Annealed deviations of random walk in random scenery

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Annealed deviations of random walk in random scenery

Nina Gantert

^a

, Wolfgang König

^b,^∗

, Zhan Shi

^c

aFachbereich Mathematik und Informatik der Universität Münster, Einsteinstraße 62, 48149 Münster, Germany bMathematisches Institut, Universität Leipzig, Augustusplatz 10/11, 04109 Leipzig, Germany

cLaboratoire de Probabilités et Modèles Aléatoires, Université Paris VI, 4, place Jussieu, 75252 Paris Cedex 05, France

Received 4 April 2005; received in revised form 21 November 2005; accepted 20 December 2005 Available online 7 July 2006

Abstract

Let(Z_n)_n_∈Nbe ad-dimensionalrandom walk in random scenery, i.e.,Z_n=_n₋₁

k=0Y (S_k)with(S_k)_k_∈N₀ a random walk in Z^d and(Y (z))_z_∈Zd an i.i.d. scenery, independent of the walk. The walker’s steps have mean zero and some finite exponential moments. We identify the speed and the rate of the logarithmic decay ofP(¹_nZn> bn)for various choices of sequences(bn)nin [1,∞). Depending on(b_n)_nand the upper tails of the scenery, we identify different regimes for the speed of decay and different variational formulas for the rate functions. In contrast to recent work [A. Asselah, F. Castell, Large deviations for Brownian motion in a random scenery, Probab. Theory Related Fields 126 (2003) 497–527] by A. Asselah and F. Castell, we consider sceneries unboundedto infinity. It turns out that there are interesting connections to large deviation properties of self-intersections of the walk, which have been studied recently by X. Chen [X. Chen, Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks, Ann. Probab. 32 (4) 2004].

Résumé

Soit(Zn)_n_∈Nune marche aléatoire en paysage aléatoire surZ^d; il s’agit du processus défini parZn=_n₋₁

k=0Y (S_k), où(S_k)_k_∈N₀ est une marche aléatoire à valeurs dansZ^d, et le paysage aléatoire(Y (z))_z_∈Zdest une famille de variables aléatoires i.i.d. indépen- dante de la marche. On suppose queS₁est centrée et admet certains moments exponentiels finis. Nous identifions la vitesse et la fonction de taux deP(¹_nZ_n> b_n), pour diverses suites(b_n)_nà valeurs dans[1,∞[. Selon le comportement de(b_n)_net de la queue de distribution du paysage aléatoire, nous découvrons différents régimes ainsi que différentes formules variationnelles pour les fonctions de taux. Contrairement au travail récent de A. Asselah et F. Castell [A. Asselah, F. Castell, Large deviations for Brownian motion in a random scenery, Probab. Theory Related Fields 126 (2003) 497–527], nous étudions le cas où le paysage aléatoire n’est pas borné. Finalement, nous observons des liens intéressants avec certaines propriétés d’auto-intersection de la marche(S_k)_k_∈N₀, récemment étudiées par X. Chen [X. Chen, Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks, Ann. Probab. 32 (4) 2004].

MSC:60K37; 60F10; 60J55

Keywords:Random walk in random scenery; Local time; Large deviations; Variational formulas

* Corresponding author.

E-mail addresses:gantert@math.uni-muenster.de (N. Gantert), koenig@math.uni-leipzig.de (W. König), zhan@proba.jussieu.fr (Z. Shi).

doi:10.1016/j.anihpb.2005.12.002

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1. Introduction

1.1. Model and motivation

Let S=(S_n)_n_∈N₀ be a random walk on Z^d starting at the origin. We denote by P the underlying probability measure and byEthe corresponding expectation. We assume thatE[S₁] =0 andE[|S₁|²]<∞. Defined on the same probability space, letY =(Y (z))_z_∈Zd be an i.i.d. sequence of random variables, independent of the walk. We refer to Y as therandom scenery.Then the process(Zn)_n_∈Ndefined by

Zn=

n−1

k=0

Y (Sk), n∈N,

whereN= {1,2, . . .}, is called arandom walk in random scenery, sometimes also referred to as theKesten–Spitzer random walk in random scenery, see [19]. An interpretation is as follows. If a random walker has to payY (z)units at any time he/she visits the sitez, thenZ_nis the total amount he/she pays by timen−1.

The random walk in random scenery has been introduced and analyzed for dimensiond =2 by H. Kesten and F. Spitzer [19] and by E. Bolthausen [6] ford=2. The cased=1 was treated independently by A.N. Borodin [7,8].

Under the assumption thatY (0)has expectation zero and varianceσ²∈(0,∞), their results imply that 1

nZ_n≈a_n⁽⁰⁾=

⎧⎨

⎩

n⁻^1/4 ifd=1, (_lognⁿ )⁻^1/2 ifd=2, n⁻^1/2 ifd3.

(1.1) More precisely,(1/na⁽⁰⁾n )Z_n converges in distribution towards some non-degenerate random variable. The limit is Gaussian ind2 and a convex combination of Gaussians (but not Gaussian) ind=1. This can be roughly explained as follows. In terms of the so-calledlocal timesof the walk,

_n(z)=

n−1

k=0

1_{Sk=z}, n∈N, z∈Z^d, (1.2)

the random walk in random scenery may be identified as Z_n=

z∈Z^d

Y (z)_n(z). (1.3)

The number of effective summands in (1.3) is equal to therangeof the walk, i.e., the number of sites visited by time n−1. Hence, conditional on the random walk,Z_n is, for dimensiond 3, a sum of O(n)independent copies of finite multiples ofY (0), and hence it is plausible thatZ_n/n^1/2 converges to a normal variable. The same assertion with logarithmic corrections is also plausible ind=2. However, in d=1,Z_n is roughly a sum ofO(n^1/2)copies of independent variables with variances of orderO(n), and this suggests the normalization in (1.1) as well as a non- normal limit.

In this paper, we analyze deviations{¹_nZn> bn}for various choices of sequences(bn)n∈Nin[1,∞). We determine the speed and the rate of the logarithmic asymptotics of the probability of this event asn→ ∞, and we explain the typical behavior of the random walk and the random scenery on this event.

This problem has been addressed in recent work [11,1] and [10] by F. Castell in partial collaboration with F. Pradeilles and A. Asselah for Brownian motion instead of random walk. While [11] and [10] treat the case of a continuous Gaussian scenery forb_n=n^1/2 and cst.b_nn^1/2, respectively, the case of an arbitrary bounded scenery (constant on the unit cubes) andb_n=cst.is considered in [1]. See also [1] for further references on this topic and [2] and [16] for recent results on the random walk case.

The main novelty of the present paper is the study of arbitrary sceneriesunbounded to +∞and general scale functionsbncst.in the discrete setting. On the technical side, in particular the proof of the upper bound is rather demanding and requires new techniques. We solve this part of the problem by a careful analysis of high integer moments, a technique which has been recently established in the study of intersection properties of random motions.

A very rough, heuristic explanation of the interplay between the deviations of the random walk in random scenery and the tails of the scenery at infinity and the dimensiondis as follows. In order to realize the event{¹_nZ_n> b_n}, it is

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clear that the scenery has to assume larger values on the range of the walk than usual. In order to keep the probabilistic cost for this low, the random walker has to keep its range small, i.e., it has to concentrate on less sites by timenthan usual. Theoptimaljoint strategy of the scenery and the walk is determined by a balance between the respective costs.

The optimal strategies in the cases considered in the present paper are homogeneous. More precisely, the scenery and the walk each approximate optimal (rescaled) profiles in a large, n-dependent box. These optimal profiles are determined by a (deterministic) variational problem.

The topic of the present paper has deep connections to large deviation properties of self-intersections of the walk.

This is immediate in the important special case of a standard Gaussian sceneryY. Indeed, the conditional distribution ofZ_ngiven the random walkSis a centered Gaussian with variance equal to

Λ_n=

z∈Z^d

_n(z)²= _n ²₂, (1.4)

which is often called theself-intersection local time. Hence, large deviations for the random walk in Gaussian scenery would be a consequence of an appropriate large deviation statement for self-intersection local times. However, the latter problem is notoriously difficult and is, up to the best of our knowledge, open in the precision we would need in the present paper. (However, compare to interesting and deep work on self-intersections and mutual intersections by X. Chen [12].) Recent results for self-intersection local times for random walks in dimensiond5 and applications to random walk in random scenery are given in [3].

The remainder of Section 1 is organized as follows. Our main results are in Section 1.2, a heuristic derivation may be found in Section 1.3, a partial result for Gaussian sceneries for dimensiond=2 is in Section 1.4. The structure of the remainder of the paper is as follows. In Section 2 we analyze the variational formulas, in Section 3 we present the tools for our proofs of the main results, in Sections 4 and 5 we give the proofs of the upper and the lower bounds, respectively, and finally in Appendix A, we provide the proof of a large deviation principle that is needed in the paper.

1.2. Results

Our precise assumptions on the random walk,S, are the following. The walker starts atS0=0, and the steps have mean zero and some finite exponential moments, more precisely,

E e^t^|^S¹^|

<∞ for somet >0. (1.5)

ByΓ ∈R^d^×^dwe denote the covariance matrix of the walk’s step distribution. Hence,Slies in the domain of attraction of the Brownian motion with covariance matrixΓ. We assume thatΓ is a regular matrix. Furthermore, we assume thatSis strongly aperiodic, i.e., for anyz∈Z^d, the smallest subgroup ofZ^dthat contains{z+x: P(S₁=x) >0}is Z^d itself. Finally, to avoid technical difficulties, we also assume that the transition function of the walk is symmetric, i.e.,p(0, z)=p(0,−z)forz∈Z^d, wherep(z,z)˜ denotes the walker’s one-step probability fromz∈Z^dtoz˜∈Z^d.

Our assumptions on the scenery are the following. LetY =(Y (z))_z_∈Zd be a family of i.i.d. random variables, not necessarily having finite expectation, such that

E e^{t Y (0)}

<∞ for everyt >0. (1.6)

In particular, thecumulant generating function ofY (0), is finite:

H (t )=logE e^{t Y (0)}

<∞, t >0. (1.7)

In some of our results, we additionally suppose the following.

Assumption (Y).There are constantsD >0 andq >1 such that logP Y (0) > r

∼ −Dr^q, r→ ∞.

According to Kasahara’s exponential Tauberian theorem (see [5, Theorem 4.12.7]), Assumption (Y) is equivalent to

H (t )∼Dt^p, ast→ ∞, whereD=(q−1) Dq^q1/(1−q)

and 1 q +1

p =1. (1.8)

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In our first main result, we consider the case of sequences(bn)n tending to infinity slower thann^1/q. By∇ we denote the usual gradient acting on sufficiently regular functionsR^d→R. ByH¹(R^d)we denote the usual Sobolev space, and we write ∇ψ ²₂=

R^d|∇ψ (x)|²dx. We use the notationb_nc_nif limn→∞b_n/c_n= ∞.

Theorem 1.1(Very large deviations). Suppose that Assumption(Y)holds with someq >^d₂. Pick a sequence(bn)n∈N

satisfying1bnn^1/q. Then

nlim→∞n⁻^d/(d⁺²⁾b⁻_n^2q/(d⁺²⁾logP 1

nZn> bn

= −KD,q, (1.9)

where

K_D,q≡inf 1

2Γ^1/2∇ψ²

2+Dψ²⁻^q

p : ψ∈H¹ R^d

, ψ ₂=1

, (1.10)

(we recall that_p¹+_q¹=1), andK_D,qis positive.

Remark 1.2. Forq ∈(1,^d₂), (1.9) also holds true, but K_D,q =0. Indeed, this follows from Proposition 1.6 below together with our proof of Theorem 1.1. One can also see this directly by giving an explicit lower bound for logP(_n¹Z_n> b_n)which runs on a strictly smaller scale thann^d/(d⁺²⁾b^2q/(d_n ⁺²⁾. It remains an open problem in this paper to determine thepreciselogarithmic rate ofP(¹_nZ_n> b_n)in the caseq∈(1,^d₂). The caseq=^d₂ seems even more delicate and is also left open in the present paper. The caseq∈(0,1)has been studied in [16].

Note that the variational problem in (1.10) is of independent interest; it also appeared in [4, Theorem 1.1] in the context of heat kernel asymptotics. In Proposition 1.6 below it turns out thatKD,qis positive if and only ifq^d₂.

Our next result essentially extends [1, Theorem 2.2] from the case of bounded sceneries to the case in (1.6).

Theorem 1.3(Large deviations). Suppose that(1.6)holds. Assume thatE[Y (0)] =0, and setp¯≡lim sup_t_→∞^log_logt^{H (t )}. Assume thatp <¯ ∞ind2respectivelyp <¯ _d₋^d₂ind3. Then, for anyu >0satisfyingu∈supp(Y (0))^◦,

nlim→∞n⁻^d/(d⁺²⁾logP 1

nZn> u

= −KH(u), (1.11)

where

K_H(u)≡inf 1

2Γ^1/2∇ψ²

2+Φ_H ψ², u

: ψ∈H¹ R^d

, ψ 2=1

, (1.12)

and

Φ_H ψ², u

= sup

γ∈(0,∞)

γ u−

R^d

H γ ψ²(y) dy

. (1.13)

The constantK_H(u)is positive.

Switching to the scenery−Y, one may, under appropriate conditions, use Theorem 1.3 to obtain the ‘other half’ of a full large deviation principle for(_n¹Z_n)_n. This was carried out in [1] for bounded sceneries. For Brownian motion in a Gaussian scenery, a result analogous to Theorems 1.1 and 1.3 is [10, Theorem 2].

Note that the constantK_H(u)depends on the entire scenery distribution, whileK_D,qin (1.10) only depends on its upper tails.

Remark 1.4.A statement analogous to Remark 1.2 also applies here: for dimensionsd3, when lim inft→∞logH (t ) logt >

d

d−2, (1.11) also holds true, butK_H(u)=0 for anyu >0. It was shown recently in [2] that under Assumption (Y) withq∈(1,^d₂)and an additional symmetry assumption, logP(_n¹Z_n> b_n)is of the ordern^q/(q⁺²⁾. The caseq∈(0,1) has been studied in [16].

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Remark 1.5(Large deviations and non-convexity).It is easy to see that, in the special case whereH (t )=Dt^p (see (1.8)),K_H(u)=u^2q/(d⁺²⁾K_D,q, for anyu >0. (For asymptotic scaling relations see Lemma 1.7.) In particular,_n¹Z_n satisfies a large deviation principle on(0,∞)with speedn^d/(d⁺²⁾and rate functionu→u^2q/(d⁺²⁾KD,q. This function is strictly convex forq >^d₂+1 and strictly concave forq <^d₂+1. In the important special case of a centered Gaussian scenery, Theorem 1.3 contains non-trivial information only in the case d ∈ {1,2,3}, in which the rate function is strictly convex, linear and strictly concave, respectively; see also [11] and [10].

The non-convexity around zero for bounded sceneries in d∈ {3,4}was found in [1] by proving thatK_H(u) Cu^4/(d⁺²⁾asu→0 for some positive constantC.

The upper bounds in Theorems 1.1 and 1.3 are proved in Section 4, and the lower bounds in Section 5. We consider only sequencesbn1 there. The casea_n⁽⁰⁾bn1 seems subtle and is left open in the present paper; however see Section 1.4 for a partial result.

Our next proposition gives almost sharp criteria for the positivity of the constantsK_D,q andK_H(u)appearing in Theorems 1.1 and 1.3.

Proposition 1.6(Positivity of the constants). Fixd∈Nandp, q >1satisfying _p¹+_q¹=1.

(i) For anyD >0, K_D,q=(d+2)

D 2

2/(d+2) χ_d,p

d

d/(d+2)

, (1.14)

where

χ_d,p=inf 1

2Γ^1/2∇ψ²

2: ψ∈H¹ R^d

: ψ ₂=1= ψ _2p

. (1.15)

The constantχ_d,pis positive if and only ifd_p^2p₋₁=2q. Hence,K_D,q is positive if and only ifd_p^2p₋₁=2q.

(ii) The constantK_H(u)is positive for anyu >E[Y (0)] =0if lim sup

t→∞

logH (t ) logt <

∞ ifd2,

d

d−2 ifd3.

Ford3, iflim inft→∞logH (t )

logt >_d₋^d₂, thenK_H(u)=0for anyu >0.

The proof of Proposition 1.6 is in Section 2. There we also clarify the relation betweenχ_d,p and the so-called Gagliardo–Nirenbergconstant.

Now we formulate asymptotic relations between the rates obtained in Theorems 1.1 and 1.3.

Lemma 1.7(Asymptotic scaling relations). FixD >0andq >1, and recall(1.8).

(i) Assume thatH (t )∼Dt ^past→ ∞, then

K_H(u)∼u^2q/(d⁺²⁾K_D,q asu→ ∞. (1.16)

(ii) Assume thatE[Y (0)] =0andE[Y (0)²] =1, then K_H(u)u^4/(d⁺²⁾

K1

2,2+o(1)

asu↓0. (1.17)

The proof of Lemma 1.7 is in Section 2.4.

Remark 1.8.We conjecture that the lower bound in (1.17) also holds under an appropriate upper bound onH. It is clear (see Remark 1.5 and note the monotonicity ofK_H(u) inH) thatu⁻^4/(d⁺²⁾K_H(u)K_D,2for everyu >0 if H (t )Dt ²for everyt0. The positivity of lim infu↓0u⁻^4/(d⁺²⁾K_H(u)(for cumulant generating functions H of boundedvariables) is contained in [1] as part of the proof for non-convexity of the rate functionK_H ind ∈ {3,4}.

SinceK_D,2=0 ind >4, it is clear that this proof must fail ind >4.

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Lemma 1.7(i) is consistent with Theorems 1.1 and 1.3.

1.3. Heuristic derivation of Theorems 1.1 and 1.3

The asymptotics in (1.9) and (1.11) are based on large deviation principles for scaled versions of the walker’s local times_n and the scenery Y. A short summary of the joint optimal strategy of the walker and the scenery is the following. Let us first explain the exponential decay rate of the probabilities under consideration. Assume that 1b_nn^1/q. In order to contribute optimally to the event{_n¹Z_n> b_n}, the walker spreads out over a region whose diameter is of orderα_n(for a particular choice ofα_n, depending on the sequence(b_n)_n). The cost for this behavior is e^O^(nα⁻²ⁿ ⁾. The scenery assumes extremely large values within that region, more precisely: values of the orderb_n. The cost for doing that is exp{O(b^q_nα^d_n)}, under Assumption (Y). The choice ofα_nis now determined by putting

n

α_n²=α_n^db^q_n. (1.18)

A calculation shows that for this choice ofα_n both sides of (1.18) are equal to the logarithmic decay order of the probabilityP(¹_nZ_n> b_n)in Theorem 1.1.

Next we give a more precise argument for the very large deviations (Theorem 1.1) which also explains the constants on the right-hand side of (1.9). Introduce the scaled and normalized version of the walker’s local times,

L_n(x)=α^d_n

n _n xα_n

, x∈R^d. (1.19)

ThenL_nis a random element of the set F=

ψ²∈L¹ R^d

: ψ ₂=1

(1.20) of all Lebesgue probability densities onR^d. Furthermore, introduce the scaled version of the field,

Y_n(x)= 1

b_nY xα_n

, x∈R^d. (1.21)

Then we have, writing·,·for the inner product onL²(R^d), 1

nZn=1 n

z∈Z^d

n α^d_nLn

z α_n

bnYn

z α_n

=bn

Ln,Yn

. (1.22)

Hence, the logarithmic asymptotics of the probabilityP(¹_nZ_n> b_n)=P(L_n,Y_n>1)will be determined by a combination of large deviation principles forL_nandY_n.

In the spirit of the celebrated large deviation theorem of Donsker and Varadhan, the distributions ofL_n satisfy a weak large deviation principle in the weakL¹-topology onF with speednα⁻_n²and rate functionI:F→ [0,∞]

given by I(ψ²)=

1

2 Γ^1/2∇ψ ²₂ ifψ∈H¹(R^d),

∞ otherwise. (1.23)

Roughly speaking, this principle says that, forψ²∈F, P L_n≈ψ²

≈exp

−n

α²_nI ψ²

, n→ ∞. (1.24)

Using Assumption (Y), we see that the distributions ofYn should satisfy, for any R >0, a weak large deviation principle on some appropriate set of sufficiently regular functions[−R, R]^d →(0,∞) with speed α_n^dbn^q and rate function

Φ_D,q(ϕ)=D

[−R,R]^d

ϕ^q(x)dx,

as the following heuristic calculation suggests:

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P Y_n≈ϕon[−R, R]^d

≈P

Y (z) > b_nϕ z

α_n

forz∈ [−Rα_n, Rα_n]^d∩Z^d

≈

z∈[−Rαn,Rαn]^d∩Z^d

exp

−D

b_nϕ z

αn

q

≈exp

−Dα_n^db^qn

[−R,R]^d

ϕ^q(x)dx

. (1.25)

Note that the speeds of the two large deviation principles in (1.24) and (1.25) are equal because of (1.18). Using the two large deviation principles and (1.22), we see that

P 1

nZ_n> b_n

≈exp

−n α_n²K_D,q

,

where

K_D,q=inf I ψ²

+D ϕ ^q_q: ψ²∈F, ϕ∈C+ R^d ,

ψ², ϕ

=1

. (1.26)

It is an elementary task to evaluate the infimum onϕ and to check that indeedKD,q=KD,q. This ends the heuristic explanation of Theorem 1.1.

The situation in the large deviation case, Theorem 1.3, is similar, when we put bn=1. See [1] for a heuristic argument in this case.

We distinguish the two cases of very large deviations (V) and large deviations (L). The choices ofbnandαnin the respective cases are the following.

case (V): Hypothesis of Theorem 1.1, 1b_nn^1/q, α_n=n^1/(d⁺²⁾b_n⁻^q/(d⁺²⁾,

case (L): Hypothesis of Theorem 1.3, b_n=1, α_n=n^1/(d⁺²⁾. (1.27) 1.4. Small deviations for Gaussian sceneries

Theorems 1.1 and 1.3 do not handle sequences(b_n)_nsatisfying a_n⁽⁰⁾b_n1, where we recall from (1.1) that a⁽⁰⁾_n is the scale of the convergence in distribution. In this regime, we present a partial result for Gaussian sceneries and simple random walk ind=2. This result is based on a deep result by Brydges and Slade [9] about exponential moments of the renormalized self-intersection local time of simple random walk.

Lemma 1.9(Small deviations for Gaussian sceneries). Assume thatY (0)is a standard Gaussian random variable and that(Sn)nis the simple random walk, and assume thatd=2. Letn⁻^1/2(logn)^1/2=an⁽⁰⁾bnan⁽¹⁾≡n⁻^1/2logn, then

nlim→∞

logn b²_nn logP

1

nZ_n> b_n

= −π

4. (1.28)

Proof. As we mentioned in Section 1.1, the distribution of the random walk in random scenery,Zn, is easily identified in terms of the walk’s self-intersection local timeΛndefined in (1.4). More precisely, the conditional distribution of Zn given the walk S is N ×√

Λn, whereN is a standard normal variable, independent of the walk. The typical behavior of the self-intersection local time is as follows [9]

E[Λ_n] ∼ 2

π na⁽⁰⁾_n 2

= 2

πnlogn, n→ ∞. (1.29)

We prove now the upper bound in (1.28). Recall thatd=2 and introduce the centered and normalized self-intersection local time,

γ_n=1

n Λ_n−E[Λ_n] .

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Use Chebyshev’s inequality to obtain, for anyθ >0 and anyn∈N, P

1

nZ_n> b_n

E

e^{θ Z}ⁿ

e⁻^{θ b}ⁿⁿ. (1.30)

Using the above characterization of the distribution ofZ_n, we see that E

e^{θ Z}ⁿ

=E E

e^{θ Z}ⁿS

=E E

exp θN

ΛnS

=E e¹²^θ²^Λⁿ

=E e¹²^θ²^nγⁿ

e¹²^θ²^E[^Λⁿ^]. (1.31) According to Theorem 1.2 in [9], limn→∞E[e^cγⁿ]exists and is finite for anyc < c₀, wherec₀>0 is some positive constant. Now pickθ=θ_n=π b_n/(2 logn). Note thatθ_n²n→0 because ofb_nn⁻^1/2logn, and therefore the first factor on the right-hand side of (1.31) is bounded, according to the above mentioned result of Brydges and Slade. Use (1.29) on the right-hand side of (1.31) and substitute in (1.30) to obtain

logP 1

nZ_n> b_n

− 1+o(1)π 4

b²_nn logn. This is the upper bound in (1.28).

Now we prove the lower bound in (1.28). Using the above characterization of the distribution ofZ_n, we obtain, for anyθ >0,

P 1

nZn> bn

P(N> θ )P

Λn>n²b²_n θ²

. (1.32)

Fix an arbitraryc∈(0,_π²). We apply (1.32) toθ=b_n(_c_lognⁿ )^1/2and obtain logP

1

nZ_n> b_n

−1

2b²_n n

clogn 1+o(1)

+logP(Λ_n> cnlogn), n→ ∞.

By the Paley–Zygmund inequality (Kahane [18] p. 8) stating thatP(X > rE[X])(1−r)²E[X]²/E[X²]for all r∈(0,1)and all square-integrable random variablesX, we obtain that

P(Λ_n> cnlogn)

1− cπ

2 2

E[Λn]² E[Λ²_n].

Recall from (1.29) thatE[Λ_n] ∼_π²nlognasn→ ∞. On the other hand, Bolthausen [6] proved that Var[Λ_n] =O(n²).

Therefore,E[Λ²_n] ∼E[Λ_n]², and, consequently, lim inf

n→∞ P(Λn> cnlogn) >0.

Therefore, lim inf

n→∞

logn b²_nn logP

1

nZn> bn

−1

2c.

Lettingc↑_π², this yields the lower bound in (1.28). 2 2. Variational formulas

In this section we prove Proposition 1.6 and Lemma 1.7. In Section 2.1 we prove a necessary and sufficient criterion for positivity of the constantχ_d,pdefined in (1.15). The relation to the Gagliardo–Nirenberg constant is discussed in Section 2.2, and the relation to the constantK_D,qdefined in (1.10) is proved in Section 2.3, where we also finish the proof of Proposition 1.6. Finally, Lemma 1.7 is proved in Section 2.4.

2.1. Positivity ofχ_d,p

Lemma 2.1.The constantχ_d,pis positive if and only ifd_p^2p₋₁.

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Proof. Certainly, it suffices to do the proof only in the case where¹₂Γ is the identity matrix.

See [12, Section 2] for an alternate proof of the positivity ofχ_d,pin the subcritical dimensions,d < _p^2p₋₁, using the relation to the Gagliardo–Nirenberg constant, which we explain in Section 2.2.

Let us recall standard Sobolev inequalities (see [21, Theorems 8.3, 8.5]). There are positive constantsS_d ford3 andS2,r forr >2 such that

Sd ψ ²_2d/(d₋₂₎ ∇ψ ²₂, ford3, ψ∈D¹ R^d

∩L² R^d , S_2,r ψ ²_r ∇ψ ²₂+ ψ ²₂, ford=2, ψ∈H¹ R^d

, r >2. (2.1)

HereD¹(R^d)denotes the set of locally integrable functionsR^d→Rwhich vanish at infinity and possess a distribu- tional derivative inL²(R^d).

Let us first do the proof for the case 3d_p^2p₋₁. For anyψ∈H¹(R^d)that satisfies ψ 2=1= ψ 2p, we may use the above Sobolev inequality and obtain that ∇ψ ²₂cst. ψ ²_2d/(d₋₂₎. We now rewrite

R^d

ψ^2d/(d⁻²⁾(t )dt=

R^d

ψ^2p⁻²(t )2/((d−2)(p−1))

ψ²(t )dt.

Recall that ψ² is a probability density. Therefore, an application of Jensen’s inequality to the convex map x→ x^2/^[^(d⁻^2)(p⁻¹⁾^]yields that ψ _2d/(d₋₂₎satisfies a lower bound in terms of a power of ψ _2p, which is equal to one.

Hence, on the set of thoseψ∈H¹(R^d)that satisfy ψ ₂=1= ψ _2p, the mapψ→ ∇ψ ²₂is bounded away from zero. Now compare to (1.15) to see that this implies the assertion in the case 3d_p^2p₋₁.

Now we turn tod=2 withp >1 arbitrary. By a scalingψβ=β^d/2ψ (·β), we can find, for anyδ >0, ac(δ) >0 such that

χ_2,p=c(δ)inf

∇ψ ²₂: ψ∈H¹ R^d

, ψ ₂=1, ψ _2p=δ

. (2.2)

Now we chooseδsuch that 2δ⁻²=S_2,2p, the Sobolev constant in (2.1) ford=2 andr=2p. Then we have, for any ψin the set on the right-hand side of (2.2),

2= 2

δ² ψ ²_2p=S_2,2p ψ ²_2p ∇ψ ²₂+ ψ ²₂= ∇ψ ²₂+1, and hence it follows thatχ2,pc(δ) >0.

Now we show thatχ2d,p2χd,pfor anyd∈Nandp∈(0,∞). This simply follows from the observation that, for anyψ∈H¹(R^d), the functionψ⊗ψ∈H¹(R^2d)satisfies

∇(ψ⊗ψ )²

2=2 ∇ψ ²₂.

Using this, the estimateχ2d,p2χd,peasily follows, since ψ⊗ψ 2= ψ ²₂and ψ⊗ψ 2p= ψ ²_2p. In particular, this shows thatχ1,p>0 for anyp >1.

It remains to show thatχd,p=0 ford >_p^2p₋₁. It is sufficient to construct a sequence of sufficiently regular functions ψn:R^d → [0,∞)such that ψn 2 and ψn 2p both converge towards some positive numbers, but ∇ψn 2

vanishes asn→ ∞. In order to do this, pick some rotationally invariant functionψ²=f◦ | · | ∈Fwhose radial part f:(0,∞)→(0,∞)satisfies

f (r)=D×

⎧⎨

⎩

r⁻^γ ifr∈(0,1), 1 ifr∈ [1, A], A^2dr⁻^2d ifr > A,

whereA, D, γ >0 are constants to be determined. Letω_d denote the surface of the unit ball inR^d. The following statements can be easily verified by some tedious but elementary calculations:

γ < d ⇒ ψ ²₂=ωd

d D

2A^d+ γ d−γ

<∞, (2.3)

γ < d

p ⇒ ψ ^2p_2p=ω_dD^pp d

γ

d−pγ +A^d 2 2p−1

<∞, (2.4)

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γ < d−2 ⇒ ∇ψ ²₂=1 4ω_dD

γ²

d−γ−2 +A^d⁻² 4d² 2+d

<∞. (2.5)

Sincep >1 and ^d_p< d−2, we only have to assume thatγ <^d_p. Now we pick sequencesD_n,A_nandγ_nsuch that all the following conditions are satisfied asn→ ∞:

D_n→0, A_n→ ∞, γ_n↑ d

p, D_nA^d_n→1, D_n^p

d−pγ_n→1.

Letψ_nbe defined as theψabove with these parameters. Then we have, asn→ ∞, ψ_n ²₂→2ω_d

d , ψ_n ^2p_2p→ω_d, ∇ψ_n ²₂→0.

This ends the proof. 2

2.2. Relation to the Gagliardo–Nirenberg constant

Actually, for dimensionsd2 in the special case that¹₂Γ is the identity matrix, the constantχ_d,pin (1.15) can be identified in terms of theGagliardo–Nirenbergconstant,κ_d,p, as follows. Assume thatd2 and 1< p <_d^d₋₂. Then κd,pis defined as the smallest constantC in theGagliardo–Nirenberg inequality

ψ _2pC ∇ψ

d(p−1) 2p

2 ψ ¹⁻

d(p−1) 2p

2 , ψ∈H¹ R^d

. (2.6)

This inequality received a lot of interest from physicists and analysts, and it has deep connections to Nash’s inequality and logarithmic Sobolev inequalities. Furthermore, it also plays an important role in recent work of Chen [12] on self-intersections of random walks. See [12, Section 2] for more on the Gagliardo–Nirenberg inequality.

It is clear that κd,p= sup

ψ∈H¹(R^d),ψ=0

ψ _2p ∇ψ

d(p−1) 2p

2 ψ ¹⁻

d(p−1) 2p

2

=

inf

ψ∈H¹(R^d): ψ ₂=1

ψ ⁻

4q d

2p ∇ψ ²₂ ₋^d

4q. (2.7)

Clearly, the term over which the infimum is taken remains unchanged ifψ is replaced byψ_β(·)=β^d/2ψ (·β) for any β >0. Hence, we can freely add the condition ψ 2p=1 and obtain that κ_d,p=χ_d,p⁻^d/(4q). In particular, the variational formulas forκd,p in (2.7) and forχd,p in (1.15) have the same maximizer(s) respectively minimizer(s).

It is known that (2.7) does possess a maximizer, and this is an infinitely smooth, positive and rotationally invariant function (see [26]). Uniqueness of the minimizer holds ind∈ {2,3,4}for anyp∈(1,_d₋^d₂), and ind∈ {5,6,7}for anyp∈(1,_d⁸), see [22].

2.3. Relation betweenK_D,qandχ_d,p(Proposition 1.6)

Now we prove the remaining assertions of Proposition 1.6.

(i) The relation (1.14) is proved by an elementary scaling argument and optimization. Indeed, replaceψbyψβ(·)= β^d/2ψ (·β)in (1.10) and optimize explicitly onβ >0. Afterwards the additional constraint ψ 2p=1 may freely be added. From (1.14) and Lemma 2.1 the last assertion follows.

(ii) We only show the positivity ofK_H(u)ford3 andp <¯ _d₋^d₂; the argument ford2 and anyp >¯ 1 is the same.

Since we assumed thatE[Y (0)] =0, we may pick someδ >0 such that H (t )ut /2 fort∈ [0, δ]. Pickε >0 such thatp¯+ε < _d₋^d₂, then there isc(δ, ε) >0 depending onδ, εandH only, such thatH (t )c(δ, ε)t^p^¯⁺^εfor any t∈ [δ,∞). ThenH (t )^u₂t+c(δ, ε)t^p^¯⁺^εfor anyt0, which implies that, for anyψ∈H¹(R^d)satisfying ψ 2=1,

Φ_H ψ², u

sup

γ >0

γ u−

uγ

2 ψ²(x)dx−

c(δ, ε) γ ψ²(x)p_¯+ε

dx

=sup

γ >0

u

2γ−c(δ, ε)γ^p^¯⁺^εψ²^p^¯⁺^ε

¯ p+ε

.