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El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y} Vol. 15 (2010), Paper no. 21, pages 654–683.

Journal URL

http://www.math.washington.edu/~ejpecp/

On the critical point of the Random Walk Pinning Model in

dimension

d

=

3

∗

Quentin Berger

Laboratoire de Physique, ENS Lyon Université de Lyon

46 Allée d’Italie, 69364 Lyon, France e-mail: quentin.berger@ens.fr

Fabio Lucio Toninelli

CNRS and Laboratoire de Physique, ENS Lyon Université de Lyon

46 Allée d’Italie, 69364 Lyon, France e-mail: fabio-lucio.toninelli@ens-lyon.fr

Abstract

We consider the Random Walk Pinning Model studied in[3]and[2]: this is a random walkXon

Zd_{, whose law is modified by the exponential of}β timesLN(X,Y), the collision local time up to

timeNwith the (quenched) trajectoryY of anotherd-dimensional random walk. Ifβexceeds a certain critical valueβ_c, the two walks stick together for typicalY realizations (localized phase). A natural question is whether the disorder is relevant or not, that is whether thequenched and annealed systems have the same critical behavior. Birkner and Sun[3]proved thatβ_c coincides with the critical point of theannealed Random Walk Pinning Model if the space dimension is d=1 ord=2, and that it differs from it in dimensiond_≥4 (ford_≥5, the result was proven also in[2]). Here, we consider the open case of themarginal dimensiond=3, and we prove non-coincidence of the critical points.

Key words:Pinning Models, Random Walk, Fractional Moment Method, Marginal Disorder. AMS 2000 Subject Classification:Primary 82B44, 60K35, 82B27, 60K37.

Submitted to EJP on December 15, 2009, final version accepted April 11, 2010.

1

Introduction

We consider the Random Walk Pinning Model (RWPM): the starting point is a zero-drift random walkX on Zd (d _≥1), whose law is modified by the presence of a second random walk, Y. The trajectory of Y is fixed (quenched disorder) and can be seen as the random medium. The modifi-cation of the law of X due to the presence of Y takes the Boltzmann-Gibbs form of the exponen-tial of a certain interaction parameter, β, times the collision local time of X andY up to time N,

LN(X,Y):=

P

1≤n≤N1{Xn=Yn}. Ifβ exceeds a certain threshold valueβc, then for almost every

real-ization ofY the walkX sticks together withY, in the thermodynamic limitN _{→ ∞}. If on the other handβ < βc, then LN(X,Y)iso(N)for typical trajectories.

Averaging with respect to Y the partition function, one obtains the partition function of the so-called annealed model, whose critical pointβ_cannis easily computed; a natural question is whether

βc 6= βcann or not. In the renormalization group language, this is related to the question whether

disorder isrelevantor not. In an early version of the paper[2], Birkneret al. proved thatβ_c6=βann c

in dimensiond_≥5. Around the same time, Birkner and Sun[3]extended this result tod=4, and also proved that the two critical pointsdo coincidein dimensionsd=1 andd=2.

The dimensiond=3 is themarginal dimensionin the renormalization group sense, where not even heuristic arguments like the “Harris criterion” (at least its most naive version) can predict whether one has disorder relevance or irrelevance. Our main result here is that quenched and annealed critical points differ also ind=3.

For a discussion of the connection of the RWPM with the “parabolic Anderson model with a single catalyst”, and of the implications ofβc 6=βcann about the location of the weak-to-strong transition

for the directed polymer in random environment, we refer to[3, Sec. 1.2 and 1.4].

Our proof is based on the idea of bounding the fractional moments of the partition function, together with a suitable change of measure argument. This technique, originally introduced in[6; 9; 10]for the proof of disorder relevance for the random pinning model with tail exponentα≥1/2, has also proven to be quite powerful in other cases: in the proof of non-coincidence of critical points for the RWPM in dimension d ≥ 4 [3], in the proof that “disorder is always strong” for the directed polymer in random environment in dimension(1+2)[12]and finally in the proof that quenched and annealed large deviation functionals for random walks in random environments in two and three dimensions differ[15]. Let us mention that for the random pinning model there is another method, developed by Alexander and Zygouras [1], to prove disorder relevance: however, their method fails in the marginal situationα=1/2 (which corresponds tod=3 for the RWPM). To guide the reader through the paper, let us point out immediately what are the novelties and the similarities of our proof with respect to the previous applications of the fractional moment/change of measure method:

• the change of measure chosen by Birkner and Sun in [3] consists essentially in correlating positively each increment of the random walk Y with the next one. Therefore, under the modified measure,Y is more diffusive. The change of measure we use in dimension three has also the effect of correlating positively the increments of Y, but in our case the correlations have long range (the correlation between theithand thejthincrement decays like_|i₋j_|−1/2). Another ingredient which was absent in[3]and which is essential ind=3 is a coarse-graining step, of the type of that employed in[14; 10];

• while the scheme of the proof of our Theorem 2.8 has many points in common with that of

[10, Th. 1.7], here we need new renewal-type estimates (e.g. Lemma 4.7) and a careful application of the Local Limit Theorem to prove that the average of the partition function under the modified measure is small (Lemmas 4.2 and 4.3).

2

Model and results

2.1

The random walk pinning model

Let X =_{Xn}n≥0 andY ={Yn}n≥0 be two independent discrete-time random walks onZd, d ≥1,

starting from 0, and letPX _andPY _{denote their respective laws. We make the following assumption:}

Assumption 2.1. The random walkX is aperiodic. The increments(Xi−Xi−1)i≥1 are i.i.d.,

sym-metric and have a finite fourth moment (EX_kX1k4_{<∞, wherek · kdenotes the Euclidean norm}

onZd). Moreover, the covariance matrix ofX₁, call itΣ_X, is non-singular.

The same assumptions hold for the increments ofY (in that case, we callΣ_Y the covariance matrix ofY₁).

For β ∈ R_{,N ∈} N_{and for a fixed realization of Y we define a Gibbs transformation of the path}

measurePX_{: this is the polymer path measure}Pβ

N,Y, absolutely continuous with respect toPX, given

by

dPβ

N,Y

dPX (X) =

eβLN(X,Y)₁

{XN=YN} Z_Nβ_,Y

, (1)

where L_N(X,Y) =

N

P

n=1 1_{X

n=Yn}, and where

Z_Nβ_,Y =EX[eβLN(X,Y)₁

{XN=YN}] (2)

is the partition function that normalizesPβ

N,Y to a probability.

Thequenchedfree energy of the model is defined by

F(β):= lim

N→∞

1

N logZ

β

N,Y =_N→∞lim

1

NE

Y_[logZβ

N,Y] (3)

(the existence of the limit and the fact that it isPY-almost surely constant and non-negative is proven in[3]). We define also theannealedpartition functionEY[Z_Nβ_,Y], and theannealedfree energy:

Fann(β):= lim

N→∞

1

N logE

Y_[Zβ

N,Y]. (4)

We can compare thequenchedandannealedfree energies, via the Jensen inequality:

F(β) = lim

N→∞

1

NE

Y_[logZβ

N,Y]6 _Nlim_→∞

1

NlogE

Y_[Zβ

N,Y] =F

ann_{(β). (5)}

The properties ofFann(_·)are well known (see the Remark 2.3), and we have the existence of critical points[3], for bothquenchedandannealedmodels, thanks to the convexity and the monotonicity of

Definition 2.2(Critical points). There exist06β_cann6βc depending on the laws of X and Y such

that: Fann(β) =0if β 6β_cann and Fann(β)> 0ifβ > β_cann; F(β) =0if β 6βc and F(β) >0if

β > β_c.

The inequalityβ_cann6βc comes from the inequality (5).

Remark 2.3. As was remarked in[3], theannealedmodel is just the homogeneous pinning model

[8, Chapter 2]with partition function

EY_[Zβ

N,Y] =EX−Y

 exp β

N

X

n=1

1_{(X−Y)n=0} !

1_{(X−Y)N=0}  

which describes the random walkX₋Y which receives the rewardβ each time it hits 0. From the well-known results on the homogeneous pinning model one sees therefore that

• Ifd =1 or d =2, theannealedcritical pointβann

c is zero because the random walk X−Y is

recurrent.

• Ifd_≥3, the walkX₋Y is transient and as a consequence

β_cann=₋log”1₋PX−Y (X −Y)_n6=0 for every n>0—>0.

Remark 2.4. As in the pinning model[8], the critical pointβ_c marks the transition from a delocal-ized to a localdelocal-ized regime. We observe that thanks to the convexity of the free energy,

∂βF(β) = lim

N→∞

Eβ

N,Y

 1

N

N

X

n=1

1_{XN=YN} 

, (6)

almost surely in Y, for every β such that F(_·) is differentiable at β. This is the contact fraction betweenX andY. Whenβ < β_c, we haveF(β) =0, and the limit density of contact betweenX and

Y is equal to 0: Eβ

N,Y

PN

n=11{XN=YN} = o(N), and we are in the delocalized regime. On the other

hand, ifβ > βc, we haveF(β)>0, and there is a positive density of contacts betweenX andY: we

are in the localized regime.

2.2

Review of the known results

The following is known about the question of the coincidence of quenched and annealed critical points:

Theorem 2.5. [3]Assume that X and Y are discrete time simple random walks onZd_. If d=1or d=2, the quenched and annealed critical points coincide:βc=βcann=0.

If d_≥4, the quenched and annealed critical points differ:βc> βcann>0.

Actually, the result that Birkner and Sun obtained in[3]is valid for slightly more general walks than simple symmetric random walks, as pointed out in the last Remark in [3, Sec.4.1]: for instance, they allow symmetric walksX andY with common jump kernel and finite variance, provided that

PX(X₁=0)_≥1/2.

In dimensiond ≥5, the result was also proven (via a very different method, and for more general random walks which include those of Assumption 2.1) in an early version of the paper[2].

Remark 2.6. The method and result of[3]in dimensions d =1, 2 can be easily extended beyond the simple random walk case (keeping zero mean and finite variance). On the other hand, in the case d ≥4 new ideas are needed to make the change-of-measure argument of [3]work for more general random walks.

Birkner and Sun gave also a similar result ifX andY are continuous-time symmetric simple random walks onZd, with jump rates 1 andρ≥0 respectively. With definitions of (quenched and annealed) free energy and critical points which are analogous to those of the discrete-time model, they proved:

Theorem 2.7. [3]In dimension d =1and d=2, one hasβc=βcann=0. In dimensions d≥4, one

has0< β_cann< βc for eachρ >0. Moreover, for d =4and for eachδ >0, there exists aδ >0such

thatβc−βcann≥aδρ1+δ for allρ∈[0, 1]. For d≥5, there exists a>0such thatβc−βcann≥aρfor

allρ∈[0, 1].

Our main result completes this picture, resolving the open case of the critical dimensiond=3 (for simplicity, we deal only with the discrete-time model).

Theorem 2.8. Under the Assumption 2.1, for d=3, we haveβc> βcann.

We point out that the result holds also in the case whereX (orY) is a simple random walk, a case which a priori is excluded by the aperiodicity condition of Assumption 2.1; see the Remark 2.11. Also, it is possible to modify our change-of-measure argument to prove the non-coincidence of quenched and annealed critical points in dimensiond=4 for the general walks of Assumption 2.1, thereby extending the result of[3]; see Section 4.4 for a hint at the necessary steps.

NoteAfter this work was completed, M. Birkner and R. Sun informed us that in[4]they indepen-dently proved Theorem 2.8 for the continuous-time model.

2.3

A renewal-type representation for

Z

_Nβ_,Y

From now on, we will assume thatd≥3.

As discussed in[3], there is a way to represent the partition function Z_Nβ_,Y in terms of a renewal processτ; this rewriting makes the model look formally similar to the random pinning model[8]. In order to introduce the representation of[3], we need a few definitions.

Definition 2.9. We let

1. pX_n(x) =PX(X_n= x)and p_nX−Y(x) =PX−Y (X₋Y)_n=x;

2. Pbe the law of a recurrent renewal τ =_{τ0,τ1, . . .} withτ0 =0, i.i.d. increments and

inter-arrival law given by

K(n):=P(τ1=n) =

pX_n−Y(0)

GX−Y where G

X−Y _:=X∞ n=1

p_nX−Y(0) (7)

(note that GX−Y <∞in dimension d _≥3); 3. z′= (eβ₋1)and z=z′GX−Y;

4. for n_∈Nand x_∈Zd,

w(z,n,x) =z p

X n(x)

pX−Y n (0)

; (8)

5. Zˇ_Nz_,Y := z′

1+z′Z

β

N,Y.

Then, via the binomial expansion ofeβLN(X,Y)_{= (1+z}′₎LN(X,Y)_{one gets[3]}

ˇ

Z_Nz_,Y =

N

X

m=1

X

τ0=0<τ1<...<τm=N

m

Y

i=1

K(τ_i−τ_i−1)w(z,τi−τi−1,Yτi−Yτi−1) (9)

= EW(z,τ∩ {0, . . . ,N},Y)1N∈τ

,

where we defined for any finite increasing sequences=_{s0,s1, . . . ,sl}

W(z,s,Y) =

EX• Ql

n=1z1{X_sn=Y_sn}

X_s

0=Ys0

˜

EX−Y• Ql

n=11{X_sn=Y_sn}

Xs0=Ys0

˜=

l

Y

n=1

w(z,s_n−s_n−1,Y_sn−Y_sn

−1). (10)

We remark that, taking theEY₋expectation of the weights, we get

EY”_{w(z,τi−τi−1,Yτ}

i −Yτi−1)

— =z.

Again, we see that theannealedpartition function is the partition function of a homogeneous pinning model:

ˇ

Z_Nz,_,ann_Y =EY[Zˇ_Nz_,Y] =E”zRN₁

{N∈τ}

—

, (11)

where we definedRN :=|τ∩ {1, . . . ,N}|.

Since the renewalτis recurrent, theannealedcritical point isz_cann=1.

In the following, we will often use the Local Limit Theorem for random walks, that one can find for instance in[5, Theorem 3](recall that we assumed that the increments of bothX andY have finite second moments and non-singular covariance matrix):

Proposition 2.10(Local Limit Theorem). Under the Assumption 2.1, we get PX_{(Xn=x) =} 1

(2πn)d/2_(detΣ

X)1/2

exp

− 1

2nx· €

Σ−_X1xŠ+o(n−d/2), (12)

where o(n−d/2)is uniform for x ∈Zd_.

Moreover, there exists a constant c>0such that for all x∈Zd _{and n∈}N

PX(Xn= x)6cn−d/2. (13)

(We use the notation x_· y for the canonical scalar product inRd.)

In particular, from Proposition 2.10 and the definition of K(_·) in (7), we get K(n) _∼ cKn−d/2 as

n_{→ ∞}, for some positivec_K. As a consequence, ford=3 we get from[7, Th. B]that

P(n∈τ)n→∞_∼ 1

2πcKpn

. (14)

Remark 2.11. In Proposition 2.10, we supposed that the walkX is aperiodic, which is not the case for the simple random walk. IfXis the symmetric simple random walk onZd_{, then[13, Prop. 1.2.5]}

PX(X_n=x) =1_{n↔x} 2

(2πn)d/2_(detΣ

X)1/2

exp

− 1

2nx· €

Σ−_X1xŠ+o(n−d/2), (15)

where+o(n−d/2)is uniform for x∈Zd_{, and wheren↔x means thatnandx have the same parity}

(so thatx is a possible value forX_n). Of course, in this caseΣ_X is just 1/dtimes the identity matrix. The statement (13) also holds.

Via this remark, one can adapt all the computations of the following sections, which are based on Proposition 2.10, to the case whereX (orY) is a simple random walk. For simplicity of exposition, we give the proof of Theorem 2.8 only in the aperiodic case.

3

Main result: the dimension

d

=

3

With the definition ˇF(z):= lim_N→∞ 1

Nlog ˇZ z

N,Y, to prove Theorem 2.8 it is sufficient to show that

ˇ

F(z) =0 for somez>1.

3.1

The coarse-graining procedure and the fractional moment method

We consider without loss of generality a system of size proportional toL= _z1

−1 (the coarse-graining

length), that isN=mL, withm_∈N. Then, for_{I ⊂ {}1, . . . ,m_}, we define

Z_zI_,Y :=EW(z,τ∩ {0, . . . ,N_},Y)1_N∈τ1_E

I(τ)

, (16)

where E_I is the event that the renewalτintersects the blocks (B_i)_i∈I and only these blocks over

{1, . . . ,N_}, B_i being theithblock of sizeL:

B_i:=_{(i₋1)L+1, . . . ,i L_}. (17) Since the eventsE_I are disjoint, we can write

ˇ

Z_Nz_,Y := X

I ⊂{1,...,m}

Z_zI_,Y. (18)

Note that Z_zI_,Y = 0 if m ∈ I/ . We can therefore assume m_{∈ I}. If we denote _I = _{i₁,i₂, . . . ,i_l}

Z_zI_,Y := X

a1,b1∈Bi1

a16b1

X

a2,b2∈Bi2

a26b2

. . . X

al∈B_il

K(a1)w(z,a1,Ya1)Z

z

a1,b1 (19)

. . .K(a_l−b_l−1)w(z,al−bl−1,Yal −Ybl−1)Z

z al,N,

where

Zz_j,k:=EW(z,τ∩ {j, . . . ,k},Y)1k∈τ

j∈τ (20)

is the partition function between j andk.

0 L 2L 3L 4L 5L 6L 7L 8L=N

a₁ b₁ a₂b₂ a₃ b₃ a₄ b₄=N

Figure 1: The coarse-graining procedure. Here N = 8L (the system is cut into 8 blocks), and

I =_{2, 3, 6, 8_} (the gray zones) are the blocks where the contacts occur, and where the change of measure procedure of the Section 3.2 acts.

Moreover, thanks to the Local Limit Theorem (Proposition 2.10), one can note that there exists a constantc>0 independent of the realization ofY such that, if one takesz62 (we will takezclose to 1 anyway), one has

w(z,τi−τi−1,Yτi−Yτi−1) =z p_τX

i−τi−1(Yτi−Yτi−1) pX_τ−Y

i−τi−1(0)

≤c.

So, the decomposition (19) gives

Z_zI_,Y 6c|I | X

a1,b1∈Bi1

a16b1

X

a2,b2∈Bi2

a26b2

. . . X

al∈B_il

K(a₁)Z_az

1,b1K(a2−b1)Z

z

a2,b2. . .K(al−bl−1)Z

z

al,N. (21)

We now eliminate the dependence onz in the inequality (21). This is possible thanks to the choice

L= 1

z−1. As each Z

z

ai,bi is the partition function of a system of size smaller than L, we getW(z,τ∩

{a_i, . . . ,b_i},Y) 6zLW(z = 1,τ∩ {a_i, . . . ,b_i},Y) (recall the definition (10)). But with the choice

L= 1

z−1, the factorz

L_{is bounded by a constantc, and thanks to the equation (20), we finally get}

Z_az

i,bi 6c Z

z=1

ai,bi. (22)

Notational warning: in the following, c,c′, etc. will denote positive constants, whose value may change from line to line.

We note Z_ai,bi := Z_az=1

i,bi and W(τ,Y) := W(z = 1,τ,Y). Plugging this in the inequality (21), we

finally get

Z_zI_,Y 6c′|I | X

a1,b1∈Bi1

a16b1

X

a2,b2∈Bi2

a26b2

. . . X

al∈B_il

where there is no dependence onzanymore.

The fractional moment method starts from the observation that for anyγ6=0

ˇ

F(z) = lim

N→∞

1

γNE

Y •_logZˇz N,Y

γ˜

6 lim inf

N→∞

1

NγlogE

Y•_Zˇz N,Y

γ˜

. (24)

Let us fix a value ofγ∈(0, 1) (as in[10], we will chooseγ= 6/7, but we will keep writing it as

γto simplify the reading). Using the inequality Pa_nγ 6 Paγ_n (which is valid for a_{i ≥}0), and combining with the decomposition (18), we get

EY •

ˇ

Z_Nz_,Y㠘

6 X

I ⊂{1,...,m}

EY •

Z_zI_,Y㠘

. (25)

Thanks to (24) we only have to prove that, for somez>1, lim sup_N→∞EY •

ˇ

Z_Nz_,Y㠘

<∞. We deal with the termEY

h

(Z_zI_,Y)γivia a change of measure procedure.

3.2

The change of measure procedure

The idea is to change the measurePY on each block whose index belongs to_I, keeping each block independent of the others. We replace, for fixedI, the measurePY(dY)withg_I(Y)PY(dY), where the function g_I(Y) will have the effect of creating long range positive correlations between the increments ofY, inside each block separately. Then, thanks to the Hölder inequality, we can write

EY •

Z_zI_,Y㠘

=EY _g

I(Y)γ

g_I(Y)γ

Z_zI_,Yγ

6 EY h

g_I(Y)−

γ

1−γ

i1−γ

EY h

g_I(Y)Z_zI_,Yiγ. (26)

In the following, we will denote∆_i =Yi−Yi−1 theithincrement of Y. Let us introduce, for K>0

andǫK to be chosen, the following “change of measure”:

g_I(Y) =Y

k∈I

(1_F

k(Y)6K+ǫK1Fk(Y)>K)≡ Y

k∈I

gk(Y), (27)

where

F_k(Y) =₋ X

i,j∈Bk

M_{i j}∆_i·∆_j, (28)

and _

 

M_{i j}= p 1

LlogL

1

Æ

|j−i_| if i6= j

M_ii=0.

(29)

Let us note that from the form of M, we get that kMk2 := P_i,j∈B

1M

2

i j 6 C, where the constant

C <∞ does not depend on L. We also note that F_k only depends on the increments of Y in the block labeledk.

Let us deal with the first factor of (26):

EY h

g_I(Y)−

γ

1−γ

i

=Y

k∈I

EY h

g_k(Y)−

γ

1−γ

i =

PY_(F1(Y)6K) +ǫ−

γ

1−γ

K P

Y_(F

1(Y)>K)

|I |

. (30) We now choose

ǫK :=PY(F1(Y)>K)

1−γ

γ ₍₃₁₎

such that the first factor in (26) is bounded by 2(1−γ)|I |62|I |. The inequality (26) finally gives

EY •

Z_zI_,Y γ˜

62|I |EY h

g_I(Y)Z_zI_,Y iγ

. (32)

The idea is that whenF1(Y) is large, the weight g1(Y) in the change of measure is small. That is

why the following lemma is useful:

Lemma 3.1. We have

lim

K→∞lim supL→∞

ǫK = lim

K→∞lim supL→∞

PY(F₁(Y)>K) =0 (33)

Proof. We already know thatEY[F1(Y)] =0, so thanks to the standard Chebyshev inequality, we

only have to prove thatEY_[F1(Y)2_{]is bounded uniformly inL. We get} EY[F₁(Y)2] = X

i,j∈B1

k,l∈B1

M_{i j}M_klEY”(∆_i·∆_j)(∆_k·∆_l)—

= X

{i,j}

M_{i j}2EY”(∆_i·∆_j)2— (34)

where we used thatEY”_{(∆i·∆j)(∆k·∆l)}—_{= 0 if {i,j} 6= {k,l}. Then, we can use the}

Cauchy-Schwarz inequality to get

EY[F1(Y)2]6

X

{i,j}

M_{i j}2EYh∆_i2∆_j2i 6_kMk2σ4_Y :=_kMk2”EY(_||Y1||2)

—2

. (35)

We are left with the estimation ofEY h

g_I(Y)Z_zI_,Yi. We setP_I :=P E_I,N_∈τ, that is the prob-ability forτ to visit the blocks (B_i)_i∈I and only these ones, and to visit also N. We now use the following two statements.

Proposition 3.2. For anyη >0, there exists z > 1sufficiently close to1(or L sufficiently big, since L= (z₋1)−1) such that for every_{I ⊂ {}1, . . . ,m_}with m_{∈ I}, we have

EY h

g_I(Y)Z_zI_,Yi 6η|I |P_I. (36)

Lemma 3.3. [10, Lemma 2.4] There exist three constants C₁ = C₁(L), C₂ and L₀ such that (with i0:=0)

P_I 6C₁C₂|I |

|I |

Y

j=1

1

(i_j−i_j−1)7/5 (37)

for L_≥ L₀and for every_{I ∈ {}1, . . . ,m_}.

Thanks to these two statements and combining with the inequalities (25) and (32), we get

EY •

ˇ

Z_Nz_,Y㠘

6 X

I ⊂{1,...,m}

EY •

Z_zI_,Y㠘

6C₁γ X

I ⊂{1,...,m} |I |

Y

j=1

(3C2η)γ

(i_j−i_j−1)7γ/5. (38)

Since 7γ/5=6/5>1, we can set

e

K(n) = 1 e

cn6/5, where ec=

+∞

X

i=1

i−6/5<+_∞, (39)

and Ke(_·) is the inter-arrival probability of some recurrent renewal τe. We can therefore interpret the right-hand side of (38) as a partition function of a homogeneous pinning model of sizem(see Figure 2), with the underlying renewalτ_e, and with pinning parameter log[_ec(3C2η)γ]:

EY •

ˇ

Z_Nz_,Y㠘

6C₁γEτ_e

h

ec(3C₂η)γ|τe∩{1,...,m}|i. (40)

0 1 2 3 4 5 6 7 8=m

Figure 2: The underlying renewalτeis a subset of the set of blocks(B_i)₁6i6m (i.e the blocks are

reinterpreted as points) and the inter-arrival distribution isKe(n) =1/€ecn6/5Š.

Thanks to Proposition 3.2, we can takeηarbitrary small. Let us fixη:=1/((4C2)ec1/γ). Then,

EY •

ˇ

Z_Nz_,Y γ˜

6C₁γ (41)

for everyN. This implies, thanks to (24), that ˇF(z) =0, and we are done.

Remark 3.4. The coarse-graining procedure reduced the proof of delocalization to the proof of Proposition 3.2. Thanks to the inequality (23), one has to estimate the expectation, with respect to the g_I(Y)₋modified measure, of the partition functions Zai,bi in each visited block. We will show

(this is Lemma 4.1) that the expectation with respect to this modified measure ofZ_ai,bi/P(bi−ai ∈τ)

can be arbitrarily small if Lis large, and ifb_i−a_i is of the order of L. If b_i−a_i is much smaller, we can deal with this term via elementary bounds.

4

Proof of the Proposition 3.2

As pointed out in Remark 3.4, Proposition 3.2 relies on the following key lemma:

Lemma 4.1. For everyǫandδ >0, there exists L>0such that

EY”_g1(Y)Za,b—6δP(b₋a_∈τ) (42)

for every a6b in B1such that b−a≥ǫL.

Given this lemma, the proof of Proposition 3.2 is very similar to the proof of[10, Proposition 2.3], so we will sketch only a few steps. The inequality (23) gives us

EY h

g_I(Y)Z_zI_,Y i 6 c|I | X

a1,b1∈Bi1

a16b1

X

a2,b2∈Bi2

a26b2

. . . X

al∈B_il

K(a1)EY

”

g_i1(Y)Z_a1,b1

—

K(a2−b1)EY

”

g_i2(Y)Z_a2,b2

—

. . .

. . .K(a_l−b_l−1)EY ”g_i

l(Y)Zal,N — = c|I | X

a1,b1∈Bi₁

a16b1

X

a2,b2∈Bi₂

a26b2

. . . X

al∈B_il

K(a₁)EY”g₁(Y)Z_a1−L(i1−1),b1−L(i1−1)

—

K(a₂₋b₁). . . (43)

. . .K(al−bl−1)EY

”

g1(Y)Zal−L(m−1),N−L(m−1) —

.

The terms with bi−ai ≥ ǫL are dealt with via Lemma 4.1, while for the remaining ones we just

observe thatEY[g1(Y)Za,b]≤P(b−a∈τ)since g1(Y)≤1. One has then

EY h

g_I(Y)Z_zI_,Yi 6 c|I | X

a1,b1∈Bi₁

a16b1

X

a2,b2∈Bi₂

a26b2

. . . X

al∈B_il

K(a₁)€δ+1_{b

1−a16ǫL}

Š

P(b₁₋a_1∈τ)

. . .K(a_l−b_l−1)

€

δ+1_{N−al6ǫL}

Š

P(N−a_l∈τ). (44) From this point on, the proof of Theorem 3.2 is identical to the proof of Proposition 2.3 in[10](one needs of course to chooseǫ=ǫ(η)andδ=δ(η)sufficiently small).

4.1

Proof of Lemma 4.1

Let us fixa,binB₁, such thatb₋a_≥ǫL. The small constantsδandǫare also fixed. We recall that for a fixed configuration ofτ such that a,b ∈τ, we have EY_W(τ∩ {a, . . . ,b},Y) = 1 because

z=1. We can therefore introduce the probability measure (always for fixedτ)

dP_τ(Y) =W(τ∩ {a, . . . ,b},Y)dPY(Y) (45) where we do not indicate the dependence ona andb. Let us note for later convenience that, in the particular casea=0, the definition (10) ofW implies that for any function f(Y)

E_τ[f(Y)] =EXEY_f(Y)_|Xi=Yi∀i∈τ∩ {1, . . . ,b}

With the definition (20) ofZ_a,b:=Z_az_,=_b1, we get

EY”_g1(Y)Za,b

—

=EY_Eg1(Y)W(τ∩ {a, . . . ,b},Y)1b∈τ|a∈τ

=bEE_τ[g1(Y)]P(b−a∈τ), (47)

wherebP(_·):=P(_·|a,b∈τ), and therefore we have to show thatbEE_τ[g1(Y)]6δ.

With the definition (27) of g1(Y), we get that for anyK

b

EE_τ[g₁(Y)]6ǫK+bEPτ F1<K

. (48)

If we chooseKbig enough,ǫK is smaller thanδ/3 thanks to the Lemma 3.1. We now use two lemmas

to deal with the second term. The idea is to first prove thatE_τ[F₁]is big with abP₋probability close to 1, and then that its variance is not too large.

Lemma 4.2. For every ζ > 0 and ǫ > 0, one can find two constants u = u(ǫ,ζ) > 0 and L0 =

L0(ǫ,ζ)>0, such that for every a,b∈B1such that b−a≥ǫL,

b

PE_τ[F₁]_≤uplogL_≤ζ, (49)

for every L_≥L₀.

Chooseζ=δ/3 and fixu>0 such that (49) holds for every Lsufficiently large. If 2K =uplogL

(and therefore we can makeǫK small enough by choosing Llarge), we get that

b

EP_{τ F1<K} 6 bEP_τF1−E_τ[F1]6 −K

+Pb E_τ[F1]62K

(50)

6 1

K2bEEτ ”

F1−Eτ[F1]

2—

+δ/3. (51)

Putting this together with (48) and with our choice ofK, we have

b

EE_τ[g1(Y)]62δ/3+ 4 u2logLEbEτ

”

F₁₋E_τ[F1]2— ₍₅₂₎

forL≥L0. Then we just have to prove thatEbEτ

”

F1−Eτ[F1]

2—

=o(logL). Indeed,

Lemma 4.3. For everyǫ >0there exists some constant c=c(ǫ)>0such that b

EE_τ” F₁₋E_τ[F₁]2— 6c logL3/4 (53)

for every L>1and a,b_∈B₁ such that b₋a_≥ǫL.

We finally get that

b

EE_τ[g1(Y)]62δ/3+c(logL)−1/4, (54)

and there exists a constantL1>0 such that forL>L1

b

4.2

Proof of Lemma 4.2

Up to now, the proof of Theorem 2.8 is quite similar to the proof of the main result in[10]. Starting from the present section, instead, new ideas and technical results are needed.

Let us fix a realization of τsuch that a,b∈τ(so that it has a non-zero probability underbP) and let us note τ∩ {a, . . .b_} = _{τRa = a,τRa+1, . . . ,τRb = b} (recall that Rn = |τ∩ {1, . . . ,n}|). We

observe (just go back to the definition of P_τ) that, if f is a function of the increments of Y in

{τn−1+1, . . . ,τn}, gof the increments in {τm−1+1, . . . ,τm}withRa <n6= m≤Rb, and ifhis a

function of the increments ofY not in_{a+1, . . . ,b_}then

E_{τf {}∆_i}i∈{τ

n−1+1,...,τn}

g {∆_i}i∈{τ

m−1+1,...,τm}

h {∆_{i}i∈{a/ +1,...,b}} (56)

= E_{τf {}∆_i}i∈{τ

n−1+1,...,τn}

E_{τg {}∆_i}i∈{τ

m−1+1,...,τm}

EY_{h {}∆_{i}i∈{/ a+1,...,b}}, and that

E_τf _{∆_i}i∈{τ

n−1+1,...,τn}

=EXEYf _{∆_i}i∈{τ

n−1+1,...,τn}

|X_τ

n−1=Yτn−1,Xτn=Yτn =EXEY_{f {∆i−τ}

n−1}i∈{τn−1+1,...,τn}

|Xτ_n−τ_n−1=Yτn−τn−1

. (57) We want to estimateE_τ[F1]: since the increments∆i fori∈B1\{a+1, . . . ,b}are i.i.d. and centered

(like underPY_{), we have}

E_τ[F₁]:=

b

X

i,j=a+1

M_{i j}E_τ[₋∆_i·∆_j]. (58)

Via a time translation, one can always assume thata=0 and we do so from now on. The key point is the following

Lemma 4.4. 1. If there exists1≤n≤Rb such that i,j∈ {τn−1+1, . . . ,τn}, then

E_τ[₋∆_i·∆_j] =A(r)r→∞_∼ CX,Y

r (59)

where r =τn−τn−1 (in particular, note that the expectation depends only on r) and CX,Y is a

positive constant which depends onPX_,PY_; 2. otherwise,E_τ[₋∆_i·∆_j] =0.

Proof of Lemma 4.4Case (2). Assume thatτn−1<i≤τn andτm−1< j≤τm withn6=m. Thanks

to (56)-(57) we have that

E_τ[∆_i·∆_j] =EXEY[∆_i|X_τ

n−1=Yτn−1,Xτn=Yτn]·E

X_EY_[∆

j|Xτm−1=Yτm−1,Xτm=Yτm] (60)

and both factors are immediately seen to be zero, since the laws of X and Y are assumed to be symmetric.

Case (1).Without loss of generality, assume thatn=1, so we only have to compute

wherer =τ1. Let us fixx ∈Z3, and denoteEYr,x[·] =E Y_[

·Y_r=x].

EY_{[∆i·∆jYr=x] =} EY

r,x

h ∆_i·EY

r,x

”

∆_j∆_i—i = EY

r,x

∆_i· x−∆i r₋1

= x

r₋1·E

Y r,x

∆_i− 1 r₋1E

Y r,x

h ∆_i2i

= 1

r₋1

‚

kx_k2 r −E

Y r,x

h ∆₁2

iŒ

, where we used the fact that underPY

r,x the law of the increments{∆i}i≤ris exchangeable. Then, we

get

E_τ[∆_i·∆_j] =EXEY”∆_i·∆_j1_{Y

r=Xr} —

PX−Y(Y_r=X_r)−1 = EX

h

EY”∆_i·∆_jYr=Xr

—

PY(Yr=Xr)

i

PX−Y(Yr=Xr)−1

= 1

r₋1

 EX

  X_r2

r P

Y_(Y r=Xr)



PX−Y(Y_r=X_r)−1

−EXEYh_∆12_1{Y r=Xr}

i

PX−Y_(Yr=Xr)−1

= 1

r−1

 EX

  Xr

2

r P

Y_(Y r=Xr)



PX−Y(Y_r=X_r)−1₋EXEYh∆₁2Y_r=X_ri  . Next, we study the asymptotic behavior of A(r) and we prove (59) with C_X,Y = t r(Σ_Y) ₋ t r€(Σ−_X1+ Σ_Y−1)−1Š_{. Note thatt r(Σ}

Y) =EY(||Y1||2) =σ2Y. The fact thatCX,Y >0 is just a

conse-quence of the fact that, ifAand Bare two positive-definite matrices, one has thatA₋B is positive definite if and only ifB−1−A−1 is[11, Cor. 7.7.4(a)].

To prove (59), it is enough to show that

EXEYh∆₁2_Yr=Xr

i_r→∞

→ EXEYh∆₁2 i

=σ2_Y, (62)

and that

B(r):= EX

kXrk

2

r P Y_(Y

r=Xr)

PX−Y(X_r=Y_r)

r→∞

→ t r€(Σ−_X1+ Σ_Y−1)−1Š. (63)

To prove (62), write

EXEYh∆₁2Y_r=X_ri = EYh∆₁2PX(X_r=Y_r)iPX−Y(X_r=Y_r)−1

= X

y,z∈Zd

kyk2PY(Y1= y)

PY_(Yr−1=z)PX_{(Xr= y+z)}

PX−Y(X_r−Y_r=0) . (64)

We know from the Local Limit Theorem (Proposition 2.10) that the term PX(Xr=y+z)

PX−Y(Xr−Yr=0) is uniformly

bounded from above, and so there exists a constantc>0 such that for all y _∈Zd X PY_(Yr−1=z)PX_{(Xr= y+z)}

If we can show that for every y fixed Zd the left-hand side of (65) goes to 1 as r goes to infinity, then from (64) and a dominated convergence argument we get that

EXEYh∆₁2Yr=Xr

i_r→∞

−→ X

y∈Zd

kyk2PY(Y1= y) =σ2Y. (66)

We use the Local Limit Theorem to get

X

z∈Zd

PY(Yr−1=z)PX(Xr= y+z) =

X

z∈Zd c_Xc_Y

rd e

−2(r1−1)z·(Σ−

1

Y z)_e−₂1_r(y+z)·(ΣX−1(y+z)) +_o(_r−d/2)

= (1+o(1))X

z∈Zd c_Xc_Y

rd e

−₂1_rz·(Σ−1

Y z)_e−

1 2rz·(Σ−

1

X z) +_o(_r−d/2) ₍₆₇₎

where c_X = (2π)−d/2(detΣ_X)−1/2 and similarly for c_Y (the constants are different in the case of simple random walks: see Remark 2.11), and where we used that y is fixed to neglect y/pr. Using the same reasoning, we also have (with the same constantsc_X andc_Y)

PX−Y_{(Xr=Yr) =} X

z∈Zd

PY_(Yr=z)PX_(Xr=z)

= X

z∈Zd c_Xc_Y

rd e

−₂1_rz·(Σ−1

Y z)_e−

1 2rz·(Σ−

1

X z) +_o(_r−d/2)_{. (68)}

Putting this together with (67) (and considering that PX−Y(X_r = Y_r) _∼ c_X,Yr−d/2), we have, for every y ∈Zd

X

z∈Zd

PY(Yr−1=z)PX(Xr= y+z)

PX−Y_(Xr−Yr=0)

r→∞

−→1. (69)

To deal with the termB(r)in (63), we apply the Local Limit Theorem as in (68) to get

EX   Xr

2

r P

Y_(Y r=Xr)



= cYcX rd

X

z∈Zd

kzk2

r e

−21rz·(Σ−

1

Y z)_e−

1 2rz·(Σ−

1

X z) +_o(r−d/2_{). (70)}

Together with (68), we finally get

B(r) =

cYcX

rd P

z∈Zd k

zk2 r e

−₂1_rz·((Σ−1_Y +Σ−1_X )z_{) +o(r−}d/2₎

cYcX

rd P

z∈Zde−

1 2rz·((Σ−

1

Y +Σ−

1

X )z) +_o(r−d/2₎

= (1+o(1))E”_{kN k}2—, (71)

where _{N ∼ N} €0,(Σ−_Y1+ Σ−_X1)−1Š is a centered Gaussian vector of covariance matrix

(Σ−_Y1+ Σ−_X1)−1. Therefore,E”_{kN k}2—=t r€(Σ−_Y1+ Σ−_X1)−1Šand (63) is proven.

Remark 4.5. For later purposes, we remark that with the same method one can prove that, for any given k_{0 ≥} 0 and polynomials U and V of order four (so that EY_[|U€_{∆k}k6k

0

Š

|] < ∞and

EX[V(_||X_r||/pr)]<∞), we have

EXEY 

U€{∆k

_}k6k

0

Š V

X_r

p

r !

Yr=Xr 

r→∞_→ EY_U€_{k∆_kk}k6k

0

Š

where_N is as in (71).

Let us quickly sketch the proof: as in (64), we can write

EXEY –

U€{k∆_kk}k6k

0

Š V

kX_rk

p

r

Yr=Xr

™

= (73)

X

y1,...,yk₀∈Zd

U€_{ky_kk}k6k0

Š X

z∈Zd V

kz_k

p

r

PX(X_r=z)

PY(Yr−k0=z−y1−. . .− yk0) PX−Y(X_r−Y_r=0)

×PY(∆_i= yi,i≤k0).

Using the Local Limit Theorem the same way as in (68) and (71), one can show that for any

y₁, . . . ,y_k 0

X

z∈Zd V

kzk p

r

PX(X_r=z)

PY(Y_r−k

0=z−y1−. . .− yk0) PX−Y(X_r−Y_r=0)

r→∞

→ EV(_{kN k}). (74)

The proof of (72) is concluded via a domination argument (as for (62)), which is provided by uniform bounds onPY(Y_r−k0=z−y1−. . .−yk0) andP

X−Y_(X

r−Yr =0)and by the fact that the

increments ofX andY have finite fourth moments.

Given Lemma 4.4, we can resume the proof of Lemma 4.2, and lower bound the averageE_τ[F1].

Recalling (58) and the fact that we reduced to the casea=0, we get

E_τ[F1] =

Rb X

n=1

  

X

τn−1<i,j≤τn M_{i j}

 

A(∆τ_n), (75)

where∆τ_n:=τ_n−τ_n−1. Using the definition (29) of M, we see that there exists a constantc>0

such that for 1<m_≤L

m

X

i,j=1

M_{i j≥ p} c LlogL

m3/2. (76)

On the other hand, thanks to Lemma 4.4, there exists some r0>0 and two constants candc′such

thatA(r)_≥ c_r forr_≥r₀, andA(r)_{≥ −}c′for everyr. Plugging this into (75), one gets

p

LlogLE_τ[F1]≥c

Rb X

n=1

p

∆τn1{∆τn≥r0}−c

′ Rb X

n=1

(∆τn)3/21{∆τn6r0}≥c

Rb X

n=1

p

∆τn−c′Rb. (77)

Therefore, we get for any positiveB>0 (independent of L)

b

PE_τ[F₁]6gplogL 6bP

 _p 1

LlogL c

Rb X

n=1

p

∆τn−c′Rb

!

6uplogL  

6 bP

 _p 1

LlogL c

Rb X

n=1

p

∆τn−c′

p

LB !

6uplogL 

+bP€R_b>BpLŠ

6 bP

  

Rb/2

Xp

∆τn≤(1+o(1))

u c

p

LlogL  

+bP(Rb>B

p

Now we show that forB large enough, andL_≥L₀(B),

b

P(Rb>B

p

L)6ζ/2, (79)

whereζis the constant which appears in the statement of Lemma 4.2. We start with getting rid of the conditioning inbP(recall bP(_·) = P(_·|b∈τ)since we reduced to the case a=0). IfRb> B

p

L, then either _|τ∩ {1, . . . ,b/2_}| or _|τ∩ {b/2+1, . . . ,b_}| exceeds B₂pL. Since both random variables have the same law underbP, we have

b

P(R_b>BpL)62bP

R_b/2>

B

2

p

L

≤2cP

R_b/2>

B 2 p L , (80)

where in the second inequality we applied Lemma A.1. Now, we can use the Lemma A.3 in the Appendix, to get that (recall b_≤L)

P

Rb/2>

B 2 p L ≤P RL/2>

B 2 p L L→∞ → P |Z | p

2π ≥B c_K

p

2

, (81)

with_Z a standard Gaussian random variable andc_K the constant such that K(n)_∼ c_Kn−3/2. The inequality (79) then follows forB sufficiently large, andL≥ L0(B).

We are left to prove that forLlarge enough andusmall enough

b

P

  

Rb/2

X

n=1

p ∆τn6

u c

p

LlogL  

 6ζ/2. (82)

The conditioning inbPcan be eliminated again via Lemma A.1. Next, one notes that for any given

A>0 (independent ofL)

P

  

Rb/2

X

n=1

p ∆τn6

u c

p

LlogL    6P

  

ApL

X

n=1

p ∆τn6

u c

p

LlogL  

+P€Rb/2<A

p

LŠ. (83)

Thanks to the Lemma A.3 in Appendix and tob≥ǫL, we have

lim sup L→∞ P _R b/2 p

L <A

≤P _p|Z |

2π <AcK

r

2

ǫ

!

,

which can be arbitrarily small ifA=A(ǫ)is small enough, for Llarge. We now deal with the other term in (83), using the exponential Bienaymé-Chebyshev inequality (and the fact that the∆τn are

i.i.d.):

P

  p 1

LlogL

A_XpL

n=1

p ∆τn<

u c p logL  

 6e(u/c)plogLE

– exp ‚ − r τ1

LlogL ΪApL

. (84)

To estimate this expression, we remark that, for Llarge enough,

E

–

1−exp

‚

−

r

τ1

LlogL Ϊ

=

∞

X

n=1

K(n) 

1−e−

Æ _n

LlogL ‹

≥ c′

∞

X

n=1

1₋e−

Æ _n

LlogL n3/2 ≥c

′′

r

logL

where we used the fact that, under bP(_·|R_b = p) for a fixed p, the law of the jumps_{∆τn}n≤p is

exchangeable. We first bound (120) whenRb is large:

b E h

R_bpτ1 p

τ2−τ11_{Rb≥κ

p

LlogL} i

6 L2bP

R_b≥κpLlogL

6 L2P(b∈τ)−1P

Rb≥κ

p LlogL

. (121) In view of (14), we haveP(b_∈τ)−1=O(pL). Thanks to Lemma A.2 in the Appendix, and choosing κlarge enough, we get

P

R_b≥κpLlogL

6e−cκ2logL+o(logL)=o(L−5/2), (122) and therefore

b E

h Rbpτ1

p

τ2−τ11_{R

b≥κ

p

LlogL} i

=o(1). (123)

As a consequence, b E h

Rbpτ1 p

τ2−τ11{Rb≥2}

i

=Eb h

Rbpτ1 p

τ2−τ11_{2≤R

b<κ

p

LlogL} i

+o(1) 6 pL(logL)1/12bEhpτ1

p

τ2−τ11{Rb≥2}

i

+κpLlogLbEhpτ1 p

τ2−τ11_{Rb>

p

L(logL)1/12_} i

+o(1). (124) Let us deal with the second term:

b E

h 1_{Rb>p

L(logL)1/12_}pτ₁ p

τ2−τ1 i

= 1

P(b_∈τ) b

X

i=1

b−i

X

j=1 p

ipjP€τ1=i,τ2−τ1= j,b∈τ,Rb>

p

L(logL)1/12Š

= 1

P(b_∈τ) b

X

i=1

b−i

X

j=1 p

ipjK(i)K(j)P€b−i− j∈τ,R_b−i−j>pL(logL)1/12₋2Š. (125) But we have

P€Rb−i−j>

p

L(logL)1/12₋2b−i−j∈τŠ 62P

R(b−i−j)/2> 1 2

p

L(logL)1/12₋1b−i−j∈τ

6 cP

R(b−i−j)/2> 1 2

p

L(logL)1/12₋1

6 cP

RL>

1 2

p

L(logL)1/12₋1

6c′e−c(logL)1/6 (126) where we first used Lemma A.1 to take the conditioning off, and then Lemma A.2. Putting (125) and (126) together, we get

b Eh1_{R

b>

p

L(logL)1/12_}pτ₁ p

τ2−τ1 i

6 c′e−c(logL)1/6 1 P(b_∈τ)

b

X

i=1

b−i

X

j=1 p

ipjK(i)K(j)P b−i− j∈τ

= c′e−c(logL)1/6Ebhpτ1 p

τ2−τ11{Rb≥2}

i

So, recalling (124), we have b

EhR_bpτ1 p

τ2−τ11{Rb≥2}

i

62pL(logL)1/12bEhpτ1 p

τ2−τ11{Rb≥2}

i

+o(1) (128) and we only have to estimate (recall (14))

b E

h_p τ1

p

τ2−τ11{Rb≥2} i

= b−1 X

p=1

b−p

X

q=1

p_pp_qK(p)K(q)P(b−p−q∈τ)

P(b∈τ)

6 c

p b

b−1 X

p=1

b−p

X

q=1 1 p q

1 p

b+1−p−q

. (129)

Using twice the elementary estimate

M_X−1

k=1 1 k

1 p

M₋k ≤c 1 p

M logM, we get

b E

h_p τ1

p

τ2−τ11{Rb≥2} i

6cpb

b−1 X

p=1 1 p

1 p

b−p+1

log(b₋p+1)6cpb_p1

b(logL)

2_{. (130)}

Together with (128), this proves the desired estimate (114).

4.4

Dimension

d

=

4

(a sketch)

As we mentioned just after Theorem 2.8, it is possible to adapt the change-of-measure argument to prove non-coincidence of quenched and annealed critical points in dimensiond _≥4 for the general walks of Assumption 2.1, while the method of Birkner and Sun[3]does not seem to adapt easily much beyond the simple random walk case. In this section, we only deal with the cased=4, since the Theorem 2.8 is obtained ford_≥5 in[2], with more general condition than Assumption 2.1. We will not give details, but for the interested reader we hint at the “right” change of measure which works in this case.

The “change of measure function” g_I(Y)is still of the form (27), factorized over the blocks which belong toI, but this timeM is a matrix with a finite bandwidth:

F_k(Y) =_−p1

L

k LX−p0

i=L(k−1)+1

∆_i·∆_i+p

0, (131)

where p₀ is an integer. The role of the normalization L−1/2 is to guarantee that_kM_k< ∞. The integerp₀is to be chosen such thatA(p₀)>0, whereA(_·)is the function defined in Lemma 4.4. The existence of suchp0is guaranteed by the asymptotics (59), whose proof ford=4 is the same as for d=3.

For the rest, the scheme of the proof ofβc 6=βcann (in particular, the coarse-graining procedure) is

A

Some technical estimates

Lemma A.1. (Lemma A.2 in [9]) Let P be the law of a recurrent renewal whose inter-arrival law satisfies K(n)n→∞_∼ cKn−3/2 for some cK >0. There exists a constant c>0, that depends only on K(·),

such that for any non-negative function f_N(τ)which depends only onτ∩ {1, . . . ,N_}, one has sup

N>0

E[f_N(τ)_|2N_∈τ]

E[f_N(τ)] 6c. (132)

Lemma A.2. Under the same assumptions as in Lemma A.1, and with RN := |τ∩ {1, . . . ,N}|, there

exists a constant c > 0, such that for any positive function α(N) which diverges at infinity and such thatα(N) =o(pN), we have

P€R_{N ≥}pNα(N)Š 6e−cα(N)2+o(α(N)2). (133)

Proof.For everyλ >0 P€RN ≥

p

Nα(N)Š = P€τp

Nα(N)6N

Š

=P

λα(N)2τ

p

Nα(N)

N 6λα(N) 2

(134)

6 eλα(N)2E

e−λα(N) 2

N τpNα(N)

=eλα(N)2Ehe−λα(N)2τN1

ipNα(N)

. The asymptotic behavior ofEhe−λα(N)2τN1

i

is easily obtained: 1−E

h

e−λα(N)2τN1

i

= X

n∈N

K(n)1−e−nλα(N)2/N

N→∞

∼ c

p λα(N)

p

N , c=cK Z ∞

0

1₋e−x

x3/2 dx, (135) where the conditionα(N)2/N →0 was used to transform the sum into an integral. Therefore, we get

Ehe−λα(N)2τN1

ip

Nα(N)

=

‚ 1₋c

p λα(N)

p

N +o

_α(N) p

N

ŒpNα(N)

= e−c p

λα(N)2_+o(α(N)2₎

. (136)

Then, for anyλ >0,

P€R_{N ≥}pNα(N)Š 6e(λ−cpλ)α(N)2+o(α(N)2) (137) and takingλ=c2/4 we get the desired bound.

We need also the following standard result (cf. for instance[10, Section 5]):

Lemma A.3. Under the same hypothesis as in Lemma A.1, we have the following convergence in law: c_K

p NRN

N→∞ ⇒ p1

A.1

Proof of

(100)

We wish to show that for distincti,j,l smaller thanr,

EXEY_{[(∆i·∆j)(∆i·∆l)|Xr=Yr]≤} c

r. (139)

We use the same method as in Section 4.2: we fix x _∈ Zd_{, and we use the notation} EY r,x[·] = EY[_·Yr=x]. Then,

EY r,x

”

(∆_i·∆_j)(∆_i·∆_l)— = EY r,x

h

(∆_i·∆_j)∆_i·EY r,x

”

∆_l∆_i,∆_j—i

= 1

r−2E

Y r,x

”

(∆_i·∆_j)€∆_i·(x₋∆_i−∆_j)Š—

= 1

r₋2E

Y r,x

h

(∆_i·∆_j)(x·∆_i)₋∆_i2₋(∆_i·∆_j)2i

6 1

r−2E

Y r,x

h

(x_·∆_i)₋∆_i2 ∆_i·EY r,x

”

∆_j∆_i—i

= 1

(r₋1)(r₋2)E Y r,x

(x_·∆_i)₋∆_i2

2

6 2

(r₋1)(r₋2)E Y r,x

h

kxk2∆_i2+∆_i4i

and we can take by symmetryi=1. Therefore,

EXEY(∆_i·∆_j)(∆_i·∆_l)Xr=Yr

=

EXEY”(∆_i·∆_j)(∆_i·∆_l)Y_r=X_r—PY(Y_r=X_r)

PX−Y(Y_r=X_r) (140)

≤ c r

EX

h kXrk2

r E

Y

k∆_1k2_|Y_r=X_r+1_rEY_(k∆1k4_|Yr=Xr)

PY_(Yr=Xr)

i

PX−Y(Y_r=X_r)

= c

rE

X_EY

– Q

kX_rk

p

r ,k∆1k

Yr=Xr

™ , where

Q

kX_rk p

r ,k∆1k

= kXrk

2 r k∆1k

2₊k∆1k4

r . (141)

At this point, one can apply directly the result of Remark 4.5.

A.2

Proof of

(101)

We wish to prove that, for distincti,j,k,l_≤r,

E_{τ[(∆i·∆j)(∆k·∆l)]}6 c

r2. (142)

The proof is very similar to that of (139), so we skip details. What one gets is that

E_τ”(∆_i·∆_j)(∆_k·∆_l)— 6 c

r2

EX

–

EY

– Q′

 kXrk

r1/2 ,{k∆ik}i=1,2,3 ‹

Yr=Xr

™

PY_(Yr=Xr)

™

whereQ′is a polynomial of degree four. Again, like after (140), one uses the Remark 4.5 to get the desired result.

A.3

Proof of

(102)-(103)

In view of (56), in order to prove (102) it suffices to prove that for 0<i₆=k_≤r, 0< j₆=l_≤s 3

X

ν,µ=1

EXEY[∆(_iν)∆_k(µ)_|X_r=Y_r]EXEY[∆(_jν)∆_l(µ)_|X_s=Y_s]_≤ c

rs. (144)

Both factors in the left-hand side have already been computed in (96)-(97). Using these two ex-pressions and once more the Local Limit Theorem, one arrives easily to (144). The proof of (103) is essentially identical.

Acknowledgments

We would like to thank Rongfeng Sun for showing us the preprint[4] before its publication and for several enlightening discussions and comments, which by the way allowed us to considerably weaken the conditions of Theorem 2.8.

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