P OIS S O N – B O O L E A N P E R C O L AT IO N A N D I T S VA C A N T S E T

H U G O D U M I N IL - C O PI N A R A N R A O U F I

V I N C E N T TA S SIO N

S U B C R I T IC A L P H A S E O F d - DI M E N SIO N A L

P OIS S O N – B O O L E A N P E R C O L AT IO N A N D I T S VA C A N T S E T

P H A S E S O U S - C R I T IQ U E D U M O D È L E D E P E R C O L AT IO N P OIS S O N – B O O L É E N E T D E S O N C O M P L É M E N TA IR E E N DI M E N SIO N d .

Abstract. — We prove that the Poisson–Boolean percolation onR^d undergoes a sharp phase transition in any dimension under the assumption that the radius distribution has a 5d−3 finite moment (in particular we do not assume that the distribution is bounded). To the best of our knowledge, this is the first proof of sharpness for a model in dimensiond>3 that does not exhibit exponential decay of connectivity probabilities in the subcritical regime.

More precisely, we prove that in the whole subcritical regime, the expected size of the cluster of the origin is finite, and furthermore we obtain bounds for the origin to be connected to distancen: when the radius distribution has a finite exponential moment, the probability decays exponentially fast inn, and when the radius distribution has heavy tails, the probability is Keywords:continuum percolation, sharp threshold, phase transition, subcritical phase.

2020Mathematics Subject Classification:82B43.

DOI:https://doi.org/10.5802/ahl.43

(*) This research was supported by an IDEX grant from Paris–Saclay, the ERC grant CriBLaM, and the NCCR SwissMAP.

equivalent to the probability that the origin is covered by a ball going to distancen(this result is new even in two dimensions). In the supercritical regime, it is proved that the probability of the origin being connected to infinity satisfies a mean-field lower bound. The same proof carries on to conclude that the vacant set of Poisson–Boolean percolation onR^dundergoes a sharp phase transition.

Résumé. — Nous prouvons que la percolation Poisson–Booléenne surR^d a une transition de phase rapide en toute dimension sous l’hypothèse que la distribution pour le rayon des boules a un moment d’ordre 5d−3 qui est fini (en particulier, nous ne supposons pas que cette distribution est à support compact). Ceci représente la première preuve de ce fait en dimensiond>3 pour un modèle qui n’exhibe pas de décroissance exponentielle du rayon de la composante connexe de l’origine dans le régime sous-critique. Plus précisément, nous montrons que dans tout le régime sous-critique, l’expérance de la taille de la composante connexe de l’origine est finie, et de plus nous obtenons des estimées sur la probabilité que cette composante connexe ait un rayon plus grand que n : quand la distribution pour le rayon des boules a un moment exponentiel fini, cette probabilité décroit exponentiellement vite enn, et quand cette distribution a une queue lourde, la probabilité devient équivalente à la probabilité qu’il existe une boule de diamètre au moins n contenant l’origine. Le même résultat s’étend au complémentaire de la percolation Poisson–Booléenne (aussi appelé ensemble vacant).

1. Introduction

1.1. Definition of the model

Bernoulli percolation was introduced in [BH57] by Broadbent and Hammersley to model the diffusion of a liquid in a porous medium. Originally defined on a lattice, the model was later generalized to a number of other contexts. Of particular interest is the development of continuum percolation (see [MR96] for a book on the subject), whose most classical example is provided by the Poisson–Boolean model (introduced by Gilbert [Gil61]). It is defined as follows.

Fix a positive integer d > 2 and let R^d be the d-dimensional Euclidean space endowed with the`²normk·k. Forr >0 andx∈R^d, setB^x_r :={y∈R^d :ky−xk6r}

and∂B^x_r :={y∈R^d :ky−xk=r} for the ball and sphere of radius r centered at x.

When x= 0, we simply write B_r and ∂B_r. For a subset η of R^d×R+, set O(η) := ^[

(z,R)∈η

B^z_R.

Letµbe a measure on R+ (below, we use the notationµ[a, b] to refer to µ([a, b)), where a, b∈R∪ {∞}) and λbe a positive number. Let η be a Poisson point process of intensity λ·dz ⊗µ, where dz is the Lebesgue measure on R^d. Write Pλ for the law ofη and Eλ for the expectation with respect to Pλ. The random variableO(η), where η has law Pλ is called the Poisson–Boolean percolation of radius law µ and intensityλ. A natural hypothesis in the study of Poisson–Boolean percolation is to assumed^th moment on the radius distribution:

(1.1)

Z ∞ 0

r^ddµ(r)<∞.

Indeed, as observed by Hall [Hal85], the condition (1.1) is necessary in order to avoid the entire space to be almost surely covered, regardless of the intensity (as long as positive) of the Poisson point process.

Remark 1.1. — The techniques of this paper do not rely heavily on the round shape of the balls used to constructO(η). We think that the results below extend to general (even random) shapes under proper assumptions (such as non-empty interior and boundedness).

1.2. Main result

Two pointsxandyofR^dare said to beconnected (byη) if there exists a continuous path in O(η) connecting x to y or x = y. This event is denoted by x ←→ y. For X, Y ⊂ R^d, the event {X ←→ Y} denotes the existence of x∈ X and y∈ Y such that x is connected to y (when X = {x}, we simply write x ←→ Y). Define, for every r >0, the two functions of λ

θ_r(λ) :=Pλ[0←→∂B_r] and θ(λ) := lim

r→∞θ_r(λ).

We define the critical parameterλ_c=λ_c(d) of the model by the formula λ_c:= inf{λ>0 :θ(λ)>0}.

Another critical parameter is often introduced to discuss Poisson–Boolean perco- lation. Define λ^e_c=λ^e_c(d) by the formula

λe_c:= inf

λ>0 : inf

r>0Pλ[B_r←→∂B_2r]>0

.

This quantity is of great use as it enables one to initialize renormalization arguments;

see e.g. [Gou08, GT18] and references therein. As a consequence, a lot is known for Poisson–Boolean percolation with intensity λ < λ^e_c. We refer to Theorems 1.4 and 1.5 and to [MR96] for more details on the subject.

Under the minimal assumption (1.1), Gouéré [Gou08] proved that λc and λ^ec are nontrivial. More precisely he proved 0<λ^e_c6λ_c<∞. The main result of this paper is the following.

Theorem 1.2 (Sharpness for Poisson–Boolean percolation). — Fix d > 2 and assume that

(1.2)

Z

R+

r^5d−3dµ(r)<∞.

Then, we have thatλ_c=λ^e_c. Furthermore, there existsc > 0such thatθ(λ)>c(λ−λ_c) for any λ>λ_c.

The case of bounded radius is already proved in [MRS94, ZS85] (see also [MR96]), and we refer the reader to [Zie16] for a new proof. Theorems stating sharpness of phase transitions for percolation models in general dimension d were first proved in the eighties for Bernoulli percolation [AB87, Men86] and the Ising model [ABF87].

The proofs of sharpness for these models (even alternative proofs like [DCT16]) har- vested independence for Bernoulli percolation and special structures of the random- current representation of the Ising model. In particular, they were not applica- ble to other models of statistical mechanics. In recent years, new methods were developed to prove sharpness for a large variety of statistical physics models in two

dimensions [ATT16, BDC12, BR06]. These methods rely on general sharp threshold theorems for Boolean functions, but also on planar properties of crossing events.

In particular, the proofs use planarity in an essential way and seem impotent in higher dimensions. Recently, the authors proved the sharpness of phase transition for random-cluster model [DCRT17b] and Voronoi percolation [DCRT17a] in arbitrary dimensions. The shared theme of the proofs is the use of randomized algorithms to prove differential inequalities for connection probabilities.

Here, we do not prove that connectivity probabilities decay exponentially fast in the distance below criticality since this is simply not true in general for Poisson–Boolean percolation. Indeed, if the tail of the radius distribution is slower than exponential, then the event that a large ball covers two given points has a probability larger than exponential in the distance between the two points. Instead, we show by contradiction thatλc=λ^ec, without ever referring to exponential decay, by controlling the derivative of θ_r(λ) in terms of “pivotality events” for λ in the “fictitious” regime (λ^e_c, λ_c) and by deriving a differential inequality on θ_r(λ) (using randomized algorithms) which is validonly in this regime. The regime (λ^e_c, λ_c) is referred to as the fictitious regime since one eventually concludes from our proof that the interval is empty.

We wish to highlight that while the proof below also harvests randomized algorithms, we consider that the main novelty of the paper lies in the comparison between the derivative ofθ_r(λ) and pivotal events. In many percolation models con- structed from disordered systems with long range interaction, the pivotality events that typically appear in Russo’s formula for Bernoulli percolation get replaced by more complicated quantities. For instance, in our model probability of large ball being pivotal also enter into the expression of the derivative, in percolation models built out of Gaussian processes such as Gaussian Free Field or Bargmann–Fock, the probability of pivotalityconditioned on the field being equal to a certain value are relevant, etc. It is not true that these quantities are comparable to the pivotality probabilities for arbitrary values ofλ, but one may show with some work that this is indeed the case in the fictitious regime. We believe that the introduction of this fictitious regime is therefore a powerful tool towards a better understanding of these models. To illustrate recent applications of similar techniques that were inspired by the present paper, we refer e.g. to [DCGRS19, DCGR⁺18].

1.3. Decay of θ_r(λ) when λ <λ^e_c

For standard percolation, sharpness of the phase transition refers to the exponen- tial decay of connection probabilities in the subcritical regime. In Poisson–Boolean percolation with arbitrary radius lawµ, we mentioned above that one cannot expect such behavior to hold in full generality. In order to explain why the theorem above is still called “sharpness” in this article, we provide below some new results concerning the behavior of Poisson–Boolean percolation whenλ <λ^e_c.

Let us start with the following easy Proposition 1.3.

Proposition 1.3. — The following properties hold:

• For λ <^eλ_c, lim

r→∞P^λ[B_r ←→∂B_2r] = 0.

• There existsc > 0such that for everyλ>λ^e_candr>1,Pλ[B_r ←→∂B_2r]>c.

Notice that the previous proposition immediately implies that for every λ < λ^ec, the expected size of the cluster of the origin is finite (see Remark 4.3).

Now, remark that for every λ >0, θ_r(λ) is always bounded from below by (1.3) φr(λ) :=P^λ[∃(z, R)∈η such that B^z_R contains 0 and intersects∂Br], whose decay may be arbitrarily slow. Nevertheless, one may expect the following phenomenology when λ <λ^ec:

• If µ[r,∞] decays exponentially fast, then so does θ_r(λ) (but not necessarily at the same rate of exponential decay).

• Otherwise, the decay of θ_r(λ) is governed by φ_r(λ), in the sense that it is roughly equivalent to it.

The first item above is formalized by the following theorem.

Theorem 1.4. — If there exists c > 0 such that µ[r,∞] 6 exp(−cr) for every r>1, then, for everyλ <λ^e_c, there exists c_λ >0 such that for every r>1,

θr(λ)6exp(−cλr).

Giving sense to the second item above is not easy in full generality, for instance when the law ofµis very irregular (one can imagine distributionsµthat do not decay exponentially fast, but for which large range of radii are excluded). In Section 4.5, we give a general condition under which a precise description ofθ_r(λ) can be obtained.

To avoid introducing technical notation here, we only give two applications of the results proved in Section 4.5. We believe that these applications bring already a good idea of the general phenomenology. The proof mostly relies on new renormalization inequalities. We believe that these renormalization inequalities can be of great use to other percolation models such as long-range Bernoulli percolation models. The Theorem 1.4 claims that the cheapest way for 0 to be connected to distancer is if a single huge ball covers 0 and intersects the boundary ofB_r.

Theorem 1.5. — Fixd >2. Let µ satisfy one of the two following cases:

(C1) There exists c > 0such that µ[r,∞] = 1/r^d+c for every r>1,

(C2) There exist c > 0 and 0 < γ < 1 such that µ[r,∞] = exp(−cr^γ) for every r >1.

Then, for every λ <^eλ_c,

r→∞lim θ_r(λ) φ_r(λ) = 1.

This theorem is new even in two dimensions. The proofs of Theorem 1.4 and Theorem 1.5 are independent of the proof of Theorem 1.2 and can be read separately.

1.4. Vacant set of the Poisson–Boolean model

Another model of interest can also be studied using the same techniques. In this model, the connectivity of the points is given by continuous paths in the complement of O(η). Write x ←→^∗ y for the event that x and y are connected by a continuous path inR^d\ O(η), and X ←→^∗ Y if there exist x∈X and y∈Y such thatx and y are connected. Forλ and r >0, define

θ_r^∗(λ) :=Pλ[0←→^∗ ∂B_r] and θ^∗(λ) = lim

r→∞θ^∗_r(λ).

(Note that θ^∗(λ) is decreasing in λ.) We define the critical parameter λ^∗_c (see e.g.

[Pen17] or [ATT17] for the fact that it is positive) by the formula λ^∗_c := sup{λ>0 :θ^∗(λ)>0}= inf{λ>0 :θ^∗(λ) = 0}.

We have the following Theorem 1.6.

Theorem 1.6 (Sharpness of the vacant set of the Poisson–Boolean model). — Fixd>2 and assume that the radius distribution µis compactly supported. Then, for all λ < λ^∗_c, there exists cλ >0 such that for every r>1,

θ_r^∗(λ)6exp(−c_λr).

Furthermore, there exists c >0such that for every λ>λ^∗_c,θ^∗(λ)>c(λ−λ^∗_c).

Since the proof follows the same lines as in Theorem 1.2, we omit it in the article and leave it as an exercise to the reader.

1.5. Strategy of the proof of Theorem 1.2

Let us now turn to a brief description of the general strategy to prove our main theorem. Theorem 1.2 is a consequence of the following Lemma 1.7.

Lemma 1.7. — Assume the moment condition (1.2) on the radius distribution.

Fixλ₀ >λ^e_c. There existsc₁ >0such that for every r >0 and λ^e_c < λ < λ₀, (1.4) θ_r⁰(λ) > c₁ r

Σ_r(λ)θ_r(λ)(1−θ_r(λ)), where Σ_r(λ) :=^R₀^rθ_s(λ)ds.

The whole point of Section 3 will be to prove Lemma 1.7. Before that, let us mention how it implies Theorem 1.2.

Proof of Theorem 1.2. — Fix λ₀ >λ^e_c. As in [DCRT17b, Lemma 3.1], Lemma 1.7 implies that there exists λ₁ ∈[λ^e_c, λ₀] such that

• for any λ>λ₁,θ(λ)>c(λ−λ₁).

• for anyλ∈(λ^e_c, λ₁), there existsc_λ >0 such thatθ_r(λ)6exp(−c_λr) for every r >0.

The two items imply that λ₁ =λ_c. Yet, the second item implies that λ₁ 6λ^e_c, since clearly exponential decay would imply that for λ∈(λ^e_c, λ₁),

r→∞lim P^λ[Br ←→∂B2r] = 0.

Since λ₁ = λ_c > λ^e_c, we deduce that λ₁ = λ_c = λ^e_c and the proof of Theorem 1.2

is finished.

Remark 1.8. — Note that we did not deduce anything from Lemma 1.7 about exponential decay since eventually λ₁ = λ^e_c. It is therefore not contradictory with the case in which µ[r,∞] does not decay exponentially fast.

The proof of Lemma 1.7 relies on the (OSSS) inequality, first proved in [OSSS05], connecting randomized algorithms and influences in a product space. Let us briefly describe this inequality. Let I be a finite set of coordinates, and let Ω = ^Q_i∈IΩ_i be a product space endowed with product measure π = ⊗i∈Iπi. An algorithm T determiningf : Ω→ {0,1} takes a configuration ω = (ω_i)_i∈I ∈Ω as an input, and reveals the value of ω in different i∈I one by one. At each step, which coordinate will be revealed next depends on the values of ωrevealed so far. The algorithm stops as soon as the value of f is the same no matter the values of ω on the remaining coordinates. Define the functionsδ_i(T) and Inf_i(f), which are respectively called the revealment and theinfluence of the i^th coordinate, by

δ_i(T) :=π[Treveals the value of ω_i], Inf_i(f) :=π[f(ω)6=f(ω)]_e ,

whereω_e denotes the random element in Ω^I which is the same asωin every coordinate except the i^th coordinate which is resampled independently.

Theorem 1.9 ([OSSS05]). — For every function f : Ω → {0,1}, and every algorithm T determining f,

(OSSS) Var_π(f)6^X

i∈I

δ_i(T) Inf_i(f), where Var_π is the variance with respect to the measure π.

This inequality is used as follows. First, we write Poisson–Boolean percolation as a product space. Second, we exhibit an algorithm for the event{0↔∂B_r}for which we control the revealments. Then, we use the assumption λ > λ^ec to connect the influences of the product space to the derivative of θ_r. Altogether, these steps lead to (1.4).

Organization of the article

The next Section 2 contains some preliminaries. Section 3 contains the proof of Theorem 1.2 while the last Section 4 contains the proofs of Theorems 1.4 and 1.5.

2. Background

We introduce some notation and recall three properties of the Poisson–Boolean percolation that we will need in the proofs of the next sections.

2.1. Further notation

Forx∈Z^d, introduce the squared box S^x :=x+ [−1/2,1/2)^d around x. In order to apply the OSSS inequality, we wish to write our probability space as a product space. To do this, we introduce the following notation. For any integer n > 1 and x∈Z^d, let

η_(x,n) :=η∩S^x×[n−1, n),

which corresponds to all the balls of η centered at a point in S^x with radius in [n−1, n). All the constants c_i below (in particular in the lemmata) are independent of all the parameters.

2.2. Insertion tolerance

We will need the following insertion tolerance property. Consider r∗ and r^∗ such that

(IT) c_IT=c_IT(λ) := Pλ[D_x]>0,

where D_x is the event that there exists (z, R)∈η with z ∈S^x and B^x_r∗ ⊂B^z_R ⊂B^x_r∗. Without loss of generality (the radius distribution may be scaled by a constant factor), we further assume thatr_∗ andr^∗ satisfy the following conditions (these will be useful at different stages of the proof):

(2.1) 1 + 2√

d6r∗ 6r^∗ 62r∗ −2√ d.

While the quantityc_ITvaries withλ, the dependency will be continuous and therefore irrelevant for our arguments. We will omit to refer to this subtlety in the proofs to avoid confusion.

2.3. FKG inequality

Anincreasing event A is an event such thatη ∈Aand η⊂η⁰ implies η⁰ ∈A. The FKG inequality for Poisson point processes states that for every λ > 0 and every two increasing eventsA and B,

(FKG) Pλ[A∩B]>Pλ[A]Pλ[B].

2.3.1. Russo’s formula

Forx∈Z^d and an increasing event A, define the random variable (2.2) Piv_x,A(η) := 1_{η /∈A}

Z

S^x

Z

R+

1_{η∪(z,r)∈A}dz µ(dr).

Russo’s formula [LP17, Equation 19.2] yields that

(Russo) d

dλPλ[A] = ^X

x∈Z^d

Eλ[Piv_x,A].

3. Proof of Lemma 1.7

The next subsection contains the proof of Lemma 1.7 conditioned on two lemmatas, Lemmas 3.3 and 3.4 which are proved in the next two Subsections 3.1 and 3.2.

3.1. Proof of Lemma 1.7

As mentioned above, the proof of the Lemma 1.7 is obtained by applying the (OSSS) inequality to a truncated version of the probability space generated by the inde- pendent variables (η_(x,n))_x∈Zd,n>1. In this section, we fix L > 2r > 0. Set A :=

{0←→∂B_r} and f =1A. Define

I_L :=ⁿ(x, n)∈Z^d×N such that kxk6Land 16n 6L^o.

Also, for r > 0, let η_g denote the union of the η_(x,n) for (x, n) ∈/ I_L. For i being either (x, n) org, set Ω_i to be the space of possible η_i and π_i the law ofη_i under Pλ.

For 06s6r, apply the (OSSS) inequality (Theorem 1.9) to

• the product space (^Q_i∈IΩ_i,⊗i∈Iπ_i) whereI :=I_L∪ {g},

• the indicator function f = 1A considered as a function from ^Q_i∈IΩ_i onto {0,1},

• the algorithm T_s,L defined below.

Definition 3.1(AlgorithmT_s,L). — Fix a deterministic ordering ofI. Seti₀ =g, and revealη_g. At each stept, assume that{i₀, . . . , i_t−1} ⊂I has been revealed. Then,

• If there exists (x, n) ∈ I \ {i₀, . . . , it−1} such that the Euclidean distance between S^x and the connected component of∂B_s in ∂B_s∪ O(η_i1∪ · · · ∪η_it−1) is smaller than n, then set it+1 to be the smallest such(x, n) for some fixed ordering.

• If such an (x, n) does not exist, halt the algorithm.

Remark 3.2. — Roughly speaking, the algorithm checksO(η_g) and then discovers the connected components of∂B_s.

Theorem 1.9 implies that

(3.1) θ_r(1−θ_r)6^X

i∈I

δ_i(T_s,L)Inf_i(f).

By construction, the random variableη_g is automatically revealed so its revealment is 1. Also,

(3.2) Inf_g(f)62Pλ[∃(z, R)∈η with R >L and B^z_R∩B_r 6=∅]

so that this quantity tends to 0 as L tends to infinity (thanks to the moment assumption (1.1) on µ).

Let us now bound the revealment for (x, n) ∈ I_L. The random variable η_(x,n) is revealed byT_s,L if the Euclidean distance betweenS^x and the connected component of ∂B_s is smaller than n. Let S^x_n be the union of the boxes S^y that contain a point at distance exactly n from S^x. We deduce that

(3.3) δ_(x,n)(T_s,L)6Pλ[S^x_n ←→∂B_s].

Overall, plugging the previous bounds (3.2) and (3.3) on the revealments of the algorithmsT_s,L into (3.1), we obtain

θr(1−θr)6 ^X

(x,n)∈I_L

P^λ[S^x_n←→∂Bs]·Inf_(x,n)(f) +o(1).

(Above,o(1) denotes a quantity tending to 0 as L tends to infinity.) Letting Ltend to infinity (note that the influence Inf_(x,n) does not depend on the algorithms T_s,L), we find

θ_r(1−θ_r)6 ^X

x∈Z^d n>1

P^λ[S^x_n←→∂B_s]·Inf_(x,n)(f).

(3.4)

In order to conclude the proof, we need the following two lemmata 3.3 and 3.4. First, a simple union bound argument allows us to bound Pλ[S^x_n ←→∂B_s] in terms of the one-arm probability. More precisely, consider the following Lemma 3.3, which will be proved at the end of the Section 3.1.

Lemma 3.3. — Fixλ₀ >0. There existsc₂ >0such that for everyλ>λ₀,x∈Z^d and n >1,

Z r

0 Pλ[S^x_n ←→∂B_s]ds6c₂n^d−1Σ_r(λ).

Integrating (3.4) for radii between 0 and r and using the Lemma 3.3 above gives (3.5) rθ_r(1−θ_r)6c₂Σ_r ^X

x∈Z^d n>1

n^d−1Inf_(x,n)(f).

The most delicate step is to relate the influences in the equation above with the pivotal probabilities appearing in the derivative formula (Russo). This is the content of the following Lemma 3.4, which will be proved in Section 3.3.

Lemma 3.4. — There existsc₃ >0such that for every x∈Z^d, every n >1and every λ >λ^ec,

X

x∈Z^d

Inf_(x,n)(f)6c₃λ n^4d−2µ[n−1, n] ^X

x∈Z^d

Eλ[Piv_x,A].

Dividing (3.5) by Σ_r, then applying the lemma above, and then usingλ < λ₀ gives r

Σ_rθ_r(1−θ_r)6c₂c₃λ ^X

n>1

n^5d−3µ[n−1, n] ^X

x∈Z^d

Eλ[Piv_x,A] 6c₁ ^X

x∈Z^d

Eλ[Piv_x,A]^(Russo)= c₁θ_r⁰.

This implies Lemma 1.7. Before proving Lemmatas 3.3 and 3.4, we would like to make several remarks concerning the proof above.

Remark 3.5. — The role of L is the following. The interest of working with T_s,L is to have a finite number (depending on L) of coordinates for the algorithm. We could have stated an OSSS inequality valid for countably many states and get (3.4) directly, but we believe the previous strategy to be shorter and thriftier. The role of s on the other hand is purely technical and one could have avoided it by working with a randomized algorithm.

Remark 3.6. — Lemma 3.3 may a priori be improved. Indeed, the union bound in the first inequality is quite wasteful. Nonetheless, with the moment assumption on the radius distribution, the claim above is sufficient.

Remark 3.7. — We refer to λ>λ₀ in Lemma 3.3 instead of λ >λ^e_c (even though the lemma is anyway used in this context) to illustrate that the only place where λ >λ^e_c is used is in Lemma 3.4.

We finish this section with the proof of Lemma 3.3.

Proof of Lemma 3.3. — Let Y be the subset of Z^d such that S^x_n = ^[

y∈Y

S^y.

We have that

Pλ[S^x_n ←→∂B_s]6 ^X

y∈Y

Pλ[S^y ←→∂B_s]6 1 c_IT

X

y∈Y

Pλ[y←→∂B_s].

where in the last inequality, we used the FKG inequality, (IT) and the fact that if S^y ←→∂B_s and the event D_y defined below (IT) occur, then y is connected to∂B_s.

Integrating ons between 0 and r and observing thaty is at distance |s− kyk| of

∂B_s, we deduce that

Z r

0 P^λ[y ←→∂Bs]ds6

Z r 0

θ|s−kyk|(λ)ds62Σr(λ).

Since the cardinality ofY is bounded by a constant timesn^d−1, the result follows.

3.2. A technical statement regarding connection probabilities above λ^e_c

The following Lemma 3.8 will be instrumental in the proof of Lemma 3.4. It is the unique place where we use the assumption λ >λ^e_c.

Below, X ←→^Z Y means that X is connected to Y in O(η^Z), where η^Z is the set of (z, R) ∈ η such that B_z(R) ⊂ Z. We highlight that this is not the same as the

existence of a continuous path inO(η)∩Z connecting x and y, since such a path could a priori pass through regions inZ which are only covered by balls intersecting R^d\Z.

Lemma 3.8. — There exists a constant c₄ > 0 such that for every λ > λ^e_c and r>r^∗,

(3.6) P^λ

0←→^Br B^x_r^∗

> c₄

r^2d−2 for every x∈∂Br.

Remark 3.9. — Before diving into the proof, let us first explain how we will use the assumptionλ >λ^e_c. Fix r >0. If B_r is connected to ∂B_2r, then there must exist x ∈ Z^d with B^x₁ ∩∂B_r 6=∅ such that B^x₁ is connected to ∂B_2r. Since above λ^e_c, the probability of the former is bounded away from 0 uniformly in r and λ thanks to Proposition 1.3, and since the probabilities of the latter events are all smaller than Pλ[B₁ ←→∂B_r], the union bound gives the existence of c₅ >0 such that for every r>1 and λ>λ^ec,

(3.7) Pλ[B₁ ←→∂B_r]> c5

r^d−1.

The argument will rely on the following observation. As explained in (3.7), the probability that B₁ is connected to ∂B_r does not decay quickly. This event implies the existence of y such that B₁ is connected in B_r to B^y_r∗, and one ball B^z_R (with (z, R)∈η) coveringB^y_r∗ and intersecting the complement of B_r. The problem is that this site y may be quite far from ∂B_r. Nonetheless, this seems unlikely since the cost (by the moment assumption on µ) of having a large ball B^z_R intersecting B^y_r∗ is overwhelmed by the probability that the latter is connected to ∂B_r in B^y_r−kyk by a path inO(η) using a priori smaller balls (and maybe even only balls included inB_r).

The proof below will harvest this idea though with some important variations due to the fact that all the balls under consideration must remain insideB_r. One of the key idea is the introduction of a new scale r^∗∗ and an induction on the probability that B₁ is connected to B^x_r∗∗ inB_r, wherex∈∂B_r.

Proof. — We will prove that there exist r^∗∗>0 and c₆ >0 such that

(3.8) Pλ

hBr∗

Br

←→B^x_r∗∗

i> c₆

r^2d−2 for every x∈∂B_r.

This is amply sufficient to prove the Lemma 3.8 since, if T denotes the set oft∈Z^d such thatB^t_r∗∩B^x_2r^∗∗ 6=∅ and B^t_r^∗ ⊂Br, then

Pλ

h0←→^Br B^x_r∗

i^(FKG)

> Pλ[D₀] ^Y

t∈T

Pλ[D_t]Pλ

hB_r∗ ←→^Br B^x_r∗∗

i^(IT)

> c_IT^|T|+1· c₆ r^2d−2. We therefore focus on the proof of (3.8). We will fix r^∗∗>2r^∗ sufficiently large (see the end of the proof). Introduce Y := Z^d∩B_r−r^∗∗ and a finite set Z ⊂ ∂B_r such that the union of the balls B^y_r∗ and B^z_r∗∗ with y ∈ Y and z ∈ Z cover the ball B_r. Note that ifB_r∗ is connected to∂B_r, then either one of the z ∈Z is such thatB_r∗ is connected toB^z_r∗∗ in B_r, or there exists y∈Y such that the event

A(y) := ⁿB_r∗ ←→^Br B^y_r∗^o∩ⁿ∃(u, R)∈η such that B^u_R intersects both B^y_r∗and ∂B_r^o

occurs. The union bound therefore implies that

(3.9) ^X

z∈Z

Pλ

B_r∗ ←→^Br B^z_r∗∗

+ ^X

y∈Y

Pλ[A(y)]>Pλ[B_r∗ ←→∂B_r]

(3.7)

> c₅ r^d−1. Fory∈Y, introduce x:= (r/kyk)y∈∂B_r and r=r− kyk. The event on the right of the definition of A(y) is independent of the event on the left. Since any point in B^y_r∗ is at a distance at least r−r∗ of ∂B_r, we deduce that

Pλ[A(y)]6

c₇

Z ∞ r−r∗

r^d−1µ(dr)

·Pλ

B_r∗ ←→^Br B^y_r∗

6 c₈

r^4d−2Pλ

B_r∗ ←→^Br B^y_r∗

.

In the second inequality, we used the moment assumption on µ and the fact that r−r_∗ >r/2 since r^∗∗>2r^∗. Using this latter assumption one more time implies that

(3.10) Pλ

B_r∗ ←→^Br B^x_r∗∗

_(FKG)

> Pλ

B_r∗ ←→^Br B^y_r∗

·Pλ[D_y]·Pλ

"

B^y_r∗ ^B

y

←→r B^x_r∗∗

#

> c_ITr^4d−2

c₈ ·P^λ[A(y)]·P^λ

"

B^y_r∗ ^B

y

←→r B^x_r^∗∗

#

.

From now on, define U(r) := P^λ[Br∗

Br

←→ B^x_r^∗∗] for x ∈ ∂Br (the choice of x is irrelevant by invariance under the rotations). Now, one may choose Z in such a way that |Z|6c₉r^d−1. Plugging (3.10) in (3.9), we obtain the following inequality

c₉r^d−1U(r) +c₁₀ ^X

y∈Y

U(r)

U(r− kyk)(r− kyk)^4d−2 > c₅ r^d−1.

Using that there exists c₁₁ > 0 such that U(r) > c₁₁ > 0 for every r 6 r^∗∗, we deduce (3.8) by induction on r ∈ Z+, provided r^∗∗ is chosen large enough to

start with.

3.3. Proof of Lemma 3.4

We are now in a position to prove Lemma 3.4. Fix x∈Z^d and r, n>1. Define P_x(n) := {0←→B^x_n} ∩ {B^x_n←→∂B_r} ∩ {0←→∂B_r}^c.

IntroduceC=C(η) :={z ∈R^d:z ←→0}andC⁰ =C⁰(η) :={z ∈R^d:z ←→∂B_r} the connected components of 0 and ∂B_r in O(η). Our first goal is to prove that, conditionally onP_x(n), the probability that CandC⁰ are close to each other inB^x_3n is not too small.

Claim 3.10. — We have that

P^λ[P_x(n) and d(C∩B^x_3n,C⁰)<r^∗]> c₁₁

n^3d−2 ·P^λ[P_x(n)].

Before proving this claim, let us conclude the proof of the Lemma 3.4. If the event on the left occurs, there must exist y ∈ Z^d within a distance at most r_∗ of both C∩ B^x_3n and C⁰: simply pick a point z ∈ R^d at a distance smaller than r^∗/2 of both C∩B^x_3n and C⁰, and then take y ∈Z^d within distance √

d of it (we used the

inequality on the right in (2.1)). In particular,P_y(r_∗) must occur for this y and we deduce that

(3.11) Pλ[P_x(n)]6c₁₂n^{3d−2 X}

y∈Z^d ky−xk63n+r∗

Pλ[P_y(r∗)].

To conclude, observe that the definition of r∗ given by (IT) implies that for all y, Piv_x,A(η)>1_{η∈Py(r∗)}

Z

S^x

Z

R+

1_B^x_r∗⊂B^z_r dz µ(dr)>1_{η∈Py(r∗)}Pλ[D_y].

which, after integration, gives

(3.12) c_ITP^λ[P_y(r_∗)]6E^λ[Piv_y,A].

Now, one easily gets that

(3.13) Inf_(x,n)(f)6λ µ[n−1, n]×Pλ

hP_xn+√ dⁱ.

(Simply observe that P_x(n+√

d) must occur in η, and that the resampled Poisson point process η_e(x,n) must contain at least one point). Plugging (3.11) (applied to n+√

d) in (3.13), then using (3.12), and finally summing over every x ∈Z^d gives the Lemma 3.4.

Proof of the Claim 3.10. — The proof consists in expressing the probability that Cand C⁰ come within a distance r^∗ of each other in terms of the probability that they remain at a distance at least r^∗ of each other.

Fix y ∈ Z^d. For a non-empty subset C of R^d, let u_C = u_C(y) be the point of S^y furthest to C, and v_C = v_C(y) the point of C closest to u_C (in case there is more than one, select one according to a predetermined rule). Consider the event

E_y :={C∩B^x_n6=∅ and d(u_C,C)>r^∗}.

Note that the eventE_y is measurable in terms ofC. Let us study, on E_y, the condi- tional expectation with respect toC. Introduce the three events

F_y :=D_uC G_y :=

u_C^R

d\C

←→B_vC(r^∗)

H_y :=

B^x_n∩S^{y R}

d\C

←→∂B_r

,

whereDyis extended to eachy∈R^dby taking the translate byyofD0. See Figure 3.1 for an example of a simultaneous occurrence of all the above events.

On the event E_y, conditioned on C, η^Rd\C has the same law as η_e^Rd\C for some independent realization η_e of the Poisson point process (recall the definition of η^Z from the previous Section 3.2). Also, the distance betweenu_Candv_C(or equivalently C) is at least r^∗ so thatF_y is measurable with respect to η^Rd\C. Therefore, we may use (IT), Lemma 3.8 and the FKG inequality to get the following inequality between conditional expectations:

(3.14) Pλ[F_y ∩ G_y ∩ H_y|C]

>Pλ[F_y|C]·Pλ[G_y|C]·Pλ[H_y|C]>c_IT c₄

n^2d−2Pλ[H_y|C] a.s. onE_y. Now, observe that ifE_y andH_y occur simultaneously, then P_x(n) occurs (we use r^∗ > √

d). Furthermore if F_y and G_y also occur simultaneously with them, then

0

∂Br

0

C x

vc(y) y uc(y) n

Figure 3.1. A simultaneous occurrence of the events E_y, F_y, G_y, and H_y.

B^v_r∗^C∩C⁰ 6=∅. Since by construction v_C ∈B^x_3n, we deduce that d(C∩B^x_3n,C⁰)<r^∗. Integrating (3.14) on E_y, we deduce that

Pλ[P_x(n) and d(C∩B^x_3n,C⁰)<r^∗]>Pλ[E_y∩ F_y ∩ G_y ∩ H_y]> c_ITc₄

n^2d−2 Pλ[E_y∩ H_y].

Observe that if P_x(n) and d(C∩B^x_n,C⁰)>r^∗ occur, then there exists y ∈Z^d such that the event on the right-hand side occurs. In particular,S^y must intersect B^x_n for H_y to occur, so that there are c₆n^d possible values for y. Summing over all these values, we therefore get that

c₆n^dPλ[P_x(n) and d(C∩B^x_3n,C⁰)<r^∗]> c_ITc₄

n^2d−2 Pλ[P_x(n) and d(C∩B^x_n,C⁰)>r^∗],

which implies the claim 3.10 readily.

4. Renormalization of crossing probabilities

In this section, we prove Theorems 1.4 and 1.5. The proof is quite different in the regime whereµ[r,∞] decays exponentially fast (light tail), and in the regime where it does not (heavy tail). Also, since in the heavy tail regime the renormalization argument performed below may be of some use in the study of other models, or for different distributionsµ, we prove a quantitative lemma which we believe to be of independent interest.

In this section, we reboot the count for the constants c_i starting from c₁. The section is organized as follows. In Section 4.1, we prove a renormalization lemma,