Phase transition and level-set percolation for the Gaussian free field

0.37 MB·English

by  Pierre-François Rodriguez

PHASE TRANSITION AND LEVEL-SET PERCOLATION
FOR THE GAUSSIAN FREE FIELD
Preliminary Draft
Pierre-Franc¸ois Rodriguez1 and Alain-Sol Sznitman1
2
1
0
2
Abstract

b
e
F We consider level-setpercolationfor the Gaussianfree field onZd, d 3, andprove that, as h
≥
varies, there is a non-trivial percolation phase transition of the excursion set above level h for
3
2 all dimensions d 3. So far, it was known that the corresponding critical level h∗(d) satisfies
≥
h∗(d) 0 for all d 3 and that h∗(3) is finite, see [2]. We prove here that h∗(d) is finite
≥ ≥
] for all d 3. In fact, we introduce a second critical parameter h∗∗ h∗, show that h∗∗(d) is
R ≥ ≥
finite for all d 3, and that the connectivity function of the excursion set above level h has
P ≥
stretched exponential decay for all h > h∗∗. Finally, we prove that h∗ is strictly positive in
.
h highdimension. It remains openwhether h∗ andh∗∗ actually coincide andwhether h∗ >0 for
t all d 3.
a
≥
m
[

1
v
2
7
1
5
.
2
0
2
1
:
v
i
X
r
a
February 2012
1Departement Mathematik, ETH Zu¨rich, CH-8092 Zu¨rich, Switzerland.
This research was supported in part bythe grant ERC-2009-AdG 245728-RWPERCRI.

0 Introduction
In the present work, we investigate level-set percolation for the Gaussian free field on Zd, d 3.
≥
Thisproblemhasalready received muchattention inthepast, seeforinstance[11], [2], andmore
recently[5],[14]. Thelong-rangedependenceofthemodelmakesthisproblemparticularlyinter-
esting, but also harder to analyze. Here, we prove the existence of a non-trivial critical level for
alld 3, andthepositivity ofthiscriticallevelwhendislargeenough. Someofourmethodsare
≥
inspired by the recent progresses in the study of the percolative properties of random interlace-
ments, where a similar long-range dependence occurs, see for instance [18], [19], [22], [25].
We now describe our results and refer to Section 1 for details. We consider the lattice Zd,
d 3, endowed with the usual nearest-neighbor graph structure. Our main object of study is
th≥e Gaussian free field on Zd, with canonical law P on RZd such that,
under P, the canonical field ϕ = (ϕ ) is a centered Gaussian
x x Zd
(0.1) ∈
field with covariance E[ϕ ϕ ]= g(x,y), for all x,y Zd,
x y
∈
where g(, ) denotes the Green function of simple random walk on Zd, see (1.1). Note in
· ·
particular the presence of strong correlations, see (1.9). For any level h R, we introduce the
∈
(random) subset of Zd
(0.2) E h = x Zd ; ϕ h ,
ϕ≥ { ∈ x ≥ }
sometimes called excursion set (above level h). We are interested in the event that the origin
lies in an infinite cluster of Eϕ≥h, which we denote by {0 ←≥→h ∞}, and ask for which values of h
this event occurs with positive probability. Since
(0.3) η(h) d=ef. P[0 ≥h ]
←→ ∞
is decreasing in h, it is sensible to define the critical point for level-set percolation as
(0.4) h (d) = inf h R ; η(h) = 0 [ , ]
∗ { ∈ } ∈ −∞ ∞
(with the convention inf = ). A non-trivial phase transition is then said to occur if h is
∅ ∞ ∗
finite. It is known that h (d) 0 for all d 3 and that h (3) < (see [2], Corollary 2 and
∗ ≥ ≥ ∗ ∞
Theorem 3, respectively; see also the concluding Remark 5.1 in [2] to understand why the proof
does not easily generalize to all d 3). It is also known that when d 4, for large h, there is no
≥ ≥
directed percolation inside E h, see [5], p. 281 (note that this reference studies the percolative
ϕ≥
properties of the excursion sets of ϕ in place of ϕ).
| |
It is not intuitively obvious why h should be finite, for it seems a priori conceivable that
infinite clusters of E h could exist for∗all h > 0 due to the strong nature of the correlations.
ϕ≥
We show in Corollary 2.7 that this does not occur and that
(0.5) h (d) < , for all d 3.
∗ ∞ ≥
In fact, we prove a stronger result in Theorem 2.6. We define a second critical parameter
h (d) = inf h R ; α(h) > 0 , with
∗∗ { ∈ }
(0.6)
α(h) = sup α 0 ; lim Lα P B(0,L) ≥h S(0,2L) = 0 , for h R
≥ L ←→ ∈
→∞
(cid:8) (cid:2) (cid:3) (cid:9)
h
(with the convention sup = 0), where the event B(0,L) ≥ S(0,2L) refers to the existence
∅ ←→
of a (nearest-neighbor) path in E h connecting B(0,L), the ball of radius L around 0 in the
ϕ≥ (cid:8) (cid:9)
1
ℓ -norm, to S(0,2L), the ℓ -sphere of radius 2L around 0. It is an easy matter (see Corollary
∞ ∞
2.7 below) to show that
(0.7) h h .
∗ ≤ ∗∗
Now, we prove in Theorem 2.6 the stronger statement
(0.8) h < , for all d 3,
∗∗ ∞ ≥
and then obtain as a by-product that
(0.9) the connectivity function of E h has stretched exponential decay for all h> h
ϕ≥
∗∗
(see Theorem 2.6 below for a precise statement). This immediately leads to the important
question of whether h and h actually coincide. In case they differ, our results imply a marked
transition in the decay∗ of the∗∗connectivity function of E h at h = h , see Remark 2.8 below.
ϕ≥
∗∗
Our second result concerns the critical level h in high dimension. We are able to show in
∗
Theorem 3.3 that
(0.10) h is strictly positive when d is sufficiently large.
∗
This is in accordance with recent numerical evidence, see [14], Chapter 4. We actually prove a
stronger result than (0.10). Namely, we show that one can find a positive level h , such that
0
for large d, the restriction of E h0 to a thick two-dimensional slab percolates, see above (0.12).
ϕ≥
Let us however point out that, by a result of [7] (see p. 1151 therein), the restriction of E 0 to
ϕ≥
Z2 (viewed as a subset of Zd), and, a fortiori, the restriction of E h to Z2, when h is positive,
ϕ≥
only contain finite connected components: excursion sets above any non-negative level do not
percolate in planes. We refer to Remark 3.6 1) for more on this.
Wenow commentontheproofs. We beginwith(0.8). Thekey ingredientis acertain (static)
renormalization scheme very similar to the one developed in Section 2 of [19] for the problem
of percolation of the vacant set left by random interlacements (for a precise definition of this
model, see [21], Section 1; we merely note that the two “corresponding” quantities are E h and
ϕ≥
u, the vacant set at level u 0). We will be interested in the probability of certain crossing
V ≥
events viewed as functions of h R,
∈
f (h) “=” P E h contains a path from a given block of
n ϕ≥
side length L to the complement of its L -neighborhood
(cid:2) n n
(see (2.9) for the precise definition), where (L ) is a geometrically increasing(cid:3)sequence of
n n 0
≥
length scales, see (2.1). Note that by (0.2), f is decreasing in h. We then explicitly construct
n
an increasing but bounded sequence (h ) , with (finite) limit h , such that
n n 0
≥ ∞
(0.11) lim f (h ) =0.
n n
n
→∞
This readily implies (0.5), since η(h ) f (h ) f (h ) for all n 0, hence η(h ) vanishes.
n n n
∞ ≤ ∞ ≤ ≥ ∞
By separating combinatorial complexity estimates from probabilistic bounds in f (), see (2.8)
n
·
and Lemma 2.1, we are led to investigate the quantity
p (h) “=” P E h contains paths connecting each of 2n “well-separated”
n ϕ≥
boxes of side length L (within a given box of side length L )
(cid:2) 0 n
∼
to the complement of their respective L -neighborhoods
0
(cid:3)
2
(see (2.8) for the precise definition), where the 2n boxes are indexed by the “leaves” of a dyadic
tree of depth n. The key to proving (0.11) is to provide a suitable induction step relating
p (h ) to p (h ), for all n 0, where the increase in parameter h h allows to dom-
n+1 n+1 n n n n+1
≥ →
inate the interactions (“sprinkling”). This appears in Proposition 2.2. The resulting estimates
are fine enough to imply not only that h h , but even the stretched exponential decay of
the connectivity function of E h∞, thus y∞iel≥ding∗(0.8). The proof of (0.9) then only requires a
ϕ≥
small refinement of this argument. Note that the strategy we have just described is precisely
the one used in [19] for the proof of a similar theorem in the context of random interlacements.
We actually also provide a generalization of Proposition 2.2, which is of independent interest,
but goes beyond what is directly needed here, see Proposition 2.2’. It has a similar spirit to the
main renormalization step leading to the decoupling inequalities for random interlacements in
[22], see Remark 2.3.
Wenowcommentontheproofof (0.10),whichhastwomainingredients. Thefirstingredient
is a suitable decomposition of the field ϕ restricted to the subspace Z3 into the sum of two
independent Gaussian fields. The first field has independent components and the second field
only acts as a “small” perturbation when d becomes large, see Lemmas 3.1 and 3.2 below. The
second ingredient is a Peierls-type argument, which actually enables us to deduce a stronger
result than (0.10). Namely, we show in Theorem 3.3 that one can find a level h > 0 and a
0
positiveinteger L ,suchthatforlargedandallh h ,theexcursionsetE h already percolates
0 ≤ 0 ϕ≥
in the two-dimensional slab
(0.12) Z2 [0,2L ) 0 d 3 Zd.
0 −
× ×{ } ⊂
As already pointed out, some of our proofs employ strategies similar to those developed in
the study of the percolative properties of the vacant set left by random interlacements, see [19],
[22]. Thisisnotamerecoincidence, as wenowexplain. Continuous-timerandominterlacements
onZd,d 3, correspondtoacertainPoisson pointprocessofdoublyinfinitetrajectories modulo
≥
time-shift, governed by a probability P, with a non-negative parameter u playing the role of
a multiplicative factor of the intensity measure pertaining to this Poisson point process (the
bigger u, the more trajectories “fall” on Zd), see [23], [21]. This Poisson gas of doubly infinite
trajectories (modulo time-shift) induces a random field of occupation times (L ) (so that
x,u x Zd
the interlacement at level u coincides with x Zd;L >0 , whereas the vacant s∈et at level u
x,u
{ ∈ }
equals x Zd;L = 0 ). This field is closely linked to the Gaussian free field, as the following
x,u
{ ∈ }
isomorphism theorem from [23] shows:
1 1
(0.13) L + ϕ2 , under P P, has the same law as (ϕ +√2u)2 , under P.
x,u 2 x x Zd ⊗ 2 x x Zd
∈ ∈
It is tem(cid:0)pting to use(cid:1)this identity as a transfer mechanism, an(cid:0)d we hope to re(cid:1)turn to this point
elsewhere.
We conclude this introduction by describing the organization of this article. In Section 1,
we introduce some notation and review some known results concerning simple random walk on
Zd and the Gaussian free field. Section 2 is devoted to proving that excursion sets at a high
level do not percolate. The main results are Theorem 2.6 and Corollary 2.7. The positivity of
the critical level in high dimension and the percolation of excursion sets at low positive level in
large enough two-dimensional slabs is established in Theorem 3.3 of Section 3.
One final remark concerning our convention regarding constants: we denote by c,c,...
′
positive constants with values changing from place to place. Numbered constants c ,c ,... are
0 1
defined at the place they first occur within the text and remain fixed from then on until the
end of the article. In Sections 1 and 2, constants will implicitly depend on the dimension d. In
Section 3 however, constants will be purely numerical (and independent of d). Throughout the
entire article, dependence of constants on additional parameters will appear in the notation.
3
1 Notation and some useful facts
In this section, we introduce some notation to be used in the sequel, and review some known
results concerning both simple random walk and the Gaussian free field.
We denote by N = 0,1,2,... the set of natural numbers, and by Z = ..., 1,0,1,...
{ } { − }
the set of integers. We write R for the set of real numbers, abbreviate x y = min x,y and
∧ { }
x y = max x,y for any two numbers x,y R, and denote by [x] the integer part of x, for
∨ { } ∈
any x 0. We consider the lattice Zd, and tacitly assume throughout that d 3. On Zd, we
≥ ≥
respectively denote by and the Euclidean and ℓ -norms. Moreover, for any x Zd
∞
and r 0, we let B(x,|r·)|= y|·|∞Zd; y x r and S(x,r) = y Zd; y x =∈ r
≥ { ∈ | − |∞ ≤ } { ∈ | − |∞ }
stand for the the ℓ -ball and ℓ -sphere of radius r centered at x. Given K and U subsets of
∞ ∞
Zd, Kc = Zd K stands for the complement of K in Zd, K for the cardinality of K, K Zd
\ | | ⊂⊂
means that K < , and d(K,U) = inf x y ; x K,y U denotes the ℓ -distance
∞
| | ∞ {| − |∞ ∈ ∈ }
between K and U. If K = x , we simply write d(x,U). Finally, we define the inner boundary
{ }
of K to be the set ∂iK = x K; y Kc, y x = 1 , and the outer boundary of K
{ ∈ ∃ ∈ | − | }
as ∂K = ∂i(Kc). We also introduce the diameter of any subset K Zd, diam(K), as its
⊂
ℓ -diameter, i.e. diam(K) = sup x y ; x,y K .
∞
We endowZd with thenearest{-n|ei−ghb|∞or graph∈stru}cture, i.e. theedge-set consists of allpairs
of sites x,y , x,y Zd, such that x y = 1. A (nearest-neighbor) path is any sequence of
{ } ∈ | − |
vertices γ = (x ) , where n 0 and x Zd for all 0 i n, satisfying x x = 1 for
i 0 i n i i i 1
≤≤ ≥ ∈ ≤ ≤ | − − |
all 1 i n. Moreover, two lattice sites x,y will be called -nearest neighbors if x y = 1.
A -p≤ath≤is defined accordingly. Thus, any site x Zd ha∗s 2d nearest neighbor|s a−nd|∞3d 1
∗ ∈ −
-nearest neighbors.
∗
We now introduce the (discrete-time) simple random walk on Zd. To this end, we let W
be the space of nearest-neighbor Zd-valued trajectories defined for non-negative times, and let
, (X ) , stand for the canonical σ-algebra and canonical process on W, respectively. Since
n n 0
W ≥
d 3, the random walk is transient. Furthermore, we write P for the canonical law of the walk
x
≥
starting at x Zd and E for the corresponding expectation. We denote by g(, ) the Green
x
∈ · ·
function of the walk, i.e.
(1.1) g(x,y) = P [X = y], for x,y Zd,
x n
∈
n 0
X≥
def.
which is finite (since d 3) and symmetric. Moreover, g(x,y) = g(x y,0) = g(x y)
≥ − −
due to translation invariance. Given U Zd, we further denote the entrance time in U by
⊂
H = inf n 0;X U , the hitting time of U by H = inf n 1;X U , and the exit
U n U n
{ ≥ ∈ } { ≥ ∈ }
time from U by TU = inf n 0;Xn / U = HUc. This allows us to define the Green function
{ ≥ ∈ }
g (, ) killed outside U as e
U
· ·
(1.2) g (x,y) = P [X = y, n < T ], for x,y Zd.
U x n U
∈
n 0
X≥
It vanishes if x / U or y / U. The relation between g and g for any U Zd is the following
U
∈ ∈ ⊂
(we let K = Uc):
(1.3) g(x,y) = g (x,y)+E [H < , g(X ,y)], for x,y Zd.
U x K ∞ HK ∈
The proof of (1.3) is a mere application of the strong Markov property (at time H ).
K
We now turn to a few aspects of potential theory associated to simple random walk. For any
K Zd, we write
⊂⊂
(1.4) e (x) = P [H = ], x K,
K x K
∞ ∈
e
4
for the equilibrium measure (or escape probability) of K, and
(1.5) cap(K) = e (x)
K
x K
X∈
for its capacity. It immediately follows from (1.4) and (1.5) that the capacity is subadditive, i.e.
(1.6) cap(K K ) cap(K)+cap(K ), for all K,K Zd.
′ ′ ′
∪ ≤ ⊂⊂
Moreover, the entrance probability in K may be expressed in terms of e () (see for example
K
·
[20], Theorem 25.1, p. 300) as
(1.7) P [H < ] = g(x,y) e (y),
x K K
∞ ·
y K
X∈
from which, together with classical bounds on the Green function (c.f. (1.9) below), one easily
obtains (see [19], Section 1 for a derivation) the following usefulboundfor the capacity of a box:
(1.8) cap(B(0,L)) cLd 2, for all L 1.
−
≤ ≥
We next review some useful asymptotics of g(). Given two functions f ,f :Zd R, we write
1 2
· −→
f (x) f (x), as x , if they are asymptotic, i.e. if lim f (x)/f (x) =1.
1 2 x 1 2
∼ | | → ∞ | |→∞
Lemma 1.1. (d 3)
≥
(1.9) g(x) cx 2 d, as x .
−
∼ | | | |→ ∞
1
(1.10) g(0) = 1+ +o(d 1), as d .
−
2d → ∞
7
(1.11) P0[HZ3 = ]= 1 +o(d−1), as d ,
∞ − 2d → ∞
where Z3 is viewed as Z3 e 0 d 3 Zd in (1.11).
−
× { } ⊂
Proof. For (1.9), see [1(cid:0)0], Theorem(cid:1)1.5.4, for (1.10), see [15], pp. 246-247. In order to prove
(1.11), we assume that d 6 and define π : Zd Zd 3 : (x1,...,xd) (x4,...,xd). Then,
−
≥ −→ 7→
under P ,
0
def.
Y = π X , for all n 0,
n n
◦ ≥
is a “lazy” walk on Zd−3 starting at the origin. Clearly, {HZ3 = ∞} = {H0(Y) = ∞}, where
(Y)
H refers to the first return to 0 for the walk Y. Hence,
0
e e
e P0[HZ3 = ]= g(Y)(0) −1 = d g(d−3)(0) −1 (1=.10) 1 7 +o(d−1),
∞ d 3 · − 2d
h − i
(cid:2) (cid:3)
as d , wheere g(Y)() denotes the Green function of Y and g(d 3)() that of simple random
−
→ ∞ · ·
walk on Zd 3.
−
We now turn to the Gaussian free field on Zd, as defined in (0.1). Given any subset K Zd,
⊂
we frequently write ϕ to denote the family (ϕ ) . For arbitrary a R and K Zd, we
K x x∈K ∈ ⊂⊂
also use the shorthand ϕ > a for the event min ϕ ; x K > a and similarly ϕ < a
x
{ |K } { { ∈ } } { |K }
instead of max ϕ ; x K < a . Next, we introduce certain crossing events for the Gaussian
x
{ { ∈ } }
Zd
free field. To this end, we first consider the space Ω = 0,1 endowed with its canonical
{ }
σ-algebra and define, for arbitrary disjoint subsets K,K Zd, the event (subset of Ω)
′
⊂
(1.12) K K = there exists an open path connecting K and K .
′ ′
{ ←→ } { }
5
For any level h R, we write Φh for the measurable map from RZd into Ω which sends ϕ RZd
∈ ∈
to 1 ϕ h Ω, and define
{ x ≥ } x Zd ∈
∈
(cid:0) (cid:1)
(1.13) K ≥h K′ = (Φh)−1 K K′
{ ←→ } { ←→ }
(a measurable subset of RZd endowed with its canon(cid:0)ical σ-algebra(cid:1) ), which is the event that K
F
andK areconnected bya(nearest-neighbor) pathin E h, c.f. (0.2). Denoting by Qh theimage
′ ϕ≥
of P under Φh, i.e. the law of 1{ϕx ≥ h} x Zd on Ω, we have that P[K ←≥→h K′] = Qh[K ←→
K′]. Note that K ≥h K′ is(cid:0)an increasin(cid:1)g∈event upon introducing on RZd the natural partial
{ ←→ }
order (i.e. f f when f f for all x Zd).
≤ ′ x ≤ x′ ∈
We proceed with a classical fact concerning conditional distributions for the Gaussian free
field on Zd. We could not find a precise reference in the literature, and include a proof for the
Reader’s convenience. We first define, for U Zd, the law PU on RZd of the centered Gaussian
⊂
field with covariance
(1.14) EU[ϕ ϕ ] = g (x,y), for all x,y Zd,
x y U
∈
with g (, ) given by (1.2). In particular, ϕ = 0, PU-almost-surely, whenever x K = Uc. We
U x
· · ∈
then have
Lemma 1.2.
Let = K Zd, U = Kc and define (ϕ ) by
∅ 6 ⊂⊂ x x∈Zd
(1.15) ϕ = ϕ +µ , for x Zd,
x x x
e ∈
where µ is the σ(ϕ ;x K)-measurable map defined as
x x
∈ e
(1.16) µ = E [H < ,ϕ ]= P [H < ,X = y] ϕ , for x Zd.
x x K ∞ XHK x K ∞ HK · y ∈
y K
X∈
Then, under P,
(1.17) (ϕ ) is independent from σ(ϕ ;x K), and distributed as (ϕ ) under PU.
x x∈Zd x ∈ x x∈Zd
Proof. Note that for all x K, ϕ = 0 (since µ = ϕ for x K, by (1.16)) and that for
e ∈ x x x ∈
all x K, ϕ = 0, PU-almost surely. Hence, it suffices to consider (ϕ ) . We first show
x x x U
∈ ∈
independence. By (1.16), (ϕx)x U,e(ϕy)y K, are centered and jointly Gaussian. Moreover, they
∈ ∈
are uncorrelated, since for x U, y K, e
∈ ∈
e
E[ϕ ϕ ] =E[ϕ ϕ ] E[µ ϕ ] (0.1)=,(1.16) g(x,y) P [H < ,X = z]g(z,y) (1=.3) 0.
x y x y − x y − x K ∞ HK
z K
X∈
e
Thus, (ϕ ) , (ϕ ) , are independent. To conclude the proof of Lemma 1.2, it suffices to
x x U y y K
∈ ∈
show that
e
(1.18) E 1 (ϕ ) = EU 1 (ϕ ) , for all A ,
A x x U A x x U U
∈ ∈ ∈ F
where stands fo(cid:2)r th(cid:0)e canoni(cid:1)c(cid:3)al σ-alg(cid:2)ebr(cid:0)a on RU(cid:1).(cid:3) Furthermore, choosing some ordering
FU e
(x ) of U, by Dynkin’s Lemma, it suffices to assume that A has the form
i i 0
≥
(1.19) A = A A RU x0,...,xn , for some n 0 and A (R), i= 0,...,n.
x0 ×···× xn × \{ } ≥ xi ∈ B
WefixsomeAoftheform(1.19),andconsiderasubsetV suchthatK x ,...,x V Zd
0 n
∪{ } ⊆ ⊂⊂
(we will soon let V increase to Zd). We let PV, x V, denote the law of simple random walk on
x ∈
6
V starting at x killed when exiting V (its Green function corresponds to g (, )), and define ϕV
V · · x
for x V as in (1.15) but with PV replacing P in the definition (1.16) of µ . It then follows
∈ x x x
from Proposition 2.3 in [24] (an analogue of the present lemma for finite graphs) that e
(1.20) EV 1 (ϕV) = EV K 1 (ϕ ) ,
AV x x V K \ AV x x V K
∈ \ ∈ \
withPV,PV K asdefinedi(cid:2)n(1.1(cid:0)4)andAV =(cid:1)(cid:3)A (cid:2)A (cid:0) RV (K x0(cid:1),.(cid:3)..,xn ). LettingV Zd,
\ e x0×···× xn× \ ∪{ } ր
it follows that g (x,y) g(x,y), g (x,y) g (x,y), hence by dominated convergence
V V K U
ր \ ր
that both sides of (1.20) converge towards the respective sides of (1.18), thus completing the
proof.
Remark 1.3.
Lemma 1.2 yields a choice of regular conditional distributions for (ϕ ) conditioned on the
x x Zd
variables (ϕ ) , which is tailored to our future purposes. Namely, P-a∈lmost surely,
x x K
∈
(1.21) P (ϕ ) (ϕ ) = P (ϕ +µ ) ,
x x∈Zd ∈ · x x∈K x x x∈Zd ∈ ·
wGhauersesiaµnx,fixeld∈uZnddeisr Pgi,v(cid:2)wenithbyϕ(1=.160),,(cid:12)(cid:12)PP-adlmoeosstn(cid:3)osutraeecl(cid:2)yt eofonr (xµx)xK∈Z.dL, eamndm(cid:3)a(ϕ1x.)2x∈aZlsdoipsraovciednetsertehde
x
∈
covariance structure of this field (nameely gU(, ), with U = Kc), but its perecise form will be of
· ·
no importance in whaet followse. Note tehat conditioning on (ϕx)x K produces the (random) shift
µ , which is linear in the variables ϕ , y K. ∈ (cid:3)
x y
∈
Theexplicit form of the conditional distributions in (1.21) readily yields the following result,
which can be viewed as a consequence of the FKG-inequality for the free field (see for example
[6], Chapter 4).
Lemma 1.4.
Let α R, = K Zd, and assume A (the canonical σ-algebra on RZd) is an increasing
∈ ∅ 6 ⊂⊂ ∈ F
event. Then
(1.22) P A ϕ = α P A ϕ α ,
|K ≤ |K ≥
where the left-hand side is defin(cid:2)ed(cid:12)by the ver(cid:3)sion of(cid:2)th(cid:12)e conditio(cid:3)nal expectation in (1.21).
(cid:12) (cid:12)
Intuitively, augmenting the field can only favor the occurrence of A, an increasing event.
Proof. On the event ϕ α , we have, for µ , x Zd, as defined in (1.16),
x
|K ≥ ∈
(1.23) µ = ϕ(cid:8)P [H <(cid:9) ,X =y] αP [H < ]d=ef. m (α), for all x Zd,
x y x K ∞ HK ≥ x K ∞ x ∈
y K
X∈
with equality instead on the event ϕ = α . Since A is increasing, this yields, with a slight
K
|
abuse of notation,
(cid:8) (cid:9)
P[Aϕ = α] 1 (1=.21) P A (ϕ +m (α)) 1
| |K · {ϕ|K≥α} x x x∈Zd · {ϕ|K≥α}
P A (ϕ +µ ) 1
(1=≤.21) Pe(cid:2)(cid:2)A(cid:0)(cid:0)ϕex 1 x x∈Zd(cid:1).(cid:3)·(cid:1) {(cid:3)ϕ|K≥α}
e e|K · {ϕ|K≥α}
(cid:2) (cid:12) (cid:3)
Integrating both sides with respect to the probab(cid:12)ility measure ν() d=ef. P[ ϕ α], we obtain
· · | |K ≥
P[Aϕ = α] E P Aϕ 1 = P A ϕ α .
| |K ≤ ν | |K · {ϕ|K≥α} |K ≥
This completes the proof of Lemma 1(cid:2).4.(cid:2) (cid:3) (cid:3) (cid:2) (cid:12) (cid:3)
(cid:12)
7
We now introduce the canonical shift τ on RZd, such that τ (f)() = f( +z), for arbitrary
z z
· ·
f RZd and z Zd. The measure P is invariant under τ , i.e. P[τ 1(A)] = P[A], for all A
∈ ∈ z z− ∈ F
(the canonical σ-algebra on RZd), by translation invariance of g(, ) (see below (1.1)), and has
· ·
the following mixing property:
(1.24) lim P[A τ 1(B)]= P[A]P[B], for all A,B
z ∩ z− ∈ F
→∞
(one first verifies (1.24) for A,B depending on finitely many coordinates with the help of (1.9)
and the general case follows by approximation, see [3], pp.157-158). The following lemma gives
a 0-1 law for the probability of existence of an infinite cluster in E h, the excursion set above
ϕ≥
level h R, c.f. (0.2).
∈
Lemma 1.5.
Let Ψ(h) = P[E h contains an infinite cluster ], for arbitrary h R. One then has the following
ϕ≥ ∈
dichotomy:
0, if η(h) = 0,
(1.25) Ψ(h) =
1, if η(h) > 0,
(cid:26)
where η(h) = P[0 ≥h ]. In particular, recalling the definition (0.4) of h , (1.25) implies that
←→ ∞ ∗
Ψ(h) = 1 for all h< h , and Ψ(h) = 0 for all h > h .
∗ ∗
Proof. Thisfollowsbyergodicity, whichisitselfaconsequenceofthemixingproperty(1.24).
Remark 1.6.
When Ψ(h) = 1, in particular in the supercritical regime h < h , the infinite cluster in E h is
ϕ≥
P-almost surely unique. This follows by Theorem 12.2 in [9] (Bu∗rton-Keane theorem), because
the field 1 ϕ h is translation invariant and has the finite energy property (see [9],
{ x ≥ } x Zd
Definition 12.1). ∈ (cid:3)
(cid:0) (cid:1)
2 Non-trivial phase transition
Themaingoal ofthissection is theproofofTheorem2.6below, whichroughlystates thath (d)
∗∗
(and hence h (d), c.f. Corollary 2.7) is finite for all d 3, and that the connectivity function
of E h, c.f. (∗0.2), has stretched exponential decay for ≥arbitrary h > h . The proof involves a
ϕ≥
∗∗
certainrenormalizationschemeakintotheonedevelopedin[19]and[22]inthecontextofrandom
interlacements. This scheme will be used to derive recursive estimates for the probabilities of
certain crossing events, c.f. Proposition 2.2, which can subsequently be propagated inductively,
c.f. Proposition 2.4. The proper initialization of this induction requires a careful choice of the
parameters occurring in the renormalization scheme. The resulting bounds constitute the main
tool for the proof of the central Theorem 2.6. In addition, an extension of Proposition 2.2 can
be found in Remark 2.3 2).
We begin by defining on the lattice Zd a sequence of length scales
(2.1) L = lnL , for n 0,
n 0 0 ≥
where L 1 and l 100 are both assumed to be integers and will be specified below. Hence,
0 0
≥ ≥
L represents the finest scale and L < L < ... correspond to increasingly coarse scales. We
0 1 2
further introduce renormalized lattices
(2.2) L = L Zd Zd, n 0,
n n
⊂ ≥
8