the Cayley graph of ( Z , +) by the Cayley graph of Z Z with ¯1as generator. By the Theorem 1 in [5] (or Proposition 3.2.1 in [6]) we immediatlydeduce that the Folner fonction of this wreath product is like e n . So this graphanswers to proposition 1.2 in the case α − α = 1. ie : α = 1 / α = 1 /
3, it would be rather natural to think that we can get the expectedgraph, by considering the wreath product of Z by fibers with variable sizes. CL´EMENT RAU • If α ≥ /
3, the return probability in the graph D F should be in e − n α so lessthan in the graph Z ≀ Z Z ( in e − n / ) . Thus to force the walk to come back rarelyat the origin, an idea is to make the size of the fibers grow when we move away theorigin in order to force the walk to loose time in the fiber.Note that for α ≥ • If α ≤ /
3, the return probability in the graph D F should be larger than in e − n / . The idea is to add some links (some edges by example) to force the walkto come back often to the origin. Suppose all lamps are identified then we get adecay in n − / and if all lamps are independent we get a decay in e − n / , so itremains to find an identification of lamps which implies an intermediate decay. Weare going to construct a wreath product where the walker (at a certain point) isallowed to change the value of the configuration at differents points. Such graphsare sometimes called generalized wreath products.To prove isoperimetric inequality on wreath product ( point (i) of the proposition1.2) we use idea of Erschler and the concept of ”satisfactory” points. We begin tointroduce this notion in section 2.2. At the beginning of section 2.3, we explain whyan improvement is needed in the definition of ”satisfactory” points. The improve-ment takes place through the introduction of a new and more theoretical way ofdefining the notion of ”satisfactory” points than in section 2.2. For simplicity, weuse the same words for this concept in the two sections but notions which appearin sections 2.3 and 2.2 are independent.2.2. case ≤ α < . Construction of the graph and preliminary notions and lemmas.
Let A ′ =( Z , E ( Z )) where E ( Z ) = { ( x, y ); | x − y | = 1 } and ( B ′ z ) z ∈ Z be the Cayley graph ofthe groups ( Z l ( z ) Z , +) with { ¯1 } as generators where l ( z ) = | V ( B ′ z ) | = F ( | z | +1) F ( | z | ) , ( F isdefined at proposition 1.2).Notice that since α ∈ [1 / , , the fonction z l ( z ) is increasing on R + .Finally put D F = A ′ ≀ B ′ z . Let us prove that this graph answers to propostion 1.2.We begin by proving (i).The proof is similar to the Theorem 1 in [5] or proposition 3.2.1 in [6].Let ψ ( n ) = F ol A ′ ( n ) = min U ⊂ Z | ∂A ′ U || U | ≤ /n | U | = 2 n .Take U ⊂ V ( D F ) = V ( A ′ ≀ B ′ z ) such that | ∂ DF U || U | ≤ /n for some n . We want tofind a lower bound on | U | . • For each set U , we attach an hypergraph K U = (cid:16) V ( K U ) , ξ ( K U ) (cid:17) such that:- the vertices of K U are the configurations f which belong to the set { f ; ∃ a ∈ Z ( a, f ) ∈ U } ,- let us now define the edges of K U : for all f ∈ V ( K U ) and a ∈ Z , we link f to allconfigurations g satisfying: ( a, g ) ∈ Uand ∀ x = a f ( x ) = g ( x ) , by a multidimensional edge l of dimension d where d = dim a f := { g ; ( a, g ) ∈ U and ∀ x = a f ( x ) = g ( x ) } . We say that the edge l is associated to point a . • To each hypergraph K U we associate a graph called the ” one dimensional skele-ton”, noted by Γ( K U ) = Γ U = ( V (Γ U ) , E (Γ U )) and defined by:- V (Γ U ) = V ( K U ) , - two configurations f and f are linked by an edge if they belong to a same mul-tidimensional edge in K U .Let w be the weight defined by w ( e ) = 1 /d for e belonging to E (Γ U ) and comingfrom a multidimensional edge in K U of dimension d . Notice that this choice ofweights gives : | U | ≥ X e ∈ E (Γ U ) w ( e ) , (4)and if we assume moreover that for all ( x, f ) ∈ U, dim x f ≥ U has no separetedpoints) then the equality holds in 4 Let p be the projection Z ≀ B ′ z → Z . Let us nowintroduce some notations. Denote λ = ( λ a ) a ∈ p ( U ) ∈ R p ( U ) and b ≥ • For f ∈ V ( K U ), we say that f is ( λ, b ) − satisf actory if : { a ∈ p ( V ); dim a f ≥ λ a } ≥ b. ie : f is ( λ, b ) − satisf actory if there exists at least b multidimensional edgesattached to f in K U of dimension at least λ a at point a . We denote by S U ( λ, b ) theset of these points. Most of the time, in order to simplify notations we will dropthe subscript U when there is no ambiguity. • Otherwise we tell that f is ( λ, b ) − nonsatisf actory and we denote by N S ( λ, b )the set of nonsatisfactory points. • An edge of Γ U is ( λ, b ) − satisf actory if it links two ( λ, b ) − satisf actory con-figurations otherwise it is said ( λ, b ) − nonsatisf actory . We denote S e ( λ, b ) [resp N S e ( λ, b )] the set of ( λ, b ) − satisf actory [resp ( λ, b ) − nonsatisf actory ] edges. • A point u = ( x, f ) ∈ U is ( λ, b ) − satisf actory [resp ( λ, b ) − nonsatisf actory ] if f ∈ S ( λ, b ) [resp N S ( λ, b )]. We denote by S p ( λ, b ) and N S p ( λ, b ) the set of pointswhich are (or are not ) ( λ, b ) − satisf actory . • A point u = ( a, f ) ∈ U is said b − good if dim a f ≥ b otherwise it is b − bad. Let us now explain the main steps of the proof. We take U ⊂ V ( D F ) such that | ∂ DF U || U | ≤ n . We begin to prove that there exists some value of b and some se-quence λ such that there are few points ( λ, b ) − nonsatisf actory . Then, we extracta subgraph of Γ U where all points are ( λ , b ) − satisf actory and this allows us to CL´EMENT RAU obtain a lower bound of | U | . We begin by the following lemma. Lemma 2.2.
Let U ∈ V ( A ′ ≀ B ′ z ) such that | ∂ DF U || U | ≤ n then ( i ) { u =( x,f ) ∈ U ; u is λ x ( n ) − bad } U ≤ n ( ii ) { u =( x,f ) ∈ U ; u ∈ NS p ( λ ( n ) / ,ψ ( n ) / } U ≤ , where λ = ( λ x ) x with λ x ( n ) = F ol B ′ x ( n ) and ψ ( n ) = F ol A ′ ( n ) .Proof. For (i) we notice that we can associate to certain bad points, some point of theboundary of U . Indeed, for ( x, f ) a point, we call:˜ P x,f = { g ( x ); ( x, g ) ∈ U and ∀ y = x g ( y ) = f ( y ) } and P x,f = { ( x, g ); g ( x ) ∈ ˜ P x,f } . Note that | ˜ P x,f | = | P x,f | .F stands for a set of configurations such that:˙ [ x ∈ A ′ ,f ∈ F P x,f = { u = ( x, g ) ∈ U ; u is F ol B ′ x ( n ) − bad } . Take note that, for a point u = ( x, f ) which is F ol B ′ x ( n ) − bad, by the definition ofa Folner function, we have: | ˜ P x,f | < F ol B ′ x ( n ) . So, | ∂ B x ˜ P x,f | ≥ n | ˜ P x,f | Now the application ˙ S x ∈ A ′ ,f ∈ F ∂ B x ˜ P x,f −→ ∂ D F U is injective,( g , g ) (cid:16) ( x, f x,g ) , ( x, f x,g ) (cid:17) where ( g , g ) ∈ ∂ B x ˜ P x,f and f a,h : v → f ( v ) for v = a.a → h Hence, we have : | U | n ≥ | ∂ D F U | ≥ X x ∈ A,f ∈ F | ∂ B ˜ P x,f |≥ n X x ∈ A,f ∈ F | ˜ P x,f | = 1 n { u = ( a, f ) ∈ U ; u is F ol B ′ a ( n ) − bad } . For (ii), the proof splits into three parts.
A. Let,
N eud = { u ∈ U ; u ∈ N S p ( λ , F ol A ′ ( n )3 ) } = { u = ( x, f ) ∈ U ; f ∈ N S ( λ , F ol A ′ ( n )3 ) } , and let: N eud ( f ) = { ( x, f ); ( x, f ) ∈ U } . Notice that p ( N eud ( f )) = { x ; ( x, f ) ∈ U } . For F a set of configurations, we call N eud ( F ) = ∪ f ∈ F N eud ( f ) . Note well that it is a disjointed union.B. Now take f ∈ N S ( λ , F ol A ′ ( n )3 ), and look at the set p ( N eud ( f )). There are onlytwo possibilties:-either, it gives a large part of boundary in ’base’,-either, it gives a few part of boundary in ’base’. If this is the case, taking intoaccount that f is not satisfactory, we retrieve boundary in ’configuration’.Anyway, we get some boundary of U , but our assumptions restrict this contri-bution.So we differentiate two cases:First case : f ∈ F := { f ∈ N S ( λ , F ol A ′ ( k )3 ); ∂ A ′ p ( Neud ( f )) p ( Neud ( f )) > n } . The application ˙ S f ∈ F ∂ A ′ p ( N eud ( f )) −→ ∂ D F U is injective.( x, y ) (cid:16) ( x, f ) ; ( y, f ) (cid:17) So, we get:(5) | ∂ D F U | ≥ X f ∈ F | ∂ A ′ p ( N eud ( f )) | ≥ n X f ∈ F | p ( N eud ( f )) | ≥ n | N eud ( F ) | . Second case : f ∈ F := { f ∈ N S ( λ , F ol A ′ ( n )3 ); ∂ A ′ p ( Neud ( f )) p ( Neud ( f )) ≤ n } . Since f ∈ N S ( λ , F ol A ′ ( n )3 ) it follows that : { x ∈ p ( N eud ( f )); dim x f ≥ λ x } < F ol A ′ ( k ) . Hence, { x ∈ p ( N eud ( f )); dim x f < λ x } ≥ | N eud ( f ) | − F ol A ′ ( n )(We use that | p ( N eud ( f ) | = | N eud ( f ) | . )Since f ∈ F and by definition of a Folner fonction: | N eud ( f ) | ≥ F ol A ′ ( n ) . CL´EMENT RAU
As a result, we have: { x ∈ p ( N eud ( f ); dim x f < λ x } ≥ | N eud ( f ) | .ie : | P f | ≥ | N eud ( f ) | , (6) with P f = { x ∈ p ( N eud ( f ); dim x f < λ x } . Let ˜ P x,f = { g ( x ); ( x, g ) ∈ U and ∀ y = x g ( y ) = f ( y ) } . To each point of ∂ B ′ x ˜ P x,f we can associate, by the same way as before, a point of ∂ D F U . So, wehave: | ∂ D F U | ≥ X x ∈ P f ,f ∈ F | ∂ B ′ x ˜ P x,f | . Now for x in P f , dim x f = | ˜ P x,f | < λ x = F ol B ′ x ( n ) < F ol B ′ x ( n ) . So | ∂ B ′ x ˜ P x,f | > n | ˜ P x,f | , ie: | ∂ B ′ x ˜ P x,f | ≥ . Then, X x ∈ P f ,f ∈ F | ∂ B ˜ P x,f | ≥ X f ∈ F | N eud ( f ) | by (19) , ≥ | N eud ( F ) | We have thus | ∂ D F U | ≥ n | N eud ( F ) | for n ≥ | ∂ DF U || U | < n , weobtain : | N eud || U | < . (cid:3) Lemma 2.3.
Let (Γ U , w ) be the one dimensional skeleton with weights w , con-structed from K U . Let η = ( η a ) a ∈ p ( U ) . Assume that E (Γ U ) = ∅ and ∀ ( a, f ) ∈ U dim a f ≥ η a > . If the following conditionis satisfied : P e ∈ NS eU ( η,b ) w ( e ) P e ∈ E (Γ U ) w ( e ) < / , then there exists a not empty subgraph Γ ′ = (cid:16) V (Γ ′ ) , E (Γ ′ ) (cid:17) of Γ U such that alledges are S eU ( η/ , b/ . Proof.
In the gaph (cid:16) V (Γ U ) , E (Γ U ) (cid:17) , we remove all points N S pU ( η/ , b/
10) and theadjacent edges. After this step, it may appear new points which are
N S PU ( η/ , b/ U = U − N S pU ( η/ , b/ U i be the set of points still present at step i . (cid:26) U = U, for i ≥ U i +1 = U i − N S pU i ( η/ , b/ . It is sufficient to prove that this process stops before the graph becomes empty.Let C = P e ∈ NS U ( η,b ) w ( e ) , C = P e ∈ S eU ( η,b ); e removed at the end of the process w ( e ) , et C = X e ∈ E (Γ U )); e removed at the end of the process w ( e ) . If we show that C ≤ C , the propostion is proved, since : C ≤ C + C ≤ C < X e ∈ E (Γ U ) w ( e ) . Indeed, this means that it remains point(s) not removed. ie: ∃ k ∈ N such that allvertices of the graph we get at step k , are S pU k ( η/ , b/ S pU ( η/ , b/ . In order to see this, let us introduce an orientation on edges removed: if L and Q are points of the graph, we orient the edge from L to Q if L is removed before Q , and we choose an arbitrary orientation if they are removed together. We denoteby L ↓ the set of edges leaving the point L and L ↑ the set of edges ending at point L ,both at step 0. Sublemma 2.4.
Let k ∈ N and let L stands for a point of the graph Γ U (satisfyingassumptions of lemma 2.3), removed after k + 1 steps. Suppose that L is initially S pU ( η, b ) , then X e ∈ L ↓ w ( e ) ≤ X e ∈ L ↑ w ( e ) . . L L STATE JUST BEFORE L WAS REMOVED INITIAL STATE There are at least b multidimensional edges of dimension at least L is S( ,b) ηη There are less than b/10 multidimensional edges associated to point x. x η x of dimension at least /10 associated to point x.. These pictures are represented at step 0 on the left side and at step k on the rightside. Proof.
It would be useful to notice that for a multidimensional edge e , the sum ofthe weights (in the skeleton) of edges coming from e and adjacent to a point, isalways equal to 1. This is implied by our choice of the weight.The proof is divideds into five parts.A. Let N the number of multidimensional edges at step 0. Since L is S pU ( η, b ) , there are at least b multidimensional edges attached to L . So, N ≥ b. (7) Note that: X e ∈ E (Γ U ) e contains L w ( e ) = N . B. Let : L ↓ = { e ∈ L ↓ , e coming from a multidimensional edge of K U k , associated to a point x, of dim ≥ η x / } , and L ↓ = { e ∈ L ↓ , e coming from a multidimensional edge of K U k , associated to a point x, of dim < η x / } . We have: L ↓ = L ↓ ∪ L ↓ , because edges of L ↓ , are edges leaving L at step k .C. Since L becomes N S pU k ( η/ , b/ b/
10 multidimensionaledges associated to each point x , of dimension at least η x /
10. Call them f , ..., f q , with q < b/ . X e ∈ L ↓ w ( e ) = X k =1 ..q X e coming from fk w ( e ) | {z } ≤ ≤ q. (8) (Initially this last sum was equal to 1, but after removing some edges, this sumvalue becomes less than 1.)D. Let g , ..., g h be the other multidimensional edges attached to L at step k as-sociated to a point x , and with dimension strictly less than η x /
10. We have h ≤ N − q. Consider an edge e coming from a multidimensional edge associated to a point x . For all k = 1 ...h we have: X e coming from gk w ( e ) ≤ η x η x ≤ . (9) Indeed, firstly since all configurations (relatively to this edge e ) have initiallydimension at least η x we deduce that w ( e ) ≤ /η x . And secondly a multidi-mensional edge of dimension less than η x /
10 gives less than η x /
10 edges in theskeleton.E. Finaly by (8) and (9), we get: X e ∈ L ↓ w ( e ) = X e ∈ L ↓ w ( e ) + X e ∈ L ↓ w ( e ) ≤ q + ( N − q ) 110= 110 N + 910 q = 19100 N . ( q < b/ ≤ N /
10 by (7).)So, X e ∈ A ↓ w ( e ) ≤ N and X e ∈ A ↑ w ( e ) ≥ N − N = 81100 N . So, X e ∈ A ↓ w ( e ) ≤ X e ∈ A ↑ w ( e ) ≤ X e ∈ A ↑ w ( e ) . (cid:3) To finish the proof, let us consider: D = { vertices removed at step 1 } , and for i ≥ D i = { vertices S pU ( η, b ) removed at step i } ,F i = { edges between D i and D i − } ,F ′ i = { edges leaving D i − } .Note that F i ⊂ F ′ i and that the edges of F ′ i are removed. D1D2 } F2F3 } D3 The proof ends up in four parts:A. Apply sublemma 2.13 to each point of D i , in the graph staying at step i − D i is S ( η, b ).) We get : ∀ i ≥ X e ∈ F ′ i +1 w ( e ) ≤ X e ∈ F i w ( e ) . So, X e ∈ F ′ i +1 w ( e ) ≤ ( 12 ) i − X e ∈ F w ( e ) . (We use that F i ⊂ F ′ i . )Hence, X e ∈ ∪ i ≥ F ′ i w ( e ) ≤ ( X i ≥ ( 12 ) i ) X e ∈ F w ( e )= X e ∈ F w ( e ) . B. Now, an edge of F is N S eU ( η, b ) since if it was S eU ( η, b ), it would link twopoints S pU ( η, b ) and in particular points of D would have been S pU ( η, b ) , then S pU ( η/ , b/
10) and so would not have been removed. In consequence : X e ∈ F w ( e ) ≤ X e ∈ NS e ( η,b ) w ( e ) = C . C. Besides, all removed edges S eU ( η, b ) are in some F ′ i with i ≥
3, so C = X e removed at the end of the process e ∈ SeU ( η,b ) w ( e ) ≤ X e ∈ ∪ i ≥ F ′ i w ( e ) . D. Hence, C ≤ C , which achieves the proof. (cid:3) Now, we use the following lemma to get a lower bound of the volume of U . Lemma 2.5.
Let N : R + −→ R + , a non decreasing function.Let us take b ∈ N ∗ and A a not empty set of configurations such that : ∀ f ∈ A ∃ x , x , ..., x b ∈ Z such that ∀ i ∈ [ | b | ] g i ∈ A where g i is one of the following functions, defined from f by : g i ( x ) = ( f ( x ) if x = x i , there are N ( | x i | ) possibilities for g i ( x i ) if x = x i , then |A| ≥ N (0) (cid:16) N (1) N (2) ...N ( b − ) (cid:17) if b is odd, N (0) (cid:16) N (1) N (2) ...N ( b − ) (cid:17) N ( b ) if b is even.Proof. We will proceed by induction on b .If b = 1 it is true, since N is non decreasing on R + .Assume b ≥ x in the base such that: •| x | ≥ b − if b is odd and | x | ≥ b if b is even. • And there exists f , ..., f N ( | x | ) ∈ A satisfying ∀ i ∈ [ | N ( | x | ) | ] f i ( x ) rangeamong the N ( | x | ) possible images.For i ∈ [ | N ( | x | ) | ], we denote by A i the set { f ∈ A ; f ( x ) = f i ( x ) } , which is notempty.We have A = ˙ S ≤ i ≤ N ( | x | ) A i . Besides, the A i satisfies the induction assumption with constant b − b is odd, N ( | x | ) ≥ N ( b − ) and we have: |A| = X ≤ i ≤ N ( | x | ) |A i |≥ X ≤ i ≤ N ( | x | ) N (0) (cid:16) N (1) ...N ( b −
32 ) (cid:17) N ( b −
12 ) ≥ N (0) (cid:16) N (1) ...N ( b −
32 ) (cid:17) N ( b −
12 ) N ( x ) ≥ N (0) (cid:16) N (1) ...N ( b −
12 ) (cid:17) . The proof unfolds the same way when b is an even number. (cid:3) Proof of (i) of the proposition 1.2 : • Lower bound of Folner function.
For the lower bound of
F ol D F , take U ⊂ V ( A ′ ≀ B ′ z ) such that | ∂ DF U || U | ≤ n Let˜ K = (cid:16) V ( ˜ K ) , ξ ( ˜ K ) (cid:17) the subhypergraph of K U constructed with points ( x, f ) whichare F ol B ′ x ( n ) / − good . ˜ K is not empty, since by the part (i) of the lemma 2.2 | V ( ˜ K ) | ≥ (1 − n ) | U | . Then we have: X e ∈ E (Γ( ˜ K )) ∩ NS e ( λ ( n )3 , ψ ( n )3 ) w ( e ) ≤
12 { u ∈ U ; N S p (cid:16) λ ( n )3 , ψ ( n )3 (cid:17) } by remark (4) ≤ | U | by lemma 2.2( ii ) ≤ − n { u = ( x, f ) ∈ U, λ x ( n )3 − good } by lemma 2.2( i )= 21000 − k X e ∈ E (Γ( ˜ K )) w ( e ) ≤ θ X e ∈ E (Γ( ˜ K )) w ( e ) . with θ = < , so lemma 2.3 can be applied to ˜ K , to deduce there exists asubgraph K ′ = ( V ( K ′ ) , E ( K ′ )) of ˜ K such that all edges are S e ( λ ( n ) / , ψ ( n ) / N ( | x | ) = F ol B x ( n ) /
30 to the set of configurationsrelatively to K ′ , we deduce for large enough n : | U | ≥ l (0) (cid:16) l (1) ...l ( ψ ( n )40 ) (cid:17) = F (1) F (0) (cid:16) F (2) F (1) ... F ( n/
40 + 1) F ( n/ (cid:17) . (We use that for k ≥ , F ol B ′ x ( k ) = | B ′ x | = l ( | x | ) = F ( | x | +1) F ( | x | ) .)So, | U | ≥ cF ( n/ (cid:23) F ( k ) . (Since F ( x ) = e cx α − α we have F ≈ F . )ie : F ol D F ( k ) (cid:23) F ( k ) . • Upper bound of Folner function.
For the upper bound of the Folner fonction of D F , we take: U = { ( a, f ); 0 ≤ a ≤ n ; supp ( f ) ⊂ [ | n | ] } . On a | U | = nF ( n ) et | ∂ D F U | / | U | ≤ c/n, so, F ol D F ( n ) ≤ nF ( n ) (cid:22) F ( n ) . • So the graph D F has the expected Folner function on the case α > / . Proof of (ii) of the proposition 1.2 :
We proceed in 5 steps.A. Let d = (0 , f ) where f is the null configuration.Let H n = ( K n , g n ) the random walk on D F starting from d which jumps uni-formly on the set of points formed by the point where the walker is and itsneighbors.This random walk admits a reversible measure µ defined by µ ( x ) = ν D F ( x ) + 1.Note that for all x ∈ V ( D F ) , µ ( x ) ≤ . B. Using reversiblity, we can write, p D F n ( d , d ) = X z p D F n ( d , z ) p D F n ( z, d ) ≥ X z ∈ A p D F n ( d , z ) µ ( d ) µ ( z ) ≥ µ ( d ) µ ( A ) [ X z ∈ A p D F n ( d , z )] ≥ µ ( d ) µ ( A ) [ P D F d ( H n ∈ A )] , where A is some subset of V ( D F ) . Choose A = A r = { ( a, f ) ; | a | ≤ r and supp ( f ) ⊂ [ − r, r ] } . C. The structure of edges on D F implies: P D F d ( H n ∈ A r ) ≥ P D F d ( ∀ i ∈ [ | , n | ] | K i | ≤ r ) ≥ P K ( ∀ i ∈ [ | , n | ] | K i | ≤ r ) , where P K is the law of ( K i ) which is again a random walk with probabilitytransitions that can be represented for n large enough by : n−1 n n+1 Indeed, as soon as l ( | n | ) >
3, the point ( n, f ) has 2 neighbors in ”configuration”,2 neighbors in ”base” and itself as neighbor. For this walk we can prove (as inproposition 5.2 in [6]) that : ∃ c > , ∀ n ≥ P K ( ∀ i ∈ [ | , n | ] | K i | ≤ r ) ≥ e − c ( n/r + r ) . In fact, a better bound holds P K ( ∀ i ∈ [ | , n | ] | K i | ≤ r ) ≥ e − cn/r (see lemma7.4.3 of [7]) but it is not necessary here.Thus, P D F d ( H n ∈ A r ) ≥ e − c ( n/r + r ) . (10) D. Compute now µ ( A r ), we have: µ ( Ar ) ≤ | Ar | max A r µ ≤ (2 r + 1) F (1) F (0) ( Y k =1 ..r F ( k + 1 F ( k ) ) × ≤ CrF ( r + 1) (cid:22) F ( r ) . ( This last inequality comes from the form of F ( r ) in e cr α − α .)E. Gathering the results, by inequality (10) and the fact that α − α ≥
1, we deducethat it exists c > p D F n ( d , d ) ≥ e − c ( nr + r α − α ) . The function r nr + r α − α is minimal for r like n − α .So , it exists c > p D F n ( d , d ) ≥ e − cn α . Remark 2.6.
Note that by proposition 1.1 and with our estimate of
F ol D F , wehave for all x, y in D F , p D F n ( x, y ) (cid:22) e − n α . So p D F n ( d , d ) ≈ e − n α case ≤ α ≤ . Construction of the graph and preliminary lemmas.
Consider the general fol-lowing context: let A and B two graphs and φ an application A → A ′ . Now welook at the graph such that:- the points are elements of ( A × B A ′ ),- edges are couple (( a, f ); ( b, g )) such that :(i) either ∀ x ∈ A ′ , f ( x ) = g ( x ) and a is neighbor of b in A .(ii) either a = b and ∀ x = φ ( a ) f ( x ) = g ( x ) and f ( φ ( a )) is neighbor of g ( φ ( a )) in B .Such graphs are called generalized wreath products.If A ′ = A and φ = id we retrieve our ordinary wreath products.Case which interest us is when A = A ′ = ( Z , E ( Z )) and B is the Cayley graphof Z Z with ¯1 as generator.To define φ : Z → Z , it is sufficient to give the following sets A i = { x ; φ ( x ) = i } ,which should form a partition of φ ( Z ) (which is here Z ). Let A = { A i } , we note A ≀ A B the generalized wreath product considered.Let β = α − α < . If we want a Folner function like e n β , we should construct φ (or the partition A )with some redundancies. Suppose for example that Folner sets are : U n = { ( a, f ); a ∈ [ − n ; n ] et supp ( f ) ∈ [ − n ; n ] } , (11)we should have φ ([ | − n ; n | ]) = { i ; A i ∩ [ − n ; n ] = ∅} ≈ n β . For Ω ⊂ A , it would be useful to introduce: N A (Ω) = { i ; A i ∩ Ω = ∅} , and S j (Ω) = A j ∩ Ω) . In particular, let: N A ( k, k + m ) = N A ([ k, k + m ]) et S j ( k, k + m ) = S j ([ k, k + m [) . The following lemma gives us the construction of the partition which answers toour problem.
Lemma 2.7.
Let g : N → N increasing with g (1) = 1 such that for all n in N , g (2 n ) ≤ g ( n ) . Then there exists a partition A g = { A i } of Z satisfying:(i) for all m ≥ and for all k in Z , N A g ( k, k + m ) ≈ g ( m ) , (ii) there exists K > such that for all m ≥ , for all k in Z and for all i, j in S j ( k, k + m ) = 0 : S i ( k, k + m ) S j ( k, k + m ) ≤ K. Proof.
A. We first define partition on intervals [1 , s ] ( s ≥
0) by induction on s , such that: ( P s ) ( N A g (1 , s ) = g (2 s ) , S i (1 , s ) S j (1 , s ) ≤ S j (1 , s ) = 0 . • For s = 0, we put the point 1 in some A i , since g (1) = 1 (for example A ). • Let s ≥ , s ]. We extend thispartition to ]2 s , s +1 ].Let A , A , ..., A g (2 s ) the partition on [1 , s ] given by induction assumption.Rank by decreasing cardinal these sets: A i , A i , ..., A i g (2 s ) . (*)ie: A i ∩ [1 , s ]) ≥ A i ∩ [1 , s ]) ≥ ... ≥ A i g (2 s ) ∩ [1 , s ]).(*) is only to get (ii).Let j ∈ ]2 s , s +1 ], there exists i k such that j − s ∈ A i k , -if k > g (2 s +1 ) − g (2 s ), we put j in A i k ,-otherwise, we put j in a ”new ” class, j ∈ A g (2 s )+ k .Thus we have : N A g (1 , s +1 ) = N A g (1 , s ) + { k ∈ [1 , g (2 s )]; k ≤ g (2 s +1 ) − g (2 s ) } = g (2 s ) + g (2 s +1 ) − g (2 s )= g (2 s +2 ) . Besides, note that by construction either S i (1 , s +1 ) = S i (1 , s ) or either S i (1 , s +1 ) =2 S i (1 , s ). So the second assertion of ( P ) is well satisfied at the rank s + 1, ex-cept when S i (1 , s +1 ) has doubling and S j (1 , s +1 ) is unchanged. But in thiscase, by (*) we have A i ∩ [1 , s ]) ≤ A j ∩ [1 , s ]), that could be written S i (1 , s ) ≤ S j (1 , s ). So, S i (1 , s +1 ) S j (1 , s +1 ) = 2 S i (1 , s ) S j (1 , s ) ≤ . B. We end up the construction of the partition on Z as follow: for j ≤ , we put j ∈ A i where − j + 1 ∈ A i . we call A g this partition.C. Let us check conditions (i) and (ii).First, notice that for all integers A and for all s ≥
0, partitions on[1 , s ] and[ A s + 1 , ( A + 1)2 s +1 ] are equivalents. And in particular we have: N A g (0 , s ) = N A g (2 s A, s ( A + 1)) , (12) et S i (2 s A, s ( A + 1)) S j (2 s A, s ( A + 1)) ≤ . (13)Consider k ∈ Z and m ≥ s ≥ s − < m ≤ s − and let A = min { D ; k ≤ D s − } . Wehave [ A s − , ( A + 1)2 s − ] ⊂ [ k, k + m ] and then N A g ( k, k + m ) ≥ N A g (2 s − A, s − ( A + 1))= N A g (0 , s − )= g (2 s / ≥ g ( m/ (cid:23) g ( m ) . Let B = max { D ; D s − ≤ k } , we have [ k, k + m ] ⊂ [ B s − , ( B + 2)2 s − ].So, N A g ( k, k + m ) ≤ N A g ( B s − , ( B + 2)2 s − )= N A g ( B s − , ( B + 1)2 s − ) + N A g (( B + 1)2 s − , ( B + 2)2 s − )= 2 g (2 s − ) ≤ g (2 m ) (cid:22) g ( m ) . That proves (i).Let now C = max { D ; D s − ≤ k } , by the definition of s , it is easy to ver-ify that :[( C + 1)2 s − , ( C + 2)2 s − ] ⊂ [ k, k + m ] ⊂ [ C s − , ( C + 5)2 s − ] . (14) k k+m s−3 s−3 c2 (c+1)2 (c+2)2 s−3 s−3 s−3 (c+5)2 s−3 Let i, j be the subscript which index the partition such that S i ( k, k + m ) = 0and S j ( k, k + m ) = 0, we can write, S i ( k, k + m ) ≤ S i ( C s − , ( C + 5)2 s − ) ≤ S j ( C s − , ( C + 5)2 s − ) par (13)= 2[ S j ( C s − , k ) + S j ( k, k + m ) + S j ( k + m, ( C + 5)2 s − )] . (15)Consider the terms S j ( C s − , k ) and S j ( k + m, ( C + 5)2 s − ).First we have S j ( C s − , k ) ≤ S j ( C s − , ( C + 1)2 s − ).Besides, there exists j such that S j ( C s − , ( C + 1)2 s − ) = S j (( C + 1)2 s − , ( C + 2)2 s − ) . We deduce S j ( C s − , ( C + 1)2 s − ) = S j (( C + 1)2 s − , ( C + 2)2 s − ) ≤ S j (( C + 1)2 s − , ( C + 2)2 s − ) by (13) ≤ S j ( k, k + m ) by the first inclusion of (14)By using the same approach, we prove, S j ( k + m, ( C + 5)2 s − ) ≤ S j ( k, k + m ) . Finaly with (15) we get, S i ( k, k + m ) ≤ KS j ( k, k + m ) with K = 10 . That proves (ii). (cid:3)
Remark 2.8.
The property (ii) of lemma 2.7, can be extend immediatly for all finiteset Ω . Indeed, we have for each connected component Ω s of Ω , S i (Ω s ) ≤ KS j (Ω s ) .Then summing on s , we get S i (Ω) ≤ KS j (Ω)Before showing that the graph A ≀ A g B is solution of our problem, let us noticethe following property of the partition A g , that will be useful in the next. Lemma 2.9.
Let g satisfying assumptions of property 2.7 and A g = { A i } theassociated partition. There exists constants c , c > such that for all Ω ⊂ Z ,satisfying | ∂ A ′ Ω || Ω | ≤ k , for all Ω δ ⊂ Ω such that | Ω δ | ≥ δ | Ω | , ( δ > ) we have: { i ; A i ∩ Ω δ = ∅} ≥ c δ K g ( c F ol A ( k )) , where K is the constant which appears in the item (ii) of lemma 2.7.Proof. (1) Let Ω ⊂ Z such that | ∂ A ′ Ω || Ω | ≤ k . There exists at least one connectedcomponent Ω s of Ω such that | ∂ A ′ Ω s || Ω s | ≤ k and so | Ω s | ≥ F ol A ( k ).(2) Take for c et c the constants verifying N A g ( k, k + m ) ≥ c g ( c m ) , for all k in Z and m in N .(3) There exists i such that 0 < | A i ∩ Ω | ≤ | Ω | c g ( c F ol A ( k )) . Indeed, if for all j such that | A j ∩ Ω | > | A j ∩ Ω | > | Ω | c g ( c F ol A ( k ))0 CL´EMENT RAU then we would have had , | Ω | = X j | A j ∩ Ω | > N A g (Ω) | Ω | c g ( c F ol A ( k )) > N A g (Ω s ) | Ω | c g ( c F ol A ( k )) > | Ω | (by the choice of c et c . )Absurd.(4) We deduce that for all i, | A i ∩ Ω | ≤ K | Ω | c g ( c F ol A ( k )) . Indeed, by remark 2.3.1, for all i we can write : | A i ∩ Ω | = S i (Ω) ≤ KS i (Ω) = K | A i ∩ Ω | ≤ K | Ω | c g ( c F ol A ( k )) . (5) Assume now that { i ; A i ∩ Ω δ = ∅} ≤ c δ K g ( c F ol A ( k )) . Then we havesuccessively, δ | Ω | ≤ | Ω δ | = X i ; A i ∩ Ω δ = ∅ | A i ∩ Ω δ |≤ { i ; A i ∩ Ω δ = ∅} × max i | A i ∩ Ω δ |≤ { i ; A i ∩ Ω δ = ∅} × max i | A i ∩ Ω |≤ c δ K g ( c F ol A ( k )) × K | Ω | c g ( c F ol A ( k )) = δ | Ω | . Absurd. (cid:3)
Take now g : x → x β . Since β <
1, assumptions of lemma 2.7 are satisfying.Let D F = A ≀ A g B , in the following lines we are going to prove that this graph issolution of propostion 1.2.2.3.2. proof of (i) of proposition 1.2. • Upper bound of Folner function
Using the sets U n defined by (11), we get upper bound of Folner function. . F ol D F ( n ) (cid:22) | U n | = (2 n + 1)2 N A g ( − n,n ) ≈ e n β . • Lower bound of Folner function
We get the lower bound by the same ideas as in the case α > /
3, but we haveto improve the definition of satisfactory points. Let M a set of part of V ( A ) andlet ǫ > y >
0. Given U ⊂ V ( A ≀ A g B ) and f a configuration of U , we saythat the configuration f is (1 − ǫ, y ) M satisfactory if there exists M ∈ M such that M ′ ⊂ M and (1 − ǫ ) | M | ≤ | M ′ | , where M ′ = { a ∈ V ( A ); dim φ ( a ) f ≥ y } .Then the proof falls into 3 steps.(1) Let U ⊂ V ( D F ) such that | ∂ DF U || U | ≤ k . (**) (2) For W ⊂ V ( D F ), we call W c = { f ; ∃ a ∈ V ( A ) ( a, f ) ∈ W } . By the sameway as in the proof of propostion 1.2 in the case α > /
3, we prove thatthere exists ǫ > U verifying (**), there exists W ⊂ U such that all f of W c is (1 − ǫ, F ol B ( k ) / M satisfactory, with M = { D ⊂ V ( A ); | ∂ A D || D | ≤ k } . This result is analogous to lemma 2.2 et 2.3 is proved in the next section2.4.(3) Take now f ∈ W c , there exists M ∈ M such that, M ′ = { a ∈ V ( A ); dim φ ( a ) f ≥ F ol B ( k ) / } ⊂ M and | M ′ | ≥ (1 − ǫ ) | M | . Lemma 2.9 apply with δ = 1 − ǫ, M = Ω and M ′ = Ω δ . We deduce thatfor all f in W c , we can change the value of the configuration f in at least c − ǫ K g ( c F ol A ( k )) points in F ol B ( k ) /
30 ways by staying in W c . Then weconclude by the following lemma: Lemma 2.10.
Let
Y > and X > . Let A a non empty set of configu-rations, such that for all configurations of A , there exists at least Y pointswhere we can change the value of the configuration in X way without leav-ing A . Then : |A| ≥ X Y . ie: ( ∀ f ∈ A ∃ a , a , ..., a Y ∈ A such that g ∈ A ) = ⇒ |A| ≥ X Y , where g is defined from f by : g ( x ) = ( f ( x ) if x = a i ,X possibilities f or g ( a i ) if x = a i . Proof.
We proceed by induction on Y .If Y = 1, it is exact.Suppose Y ≥ x in the base such that there exists X distinct configurations f , ..., f X ∈ A such that ∀ y = x f ( y ) = f ( y ) = ... = f X ( y ) . For all i = 1 ...X , let A i = { f ∈ A ; f ( x ) = f i ( x ) } , which are not empty. A = ˙ S i =1 ...X A i and the A i satisfy induction hypothesis with constant Y − |A| = P i =1 ...X | A i | ≥ X.X Y − = X Y . (cid:3) Finally, lemma 2.10 gives, | U | ≥ | W c | ≥ ( F ol B ( k )30 ) c ′ g ( c F ol A ( k )) (cid:23) e g ( k ) , since first F ol B ( n ) = 2 and secondly F ol A ( k ) = 2 k .2.3.3. proof of (ii) of proposition 1.2. We follow idea of the case α ≥ / d = (0 , f ) where f is the configuration which is null every where. Let X n = ( K n , g n ) be the random walk on D F defined above. X starts from d and jumps uniformly on the set of points formed by the point where the walk is and its neighbor. On this generalized wreath product, this walk isstill reversible for the uniform measure since the number of neighbor in D F is constant, equal to 4. Now write: p D F n ( d , d ) = X z p D F n ( d , z ) p D F n ( z, d ) ≥ X z ∈ G p D F n ( d , z ) ≥ | G | [ X z ∈ G p D F n ( d , z )] ≥ | G | [ P D F d ( X n ∈ G )] , where G is some finite set of V ( D F ) . (2) Take G = G r = { ( a, f ) ; | a | ≤ r and supp ( f ) ⊂ φ ([ | − r, r | ]) } . By the structure of edges on D F , we have : P D F d ( X n ∈ G r ) ≥ P D F d ( ∀ i ∈ [ | , n | ] | K i | ≤ r ) ≥ P K ( ∀ i ∈ [ | , n | ] | K i | ≤ r ) , where P K is the law of ( K i ) i which is still a random walk with transitionsprobability which can be represented by : n−1 n n+1 (3) Now we have to find a lower bound for P K ( ∀ i ∈ [ | , n | ] | K i | ≤ r ). It isnot sufficient to use P K ( ∀ i ∈ [ | , n | ] | K i | ≤ r ) ≥ e − c ( n/r + r ) as in the case α > /
3, because β = α − α < ∃ c > , ∀ n ≥ P K ( ∀ i ∈ [ | , n | ] | K i | ≤ r ) ≥ e − cn/r . One can find this result in the lemma 7.4.3 of [7]. It is known for a simplerandom walk on Z d and we can deduce it in this particular case with acoupling. Consider K ′ i which takes values in Z . K ′ i follows the horizontaljumps of K i if K i moves and jumps uniformly on its 2 vertical neighborsif K i stays at its place. On the first hand we have { sup ≤ i ≤ n | K ′ i | ≤ r } ⊂{ sup ≤ i ≤ n | K i | ≤ r } and on other hand K ′ i is a simple random walk on Z .(For x = ( a, b ) ∈ Z , we note | x | := max( a, b ).) Then the result for K i in Z follows from the result for K ′ i in Z .(4) We can end up the proof. From | G r | = (2 r + 1)2 N A g ( − r,r ) (cid:22) e r β , we deducethere exists c > p D F n ( d , d ) ≥ e − c ( nr + r β ) . But the function r nr + r β , is minimal for r like n β +2 .So there exists c > p D F n (0 , ≥ e − cn ββ +2 = e − cn α . Complement on satisfactory points.
In this section we improve the notionof satisfactory point used in subsection 2.3 which is more abstract that the notionintroduced in subsection 2.2. The reasons of this improvement will be explain inthe next.We still consider a wreath product A ≀ B of two graphs A and B or a generalizedwreath product A ≀ A B associated to some partition A . We take U ⊂ V ( A ≀ B )and as before to each U we associate an hypergraph K U and its one dimensionalskelelton Γ U with weight w , built as the same way that in section 2.2.Let ǫ > a ≥
0. Let M a set of parts of V ( A ). To light the way of thisdefinition and to link it with the old definition of satisfactory points (section 2.2),one can think to take for M set of the form { D ⊂ V ( A ); | ∂ A D || D | ≤ k } . • A configuration f of V ( K U ) is said (1 − ǫ, a ) M satisfactory if :there exists M ∈ M such that M ′ ⊂ M and(1 − ǫ ) | M | < | M ′ | (16)where M ′ = { m ∈ V ( A ); dim m f ≥ a } . Once again, we denote by S U (1 − ǫ, a ) M (or S (1 − ǫ, a ) M ) the set of satisfactoryconfigurations. • Otherwise f is not satisfactory and we note N S (1 − ǫ, a ) M ) the set of notsatisfactory configurations. • If Γ ′ is a subgraph of Γ U , we say that f is S (1 − ǫ, a ) M in respect to Γ ′ if f satisfies the same condition as in (16) but where dimension of f is counted onlywith edges in Γ ′ . More precisely : dim m, Γ ′ f = { g ; ( f, g ) ∈ E (Γ ′ ) and ( x, g ) ∈ U and ∀ y = x f ( y ) = g ( y ) } . • An edge of Γ U is said (1 − ǫ, a ) M satisfactory if it joins two (1 − ǫ, a ) M satisfac-tory configurations, otherwise it is said (1 − ǫ, a ) M not satisfactory. As before wedenote by S e (1 − ǫ ; a ) M [resp N S e (1 − ǫ, a ) M ] the set of satifactory edges [respnot satisfactory ]. • A point u = ( x, f ) ∈ U is said (1 − ǫ, a ) M satisfactory [resp (1 − ǫ, a ) M not sat-isfactory ] if f ∈ S (1 − ǫ, a ) M [resp N S (1 − ǫ, a ) M ]. We denote by S p (1 − ǫ, a ) M and N S p (1 − ǫ, a ) for the set of points which are (or are not ) satisfactory. • We keep the same defintion for good points, u = ( x, f ) ∈ U is said a − good if dim x f ≥ a otherwise it is said a − bad. The interest of this new definition of satisfactory points is the following. Con-sider a set U c of ( λ, b ) − satisf actory configurations. With the ”old” definition weknow that we can change the value of f in at least b points in λ x ways (at point x )without leaving U c but we do not know exactly where are these b points whereaswith the ”new” definition, for a set U c of S (1 − ǫ, a ) M configurations, we know that we can change the value of f in at least (1 − ǫ ) min M ∈M | M | points in a ways withoutleaving U c et moreover we know that these points are contained in some M ∈ M .This would be useful for our generalized wreath products since this property con-centrate points where we can change value of f . By the properties of partition, itremains only to get lower bound of φ ( M ).Let U ∈ V ( A ≀ B ) such that | ∂ A ≀ B U || U | ≤ k , the two following lemmas are similarto lemmas 2.2 et 2.3. Lemma 2.11.
Let M = { D ⊂ V ( A ); | ∂ A D || D | ≤ k } then we have : ( i ) { u ∈ U ; u is F ol B ( k ) − bad }| U | ≤ , ( ii ) there exists ǫ > such that { u ∈ U ; (1 − ǫ,F ol B ( k ) / M − not satisfactory }| U | ≤ . Proof.
For (i), it is the same argument that in part (i) of lemma 2.2.For (ii) let,
N eud = { u ∈ U ; u ∈ N S p (1 − ǫ, F ol B ( k )3 ) M } = { u = ( x, f ) ∈ U ; f ∈ N S (1 − ǫ, F ol B ( k )3 ) } , and let: N eud ( f ) = { ( x, f ); ( x, f ) ∈ U } . Note that p ( N eud ( f )) = { x ; ( x, f ) ∈ U } . For F a set of configurations, let N eud ( F ) = ∪ f ∈ F N eud ( f ) . Note the union is disjointed.Take now f ∈ N S (1 − ǫ, F ol B ( k )3 ) M , and consider the set p ( N eud ( f )).Two cases appear. Either p ( N eud ( f )) gives a large part of boundary in ”base”either not and this case by assumptions on f we will prove that p ( N eud ( f )) givesboundary in ”configurations”First case : f ∈ F := { f ∈ N S (1 − ǫ, F ol B ( k )3 ) M ; ∂ A p ( Neud ( f )) p ( Neud ( f )) > k } . The application ˙ S f ∈ F ∂ A p ( N eud ( f )) −→ ∂ A ≀ B U is injective.( x, y ) (cid:16) ( x, f ) ; ( y, f ) (cid:17) So, we can write :(17) | ∂ A ≀ B U | ≥ X f ∈ F | ∂ A p ( N eud ( f )) | ≥ k X f ∈ F | p ( N eud ( f )) | ≥ k | N eud ( F ) | . Second case : f ∈ F := { f ∈ N S (1 − ǫ, F ol B ( k )3 ) M ; ∂ A p ( Neud ( f )) p ( Neud ( f )) ≤ k } . Since f ∈ N S (1 − ǫ, F ol B ( k )3 ) M we have :for all M ∈ M ∃ m ′ ∈ M ′ − M, or | M ′ | ≤ (1 − ǫ ) | M | , (18)where M ′ stands for { m ∈ V ( A ); dim m f ≥ F ol B ( k )3 } . Choose M = p ( N eud ( f )) since f ∈ F we have M ∈ M and M ′ ⊂ M . So it is thesecond item of assertion (18) which is satisfied. ie : | M ′ | ≤ (1 − ǫ ) | M | . So, { x ∈ p ( N eud ( f )); dim x f ≥ F ol B ( k )3 } < (1 − ǫ ) | M | = (1 − ǫ ) | N eud ( f ) | . (We have used that | p ( N eud ( f ) | = | N eud ( f ) | . )So { x ∈ p ( N eud ( f )); dim x f < F ol B ( k )3 } ≥ ǫ | N eud ( f ) | ie : | P f | ≥ ǫ | N eud ( f ) | , (19)with P f = { x ∈ p ( N eud ( f ); dim x f < F ol B ( k )3 } . To each point of P f ( for f in F ), we can associate in an injective way a point ofthe boundary (in configuration ) of U . Indeed, as before :for x ∈ P f and f ∈ N eud ( F ), we have : | ˜ P x,f | ≤ F ol B ( k )3 < F ol B ( k ) . where ˜ P x,f = { g ( x ); ( x, g ) ∈ U and ∀ y = x g ( y ) = f ( y ) } .Thus, | ∂ B ˜ P x,f | > k | ˜ P x,f | ≥ , and then | ∂ B ˜ P x,f | ≥ . Finally, | ∂ A ≀ B U | ≥ X x ∈ P f ,f ∈ F | ∂ B ˜ P x,f |≥ X f ∈ F ǫ | N eud ( f ) | by (19) , ≥ ǫ | N eud ( F ) |≥ k | N eud ( F ) | by choising ǫ < /k .By adding (17) and this last inequality and using the fact that | ∂ A ≀ B U || U | < k , weget : | N eud || U | < . (cid:3) Lemma 2.12.
Let ǫ > and x > . Consider Γ U the one dimensional skeleton withweight w , constructed from K U . Assume that E (Γ U ) = ∅ and ∀ f ∈ K U dim x f ≥ a and M does not contain the empty set. If we have : P e ∈ NS eU (1 − ǫ,a ) M w ( e ) P e ∈ E (Γ U ) w ( e ) < / , then, there exists a not empty subgraph Γ ′ of Γ U such that all edges are S U (1 − ǫ , a ) M satisfactory in respect to Γ ′ .Proof. In the graph Γ U , we remove all points N S pU (1 − ǫ , a ) M and adjacentsedges. After this first step, it may appear some new points N S PU (1 − ǫ , a ) M ,where U = U − N S pU (1 − ǫ , a ) M .We remove again all adjacent edges and points and we iterate this process.Let U i the set of vertices staying at step i . (cid:26) U = U, for i ≥ U i +1 = U i − N S pU i (1 − ǫ , a ) M . it is sufficient to prove that this process ends up before the graph becomes empty.Let C = P e ∈ NS U (1 − ǫ,a ) M w ( e ) , C = P e ∈ S eU (1 − ǫ,a ) M ; e removed at the end of the process w ( e ) , and C = X e ∈ E (Γ U )); e removed at the end of the process w ( e ) . If we show that C ≤ C , the result is proved since: C ≤ C + C ≤ C < X e ∈ E (Γ U ) w ( e ) . That would mean that it stays at least one point not removed. ie: ∃ k ∈ N such thatall points of the graph get at step k , are S pU k (1 − ǫ , a ) M , so S pU (1 − ǫ , a ) M . To see this, let us introduce an orientation on removed edges : if L and Q arepoints of the graph, we orient the edge from L to Q if L s removed before Q oth-erwise we choose an arbitrary orientation. We note L ↓ the set of edges leaving thepoint L and L ↑ for the set of edge ending in L at step 0. Sublemma 2.13.
Let k ∈ N and let L be a point of the graph Γ U (satisfyingassumptions of lemma 2.12) removed after k + 1 steps. Assume that L is initially S pU (1 − ǫ, a ) M , then X e ∈ L ↓ w ( e ) ≤ X e ∈ L ↑ w ( e ) . Proof.
It would be useful to notice that for a multidimensional edge e , the sum ofthe weight (in the skeleton) of edges coming from e and adjacent to a point, is equalto 1. This is implied by our choice of the weight.(1) Let now N be the number of multidimensional edges at step 0. Since L is initially S pU (1 − ǫ, a ) M , there exists M ∈ M such that (1 − ǫ ) | M | multidimensional edges are attached to L . So, N ≥ (1 − ǫ ) | M | . (20) Besides notice that : X e ∈ E (Γ U ) e contains L w ( e ) = N . Let : L ↓ = { e ∈ L ↓ , e coming from a multidimensionnal edge of K U k , of dim ≥ a/ } , and L ↓ = { e ∈ L ↓ , e coming from a multidimensionnal edge K U k , of dim < a/ } . We have L ↓ = L ↓ ∪ L ↓ , because edges of L ↓ correspond to edges leaving L at step k .(2) Since L becomes N S pU k (1 − ǫ , a ) M , we have:for all M in M M ′′ M or | M ′′ | ≤ (1 − ǫ ) | M | where M ′′ = { m ∈ V ( A ); dim m,U k L ≥ a } .Take M = M , observe that M ′′ ⊂ M so that implies | M ′′ | ≤ (1 − ǫ ) | M | . Finally L has less than (1 − ǫ ) | M | multidimensional edges ofdimension at least a/
10. call them f , ..., f q , with q < (1 − ǫ ) | M | . X e ∈ L ↓ w ( e ) = X k =1 ..q X e coming from fk w ( e ) | {z } ≤ ≤ q. (21) (Initially this last sum was equal to 1, but after removing some edges, thissum is less than 1.)Besides, call g , ..., g h the other multidimensional edges of dimension strictlyless than a/ , attached to L at step k , with h ≤ N − q .For all k = 1 ...h, X e coming from gk w ( e ) ≤ a a ≤ . (22) (Indeed, first all point have initially dimension at least a so we deduce ∀ e ∈ E (Γ U i ) w ( e ) ≤ /a and secondly an edge of dimension less than a/ a/
10 edges attached to one point, in the skeleton. )(3) Finally with (21) and (22), we get : X e ∈ L ↓ w ( e ) = X e ∈ L ↓ w ( e ) + X e ∈ L ↓ w ( e ) ≤ q + ( N − q ) 110= 110 N + 910 q = 19100 N . ( q < (1 − ǫ ) | M | ≤ N − ǫ − ǫ ≤ N by (20).)So, X e ∈ L ↓ w ( e ) ≤ N et X e ∈ L ↑ w ( e ) ≥ N − N = 81100 N . And then, X e ∈ L ↓ w ( e ) ≤ X e ∈ L ↑ w ( e ) ≤ X e ∈ L ↑ w ( e ) . (cid:3) The proof ends up by the same way as proposition 2.3, let: D = { vertices removed at step 1 } , and for i ≥ D i = { vertices S pU (1 − ǫ, a ) removed at step i } ,F i = { edges between D i and D i − } ,F ′ i = { edges leaving D i − } .Notice that F i ⊂ F ′ i and that edges of F ′ i are removed. D1D2 } F2F3 } D3 By sublemma2.13 applied in each point of D i in the graph get at step i −
2. (Each point of D i is at this moment, at least S (1 − ǫ, a ) M .) We get : ∀ i ≥ X e ∈ F ′ i +1 w ( e ) ≤ X e ∈ F i w ( e ) . so, X e ∈ F ′ i +1 w ( e ) ≤ ( 12 ) i − X e ∈ F w ( e ) . (We have used that F i ⊂ F ′ i . )Thus, X e ∈ ∪ i ≥ F ′ i w ( e ) ≤ ( X i ≥ ( 12 ) i ) X e ∈ F w ( e )= X e ∈ F w ( e ) . Now, an edge of F is N S eU (1 − ǫ, a ) because if this edge was S eU (1 − ǫ, a ), this edgewould have linked two points S pU (1 − ǫ, a ) and in particular, points of D wouldhave been S pU (1 − ǫ, a ), so S pU (1 − ǫ , a/
10) and so would not have removed. Thus: X e ∈ F w ( e ) ≤ X e ∈ NS e ( a,b ) w ( e ) = C . Besides, all removed edge S eU (1 − ǫ, a ) is in some F ′ i with i ≥
3, so C = X e removed at the of the process e ∈ SeU (1 − ǫ,a ) w ( e ) ≤ X e ∈ ∪ i ≥ F ′ i w ( e ) . That ends the proof. (cid:3)
Now we are able to explain the fact that we used in section 2.3.2 in order to provethe lower bound of
F ol D F . We recall that U ⊂ V ( D F ) is such that | ∂ A ≀ B U || U | ≤ k . Let ˜ K be the sub hypergraph of K U which contains only F ol B ( k ) / − good points.As in the proof of (i) in the case α > / θ < / P e ∈ NS eU (1 − ǫ,F ol B ( k ) / M e ∈ E (Γ ˜ K ) w ( e ) P e ∈ E (Γ ˜ K ) w ( e ) < θ, for some ǫ > M = { D ⊂ V ( A ); | ∂ A D || D | ≤ k } .Lemma 2.12 gives us a sub graph where all edges are S e (1 − δ, F ol B ( k ) / M for δ = 1 − ǫ . By definition of satisfactory points, this proves the fact that we haveused. 3. Applications: study of some functionals
Kind of problems, case of the lattice Z d . Recall that for G a graph and X is a simple random walk on G , we note L x,n = { k ∈ [0; n ]; X k = x } . Thequestion is to estimate functional of type E ω ( e − λ P z ; Lz ; n> F ( L z ; n ,z ) ) , (23)where F is a two variables non negative function. The method developped here isdue to Erschler and can be applied on general graph G provided the isoperimetricprofile on the graph G is known and the function F has some ”good” properties.For the case of the simple random walk on Z d , in [4] it is proved that ∀ α ∈ [0 , E ω ( e − λ P z ; Lz ; n> L αz ; n ) ≈ e − n η , (24) ∀ α ≥ / E ω ( Y z ; L z ; n > L − αz ; n ) ≈ e − n dd +2 ln ( n ) d +2 , (25)where η = d + α (2 − d )2+ d (1 − α ) . This section is devoted to extend these estimates to an infintecluster of the percolation model.3.2.
In an infinite cluster of the percolation model.
Percolation context.
Consider the graph L d = ( Z d , E d ) where E d are the couple of points of Z d atdistance 1 for the N norm. Now pick a number p ∈ ]0 , p [resp 1 − p ] in an independant way. We get a graph ω and we call C the connected component that contains the origin and C n theconnected component of C ∩ [ − n, n ] d that contains the origin.We still use the notation ω for the application E d → { , } such that ω ( e ) = 0if e is a removed edge and 1 otherwise. Let Q be the probability measure underwhich the variable ( ω ( e ) , e ∈ E d ) are Bernouilli(p) independent variables. If p islarger than some critical value p c , the Q probability that C is infinite, is strictlypositive and so we can work on the event { C = + ∞} .We denote by C g the graph such that V ( C g ) = C and E ( C g ) = { ( x, y ) ∈ E d ; ω ( x, y ) = 1 } and C gn the graph such that V ( C gn ) = C and E ( C gn ) = { ( x, y ) ∈ E d ; x, y ∈ C n and ω ( x, y ) = 1 } From now on and until the end, X will design the simple random walk on thegraph C g . We are going to prove estimate (24) and (25) for the walk X .3.2.2. Sketch plan.
Let ( B x ) x ∈C be a family of graphs and let 0 x an arbitrary point in each B x thatwe call the origin. For all x ∈ C , consider the random walk ( Y xn ) n on B x startingfrom point 0 x , and jumping uniformly on the set of points formed by the pointwhere the walk is and its neighbors. Let P B x x be the law of ( Y xn ) n .Transition kernels of Y x satisfy : p B x ( a, b ) = 1 ν x ( a ) + 1 (1 { a = b } + 1 { ( a,b ) ∈ E ( B x ) } ) , where ν x ( a ) stands for the number of neighbors of a in graph B x .Consider now the graph W = W C = C g ≀ ( B z ) z ∈C . (26)Let f be the nulle configuration, such that , for all x ∈ C , f ( x ) = 0 x , and let o = (0 , f ). And we look at the random walk ( Z n ) n on the graph W C starting from o , defined by the following: suppose that the walk is at point z = ( x, f ), then inone unit of time the walk makes three independent steps. First, the value of f atpoint x jumps in graph B x in respect to the walk Y x starting from f ( x ). Secondly,we make the walker in C jump on his neighbors in respect to uniformly law on hisneighbor, so the walker in C (projection on C of walk on W C ) arrives at point y ∈ C .And thirdly, the value of f at point y jumps in graph B y in respect to the walk Y y starting from f ( y ).Thus, calling ˜ p transitions kernel of Z , we have:for all (( a, f ); ( b, g )) ∈ ( V ( C g ≀ B z )) :˜ p [( a, f )( b, g )] = χ [( a, f ) , ( b, g )] ν ( a ) [ ν a ( f ( a )) + 1] [ ν b ( f ( b )) + 1] , (27)where χ [( a, f ) , ( b, g )] is equal to 1 if the walk is able to jump from ( a, f ) to ( b, g )and 0 otherwise.More precisely, χ [( a, f ) , ( b, g )] = ω ( a, b ) ( χ [( a, f ) , ( b, g )] + χ [( a, f ) , ( b, g )]+ χ [( a, f ) , ( b, g )] + χ [( a, f ) , ( b, g )]) , with χ [( a, f ) , ( b, g )] = 1 {∀ x f ( x )= g ( x ) } , χ [( a, f ) , ( b, g )] = 1 { ( f ( a ) ,g ( a )) ∈ E ( B a ) ∀ x = a f ( x )= g ( x ) } ,χ [( a, f ) , ( b, g )] = 1 { ( f ( b ) ,g ( b )) ∈ E ( B b ) ∀ x = b f ( x )= g ( x ) } , χ [( a, f ) , ( b, g )] = 1 {∀ x ∈{ a,b } ( f ( x ) ,g ( x )) ∈ E ( B x ) ∀ x = a,b f ( x )= g ( x ) } . Notice that ˜ m defined by, ˜ m ( a, f ) = ν ( a ) , (28)is a reversible measure for the walk Z . We note ˜ a the following kernels:˜ a ( x, y ) = ˜ m ( x )˜ p ( x, y )(29)Let ˜ P ωo be the law of Z starting from o. The key for our problem is the followinginterpretation of the return probability of Z : Proposition 3.1. ˜ P ωo ( Z n = o ) = E ω ( Y x ; L x ; n > P B x x ( Y xL x ; n = 0 x ) 1 { X n =0 } ) . Proof. :˜ P ωo ( Z n = o ) = ˜ P ωo (cid:16) ( X n , f n ) = (0 , f ) (cid:17) = X ( k ,k ,...,k n ) ∈ Z dk kn =0 ˜ P ωo ( X = k , X = k , ..., X n = k n et f n = f )= X ( k ,k ,...,k n ) ∈ Z dk kn =0 ˜ P ωo ( X = k , X = k , ..., X n = k n ) × ˜ P ωo ( f n = f | X = k , ...X n = k n )= X ( k ,k ,...,k n ) ∈ Z dk kn =0 P ω ( X = k , X = k , ..., X n = k n ) × Y x ; L x ; n > P B x x ( Y xL x ; n = 0 x )= E ω ( Y x ; L x ; n > P B x x ( Y xL x ; n = 0 x ) 1 { X n =0 } ) . (cid:3) In order to estimate functional such that (23) and in view of propostion 3.1, wehave to find graphs B x such that for all m ∈ N : P B x x ( Y xm = 0) ≈ e − λF ( m,x ) . Moreover, since we know that an isoperimetric inequality with volume counted inrespect to measure m and boundary counted in respect to kernels ˜ a , gives an upperbound of the decay of the probability transitions of walk Z , in a first time we haveto estimate the Folner function of W C and so (by similar results of section 1, see [4][5] and [6]) we should know Folner function of each B x .The graph formed by the possible jumps of walk Z is not W C = C ≀ ( B z ) z ∈C , sowe introduce the graph with same set of points of W C but different set of edges.We call it C ≀ ≀ ( B z ) z ∈C or shortly W ′C (or W ′ ), the graph such that : V ( W ′C ) = V ( W C ) and , (30) (( a, f ); ( b, g )) ∈ E ( W ′C ) ⇐⇒ χ [( a, f ); ( b, g )] = 1 . Thus, in the graph W ′C , the random walk Z is a nearest neighbor walk. Propertiesof Z are linked to geometry of W ′C but as we will see later W C and W ′C are roughlyisometric, so we can study isoperimetric profile of W C .3.2.3. Study of E ω ( e − λ P L αz ; n ) . Upper bound
Let α ∈ ]0 ,
1[ and β = α − α and let F ( x ) = e x β . Let D F be thegraph given by proposition 1.2. We put for all x ∈ C , B x = D F .First we want to obtain a lower bound of F ol C g ≀≀ D F C gn ≀≀ D F ( k ) . We proceed in 3 steps: A. By using general results on wreath product, see [4], [5] and [6], we have :
F ol C g ≀ D F C gn ≀ D F ( k ) ≈ ( F ol D F ( k )) F ol C g C gn ( k ) . B. By proposition 1.4 of [6], we get:for all γ >
0, there exists β > c > Q a.s for large enough n , we have : F ol C g ≀ D F C gn ≀ D F ( k ) (cid:23) ( F ( k ) k if k < cn γ , ( F ( k ) βk d if k ≥ cn γ . (31)C. We want to carry (31 ) on F ol C g ≀≀ D F C gn ≀≀ D F . Let δ a imaginary point and consider thefollowing graphs: W n = C gn ≀ D F , (32)and W ′ n = C gn ≀ ≀ D F , (33)defined by : V ( C gn ≀ D F ) = V ( C gn ≀ ≀ D F ) = V ( C gn ≀ D F ) ∪ { δ } and set of edges are given by E ( W n ) = E ( C gn ≀ D F ) ∪ { ( x, δ ); x ∈ V ( C gn ≀ D F ) and ∃ y ∈ V ( W ) ( x, y ) ∈ E ( W ) } and E ( W ′ n ) = E ( C gn ≀≀ D F )) ∪{ ( x, δ ); x ∈ V ( C gn ≀≀ D F ) and ∃ y ∈ V ( W ) ( x, y ) ∈ E ( W ′ ) } . Let respectively d and d ′ be the distances on W and W ′ , given by edges of thesegraphs. W n are W ′ n are rough isometric with constants independant of n . Withthe notations of definition 3.7 in [8], we have A = 3 and B = 0 . Indeed, consider id : ( V ( W n ) , d ) → ( V ( W ′ n ) , d ) . For all x, y ∈ V ( W n ) = V ( W ′ n ) , we have:13 d ( x, y ) ≤ d ′ ( x, y ) ≤ d ( x, y ) . Thus the respective Dirichlet forms E and E ′ for simple random walks on W n and W ′ n satisfy: there exists c , c > f : V ( W n ) → R wehave, c E ( f, f ) ≤ E ′ ( f, f ) ≤ c E ( f, f ) , with E ( f ) = X ( x,y ) ∈ E ( W n ) ( f ( x ) − f ( y )) , and E ′ ( f ) = X ( x,y ) ∈ E ( W ′ n ) ( f ( x ) − f ( y )) . Now, let U ⊂ V ( C gn ≀ ≀ D F ) and take f = 1 U , we get : c | ∂ W U | ≤ | ∂ W ′ U | ≤ c | ∂ W U | . Hence, we have proved that (31) carry to
F ol C g ≀≀ D F C gn ≀≀ D F , so we deduce: Proposition 3.2.
For all γ > , there exists β > such that for all c > , Q a.son the set |C| = + ∞ and for large enough n , we have : F ol C g ≀≀ D F C gn ≀≀ D F ( k ) (cid:23) ( F ( k ) k if k < cn γ , ( F ( k ) βk d if k ≥ cn γ . (34)Now we are able to get an upper bound of ˜ P ωo ( Z n = o ) and then an upper boundof our functional. Let τ n = inf { s ≥ Z s V ( C gn ≀ ≀ D F ) } .We have,˜ P ωo ( Z n = o ) = ˜ P ωo ( Z n = o and τ n ≤ n ) + ˜ P ωo ( Z n = o and τ n > n ) . The first term is zero since the walk can not go out the box V ( C gn ≀ ≀ D F ) beforetime n .The second term can be bounded with proposition 3.2. Let : H ( k ) = ( F ( k ) k if k < cn γ , ( F ( k ) βk d if k ≥ cn γ . (35)- H is increasing and we can define an inverse function by H − ( y ) = inf { x ; H ( x ) ≥ y } . - Besides, with the help of (34), F ol C g ≀≀ D F C gn ≀≀ D F (cid:23) H . - C and D F have bounded valency and from formula of ˜ m and ˜ a (see (29) and (28))we have : inf V ( W ′ ) ˜ m ≥ d > inf E ( W ′ ) ˜ a > . Thus, (see theorem 14.3 in[8] for example) there exists constants c , c and c > P ωo ( Z n = o et τ n > n ) (cid:22) u ( n )where u is solution of the differential equation : (cid:26) u ′ = − uc ( H − ( c /u )) ,u (0) = c . Replacing F ( k ) by e k β into H , we get the expression of H − : H − ( y ) = c ( ln ( y )) − α α if 1 ≤ y < e cn γ (1+ α )1 − α ,cn γ if e cn γ (1+ α )1 − α ≤ y < e cn γ ( d + α (2 − d ))1 − α ,c ( ln ( y )) − αd + α (2 − d ) if e cn γ ( d + α (2 − d ))1 − α ≤ y. (36)Resolving the differential equation in the different cases, we get : u ( t ) = ce − ct α − α if t ≤ cn γ (3 − α )1 − α ,ce cn γ α − α e − ct/n γ if cn γ (3 − α )1 − α < t ≤ cn γ ( d +2 − dα )1 − α + n γ (3 − α )1 − α ,ce − ( ct − c ′ n γ ( d +2 − dα )1 − α − cn γ (3 − α )1 − α ) d + α (2 − d )2+ d − dα if cn γ ( d +2 − dα )1 − α + n γ (3 − α )1 − α ≤ t. (Each c design a different constant.) Now whe choose γ such that 0 < γ < min( − αd +2 − dα , − α − α ), then we get :there exists c = c ( p, d, α, λ ) > u (2 n ) ≤ e − cn η , with η = d + α (2 − d )2+ d (1 − α ) . So, Q a.s on the set |C| = + ∞ , and for large enough n (which depends on thecluster ω ), ˜ P ωo ( Z n = o ) (cid:22) e − n η . By proposition 3.1, we deduce that Q a.s on the event |C| = + ∞ and for largeenough n , E ω ( Y x ; L x ;2 n > P D F d ( Y D F L x ;2 n = d ) 1 { X n =0 } ) (cid:22) e − n η . (37)By our choice of graph D F , there exists C , C > n ≥ P D F d ( Y D F n = d ) ≥ C e − ( C n ) α , (38) ≥ e − λ n α , (39)for some λ > λ > Q a.s on the set C| = + ∞ and for large enough n , E ω ( e − λ P x ; Lx ;2 n> L αx ;2 n { X n =0 } ) (cid:22) e − n η . (40)To conclude, it remains only to prove that we can suppress the indicatrice functionand that we can extend the inequality (40) to all λ >
0. We explain this in 3 steps.1. First of all, notice that is is sufficient to prove (3) only for one value of λ . Indeed,let λ >
0, assume that for λ = λ , we have: E ω ( e − λ P x ; Lx ; n> L αx ; n ) (cid:22) e − n η . (41)-If λ ≥ λ , (41) is true because we can replace λ by λ using merely the decrease.-If λ < λ , we write E ω [ e − λ P x ; Lx ; n> L αx ; n ] = E ω [( e − λ P x ; Lx ; n> L αx ; n ) λλ ] ≤ ( E ω [ e − λ P x ; Lx ; n> L αx ; n ]) λλ (Jensen inequality applied to concave function x → x λλ . ) (cid:22) e − n η .
2. To take out the indicatrice function, we use the following lemma:
Lemma 3.3.
For all m ≥ , we have : P ω ( X x L αx ; n = m ) ≤ d (2 m + 1) d P ω ( X x L αx ;2 n ≤ m et X n = 0) . Proof. [ P ω ( X x L αx ; n = m )] = (cid:16) X h ∈ B m ( C ) P ω ( X x L αx ; n = m ; X n = h ) (cid:17) = (cid:16) X h ∈ B m ( C ) p ν ( h ) × / p ν ( h ) × P ω ( X x L αx ; n = m ; X n = h ) (cid:17) ≤ ν ( B m ( C )) X h ∈ B m ( C ) (1 /ν ( h )) P ω ( X x L αx ; n = m ; X n = h ) (Cauchy-Schwarz inequality) ≤ d (2 m + 1) d X h ∈ B m ( C ) P ω ( X x L αx ; n = m ; X n = h ) × P ωh ( X x L αx ; n = m ; X n = 0)(1 /ν (0))(by reversibility ) ≤ d (2 m + 1) d X h ∈ B m ( C ) P ω ( X x L αx ; n = m ; X n = h ) × P ω ( X x L αx ;[ n ;2 n ] = m ; X n = h ; X n = 0)(where L x ;[ n ;2 n ] = { i ∈ [ n ; 2 n ]; X i = x } ) ≤ d (2 m + 1) d P ω ( X x L αx ;2 n ≤ m ; X n = 0) . because { P x L αx ; n = m et P x L αx ;[ n ;2 n ] = m } ⊂ { P x L αx ;2 n ≤ m } , since for α ∈ [0 , , we have : L αx ;2 n ≤ ( L x ; n + L x ;[ n ;2 n ] ) α ≤ L αx ; n + L αx ;[ n ;2 n ] . (cid:3) Then we write, E ω ( e − λ P x L αx ;2 n { X n =0 } ) = X m ≥ e − λ m P ω ( X x L αx ;2 n = m ; X n = 0)= (1 − e − λ ) X m ≥ e − λ m P ω ( X x L αx ;2 n ≤ m ; X n = 0) , since { P x L αx ;2 n = m } = { P x L αx ;2 n ≤ m } − { P x L αx ;2 n ≤ m − } . Thus, wehave, E ω ( e − λ X x L αx ;2 n { X n =0 } ) ≥ (1 − e − λ ) X m ≥ e − λ m P ω ( X x L αx ;2 n ≤ m ; X n = 0)(we add only the even m ) ≥ (1 − e − λ ) X m ≥ d (2 m + 1) d e − λ m [ P ω ( X x L αx ; n = m )] (by lemma 3.3) ≥ X m ≥ e − λ m [ P ω ( X x L αx ; n = m )] (for some λ > λ ) ≥ (cid:16) X m ≥ e − λ m (cid:17) − × (cid:16) X m ≥ e − mλ P ω ( X x L αx ; n = m ) (cid:17) (By Cauchy-Schwarz inequality) ≥ c E ω [ e − λ P x L αx ; n ] .
3. We can now conclude. By previous inequality and by (40), there exists λ suchthat : E ω [ e − λ P x L αx ; n ] (cid:22) e − n η . Then by step 1, we can extend this inequality to all λ .Finaly we have proved : Proposition 3.4.
Q a.s on |C| = + ∞ for large enough n and for all λ > wehave, for all α ∈ [0 , , E ω ( e − λ P x ; Lx ; n> L αx ; n ) (cid:22) e − n η , where η = d + α (2 − d )2+ d (1 − α ) . Remark 3.5.
1) If α = 0 , we retrieve the Laplace transform of the number of visited points bythe simple random walk on an infinite cluster.2)For α = 1 , inequality is satisfied since P x ; L x ; n > L x ; n = n and η = 1 when α = 1 . Lower bound
The proof falls into 4 steps.1. By concavity of the function x x α for α ∈ [0 , , we have : X x ; L x ; n > L αx ; n ≤ N n ( X x ; L x ; n > L x ; n N n ) α = N − αn n α . So, E ( e − λ P x ; Lx ; n> L αx ; n ) ≥ E ( e − λn α N − αn ) ≥ P ( sup ≤ i ≤ n D (0 , X i ) ≤ m ) e − λV ( m ) − α n α . where V ( m ) = | B m ( C ) | stands for the volume of the ball of C centred at theorigin with radius m .2. By proposition 5.2 of [6], we have : P ( sup ≤ i ≤ n D (0 , X i ) ≤ m ) ≥ e − c ( m + nm ) (42)3. By lemma 5.3 of [6], there exists c > Q a.s on |C| = + ∞ and forlarge enough n , V ( m ) ≥ cm d .
4. So, we deduce, there exists
C > Q a.s on |C| = + ∞ and for largeenough n , E ( e − λ P x ; Lx ; n> L αx ; n ) ≥ e − C ( m + nm + λn α m d (1 − α ) ) Taking m = n − α d (1 − α ) , we get : E ( e − λ P x ; Lx ; n> L αx ; n { X n =0 } ) ≥ e − cn η , with η = d + α (2 − d ) d (1 − α )+2 and for all α ∈ [0 , . Hence, we have proved :
Proposition 3.6.
For all α ∈ [0 , , Q a.s on |C| = + ∞ and for large enough n , E ( e − λ P x ; Lx ; n> L αx ; n ) (cid:23) e − cn η , with η = d + α (2 − d ) d (1 − α )+2 . Thus, the first assertion of Theorem 1.3 comes from proposition 3.4 and propo-sition 3.6.3.2.4.
Study of E ω ( Q L − αz ; n ) . We assume α > / . Upper bound
For this functional, one can take for all x ∈ C , B x = L = ( Z , E ) ( if we take some L r , we get the same bound). We have : F ol L ( k ) = 2 k. We still use a random walk Y which jumps can be represented by: n−1 n n+1 Let P L be the law of random walk Y . As before, let
W, W ′ , W n and W ′ n be the graphs defined respectively by (26) (30)(32) and (33) with D F = L = ( Z , E ). With the help of proposition 1.4 in [6] andgeneral properties of isoperimetry on wreath product, we deduce:for all γ >
0, there exists c, β > Q a.s on |C| = + ∞ we have : F ol WW n ( k ) (cid:23) ( k k if k < cn γ ,k βk d if k ≥ cn γ . (43)With the same argument as in the upper bound of section 3.2.3, we carry (43)to F ol W ′ W ′ n by rough isometry between graphs W n and W ′ n . We get:for all γ >
0, there exists β > c > , Q a.s on |C| = + ∞ we have: F ol W ′ W ′ n ( k ) (cid:23) ( k k if k < cn γ ,k βk d if k ≥ cn γ . (44)In order to get an upper bound of ˜ P ωo ( Z n = o ), let again: τ n = inf { s ≥ Z s V ( W ′ n ) } . We still have ˜ P ωo ( Z n = o ) = ˜ P ωo ( Z n = o and τ n > n ). We use the same way toget the upper bound from (44).Inequality (44) implies: ∀ k ≥ F ol W ′ W ′ n ( k ) (cid:23) J N ( k ) = ( k < cn γ , N βd ′ k d if k ≥ cn γ , (45)where N ≤ cn γ . J N is increasing and we can compute J − N : J − N = inf { x ; J N ( x ) ≥ y } = cn γ if 1 ≤ y < N cn dγ ,c (cid:16) ln( y )ln( N ) (cid:17) /d if N cn dγ ≤ y. Remark 3.7.
Let J ( k ) = ( k k if k < cn γ ,k βk d if k ≥ cn γ . (46) Inequality (44) can be read
F ol W ′ W ′ n ( k ) (cid:23) J ( k ) . J is increasing but the form of J does not enable us to compute an inverse and for this reason we use J N for thelower bound of F ol W ′ W ′ n ( k ) instead of J . C and L have bounded valency so we still have inf V ( W ′ ) ˜ m ≥ > inf E ( W ′ ) ˜ a > . Thus with the same tools as in section 3.2.3 we get, there exists constants c , c and c > P ωo ( Z n = o and τ n > n ) (cid:22) u ( n )where u is solution of the differential equation: ( u ′ = − uc ( J − N ( c /u )) ,u (0) = 1 / . Solving this equation, we obtain: u ( t ) = ( e − ct/n γ if t ≤ t := cn γ ( d +2) ln( N ) ,e − ( c (ln( N ) /d ( t − t )+ln(1 /u ( t )) d +2 d ) dd +2 if t > t . Chosing γ < d +2 and taking N = cn γ , we obtain in t = n : Q a.s on the event |C| = + ∞ and for large enough n ,˜ P ωo ( Z n = o ) (cid:22) e − n dd +2 ln( n ) d +2 . So with proposition 3.1, we deduce :
Proposition 3.8.
There exists a constant
C > such that Q a.s on |C| = + ∞ and for large enough n , E ω ( Y x ; L x ;2 n > P L ( Y L x ;2 n = 0) 1 { X n =0 } ) ≤ e − Cn dd +2 ln( n ) d +2 . (47)For the walk Y, we know that there exists c > P L ( Y n = 0) ∼ c n / . In particular, ∃ c > ∀ n ≥ P L ( Y n = 0) ≥ c n α , (48)with c ≤ . So, for α > / A > c > ∀ n ≥ P L ( Y n = 0) ≥ ( n α if n ≥ A, c n α if n < A, (49)with c ≤
1. If we directly use the lower bound (49) in (47) at time L x ;2 n , it appearsa supplementary factor c { x ; 0
1. For the term corresponding to A , we write : E ω ( Y x ; L x ;2 n > L − αx ;2 n { X n =0 } A ) = E ω ( Y x ; L x ;2 n > c L αx ;2 n × Y x ; L x ;2 n > c × { X n =0 } A ) ≤ E ω ( Y x ; L x ;2 n > P L ( Y L x ;2 n = 0) × ( 1 c ) N n × { X n =0 } A )(par 48) ≤ E ω ( Y x ; L x ;2 n > P L ( Y L x ;2 n = 0)1 { X n =0 } ) × ( 1 c ) ε n dd +2 ln( n ) d +2 ≤ e − ( C + ε ln( c )) n dd +2 ln( n ) d +2 . (by proposition 3.8)Now, choosing ε small enough ( recall that ln( c ) ≤ C > Q a.s on |C| = + ∞ , we have, E ω ( Y x ; L x ;2 n > L − αx ;2 n { X n =0 } A ) ≤ e − C n dd +2 ln( n ) d +2 . (51)2. For the second term, we notice that on the event A the product Q x ; L x ;2 n > L − αx ;2 n is less than (1 / ε n dd +2 ln( n ) d +2 . Thus there exists a constant C > E ω ( Y x ; L x ;2 n > L − αx ;2 n { X n =0 } A ) ≤ e − C n dd +2 ln( n ) d +2 . (52)3. For the last term, we use the following lemma: Lemma 3.9.
There exists ε ′ > such that for all ε > , there exists a constant C > such that, for all n, N ≥ , P ω ( N n ≥ εN et N n, ≤ ε ′ N ) ≤ e − C N . (53) Proof. • Let τ = 0 and for k ≥ τ k = min { s ≥ τ k − ; X s
6∈ { X , X , ..., X s − } } . The τ k represent instants when the walk X visits a new point. Consider now,the variables ǫ k defined by : ǫ k = ( X τ k = X τ k +2 , ǫ k is equal to 1 only whenthe new visited point X τ k is immediatly re visited after a backward and foward.The ǫ k are not independent but their laws are all some Bernouilli with differentparameters. Besides, these parameters have a same lower bound δ >
0, sincethe graph C g has bounded valency. • Consider the following filtrations, G m = σ ( X j ; 0 ≤ j ≤ m ) , F m = σ ( X j ; 0 ≤ j ≤ τ m ) .ǫ k are G τ k measurable and so F k +2 measurable. For all λ > L >
0, we can write, E ω ( e − λ P Lk =1 ǫ k ) = E ω ( e − λ P L − k =1 ǫ k E ω ( e − λ ( ǫ L − + ǫ L ) |F L ) ) ≤ E ω ( e − λ P L − k =1 ǫ k E ω ( e − λǫ L |F L ) ) . (55) • For the term E ω ( e − λǫ L |F L ) , we have: E ω ( e − λǫ L |F L ) = e − λ P ω ( ǫ L = 1 |F L ) + P ω ( ǫ L = 0 |F L )= 1 + ( e − λ − P ω ( ǫ L = 1 |F L ) . (56) Now, we want a lower bound of P ω ( ǫ L = 1 |F L ). We have successively: P ω ( ǫ L = 1 |F L ) = P ω ( ǫ L = 1 | X τ L )( Markov property)= X x ; P ω ( X τL = x ) > { X τL = x } P ω ( ǫ L = 1 | X τ L = x ) ≥ δ . (57) Last inequality comes from the fact that the graph C g has bounded valency,so in each point x the probability to do a backward and foward is greater than δ (with δ ≥ / d ). • So, we deduce from (56) and (57) that, E ω ( e − λǫ L |F L ) ≤ e − λ − δ . Iterating (55), we get, E ω ( e − λ P Lk =1 ǫ k ) ≤ (1 + ( e − λ − δ ) ⌊ L/ ⌋ , (58) where ⌊ a ⌋ stands for the whole number portion of a . Let: a λ = − ln(1 + ( e − λ − δ ) > . By Bien-aym´e inequality, we deduce, P ω ( L X k =1 ǫ k ≤ ε ′ L ) ≤ e ε ′ λL − a λ ⌊ L/ ⌋ . Using ⌊ L/ ⌋ for L ≥ , L ≤
3, we get : P ω ( L X k =1 ǫ k ≤ ε ′ L ) ≤ e −⌊ L/ ⌋ ( a λ − λε ′ ) . Note that this last inequality is still valid for L = 1.Fix λ >
0, ( by example λ = 1) then we can choose ε ′ small enough such that a λ − ε ′ >
0. We deduce the existence of constant b such that : P ω ( L X k =1 ǫ k ≤ ε ′ L ) ≤ e − bL . (59) • Now, notice that { N n ≥ εN et N n, ≤ ε ′ N } ⊂ { εN X k =1 ǫ k ≤ ε ′ N } . Indeed, first if N n ≥ εN that means that at least εN new points have beenvisited. Secondly if there are less than ε ′ N points visited more than twicethen there are less than ε ′ N points which have been immediatly visited aftertheir first visit. Finaly we have: P ω ( N n ≥ εN and N n, ≤ ε ′ N ) ≤ e − εbN . (cid:3) We can now get an upper bound of the term corresponding to A The productis less than 1, so we can write: E ω ( Y x ; L x ;2 n > L − αx ;2 n { X n =0 } A ) ≤ P ( A )Let ε small enough satisfying the first point (event A ), lemma 3.9 with ε = ε give us the existence of ε ′ such that (53). Then we take ε = ε ′ and we deducethere exists a constant C > P ω ( A ) ≤ e − C n dd +2 ln( n ) d +2 . So, E ω ( Y x ; L x ;2 n > L − αx ;2 n { X n =0 } A ) ≤ e − C n dd +2 ln( n ) d +2 . (60)Finaly, we deduce from (51) (52) and (60), the following property. Proposition 3.10. Q a.s on |C| = + ∞ and for large enough n , for all α > / , E ω ( Y x ; L x ;2 n > L αx ;2 n { X n =0 } ) (cid:22) e − n dd +2 ln( n ) d +2 . To get the upper bound of the second point of Theorem 1.3, it remains to takeout the indicatrice 1 { X n =0 } . We use the same way as in the section 3.2.3. Weprove: Lemma 3.11.
For all m ≥ , we have: P ω ( X x ln( L x ; n ) = m ) ≤ d (2 m + 1) d P ω ( X x ln( L x ;2 n ) ≤ m et X n = 0) . The proof is similar to lemma 3.3. We use in particular :ln( L x ;2 n ) ≤ ln( L x ; n + L x ;[ n ;2 n ] ) ≤ ln( L x ; n ) + ln( L x ;[ n ;2 n ] ) . Then E ω ( Y x L − αx ;2 n { X n =0 } ) = E ω ( e − α P x ln( L x ;2 n ) { X n =0 } )= X m ≥ e − αm P ω ( X x ln( L x ;2 n ) = m ; X n = 0)= (1 − e − α ) X m ≥ e − αm P ω ( X x ln( L x ;2 n ) ≤ m ; X n = 0) . ≥ (1 − e − α ) X m ≥ e − αm P ω ( X x ln( L x ;2 n ) ≤ m ; X n = 0) ≥ (1 − e − α ) X m ≥ d (2 m + 1) d e − αm [ P ω ( X x ln( L x ; n ) = m )] (by lemma 3.11) ≥ X m ≥ e − α m [ P ω ( X x ln( L x ; n ) = m )] (for some α > α ) ≥ (cid:16) X m ≥ e − α m (cid:17) − × (cid:16) X m ≥ e − α m P ω ( X x ln( L x ; n ) = m ) (cid:17) (by Cauchy-Schwarz inequality) ≥ c E ω [ e − α P x ln( L x ; n ) ]= c E ω ( Y x L − α x ; n ) . So, with this last inequality and with proposition 3.10, we obtain the expectedupper bound for some value α : E ω ( Y x L − α x ; n ) (cid:22) e − n dd +2 ln( n ) d +2 . (61)From this inequality at point α , we extend this relation for all α > /
2. Let α > / α ≥ α , we can replace in (61) α by α , by monotony in α .-If α < α , we write E ω [ e − α P x ; Lx ; n> ln( L x ; n ) ] = E ω [( e − α P x ; Lx ; n> ln( L x ; n ) ) αα ] ≤ ( E ω [ e − α P x ; Lx ; n> ln( L x ; n ) ]) αα ( Jensen inequality to concave function x → x αα . ) (cid:22) e − n η . So we have proved : Proposition 3.12. Q a.s on the set |C| = + ∞ and for large enough n , for all α > / , E ω ( Y x ; L x ; n > L αx ; n ) (cid:22) e − n dd +2 ln( n ) d +2 . Lower bound
By concavity of the function ln, we get: Y z ; L z ; n > L − αz ; n = e − αN n P z ; Lz ; n> Nn ln ( L z ; n ) ≥ e − αN n ln ( P z ; Lz ; n> Lz ; nNn ) = e − αN n ln ( nNn ) . On the event { sup ≤ i ≤ n D (0 , X i ) ≤ m } , it comes that : N n ≤ | B m ( C ) | ≤ c m d , and nN n ≥ ncm d . Since function x ln ( x ) x is decreasing on [ e, + ∞ ], if we choose m such that ncm d ≥ e, (62)then we can write : E ( Y x ; L x ; n > L − αx ; n ) ≥ e − αcm d ln ( ncmd ) P ( sup ≤ i ≤ n | X i | ≤ m ) . Then by using (42), we deduce : E ( Y x ; L x ; n > L − αx ; n ) ≥ e − αcm d ln ( ncmd ) e − c ( m + nm ) . Taking m = ( nln ( n ) ) d +2 , inquality (62) is well satisfied for large enough n .Finaly, for large enough n we obtain, E ( Y x ; L x ; n > L − αx ; n ) (cid:23) e − n dd +2 ln ( n ) d +2 . So,
Proposition 3.13.
For all α > / , Q a.s on the set |C| = + ∞ and for largeenough n , E ( Y x ; L x ; n > L − αx ; n ) (cid:23) e − n dd +2 ln ( n ) d +2 . Remark 3.14.
In the proof of the lower bound, we have only used the assumptionthat α ≥ , so this bound is valid for all α ≥ . So the second assertion of Theorem 1.3 follows from propositions 3.12 and 3.13.
Acknowledgment
The author would like to thank Pierre Mathieu and Anna Erschler for usefullremarks.
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