Mobile Geometric Graphs: Detection, Isolation and Percolation Perla - - PowerPoint PPT Presentation

Mobile Geometric Graphs: Detection, Isolation and Percolation

Perla Sousi 1

Based on joint works with Yuval Peres, Alistair Sinclair, Alexandre Stauffer

1Statistical Laboratory, 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 2 time 0 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≤tB(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≤tB(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≤tB(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≤tB(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≤tB(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≤tB(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≤tB(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≤tB(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≤tB(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≤tB(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≤tB(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

t log t (1 + o(1)),

d = 2 cdr d−2t(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

t log t (1 + o(1)),

d = 2 cdr d−2t(1 + o(1)), d ≥ 3 T f

det: 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

t log t (1 + o(1)),

d = 2 cdr d−2t(1 + o(1)), d ≥ 3 T f

det: detection time when target motion is f deterministic.

Theorem (Peres, Sinclair, S., Stauffer (2010)) 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

t log t (1 + o(1)),

d = 2 cdr d−2t(1 + o(1)), d ≥ 3 T f

det: detection time when target motion is f deterministic.

Theorem (Peres, Sinclair, S., Stauffer (2010)) Let f be a continuous motion of the target. In dimension 1 P(T f

det > t) ≤ P(T 0 det > t), for all t.

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Detection in a mobile geometric graph

E[vol(Wr(t))] =

• 2π

t log t (1 + o(1)),

d = 2 cdr d−2t(1 + o(1)), d ≥ 3 T f

det: detection time when target motion is f deterministic.

Theorem (Peres, Sinclair, S., Stauffer (2010)) Let f be a continuous motion of the target. In dimension 1 P(T f

det > t) ≤ P(T 0 det > t), for all t.

In dimension 2 as t → ∞ P(T f

det > t) ≤ exp

• −2πλ

t log t (1 + o(1))

• .

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Detection in a mobile geometric graph

E[vol(Wr(t))] =

• 2π

t log t (1 + o(1)),

d = 2 cdr d−2t(1 + o(1)), d ≥ 3 T f

det: detection time when target motion is f deterministic.

Theorem (Peres, Sinclair, S., Stauffer (2010)) Let f be a continuous motion of the target. In dimension 1 P(T f

det > t) ≤ P(T 0 det > t), for all t.

In dimension 2 as t → ∞ P(T f

det > t) ≤ exp

• −2πλ

t log t (1 + o(1))

• .

In dimension 3 and above as t → ∞ P(T f

det > t) ≤ exp

• −λα(d)cdr d−2t(1 + o(1))
• .

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Wiener sausage with drift

It turned out that the right way to look at that was through the detection problem. Using this connection we showed

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Wiener sausage with drift

It turned out that the right way to look at that was through the detection problem. Using this connection we showed Theorem (Peres, S. (2011)) Let (ξ(s))s be a standard Brownian motion in d dimensions and f a deterministic function, f : R+ → Rd. Then for all r > 0 and all t we have that E[vol(∪s≤tB(ξ(s) + f (s), r))] ≥ E[vol(∪s≤tB(ξ(s), r))].

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Π = {Xi}:Poisson process(λ) in Rd Xi performs an independent random walk ζi with transition kernel

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Π = {Xi}:Poisson process(λ) in Rd Xi performs an independent random walk ζi with transition kernel p(x, y) = 1(x − y < 1) c(d) .

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Π = {Xi}:Poisson process(λ) in Rd Xi performs an independent random walk ζi with transition kernel p(x, y) = 1(x − y < 1) c(d) . f : deterministic motion of target particle Dk = B(f (k), r)

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Π = {Xi}:Poisson process(λ) in Rd Xi performs an independent random walk ζi with transition kernel p(x, y) = 1(x − y < 1) c(d) . f : deterministic motion of target particle Dk = B(f (k), r) T f

det = inf{n ≥ 0 : ∃i s.t. Xi + ζi(n) ∈ Dn}.

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Π = {Xi}:Poisson process(λ) in Rd Xi performs an independent random walk ζi with transition kernel p(x, y) = 1(x − y < 1) c(d) . f : deterministic motion of target particle Dk = B(f (k), r) T f

det = inf{n ≥ 0 : ∃i s.t. Xi + ζi(n) ∈ Dn}.

We will show that P(T f

det > n) ≤ P(T 0 det > n).

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Π = {Xi}:Poisson process(λ) in Rd Xi performs an independent random walk ζi with transition kernel p(x, y) = 1(x − y < 1) c(d) . f : deterministic motion of target particle Dk = B(f (k), r) T f

det = inf{n ≥ 0 : ∃i s.t. Xi + ζi(n) ∈ Dn}.

We will show that P(T f

det > n) ≤ P(T 0 det > n).

Connection to E[vol(sausage around the walk)] and Donsker’s invariance principle give the result.

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Points outside this ball cannot have reached any of the sets (Di) by time n.

b b b

D1 Dn 2n

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

• Rd . . .
• Rd
• 0≤i≤n

1(xi ∈ Ai)

• 1≤i≤n

ψ(xi−1, xi) dx0 . . . dxn ≤

• Rd . . .
• Rd
• 0≤i≤n

1(xi ∈ A∗

i )

• 1≤i≤n

ψ(xi−1, xi) dx0 . . . dxn.

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Hence we have P(T f

det > n) =

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Hence we have P(T f

det > n) = E[P(∀k = 0, . . . , n : ζ(k) /

∈ Dk)N],

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Hence we have P(T f

det > n) = E[P(∀k = 0, . . . , n : ζ(k) /

∈ Dk)N], where N ∼Poisson(λ × vol(ball)) and

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Hence we have P(T f

det > n) = E[P(∀k = 0, . . . , n : ζ(k) /

∈ Dk)N], where N ∼Poisson(λ × vol(ball)) and ζ is a random walk started uniformly at random in the ball with transition kernel p.

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Hence we have P(T f

det > n) = E[P(∀k = 0, . . . , n : ζ(k) /

∈ Dk)N], where N ∼Poisson(λ × vol(ball)) and ζ is a random walk started uniformly at random in the ball with transition kernel p. P(∀k : ζ(k) / ∈ Dk) =

• Rd · · ·
• Rd

1(x0 /

∈ D0) vol(ball)

n

• i=1

p(xi−1, xi)1(xi / ∈ Di) dx

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Hence we have P(T f

det > n) = E[P(∀k = 0, . . . , n : ζ(k) /

∈ Dk)N], where N ∼Poisson(λ × vol(ball)) and ζ is a random walk started uniformly at random in the ball with transition kernel p. P(∀k : ζ(k) / ∈ Dk) =

• Rd · · ·
• Rd

1(x0 ∈ Dc

0)

vol(ball)

n

• i=1

p(xi−1, xi)1(xi ∈ Dc

i ) dx

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

R

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Let S denote a sphere in d dimensions and fix x∗ ∈ S.

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

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

Main ideas of the proof

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 Schmuckenschl¨ ager(2001)) Let A1, . . . , An ⊂ S and ψ : S × S → R+ a nonincreasing function of

• distance. Then
• S

. . .

• S
• 0≤i≤n

1(xi ∈ Ai)

• 1≤i≤n

ψ(xi−1, xi) dµ(x0) . . . dµ(xn) ≤

• S

. . .

• S
• 0≤i≤n

1(xi ∈ A∗

i )

• 1≤i≤n

ψ(xi−1, xi) dµ(x0) . . . dµ(xn).

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Let S be a sphere of radius R in d + 1 dimensions centered at 0.

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Let S be a sphere of radius R in d + 1 dimensions centered at 0. M = {Yi}: Poisson process(λ) on S

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Let S be a sphere of radius R in d + 1 dimensions centered at 0. M = {Yi}: Poisson process(λ) on S Yi moves independently as a random walk ˜ ζi (i.e. ˜ ζi(0) = Yi) on S with transition kernel

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Let S be a sphere of radius R in d + 1 dimensions centered at 0. M = {Yi}: Poisson process(λ) on S Yi moves independently as a random walk ˜ ζi (i.e. ˜ ζi(0) = Yi) on S with transition kernel ψ(x, y) = 1(ρ(x, y) < 1) µ(cap)

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Let S be a sphere of radius R in d + 1 dimensions centered at 0. M = {Yi}: Poisson process(λ) on S Yi moves independently as a random walk ˜ ζi (i.e. ˜ ζi(0) = Yi) on S with transition kernel ψ(x, y) = 1(ρ(x, y) < 1) µ(cap) Detection time on the sphere T D

det = inf{n ≥ 0 : ∃i s.t. ˜

ζi(n) ∈ Dn}.

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Then like before we have P(T D

det > n) =

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Then like before we have P(T D

det > n) = E[P(∀k = 0, . . . , n : ˜

ζ(k) / ∈ Dk)N′],

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Then like before we have P(T D

det > n) = E[P(∀k = 0, . . . , n : ˜

ζ(k) / ∈ Dk)N′], where N′ ∼Poisson(λ × µ(cap)) and

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Then like before we have P(T D

det > n) = E[P(∀k = 0, . . . , n : ˜

ζ(k) / ∈ Dk)N′], where N′ ∼Poisson(λ × µ(cap)) and ˜ ζ is a random walk started uniformly at random in the cap with transition kernel ψ.

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Then like before we have P(T D

det > n) = E[P(∀k = 0, . . . , n : ˜

ζ(k) / ∈ Dk)N′], where N′ ∼Poisson(λ × µ(cap)) and ˜ ζ is a random walk started uniformly at random in the cap with transition kernel ψ. P(∀k : ˜ ζ(k) / ∈ Dk) =

• S

· · ·

• S

1(x0 /

∈ D0) vol(cap)

n

• i=1

ψ(xi−1, xi)1(xi / ∈ Di) dµ(x)

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Then like before we have P(T D

det > n) = E[P(∀k = 0, . . . , n : ˜

ζ(k) / ∈ Dk)N′], where N′ ∼Poisson(λ × µ(cap)) and ˜ ζ is a random walk started uniformly at random in the cap with transition kernel ψ. P(∀k : ˜ ζ(k) / ∈ Dk) =

• S

· · ·

• S

1(x0 ∈ Dc

0)

vol(cap)

n

• i=1

ψ(xi−1, xi)1(xi ∈ Dc

i ) dµ(x)

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Applying the rearrangement inequality, we get

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Main ideas of the proof

Applying the rearrangement inequality, we get P(T D

det > n) ≤ P(T C det > n),

where Ck is a cap centered at (0, . . . , 0, −R) with µ(Ck) = µ(Dk).

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. (2011)) 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≤tB(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. (2011)) 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≤tB(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 Wiener sausage with disks

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))] t → Cap(A) as 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))] t → Cap(A) as t → ∞. Our theorem is a refinement of a classical inequality due to P´

• lya and

Sz¨ ego:

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))] t → Cap(A) as t → ∞. Our theorem is a refinement of a classical inequality due to P´

• lya and

Sz¨ ego: 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 2

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 2 time 0 time Tisol

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Isolation in a mobile geometric graph

Define Ψd(t) =    √t, for d = 1 log t, for d = 2 1, for d ≥ 3. (1)

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Isolation in a mobile geometric graph

Define Ψd(t) =    √t, for d = 1 log t, for d = 2 1, for d ≥ 3. (1) Theorem (Peres, S., Stauffer (2011)) For all d ≥ 1 as t → ∞ P(Tisol > t) ≤ exp

• −c

t Ψd(t)

• .

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Isolation in a mobile geometric graph

Define Ψd(t) =    √t, for d = 1 log t, for d = 2 1, for d ≥ 3. (1) Theorem (Peres, S., Stauffer (2011)) For all d ≥ 1 as t → ∞ P(Tisol > t) ≤ exp

• −c

t 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, for d = 1 log t, for d = 2 1, for d ≥ 3. (1) Theorem (Peres, S., Stauffer (2011)) For all d ≥ 1 as t → ∞ P(Tisol > t) ≤ exp

• −c

t 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 (2011)) 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. Write T f

isol 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. Write T f

isol 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 P(T f

isol > 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. Write T f

isol 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 P(T f

isol > 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 (2010)) As R → ∞, we have that ETcov(QR) ∼ 2 2πλ log R log log R and Tcov(QR) ETcov(QR) → 1 in probability

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Coverage

Theorem (Peres, Sinclair, S., Stauffer (2010)) As R → ∞, we have that ETcov(QR) ∼ 2 2πλ log R log log R and Tcov(QR) ETcov(QR) → 1 in probability Theorem ((General result), Peres, Sinclair, S., Stauffer (2010)) For a set A and R > 0, let RA = {Ra : a ∈ A}. If A has Minkowski dimension α, then as R → ∞ ETcov(RA) ∼ α 2πλ log R log log R and Tcov(RA) ETcov(RA) → 1 in probability

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

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 ∈ 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 ∈ 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′′t/ log t

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation

Percolation

Upper bound: P(Tperc > t) ≤ exp

• −c√t
• (Sinclair, Stauffer 2010)

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

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

Before starting the proof... Observe graphs in discrete time steps only i = 0, 1, 2, . . . Hi = {target ∈ infinite component at step i} P(Tperc > t) ≤ P(∩t

i=0Hi)

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. to appear in PTRF.

• Y. Peres and P. Sousi.

An isoperimetric nequality for the Wiener sausage. to appear in GAFA.

• Y. Peres, P. Sousi and A. Stauffer.

The Isolation Time of Poisson Brownian motions. Arxiv e-prints.

Perla Sousi Mobile Geometric Graphs:Detection, Isolation and Percolation