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={ωh :h∈R}, where ωh :=Tδ{ϕL≥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 = {Ux : x ∈ Zd} independent of the process Z (recall the definition from Section 5.3.1) underP such that, for h∈R,
ωh(x) =
0 if Ux ≤δ/2,
1ϕLx≥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 ωh is invariant with respect to translations ofZd, reflections with respect to hyperplanes and rotations by π/2.
(b) Positive association. For all h ∈ R, the law of ωh is positively associated, i.e., any pair of increasing events satisfies the FKG-inequality.
(c) Finite-energy. There exists cFE ∈(0,1) such that for any h∈R, P[ωh(x) = 0|ωh(y) for all y6=x]∈(cFE,1−cFE).
(d) Bounded-range i.i.d. encoding. Let{V(x) :x∈Zd}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 = gh : RZd → {0,1}Zd such that the law of g((V(x))x∈Zd) is the same as that of ωh and, for any x ∈ Zd, gx((vy)y∈Zd) depends only on {vy :y ∈BL(x)}. Thus, in particular,ωh is an L-dependent process.
Property (a) is inherited from corresponding symmetries of the laws of U and ϕL. 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 Ux and Z`(˜z), for all 0≤`≤L and ˜z ∈ {x, x+12ei,1≤i≤d}. We will use another property of ωh, 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 < h0 ∈ R, there exists ε = ε(h, h0) > 0 such thatωh stochastically dominatesωh0∨ηε, whereηε is a Bernoulli percolation with density ε independent of ωh0. Henceforth we will denote this domination byωh ωh0 ∨ηε.
Proof. We assume by suitably extending the underlying probability space that there existsη={ηε :ε∈(0,1)}withηindependent ofωandηεdistributed as i.i.d. Bernoulli variables with density ε. We will progressively replace the threshold h with h0 in a finite number of steps. To this end we claim that for anyκ >0 there existsε >0 such that for anyh∈[h, h0] andL∈ L, whereLcomprises all the (finitely many) translates of the sub-lattice 10LZd, we have
ωhωh+κ∨ηεL, (5.6.3)
where ηεL(x) = ηε(x) if x ∈ L and ηεL(x) = 0 otherwise. Let us first explain how to derive property (e) from (5.6.3): choosingκ:= h|L|0−h givesωh ωh+κ∨ηεL, for suitable ε = ε(h, h0, L, d) > 0 and any choice L, so that iterating this over the lattices in L gives
ωh ωh+|L|κ∨ _
L∈L
ηLε =ωh0 ∨ηε, as desired.
By approximation, it suffices to verify (5.6.3) for all fields restricted to a finite set Γ ⊂ Zd. Let ωh(S) := {ωh(x) :x ∈ S} for S ⊂Zd and define similarly ηε(S), ηεL(S).
Notice thatωh =ωh+κ ∨ωh and ωh+κ∨ηLε are both increasing functions of (ωh+κ, ωh) and (ωh+κ, ηεL) respectively. Therefore it suffices to show that, conditionally on any realization ofωh+κ, the field ωh(Γ) stochastically dominates ηLε(Λ). To this end we fix an ordering{x1, x2, . . .} of the vertices in Γ such that all the vertices inL∩Γ appear before all the vertices inLc∩Γ. In view of [101, Lemma 1.1], it then suffices to show that for any k ≥1, with Λk ={x1, . . . , xk−1} (Λ0 =∅),
P-a.s., P[ωh(xk) = 1|A, A0]≥ε·1xk∈L, (5.6.4) where A := {ωh(x) = σ(x), x ∈ Λk}, A0 := {ωh+κ(x) = σ0(x), x ∈ Λ}, for arbitrary σ, σ0 ∈ {0,1}Zd with σ ≥ σ0 (as ωh ≥ ωh+κ). We now show (5.6.4) and assume that xk∈L (the other cases are trivial). To this end let us first define, for any x∈Zd, the pair of events
T(x) :={Ux ∈/ (δ/2,1−δ/2)} and U(x) := \
y∈BL(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(xk),
P[ωh(xk) = 1|A, A0]≥P[ωh(xk) = 1| U, A, A0]·P[U |A, A0]. (5.6.6) We will bound the two probabilities on the right-hand side separately from below. It follows from the definition ofωhin (5.6.2) thatωh+κ(y) =ζ(y) on the eventT(y), where
ζ(y) = 1Uy≥1−δ/2. Consequently, decomposing A0, we obtain, with A01 := {ωh+κ = σ0 on Γ∩BL(xk)c} and A02 :={ζ =σ0 onBL(xk)\ {xk}}, that
p:= P[ωh(xk) = 1| U, A, A0] =P[ωh(xk) = 1| U, A, ωh+κ(xk) =σ0(xk), A01, A02]. However, in view of definition (5.6.2), (5.6.5) and the fact that L is 10L-separated, it follows that both {ωh(xk) = 1, ωh+κ(xk) = σ0(xk)} and {ωh+κ(xk) = σ0(xk)} are where the equality in the second line follows by (5.6.2) and (5.6.5) upon conditioning on ωh(x), x∈ Λk, ωh+κ(y), y∈ (Λ\BL(xk))∪ {xk} and 1ϕL
y≥h+κ, y ∈BL(xk)\ {xk}, and the final lower bound follows by distinguishing whether ρ(y) 6= σ0(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 := ε0(δ/2)|BL|, 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 ωh 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 ∂BR]≤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 γε := ωh ∨ηε indexed by ε ∈ [0,1] where ηε is a Bernoulli percolation with density ε independent of ωh. Let εc=εc(h)∈[0,1] denote the critical parameter of the family of percolation processes {γε : ε ∈ [0,1]} and suppose for a moment that for every 0 ≤ ε < εc(h), there exists c=c(h, ε)>0 such that for everyR ≥1,
θR(ε) :=P[0←→γε ∂BR]≤exp(−cR). (5.6.9) Then (5.6.8) follows immediately by takingε= 0 in caseεc(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), choosingh0 = (h∗(δ, L) +h)/2, one hasωh0 ωh∨ηε0 for some ε0 > 0, whence ε0 ≤ εc(h) as ωh0 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 varianceθR(1−θR) 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 ofθR(ε) with respect toεto deduce a system of differential inequalities of the following form:
θR0 ≥β R
ΣRθR(1−θR), (5.6.10)
where ΣR :=PR−1
r=0 θr 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 < εc. 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 ωh in terms of {V(x) :x∈Zd}provided by Property (d). In view of this,1ER with ER={0←→γε ∂BR} can now be written as a function of independent random variables (V(x) :x∈BR+L) and (ηε(x) : x ∈ BR). 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∈BR+N
InfV(x)+ X
x∈BR
Infηε(x) ≥cL−(d−1) R
ΣRθR(1−θR), (5.6.11) where InfV(x) := P[1ER(V, ηε) 6= 1ER( ˜V, ηε)] with ˜V being the same as V for every vertex except atxwhere it isresampled independently, and Infηε(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,
θR0 (ε)≥α(ε) X
x∈BR+L
InfV(x)+ X
x∈BR
Infηε(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
θ0R=P[ωh(0) = 0] X
x∈BR
P[Pivx].
where Pivx is the same event as in Section 5.5 (see below (5.5.6)) with χt replaced by γε. Now, on the one hand, Infηε(x) ≤ P[Pivx]. On the other hand, since V(x) affects the states of vertices only in BL(x) by Property (d), one immediately gets InfV(x) ≤ P[Piv(BL(x))], where Piv(BL(x)) – called the pivotality of the box BL(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 ωh, we can write
P[Piv(BL(x))]≤(cFE(1−ε))−c0LdP[Piv(BL(x)), γε(y) = 0 for all y∈BL(x)]. If we start on the event on the right-hand side and then open the vertices insideBL(x) one by one inγε until 0 gets connected to ∂BR, which must happen by the definition of Piv(BL(x)). The last vertex to be opened is pivotal for the resulting configuration.
This implies that
P[Piv(BL(x)), ηε(y) = 0 for all y∈BL(x)]≤ε−c0Ld X
y∈BL(x)
P[Pivy]. 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 ∈ BR+L) and (ηε(x) : x∈ BR) as inputs, and determines the event ER by revealing V(x), ηε(x) one by one as follows:
Definition 5.6.3 (Algorithm T). Fix a deterministic ordering of the vertices in BR and choose j ∈ {1, . . . , R} uniformly and independently of (V, ηε). Now set x0 to be the smallest x ∈ ∂Bj in the ordering and reveal ηε(x0) as well as V(y) for all y ∈ BL(x0). Set E0 := {x0} and C0 := {x0} if γε(x0) = 1 and C0 := ∅ otherwise. At each stept ≥1, assume that Et−1 and Ct−1 have been defined. Then,
— If the intersection of BR with the set (∂outCt−1 ∪∂Bj)\Et−1 is non-empty, let x be the smallest vertex in this intersection and set xt := x, Et := Et−1 ∪ {xt}.
Reveal ηε(xt) as well as V(y) for all y ∈ BL(xt) and set Ct := Ct−1 ∪ {xt} if γε(xt) = 1 and Ct :=Ct−1 otherwise.
— If the intersection is empty, halt the algorithm.
In words, the algorithm T explores the clusters in γε of the vertices in ∂Bj. By applying the OSSS inequality [125] to 1ER and the algorithmT, we now get
Var[1ER] =θR(1−θR)≤ X
x∈BR+L
θV(x)(T) InfV(x)+ X
x∈BR
θηε(x)(T) Infηε(x), (5.6.13) where the functionθ·(T), called the revealment of the respective variable, is defined as θV(x)(T) :=π[Treveals the value of V(x)], (5.6.14) and θηε(x)(T) is defined in a similar way. Here π denotes the probability governing the extension of (V, ηε) which accommodates the random choice of layer j, which is independent of (V, ηε).
Let us now bound the revealments for V(x) and ηε(x). Notice that since V(x) affects the state of vertices only in BL(x) by Property (d), V(x) is revealed only if there is an explored vertex y ∈ BL(x)∩ BR. The vertex y, on the other hand, is explored only ify is connected to ∂Bj in γε. We deduce that
θV(x)(T)≤ 1 R
R
X
j=1
P[(BL(x)∩BR)←→γε ∂Bj]≤ 1 R
R
X
j=1
X
y∈BL(x)∩BR
P[y←→γε ∂Bj], where in the last step we used a naive union bound. Forηε(x), it is even simpler:
θηε(x)(T)≤ 1 R
R
X
j=1
P[x←→γε ∂Bj].
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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