Shape transition under excess self-intersections for transient random walk

www.imstat.org/aihp 2010, Vol. 46, No. 1, 250–278

DOI:10.1214/09-AIHP318

© Association des Publications de l’Institut Henri Poincaré, 2010

Shape transition under excess self-intersections for transient random walk

Amine Asselah

Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris-Est, UMR CNRS 8050, France. E-mail:amine.asselah@univ-paris12.fr Received 3 May 2008; revised 10 February 2009; accepted 13 March 2009

Abstract. We reveal a shape transition for a transient simple random walk forced to realize an excessq-norm of the local times, as the parameterqcrosses the valueqc(d)=_d−^d₂. Also, as an application of our approach, we establish a central limit theorem for theq-norm of the local times in dimension 4 or more.

Résumé. Nous décrivons un phénomène de transition de forme d’une marche aléatoire transiente forcée à réaliser une grande valeur de la norme-qdu temps local, lorsque le paramètreqtraverse la valeur critiqueqc(d)=_d^d₋₂. Comme application de notre approche, nous établissons un théorème de la limite centrale pour la norme-qdu temps local en dimension 4 et plus.

MSC:60K35; 82C22; 60J25

Keywords:Self-intersection local times; Large deviations; Random walk

1. Introduction

We consider a simple random walk{S(n), n∈N}onZ^d, starting at the origin. For any setA, we denote by1A the indicator ofA, and consider the local times of the walk{l_n(z), z∈Z^d}given by

l_n(z)=1_{S(0)=z}+ · · · +1_{S(n−1)=z}. (1.1) For a realq >1, we form the sum of theqth power of the local times

l_n^qq=

z∈Z^d

l_n(z)^q. (1.2)

Whenq is integer, l_n^qq can be written in terms of the q-fold self-intersection local times of a random walk. For instance, whenq=2

l_n²₂=n+2

0≤i<j <n

1_{S(i)=S(j )}.

Forq positive real, we still calll_n^qqtheq-fold self-intersection local times.

In dimension three and more, Becker and König [6] have shown that there are positive constants, sayκ(q, d), such that almost surely

nlim→∞

l_n^qq

n =κ(q, d). (1.3)

Here, we are concerned with estimating the deviations ofl_n^qq away from its mean. That is, ifP denotes the law of the walk started at 0, we give estimates for

P

l_n^qq−E l_n^qq

≥ξ n

(1.4) forξ positive, andngoing to infinity.

There is a rich literature concerning the two-fold self-intersection local times. The reason being thatln2 is a natural object in quantum-field theory (see [1,14,22], for instance), as well as in the statistical physics of polymers (see [8,9,13], for instance). Howeverl_nqforq∈R\Nhas no such direct links with physics. It comes up naturally in studying large and moderate deviations forrandom walk in random sceneries(see [5] and [15]).

Now, in the large deviations results for the two-fold self-intersection of a transient random walk (see [2,3,5,11]) two strategies have a distinguished role:

• Strategy A: the walk visits of the order of (ξ n)^1/q-times, finitely many sites in a ball of bounded radius. For a transient walk, the number of visits of a bounded domain is bounded by a geometric variable. Thus, strategy A costs of the order of exp(−O((nξ )^1/d)), where we use the notationyn=O(xn)for two positive sequences{xn, yn, n∈N}, to mean that there isK >0 such that 0≤y_n≤Kx_n.

• Strategy B: the walk visits of the order of ξ^1/(q−1)-times, about n/ξ^1/(q−1) sites. Presumably, the walk stays, a timen, in a ball of volume n/ξ^1/(q−1). The cost of staying a time n within a ball of radiusr_n√

n is about exp(−O(n/r_n²)), so that strategy B costs of the order of exp(−O(n^1−2/dξ^2/(d(q−1)))).

Whenq=2, [2,5] have shown that strategy A is adopted ind≥5, whereas [3] (see also Chapter 8.4 of [11]) suggests that strategy B is adopted ind=3.

To summarize in words our main finding, assumed ≥3, fixξ >0 and look at typical paths realizing {ln^qq− E[l_n^qq] ≥ξ n}. As we increaseq, we step on a value,q_c(d), above which our large deviation event is realized by strategy A, and below which it is realized by strategy B. The critical valueq_c(d)=_d−^d₂ is obtained as we equal the costs of strategies A and B.

Note thatqc(d)is a well known number: ifqis integer, thenqindependent simple random walks, onZ^d, intersect infinitely often if and only ifq < q_c(d)(see, for instance, [18], Proposition 7.1 and [17], Section 4.1).

Let us now describe, in mathematical terms, this shape transition. The first theorem deals with the sub-critical regimeq < q_c(d).

Theorem 1.1. Assume dimensiond≥3.Then,forq anddsuch that1< q <_d−^d₂,there are constantsc^±₁(q, d) >0 such that forξ≥1,andnlarge enough

exp

−c⁻₁(q, d)ξ^{(2/d)(1/(q−1))}n^1−2/d

≤P

l_n^qq−E l_n^qq

≥ξ n

≤exp

−c⁺₁(q, d)ξ^{2/d(1/(q−1))}n^1−2/d

. (1.5)

Moreover,in this regime the sites visited more than some large constant do not contribute to realizing the excess self-intersection.In other words,

lim sup

A→∞ lim sup

n→∞

1

n^1−2/d logP

z∈Z^d

1_{l_n(z)>A}ln(z)^q≥ξ n

= −∞. (1.6)

Our second theorem deals with thesuper-critical regimeq > q_c(d).

Theorem 1.2. Assume dimensiond≥3.Forqandd such thatq >_d^d₋₂,there are constantsc^±₂(q, d) >0such that forξ≥1,andnlarge enough

exp

−c⁻₂(q, d)(ξ n)^1/q

≤P

ln^qq−E ln^qq

≥ξ n

≤exp

−c⁺₂(q, d)(ξ n)^1/q

. (1.7)

Moreover,the sites visited much less thann^1/q do not contribute to realizing the excess self-intersection.In other words,

lim sup

ε→0

lim sup

n→∞

1 n^1/qlogP

z∈Z^d

1_{ln(z)<εn^1/q_}ln(z)^q≥E ln^qq

+ξ n

= −∞. (1.8)

Remark 1.3. In Theorems1.1and1.2,we could takeξ to grow withn.The only(necessary)bound onξ_ncomes from the boundl_nq≤nwhich imposes thatξ_n≤n^q−1.The proofs are written with generalξ_n≥1.

The next result deals with the contribution of some level sets of the local times to deviation on a much larger scale than the mean, and can be obtained by the same approach yielding Theorem1.2. We include it in this form since it can be of independent interest, while showing the possibilities offered by our approach. Also, it generalizes Lemma 1.8 of [5].

Lemma 1.4. Assumed≥3andq≥q_c(d).Choosea, b >0such that1< a <1+b(q−1).Then,for anyε >0,and nlarge enough

P

z∈Z^d

1_{ln(z)<nb}l_n(z)^q≥n^a

≤e^{−nζ (q,a,b)−ε} withζ (q, a, b)=b+ 1

qc(d)(a−qb). (1.9)

Remark 1.5. Our approach is not suited to studying smallξn for reasons explained later in Remark1.7.However, when1> ξ_n≥n^−δ,for some positiveδsmall enough,our approach yields a constantc₁such that forq < q_c(d)

P

l_n^qq−E l_n^qq

≥ξ_nn

≤exp

−c₁ξ_n^{2/d(q/(q−1))}n^1−2/d

. (1.10)

Whenq > q_c(d),we have a constantc₂such that P

l_n^qq−E l_n^qq

≥ξ_nn

≤exp

−c₂ξ_n^1/q+2/dn^1/q

. (1.11)

We believe that the powers ofξnin(1.10)and(1.11)are not optimal.However, (1.10)and(1.11)are useful in deriving a central limit theorem stated in Theorem1.9.

Our initial goal was to improve the main result of [3], which states that in dimension 3, there isχ >0 andε >0 such that forξ >0, andnlarge

P

z∈Z³

1_{ln(z)>log(n)^χ}l²_n(z) > nξ

≤exp

−n^1/3log(n)^ε

. (1.12)

Note that (1.6) improves (1.12). One reason to studyl_nq forq >2, is that the upper bound (1.5) forq >2, yields (1.6) at once. More precisely, forq < q_c(d), chooseqwithq < q< q_c(d), and for anyA >0, the obvious inequality

z∈Z^d

1_{ln(z)>A}l^q_n(z)≤l_n^q_q

A^q−q, (1.13)

implies that P

z∈Z^d

1_{ln(z)>A}l_n^q(z)≥nξ

≤P

l_n^q_q≥A^q−qnξ .

ForAandnlarge enough,A^q−qnξ≥2E[ln^q_q]. Thus, if we setβ=²_d^q_q⁻−^q1>0, then from (1.5), we have a constant c₁(d, q)such that

P

z∈Z^d

1_{ln(z)>A}l_n^q(z)≥nξ

≤exp

−c₁ d, q

ξ^{(2/d)(1/(q−1))}A^βn^1−2/d

. (1.14)

Thus, in order to improve (1.12) ind=3, we were left with studyingq-fold self-intersections with 2< q <3=q_c(3).

In most works on two-fold self-intersection, a starting point, which we trace back to the work of Westwater [21]

and Le Gall [19], is a decomposition ofl_n²₂in terms ofintersection local timesof two independent random walks starting at the origin. However, such a decomposition is restricted toq-fold self-intersection local times withq∈N:

Whenq =2 and d =3 (in the sub-critical regime) Le Gall’s decomposition is a first step in obtaining, in [11], a moderate and large deviations principles. Whenq=3 andd≥4 (in thesuper-critical regime), [15] uses a type of Le Gall’s decomposition to obtain moderate and large deviations estimates.

Here, our starting point is an approximate decompositionobtained by slicing l_n^qq over level sets of the local times, for any realq >1. This is based on the following simple inequality. Let{b_n, n∈N}be a subdivision of[1,∞), and letl₁andl₂be positive integers (which we think of as the local times of a given site in each half time-period).

Then, forq >1, we have the upper bound (l₁+l₂)^q≤l₁^q+l₂^q+2^q

∞ i=0

b_i^q₊⁻₁²1_{b_i≤max(l₁,l₂)<b_i+1}l₁×l₂, (1.15)

as well as the obvious lower bound:(l1+l2)^q≥l₁^q+l₂^q. The desirable feature of (1.15) is that on its right-hand side, theqth power ofl1andl2comes without penalty, whereas the terml1×l2yields an intersection local times. Thus, (1.15) leads to the following result which plays here the role of Le Gall’s decomposition of [19].

Proposition 1.6. For any integersnandl,with2^l< n,let{n_i, i=1, . . . ,2^l}be positive integers summing up ton.

Let{l_·⁽ⁱ⁾, i=1, . . . ,2^l}be the local times of2^l independent random walks starting at0.If{b_i, i∈N}is a subdivision of[1, n],then,

S_q^(l)≤ ln^qq≤S_q^(l)+ l j=1

Ij, whereS^(l)_q ^law=

2^l

i=1

l⁽ⁱ⁾_n

i ^q

q, (1.16)

and,forj=1, . . . , l,andm_k=n_(k−1)2l−j+1+ · · · +n_k2l−j fork=1, . . . ,2^j Ij

law=

2^j−1

k=1

i

2^qb_i^q₊⁻₁¹

z:b_i≤l_m^(2k)

2k(z)<b_i+1

l^(2k_m ⁻¹⁾

2k−1 (z)+

z:b_i≤l^(2k_m2k−1⁻¹⁾(z)<b_i+1

l^(2k)_m

2k(z)

. (1.17)

Remark 1.7. We first note some natural limitations in using the approximate decomposition (1.16).When we deal with{l_n^qq−E[l_n^qq] ≥ξ_nn}for smallξ_n,we need to bound the difference betweenE[l_n^qq]and the expectation of the upper bound in(1.16).When,we takelsuch that2^l∼n^1−δ0,then this difference turns out to be of order smaller thann^1−δ0/2,allowing us to write

l_n^qq−E l_n^qq

≥ξ_nn

⊂

S_q^(l)−E S_q^(l)

≥ξ_n 2n

∪ _l

j=1

Ij−E[Ij] ≥ξ_n

2n−n^1−δ0/2

. (1.18)

(1.18)requires thatξ_n≥n^−δ0/2.

Proposition1.6is our initial step in the proof of Theorems1.1and1.2, and leads to a central limit theorem (CLT) forln^qqin dimension 4 or more, as well as a characterization of the variance ofln^qq.

Before stating results concerning the typical behavior of l_n^qq, we give a heuristic discussion of the proof of Theorem1.1assuming Proposition1.6. More precisely, we wish to sketch the reasons why theapproximate decompo- sition(1.16), reduces large deviations for theq-norm of the local times, to large deviations for a sum of independent geometric variables.

Consider a choice oflsuch that 2^lis close tonin (1.16), andn_i∼n/2^l. Then,S_q^(l)is a sum of aboutnindependent terms, each one bounded by its time-span,n/2^l, to the powerq. Recall that the probability of deviating from the mean, for a sum ofnindependent and essentially bounded variables, is of order exp(−O(n))(see Lemma2.4for a precise statement). We can therefore neglect the contribution ofS^(l)_q to the excessq-norm, though the mean ofS^(l)_q is close toE[l_n^qq], as easily seen from Lemma1.8below. Now, for a fixedj in{1, . . . , l}, letm=n/2^j, and note thatIj

is a finite sum of independent terms distributed asb^q−1l_m(D˜(b)), whereD˜(b)is the set{z: l˜_m(z)∼b}, withl˜_man independent copy ofl_m, andbspans a subdivision of[1, n]. Since we consider transient random walks,l_m(D˜(b))is bounded by a geometric variable (when fixingD˜(b), as shown in Lemma 1.2 of [4]). At this point, one normalizes l_m(D˜(b)). IfP₀andP˜₀are the law of the two independent copies of the (transient) walk started at 0, define

X= l_m(D˜(b))

E₀[l_m(D˜(b))], so that for someκ >0, P˜0⊗P0(X > t)≤e^−κt. (1.19) Now, it is well known that for anym,E0[l_m(D˜(b))] ≤C| ˜D(b)|^2/d, (see LemmaA.2). Thus, an estimate of the large deviation probability requires an estimate on the volume of level sets of the local times. Now, in obtaining a bound on the volume ofD˜(b), assume for simplicity that we only have two types ofb: that is, we distinguishoften visited sites, say sites visitedn^x-times withxclose to 1/q, whose level sets are part of what we calltop levels, and say the once-visited sites, whose level sets are part of what we callbottom levels. Thebottom levelsare the easiest to treat (see Section3.2and Lemma3.1). Indeed, we use essentially that| ˜D(b)| ≤nforb∼1, so that we expect (when we restrictIj only tobottom levelsand usingXk=Xin law)

P

Ij−E[Ij] ≥nξ

∼P ₂j

k=1

X_k−E[X_k] ≥ nξ n^2/d

∼exp

−O

n^1−2/d .

Thetop levels, treated in Section3.3, require a type ofbootstrap argument, using that ifD˜(b)has a large volume, it implies that theq-norm of the local timel˜mis large. The bootstrap argument yields Corollary2.2. It allows us to normalize properly our geometric-like random variablesb^q−1lm(D˜(b)).

We turn now to the typical behavior of the q-norm. Chen has provided in [10] asymptotics for the variance of l_n²₂ind≥3. He shows that (i) ind=3, var(l_n²₂)∼λ₃nlog(n), and (ii) in d≥4, var(l_n²₂)∼λ_dn, whereλ_d are constants expressed in terms of the Green’s function of the walk. Following ideas of Jain and Pruitt [16], and of Le Gall and Rosen [20], Chen obtains a CLT in dimension 3 or more for l_n²₂. Finally, Becker and König in [6]

have shown that forqinteger, (i) ind=3, var(l_n^qq)≤n^3/2, (ii) ind=4, var(l_n^qq)≤nlog(n), and (iii) ind≥5, var(l_n^qq)≤c_dn. Our result deals with the general case (q >1 real), where no representation ofl_n^qq is possible in terms of multiple time-intersections. We transform Lindeberg’s condition into alarge deviationevent forl_Tn²₂on the scale of time of the CLT, that isT_n≈√

n.

We start with an estimate for the expectation ofl_n^qq, of the same type as Theorem 1 of Dvoretzky and Erdös [12]

for the range of a transient random walk. Thus, ifγdis the probability of never returning to its original position, it is shown in [12] that for positive constantscd, whenRnis the set of visited sites before timen,

E

|R_n|

−nγ_d≤c_dψ_d(n), withψ_d(n)=

⎧⎨

⎩

n^1/2 for d=3, log(n) for d=4,

1 for d≥5,

(1.20)

Jain and Pruitt [16] obtain the asymptotics var(|Rn|)∼alog(n)n for somea >0 ind =3, and var(|Rn|)∼c_dn in d >3, for some positive constantsc_d. The corresponding CLT (ind≥3) was shown by Jain and Pruitt [16] for the simple random walk, and by Le Gall and Rosen [20] for stable random walks. Note that the limiting law is Gaussian, ind≥3 but fails to be so ind=2, as shown by Le Gall in [18].

Lemma 1.8. Assume thatd≥3andq >1.There is a constantC_d,such that 0≤κ(q, d)n−E

l_n^qq

≤C_dψ_d(n), withκ(q, d)=γ_dE l_∞(0)^q

. (1.21)

Also,ifd=3,then,there is a constantc₃such that var

l_n^qq

≤c3log(n)²n. (1.22)

Ifd≥4,then there are positive constantsv(q, d)andc(q, d),such that var(l_n^qq)

n −v(q, d)

≤c(q, d)log(n)

√n . (1.23)

Finally, we have the following central limit theorem.

Theorem 1.9. IfZis a standard normal variable,then ln^qq−nκ(q, d)

√nv(q, d)

−→law Z. (1.24)

A challenging open question is to understand the strategy which realizes {l_n^qq−E[l_n^qq] ≥ξ n}, right at the critical valueq=qc(d)=_d−^d₂.

The paper is organized as follows. The approximate decomposition ofl_n^qq is given in Section2.1. The sub- critical regime is studied in Section3: The upper bound in (1.5) is proved in Section3.5, and the lower bound is given in Section3.4. The super-critical regime is studied in Section4. Theorem1.2is proved in Section4. The proof of (1.8) is given in Section4.1. The proof of Lemma1.4is given in Section4.2. Lemma1.8, as well as the CLT are proved in Section5. In theAppendix, we recall Lemma 5.1 of [3], and improve Lemma 5.3 of [3], used to control intersection local times-type quantities.

2. General considerations (q >1)

In this section, we deal with the general caseq >1. In Section2.1, we develop a approximation ofl_n^qqas sums of two types of independent variables:

1. Intersection local times of independent walks.

2. Self-intersection local times, on a much shorter time-period.

In Section2.2, we treat the sums of self-intersection local times.

2.1. Approximate decomposition forl_n^qq

Before we prove Proposition1.6, we present a useful corollary which requires more notations.

For integersnandl, with 2^l< n, we recall the “almost” dyadic decomposition ofnof Remark 2.1 of [5]. We divide ninto 2^lintegersn^(l)₁ , . . . , n^(l)

2^l withn=n^(l)₁ + · · · +n^(l)

2^l and maxi

n^(l)_i

−min

i

n^(l)_i

≤1, n

2^l −1≤n^(l)_i ≤ n

2^l +1 and n^(l_k⁻¹⁾=n^(l)_2k−1+n^(l)_2k. (2.1) We run 2^lindependent random walks starting at the origin. Theith walk runs for a time-period[0, n^(l)_i [, and we denote byl_i^(l):Z^d→Nits local times during time-period[0, n^(l)_i [. Also, we introduce, fork=1, . . . ,2^l, the following sets

D^(l)_k,i=

z∈Z^d: bi≤l_k^(l)(z) < bi+1

. (2.2)

Now, for anyM >0, let{bi, i∈N}be a subdivision of[1, M], and denote byΘM(x)=x1_{x≤M}.

Remark 2.1. We could restrict the sum overZ^dwhich entersl_n^qq in(1.16)over{z: l_n(z)≤M}for any positiveM.

The proof of Proposition1.6yields,for any{b_i, i∈N}subdivision of[1, M],

z∈Z^d

ΘM

ln(z)q

≤

2^l

k=1

z∈Z^d

ΘM

l_k^(l)(z)q

+ l j=1

Ij. (2.3)

The only difference with(1.16)is the subdivision which enters into the definition ofIj.The proof of Proposition1.6 is written in view of(2.3) (see the key step(2.12)).

As a corollary of (2.3), we obtain the following result.

Corollary 2.2. For anyM >0,let{bi, i∈N}be a subdivision of[1, M].For any integersnandL,with2^L< n,and for any sequence of positive numbers{mn, εn, n∈N},we have

P Θ_M(l_n) ^q

q≥m_n+ε_n

≤2^L+1P ₂L

j=1

Θ_M l_j^{(L) q}

q≥m_n

+ L h=1

2^hP _L

l=h

1_{G(l)

1 ∩G2^(l)}Il≥ε_n

, (2.4)

where forl≤L,k=1, . . . ,2^l,andi∈N G^(l)_k,i=D_k,i^(l)≤m_n+ε_n

b^q_i

, G₁^(l)=

2^l

k=1

i

G_2k,i^(l) and G₂^(l)=

2^l

k=1

i

G_2k^(l)_−1,i. (2.5)

Remark 2.3. The symbols,εnandmn,are suggestive of the fact that whenLis large enough,the sum of2^Lindepen- dentq-fold self-intersections,that we calledSq^(L),stays close to its mean,which is also close to the mean ofln^qq. This is shown in Section2.2.So,mn stands for mean,andεnstands for excess.To estimate how small canεnbe,we now compute the expectation ofL

l=1Il.We use LemmaA.2in the worse case,that is in dimension3,to obtain for some constantsc1, c2andc3

E _L

l=1

Il

= L

l=1

2^l

i∈N

2^q(b_i+1)^q−1C_dψ_d n

2^l

e^−κdbi

≤c₁√ n

L l=1

2^l/2

i∈N

(b_i+1)^q−1e^−κdbi

≤c₂ 2^Ln

i∈N

(b_i+1)^q−1e^−κdbi. (2.6)

First,we need to choose a subdivision{b_i, i∈N}such that the last sum in(2.6)is convergent.Secondly,the right-hand side of(2.4)is small if_L

l=hE[1{G₁^(l)∩G₂^(l)}Il] ε_n.From(2.6),we see thatε_n can be chosen small if2^Ln.

On the other hand,we see in Section2.2,thatL has to be large enough,for the probability of {S_q^(L)≥mn}to be negligible,whenmn=E[ln^qq] +nε. Remark2.5shows that a choice ofL such that2^L> n^1−δ0 withqδ0<²_d, fulfills both requirements.

Proof of Proposition1.6. The proof proceeds by induction onl≥1. It is however easy to see that proving the case l=1 requires the same arguments as going froml−1 tol. We focus on the first stepl=1, and omit the easy passage froml−1 tol.

For anyx∈ [0,1], andq≥1, we have

1+x^q≤(1+x)^q≤1+x^q+2^qx. (2.7)

Thus, for any nonnegative integersl₁, l₂with 0≤l₁, l₂≤M, we have from (2.7)

l₁^q+l₂^q≤(l1+l2)^q≤l₁^q+l^q₂+2^qM^q−2l1l2. (2.8) Now, for anyM >0, let{b_i, i∈N}be a subdivision of[1, M], and recall thatΘ_M(x)=x1_{x≤M}. For any nonnegative integersl1, l2

ΘM(l1+l2)q

≤

ΘM(l1)q

+

ΘM(l2)q

+2^q n i=1

b_i^q₊⁻₁²1

bi≤max(l1, l2) < bi+1

l1×l2. (2.9)

Indeed,l₁+l₂≤Mandl₁, l₂≥0, imply (i)l₁≤M andl₂≤M, and (ii) for somei₀>0, max(l1, l₂)∈ [b_i0, b_i0+1[. Then, from (2.8)

ΘM(l1+l2)q

≤ΘM(l1)^q+ΘM(l2)^q+2^qb^q_i⁻²

0+1l1×l2.

For any integern, we consider the local timel_n, which we denote asl_[0,n[(z)to emphasize the time period. For any integern1with 0< n1< n, setn2=n−n1, and from the increments of our initial random walk, say{Yn, n∈N}, we build two independent random walks with local times

l_]^(1,1)_0,k](z)=1{Y_n1=z} + · · · +1{Y_n1+ · · · +Y_n1−k+1=z} and

l_[^(1,2)_0,k[(z)=1{0=z} +1{−Y_n1+1=z} + · · · +1{−Y_n1− · · · −Y_n1+k−1=z}. (2.10) It is obvious that on{S(n1)=y},

l_n(z)=l_]^(1,1)_0,n

1](y−z)+l_[^(1,2)_0,n

2[(y−z). (2.11)

If we denotel¯⁽¹⁾(z)=max(l_]^(1,1)_0,n

1](z), l_[^(1,2)_0,n

2[(z)), and sum (2.11) overz∈Z^d, we obtain

z

ΘM

ln(z)q

≤

z

ΘM

l_]^(1,1)_0,n

1]

S(n1)−zq

+

z

ΘM

l_[^(1,2)_0,n

2[

S(n1)−zq

+2^q

z∈Z^d

n i=1

b^q_i+⁻₁²1_{b

i≤¯l⁽¹⁾(S(n1)−z)<bi+1}l^(1,1)_]0,n

1]

S(n1)−z

×l_[^(1,2)_0,n

2[

S(n1)−z

≤

z∈Z^d

Θ_M l^(1,1)_]0,n

1](z)q

+

z∈Z^d

Θ_M l_[^(1,2)_0,n

2[(z)q

+2^q

z∈Z^d

n i=1

b^q_i+⁻₁²1_{b

i≤¯l⁽¹⁾(z)<b_i+1}l_]^(1,1)_0,n

1](z)×l_[^(1,2)_0,n

2[(z). (2.12)

Now, we rewrite (2.12) in a concise form as Θ_M(l_n) ^q

q≤ Θ_M l^(1,1)_]0,n

1] ^q

q+ Θ_M l_[^(1,2)_0,n

2[) ^q

q+ ˜I1(n₁, n₂), (2.13)

where the term dealing with intersection times of independent strands is I˜1(n₁, n₂)=2^q

z∈Z^d

n i=1

b_i^q₊⁻₁²1_{b

i≤¯l⁽¹⁾(z)<b_i+1}l^(1,1)_]0,n

1](z)×l_[^(1,2)_0,n

2[(z). (2.14)

To prove the upper bound in (1.16) forl=1, it suffices to setM=n(so that this truncation disappears), and to show thatI˜1(n₁, n₂)≤I1given in (1.17). This latter inequality follows first by noting the obvious inclusion

z: b_i≤max l_]^(1,1)_0,n

1](z), l^(1,2)_[0,n

2[(z)

< b_i+1

⊂

z: b_i≤l_]^(1,1)_0,n

1](z) < b_i+1

∪

z: b_i≤l_[^(1,2)_0,n

2[(z) < b_i+1 , and secondly, using that

z∈Z^d

1_{b

i≤¯l⁽¹⁾(z)<b_i+1}l_]^(1,1)_0,n

1](z)×l^(1,2)_[0,n

2[(z)≤

z∈Z^d

1

bi≤l_]^(1,1)_0,n

1](z) < bi+1

bi+1l^(1,2)_[0,n

2[(z) +

z∈Z^d

1

b_i≤l_[^(1,2)_0,n

2[(z) < b_i+1

b_i+1l_]^(1,1)_0,n

1](z). (2.15)

As we iterate the approximate decompositionl-times, we obtain the upper bound in (1.16), or more generally the bound (2.3).

The lower bound in (1.16) is an obvious corollary of the inequality(l₁+l₂)^q≥l₁^q+l₂^q, valid forq≥1 andl₁, l₂

nonnegative.

Proof of Corollary2.2. We write the caseM=n, that is the case with no truncation. The case with truncation is obtained as we replacel_n(z)byΘ_M(l_n(z))wherever it appears. Assume that we stop the induction in Proposition1.6 at some stepL (typically 2^L=n^1−δ0 andδ₀ small). For any sequence of positive numbersε_n, m_n, we have from (1.16),

P

l_n^qq≥m_n+ε_n

≤P

S_q^(L)≥m_n +P

_L

l=1

Il≥ε_n

, whereS_q^(l)=

2^l

k=1

l^(l)

k ^q

q. (2.16)

We introduce, as in [3,5], a bootstrap control on the volume ofD_k,i^(l). ConsiderG^(l)_k,igiven in (2.5). On the complement ofG^(l)=G₁^(l)∩G₂^(l), there isk₀, i₀such that|D_k^(l)

0,i₀|> (m_n+ε_n)/b_i^q

0 so that S_q^(l)≥

z∈D_k^(l)

0,i0

l^(l)_k

0(z)q

≥mn+εn

b^q_i

0

b^q_i

0=m_n+ε_n. (2.17)

WritingS_q⁽⁰⁾= l_n^qq, we write a more suggestive relation

P

S_q⁽⁰⁾≥mn+εn

≤P

S_q^(L)≥mn

+P _L

l=1

1_{G(l)}Il≥εn

+

L l=1

P

S_q^(l)≥mn+εn

. (2.18)

Starting the approximation withS_q^(l)withl < L, we obtain similarly

P

S_q^(l)≥m_n+ε_n

≤P

S_q^(L)≥m_n +P

_L

j=l+1

1_{G(j )}Ij≥ε_n

+ L j=l+1

P

S_q^{(j )}≥m_n+ε_n

. (2.19)

Assume now that forj > l, andj < Lwe have

P

S_q^{(j )}≥mn+εn

≤2^L−j+1P

S_q^(L)≥mn

+ L h=j+1

2^h−j−1P _L

i=h

1_{G(i)}Ii≥εn

. (2.20)