ELECTRONIC COMMUNICATIONS in PROBABILITY
A NOTE ON DIRECTED POLYMERS IN GAUSSIAN ENVIRONMENTS
YUEYUN HU
Département de Mathématiques, Université Paris XIII, 99 avenue J-B Clément, 93430 Villetaneuse, France
email: [email protected]
QI-MAN SHAO
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
email: [email protected]
SubmittedMay 28, 2009, accepted in final formSeptember 19, 2009 AMS 2000 Subject classification: Primary 60K37
Keywords: Directed polymer, gaussian environment.
Abstract
We study the problem of directed polymers in gaussian environments inZd from the viewpoint of a gaussian family indexed by the set of random walk paths. In the zero-temperature case, we give a numerical bound on the maximum of the Hamiltonian, whereas in the finite temperature case, we establish an equivalence between thevery strong disorderand the growth rate of the entropy associated to the model.
1
Introduction and main results
1.1
Finite temperature case
Let (g(i,x))i≥0,x∈Zd be i.i.d. standard real-valued gaussian variables. We denote byPandEthe corresponding probability and expectation with respect to g(·,·). Let {Sk,k ≥ 0} be a simple symmetric random walk onZd, independent of g(·,·). We denote byPx the probability measure of(Sn)n∈N starting atx∈Zd and byEx the corresponding expectation. We also writeP=P0and
E=E0.
The directed polymer measure in a gaussian random environment, denoted by〈·〉(n), is a random
probability measure defined as follows: LetΩnbe the set of nearest neighbor paths of length n:
Ωndef=
n
γ:{1, ...,n} →Zd,|γ
k−γk−1|=1,k=2, . . . ,n,γ0=0
o
. For any functionF:Ωn→R+,
〈F(S)〉(n)def= 1
ZnE
F(S)eβHn(g,S)−β
2n 2
, Hn(g,γ)
def =
n X
i=1
g(i,γi),
whereβ >0 denotes the inverse of temperature andZnis the partition function: Zn=Zn(β,g) =E
eβHn(g,S)−β
2n 2
.
We refer to Comets, Shiga and Yoshida[3]for a review on directed polymers. It is known (see e.g.[2],[3]) that the so-called free energy, the limit of1nlogZnexists almost surely and in L1:
p(β):= lim n→∞
1 nlogZn,
p(β)is some constant andp(β)≤0 by Jensen’s inequality sinceEZn=1.
A problem in the study of directed polymer is to determine the region of{β >0 :p(β)<0}, also called the region ofvery strong disorder. It is an important problem, for instance,p(β)<0 yields interesting information on the localization of the polymer itself.
By using the F-K-G inequality, Comets and Yoshida[4]showed the monotonicity ofp(β), there-fore the problem is to determine
βc:=inf{β >0 :p(β)<0}.
It has been shown by Imbrie and Spencer[8]that ford≥3,βc>0 (whose exact value remains unknown). Comets and Vargas[5]proved that (for a wide class of random environments)
βc=0, if d=1. (1.1)
Recently, Lacoin[10], skilfully used the ideas developed in pinned models and solved the problem in the two-dimensional case:
βc=0, if d=2. (1.2)
Moreover, Lacoin[10]gave precise bounds on p(β)when β → 0 both in one-dimensional and two-dimensional cases.
In this note, we study this problem from the point of view of entropy (see also Birkner[1]). Let
en(β):=E
ZnlogZn
−(EZn)log(EZn) =E
ZnlogZn
be the entropy associated toZn(recallingEZn=1). Theorem1.1. Letβ >0. The following limit exits
˜e∞(β):= lim n→∞
en(β) n =ninf≥1
en(β)
n ≥0. (1.3)
There is some numerical constantcd>0, only depending ond, such that the following assertions are equivalent:
(a) ˜e∞(β)>0 . (b) lim sup
n→∞
en(β)
The proof of the implication(b)=⇒(c)relies on a criterion ofp(β)<0 (cf. Fact 3.1 in Section 3) developed by Comets and Vargas[5]in a more general settings.
We can easily check(b)in the one-dimensional case: In fact, we shall show in the sequel (cf. (3.7)) that in any dimension and for anyβ >0,
en(β)≥
β2
2 E(Ln(S
1,S2)),
whereS1andS2are two independent copies ofSandLn(γ,γ′) = Pn
k=11(γk=γ′k)is the number of common points of two pathsγandγ′. It is well known thatE(Ln(S1,S2))is of order n1/2when d=1 and of order lognwhen d=2. Therefore(b)holds ind=1 and by the implication(b)
=⇒(c), we recover Comets and Vargas’ result (1.1) in the one-dimensional gaussian environment case.
1.2
Zero temperature case
Whenβ→ ∞, the problem of directed polymers boils down to the problem of first-passage perco-lation. Let
H∗n=Hn∗(g):=max γ∈Ωn
Hn(g,γ), Hn(g,γ) = n X
1
g(i,γi), where as beforeΩn={γ:[0,n]→Zd,|γ
i−γi−1|=1,i=2, . . . ,n,γ0=0}.
The problem is to characterize these pathsγwhich maximize Hn(g,γ). See Johansson [9]for the solution of the Poisson points case. We limit here our attention to some explicit bounds onH∗n. An easy subadditivity argument (see Lemma 2.2) shows that
H∗n n → supn≥1
EH∗n n
def
=c∗d, botha.s. and inL1. By Slepian’s inequality ([12]),
EHn∗≤pnEmax γ∈Ωn
Yγ,
where(Yγ)γ∈Ωn is a family of i.i.d. centered gaussian variables of variance 1. Since #Ωn= (2d) n, it is a standard exercise from extreme value theory that
1
pnEmax
γ∈Ωn Yγ→
p
2 log(2d).
Hence
c∗d≤p2 log(2d).
It is a natural problem to ask whether this inequality is strict; In fact, a strict inequality means that the gaussian family{Hn(g,γ),γ∈Ωn} is sufficiently correlated to be significantly different from the independent one, exactly as the problem to determine whetherp(β)<0.
We prove that the inequality is strict by establishing a numerical bound: Theorem1.2. For anyd≥1, we have
cd∗≤ r
2 log(2d)−(2d−1)
2d
1−2d−1 5πd
(1−Φ(plog(2d))),
whereΦ(x) = p1 2π
Rx
−∞e−
u2/2
2
Proof of Theorem 1.2
We begin with several preliminary results. Recall at first the following concentration of measure property of Gaussian processes (see Ibragimov and al.[7]).
Fact2.1. Consider a functionF :RM→Rand assume that its Lipschitz constant is at mostA, i.e.
|F(x)−F(y)| ≤A||x−y|| (x,y∈RM),
Lemma2.2. There exists some positive constantc∗dsuch that H∗n
Proof:We prove at first the concentration inequality. Define a functionF:Rm→Rby
F(z) =max Gaussian concentration inequality Fact 2.1, we get (2.2).
Now we prove that n → EHn∗ is superadditive: for n,k ≥ 1, let γ∗ ∈ Ω
which in view of concentration (2.2) implies (2.1).
Define
φn(λ) =log X x←-n
EeλHn∗,x !
, φn∗(a) =sup λ>0
aλ−φn(λ).
Lemma2.3. For anyn≥1, Letζndef= inf{c>0 :φ∗n(cn)>0}. We have
cd∗≤ζn≤cd∗+2 r
dlog(2n+1)
n .
Proof:Letτn,xbe the time and space shift on the environment:
g◦τn,x(·,·) =g(n+·,x+·). (2.3) We have for anyn,k,
Hn∗+k=max x←-n
n max γ∈Ωn:γn=x
Hn(g,γ) +Hk∗(g◦τn,x) o
.
Write for simplificationH∗n,x:=maxγ∈Ωn:γn=xHn(g,γ). Then for anyλ∈R, EeλH∗n+k = Eeλmaxx←-n(H∗n,x+H∗k(g◦τn,x))
≤ E X
x←-n
eλH∗n,xeλHk∗(g◦τn,x) !
= E X
x←-n eλH∗n,x
! EeλHk∗.
We get
EeλH∗jn ≤ejφn(λ), j,n≥1,λ∈R. Chebychev’s inequality implies that
PH∗jn>c jn
≤e−jφn∗(cn), c>0,
whereφ∗
n(a) =supλ>0 aλ−φn(λ)
. Then for anynandcsuch thatφ∗
n(cn)>0, lim sup j→∞
H∗jn jn ≤c, a.s. It follows that
cd∗≤ζn, ∀n≥1.
On the other hand, by using the concentration inequality (2.2) and the fact thatE(H∗n)≤nc∗d, we get
EeλH∗n≤eλEH∗neλ2n/2≤eλnc∗d+λ
2n/2
, λ >0.
It follows that for anyλ >0,
φn(λ)≤logX x←-n
EλH∗n≤λnc∗ d+
λ2
2 n+2dlog(2n+1).
Choosingλ=2 q
dlog(2n+1)
n , we getφn(λ)< λn(c
∗
d+a)for anya>2 q
dlog(2n+1)
Proof of Theorem 1.2:Letn=2 in Lemma 2.3, we want to estimateφ2(λ).
Letg,g0,g1,g2,· · · be iid standard gaussian variables. Observe that the possible choices ofγ1are
(±1, 0,· · ·, 0),(0,±1, 0,· · ·, 0),· · · ,(0,· · ·, 0,±1), and the possible choices ofγ2are(0, 0,· · ·, 0),
(±2, 0,· · ·, 0),(±1,±1, 0· · ·, 0),(±1, 0,±1, 0,· · ·, 0),· · ·,(±1, 0,· · ·, 0,±1),· · ·,(0,±1,±1, 0,· · ·, 0),
· · ·,(0,· · ·,±1,±1). Therefore X
x←-2
EeλH∗2,x
= Eeλ(g0+max1≤i≤2dgi)+2dEeλ(g0+g1)+4(d−1+d−2+· · ·+1)Eeλ(g0+max(g1,g2))
= Eeλ(g0+max1≤i≤2dgi)+2dEeλ(g0+g1)+2d(d−1)Eeλ(g0+max(g1,g2)) ≤
d X
i=1
Eeλ(g0+max(g2i−1,g2i))+2d Eeλ(g0+g1)+2d(d−1)Eeλ(g0+max(g1,g2))
= 2dEeλ(g0+g1)+d(2d−1)Eeλ(g0+max(g1,g2))
= 2d eλ2+d(2d−1)eλ2/2Eeλmax(g1,g2)
= 2d eλ2+d(2d−1)eλ2/22eλ2/2Φ(λ/p2)
= eλ2(2d+2d(2d−1)Φ(λ/p2)), (2.4)
where we use the fact that
Eeλmax(g1,g2)=2eλ2/2Φ(λ/p2).
In fact, since max(g1,g2) = (1/2)(g1+g2+|g1−g2|)andg1+g2andg1−g2are independent,
we have
Eeλmax(g1,g2)=Eeλ(g1+g2+|g1−g2|)/2=eλ2/4Eeλ|g1|/
p
2=2eλ2/2
Φ(λ/p2),
where we use
Eeλ|g|=p2
2π
Z∞
0
eλx−x2/2d x= 2e
λ2/2 p
2π
Z∞
0
e−(x−λ)2/2d x=2eλ2/2Φ(λ). We conclude from (2.4) that
φ2(λ) ≤ λ2+log2d+2d(2d−1)Φ(λ/p2)
= λ2+2 log(2d) +log1−(2d−1)
2d (1−Φ(λ/
p
2))
≤ λ2+2 log(2d)−(2d−1)
2d (1−Φ(λ/
p
2))def=h(λ)
Now consider the function
h∗(2c):=sup λ>0
(2cλ−h(λ)), c>0.
Clearlyh∗(2c)≤φ∗(2c)for anyc>0. Let us study theh∗(2c): The maximal achieves when
2c=h′(λ)
That is
2c=2λ+(2d−1)
2dp2πe
Now choosecso that
2cλ=h(λ) =λ2+2 log(2d)−(2d−1)
2d (1−Φ(λ/
p
2)) (2.5)
Let
a= (2d−1)
4dp2π e
−λ2/4
, b=(2d−1)
2d (1−Φ(λ/
p
2)).
Then
2c=2λ+2a, 2cλ=λ2+2 log(2d)−b,
which gives
λ2+2aλ=2 log(2d)−b or
c2= (λ+a)2=2 log(2d)−b+a2. It is easy to see that
(1−Φ(t))≥ 5
16e
−t2
, fort>0 Hence
16 5
2d−1
4dp2π
2
(1−Φ(λ/p2))≥a2 and
c2 = 2 log(2d)−(2d−1)
2d (1−Φ(λ/
p
2)) +a2
≤ 2 log(2d)−
(2d−1)
2d −
2d−1
4dp2π
16
5
(1−Φ(λ/p2))
= 2 log(2d)−(2d−1)
2d
1−2d−1 5πd
(1−Φ(λ/p2))
≤ 2 log(2d)−(2d−1)
2d
1−2d−1 5πd
(1−Φ(plog(2d)))def=ec2,
Here in the last inequality, we used the fact thatλ≤p2 log(2d). Recall(c,λ)satisfying (2.5). For anyǫ >0,φ2∗(2(ec+ǫ))≥h∗(2(ec+ǫ))≥2(c+ǫ)λ−h(λ) =2ǫλ >0. It followsζ2≤ec+ǫ and
hencecd∗≤ec.
3
Proof of Theorem 1.1
Let
Zm(x):=E
1(Sm=x)eβHm(g,S)−β 2m/2
, m≥1,x∈Zd. Fact3.1 (Comets and Vargas[5]). If there exists somem≥1 such that
E X x
Zm(x)logZm(x)
≥0, (3.1)
In fact, the caseE PxZm(x)logZm(x)
=0 follows from their Remark 3.5 in Comets and Vargas
(2006), whereas ifE PxZm(x)logZm(x)
>0, which means the derivative ofθ →EPxZmθ(x)
at 1 is positif, hence for someθ <1,EPxZmθ(x)<1 and again by Comets and Vargas (2006) (lines before Remark 3.4), we havep(β)<0.
We try to check (3.1) in the sequel:
Lemma3.2. There exists some constantcd>0 such that for anym≥1, E X
where the probability measureQ=Q(β)is defined by
dQ|Fg some positive constantc′
d. Assembling all the above, we get a constantcd>0 such that E X
x
Zm(x)logZm(x)≥QlogZm−cdlogm
for allm≥1, as desired.
Let µ be a Gaussian measure onRm . The logarithmic Sobolev inequality says (cf. Gross[6], Ledoux[11]): for any f :Rm→R,
Using the above inequality, we have
Lemma3.3. LetS1andS2be two independent copies ofS. We have
where〈Ln(S1,S2)〉
(n)
2 =
1
Z2
n
E eβHn(g,S1)+βHn(g,S2)−β2nLn(S1,S2). Consequently,
en(β)≥
β2
2 E(Ln(S
1,S2)). (3.7)
Proof:Taking
f(z) =
v u u
tEexp βXn i=1
X
x
z(i,x)1(Si=x)−
β2
2 n !
, z= (z(i,x), 1≤i≤n,x←-i).
Note that Zn = f2(g)with g= (g(i,x), 1≤ i ≤n,x ←- i). Applying the log-Sobolev inequality
yields the first estimate (3.5).
The another assertion follows from the integration by parts: for a standard gaussian variable g and any derivable functionψsuch that bothgψ(g)andψ′(g)are integrable, we have
E(gψ(g)) =E(ψ′(g)).
Elementary computations based on the above formula yield (3.6). The details are omitted. From (3.5) and (3.6), we deduce that the functionβ→ en(β)
β2 is nondecreasing. On the other hand, it is
elementary to check that lim β→0
en(β)
β2 =
1 2E(Ln(S
1,S2)), which gives (3.7) and completes the proof
of the lemma.
IfPandQare two probability measures on(Ω,F), the relative entropy is defined by
H(Q|P)def=
Z logdQ
d PdQ,
where the expression has to be understood to be infinite ifQ is not absolutely continuous with respect toPor if the logarithm of the derivative is not integrable with respect toQ. The following entropy inequality is well-known:
Lemma3.4. For anyA∈ F, we have
logP(A) Q(A) ≥ −
H(Q|P) +e−1 Q(A) .
This inequality is useful only ifQ(A)∼1. Recall (3.3) for the definition ofQ. Note that for any
δ >0,Q
Zn≥ δ
1+δ
≥1+1δ, it follows that
PZn≥ δ 1+δ
≥ 1
1+δe
−(1+δ)e−1
exp−(1+δ)en(β). (3.8) Now we give the proof of Theorem 1.1:
Proof of Theorem 1.1: We prove at first (1.3) by subadditivity argument: Recalling (3.2) and (2.3). By markov property ofS, we have that for alln,m≥1,
Zn+m=Zn X
x
Letψ(x) =xlogx. We have
Zn+mlogZn+m = ψ(Zn) X
x
un(x)Zm(x,g◦τn,x) +Znψ X
x
un(x)Zm(x,g◦τn,x)
!
≤ ψ(Zn)
X
x
un(x)Zm(x,g◦τn,x) +Zn X
x
un(x)ψ(Zm(x,g◦τn,x)),
by the convexity ofψ. Taking expectation gives (1.3).
To show the equivalence between(a), (b), (c), we remark at first that the implication(b)⇒(c) follows from Lemma 3.2 and (3.1). It remains to show the implication(c)⇒(a). Assume that p(β)<0. The superadditivity says that
p(β) =sup n≥1
pn(β), withpn(β):=1
nE(logZn). On the other hand, the concentration of measure (cf.[2]) says that
P1
nlogZn−pn(β) >u
≤exp− nu
2
2β2
, ∀u>0. It turns out forα >0,
E
Znα
= eαE(logZn)eα(logZn−E(logZn))
≤ eαE(logZn)eα2β2n/2
≤ eαp(β)n+α2β2n/2.
By choosingα=−p(β)/β2(note thatα >0), we deduce from the Chebychev’s inequality that
PZn>1
2
≤2αe−p2(β)n/(2β2),
which in view of (3.8) withδ=1 imply that lim infn→∞1nen(β)≥ p(β)2
4β2 .
Acknowledgements Cooperation between authors are partially supported by the joint French-Hong Kong research program (Procore, No.14549QH and F-HK22/06T). Shao’s research is also partially supported by Hong Kong RGC CERG 602608.
References
[1] Birkner, M.: A condition for weak disorder for directed polymers in random environment. Electron. Comm. Probab.9 (2004), 22–25. MR2041302
[2] Carmona, Ph. and Hu, Y.: On the partition function of a directed polymer in a Gaus-sian random environment. Probab. Theory Related Fields 124 (2002), pp. 431–457. MR1939654
[4] Comets, F. and Yoshida, N.: Directed polymers in random environment are diffusive at weak disorder.Ann. Probab.34 (2006), no. 5, 1746–1770. MR2271480
[5] Comets, F. and Vargas, V.: Majorizing multiplicative cascades for directed polymers in random media.ALEA Lat. Am. J. Probab. Math. Stat.2 (2006), 267–277 MR2249671
[6] Gross, L.: Logarithmic Sobolev inequalities.Amer. J. Math. 97 (1975), no. 4, 1061–1083. MR0420249
[7] Ibragimov,I.A., Sudakov, V. and Tsirelson, B.: Norms of Gaussian sample functions, Pro-ceedings of the Third Japan-USSR Symposium on Probability (Tashkent), Lecture Notes in Mathematics, vol. 550, 1975, pp. 20–41. MR0458556
[8] Imbrie, J.Z. and Spencer,T.: Diffusion of directed polymers in a random environment, Jour-nal of Statistical Physics52(1988), 608–626. MR0968950
[9] Johansson, K. Transversal fluctuations for increasing subsequences on the plane.Probab. Theory Related Fields 116 (2000), no. 4, 445–456. MR1757595
[10] Lacoin, H. New bounds for the free energy of directed polymers in dimension 1+1 and 1+2.Arxiv arXiv:0901.0699
[11] Ledoux, M.: Concentration of measure and logarithmic Sobolev inequalities.Sém. Probab., XXXIII120–216, Lecture Notes in Math., 1709, Springer, Berlin, 1999. MR1767995
