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Large deviation principle for the cutsets and lower large
deviation principle for the maximal flow in first passage
percolation
Barbara Dembin, Marie Théret
To cite this version:
Barbara Dembin, Marie Théret. Large deviation principle for the cutsets and lower large deviation
principle for the maximal flow in first passage percolation. 2021. �hal-03149069�
Large deviation principle for the cutsets and lower large deviation
principle for the maximal flow in first passage percolation
∗
Barbara Dembin
†, Marie Théret
‡Abstract: We consider the standard first passage percolation model in the rescaled lattice Zd/n for
d ≥ 2 and a bounded domain Ω in Rd. We denote by Γ1 and Γ2 two disjoint subsets of ∂Ω representing
respectively the sources and the sinks, i.e., where the water can enter in Ω and escape from Ω. A cutset is a set of edges that separates Γ1 from Γ2 in Ω, it has a capacity given by the sum of the capacities of
its edges. Under some assumptions on Ω and the distribution of the capacities of the edges, we already know a law of large numbers for the sequence of minimal cutsets (Enmin)n≥1: the sequence (Enmin)n≥1
converges almost surely to the set of solutions of a continuous deterministic problem of minimal cutset in an anisotropic network. We aim here to derive a large deviation principle for cutsets and deduce by contraction principle a lower large deviation principle for the maximal flow in Ω.
1
Introduction
1.1
First definitions and main results
1.1.1 The environment, minimal cutsets
We use here the same notations as in [4]. Let n ≥ 1 be an integer. We consider the graph (Zd n, Edn)
having for vertices Zd
n = Zd/n and for edges Edn, the set of pairs of points of Zdn at Euclidean distance
1/n from each other. With each edge e ∈ Edn we associate a capacity t(e), which is a random variable
with value in R+. The family (t(e))e∈Ed
n is independent and identically distributed with a common law
G. We interpret this capacity as a rate of flow, i.e., it corresponds to the maximal amount of water that
can cross the edge per second. Throughout the paper, we work with a distribution G on R+ satisfying the following hypothesis.
Hypothesis 1. There exists M > 0 such that G([M, +∞[) = 0 and G({0}) < 1 − pc(d).
Here pc(d) denotes the critical parameter of Bernoulli bond percolation on Zd.
Let Ω be a bounded domain in Rd. Let Γ1and Γ2 be two disjoint subsets of the boundary ∂Ω of Ω
that represent respectively the sources and the sinks. We aim to study the minimal cutsets that separate Γ1 from Γ2 in Ω for the capacities (t(e))
e∈Ed
n. We shall define discretized versions for those sets. For
x = (x1, . . . , xd) ∈ Rd, we define kxk2= v u u t d X i=1 x2 i, kxk1= d X i=1 |xi| and kxk∞= max |xi|, i = 1, . . . , d .
∗Research was partially supported by the ANR project PPPP (ANR-16-CE40-0016) and the Labex MME-DII (ANR
11-LBX-0023-01).
†ETH Zürich. barbara.dembin@math.ethz.ch
‡Modal’X, UPL, Univ Paris Nanterre, F92000 Nanterre France and FP2M, CNRS FR 2036.
We use the subscript n to emphasize the dependence on the lattice (Zdn, Edn). Let Ωn, Γn, Γ1n and Γ2n be
the respective discretized version of Ω, Γ, Γ1and Γ2:
Ωn= x ∈ Zdn : d∞(x, Ω) < 1 n , Γn = x ∈ Ωn : ∃y /∈ Ωn, hx, yi ∈ Edn , Γin = x ∈ Γn : d∞(x, Γi) < 1 n, d∞(x, Γ 3−i) ≥ 1 n , for i = 1, 2,
where d∞ is the L∞ distance associated with the norm k · k∞ and hx, yi represents the edge whose
endpoints are x and y. We denote by Πn the set of edges that have both endpoints in Ωn, i.e.,
Πn= e = hx, yi ∈ Edn: x, y ∈ Ωn .
Throughout the paper, Ω, Γ1, Γ2 satisfy the following hypothesis:
Hypothesis 2. The set Ω is an open bounded connected subset of Rd, it is a Lipschitz domain and
its boundary Γ = ∂Ω is included in a finite number of oriented hypersurfaces of class C1 that intersect each other transversally. The sets Γ1 and Γ2 are two disjoint subsets of Γ that are open in Γ such that
inf{kx − yk2, x ∈ Γ1, y ∈ Γ2} > 0, and their relative boundaries ∂ΓΓ1and ∂ΓΓ2 have null Hd−1measure, where Hd−1 denotes the (d − 1)-dimensional Hausdorff measure.
(Γ1
n, Γ2n)-cutset in Ωn. A set of edges En⊂ Πnis a (Γn1, Γ2n)-cutset in Ωnif for any path γ from Γ1nto Γ2n
in Ωn, γ ∩ En 6= ∅. We denote by Cn(Γ1, Γ2, Ω) the set of (Γ1n, Γ2n)-cutset in Ωn. For En ∈ Cn(Γ1, Γ2, Ω),
we define its capacity V (En) and its associated measure µn(En) by
V (En) = X e∈En t(e) and µn(En) = 1 nd−1 X e∈En t(e)δc(e) (1.1)
where c(e) denotes the center of the edge e and δc(e)the dirac mass at c(e). The set En is a discrete set,
but in the limit it is more convenient to work with a continuous set. We first define r(En) ⊂ Zdn by
r(En) = x ∈ Zdn: there exists a path from x to Γ
1
n in (Z d
n, Πn\ En) .
We define upon r(En) a continuous version R(En) by setting
R(En) = r(En) +
1 2n[−1, 1]
d.
Hence, we have R(En) ∩ Zdn = r(En).
Minimal cutsets. A set En ∈ Cn(Γ1, Γ2, Ω) is a minimal cutset in Ωn if we have
V (En) = infV (Fn) : Fn∈ Cn(Γ1, Γ2, Ω) .
We denote by Cn(0) the set of minimal cutsets in Ωn. We will often use the notation Enmin to denote an
element of Cn(0) chosen according to a deterministic rule. Minimal cutsets are the analogous in dimension
d − 1 of geodesics in the classical interpretation of first passage percolation. Indeed, geodesics minimize
the sum of times along paths that are one-dimensional objects, whereas minimal cutsets minimize the sum of capacities along surfaces that are (d − 1)-dimensional objects.
Almost minimal cutsets. Let ε > 0. A set En ∈ Cn(Γ1, Γ2, Ω) is a (Γ1n, Γ2n) ε-cutset in Ωn if for any
Fn∈ Cn(Γ1, Γ2, Ω), we have
V (En) ≤ V (Fn) + εnd−1.
We denote by Cn(ε) the set of (Γ1n, Γ
2
n) ε-cutset in Ωn. Note that the typical size of an almost minimal
Maximal flow. We define φn the maximal flow between Γ1n and Γ2n in Ωn as follows
φn(Γ1, Γ2, Ω) = infV (En) : En∈ Cn(Γ1, Γ2, Ω) .
The reason why this quantity is called a maximal flow is due to the max-flow min-cut theorem that states that the study of the minimal capacity of a cutset is the dual problem of the study of the maximal flow. Just as the study of minimal capacity is linked with the study of cutsets, the study of maximal flow is linked with the study of streams (a stream is a function that describes a stationary circulation of water in the lattice). We won’t define rigorously what a stream is and its link with maximal flow. We refer for instance to the companion paper [5] where we study large deviation principle for admissible streams to obtain an upper large deviation principle for the maximal flow.
1.1.2 Presentation of the limiting objects and main results
We want to define the possible limiting objects for R(En) and µn(En) where En∈ Cn(ε).
Continuous cutsets. We denote by C<∞ the set of subsets of Ω having finite perimeter in Ω, i.e.,
C<∞=E Borelian subset of Rd: E ⊂ Ω, P(E, Ω) < ∞ .
When E is regular enough, its perimeter in the open set Ω corresponds to Hd−1(∂E ∩ Ω). We will give a more rigorous definition of the perimeter later (see (1.7)). Let E ∈ C<∞. We want to build from
∂E a continuous surface that would be a continuous cutset between Γ1 and Γ2 (we don’t give a formal definition of what a continuous cutset is). However, a continuous path from Γ1 to Γ2 does not have to intersect ∂E in general. For regular sets E and Ω, such a path should intersect
b
E = (∂E ∩ Ω) ∪ (Γ1\ ∂E) ∪ (∂E ∩ Γ2) We define E as a more regular version of bE (see figure 1),
E = (∂∗E ∩ Ω) ∪ (∂∗Ω ∩ ((Γ1\ ∂∗E) ∪ (∂∗E ∩ Γ2)) , (1.2)
where X denotes the closure of the set X and ∂∗ is the reduced boundary, we will give a rigorous definition later (see section 1.2.2).
Figure 1 – Representation of the set E (the dotted lines) for some E ∈ C<∞.
Note that in order to obtain a lower large deviation principle for the minimal cutsets, it is not enough to keep track of the localization of the cutset. Indeed, if we only know the localization of the discrete minimal cutset it does not give information on its capacity. We will not only need the total capacity of a cutset but also its local distribution. For a given localization of the minimal cutset and a given capacity, there are different macroscopic configurations where there exists a discrete cutset at the given localization with the given capacity. However this different macroscopic configurations do not have necessarily the same cost, i.e., the same probability to be observed. This is why we need to know locally the capacity. In the continuous setting, this boils down to introducing a function f : E → R+. The local capacity at
a point x in E will be given by f (x). Naturally, the total capacity will be obtained by summing these contributions. We define Cap(E, f ) as the capacity of the cutset E equiped with a local capacity f (x) at x:
Cap(E, f ) = Z
E
f (x)dHd−1(x) .
The smaller f (x) is the bigger the cost of having this local capacity is. We will see that there is a local capacity that is in some sense costless. For x ∈ E and n(x) the associated normal exterior unit vector (of E or Ω depending whether x ∈ ∂∗E or x ∈ ∂∗Ω), the typical local capacity is ν
G(n(x)) where νG is a
called the flow constant. In other words, the probability that the local capacity is close to νG(n(x)) at
x is almost 1.
We say that (E, f ) is minimal if for any F ⊂ Ω of finite perimeter we have Cap(E, f ) ≤ Z F∩E f (y)dHd−1(y) + Z (F\E)∩∂Ω νG(nΩ(y))dHd−1(y) + Z (F\E)\∂Ω νG(nF(y))dHd−1(y)
where F is defined for F in (1.2). This condition will be useful in what follows. This condition is very natural, if we start with a minimal cutset that is close to E and with local capacity f , we expect that the limiting object inherits this property. On E the local capacity is f but everywhere else the local capacity is the typical one given by the flow constant νG. The right hand side may be interpreted as the capacity
of F in the environment where the local capacity on E is f . Let T be the following set:
T =
(E, f ) ∈ C<∞× L∞(E → R, Hd−1) :
(E, f ) is minimal, Cap(E, f ) ≤ 10d2M Hd−1(Γ1) , f (x) ≤ νG(nE(x)) Hd−1-a.e. on ∂∗E ∩ Ω, f (x) ≤ νG(nΩ(x)) Hd−1-a.e. on ∂∗Ω .
We also define TM as follow
TM= (E, f Hd−1|E) : (E, f ) ∈ T .
For two Borelian sets E and F , we define d(E, F ) = Ld(E∆F ) where ∆ denotes the symmetric difference and Ld is the d-dimensional Lebesgue measure.
Remark 1.1. Let En ∈ Cn(Γ1, Γ2, Ω). If limn→∞d(R(En), E) = 0 for some E ∈ B(Rd) and µn(En)
weakly converges towards µ, then we do not necessarily have that µ is absolutely continuous with respect to
Hd−1|
E. More generally, for any sequence (En)n≥1of Borelian sets of Rdsuch that limn→∞d(En, E) = 0,
we do not have necessarily that Hd−1|
∂En weakly converges towards H
d−1|
∂E. However, we can prove
that if instead of studying any sequence of cutsets we study a sequence of minimal cutsets, then in the limit µ will be supported on E. But, since it is too difficult to ensure we build a configuration of the capacities of the edges with a minimal cutset at a given localization (it is difficult to ensure that the cutset we have built is indeed minimal), we will work instead with almost minimal cutsets, i.e., ε-cutsets. They are more flexible than minimal cutsets and their continuous limit is in the set TM.
We denote by Sd−1the unit sphere in Rd. Let (E, f ) ∈ T. For any v ∈ Sd−1, we denote by Jvthe rate
function associated with the lower large deviation principle for the maximal flow in a cylinder oriented in the direction v (see theorem 1.13). We can interpret Jv(λ) as the cost of having a local capacity λ
which is abnormally small in the direction v (λ < νG(v)). It will be properly defined later in theorem
1.13. We define the following rate function: I(E, f ) = Z ∂∗E∩Ω JnE(x)(f (x))dH d−1(x) +Z ∂∗Ω∩((Γ1\∂∗E)∪(Γ2∩∂∗E)) JnΩ(x)(f (x))dH d−1(x) .
Roughly speaking I(E, f ) is the total cost of having a cutset E with the local capacities given by f , the overall cost is equal to the sum of the contributions of the local costs over the continuous cutset E.
We denote by M(Rd
) the set of positive measures on Rd
. We endow M(Rd) with the weak topology
O. We denote by B the σ-field generated by O. We endow the set B(Rd
) of Borelian sets of Rdwith the topology O0 associated with the distance d. We denote by B0 the σ-field associated with this distance. Let n ≥ 1 and ε > 0, Pεn denotes the following probability:
We define the following rate function eI on B(Rd) × M(Rd) as follows:
∀(E, ν) ∈ B(Rd) × M(Rd) I(E, ν) =e
+∞ if (E, ν) /∈ TM
I(E, f ) if ν = f Hd−1|E with (E, f ) ∈ T .
The following theorem is the main result of this paper.
Theorem 1.2 (Large deviation principle on cutsets). Let G that satisfies hypothesis 1. Let (Ω, Γ1, Γ2)
that satisfies hypothesis 2. The sequence (Pεn)n≥1 satisfies a large deviation principle with speed nd−1
governed by the good rate function eI and with respect to the topology O0⊗ O in the following sense: for
all A ∈ B0⊗ B − inf e I(ν) : ν ∈ ˚A ≤ lim ε→0lim infn→∞ 1 nd−1log P ε
n(A) ≤ limε→0lim sup n→∞ 1 nd−1log P ε n(A) ≤ − inf e I(ν) : ν ∈ A .
Remark 1.3. Because of the limit in ε, it is not a proper large deviation principle. Unfortunately, we
were not able to obtain a large deviation principle on Emin
n because of the lower bound (see remark 1.1).
However, this result is enough to prove a lower large deviation principle on the maximal flow.
We can deduce from theorem 1.2, by a contraction principle, the existence of a rate function governing the lower large deviations of φn(Γ1, Γ2, Ω). Let J be the following function defined on R+:
∀λ ≥ 0 J (λ) = infnI(E, ν) : (E, ν) ∈ B(Re d) × M(Rd), ν(Rd) = λ o
and define λmin as
λmin= inf {λ ≥ 0 : J (λ) < ∞} .
The real number φΩ> 0 will be defined in (1.11), it is the almost sure limit of φn(Γ1, Γ2, Ω)/nd−1. We
have the following lower large deviation principle for the maximal flow.
Theorem 1.4 (Lower large deviation principle for the maximal flow). Let G that satisfies hypothesis
1. Let (Ω, Γ1, Γ2) that satisfies hypothesis 2. The sequence (φn(Γ1, Γ2, Ω)/nd−1, n ∈ N) satisfies a large
deviation principle of speed nd−1 governed by the good rate function J .
Moreover, the map J is finite on ]λmin, φΩ], infinite on [0, λmin[∪]φΩ, +∞[, and we have
∀λ < λmin ∃N ≥ 1 ∀n ≥ N P(φn(Γ1, Γ2, Ω) ≤ λnd−1) = 0 . (1.3)
Remark 1.5. The property (1.3) is used to prove upper large deviation principle for the maximal flow
in the companion paper [5]. It also gives the precise role of λmin in our study.
1.1.3 Upper large deviations for the maximal flows
In the companion paper [5], we proved the upper large deviation principle for the maximal flow.
Theorem 1.6 (Upper large deviation principle for the maximal flow). Let G that satisfies hypothesis
1. Let (Ω, Γ1, Γ2) that satisfies hypothesis 2. The sequence (φn(Γ1, Γ2, Ω)/nd−1, n ∈ N) satisfies a large
deviation principle of speed nd with the good rate function eJu.
Moreover, there exists λmax> 0 such that the map eJuis convex on R+, infinite on [0, λmin[∪]λmax, +∞[,
e
Ju is null on [λmin, φΩ] and strictly positive on ]φΩ, +∞[.
We refer to [5] for a precise definition of eJu. Theorems 1.4 and 1.6 give the full picture of large
deviations of φn(Γ1, Γ2, Ω). The lower large deviations are of surface order since it is enough to decrease
the capacities of the edges along a surface to obtain a lower large deviations event. The upper large deviations are of volume order, to create an upper large deviations event, we need to increase the capacities of constant fraction of the edges. This is the reason why to study lower large deviations, it is natural to study cutsets that are (d − 1)-dimensional objects, whereas to study the upper large deviations, we study streams (functions on the edges that describe how the water flows in the lattice) that are d-dimensional objects. Actually, theorem 1.4 is used in [5] to prove theorem 1.6. Theorem 1.4 justifies the fact that the lower large deviations are not of the same order as the order of the upper large deviations.
1.2
Background
We now present the mathematical background needed in what follows. We present a maximal flow in cylinders with good subadditive properties and give a rigorous definition of the limiting objects.
1.2.1 Probabilistic background
Let A be a non-degenerate hyperrectangle, i.e., a rectangle of dimension d − 1 in Rd. Let v ∈ Sd−1 such that v is not contained in an hyperplane parallel to A. Let h > 0. If v is one of the two unit vectors normal to A, we denote by cyl(A, h) the following cylinder of height 2h:
cyl(A, h) = x + tv : x ∈ A, t ∈ [−h, h] .
We have to define discretized versions of the bottom half B0(A, h) and the top half T0(A, h) of the boundary of the cylinder cyl(A, h), that is if we denote by z the center of A:
T0(A, h) = x ∈ Zdn∩ cyl(A, h) : (z − x) · v > 0 and ∃y /∈ cyl(A, h), hx, yi ∈ Ed
n , (1.4)
B0(A, h) = x ∈ Zdn∩ cyl(A, h) : (z − x) · v < 0 and ∃y /∈ cyl(A, h), hx, yi ∈ Ed
n , (1.5)
where · denotes the standard scalar product in Rd. We denote by τn(A, h) the maximal flow from the
upper half part to the lower half part of the boundary of the cylinder, i.e.,
τn(A, h) = inf {V (E) : E cuts T0(A, h) from B0(A, h) in cyl(A, h)} .
The random variable τn(A, h) has good subadditivity properties since minimal cutsets in adjacent
cylin-ders can be glued together along the common side of these cylincylin-ders by adding a negligible amount of edges. Therefore, by applying ergodic subadditive theorems in the multi-parameter case, we can obtain the convergence of τn(A, h)/nd−1.
Theorem 1.7 (Rossignol-Théret [12]). Let G be a measure on R+such that G({0}) < 1−p
c(d). For any
v ∈ Sd−1, there exists a constant ν
G(v) > 0 such that for any non-degenerate hyperrectangle A normal
to v, for any h > 0, we have
lim
n→∞
τn(A, h)
Hd−1(A)nd−1 = νG(v) a.s..
The function νGis called the flow constant. This is the analogue of the time constant defined upon the
geodesics in the standard first passage percolation model where the random variables (t(e))e represent
passage times.
Remark 1.8. If G({0}) ≥ 1 − pc(d), the flow constant is null (see Zhang [14]). It follows that φΩ= 0 and there is no interest of studying lower large deviations.
We present here some result on upper large deviations for the random variable τn(A, h) that will be
useful in what follows. It states that the upper large deviation for the random variable τn(A, h) are of
volume order.
Theorem 1.9 (Théret [13]). Let v be a unit vector, A be an hyperrectangle orthogonal to v and h > 0.
Let us assume that G satisfies hypothesis 1. For every λ > νG(v), we have
lim inf n→∞ − 1 Hd−1(A)ndlog P τ n(A, h) Hd−1(A)nd−1 ≥ λ > 0 .
1.2.2 Some mathematical tools and definitions
Let us first recall some mathematical definitions. For a subset X of Rd, we denote by X the closure
of X, by ˚X the interior of X. Let a ∈ Rd
, the set a + X corresponds to the following subset of Rd
For r > 0, the r-neighborhood Vi(X, r) of X for the distance di, that can be Euclidean distance if i = 2
or the L∞-distance if i = ∞, is defined by
Vi(X, r) = y ∈ Rd: di(y, X) < r .
We denote by B(x, r) the closed ball centered at x ∈ Rd
of radius r > 0. Let v ∈ Sd−1. We define the
upper half ball B+(x, r, v) and the lower half ball B−(x, r, v) as follows:
B+(x, r, v) = {y ∈ B(x, r) : (y − x) · v ≥ 0} and B−(x, r, v) = {y ∈ B(x, r) : (y − x) · v < 0} . We also define the disc disc(x, r, v) centered at x of radius r and of normal vector v as follows:
disc(x, r, v) = {y ∈ B(x, r) : (y − x) · v = 0} .
For n ≥ 1, we define the discrete interior upper boundary ∂n+B(x, r, v) and the discrete interior lower
boundary ∂n−B(x, r, v) as follows: ∂+nB(x, r, v) = y ∈ B(x, r) ∩ Zdn: ∃z ∈ Z d n\ B(x, r), (z − x) · v ≥ 0 and kz − yk1= 1 n and ∂n−B(x, r, v) = y ∈ B(x, r) ∩ Zdn: ∃z ∈ Z d n\ B(x, r), (z − x) · v < 0 and kz − yk1= 1 n . For U ⊂ Zd
n, we define its edge boundary ∂eU by
∂eU = hx, yi ∈ Edn : x ∈ U, y /∈ U . (1.6)
For F ∈ B(Rd) and δ > 0, we denote by B
d(F, δ) the closed ball centered at F or radius δ for the topology
associated with the distance d, i.e.,
Bd(F, δ) =E ∈ B(Rd) : Ld(F ∆E) ≤ δ .
Let Cb(Rd, R) be the set of continuous bounded functions from Rd to R. We denote by Cck(A, B) for
A ⊂ Rp
and B ⊂ Rq, the set of functions of class Ck
defined on Rp, that takes values in B and whose
domain is included in a compact subset of A. The set of functions of bounded variations in Ω, denoted by BV (Ω), is the set of all functions u ∈ L1(Ω → R, Ld) such that
sup Z Ω div h dLd: h ∈ Cc∞(Ω, R d ), ∀x ∈ Ω h(x) ∈ B(0, 1) < ∞ .
Rectifiability and Minkowski content. We will need the following proposition that enables to relate
the measure of a neighborhood of a set E with its Hausdorff measure. Let p ≥ 1. Let M be a set such that Hp(M ) < ∞. We say that a set M is p-rectifiable if there exists countably many Lipschitz maps
fi: Rp→ Rd such that Hp M \[ i∈N fi(Rp) ! = 0 .
Proposition 1.10. Let p ≥ 1. Let M be a subset of Rd that is p-rectifiable. Then we have
lim r→0 Ld(V 2(M, r)) αd−prd−p = Hp(M )
where αd−p denote the volume of the unit ball in Rd−p. In particular, for p = d − 1, we have
lim r→0 Ld(V 2(M, r)) 2r = H d−1(M ) .
This proposition is a consequence of the existence of the (d − 1)-dimensional Minkowski content. We refer to Definition 3.2.37 and Theorem 3.2.39 in [7].
Sets of finite perimeter and surface energy. The perimeter of a Borel set E of Rd in an open set
Ω is defined as P(E, Ω) = sup Z E div f (x) dLd(x) : f ∈ Cc∞(Ω, B(0, 1)) , (1.7)
where Cc∞(Ω, B(0, 1)) is the set of the functions of class C∞from Rdto B(0, 1) having a compact support
included in Ω, and div is the usual divergence operator. The perimeter P(E) of E is defined as P(E, Rd). The topological boundary of E is denoted by ∂E.
The reduced boundary. For E a set of finite perimeter, we denote by χE its characteristic function.
The distributional derivative ∇χE of χE is a vector Radon measure and P(E, Ω) = k∇χEk(Ω) where
k∇χEk is the total variation measure of ∇χE. The reduced boundary ∂∗E of E is a subset of ∂E such
that, at each point x of ∂∗E, it is possible to define a normal vector nE(x) to E in a measure-theoretic
sense, that is points such that
∀r > 0 k∇χEk(B(x, r)) > 0 and lim r→0− ∇χE(B(x, r)) k∇χEk(B(x, r)) = nE(x) .
For a point x ∈ ∂∗E, we have
lim r→0 1 rdL d((E ∩ B(x, r))∆B−(x, r, n E(x))) = 0 , (1.8)
and for Hd−1almost every x in ∂∗E,
lim
r→0
1
αd−1rd−1
Hd−1(∂∗E ∩ B(x, r)) = 1 (1.9) where αd−1is the volume of the unit ball in Rd−1. Moreover, we have
∀A ∈ B(Rd) k∇χ
Ek(A) = Hd−1(∂∗E ∩ A) .
By De Giorgi’s structure theorem (see for instance theorem 15.9 in [8]), the reduced boundary is (d − 1)-rectifiable for any set of finite perimeter. This result will enable to apply proposition 1.10 to the reduced boundary. In what follows, we won’t recall this result and we will use proposition 1.10 without justification.
Let β > 0. We define the set Cβ as the sets of Borelian subsets of Ω of perimeter less than β
Cβ=
n
F ∈ B(Rd) : F ⊂ Ω, P(F, Ω) ≤ βo (1.10) endowed with the topology associated to the distance d. For this topology, the set Cβ is compact.
The following little lemma will appear several times in what follows. We refer to [1] for a proof of this lemma.
Lemma 1.11 (Lemma 6.7 in [1]). Let f1, . . . , fr be r non-negative functions defined on ]0, 1[. Then,
lim sup ε→0 ε log r X i=1 fi(ε) ! = max
1≤i≤rlim supε→0 ε log fi(ε) .
1.3
State of the art
1.3.1 Law of large numbers for minimal cutset in a domain
We work here with the same environment as in section 1.1.1. For any Borelian set F ⊂ Ω such that P(F, Ω) < ∞, we define its capacity IΩ(F ) as follows
IΩ(F ) = Z Ω∩∂∗F νG(nF(x))dHd−1(x) + Z Γ2∩∂∗F νG(nF(x))dHd−1(x) + Z Γ1∩∂∗(Ω\F ) νG(nΩ(x))dHd−1(x) .
Note that by theorem 1.7, the minimal capacity properly renormalized in a cylinder centered at a point
x ∈ ∂∗F in the direction nF(x) in the lattice (Zdn, Edn) is νG(nF(x)). The capacity IΩ(F ) corresponds
to Cap(F, f ) where f (x) = νG(nF(x)) for x ∈ ∂∗F and f (x) = νG(nΩ(x)) for x ∈ Γ1∩ ∂∗(Ω \ F ). We
can interpret the capacity IΩ(F ) as the capacity of the continuous cutset F. We can prove that, almost
surely, there exist cutsets in Cn(Γ1, Γ2, Ω) localized in some sense close to F and of capacity of order
IΩ(F )nd−1. Denote by φΩthe minimal continuous capacity, i.e.,
φΩ= inf {IΩ(F ) : F ⊂ Ω, P(F, Ω) < ∞} . (1.11)
Let us denote by Σa the set of continuous cutsets that achieves φ
Ω
Σa= {F ⊂ Ω : P(F, Ω) < ∞, IΩ(F ) = φΩ} .
In [4], Cerf and Théret proved a law of large numbers for the maximal flow and the minimal cutset in the domain Ω. The minimal cutset converges in some sense towards Σa.
Theorem 1.12. [Cerf-Théret [4]] Let G that satisfies hypothesis 1 and such that G({0}) < 1 − pc(d) (to
ensure that νG is a norm). Let (Ω, Γ1, Γ2) that satisfies hypothesis 2. Then the sequence (Enmin)n≥1 (we
recall that Enmin∈ Cn(0)) converges almost surely for the distance d towards the set Σa, that is,
a.s., lim n→∞F ∈Σinfad(R(E min n ), F ) = 0 . Moreover, we have lim n→∞ φn(Γ1, Γ2, Ω) nd−1 = φΩ> 0 .
They first prove that from each subsequence of (R(Emin
n ))n≥1 they can extract a subsequence that
converges for the distance d towards a set F ⊂ Ω such that P(F, Ω) < ∞. Using locally the law of large numbers for the maximal flow in a cylinder (theorem 1.7), they prove that
lim inf
n→∞
V (Emin n )
nd−1 ≥ IΩ(F ) ≥ φΩ.
The remaining part of the proof required working with maximal streams (which is the dual object associated with minimal cutset). We refer to the companion paper [5] for a more precise study of maximal streams.
1.3.2 Lower large deviations for the maximal flow
Lower large deviation principle in a cylinder. In [10], Rossignol and Théret proved a lower large
deviation principle for the variable τ . The rate function they obtain is going to be our basic brick to build the rate function for lower large deviation principle in a general domain.
Theorem 1.13 (Large deviation principle for τ ). Suppose that G({0}) < 1 − pc(d) and that G has an
exponential moment. Let h > 0. Then for every vector v ∈ Sd−1, for every non degenerate hyperrectangle A normal to v, the sequence
τ
n(A, h)
Hd−1(A)nd−1, n ∈ N
satisfies a large deviation principle of speed Hd−1(A)nd−1governed by the good rate function J
v.
More-over, we know that that Jvis convex on R+, infinite on [0, δGkvk1[∪]νG(v), +∞[ where δG = inf{t, P(t(e) ≤
t) > 0}, equal to 0 at νG(v), and if δGkvk1< νG(v), we also know that Jv is finite on ]δGkvk1, νG(v)],
continuous and strictly decreasing on [δGkvk1, νG(v)] and strictly positive on ]δGkvk1, νG(v)[.
Lower large deviations for the maximal flow in a domain. We here work with the same
environ-ment as in section 1.1.1. Cerf and Théret proved in [2] that the lower large deviations for the maximal flow φn through Ω are of surface order.
Theorem 1.14. If (Ω, Γ1, Γ2) satisfy hypothesis 2 and the law G of the capacity admits an exponential moment, i.e., there exists θ > 0 such that
Z
R+
exp(θx)dG(x) < ∞
and if G({0}) < 1 − pc(d), then there exists a finite constant φΩ> 0 such that
∀λ < φΩ lim sup
n→∞
1
nd−1log P(φn(Γ
1, Γ2, Ω) ≤ λnd−1) < 0 .
Remark 1.15. Note that this constant φΩis the same than in (1.11).
To prove this result, on the lower large deviation event, they consider the continuous subset En =
R(Emin
n ) where Enmin is a minimal cutset Enmin. Since we can control the number of edges in a minimal
cutset thanks to the work of Zhang [15], we can control the perimeter of En in Ω and work with high
probability with a continuous subset En with perimeter less than some constant β > 0. The set En
belongs to the compact set Cβ, this enables to localize the set Enclose to some set F ∈ Cβ. To conclude,
the idea is to say that since the capacity of Enmin is strictly smaller that IΩ(F )nd−1, then there exists
locally a region on the boundary of F where the flow is abnormally low. We can relate this event with lower large deviations for the maximal flow in a small cylinder, whose probability decay speed is of surface order. This large deviation result was used to prove the convergence of the rescaled maximal flow φn/nd−1 towards φΩ. This strategy was already using the study of a cutset but was too rough to
derive a large deviation principle.
1.4
Sketch of the proof
Step 1. Admissible limiting object. One of the main difficulty of this work was to identify the right
object to work with. The aim is that the limiting objects must be measures supported on surfaces. In particular, we want to prove that the limiting object for (R(En), µn(En)) is contained in the set TM.
Thanks to the compactness of the set Cβ, if we have a control on the perimeter of R(En) we can easily
prove that up to extraction, it converges towards a continuous Borelian subset E of Ω. If we work with general cutsets without any restriction, the limiting object for µn(En) may not be a measure supported
on the surface E. This is due to the potential presence of long thin filaments for the set En that are of
negligible volume for the set R(En) but in the limit these filaments can create measure outside of E.
Working with a minimal cutset Emin
n prevents the existence of these long filaments that are not optimal
to minimize the capacity when G({0}) < 1 − pc(d). This will ensure that the weak limit of µn(Enmin) is
supported on E. However, working with minimal cutset leads to a major difficulty for the lower bound (see the next step). One solution is to work with almost minimal cutsets, i.e., cutsets in Cn(ε). We prove
that in some sense the limiting objects for ((R(En), µn(En)), En∈ Cn(ε))n≥1are contained in the set TM.
Step 2. Lower bound. For any (E, ν) ∈ TM such that eI(E, ν) < ∞, we prove that for any
neighbor-hood U of (E, ν) the probability that there exists a cutset En∈ Cn(ε) such that that (R(En), µn(En)) ∈ U
is at least exp(−nd−1I(E, ν)). Write ν = f He d−1|E. To prove this result, we build a configuration where
the expected event occurs using elementary events as building blocks. We first cover almost all the bound-ary of E by a finite family (B(xi, ri, vi))i∈Iof disjoint closed balls such that on each ball ∂E is almost flat
and f is almost constant. Using result for lower large deviations for the maximal flow in cylinders, we can prove that the probability that there exists in the ball B(x, r, v) a cutset separating ∂n−B(x, r, v) from ∂+
nB(x, r, v) of capacity smaller than f (x)αd−1rd−1 is of probability at least exp(−αd−1rd−1Jv(f (x))).
Let us denote by E(i) this event associated with the ball B(x
i, ri, vi) and En(i) the cutset given in the
definition of the event (if there are several possible choices, we pick one according to a deterministic rule). We can build a set of edges F0 of negligible cardinal such that En:= F0∪ ∪i∈IEn(i) ∈ Cn(Γ1, Γ2, Ω). If
∩i∈IE(i), we have (R(En), µn(En)) ∈ U . Since the balls are disjoint, we have P(∃En ∈ Cn(Γ1, Γ2, Ω) : (R(En), µn(En)) ∈ U ) ≥ P \ i∈I E(i) ! =Y i∈I P(E(i)) ≥ exp −X i∈I Jvi(f (xi))αd−1r d−1 i n d−1 !
≈ exp−eI(E, ν)nd−1.
It remains to prove that En is almost minimal, this is the main difficulty of this step. To do so, we have
to ensure that everywhere outside the balls, the flow is not abnormally low. This step is very technical. Using this strategy, we did not manage to prove that the cutset we have built is minimal but only almost minimal.
Step 3. Upper bound. In the lower bound section, we build a configuration upon elementary events
on balls. Here, we do the reverse. We deconstruct the configuration into a collection of elementary events on disjoint balls. Fix (E, ν) ∈ TM, write ν = f Hd−1|E. We first cover almost all the boundary
of E by a finite family (Bi= B(xi, ri, vi))i∈I of disjoint closed balls such that on each ball ∂E is almost
flat and f is almost constant. We pick a neighborhood U of (E, ν) adapted to this covering such that for En ∈ Cn(Γ1, Γ2, Ω) such that (R(En), µn(En)) ∈ U , we have that for each i ∈ I the set En ∩ Bi
is almost a cutset between ∂−nBi and ∂n+Bi in Bi (up to adding a negligible number of edges) and
V (En∩ Bi) ≤ f (xi)αd−1rd−1i . Hence, on the event {∃En ∈ Cn(Γ1, Γ2, Ω) : (R(En), µn(En)) ∈ U }, the
event ∩i∈IE(i)occurs where E(i)was defined in the previous step. We can prove using estimates on lower
large deviations on cylinders that P(E(i)) ≤ exp(−α
d−1rid−1Jvi(f (xi))n
d−1). Since the balls are disjoint
it follows that P ∃En ∈ Cn(Γ1, Γ2, Ω) : (R(En), µn(En)) ∈ U ≤ P \ i∈I E(i) ! =Y i∈I P(E(i)) ≤ exp −X i∈I αd−1rid−1Jvi(f (xi))n d−1 ! ≈ exp(−eI(E, ν)nd−1) .
The remaining of the proof uses standard tools of large deviations theory. The proof of this large deviation principle for almost minimal cutsets enables to deduce by a contraction principle a lower large deviation principle for the maximal flow.
Organization of the paper. In section 2, we prove some useful results such as a covering theorem and
lower large deviations for the maximal flow in a ball. In section 3 (corresponding to step 1), we study the properties of the limiting objects to prove that they belong with high probability to a neighborhood of the set TM. In section 4 (corresponding to step 2) and 5 (corresponding to step 3), we prove local
estimates on the probability Pε
n(U ) for some neighborhoods U to be able to deduce a large deviation
principles for the almost minimal cutsets. Finally, in section 6, we conclude the proof of the two main theorems 1.2 and 1.4.
2
Preliminary work
2.1
Vitali covering
We will use the Vitali covering theorem for Ld. A collection of sets U is called a Vitali class for
a Borelian set Ω of Rd, if for each x ∈ Ω and δ > 0, there exists a set U ∈ U such that x ∈ U and
0 < diam U < δ where diam U is the diameter of the set U for the Euclidean distance. We now recall the Vitali covering theorem for Hd−1(Theorem 1.10 in [6])
Theorem 2.1 (Vitali covering theorem). Let F ⊂ Rd such that Hd−1(F ) < ∞ and U be a Vitali class
of closed sets for F . Then we may select a countable disjoint sequence (Ui)i∈I from U such that
either X i∈I (diam Ui)d−1= +∞ or Hd−1 F \ [ i∈I Ui ! = 0 .
We next recall the Besicovitch differentiation theorem in Rd (see for example theorem 13.4 in [1]):
Theorem 2.2 (Besicovicth differentiation theorem). Let µ be a finite positive Radon measure on Rd.
For any Borel function f ∈ L1(µ), the quotient
1
µ(B(x, r))
Z
B(x,r)
f (y)dµ(y)
converges µ-almost surely towards f (x) as r goes to 0.
We recall that for any set E ∈ C<∞, the set E denotes the continuous cutset associated with E that
was defined in (1.2). We will use at several moments in the proof the Vitali covering theorem. To avoid repeating several times the same arguments, we here present a general result for covering a surface by disjoint closed balls that satisfy a list of properties. These properties are typical for balls centered at points in the surface provided that their radius is small enough.
Proposition 2.3 (Covering E by balls). Let E ∈ C<∞. Let ε ∈]0, 1/2]. Let (P1)x,r, . . . , (Pm)x,r be a
family of logical proposition depending on x ∈ Rd, ε and r > 0. We assume that there exists R such that
Hd−1(E \ R) = 0 and
∀x ∈ R ∃rx> 0 such that ∀i ∈ {1, . . . , m} ∀ 0 < r ≤ rx (Pi)x,r holds .
Then, there exists a finite family of disjoint closed balls (B(xi, ri, vi))i∈I with vi= nΩ(xi) (respectively
nE(xi)) for xi∈ ∂∗Ω \ ∂∗E (resp. xi∈ ∂∗E) such that
Hd−1(E \ ∪i∈IB(xi, ri))) ≤ ε , ∀i ∈ I ∀ 0 < r ≤ ri 1 αd−1rd−1 Hd−1(E ∩ B(xi, r)) − 1 ≤ ε , ∀i ∈ I ∀ 0 < r ≤ ri ∀j ∈ {1, . . . , m} (Pj)xi,r holds .
At this point, the logical propositions (Pi)x,rmay seem a bit abstract. To better understand the kind
of applications, we will do, let us give an example of such a proposition:
(P1)x,r := x ∈ ∂∗Ω =⇒ Ld((Ω ∩ B(x, r))∆B−(x, r, nΩ(x))) ≤ εαdrd .
Proof of proposition 2.3. We follow the proof of lemma 14.6 in [1]. Let ε be a positive constant, with ε < 1/2.
First case: x ∈ ∂∗E. By inequality (1.9), we have for Hd−1-almost every x ∈ ∂∗E
lim
r→0
1
αd−1rd−1
Hd−1(∂∗E ∩ B(x, r)) = 1 .
We denote by R1 the set of points in ∂∗E such that the equality holds. Hence, we have
Hd−1(∂∗E \ R1) = 0 . (2.1)
It yields that for x ∈ R1, there exists a positive constant r
1(x, ε) > 0 such that for any r ≤ r1(x, ε)
1 αd−1rd−1 Hd−1(∂∗E ∩ B(x, r)) − 1 ≤ ε .
Second case: x ∈ ∂∗Ω ∩ (Γ1\ ∂∗E). By inequality (1.9), we have for Hd−1-almost every x ∈ ∂∗Ω lim r→0 1 αd−1rd−1 Hd−1(∂Ω ∩ B(x, r)) = 1 .
We denote by R2 the set of points in ∂∗Ω ∩ (Γ1\ ∂∗E) such that the previous equality holds. Hence, we
have
Hd−1 (∂∗Ω ∩ (Γ1\ ∂∗E)) \ R2 = 0 . (2.2)
It yields that for x ∈ R2, there exists a positive constant r1(x, ε) ∈]0, ε] such that for any r ≤ r1(x, ε)
1 αd−1rd−1 Hd−1(∂∗Ω ∩ B(x, r)) − 1 ≤ ε .
Extract a countable covering by balls. The family of balls
B(x, r), x ∈ (R1∪ R2) ∩ R, r < min(1, r1(x, ε), rx)
is a Vitali class for (R1∪ R2) ∩ R. By Vitali Covering theorem (theorem 2.1), we may select from this
family a countable (or finite) disjoint sequence of balls (B(xi, ri), i ∈ I) such that either
X i∈I rd−1i = +∞ or Hd−1 ((R1∪ R2) ∩ R) \ [ i∈I B(xi, ri) ! = 0 .
We know that E and Ω both have finite perimeter. Since the balls in the family (B(xi, ri), i ∈ I) are
disjoint, we have
X
i∈I
Hd−1(B(x
i, ri) ∩ E) ≤ Hd−1(E) < ∞ .
We recall that for any i ∈ I, we have αd−1rd−1i − H d−1(B(x i, ri) ∩ E) ≤ αd−1rd−1i ε . Hence, we have X i∈I rid−1≤ 1 αd−1(1 − ε) X i∈I Hd−1(B(x i, ri) ∩ E) < ∞ .
As a result, we obtain that
Hd−1 (R 1∪ R2) ∩ R \ [ i∈I B(xi, ri) ! = 0 . (2.3)
Consequently, we can extract from I a finite set I0such that
Hd−1 (R 1∪ R2) ∩ R \ [ i∈I0 B(xi, ri) ! ≤ ε .
We set vi = nΩ(xi) (respectively nE(xi)) for xi∈ ∂∗Ω \ ∂∗E (resp. xi ∈ ∂∗E). Since Hd−1(E \ ((R1∪
R2) ∩ R)) = 0, this concludes the proof.
2.2
Localization of Ω inside balls centered at the boundary of Ω
The following result will be important in what follows. Since the boundary of Ω is smooth, the intersection of Ω with a small ball B(x, r) centered at x ∈ Γ is close to B−(x, r, nΩ(x)). In the following
lemma, we localize precisely the symmetric difference between Ω ∩ B(x, r) and B−(x, r, nΩ(x)).
Lemma 2.4. Let Ω that satisfies hypothesis 1. Let x ∈ ∂∗Ω. For any δ > 0, there exists r0 depending on δ, x and Ω such that
Proof. Let x ∈ ∂∗Ω. We recall that Γ = ∂Ω is contained in the union of a finite number of oriented hypersurfaces of class C1 that intersect each other transversally. We claim that if x ∈ ∂∗Ω, it cannot be contained in the transversal intersection. Indeed, by definition for points in a transversal intersection, we cannot properly define an exterior unit normal vector at points in the intersection. It follows that Γ is locally C1 around x. Let δ > 0. There exists r
0> 0 such that
∀ 0 < r ≤ r0 ∀y ∈ B(x, r) ∩ Γ knΩ(y) − nΩ(x)k2≤ δ .
Since Ω is a Lipschitz domain, up to choosing a smaller r, there exists an hyperplane H containing x of normal vector nΩ(x) and φ : H → R a Lipschitz function such that
B(x, r) ∩ Γ = {y + φ(y)nΩ(x) : y ∈ H ∩ B(x, r)}
and
B(x, r) ∩ Ω = {y + tnΩ(x) : y ∈ H ∩ B(x, r), t ≤ φ(y)} ∩ B(x, r) Let y ∈ B(x, r) ∩ ∂Ω.
Figure 2 – Reprentation of B(x, r), H, y and z.
Let us denote by x0 and y0 the points in H such that x = x0+ φ(x0)nΩ(x) and y = y0+ φ(y0)nΩ(x)
(actually, x0= x and φ(x0) = 0, see figure 2). Let us denote by φ0 the following mapping
∀s ∈ [0, 1] φ0(s) = φ((1 − s)x0+ sy0) .
Note that since ∂Ω ∩ B(x, r) is contained in a C1 hypersurface, the mapping φ
0 ∈ C1(R, R). By the
mean value theorem, there exists s ∈]0, 1[ such that φ00(s) = φ0(1) − φ0(0) = φ(y0) − φ(x0). In other
words, the vector y0 − x0 + (φ(y0) − φ(x0)) n
Ω(x) = y − x belongs to the tangent space of the point z = ((1 − s)x0+ sy0+ φ0(s)nΩ(x)) ∈ Γ. Consequently, we have
nΩ(z) · (y − x) = 0 .
Hence, we get using Cauchy Schwartz inequality
|(y − x) · nΩ(x)| = |(y − x) · (nΩ(z) − nΩ(x))| ≤ ky − xk2knΩ(z) − nΩ(x)k2≤ δky − xk2
and
∂Ω ∩ B(x, r) ⊂ { y ∈ B(x, r) : |(y − x) · nΩ(x)| ≤ δky − xk2} .
Let w ∈ Ω ∩ B+(x, r, n
Ω(x)). There exists w0∈ H ∩ B(x, r) and 0 ≤ t ≤ φ(w0) such that w = w0+ tnΩ(x).
If φ(w0) = 0, then we have w ∈ { y ∈ B(x, r) : |(y − x) · nΩ(x)| ≤ δky − xk2}. Let us assume φ(w0) > 0.
Since w0+ φ(w0)nΩ(x) ∈ Γ, we have
It follows that |(w − x) · nΩ(x)| = |(w0− x) · nΩ(x) + t| = t = t φ(w0)φ(w 0) ≤ t φ(w0)δkw 0+ φ(w0)n Ω(x) − xk2 ≤ δ t φ(w0)kw 0− xk 2+ δt ≤ δ(kw0− xk2+ t) = δkw0+ tnΩ(x) − xk2
and w ∈ { y ∈ B(x, r) : |(y − x) · nΩ(x)| ≤ δky − xk2}. As a result, we get
Ω ∩ B+(x, r, nΩ(x)) ⊂ { y ∈ B(x, r) : |(y − x) · nΩ(x)| ≤ δky − xk2} .
By the same arguments, we can prove that
Ωc∩ B−(x, r, nΩ(x)) ⊂ { y ∈ B(x, r) : |(y − x) · nΩ(x)| ≤ δky − xk2} .
The result follows.
2.3
Lower large deviations for the maximal flow in a ball
2.3.1 Upper bound on the probability of lower large deviations for the maximal flow in a ball
Note that the starting point to understand large deviations for the maximal flow in general domain is to first understand large deviations in cylinders. Several times in this paper, we will cover surfaces by disjoint balls. Hence, we will need to understand how the maximal flow behaves in a ball. From our knowledge on large deviations in cylinders, we can deduce results for large deviations in balls that is the basic brick we need in this paper. We will need the following lemma that is an adaptation of what is done in section 6 in [2]. Let n ≥ 1. Let x ∈ Rd, v ∈ Sd−1 and r, δ, ζ be positive constants. We first define Gn(x, r, v, δ, ζ) to be the event that there exists a set U ⊂ B(x, r) ∩ Zdn such that:
card(U ∆(B−(x, r, v) ∩ Zdn)) ≤ 4δαdrdnd
and
V ((∂eU ) ∩ B(x, r)) ≤ ζαd−1rd−1nd−1
where we recall that ∂eU denotes the edge boundary of U and was defined in (1.6). The set (∂eU )∩B(x, r)
correspond to the edges in ∂eU that have both endpoints in B(x, r).
Lemma 2.5. There exists a constant κ0depending only on d and M such that for any x ∈ Rd, v ∈ Sd−1 and r, δ, ζ positive constants, we have
lim sup n→∞ 1 nd−1P (Gn(x, r, v, δ, ζ)) ≤ −g(δ)αd−1r d−1 Jv ζ + κ0 √ δ g(δ) ! where g(δ) = (1 − δ)(d−1)/2.
Proof of lemma 2.5. We aim to prove that on the event Gn(x, r, v, δ, ζ), we can build a cutset that
separates the upper half part of ∂B(x, r, v) (upper half part according to the direction v) from the lower half part that has a capacity close to ζαd−1rd−1nd−1. To do so, we build from the set U an almost flat
cutset in the ball. The fact that card(U ∆B−(x, r, v)) is small implies that ∂eU is almost flat and is close
to disc(x, r, v). However, this does not prevent the existence of long thin strands that might escape the ball and prevent U from being a cutset in the ball. The idea is to cut these strands by adding edges at a fixed height. We have to choose the appropriate height to ensure that the extra edges we needed to add to cut these strands are not too many, so that we can control their capacity. The new set of edges we create by adding to U these edges will be in a sense a cutset. The last thing to do is then to cover the disc(x, r, v) by hyperrectangles in order to use the rate function that controls the decay of the probability of having an abnormally low flow in a cylinder. Let ρ > 0 be a small constant that we will choose later depending on δ. We define
This constant γmaxwill represent the height of a cylinder of basis disc(x, r0, v). We have to choose r0 in
such a way that cyl(disc(x, r0, v), γmax) ⊂ B(x, r). We set
r0= rp1 − ρ2.
On the event Gn(x, r, v, δ, ζ), we consider a fixed set U satisfying the properties described in the definition
of the event. For each γ in {1/n, . . . , (bnγmaxc − 1)/n}, we define
D(γ) = cyl(disc(x, r0, v), γ), ∂n+D(γ) = y ∈ D(γ) ∩ Zdn: ∃z ∈ Z d n, (z − x) · v > γ and kz − yk1= 1 n and ∂n−D(γ) = y ∈ D(γ) ∩ Zdn: ∃z ∈ Z d n, (z − x) · v < −γ and kz − yk1= 1 n . The sets ∂+
nD(γ) ∪ ∂n−D(γ) are pairwise disjoint for different γ. Moreover, we have
X
γ=1/n,...,(bnγmaxc−1)/n
card(U ∩ ∂n+D(γ)) + card(Uc∩∂n−D(γ)) ≤ card(U ∆(B−(x, r, v) ∩ Zdn)) ≤ 4δαdrdnd.
By a pigeon-hole principle, there exists γ0in {1/n, . . . , (bnγmaxc − 1)/n} such that
card(U ∩ ∂n+D(γ0)) + card(Uc∩ ∂n−D(γ0)) ≤ 4δαdr dnd bnγmaxc − 1 ≤5δαdr dnd−1 γmax = 5δαdr d−1nd−1 ρ
for n large enough. If there are several choices for γ0, we pick the smallest one. We denote by X = U ∩ D(γ0). We define by X+ and X− the following set of edges:
X+= {hy, zi ∈ Edn : y ∈ ∂n+D(γ0) ∩ X, z /∈ D(γ0)} , X− = {hy, zi ∈ Edn: y ∈ ∂−nD(γ0) ∩ Xc, z /∈ D(γ0)} .
Let us control the number of edges in X+∪ X−:
card(X+∪X−) ≤ 2d card(U ∩ ∂n+D(γ0)) + card(U c ∩ ∂n−D(γ0)) ≤ 2d 5δαdrd−1nd−1 ρ = Cdδρ −1rd−1 nd−1
where Cd = 10dαd. We want to relate the event Gn(x, r, v, δ, ζ) to flows in cylinders. There exists a
constant cd depending only on the dimension such that for any positive κ, there exists a finite collection
of closed disjoint hyperrectangles (Ai)i∈J included in disc(x, r0, v) such that
X i∈J Hd−1(A i) ≥ αd−1r0d−1− κ and X i∈J Hd−2(∂A i) ≤ cdr0d−2. (2.4)
Since all the hyperrectangles are closed and disjoint, we have
ξ = min{d2(Ai, Aj) : i 6= j ∈ J } > 0 .
For any i ∈ J , we denote by Pi(n) the edges with at least one endpoint in V2(cyl(∂Ai, γ0), d/n). For
n sufficiently large, we have ξ > 4d/n and all the sets Pi(n) are pairwise disjoint. Moreover, using
proposition 1.10, we have X i∈J card(Pi(n)) ≤ 2d X i∈J Ld(V 2(cyl(∂Ai, γ0), 2d/n))nd≤ X i∈J 16d2Hd−2(∂A i)2γ0nd−1 ≤ 32d2ρrc drd−2nd−1≤ 32d2cdρrd−1nd−1. We define Ei = Pi(n) ∪ (X+∪ X−∪ (∂eU ∩ D(γ0))) ∩ cyl(Ai, γmax) .
We can check that Ei is a cutset for τn(Ai, γ0) (we don’t prove here that the set Ei is a cutset, we refer
to the proof of lemma 2.7 for a proof that a set is a cutset). The sets Ei are pairwise disjoint and
X i∈J V (Ei) ≤ V (∂eU ∩ B(x, r)) + M X i∈J card(Pi(n)) + card(X+∪ X−) ! ≤ ζαd−1+ (Cdδρ−1+ 32d2cdρ)M rd−1nd−1.
Set ρ =√δ. We obtain that
P (Gn(x, r, v, δ, ζ)) ≤ P
∃(Ei)i∈J:
∀i ∈ J Ei ⊂ Edn is a cutset in cyl(Ai, γmax) and
P i∈JV (Ei) ≤ (ζ + κ0 √ δ)αd−1rd−1nd−1 (2.5)
where κ0= M (Cd+ 32d2cd)/αd−1. Let ε0> 0. We want to sum on all possible values of
V (E i) ε0rd−1nd−1 ε0rd−1nd−1, i ∈ J .
It is easy to check that X i∈J V (E i) ε0rd−1nd−1 ε0rd−1nd−1≤ X i∈J V (Ei) + |J |ε0rd−1nd−1≤ ((ζ + κ0 √ δ)αd−1+ |J |ε0)rd−1nd−1,
where |J | denotes the cardinality of J . There are at most ((ζ + κ0
√
δ)αd−1/ε0+ |J |)|J| possible values
for the family
V (Ei) ε0rd−1nd−1 , i ∈ J .
Hence, the number of possible values is finite and does not depend on n. Let S be the following set
S = ( (βi)i∈J ∈ N|J |: X i∈J βiε0rd−1nd−1≤ ((ζ + κ0 √ δ)αd−1+ |J |ε0)rd−1nd−1 ) . By inequality (2.5), we obtain P (Gn(x, r, v, δ, ζ)) ≤ P ∃(Ei)i∈J :
∀i ∈ J Ei⊂ Edn is a cutset in cyl(Ai, γmax) and
P i∈JV (Ei) ≤ (ζ + κ0 √ δ)αd−1rd−1nd−1 ≤ X (βi)i∈J∈S P ∃(Ei)i∈J :
∀i ∈ J Ei is a cutset for τn(Ai, γmax) and
V (Ei)/ε0rd−1nd−1 = βi
. (2.6)
Since the cylinders are all disjoint, using the independence, we have ∀(βi)i∈J∈ S P
∃(Ei)i∈J : ∀i ∈ J Ei is a cutset for τn(Ai, γmax) and
V (Ei) ε0rd−1nd−1 = βi ≤Y i∈J P τn(Ai, γmax) ≤ βiε0rd−1nd−1 .
Thanks to theorem 1.13, it follows that lim sup
n→∞
1
nd−1log P
∃(Ei)i∈J : ∀i ∈ J Ei is a cutset for τn(Ai, γmax) and
V (E i) ε0rd−1nd−1 = βi ≤X i∈J lim sup n→∞ 1 nd−1log P τn(Ai, γmax) ≤ βiε0r d−1nd−1 ≤ −X i∈J Hd−1(A i)Jv βiε0rd−1 Hd−1(A i) .
Using that Jvit is a decreasing function, inequality (2.4) and the convexity of Jv, we have for (βi)i∈J∈ S, Jv ((ζ + κ0 √ δ)αd−1+ |J |ε0)rd−1 αd−1r0d−1− κ ! ≤ Jv ((ζ + κ0 √ δ)αd−1+ |J |ε0)rd−1 Hd−1(∪ i∈JAi) ! ≤ Jv 1 Hd−1(∪ i∈JAi) X i∈J Hd−1(A i) βiε0rd−1 Hd−1(A i) ! ≤X i∈J Hd−1(A i) Hd−1(∪ i∈JAi) Jv βiε0rd−1 Hd−1(A i) .
Combining the two previous inequalities and (2.4), we obtain lim sup
n→∞
1
nd−1log P
∃(Ei)i∈J : ∀i ∈ J Ei is a cutset for τn(Ai, γmax) and
V (E i) ε0rd−1nd−1 = βi ≤ −Hd−1(∪i∈JAi)Jv ((ζ + κ0 √ δ)αd−1+ |J |ε0)rd−1 αd−1r0d−1− κ ! ≤ −(αd−1r0d−1− κ)Jv ((ζ + κ0 √ δ)αd−1+ |J |ε0)rd−1 αd−1r0d−1− κ ! .
Combining this inequality with inequality (2.6) and lemma 1.11 gives
lim sup n→∞ 1 nd−1log P (Gn(x, r, v, δ, ζ)) ≤ −(αd−1r 0d−1− κ)J v ((ζ + κ0 √ δ)αd−1+ |J |ε0)rd−1 αd−1r0d−1− κ ! .
Since Jv is a good rate function, it is lower semi-continuous, hence,
lim inf κ→0 lim infε0→0 Jv ((ζ + κ0 √ δ)αd−1+ |J |ε0)rd−1 αd−1r0d−1− κ ! ≥ Jv (ζ + κ0 √ δ)αd−1rd−1 αd−1r0d−1 ! . As a result, we obtain lim sup n→∞ 1 nd−1log P(Gn(x, r, v, δ, ζ)) ≤ −αd−1r d−1(1 − δ)(d−1)/2J v ζ + κ0√δ (1 − δ)(d−1)/2 !
where we use that r0= r√1 − δ. By setting g(δ) = (1 − δ)(d−1)/2, the result follows.
2.3.2 Lower bound on the probability of lower large deviations for the maximal flow in a ball
Let n ≥ 1. Let x ∈ Rd
, v ∈ Sd−1 and r, δ, ζ be positive constants. We first define G
n(x, r, v, δ, ζ) as
the event that there exists a cutset En in B(x, r) ∩ Ωn that cuts ∂n+B(x, r, v) ∪ ((Γ1n∪ Γ2n) \ ∂−nB(x, r, v))
from ∂n−B(x, r, v) such that
En⊂ cyl(disc(x, r, v), 2δr)
and
V (En) ≤ ζαd−1rd−1nd−1.
Lemma 2.6. Let η > 0. Let δ > 0 such that
M 10d2(1 − (p1 − 4δ2(1 − 2dδ))d−1+ 4δα d−2) ≤
η
4αd−1. (2.7)
• Let x ∈ Ω and r > 0 such that B(x, r) ⊂ Ω. For any v ∈ Sd−1, we have
lim sup n→∞ 1 nd−1log P Gn(x, r, v, 1, νG(v) + η) c = −∞ . For any ζ > 0, lim inf n→∞ 1 nd−1log P Gn(x, r, v, δ, ζ + η) ≥ −αd−1r d−1J v(ζ) .
• Let x ∈ ∂∗Ω ∩ (Γ1∪ Γ2). Let r > 0 such that (Ω ∩ B(x, r, nΩ(x)))∆B−(x, r, nΩ(x)) ⊂ { y ∈ B(x, r) : |(y − x) · nΩ(x)| ≤ δky − xk2} , we have lim sup n→∞ 1 nd−1log P Gn(x, r, nΩ(x), 1, νG(nΩ(x)) + η) c = −∞ . For any ζ > 0, lim inf n→∞ 1 nd−1log P Gn(x, r, nΩ(x), δ, ζ + η) ≥ −αd−1r d−1J nΩ(x)(ζ) .
Proof of lemma 2.6. Let η > 0 and δ we will choose later depending on η.
First case: x ∈ Ω. Let v ∈ Sd−1and r > 0. We set h
max= 2δr and
r0= rp1 − 4δ2.
With such a choice of r0, we have cyl(disc(x, r0, v), hmax) ⊂ B(x, r). Let (e1, . . . , ed−1, v) be an
or-thonormal basis. Let Sδ be the hyper-square of side-length δr of normal vector v having for
ex-pression [0, δr[d−1×{0} in the basis (e1, . . . , e
d−1, v). We can pave disc(x, r0, v) with a family (Si)i∈I
of translates of Sδ such that the Si are pairwise disjoint, there are all included in disc(x, r0, v) and
disc(x, r0(1 − 2dδ), v) ⊂ disc(x, r0− dδr, v) ⊂ ∪i∈ISi. We denote by En(i) the cutset that achieves the
infimum in τn(Si, hmax). If there are several possible choices, we use a deterministic rule to break ties.
Let F0be the set of edges with at least one endpoint included in
V2 disc(x, r, v) \ disc(x, r0(1 − 2dδ), v), d/n ∪
[
i∈I
V2(∂Si, d/n) .
Note that Hd−2(∂Sδ) = 2(d − 1)(δr)d−2 where ∂Sδ denotes the relative boundary, i.e., the boundary of
Sδ in the hyperplane {x ∈ Rd: x · v = 0}. Using proposition 1.10, we have for n large enough
card(F0) ≤ 4dndLd V2(disc(x, r, v) \ disc(x, r0(1 − 2dδ), v), d/n) ∪
[ i∈I V2(∂Si, d/n) ! ≤ 10d2αd−1rd−1 1 −p1 − 4δ2(1 − 2dδ)d−1 nd−1+ 16d4|I|δd−2rd−2α2nd−2 ≤ 10d2rd−1 1 −p1 − 4δ2(1 − 2dδ)d−1 nd−1+ 16d4r d−2 δ α2αd−1n d−2.
We choose δ small enough such that
M 10d2 1 −p1 − 4δ2(1 − 2dδ)d−1 ≤η 4αd−1. Then, we choose n large enough such that
16d4Mr d−2 δ α2αd−1n d−2≤ η 4αd−1r d−1nd−1.
With this choice, we have
M card(F0) ≤ η
2αd−1r
d−1nd−1.
We notice that the set F0∪(∪i∈IEn(i)) is a cutset that cuts ∂n+B(x, r, v) from ∂n−B(x, r, v) in B(x, r)∩
Ωn. It follows that P Gn(x, r, v, 1, νG(v) + η)c ≤ P V (F0∪ (∪i∈IEn(i))) ≥ (νG(v) + η)αd−1rd−1nd−1 ≤ P X i∈I τn(Si, hmax) ≥ |I| νG(v) + η 2 (δr)d−1nd−1 ! ≤X i∈I P τn(Si, hmax) ≥ νG(v) + η 2 (δr)d−1nd−1.
By theorem 1.9, it follows that lim sup n→∞ 1 nd−1log P Gn(x, r, v, 1, νG(v) + η) c = −∞ .
Let ζ > 0. The set Fn = F0∪ (∪i∈IEn(i)) is a cutset that cuts ∂n+B(x, r, v) from ∂n−B(x, r, v) in
B(x, r) and is contained in cyl(disc(x, r, v), 2δr). On the event ∩i∈I{τn(Si, hmax) ≤ ζ(δr)d−1nd−1}, we
have V (Fn) ≤ M card(F0) + X i∈I τn(Si, hmax) ≤ ηαd−1rd−1nd−1+ |I|ζ(δr)d−1nd−1 ≤ (ζ + η)αd−1rd−1nd−1
and the eventGn(x, r, v, δ, ζ + η) occurs. Besides, we have using the independence,
P ∩i∈I{τn(Si, hmax) ≤ ζ(δr)d−1nd−1} =
Y
i∈I
P(τn(Si, hmax) ≤ ζ(δr)d−1nd−1) .
It follows by theorem 1.13 that lim inf n→∞ 1 nd−1log P(Gn(x, r, v, δ, ζ + η)) ≥X i∈I lim inf n→∞ 1 nd−1log P(τn(Si, hmax) ≤ ζ(δr) d−1nd−1) ≥ −αd−1r d−1 (δr)d−1 (δr) d−1J v(ζ) = −αd−1rd−1Jv(ζ) .
The result follows.
Second case: x ∈ ∂∗Ω. We will only prove the case where x ∈ Γ1 since the proof for x ∈ Γ2 is similar.
The proof is similar to the first case with extra technical difficulties. In particular, we cannot pave disc(x, r, nΩ(x)) directly because the cutset may exit Ω since B(x, r) 6⊂ Ω. To fix this issue, we are going
to move this cylinder slightly in the direction −nΩ(x). It is easy to check that
{ y ∈ B(x, r) : |(y − x) · nΩ(x)| ≤ δky − xk2} ⊂ { y ∈ B(x, r) : |(y − x) · nΩ(x)| ≤ δr }
⊂ cyl(disc(x, r, nΩ(x)), δr) .
Set
x0= x − 3
2δrnΩ(x) .
Using that {y ∈ B(x, r) : (y − x) · nΩ(x) ≤ −δr} ⊂ Ω (see figure 3), we have