Then, the proof follows similar lines of reasoning as the proof of (5.5.1) and (5.5.2) from Lemma 5.5.1 at the end of Section 5.5.1, choosing the prefactor δ appearing in (5.5.40) suitably small (recall that L = L(ε) is fixed) to obtain an analogue of the differential inequality (5.5.14). We omit further details.

### 5.6 Sharpness for finite-range models

In this section we prove Proposition 5.1.3. Fix L≥0 andδ ∈(0,1), we set

ω={ω_{h} :h∈R}, where ω_{h} :=T_{δ}{ϕ^{L}≥h}, h∈R (5.6.1)
(recall Tδ from the paragraph preceding the statement of Proposition 5.1.3). To be
specific, we assume thatωis sampled in the following manner. There exists a collection
of i.i.d. uniform random variables U = {U_{x} : x ∈ Z^{d}} independent of the process Z
(recall the definition from Section 5.3.1) underP such that, for h∈R,

ω_{h}(x) =

0 if Ux ≤δ/2,

1_{ϕ}^{L}_{x}≥h if Ux ∈(δ/2,1−δ/2),
1 if Ux ≥1−δ/2.

(5.6.2) We proceed to verify that ω satisfies the following properties:

(a) Lattice symmetry. For all h ∈ R, the law of ω_{h} is invariant with respect to
translations ofZ^{d}, reflections with respect to hyperplanes and rotations by π/2.

(b) Positive association. For all h ∈ R, the law of ω_{h} is positively associated, i.e.,
any pair of increasing events satisfies the FKG-inequality.

(c) Finite-energy. There exists c_{FE} ∈(0,1) such that for any h∈R,
P[ω_{h}(x) = 0|ω_{h}(y) for all y6=x]∈(c_{FE},1−c_{FE}).

(d) Bounded-range i.i.d. encoding. Let{V(x) :x∈Z^{d}}denote a family of i.i.d.
ran-dom variables (e.g. uniform in [0,1]). Then, for everyh∈R, there exists a
(mea-surable) function g = g_{h} : R^{Z}^{d} → {0,1}^{Z}^{d} such that the law of g((V(x))_{x∈}_{Z}^{d})
is the same as that of ω_{h} and, for any x ∈ Z^{d}, g_{x}((v_{y})_{y∈}_{Z}^{d}) depends only on
{v_{y} :y ∈B_{L}(x)}. Thus, in particular,ω_{h} is an L-dependent process.

Property (a) is inherited from corresponding symmetries of the laws of U and ϕ^{L}.
In view of (5.6.2), (5.3.2) and (5.3.7), ωh is an increasing function of the independent
collection (U,Z), and Property (b) follows by the FKG-inequality for independent
random variables. Still by (5.6.2), (5.3.2) and (5.3.7), Properties (c) and (d) hold: for
the former, takecFE =δ/2; for the latter one can use V(x) to generate the independent
random variables U_{x} and Z_{`}(˜z), for all 0≤`≤L and ˜z ∈ {x, x+^{1}_{2}e_{i},1≤i≤d}. We
will use another property of ω_{h}, whose proof is more involved. We therefore state it
as a separate lemma.

Lemma 5.6.1. The field ω satisfies the following:

(e) Sprinkling property. For every pair h < h^{0} ∈ R, there exists ε = ε(h, h^{0}) > 0
such thatω_{h} stochastically dominatesω_{h}^{0}∨η_{ε}, whereη_{ε} is a Bernoulli percolation
with density ε independent of ω_{h}^{0}. Henceforth we will denote this domination
byω_{h} ω_{h}^{0} ∨η_{ε}.

Proof. We assume by suitably extending the underlying probability space that there
existsη={η_{ε} :ε∈(0,1)}withηindependent ofωandη_{ε}distributed as i.i.d. Bernoulli
variables with density ε. We will progressively replace the threshold h with h^{0} in a
finite number of steps. To this end we claim that for anyκ >0 there existsε >0 such
that for anyh∈[h, h^{0}] andL∈ L, whereLcomprises all the (finitely many) translates
of the sub-lattice 10LZ^{d}, we have

ω_{h}ω_{h+κ}∨η_{ε}^{L}, (5.6.3)

where η_{ε}^{L}(x) = η_{ε}(x) if x ∈ L and η_{ε}^{L}(x) = 0 otherwise. Let us first explain how to
derive property (e) from (5.6.3): choosingκ:= ^{h}_{|L|}^{0}^{−h} givesω_{h} ω_{h+κ}∨η_{ε}^{L}, for suitable
ε = ε(h, h^{0}, L, d) > 0 and any choice L, so that iterating this over the lattices in L
gives

ωh ωh+|L|κ∨ _

L∈L

η^{L}_{ε} =ωh^{0} ∨ηε,
as desired.

By approximation, it suffices to verify (5.6.3) for all fields restricted to a finite set
Γ ⊂ Z^{d}. Let ω_{h}(S) := {ω_{h}(x) :x ∈ S} for S ⊂Z^{d} and define similarly η_{ε}(S), η_{ε}^{L}(S).

Notice thatω_{h} =ω_{h+κ} ∨ω_{h} and ω_{h+κ}∨η^{L}_{ε} are both increasing functions of (ω_{h+κ}, ω_{h})
and (ωh+κ, η_{ε}^{L}) respectively. Therefore it suffices to show that, conditionally on any
realization ofω_{h+κ}, the field ω_{h}(Γ) stochastically dominates η^{L}_{ε}(Λ). To this end we fix
an ordering{x_{1}, x_{2}, . . .} of the vertices in Γ such that all the vertices inL∩Γ appear
before all the vertices inL^{c}∩Γ. In view of [101, Lemma 1.1], it then suffices to show
that for any k ≥1, with Λ_{k} ={x_{1}, . . . , xk−1} (Λ_{0} =∅),

P-a.s., P[ω_{h}(x_{k}) = 1|A, A^{0}]≥ε·1_{x}_{k}∈L, (5.6.4)
where A := {ω_{h}(x) = σ(x), x ∈ Λ_{k}}, A^{0} := {ω_{h+κ}(x) = σ^{0}(x), x ∈ Λ}, for arbitrary
σ, σ^{0} ∈ {0,1}^{Z}^{d} with σ ≥ σ^{0} (as ω_{h} ≥ ω_{h+κ}). We now show (5.6.4) and assume that
x_{k}∈L (the other cases are trivial). To this end let us first define, for any x∈Z^{d}, the
pair of events

T(x) :={U_{x} ∈/ (δ/2,1−δ/2)} and U(x) := \

y∈B_{L}(x)\{x}

T(y). (5.6.5)

Notice that, in view of (5.6.2), T(x) corresponds to the event that the noise at x is
triggered. We then write, with U =U(x_{k}),

P[ω_{h}(x_{k}) = 1|A, A^{0}]≥P[ω_{h}(x_{k}) = 1| U, A, A^{0}]·P[U |A, A^{0}]. (5.6.6)
We will bound the two probabilities on the right-hand side separately from below. It
follows from the definition ofω_{h}in (5.6.2) thatω_{h+κ}(y) =ζ(y) on the eventT(y), where

ζ(y) = 1_{U}_{y}_{≥1−δ/2}. Consequently, decomposing A^{0}, we obtain, with A^{0}_{1} := {ω_{h+κ} =
σ^{0} on Γ∩B_{L}(x_{k})^{c}} and A^{0}_{2} :={ζ =σ^{0} onB_{L}(x_{k})\ {x_{k}}}, that

p:= P[ω_{h}(x_{k}) = 1| U, A, A^{0}] =P[ω_{h}(x_{k}) = 1| U, A, ω_{h+κ}(x_{k}) =σ^{0}(x_{k}), A^{0}_{1}, A^{0}_{2}].
However, in view of definition (5.6.2), (5.6.5) and the fact that L is 10L-separated,
it follows that both {ω_{h}(x_{k}) = 1, ω_{h+κ}(x_{k}) = σ^{0}(x_{k})} and {ω_{h+κ}(x_{k}) = σ^{0}(x_{k})} are
where the equality in the second line follows by (5.6.2) and (5.6.5) upon conditioning
on ω_{h}(x), x∈ Λ_{k}, ω_{h+κ}(y), y∈ (Λ\B_{L}(x_{k}))∪ {x_{k}} and 1_{ϕ}^{L}

y≥h+κ, y ∈B_{L}(x_{k})\ {x_{k}},
and the final lower bound follows by distinguishing whether ρ(y) 6= σ^{0}(y) (in which
case the given conditional probability equals 1) or not, and using (5.6.2). Overall,
the right-hand side of (5.6.6) is thus bounded from below by ε := ε^{0}(δ/2)^{|B}^{L}^{|}, which
implies (5.6.4) and completes the verification of (e).

The rest of this section is devoted to deriving Proposition5.1.3from Properties (a)–

(e). We prove in two parts that ˜h(δ, L) ≥h∗(δ, L) and h∗(δ, L) = h∗∗(δ, L) ≥ ˜h(δ, L) which together imply ˜h(δ, L) =h∗(δ, L) =h∗∗(δ, L), as asserted.

We first argue that ˜h(δ, L) ≥ h∗(δ, L). As a consequence of Properties (a)–(c) and (e), one can adapt the argument in [75] in the context of Bernoulli percolation

— see also [74, Chap. 7.2] — to deduce that ω_{h} percolates in “slabs”, i.e., for every
h < h∗(δ, L), there exists M ∈Nsuch that

But this implies h≤˜h(δ, L), as desired.

The proof of h∗(δ, L) =h∗∗(δ, L)≥˜h(δ, L) on the other hand follows directly from the exponential decay of ωh in the subcritical regime. More precisely, we claim that for every h > h∗(δ, L), there exists c=c(h)>0 such that

P[0←^{ω}→^{h} ∂B_{R}]≤exp(−cR), for every R ≥0, (5.6.8)
which implies that h∗(δ, L) =h∗∗(δ, L), cf. (5.1.4) (recall that h∗∗(δ, L) ≥ ˜h(δ, L), as
explained below (5.1.12) for δ = 0 and L = ∞). We therefore focus on the proof of
(5.6.8). For each h ∈ R, consider the family of processes γ_{ε} := ω_{h} ∨η_{ε} indexed by
ε ∈ [0,1] where ηε is a Bernoulli percolation with density ε independent of ωh. Let
ε_{c}=ε_{c}(h)∈[0,1] denote the critical parameter of the family of percolation processes
{γ_{ε} : ε ∈ [0,1]} and suppose for a moment that for every 0 ≤ ε < ε_{c}(h), there exists
c=c(h, ε)>0 such that for everyR ≥1,

θ_{R}(ε) :=P[0←→^{γ}^{ε} ∂B_{R}]≤exp(−cR). (5.6.9)
Then (5.6.8) follows immediately by takingε= 0 in caseε_{c}(h)>0 for allh > h∗(δ, L).

But the latter holds by the following reasoning. From Property (e), we see that
whenever h > h_{∗}(δ, L), choosingh^{0} = (h_{∗}(δ, L) +h)/2, one hasω_{h}^{0} ω_{h}∨η_{ε}^{0} for some
ε^{0} > 0, whence ε^{0} ≤ ε_{c}(h) as ω_{h}^{0} is subcritical. The remainder of this subsection is
devoted to proving (5.6.9).

For this purpose, we use the strategy developed in the series of papers [58,57, 59]

to prove subcritical sharpness using decision trees. This strategy consists of two main
parts. In the first part, one bounds the varianceθ_{R}(1−θ_{R}) using the OSSS inequality
from [125] by a weighted sum of influences where the weights are given by revealment
probabilities of a randomized algorithm (see (5.6.14) below). Then in a second part
one relates these influence terms to the derivative ofθ_{R}(ε) with respect toεto deduce
a system of differential inequalities of the following form:

θ_{R}^{0} ≥β R

Σ_{R}θ_{R}(1−θ_{R}), (5.6.10)

where Σ_{R} :=PR−1

r=0 θ_{r} and β =β(ε) : (0,1)→(0,∞) is a continuous function. Using
purely analytical arguments – see [58, Lemma 3.1] – one then obtains (5.6.9) forε < ε_{c}.
In our particular context, we apply the OSSS inequality for product measures (see
[58] for a more general version) and for that we use the encoding of ω_{h} in terms of
{V(x) :x∈Z^{d}}provided by Property (d). In view of this,1E_{R} with E_{R}={0←→^{γ}^{ε} ∂B_{R}}
can now be written as a function of independent random variables (V(x) :x∈B_{R+L})
and (η_{ε}(x) : x ∈ B_{R}). Using a randomized algorithm very similar to the one used in
[59, Section 3.1] (see the discussion after the proof of Lemma 5.6.2 below for a more
detailed exposition), we get

X

x∈B_{R+N}

Inf_{V(x)}+ X

x∈B_{R}

Inf_{η}_{ε}_{(x)} ≥cL^{−(d−1)} R

Σ_{R}θ_{R}(1−θ_{R}), (5.6.11)
where Inf_{V(x)} := P[1E_{R}(V, η_{ε}) 6= 1E_{R}( ˜V, η_{ε})] with ˜V being the same as V for every
vertex except atxwhere it isresampled independently, and Inf_{η}_{ε}_{(x)} is defined similarly.

In order to derive (5.6.10) from (5.6.11), we use the following lemma.

Lemma 5.6.2. There exists a continuous function α : (0,1) → (0,∞) such that for all R≥1,

θ_{R}^{0} (ε)≥α(ε) X

x∈B_{R+L}

Inf_{V(x)}+ X

x∈B_{R}

Inf_{η}_{ε}_{(x)}

. (5.6.12)

Proof. Since the derivative is with respect to the parameter of the Bernoulli component of the process, it follows from standard computations that

θ^{0}_{R}=P[ω_{h}(0) = 0] X

x∈B_{R}

P[Piv_{x}].

where Piv_{x} is the same event as in Section 5.5 (see below (5.5.6)) with χ^{t} replaced
by γε. Now, on the one hand, Inf_{η}_{ε}_{(x)} ≤ P[Pivx]. On the other hand, since V(x)
affects the states of vertices only in B_{L}(x) by Property (d), one immediately gets
Inf_{V}_{(x)} ≤ P[Piv(B_{L}(x))], where Piv(B_{L}(x)) – called the pivotality of the box B_{L}(x) –
is the event that 0 is connected to ∂BR in γε∪BL(x) where as it is not in γε\BL(x).

In fact, due to finite-energy property of ω_{h}, we can write

P[Piv(B_{L}(x))]≤(c_{FE}(1−ε))^{−c}^{0}^{L}^{d}P[Piv(B_{L}(x)), γ_{ε}(y) = 0 for all y∈B_{L}(x)].
If we start on the event on the right-hand side and then open the vertices insideBL(x)
one by one inγ_{ε} until 0 gets connected to ∂B_{R}, which must happen by the definition
of Piv(B_{L}(x)). The last vertex to be opened is pivotal for the resulting configuration.

This implies that

P[Piv(B_{L}(x)), η_{ε}(y) = 0 for all y∈B_{L}(x)]≤ε^{−c}^{0}^{L}^{d} X

y∈BL(x)

P[Piv_{y}].
Combining all these displays yields (5.6.12).

To conclude, we now give a full derivation of (5.6.11) for sake of completeness. To
this end let us define a randomized algorithm T which takes (V(x) : x ∈ B_{R+L}) and
(η_{ε}(x) : x∈ B_{R}) as inputs, and determines the event E_{R} by revealing V(x), η_{ε}(x) one
by one as follows:

Definition 5.6.3 (Algorithm T). Fix a deterministic ordering of the vertices in B_{R}
and choose j ∈ {1, . . . , R} uniformly and independently of (V, η_{ε}). Now set x_{0} to
be the smallest x ∈ ∂B_{j} in the ordering and reveal η_{ε}(x_{0}) as well as V(y) for all
y ∈ B_{L}(x_{0}). Set E_{0} := {x_{0}} and C_{0} := {x_{0}} if γ_{ε}(x_{0}) = 1 and C_{0} := ∅ otherwise. At
each stept ≥1, assume that Et−1 and Ct−1 have been defined. Then,

— If the intersection of B_{R} with the set (∂_{out}Ct−1 ∪∂B_{j})\Et−1 is non-empty, let
x be the smallest vertex in this intersection and set xt := x, Et := Et−1 ∪ {xt}.

Reveal η_{ε}(x_{t}) as well as V(y) for all y ∈ B_{L}(x_{t}) and set C_{t} := Ct−1 ∪ {x_{t}} if
γ_{ε}(x_{t}) = 1 and C_{t} :=Ct−1 otherwise.

— If the intersection is empty, halt the algorithm.

In words, the algorithm T explores the clusters in γ_{ε} of the vertices in ∂B_{j}. By
applying the OSSS inequality [125] to 1E_{R} and the algorithmT, we now get

Var[1E_{R}] =θ_{R}(1−θ_{R})≤ X

x∈B_{R+L}

θ_{V(x)}(T) Inf_{V(x)}+ X

x∈B_{R}

θ_{η}_{ε}_{(x)}(T) Inf_{η}_{ε}_{(x)}, (5.6.13)
where the functionθ·(T), called the revealment of the respective variable, is defined as
θ_{V(x)}(T) :=π[Treveals the value of V(x)], (5.6.14)
and θ_{η}_{ε}_{(x)}(T) is defined in a similar way. Here π denotes the probability governing
the extension of (V, η_{ε}) which accommodates the random choice of layer j, which is
independent of (V, η_{ε}).

Let us now bound the revealments for V(x) and ηε(x). Notice that since V(x)
affects the state of vertices only in B_{L}(x) by Property (d), V(x) is revealed only if
there is an explored vertex y ∈ B_{L}(x)∩ B_{R}. The vertex y, on the other hand, is
explored only ify is connected to ∂Bj in γε. We deduce that

θ_{V(x)}(T)≤ 1
R

R

X

j=1

P[(B_{L}(x)∩B_{R})←→^{γ}^{ε} ∂B_{j}]≤ 1
R

R

X

j=1

X

y∈BL(x)∩BR

P[y←→^{γ}^{ε} ∂B_{j}],
where in the last step we used a naive union bound. Forη_{ε}(x), it is even simpler:

θ_{η}_{ε}_{(x)}(T)≤ 1
R

R

X

j=1

P[x←→^{γ}^{ε} ∂B_{j}].

Now (5.6.11) follows by plugging the previous two displays into (5.6.13) combined with the translation invariance of γε implied by Property (a) and the triangle inequality.

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