**5.5 Interpolation scheme**

**5.5.2 Proof of Lemma 5.5.1**

Throughout this subsection, we fix all the parameters r, R, t, h and assume tacitly that h∈(˜h+ 2ε, h∗∗−2ε),t ≥1,

C_{5} := 10^{2}κL_{0} ≤r≤R/2 (5.5.17)

and (t, R) satisfy q_{R}(t, h) ≥ c_{7} as in Lemma 5.5.1, where L_{0} will be given by 5.5.4
below. Although they depend on t, we will write χ and ψ instead of χ^{t} and ψ^{t}. We
set T to be the smallest integer such that u(L_{T}) ≥20κL_{t} and T the smallest integer
such thatu(L_{T})≥20κL_{T}. Note that it follows directly from the definition ofu(·) that
T ≤C(L_{0})t^{3} and T ≤C(L_{0})t^{9}. We then define

u^{∗}(L_{m}) :=

(0 if m≤T ,

u(L_{m}) if m > T . (5.5.18)
The following lemma is a key step in proving Lemma5.5.1. ForN ≥1, let CoarsePiv_{x}(N)
denote the event that B_{r} and ∂B_{R} are connected in {χ ≥ h} ∪ B_{N}(x) but not in
{χ≥h}.

Lemma 5.5.4. For every ε > 0 and L0 ≥ C8(d, ε) there exist positive constants
C_{9} =C_{9}(L_{0}, ε, d), c_{10}=c_{10}(L_{0}, ε, d), ρ=ρ(d) such that for all x∈B_{R}\Br−1,

P[CoarsePiv_{x}(L_{T})]

≤e^{−c}^{10}^{(|x|/L}^{T}^{)}^{ρ} +X

n≥T

e^{C}^{9}^{(log}^{L}^{n}^{)}^{2}^{−c}^{10}^{u}^{∗}^{(L}^{n}^{)}^{ρ} X

y∈B_{10κLn}(x)

P[Piv_{y}|χ_{y} =h]

. (5.5.19) We first assume Lemma 5.5.4 to hold and give the proof of Lemma 5.5.1.

Proof of Lemma 5.5.1. LetL0 =C8(d, ε) be given by Lemma5.5.4. All constantsC, c
below may depend on L_{0} (and therefore onε and d). Assume first that R≥8L_{T}. Let
us introduce the eventE_{x} thatB_{2L}_{t}(x) and∂B_{L}_{T}−Lt(x) are not connected in{χ≥h},
and Fx the event that Br and BR are connected in {χ ≥ h} ∪ BL_{T}(x) but not in
{χ≥h} \ {x}, see Fig. 5.4.

x 2Lt

BLT−Lt(x) BLT(x) 0

Br

∂BR

χ < h χ≥h

χ≥h

Figure 5.4 – Decoupling in the proof of Lemma5.5.1. By forcing the event
Ex(in red), which is independent ofZ(Λx) and not too costly sinceh >h,˜
F_{x} and f(ψ_{x}) decouple.

Observe that, conditionally on Z(Λ_{x}) where Λ_{x} :={x} ∪B_{L}_{T}(x)^{c} (recall the
defini-tion in (5.3.8)), the eventF_{x}is decreasing in all the variables belonging toZ(Z^{d})\Z(Λ_{x}),
and so is Ex. From the FKG inequality for independent random variables, we deduce
that

E[f(ψx);Ex∩Fx|Z(Λx)] =f(ψx)P[Ex∩Fx|Z(Λx)]

≥f(ψ_{x})P[E_{x}|Z(Λ_{x})]P[F_{x}|Z(Λ_{x})]

≥c_{7}L^{−d}_{T} E[f(ψ_{x});F_{x}|Z(Λ_{x})],

(5.5.20)

where in the final step we used the lower bound q_{R}(t, h)≥ c_{7} from the hypothesis of
Lemma 5.5.1 (note that u(LT −Lt)> 2Lt by definition of T) together with the fact
that the event E_{x} is independent of Z(Λ_{x}) by (5.3.9). Since χ_{x} is measurable with
respect toZ(Λ_{x}), integrating with respect toZ(Λ_{x}) gives

E[f(ψ_{x});F_{x}|χ_{x} =h]≤c^{−1}_{7} L^{d}_{T} E[f(ψ_{x});E_{x}∩F_{x}|χ_{x}=h] (5.5.21)
(to obtain (5.5.21), one first integrates (5.5.20) against a set of the form {h ≤ χ_{x} <

h+δ}, normalizes suitably and takes the limit δ → 0). Since R−r ≥ 4L_{T} (recall
thatR ≥8L_{T} by assumption) and consequently B_{L}_{T}(x) cannot intersect bothB_{r} and

∂B_{R}, the event E_{x}∩F_{x} is independent of (χ_{x}, ψ_{x}) as the range of χ is L_{t}. From this
observation, we deduce that

E[f(ψ_{x});E_{x}∩F_{x}|χ_{x} =h] =E[f(ψ_{x})|χ_{x} =h]P[E_{x}∩F_{x}]

≤E[f(ψ_{x})|χ_{x} =h]P[CoarsePiv_{x}(L_{T})], (5.5.22)
where in the second step we used the fact that E_{x} ∩F_{x} ⊂ CoarsePiv_{x}(L_{T}) when
R−r ≥4L_{T}. Now, Lemma 5.5.4 gives that

X

x∈B_{R}\Br−1

P[CoarsePiv_{x}(L_{T})]≤L^{C}_{T}e^{−c(r/L}^{T}^{)}^{ρ} + e^{C(log}^{L}^{T}^{)}^{2} X

x∈Z^{d}

P[Piv_{x}|χ_{x} =h], (5.5.23)
where we used that (see (5.5.18) for u^{∗}(·))

X

n≥T

|B_{10κL}_{n}|e^{C}^{9}^{(log}^{L}^{n}^{)}^{2}^{−c}^{10}^{u}^{∗}^{(L}^{n}^{)}^{ρ} ≤e^{C(log}^{L}^{T}^{)}^{2},
X

x∈BR\Br−1

e^{−c}^{10}^{(|x|/L}^{T}^{)}^{ρ} ≤L^{C}_{T}e^{−c(r/L}^{T}^{)}^{ρ},

which follow by considering separately the cases T ≤ n ≤ T and n ≥ T in the first
line and the cases |x| ≤(LT ∨r) and |x|>(LT ∨r) in the second. Lemma5.5.1 now
follows in the case R ≥ 8L_{T} from (5.5.21), (5.5.22) and (5.5.23) since F_{x} ⊃ Piv_{x} and
T ≤C(L_{0})t^{3}, T ≤C(L_{0})t^{9}.

On the other hand if R < 8LT, we simply bound X

x∈Z^{d}

E[f(ψ_{x}); Piv_{x}|χ_{x} =h]≤ X

x∈B_{R}

E[f(ψ_{x})|χ_{x} =h]≤c^{0}L^{d}_{T} E[f(ψ_{0})|χ_{0} =h]. (5.5.24)
The proof is thus concluded by noting that, since r ≤ 4L_{T} and T ≤ C(L_{0})t^{3}, one
can find C_{6} large enough (depending on d, ε, L_{0} only) such that c^{0}L^{d}_{T} ≤ exp[C_{6}t^{3} −
r^{c}^{8}e^{−C}^{6}^{t}^{3}].

We now turn to the proof of Lemma 5.5.4. Roughly speaking, we would like to show
that conditionally on the event CoarsePiv_{x}(L_{T}), some Piv_{y} occurs in the box B_{L}_{T}(x)
with not too small probability. A natural strategy consists in trying to create paths
between the clusters of B_{r} and ∂B_{R}, which must necessarily intersect B_{L}_{T}(x).
How-ever, the fact that the range of dependence of χ is L_{t} presents a potential barrier for
constructing these paths, for instance by forcing the field to be quite large. In order
to poke through this barrier, we will use a good bridge – in the sense mentioned below
– connecting the clusters of B_{r} and ∂B_{R} in {χ ≥ h} ∩B_{R} so we can apply a result
akin to Lemma5.3.5 (see Remark 5.3.7 and (5.5.31) below) to construct open paths.

To begin with, let
F_{0,y} :=

ϕ^{L}_{z}^{0} −ϕ^{0}_{z} ≥ −M +ε, ϕ^{0}_{z} ≥ −M, ∀z ∈B_{L}_{0}(y) , for y∈L^{0},
Hn,y :={ϕ^{L}_{z}^{n} −ϕ^{L}_{z}^{n−1} ≥ − 6ε

(πn)^{2}, ∀z ∈B2Ln(y)}, for n ≥1 and y∈L^{n}, (5.5.25)
with M = M(L_{0}) chosen large enough (eg. M = logL_{0}) so that the bound in (C3)
holds when L0 ≥ C8(d, ε). We call a bridge from Definition 5.2.1 good if it satisfies
Definition 5.2.2, except that we only require G2 to hold for all j satisfying 1∨m ≤
j ≤ n (which is a weaker condition). With a slight abuse of notation, we define
Gn,x :=τxGn forx∈Z^{d}, where Gn denotes the event from (5.2.2) corresponding to this
weaker notion of good bridge, and with the choice Λ_{n} =B_{10κL}_{n} \B_{κL}_{n} in (5.2.1). For
later reference, we also defineG(S_{1}, S_{2}) as in (5.3.18) for any pair of admissible subsets
S1, S2 of Λn.

In view of (5.5.25) and Theorem 5.2.3, for all L_{0} ≥C_{8}(d, ε) (which we will
hence-forth tacitly assume), there exist constants c_{11} = c_{11}(L_{0}), ρ = ρ(d) > 0 such that for
every n ≥0 andx∈Z^{d},

P[G_{n,x}]≥1−e^{−c}^{11}^{L}^{ρ}^{n}. (5.5.26)
We henceforth fixL0 as above. All the constantsC, c below may depend on L0,ε and
d.

By definition, G_{n,x} guarantees the existence of a good bridge between the clusters
ofBrand∂BRin{χ≥h}∩BRprovidedthey are both admissible in Λn, i.e., they both
intersect ∂B_{10κL}_{n}(x) as well as B_{8κL}_{n}(x) – cf. above (5.2.2). But the latter condition
is satisfied on the event CoarsePiv_{x}(8κL_{n}) when B_{r} 6⊂ B_{10κL}_{n}(x). Putting these two
observations together and using (5.5.25), one ends up with the following lemma, whose
proof is postponed for a few lines.

Lemma 5.5.5 (Creating pivotals from coarse pivotals). For all x ∈ BR\Br−1 and
n≥T such that B_{r} 6⊂B_{10κL}_{n}(x), we have

P[CoarsePivx(8κLn),Gn,x]≤e^{C(log}^{L}^{n}^{)}^{2} X

y∈B_{10κLn}(x)

P[Pivy|χy =h]. (5.5.27)
In order to prove Lemma 5.5.4, we then find the first scale L_{n} at which the event
G_{n,x} occurs so that we can apply the previous lemma, which is the content of Lemma
5.5.6 below.

For x∈B_{R}\Br−1, let S_{x} denote the largest integer such that i) B_{r} 6⊂B_{10κL}_{Sx}(x),
and ii)B_{10κL}_{Sx}(x) has empty intersection with at least one ofB_{r}and ∂B_{R}. Recall that
we used a condition similar to ii) to derive (5.5.22) (this was ensured by the assumption

R ≥ 8L_{T} in the argument leading to (5.5.22)). The quantity S_{x} is well-defined, i.e.,
S_{x} ≥ 0 by condition (5.5.17). Moreover, L_{S}_{x} ≥ c|x|, as can be readily deduced from
the following: if d(x, B_{r}) > ^{|x|}_{4} , the ball B(x,^{|x|}_{5} ) does not intersect B_{r}, whereas for
d(x, Br)≤ ^{|x|}_{4} , the ball B(x,^{|x|}_{4} ) does not intersect ∂BR as R≥2r.

Lemma 5.5.6 (Finding the first good scale). For all x∈ B_{R}\B_{r} such that S_{x} > T,
the following holds:

P[CoarsePiv_{x}(L_{T})]≤

Sx

X

n=T

e^{−cu}^{∗}^{(L}^{n}^{)}^{ρ}P[CoarsePiv_{x}(8κL_{n}),G_{n,x}] +P[G_{S}^{c}_{x}_{,x}].

Lemma 5.5.4 now follows readily by combining (5.5.26) with Lemmas 5.5.5 and
5.5.6 in case S_{x} > T (this requires L_{0} to be large enough, cf. above (5.5.26)), and
simply boundingP[CoarsePiv_{x}(L_{T})] by 1 otherwise. The latter is accounted for by the
first term on the right-hand side of (5.5.19) due to the factor 1/LT appearing in the
exponent and the fact thatL_{S}_{x} ≥c|x|.

We now turn to the proofs of Lemmas 5.5.5 and 5.5.6.

Proof of Lemma 5.5.5. The proof is divided into two steps. In the first step, we prove anunconditional version of (5.5.27), namely

P[CoarsePiv_{x}(8κL_{n}),G_{n,x}]≤e^{C(log}^{L}^{n}^{)}^{2} X

y∈Λn(x)

P[Piv_{y},|χ_{y}−ϕ^{0}_{y}| ≤M^{0}], (5.5.28)
where M^{0} := h+ε+M and Λ_{n}(x) := Λ_{n}+x. In the second step, we transform the
unconditional probability into a conditional one:

P[Pivy,|χy −ϕ^{0}_{y}| ≤M^{0}]≤CP[Pivy|χy =h]. (5.5.29)
It is clear that (5.5.27) follows from these two bounds as Λ_{n}(x)⊂B_{10κL}_{n}(x).

Let us first prove (5.5.28). To this end, consider any pair of disjoint subsetsC_{1} and
C_{2} of B_{R} such that C(C_{1}, C_{2}) := {C_{B}_{r} =C_{1}, C_{∂B}_{R} =C_{2}} ⊂CoarsePiv_{x}(8κL_{n}), where
C_{A} denotes the cluster of A in B_{R}∩ {χ ≥h} (observe that CoarsePiv_{x} is measurable
relative to the pair of random sets C_{B}_{r} and C_{∂B}_{R}). Taking the union over all possible
choices of pairs (C_{1}, C_{2}) (call this collection of pairs C) yields the decomposition

[

(C1,C2)∈C

C(C1, C2) = CoarsePivx(8κLn). (5.5.30)
By (5.5.30), the sets C_{1} and C_{2} are admissible in Λ_{n}(x) for any pair (C_{1}, C_{2}) ∈ C –
cf. below (5.5.26) – and so areC_{1} andC_{2}, whereC = (C∪∂_{out}C)∩Λ_{n}(x), forC ⊂Z^{d}.
We will use a good bridge to create a (closed) pivotal pointyin (∂_{out}C_{1}∪∂_{out}C_{2})∩Λ_{n}(x),
cf. Fig. 5.5.

Now, similarly to the bound (5.3.25) derived in Section 5.3, we can prove that P

h [

y∈∂outC1

z∈∂outC2

y,z∈Λ_{n}(x)

{N(y)←−−−−−−−→^{χ≥h}

Λn(x)\(C1∪C2)

N(z)} ∩ H_{y}∩ H_{z}

C(C_{1}, C_{2}),
G(C_{1}, C_{2})

i≥e^{−C(log}^{L}^{n}^{)}^{2},

(5.5.31)

whereH_{v} ={χ_{v}−ϕ^{0}_{v} ≥ −M and ϕ^{0}_{v} ≥ −M}. We now explain the small adjustments to
the proof of Lemma5.3.5(or of (5.3.25)) needed in order to accommodate the different
setup implicit in (5.5.31). First, property (5.3.20) is replaced by the following: for each
boxB =B_{L}_{m}(y)∈ B, whereB is any good bridge in Λ_{n}(x),

χ_{z}−ϕ^{0}_{z} ≥ −M and ϕ^{0}_{z} ≥ −M ∀z ∈B_{L}_{0}(y), when m= 0,

χ_{z}−ϕ^{L}_{z}^{m} ≥ −ε ∀z ∈B˜ =B_{2L}_{m}(y), when 1≤m≤t, (5.5.32)
as follows from (5.5.25) and our (weaker) version of G2 (see below (5.5.25)). For
each B =B_{L}_{m}(y) ∈ B, one then redefines the event A_{B} (see (5.3.21)) in the proof of

and observes that, due to (5.5.32), xB and yB are connected in {χ ≥ h} whenever
B ∈ B and A_{B} occurs (the points x_{B} and y_{B} are chosen like in the paragraph above
(5.3.21)). In view of the constraint h ∈ (˜h + 2ε, h∗∗ − 2ε) and the lower bound
qR(t, h)≥c7 from the hypothesis of Lemma5.5.1, which are in force (see the beginning
of this subsection), Lemmas5.3.2and5.3.3(see also Remark5.3.4) together imply that
P[A_{B}]≥L^{−C}_{m} ^{0} for all m ≥1 satisfying L_{m} ≤R. The rest of the proof of Lemma5.3.5
then follows as before, yielding (5.5.31).

Rewriting (5.5.31) as an inequality involving the corresponding unconditional
prob-abilities, using thatG_{n,x} ⊂ G(C_{1}, C_{2}) (see (5.2.2)), and subsequently summing over all
possible choices of pairs (C1, C2)∈C, we obtain

Altogether we have

E_{1}(z)∩ H_{z} ⊂Piv_{z}∩ {|χ_{z}−ϕ^{0}_{z}| ≤M^{0}}.

Therefore a simple union bound gives P

h [

y,z∈Λ_{n}(x)

E(y, z)∩ H_{y}∩ H_{z} i

≤ |Λ_{n}| X

z∈Λ_{(}x)

P[Piv_{z},|χ_{z} −ϕ^{0}_{z}| ≤M^{0}] + X

y,z∈Λn(x)

P[E_{2}(y, z)∩ H_{y}∩ H_{z}].

(5.5.35)

Now note that E2(y, z)∩ Hy∩ Hz is independent of ϕ^{0}_{z}, therefore

P[E_{2}(y, z)∩ H_{y}∩ H_{z}] =C(L_{0})P[E_{2}(y, z)∩ H_{y} ∩ H_{z}∩ {ϕ^{0}_{z} ≥M+h}]

≤C(L_{0})P[Piv_{y},|χ_{y} −ϕ^{0}_{y}| ≤M^{0}], (5.5.36)
whereC(L_{0}) :=P[ϕ^{0}_{0} ≥M+h]^{−1}. In the last inequality we used thatE_{2}(y, z)∩ H_{y}∩
H_{z} ∩ {ϕ^{0}_{z} ≥ M +h} ⊂ Piv_{y} ∩ {|χ_{y} −ϕ^{0}_{y}| ≤ M^{0}}, which can be easily verified. The
desired inequality (5.5.28) then follows directly from (5.5.33), (5.5.35) and (5.5.36)
together.

We now turn to the proof of (5.5.29). Below, for a stationary Gaussian process Φ
indexed byZ^{d}, we write p_{Φ} for the density of Φ_{0}. The key observation is that the pair
(Piv_{y}, χ_{y} −ϕ^{0}_{y}) is independent of ϕ^{0}_{y} (Piv_{y} is measurable with respect to χ_{z}, z 6= y,
which is independent ofϕ^{0}_{y}). Consequently,

P[Piv_{y}|χ_{y}−ϕ^{0}_{y} =h_{1}, ϕ^{0}_{y} =h_{2}] =P[Piv_{y}|χ_{y}−ϕ^{0}_{y} =h_{1}]
for (h_{1}, h_{2})∈R^{2}, which leads to

P[Piv_{y},|χ_{y} −ϕ^{0}_{y}| ≤M^{0}] =
Z M^{0}

−M^{0}

P[Piv_{y}|χ_{y}−ϕ^{0}_{y} =h_{1}]p_{χ−ϕ}^{0}(h_{1})dh_{1}, (5.5.37)
and also

p_{χ}(h)P[Piv_{y},|χ_{y}−ϕ^{0}_{y}| ≤M^{0} |χ_{y} =h]

=
Z M^{0}

−M^{0}P[Pivy|χy−ϕ^{0}_{y} =h1]pχ−ϕ^{0}(h1)p_{ϕ}^{0}(h−h1)dh1.
(5.5.38)
Since ϕ^{0} is a centered Gaussian variable, we have

inf

|h1|≤M^{0}p_{ϕ}^{0}(h−h_{1})≥c(L_{0})>0

and (5.5.29) now follows from the displays (5.5.37) and (5.5.38). This completes the proof.

x

Figure 5.5 – Finding a good scale and reconstructing. On the event
CoarsePivx(8κLN)∩ G_{N,x}, one constructs a path (red) in {χ ≥ h}
con-necting the boundaries (dotted) of the clusters ofB_{r} andB_{R}. The pointz
is flipped to open andy becomes a pivotal point (Lemma 5.5.5). The
oc-currence ofG_{n,x}^{c} , decoupled by a dual surface, balances the reconstruction
cost from the bridge (Lemma5.5.6).

Proof of Lemma 5.5.6. In the proof below, we will consistently use the letters N and
n as follows: n is the largest integer such that u(L_{N}) ≥ 20κL_{n}. Together with the
definition of T (see above (5.5.18)), this implies n > T whenever N ≥ T. Now,
decomposing on the smallest good scale fromT toS_{x} gives

1 =

where we also used the monotonicity of CoarsePiv_{x}(L) in L. In view of (5.5.18),
the first sum in the right-hand side of (5.5.39) is accounted for in the statement of
Lemma5.5.6. As for the second sum, we now decouple the events G_{n,x}^{c} using a similar
technique as in the proof of Lemma5.5.1, cf. also Fig.5.4and5.5. The eventG_{N,x}∩G_{n,x}^{c}

Since our choice of (n, N) guarantees that u(L_{N}) ≥ 20κL_{n} and 10κL_{n} exceeds the
range of χ, the standing assumption q_{R}(t, h) ≥ c_{7} implies a lower bound of the form
c7L^{−d}_{N} for the above disconnection probability under P[· |Z(Λx)] for all N ≤Sx. Now,
applying the same argument as for (5.5.20) with E_{x} and F_{x} replaced by the above
disconnection and coarse pivotality events, respectively, we deduce that

P[CoarsePivx(8κLN)|Z(Λx)]

≤c^{−1}_{7} L^{d}_{N}P[CoarsePiv_{x}(8κL_{N}), B_{20κL}_{n}(x)

χ≥h

6

←→∂B_{4κL}_{N}(x)|Z(Λ_{x})].

Plugging this inequality into the previous display and integrating with respect toZ(Λ_{x})
gives

P[CoarsePiv_{x}(8κL_{N}),G_{N,x},G_{n,x}^{c} ]

≤c^{−1}_{7} L^{d}_{N}P[CoarsePiv_{x}(8κL_{N}), B_{20κL}_{n}(x)

χ≥h

6

←→∂B_{4κL}_{N}(x),G_{N,x},G_{n,x}^{c} ].

However, since N ≤ Sx (recall its definition from the discussion preceding the
state-ment of Lemma 5.5.6), B_{10κL}_{N}(x) has empty intersection with at least one of B_{r} or

∂B_{R}. We deduce from this fact that the event{CoarsePiv_{x}(8κL_{N}), B_{20κL}_{n}(x)←→6 ^{χ≥h}

∂B4κL_{N}(x), GN,x}is measurable with respect toZ(B20κLn(x)^{c}) and thus independent of
G_{n,x} by (5.3.9) (our modification of property G2, cf. below (5.5.25), is geared towards
this decoupling). Using this observation to factorize the right-hand side of the
previ-ous displayed inequality and subsequently boundingP[G_{n,x}^{c} ] by (5.5.26), we obtain for
T ≤N ≤S_{x} that

P[CoarsePiv_{x}(8κL_{N})∩ G_{N,x} ∩ G_{n,x}^{c} ]≤c^{−1}_{7} L^{d}_{N}e^{−c}^{11}^{L}^{ρ}^{n}P[CoarsePiv_{x}(8κL_{N}),G_{N,x}]

≤e^{−cL}^{ρ}^{n}P[CoarsePiv_{x}(8κL_{N}),G_{N,x}],

where in the final step we used the fact thatL_{N} ≤exp[C(logL_{n})^{3}]. Substituting this
bound into (5.5.39) and using that 20κ`_{0}L_{n} > u(L_{N}) completes the proof.