Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction

0.53 MB·English

by  Francesco Caravenna

TheAnnalsofProbability
2008,Vol.36,No.6,2388–2433
DOI:10.1214/08-AOP395
(cid:13)c InstituteofMathematicalStatistics,2008
PINNING AND WETTING TRANSITION FOR
9
0 (1+1)-DIMENSIONAL FIELDS WITH LAPLACIAN INTERACTION
0
2
By Francesco Caravenna1 and Jean-Dominique Deuschel

n
a Universita` degli Studi di Padova and Technische Universita¨t Berlin
J
0 We consider a random field ϕ:{1,...,N}→R as a model for
2 a linear chain attracted to the defect line ϕ=0, that is, the x-axis.
Thefreelawofthefieldisspecifiedbythedensityexp(− V(∆ϕ ))
i i
] withrespecttotheLebesguemeasureonRN,where∆isthediscrete
R P
Laplacian and we allow for a verylarge class of potentials V(·). The
P
interaction with the defect line is introduced by giving the field a
.
h reward ε≥0 each time it touches the x-axis. We call this model the
t pinning model. We consider a second model, the wetting model, in
a
which,inadditiontothepinningreward,thefieldisalsoconstrained
m
to stay nonnegative.
[
We show that both models undergo a phase transition as the in-

3 tensity ε of the pinning reward varies: both in the pinning (a=p)
v and in the wetting (a=w) case, there exists a critical value εac such
4 that when ε>εa the field touches the defect line a positive frac-
c
3 tion of times (localization), while this does not happen for ε<εa
c
4 (delocalization). The two critical values are nontrivial and distinct:
3 0<εp<εw<∞, and they are the only nonanalyticity points of the
c c
0
respective free energies. For the pinning model the transition is of
7
second order, hence the field at ε=εp is delocalized. On the other
0 c
hand, the transition in the wetting model is of first order and for
/
h ε=εw the field is localized. The core of our approach is a Markov
c
t renewal theory description of thefield.
a
m
: 1. Introduction and main results.
v
i
X
1.1. Definition of the models. We are going to define two distinct but
r
a related models for a (1+1)-dimensional random field. These models depend
Received March 2007; revised October2007.
1Supported in part by the German Research Foundation–Research Group 718 during
his stay at TU Berlin in January 2007.
AMS 2000 subject classifications. 60K35, 60F05, 82B41.
Key words and phrases. Pinning model, wetting model, phase transition, entropic re-
pulsion, Markov renewal theory,local limit theorem, Perron–Frobenius theorem.
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Probability,
2008,Vol. 36, No. 6, 2388–2433. This reprint differs from the original in
pagination and typographic detail.
1
2 F. CARAVENNAANDJ.-D. DEUSCHEL
onameasurablefunctionV():R R + ,thepotential.Werequirethat
· → ∪{ ∞}
x exp( V(x)) is bounded and continuous and that exp( V(x))dx<
R
7→ − −
.Sinceaglobalshifton V() isirrelevantforourpurposes,wewillactually
∞ · R
impose the stronger condition
(1.1) e V(x)dx=1.
−
R
Z
ThelastassumptionswemakeonV()arethatV(0)< ,thatis,exp( V(0))
· ∞ −
> 0, and that
(1.2) x2e V(x)dx=:σ2< and xe V(x)dx=0.
− −
R ∞ R
Z Z
AtypicalexampleisofcoursetheGaussiancaseV(x)=x2/(2σ2)+log√2πσ2,
but we stress that we do not make any convexity assumption on V(). Next
we introduce the Hamiltonian (ϕ), defined for a,b Z, with b · a 2,
[a,b]
and for ϕ: a,...,b R by H ∈ − ≥
{ }→
b 1
−
(1.3) (ϕ):= V(∆ϕ ),
[a,b] n
H
n=a+1
X
where ∆ denotes the discrete Laplacian:
(1.4) ∆ϕ :=(ϕ ϕ ) (ϕ ϕ )=ϕ +ϕ 2ϕ .
n n+1 n n n 1 n+1 n 1 n
− − − − − −
We are ready to introduce our first model, the pinning model (p-model
for short) Pp , that is, the probability measure on RN 1 defined by
ε,N −
exp( (ϕ))N 1
(1.5) Ppε,N(dϕ1···dϕN−1):= −H[−p1,N+1] − (εδ0(dϕi)+dϕi)
Zε,N iY=1
where N N, ε 0, dϕ is the Lebesgue measure on R, δ () is the Dirac
i 0
∈ ≥ p ·
mass atzeroand is thenormalization constant, usuallycalled partition
Zε,N
function.Tocompletethedefinition,inorderto makesenseof (ϕ),
[ 1,N+1]
we have to specify: H−
(1.6) the boundary conditions ϕ =ϕ =ϕ =ϕ :=0.
1 0 N N+1
−
We fix zero boundary conditions for simplicity, but our approach works for
arbitrary choices (as long as they are bounded in N).
The second model we consider, the wetting model (w-model for short)
Pw , is a variant of the pinning model defined by
ε,N
Pw (dϕ dϕ ):=Pp (dϕ dϕ ϕ 0,...,ϕ 0)
(1.7)ε,N 1··· N−1 ε,N 1··· N−1| 1≥ N−1≥
exp( (ϕ))N 1
= −HZ[−εw1,N,N+1] iY=−1(εδ0(dϕi)+ dϕi1(ϕi≥0)),
(1+1)-DIMENSIONALFIELDS WITH LAPLACIANINTERACTION 3
that is, we replace the measure dϕ by dϕ 1 and p by a new nor-
malization w . i i (ϕi≥0) Zε,N
Zε,N
Both Pp and Pw are (1+1)-dimensional models for a linear chain of
ε,N ε,N
length N which is attracted to a defect line, the x-axis, and the parameter
ε 0 tunesthe strength of the attraction. By “(1+1)-dimensional” we mean
≥
that the configurations of the linear chain are described by the trajectories
(i,ϕ ) of the field, so that we are dealing with directed models (see
i 0 i N
{ } ≤≤
Figure 1 for a graphical representation). We point out that linear chain
modelswithLaplacianinteraction appearnaturallyinthephysicalliterature
in the context of semiflexible polymers; cf. [6, 17, 21] (however, the scaling
they consider is different from the one we look at in this paper). One note
about the terminology: while “pinning” refers of course to the attraction
terms εδ (dϕ ), the use of the term “wetting” is somewhat customary in
0 i
the presence of a positivity constraint and refers to the interpretation of the
Fig. 1. A graphical representation of the pinning model Pp (top) and of the wetting
ε,N
model Pw (bottom), for N =25 and ε>0. The trajectories {(n,ϕ )} of the field
ε,N n 0≤n≤N
describe the configurations ofalinearchainattracted toadefectline,the x-axis.Thegray
circles represent the pinned sites, that is, the points in which the chain touches the defect
line, whichare energetically favored. Note that inthe pinningcase the chaincan cross the
defect line without touching it, while this does not happen in the wetting case due to the
presence of a wall, that is, of a constraint for the chain to stay nonnegative; the repulsion
effect of entropic nature that arises is responsible for the different critical behavior of the
models.
4 F. CARAVENNAANDJ.-D. DEUSCHEL
field as an effective model for the interface of separation between a liquid
above a wall and a gas; see [10] for more details.
The purpose of this paper is to investigate the behavior of Pp and Pw
ε,N ε,N
in the large N limit; in particular we wish to understand whether and when
the reward ε 0 is strong enough to pin the chain at the defect line, a
≥
phenomenon that we will call localization. We point out that these kinds of
questions have been answered in depth in the case of gradient interaction,
that is, when the Laplacian ∆ appearing in (1.3) is replaced by the discrete
gradient ϕ :=ϕ ϕ (cf. [2, 9, 10, 11, 13, 18]); we will refer to this
n n n 1
∇ − −
as the gradient case. As we are going to see, the behavior in the Laplacian
case turns out to be sensibly different.
1.2. The free energy and the main results. A convenient way to define
localization for our models is by looking at the Laplace asymptotic behavior
of the partition function a as N . More precisely, for a p,w we
Zε,N →∞ ∈{ }
define the free energy fa(ε) by
1
(1.8) fa(ε):= lim fa (ε), fa (ε):= log a ,
N N N N Zε,N
→∞
where the existence of this limit (that will follow as a by-product of our
approach) can be proven with a standard superadditivity argument. The
basic observation is that the free energy is nonnegative. In fact, setting
Ωw:=[0, ) and Ωp:=R, we have N N
∞ ∀ ∈
N 1
a = exp( (ϕ)) − (εδ (dϕ )+dϕ 1 )
Zε,N −H[−1,N+1] 0 i i (ϕi∈Ωa)
Z i=1
(1.9) Y
N 1
exp( (ϕ)) − dϕ 1 = a c1 ,
≥ −H[−1,N+1] i (ϕi∈Ωa) Z0,N ≥ Nc2
Z i=1
Y
wherec ,c arepositiveconstants andthepolynomialboundfor a (anal-
1 2 Z0,N
ogous to what happens in the gradient case; cf. [10]) is proven in (2.14).
Therefore fa(ε) fa(0)=0 for every ε 0. Since this lower bound has been
≥ ≥
obtainedbyignoringthecontributionofthepathsthattouch thedefectline,
one is led to the following
Definition 1.1. For a p,w , the a-model Pa is said to be lo-
∈{ } { ε,N}N
calized if fa(ε)>0.
The first problem is to understand for which values of ε (if any) there
is localization. Some considerations can be drawn easily. We introduce for
convenience for t R
∈
(1.10) fa (t):=fa (et), fa(t):=fa(et).
N N
e e
(1+1)-DIMENSIONALFIELDS WITH LAPLACIANINTERACTION 5
Itis easy toshow (see AppendixA)that fa () is convex, thereforealso fa()
is convex. In particular, the free energyNfa·(ε) =fa(logε) is a continuou·s
function, as long as it is finite. fa() is aelso nondecreasing, because ae is
· Zε,N
increasing in ε [cf. the first line of (1.9)]. This obseervation implies that, for
both a p,w , there is a critical value εa [0, ] such that the a-model is
localize∈d{if and} only if ε>εa. Moreover εcp∈ εw∞, since p w .
c c ≤ c Zε,N ≥Zε,N
However, it is still not clear that a phase transition really exists, that
is, that εa (0, ). Indeed, in the gradient case the transition is nontrivial
c ∈ ∞
only for the wetting model, that is, 0<εw, < while εp, =0; cf. [10, 13].
c ∇ ∞ c ∇
Ourfirsttheorem showsthat intheLaplacian case both thepinningandthe
wetting models undergo a nontrivial transition, and gives further properties
of the free energy fa().
·
Theorem 1.2 (Localization transition). The following relations hold:
εp (0, ), εw (0, ), εp<εw.
c ∈ ∞ c ∈ ∞ c c
We have fa(ε)=0 for ε [0,εa], while 0<fa(ε)< for ε (εa, ), and
∈ c ∞ ∈ c ∞
as ε
→∞
(1.11) fa(ε)=logε(1+o(1)), a p,w .
∈{ }
Moreover the function fa(ε) is real analytic on (εa, ).
c ∞
One may ask why in the Laplacian case we have εp >0, unlike in the
c
gradient case. Heuristically, we could say that the Laplacian interaction
(1.3) describes a stiffer chain, more rigid to bending with respect to the
gradient interaction, and therefore Laplacian models require a stronger re-
ward in order to localize. Note in fact that in the Gaussian case V(x)=
x2/(2σ2)+log√2πσ2 the ground state of the gradient interaction is justthe
horizontally flat line, whereas the Laplacian interaction favors rather affine
configurations, penalizing curvature and bendings.
It is worth stressing that the free energy has a direct translation in terms
of some path properties of the field. Defining the contact number ℓ by
N
(1.12) ℓ :=# i 1,...,N :ϕ =0 ,
N i
{ ∈{ } }
a simple computation (see Appendix A) shows that for every ε > 0 and
N N
∈
ℓ
(1.13) da (ε):=Ea N =(fa )(logε)=ε (fa )(ε).
N ε,N N N ′ · N ′
(cid:18) (cid:19)
Then,introducingthenonrandomquaentityda(ε):=ε (fa)′(ε)(whichiswell
·
defined for ε=εa by Theorem 1.2), a simple convexity argument shows that
6 c
da (ε) da(ε) as N , for every ε=εa. Indeed much more can be said
N → →∞ 6 c
(see Appendix A):
6 F. CARAVENNAANDJ.-D. DEUSCHEL
When ε>εa we have that da(ε)>0, and for every δ>0 and N N
• c ∈
ℓ
(1.14) Pa N da(ε) >δ exp( c N),
ε,N N − ≤ − 3
(cid:18)(cid:12) (cid:12) (cid:19)
(cid:12) (cid:12)
where c is a positive(cid:12)constant. T(cid:12)his shows that, when the a-model is
3 (cid:12) (cid:12)
localized according to Definition 1.1, its typical paths touch the defect
line a positive fraction of times, equal to da(ε). Notice that, by (1.11) and
convexity arguments, da(ε) converges to 1 as ε , that is, a strong
→∞
reward pins the field at the defect line in a very effective way (observe
that ℓ /N 1).
N
≤
On the other hand, when ε<εa we have da(ε)=0 and for every δ >0
• and N N c
∈
ℓ
(1.15) Pa N >δ exp( c N),
ε,N N ≤ − 4
(cid:18) (cid:19)
where c is a positive constant. Thus for ε<εa the typical paths of the
4 c
a-model touch the defect line only o(N) times; when this happens it is
customary to say that the model is delocalized.
What is left out from this analysis is the critical regime ε = εa. The
c
behavior of the model in this case is sharply linked to the way in which the
free energy fa(ε) vanishes as ε εa. If fa() is differentiable also at ε=εa
↓ c · c
(transition of second or higher order), then (fa)(εa)=0 and relation 1.15
′ c
holds, that is, the a-model for ε=εa is delocalized. The other possibility
c
is that fa() is not differentiable at ε=εa (transition of first order), which
· c
happenswhen the right-derivative is positive: (fa) (εa)>0. In this case the
′+ c
behavior of Pa for large N may depend on the choice of the boundary
ε,N
conditions.
We first consider the critical regime for the wetting model, where the
transition turns out to be of first order. Recall the definition 1.13 of da (ε).
N
Theorem 1.3 (Critical wetting model). For the wetting model we have
(1.16) liminfdw(εw)>0.
N c
N
→∞
Therefore (fw) (εw)>0 and the phase transition is of first order.
′+ c
Notice that (1.13) and (1.16) yield
ℓ
liminfEw N >0,
N εwc,N N
→∞ (cid:18) (cid:19)
and in this sense the wetting model at the critical point exhibits a localized
behavior. This is in sharp contrast with the gradient case, where it is well
known that the wetting model at criticality is delocalized and in fact the
(1+1)-DIMENSIONALFIELDS WITH LAPLACIANINTERACTION 7
transition is of second order; cf. [9, 10, 11, 18]. The emergence of a first-
order transition in the case of Laplacian interaction is interesting in view
of the possible applications of Pw as a model for the DNA denaturation
ε,N
transition, where the nonnegative field ϕ describes the distance between
i i
{ }
the two DNA strands. In fact for the DNA denaturation something close
to a first-order phase transition is experimentally observed; we refer to [13],
Section 1.4, for a detailed discussion (cf. also [19, 27]).
Finally we consider the critical pinning model, where the transition is of
second order.
Theorem 1.4 (Critical pinning model). For the pinning model we have
(1.17) limsuplimsupdp (ε)=0.
N
ε↓εpc N→∞
Therefore fp(ε) is differentiable at ε=εp and (fp)(εp)=0. Moreover there
c ′ c
exists c >0 such that for δ sufficiently small we have
5
δ
(1.18) fp(εp+δ,0) c ,
c ≥ 5log1/δ
that is, the transition is exactly of second order.
Although the relation (fp)(εp) = 0 yields ℓ = o(N), in a delocalized
′ c N
fashion,thepinningmodelatε=εp isactuallysomewhatborderlinebetween
c
localization and delocalization, as (1.18) suggests and as we point out in the
next paragraph.
1.3. Further path results. A direct application of the techniques that
we develop in this paper yields further path properties of the field. Let us
introduce the maximal gap
∆ :=max n N :ϕ =0,ϕ =0,...,ϕ =0 for some k N n .
N k+1 k+2 k+n
{ ≤ 6 6 6 ≤ − }
Onecanshowthat,forbotha p,w andforε>εa,thefollowingrelations
∈{ } c
hold:
∆
δ>0: lim Pa N δ =0,
∀ N ε,N N ≥
(1.19) →∞ (cid:18) (cid:19)
lim limsup max Pa (ϕ L)=0.
L N i=1,...,N 1 ε,N | i|≥
→∞ →∞ −
In particular for ε>εa each component ϕ of the field is at finite distance
c i
from the defect line and this is a clear localization path statement. On
the other hand, in the pinning case a=p we can strengthen (1.15) to the
following relation: for every ε<εp
c
(1.20) lim limsupPp (∆ N L)=0,
L N ε,N N ≤ −
→∞ →∞
8 F. CARAVENNAANDJ.-D. DEUSCHEL
that is, for ε<εp the field touches the defect line at a finite number of sites,
c
all at finite distance from the boundary points 0,N . We expect that the
{ }
same relation holds true also in the wetting case a=w, but at present we
cannot prove it: what is missing are more precise estimates on the entropic
repulsion problem; see Section 1.5 for a detailed discussion. It is interesting
to note that we can prove that the first relation in (1.19) holds true also in
the pinning case a=p at the critical point ε=εp, and this shows that the
c
pinning model at criticality has also features of localized behavior.
We do not give an explicit proof of the above relations in this paper,both
for conciseness and because in a second paper [7] we focus on the scaling
limits of the pinning model, obtaining (de)localization path statements that
aremuchmoreprecisethan(1.19)and(1.20)[understrongerassumptionson
the potential V()]. We show in particular that for all ε (0,εp) the natural
rescaling of Pp ·converges in distribution in C([0,1]) to∈the sacme limit that
ε,N
one obtains in the free case ε=0, that is, the integral process of a Brownian
bridge. On the other hand, for every ε εp the natural rescaling of Pp
≥ c ε,N
yields the trivial process which is identically zero. We stress that ε=εp is
c
included in the last statement; this is in sharp contrast with the gradient
case, where both the pinning and the wetting models at criticality have a
nontrivial scaling limit, respectively the Brownian bridge and the reflected
Brownian bridge. This shows again the peculiarity of the critical pinning
model in the Laplacian case. Indeed, by lowering the scaling constants with
suitable logarithmic corrections, we are able to extract a nontrivial scaling
limit (in a distributional sense) for the law Pp in terms of a symmetric
εpc,N
stable L´evy process.We stress that the techniques and results of the present
paper play a crucial role for [7].
1.4. Outline of the paper: approach and techniques. Although our main
results are about the free energy, the core of our approach is a precise path-
wise description of the field based on Markov renewal theory. In analogy
to [9, 10] and especially to [8], we would like to stress the power of (Markov)
renewal theory techniques for the study of (1+1)-dimensional linear chain
models. The other basic techniques that we use are local limit theorems
and an infinite-dimensional version of the Perron–Frobenius theorem. Let
us describe more in detail the structure of the paper.
In Section 2 we study the pinning and wetting models in the free case
ε = 0, showing that these models are sharply linked to the integral of a
random walk. More precisely, let Y denote a random walk starting
n n 0
at zero and with step law P(Y {dx)}=≥exp( V(x))dx [the walk has zero
1
∈ −
mean and finite variance by (1.2)] and let us denote by Z :=Y + +Y
n 1 n
···
the corresponding integrated random walk process. In Proposition 2.2 we
show that the law Pa is nothing but a bridge of length N of the process
0,N
(1+1)-DIMENSIONALFIELDS WITH LAPLACIANINTERACTION 9
Z ,withthefurtherconditioning tostay nonnegative inthewetting case
n n
{ }
a=w.Thereforewefocusontheasymptoticpropertiesoftheprocess Z ,
n n
{ }
obtaining a basic local limit theorem (cf. Proposition 2.3) and some bounds
for the probability that Z stays positive (connected to the problem of
n n
{ }
entropic repulsion that we discuss below; cf. Section 1.5).
InSection 3,whichisthecoreofthepaper,weshowthatforε>0 thelaw
Pa admits a crucial description in terms of Markov renewal theory. More
ε,N
precisely, we show that the zeros of the field i N:ϕ =0 under Pa
{ ≤ i } ε,N
are distributed according to the law of a (hidden) Markov renewal process
conditionedtohit N,N+1 ;cf.Proposition3.1.Wethusobtainanexplicit
{ }
expression for the partition function a in terms of this Markov renewal
Zε,N
process, which is the key to our main results.
Section 4 is devoted to proving some analytical results that underlie the
construction of the Markov renewal process appearing in Section 3. The
maintoolisaninfinite-dimensionalversionoftheclassicalPerron–Frobenius
theorem (cf. [33]), and a basic role is played by the asymptotic estimates
obtained in Section 2. A by-product of this analysis is an explicit formula
[cf. 4.10], that links fa() and εa to the spectral radius of a suitable integral
· c
operator and that will be exploited later.
Sections 5, 6 and 7 contain the proofs of Theorems 1.2, 1.3 and 1.4,
respectively. In view of the description given in Section 3, all the results to
provecanberephrasedinthelanguageofMarkovrenewaltheory.Theproofs
arethencarriedoutexploitingtheasymptoticestimatesderivedinSections2
and4together withsomealgebraic manipulation ofthekernelthatgives the
law of the hidden Markov renewal process. Finally, the Appendixes contain
the proof of some technical results.
We conclude the introduction by discussing briefly two interesting prob-
lems that are linked to our models, namely the entropic repulsion in Section
1.5 and the smoothing effect of disorder in Section 1.6. Finally, Section 1.7
containssomerecurrentnotations,especiallyaboutkernels,usedthroughout
the paper.
1.5. Entropic repulsion. We recall that ( Y ,P) is the random walk
n n 0
with step P(Y dx)=e V(x)dx and that Z{ =}Y≥+ +Y . The analysis
1 − n 1 n
∈ ···
of the wetting model requires estimating the decay as N of the proba-
bilities P(Ω+) and P(Ω+ Z =0, Z =0), where w→e s∞et Ω+ := Z
N N | N+1 N+2 N { 1≥
0,...,Z 0 . This type of problem is known in the literature as entropic
N
≥ }
repulsion andithas received alot of attention; see [32]for arecentoverview.
In the Laplacian case that we consider here, this problem has been solved
in the Gaussian setting (i.e., when V(x)=x2/(2σ2)+log√2πσ2) in (d+1)-
dimension with d 5; cf. [22, 28]. Little is known in the (1+1)-dimensional
≥
setting, apartfromthe following resultof Sinai’s [29]in thevery special case
10 F. CARAVENNAANDJ.-D. DEUSCHEL
when Y is the simple random walk on Z:
n n
{ }
c C
(1.21) P(Ω+) ,
N1/4 ≤ N ≤ N1/4
where c,C are positive constants. The proof of this bound relies on the
exact combinatorial results available in the simple random walk case and it
appears difficult to extend it to our situation. We point out that the same
exponent 1/4 appearsin related continuous modelsdealing withtheintegral
of Brownian motion; cf. [23, 25]. Based on Sinai’s result, which we believe
to hold for general random walks with zero mean and finite variance, we
expect that for the bridge case one should have the bound
c C
(1.22) P(Ω+ Z =0,Z =0) .
N1/2 ≤ N | N+1 N+2 ≤ N1/2
We cannot derive precise bounds as (1.21) and (1.22); however, for the pur-
pose of this paper the following weaker result suffices:
Proposition 1.5. There exist constants c,C,c >0 and c >1 such
+
that N N −
∀ ∈
c C
(1.23) P(Ω+) ,
Nc− ≤ N ≤ (logN)c+
c C
(1.24) P(Ω+ Z =0,Z =0) .
Nc− ≤ N | N+1 N+2 ≤ (logN)c+
WeprovethispropositioninAppendixC.Westressthatthemostdelicate
point is the upper bound in (1.23): the idea of the proof is to dilute the
system on superexponentially spaced times. We also point out that in the
Gaussian case, thatis, whenV(x)=x2/(2σ2)+log√2πσ2,theupperbound
can be easily strengthened to P(Ω+) (const.)/Na, for some a > 0, by
N ≤
diluting the system on exponentially spaced times and using the results in
the paper [24].
1.6. Smoothing effect of quenched disorder. The models we consider in
this work are homogeneous, in that the reward ε is deterministic and con-
stant. However, one can define in a natural way a disordered version of our
model, where the reward is itself random and may vary from site to site.
We stress that disordered pinning models have attracted a lot of attention
recently; cf. [1, 2, 16, 30, 31] (see also [13] for an overview).
Let us be more precise: we take a sequence ω= ωn n N of i.i.d. standard
{ } ∈
Gaussian random variables, defined on some probability space ( ,P), and
S
we denote by M(β):=E(exp(βω ))=exp(β2/2) the corresponding moment
1
generating function(wefocuson Gaussianvariables only forthesakeof sim-
plicity; wecouldmoregenerally take variables with zeromean,unitvariance