Mobile Geometric Graphs: Detection, Isolation and
Percolation
Perla Sousi
1
Based on joint works with
Yuval Peres, Alistair Sinclair, Alexandre Stauffer
1 Emmanuel
College, University of Cambridge
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Random Geometric Graph (Boolean Model)
Nodes: Poisson point process in Rd , intensity λ
Edges: ∃ (u, v ) ⇐⇒ d(u, v ) ≤ r (intersecting balls of radius r /2)
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Random Geometric Graph (Boolean Model)
Nodes: Poisson point process in Rd , intensity λ
Edges: ∃ (u, v ) ⇐⇒ d(u, v ) ≤ r (intersecting balls of radius r /2)
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Mobile Geometric Graph
Nodes move as independent Brownian Motions
Obtain stationary sequence of graphs (Gs )s≥0
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Mobile Geometric Graph
Nodes move as independent Brownian Motions
Obtain stationary sequence of graphs (Gs )s≥0
time 0
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Mobile Geometric Graph
Nodes move as independent Brownian Motions
Obtain stationary sequence of graphs (Gs )s≥0
time 0
time s
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Detection in a mobile geometric graph
Target particle initially at origin
Tdet = 1st time some node within distance r of target
Want to study P(Tdet > t)
P(target not detected at fixed s) = P(Tdet > 0) = e −λπr
2
time 0
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Detection in a mobile geometric graph
Target particle initially at origin
Tdet = 1st time some node within distance r of target
Want to study P(Tdet > t)
P(target not detected at fixed s) = P(Tdet > 0) = e −λπr
time 0
2
time Tdet
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Detection of a non-mobile target
Lemma (Classical result of stochastic geometry)
Let ξ be a standard Brownian motion and Wr (t) = ∪s≤t B(ξ(s), r ), the
Wiener sausage up to time t. Then
P(Tdet > t) = exp(−λE[vol(Wr (t))]).
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Detection of a non-mobile target
Lemma (Classical result of stochastic geometry)
Let ξ be a standard Brownian motion and Wr (t) = ∪s≤t B(ξ(s), r ), the
Wiener sausage up to time t. Then
P(Tdet > t) = exp(−λE[vol(Wr (t))]).
Proof.
Π = {Xi }: Poisson process(λ) in Rd
ξi : Brownian motion of Xi
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Detection of a non-mobile target
Lemma (Classical result of stochastic geometry)
Let ξ be a standard Brownian motion and Wr (t) = ∪s≤t B(ξ(s), r ), the
Wiener sausage up to time t. Then
P(Tdet > t) = exp(−λE[vol(Wr (t))]).
Proof.
Π = {Xi }: Poisson process(λ) in Rd
ξi : Brownian motion of Xi
Let Φ = {Xi : ∃s ≤ t s.t. Xi + ξi (s) ∈ B(0, r )}.
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Detection of a non-mobile target
Lemma (Classical result of stochastic geometry)
Let ξ be a standard Brownian motion and Wr (t) = ∪s≤t B(ξ(s), r ), the
Wiener sausage up to time t. Then
P(Tdet > t) = exp(−λE[vol(Wr (t))]).
Proof.
Π = {Xi }: Poisson process(λ) in Rd
ξi : Brownian motion of Xi
Let Φ = {Xi : ∃s ≤ t s.t. Xi + ξi (s) ∈ B(0, r )}.
Φ is a thinned Poisson process of intensity
Λ(x) = λP(x ∈ ∪s≤t B(ξ(s), r )).
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Detection of a non-mobile target
Lemma (Classical result of stochastic geometry)
Let ξ be a standard Brownian motion and Wr (t) = ∪s≤t B(ξ(s), r ), the
Wiener sausage up to time t. Then
P(Tdet > t) = exp(−λE[vol(Wr (t))]).
Proof.
Π = {Xi }: Poisson process(λ) in Rd
ξi : Brownian motion of Xi
Let Φ = {Xi : ∃s ≤ t s.t. Xi + ξi (s) ∈ B(0, r )}.
Φ is a thinned Poisson process of intensity
Λ(x) = λP(x ∈ ∪s≤t B(ξ(s), r )).
P(Tdet > t) =
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Detection of a non-mobile target
Lemma (Classical result of stochastic geometry)
Let ξ be a standard Brownian motion and Wr (t) = ∪s≤t B(ξ(s), r ), the
Wiener sausage up to time t. Then
P(Tdet > t) = exp(−λE[vol(Wr (t))]).
Proof.
Π = {Xi }: Poisson process(λ) in Rd
ξi : Brownian motion of Xi
Let Φ = {Xi : ∃s ≤ t s.t. Xi + ξi (s) ∈ B(0, r )}.
Φ is a thinned Poisson process of intensity
Λ(x) = λP(x ∈ ∪s≤t B(ξ(s), r )).
P(Tdet > t) = P(Φ(Rd ) = 0) =
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Detection of a non-mobile target
Lemma (Classical result of stochastic geometry)
Let ξ be a standard Brownian motion and Wr (t) = ∪s≤t B(ξ(s), r ), the
Wiener sausage up to time t. Then
P(Tdet > t) = exp(−λE[vol(Wr (t))]).
Proof.
Π = {Xi }: Poisson process(λ) in Rd
ξi : Brownian motion of Xi
Let Φ = {Xi : ∃s ≤ t s.t. Xi + ξi (s) ∈ B(0, r )}.
Φ is a thinned Poisson process of intensity
Λ(x) = λP(x ∈ ∪s≤t B(ξ(s), r )).
P(Tdet > t) = P(Φ(Rd ) = 0) = exp(−Λ(Rd )).
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Detection in a mobile geometric graph
E[vol(Wr (t))] =
2π logt t (1 + o(1)), d = 2
cd r d−2 t(1 + o(1)), d ≥ 3
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Detection in a mobile geometric graph
E[vol(Wr (t))] =
2π logt t (1 + o(1)), d = 2
cd r d−2 t(1 + o(1)), d ≥ 3
f
Tdet
: detection time when target motion is f deterministic.
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Detection in a mobile geometric graph
E[vol(Wr (t))] =
2π logt t (1 + o(1)), d = 2
cd r d−2 t(1 + o(1)), d ≥ 3
f
Tdet
: detection time when target motion is f deterministic.
Theorem (Peres, Sinclair, S., Stauffer)
Let f be a continuous motion of the target.
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Detection in a mobile geometric graph
E[vol(Wr (t))] =
2π logt t (1 + o(1)), d = 2
cd r d−2 t(1 + o(1)), d ≥ 3
f
Tdet
: detection time when target motion is f deterministic.
Theorem (Peres, Sinclair, S., Stauffer)
Let f be a continuous motion of the target.
In dimension 1
f
0
P(Tdet
> t) ≤ P(Tdet
> t), for all t.
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Detection in a mobile geometric graph
E[vol(Wr (t))] =
2π logt t (1 + o(1)), d = 2
cd r d−2 t(1 + o(1)), d ≥ 3
f
Tdet
: detection time when target motion is f deterministic.
Theorem (Peres, Sinclair, S., Stauffer)
Let f be a continuous motion of the target.
In dimension 1
f
0
P(Tdet
> t) ≤ P(Tdet
> t), for all t.
In dimension 2 as t → ∞
f
P(Tdet
t
> t) ≤ exp −2πλ
(1 + o(1)) .
log t
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Detection in a mobile geometric graph
E[vol(Wr (t))] =
2π logt t (1 + o(1)), d = 2
cd r d−2 t(1 + o(1)), d ≥ 3
f
Tdet
: detection time when target motion is f deterministic.
Theorem (Peres, Sinclair, S., Stauffer)
Let f be a continuous motion of the target.
In dimension 1
f
0
P(Tdet
> t) ≤ P(Tdet
> t), for all t.
In dimension 2 as t → ∞
f
P(Tdet
t
> t) ≤ exp −2πλ
(1 + o(1)) .
log t
In dimension 3 and above as t → ∞
f
P(Tdet
> t) ≤ exp −λα(d)cd r d−2 t(1 + o(1)) .
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Rearrangement inequalities
A classical topic in analysis started in Cambridge by Hardy and
Littlewood.
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Rearrangement inequalities
A classical topic in analysis started in Cambridge by Hardy and
Littlewood.
Later Riesz, Brascamp, Lieb, Luttinger and others continued..
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Rearrangement inequalities
A classical topic in analysis started in Cambridge by Hardy and
Littlewood.
Later Riesz, Brascamp, Lieb, Luttinger and others continued..
Definition
Let A ⊂ Rd with vol(A) < ∞.
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Rearrangement inequalities
A classical topic in analysis started in Cambridge by Hardy and
Littlewood.
Later Riesz, Brascamp, Lieb, Luttinger and others continued..
Definition
Let A ⊂ Rd with vol(A) < ∞.
The symmetric rearrangement of A, denoted A∗ , is a ball centered at
the origin with vol(A∗ ) = vol(A).
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Rearrangement inequalities
A classical topic in analysis started in Cambridge by Hardy and
Littlewood.
Later Riesz, Brascamp, Lieb, Luttinger and others continued..
Definition
Let A ⊂ Rd with vol(A) < ∞.
The symmetric rearrangement of A, denoted A∗ , is a ball centered at
the origin with vol(A∗ ) = vol(A).
Theorem (Special case of Brascamp, Lieb, Luttinger (1974))
Let A1 , . . . , An ⊂ Rd of finite volume and ψ : Rd × Rd → R+ a
nonincreasing function of distance. Then
Z
Z
Y
Y
...
1(xi ∈ Ai )
ψ(xi−1 , xi ) dx0 . . . dxn
Rd
Z
≤
...
Rd
Rd 0≤i≤n
Z
Y
Rd 0≤i≤n
1≤i≤n
1(xi ∈ A∗i )
Y
ψ(xi−1 , xi ) dx0 . . . dxn .
1≤i≤n
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Rearrangement inequalities on the sphere
Let S denote a sphere in d dimensions and fix x ∗ ∈ S.
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Rearrangement inequalities on the sphere
Let S denote a sphere in d dimensions and fix x ∗ ∈ S.
For A ⊂ S define A∗ to be a geodesic cap centered at x ∗ with
µ(A∗ ) = µ(A)
(µ is the surface area measure on the sphere).
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Rearrangement inequalities on the sphere
Let S denote a sphere in d dimensions and fix x ∗ ∈ S.
For A ⊂ S define A∗ to be a geodesic cap centered at x ∗ with
µ(A∗ ) = µ(A)
(µ is the surface area measure on the sphere).
Theorem (Burchard and Schmuckenschläger (2001))
Let A1 , . . . , An ⊂ S and ψ : S × S → R+ a nonincreasing function of
distance. Then
Z
Z Y
Y
...
1(xi ∈ Ai )
ψ(xi−1 , xi ) dµ(x0 ) . . . dµ(xn )
S
Z
≤
...
S
S 0≤i≤n
Z
Y
S 0≤i≤n
1≤i≤n
1(xi ∈ A∗i )
Y
ψ(xi−1 , xi ) dµ(x0 ) . . . dµ(xn ).
1≤i≤n
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Best strategy
Theorem (Peres, S.)
Let d ≥ 1 and f : R+ → Rd be a deterministic motion of the target.
Then for all t we have
f
0
P(Tdet
> t) ≤ P(Tdet
> t).
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Best strategy
Theorem (Peres, S.)
Let d ≥ 1 and f : R+ → Rd be a deterministic motion of the target.
Then for all t we have
f
0
P(Tdet
> t) ≤ P(Tdet
> t).
Note that f does not need to be continuous.
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
More general result
Although we started with a drift, we showed a more general theorem
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
More general result
Although we started with a drift, we showed a more general theorem
Theorem (Peres, S.)
Let (ξ(s))s≥0 be a standard Brownian motion in d ≥ 1 dimensions and
let (Ds )s≥0 be open sets in Rd with vol(Ds ) = c for all s. Then for all t
we have that
E [vol (∪s≤t (ξ(s) + Ds ))] ≥ E [vol (∪s≤t B(ξ(s), r ))] ,
where r is such that vol(B(0, r )) = c.
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
More general result
Although we started with a drift, we showed a more general theorem
Theorem (Peres, S.)
Let (ξ(s))s≥0 be a standard Brownian motion in d ≥ 1 dimensions and
let (Ds )s≥0 be open sets in Rd with vol(Ds ) = c for all s. Then for all t
we have that
E [vol (∪s≤t (ξ(s) + Ds ))] ≥ E [vol (∪s≤t B(ξ(s), r ))] ,
where r is such that vol(B(0, r )) = c.
In particular this gives that the expected volume of the Wiener sausage
with squares is bigger than the expected volume with balls.
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Squares vs disks
Wiener sausage with squares
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Squares vs disks
Wiener sausage with squares
Perla Sousi
Wiener sausage with disks
Mobile Geometric Graphs: Detection, Isolation and Percolation
A connection to capacity
Spitzer and Whitman(1964) proved that in d ≥ 3, if A ⊂ Rd is an open
set with finite volume, then
E[vol(∪s≤t (ξ(s) + A))]
→ Cap(A) as t → ∞.
t
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
A connection to capacity
Spitzer and Whitman(1964) proved that in d ≥ 3, if A ⊂ Rd is an open
set with finite volume, then
E[vol(∪s≤t (ξ(s) + A))]
→ Cap(A) as t → ∞.
t
Our theorem is a refinement of a classical inequality due to Pólya and
Szëgo:
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
A connection to capacity
Spitzer and Whitman(1964) proved that in d ≥ 3, if A ⊂ Rd is an open
set with finite volume, then
E[vol(∪s≤t (ξ(s) + A))]
→ Cap(A) as t → ∞.
t
Our theorem is a refinement of a classical inequality due to Pólya and
Szëgo:
In d ≥ 3 among all open sets of fixed volume, the ball has the smallest
Newtonian capacity.
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Isolation in a mobile geometric graph
Target particle initially at origin
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Isolation in a mobile geometric graph
Target particle initially at origin
Tisol = 1st time no node is within distance r of target
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Isolation in a mobile geometric graph
Target particle initially at origin
Tisol = 1st time no node is within distance r of target
Want to study P(Tisol > t)
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Isolation in a mobile geometric graph
Target particle initially at origin
Tisol = 1st time no node is within distance r of target
Want to study P(Tisol > t)
P(target not isolated at fixed s) = P(Tisol > 0) = 1 − e −λπr
Perla Sousi
2
Mobile Geometric Graphs: Detection, Isolation and Percolation
Isolation in a mobile geometric graph
Target particle initially at origin
Tisol = 1st time no node is within distance r of target
Want to study P(Tisol > t)
P(target not isolated at fixed s) = P(Tisol > 0) = 1 − e −λπr
time 0
2
time Tisol
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Isolation in a mobile geometric graph
Define
Ψd (t) =


√
t,
log t,

1,
Perla Sousi
for d = 1
for d = 2
for d ≥ 3.
(1)
Mobile Geometric Graphs: Detection, Isolation and Percolation
Isolation in a mobile geometric graph
Define
Ψd (t) =


√
t,
log t,

1,
for d = 1
for d = 2
for d ≥ 3.
(1)
Theorem (Peres, S., Stauffer (2011))
For all d ≥ 1 as t → ∞
P(Tisol
t
> t) ≤ exp −c
Ψd (t)
Perla Sousi
.
Mobile Geometric Graphs: Detection, Isolation and Percolation
Isolation in a mobile geometric graph
Define
Ψd (t) =


√
t,
log t,

1,
for d = 1
for d = 2
for d ≥ 3.
(1)
Theorem (Peres, S., Stauffer (2011))
For all d ≥ 1 as t → ∞
P(Tisol
t
> t) ≤ exp −c
Ψd (t)
.
Easy to see that in all dimensions
P(Tisol > t) ≥ e −Θ(t) .
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Isolation in a mobile geometric graph
Define
Ψd (t) =


√
t,
log t,

1,
for d = 1
for d = 2
for d ≥ 3.
(1)
Theorem (Peres, S., Stauffer (2011))
For all d ≥ 1 as t → ∞
P(Tisol
t
> t) ≤ exp −c
Ψd (t)
.
Easy to see that in all dimensions
P(Tisol > t) ≥ e −Θ(t) .
Lower bound matches upper bound in d ≥ 3 and up to logarithmic
factors in the exponent in d = 2.
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Better lower bound in dimension 1
A better lower bound in dimension 1 that matches up to logarithmic
factors in the exponent:
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Better lower bound in dimension 1
A better lower bound in dimension 1 that matches up to logarithmic
factors in the exponent:
Theorem (Peres, S., Stauffer)
For d = 1 as t → ∞
√
P(Tisol > t) ≥ exp(−c t log t log log t).
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Isolation for a mobile target
Let the target particle move independently of the Poisson Brownian
motions.
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Isolation for a mobile target
Let the target particle move independently of the Poisson Brownian
motions.
f
Write Tisol
for the first time target particle is isolated when it moves
according to f .
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Isolation for a mobile target
Let the target particle move independently of the Poisson Brownian
motions.
f
Write Tisol
for the first time target particle is isolated when it moves
according to f .
Theorem (Peres, S., Stauffer (2012))
Let f be continuous. Then for all d ≥ 1 and all t
f
P(Tisol
> t) ≤ P(Tisol > t).
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Isolation for a mobile target
Let the target particle move independently of the Poisson Brownian
motions.
f
Write Tisol
for the first time target particle is isolated when it moves
according to f .
Theorem (Peres, S., Stauffer (2012))
Let f be continuous. Then for all d ≥ 1 and all t
f
P(Tisol
> t) ≤ P(Tisol > t).
Proof uses rearrangement inequalities of Brascamp, Lieb, Luttinger (’74)
and a new decoupling idea.
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Coverage
QR = cube of side length R
Tcov (QR ) = 1st time all points of QR have been detected
Open problem proposed in Konstantopoulos’09.
time 0
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Coverage
QR = cube of side length R
Tcov (QR ) = 1st time all points of QR have been detected
Open problem proposed in Konstantopoulos’09.
time 0
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Coverage
QR = cube of side length R
Tcov (QR ) = 1st time all points of QR have been detected
Open problem proposed in Konstantopoulos’09.
time 0
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Coverage
QR = cube of side length R
Tcov (QR ) = 1st time all points of QR have been detected
Open problem proposed in Konstantopoulos’09.
time 0
time Tcov (QR )
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Coverage
Theorem (Peres, Sinclair, S., Stauffer)
As R → ∞, we have that
ETcov (QR ) ∼
2
log R log log R
2πλ
Perla Sousi
and
Tcov (QR )
→ 1 in probability
ETcov (QR )
Mobile Geometric Graphs: Detection, Isolation and Percolation
Coverage
Theorem (Peres, Sinclair, S., Stauffer)
As R → ∞, we have that
ETcov (QR ) ∼
2
log R log log R
2πλ
and
Tcov (QR )
→ 1 in probability
ETcov (QR )
Theorem ((General result), Peres, Sinclair, S., Stauffer)
For a set A and R > 0, let RA = {Ra : a ∈ A}. If A has Minkowski
dimension α, then as R → ∞
ETcov (RA) ∼
α
log R log log R
2πλ
Perla Sousi
and
Tcov (RA)
→ 1 in probability
ETcov (RA)
Mobile Geometric Graphs: Detection, Isolation and Percolation
Percolation on Mobile Geometric Graph
∃λc s.t. λ > λc ⇒ a.s. ∃ infinite component at fixed time s
λ < λc
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Percolation on Mobile Geometric Graph
∃λc s.t. λ > λc ⇒ a.s. ∃ infinite component at fixed time s
λ > λc
λ < λc
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Percolation on Mobile Geometric Graph
λ > λc ⇒ a.s. ∃ infinite component for every s (van den Berg, Meester,
White’97)
λ < λc
λ > λc
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Percolation
Target particle initially at origin
We assume λ > λc
Tperc = 1st time target belongs to infinite component
Want to study P(Tperc > t)
time 0
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Percolation
Target particle initially at origin
We assume λ > λc
Tperc = 1st time target belongs to infinite component
Want to study P(Tperc > t)
time 0
time Tperc
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Percolation
Lower bound in discrete time via FKG (extends to continuous time):
P(target 6∈ infinite component at time s) = P(Tperc > 0) = e −c
FKG ⇒ P(Tperc > t) ≥ e −ct
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Percolation
Lower bound in discrete time via FKG (extends to continuous time):
P(target 6∈ infinite component at time s) = P(Tperc > 0) = e −c
FKG ⇒ P(Tperc > t) ≥ e −ct
Lower bound via detection:
P(Tperc > t) ≥ P(Tdet > t) ≥ e −c
Perla Sousi
00
t/ log t
Mobile Geometric Graphs: Detection, Isolation and Percolation
Percolation
Upper bound:
√ P(Tperc > t) ≤ exp −c t
Perla Sousi
(Sinclair, Stauffer 2010)
Mobile Geometric Graphs: Detection, Isolation and Percolation
Percolation
Upper bound:
√ P(Tperc > t) ≤ exp −c t
(Sinclair, Stauffer 2010)
Theorem (Peres, Sinclair, S., Stauffer (2010))
If λ > λc , then ∃ constant c s.t.
P(Tperc > t) ≤ exp −ct/ log6 t , for all t large enough
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Percolation
Upper bound:
√ P(Tperc > t) ≤ exp −c t
(Sinclair, Stauffer 2010)
Theorem (Peres, Sinclair, S., Stauffer (2010))
If λ > λc , then ∃ constant c s.t.
P(Tperc > t) ≤ exp −ct/ log6 t , for all t large enough
Proof uses coupling and multiscale analysis.
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation
Y. Peres, A. Sinclair, P. Sousi, and A. Stauffer.
Mobile Geometric Graphs: Detection, Coverage and Percolation.
Prob. Theory and Related Fields. 156 (2013), no. 1-2, 273–305.
Y. Peres and P. Sousi.
An isoperimetric inequality for the Wiener sausage.
Geom. Funct. Anal., 22(4):1000–1014, 2012.
Yuval Peres, Perla Sousi, and Alexandre Stauffer.
The isolation time of Poisson Brownian motions.
ALEA Lat. Am. J. Probab. Math. Stat., 10(2):813–829, 2013.
Perla Sousi
Mobile Geometric Graphs: Detection, Isolation and Percolation