university of birminghamweb.mat.bham.ac.uk/t.samuel/thermodynamic... · some properties recall that...

Geometric functionals of fractal percolation

Steffen WinterKarlsruhe Institute of Technology

joint work with Michael Klatt

Thermodynamic formalism –Applications to geometry and number theory

Bremen, July 2017

1

Finsterbergen – Koserow – Friedrichroda – Greifswald –

Tabarz –

Fractal Geometry and Stochastics 6

Bad Herrenalb (Black Forest)

30 Sept – 5 Oct 2018

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Scientific committeeChristoph BandtKenneth FalconerJun KigamiMark PollicottMartina Zahle

Organizers:Uta FreibergBen HamblyMichael HinzSteffen Winter

2

Finsterbergen – Koserow – Friedrichroda – Greifswald – Tabarz –



30 Sept – 5 Oct 2018

Save the date!



2


Fractal Geometry and Stochastics

6


30 Sept – 5 Oct 2018

Save the date!



2




30 Sept – 5 Oct 2018

Save the date!



2

Geometric functionals of fractal percolation

Steffen WinterKarlsruhe Institute of Technology

joint work with Michael Klatt

Thermodynamic formalism –Applications to geometry and number theory

Bremen, July 2017

3

Fractal percolation

Let J := [0, 1]2 ⊂ R2, M ∈ N, M ≥ 2 and p ∈ [0, 1].

1. Subdivide J into squares of sidelength 1/M and decide foreach of them independently whether it is kept (with prob. p)or discarded (with prob. 1− p).

For n = 2, 3, . . .:

n. Repeat step 1 for each of the squares kept in step n − 1.

J

Let F (n) be the union of the squares kept in the n-th step.

4

Fractal percolation

Let J := [0, 1]2 ⊂ R2, M ∈ N, M ≥ 2 and p ∈ [0, 1].


For n = 2, 3, . . .:


J

1M


4

Fractal percolation

Let J := [0, 1]2 ⊂ R2, M ∈ N, M ≥ 2 and p ∈ [0, 1].


For n = 2, 3, . . .:


F (2)

1M


4

Fractal percolation

Let J := [0, 1]2 ⊂ R2, M ∈ N, M ≥ 2 and p ∈ [0, 1].


For n = 2, 3, . . .:


F (2)

1M

Let F (n) be the union of the squares kept in the n-th step.4

Fractal percolationI J ⊇ F (1) ⊇ F (2) ⊇ . . .

I Fractal percolation (Mandelbrot percolation, canonicalcurdling) is the random compact subset of J given by

F :=∞⋂n=1

F (n).

p = 0.8 p = 0.9 p = 0.97

5

Fractal percolationI J ⊇ F (1) ⊇ F (2) ⊇ . . .I Fractal percolation (Mandelbrot percolation, canonical

curdling) is the random compact subset of J given by

F :=∞⋂n=1

F (n).

p = 0.8 p = 0.9 p = 0.97

5

Some properties

Recall that F (in Rd) depends on M ≥ 2 and p ∈ [0, 1].

I [Mandelbrot 74; Chayes, Chayes, Durrett 88]:

I If Mdp ≤ 1, then F = ∅ almost surely.I If Mdp > 1, then P(F 6= ∅) > 0, and conditioned on F 6= ∅,

the Hausdorff dimension (and the Minkowski dimension) isalmost surely

dimF = D :=log(Mdp)

logM.

I There exists pc = pc(M) with 1/Md < pc < 1 such that

– for p < pc , F is a.s. totally disconnected (‘dustlike’);– for p ≥ pc , F percolates with positive probability.

I Bounds on pc (d = 2):

I [Chayes, Chayes, Durrett 88]: 1/√M ≤ pc(M) ≤ 0.9999

I [White 2000]: pc(2) > 0.8107I [Don 14]: 0.881 ≤ pc(2) ≤ 0.993 and 0.784 ≤ pc(3) ≤ 0.940.

I further properties and related models: Falconer and Grimmett,

Dekking and Meester , Orzechowski, Broman and Camia, Rams and Simon ...

6

Some properties

Recall that F (in Rd) depends on M ≥ 2 and p ∈ [0, 1].I [Mandelbrot 74; Chayes, Chayes, Durrett 88]:

I If Mdp ≤ 1, then F = ∅ almost surely.

I If Mdp > 1, then P(F 6= ∅) > 0, and conditioned on F 6= ∅,the Hausdorff dimension (and the Minkowski dimension) isalmost surely

dimF = D :=log(Mdp)

logM.








6

Some properties




dimF = D :=log(Mdp)

logM.








6

Some properties




dimF = D :=log(Mdp)

logM.

I There exists pc = pc(M) with 1/Md < pc < 1 such that– for p < pc , F is a.s. totally disconnected (‘dustlike’);

– for p ≥ pc , F percolates with positive probability.






6

Some properties




dimF = D :=log(Mdp)

logM.

I There exists pc = pc(M) with 1/Md < pc < 1 such that– for p < pc , F is a.s. totally disconnected (‘dustlike’);– for p ≥ pc , F percolates with positive probability.






6

Some properties




dimF = D :=log(Mdp)

logM.


I Bounds on pc (d = 2):I [Chayes, Chayes, Durrett 88]: 1/

√M ≤ pc(M) ≤ 0.9999




6

Some properties




dimF = D :=log(Mdp)

logM.



√M ≤ pc(M) ≤ 0.9999

I [White 2000]: pc(2) > 0.8107

I [Don 14]: 0.881 ≤ pc(2) ≤ 0.993 and 0.784 ≤ pc(3) ≤ 0.940.



6

Some properties




dimF = D :=log(Mdp)

logM.



√M ≤ pc(M) ≤ 0.9999




6

Expected intrinsic volumes

I F is a fractal (a self-similar random set).

I For each n ∈ N0, F (n) is polyconvex.

Intrinsic volumes arewell defined for F (n).

I Let r := 1/M. We are interested in the limits

Zk(F ) := limn→∞

rn(D−k)EVk(F (n)).

I compare with fractal curvatures [Zahle 11]:

Ck(F ) := limε↘0

εD−kEVk(Fε).

7


I F is a fractal (a self-similar random set).

I For each n ∈ N0, F (n) is polyconvex. Intrinsic volumes arewell defined for F (n).

I Let r := 1/M. We are interested in the limits

Zk(F ) := limn→∞

rn(D−k)EVk(F (n)).

I compare with fractal curvatures [Zahle 11]:

Ck(F ) := limε↘0

εD−kEVk(Fε).

7

Curvature measures

Theorem (Local Steiner formula) [Federer 59]

For any compact, convex set K ⊂ Rd , thereexist finite measures C0(K , ·), . . . ,Cd(K , ·)such that

ρε(K ,B) =d∑

k=0

εd−kκd−kCk(K ,B)

for all Borel sets B ⊆ K and ε ≥ 0.

B

K

ρε(K,B)

There exists an additive extension to polyconvex sets (i.e. to anyfinite union of convex sets) [Groemer 78]:

Ck(K ∪M, ·) = Ck(K , ·) + Ck(M, ·)− Ck(K ∩M, ·)

Vk(K ) := Ck(K ,Rd) k-th total curvature or intrinsic volume

8

Properties of curvature measures

I geometric interpretation:Vd – volumeVd−1 – surface areaVk – average k-volume of the projections to

k-dim. subspacesintegrals of mean curvature

V0 – Euler characteristic

I motion invariance (g Euclidean motion):

Vk(gK ) = Vk(K )

I homogeneity of degree k (λ > 0):

Vk(λK ) = λkVk(K )

I continuity (convex sets): Kn → K implies Vk(Kn)→ Vk(K )

I Hadwigers characterization theorem [Hadwiger 57]

9


Theorem (Existence of the limits)

Let F be fractal percolation in Rd with parameters M ≥ 2 andp ∈ [0, 1].

Then, for each k ∈ {0, . . . , d}, the limit

Zk(F ) = limn→∞

rn(D−k)EVk(F (n))

exists and is given by

Vk([0, 1]d) +∑

T⊂{1,...,Md},|T |≥2

(−1)|T |−1∞∑n=1

rn(D−k)EVk(⋂j∈T

F j(n)).

F j(n) is the union of the level-ncubes contained in Jj ∩ F (n).

J

1M

J1 J2 J3

J4 J5 J6

J9J8J7

10



Let F be fractal percolation in Rd with parameters M ≥ 2 andp ∈ [0, 1]. Then, for each k ∈ {0, . . . , d}, the limit

Zk(F ) = limn→∞

rn(D−k)EVk(F (n))

exists

and is given by

Vk([0, 1]d) +∑

T⊂{1,...,Md},|T |≥2

(−1)|T |−1∞∑n=1


F j(n)).


J

1M

J1 J2 J3

J4 J5 J6

J9J8J7

10



Let F be fractal percolation in Rd with parameters M ≥ 2 andp ∈ [0, 1]. Then, for each k ∈ {0, . . . , d}, the limit

Zk(F ) = limn→∞

rn(D−k)EVk(F (n))


Vk([0, 1]d) +∑

T⊂{1,...,Md},|T |≥2

(−1)|T |−1∞∑n=1


F j(n)).


J

1M

J1 J2 J3

J4 J5 J6

J9J8J7

10

The case d = 2

In R2, the general formula reduces to

Z0(F ) =1− E1 − E2 + E3 − E4,

where

E1 := 2M(M − 1)∞∑n=1

rnD

E2 := 2(M − 1)2∞∑n=1

rnD

E3 := 4(M − 1)2∞∑n=1

rnD

E4 := (M − 1)2∞∑n=1

rnD

1 2

1

4

1 2

3

11

The case d = 2


Z0(F ) =1− E1 − E2 + E3 − E4,

where

E1 := 2M(M − 1)∞∑n=1

rnDEV0(F 1(n) ∩ F 2(n))

E2 := 2(M − 1)2∞∑n=1

rnD

E3 := 4(M − 1)2∞∑n=1

rnD

E4 := (M − 1)2∞∑n=1

rnD

1 2

1

4

1 2

3

11

The case d = 2


Z0(F ) =1− E1 − E2 + E3 − E4,

where

E1 := 2M(M − 1)∞∑n=1

rnDEV0(F 1(n) ∩ F 2(n))

E2 := 2(M − 1)2∞∑n=1

rnDEV0(F 1(n) ∩ F 4(n))

E3 := 4(M − 1)2∞∑n=1

rnD

E4 := (M − 1)2∞∑n=1

rnD

1 2

1

4

1 2

3

11

The case d = 2


Z0(F ) =1− E1 − E2 + E3 − E4,

where

E1 := 2M(M − 1)∞∑n=1

rnDEV0(F 1(n) ∩ F 2(n))

E2 := 2(M − 1)2∞∑n=1

rnDEV0(F 1(n) ∩ F 4(n))

E3 := 4(M − 1)2∞∑n=1

rnDEV0

3⋂j=1

F j(n)

E4 := (M − 1)2∞∑n=1

rnD

1 2

1

4

1 2

3

11

The case d = 2


Z0(F ) =1− E1 − E2 + E3 − E4,

where

E1 := 2M(M − 1)∞∑n=1

rnDEV0(F 1(n) ∩ F 2(n))

E2 := 2(M − 1)2∞∑n=1

rnDEV0(F 1(n) ∩ F 4(n))

E3 := 4(M − 1)2∞∑n=1

rnDEV0

3⋂j=1

F j(n)

E4 := (M − 1)2

∞∑n=1

rnDEV0

4⋂j=1

F j(n)

1 2

1

4

1 2

3

11

The case d = 2


Z0(F ) =1− E1 − E2 + E3 − E4,

where

E1 := 2M(M − 1)∞∑n=1

rnDEV0(F 1(n) ∩ F 2(n))

E2 := 2(M − 1)2∞∑n=1

rnDp2n

E3 := 4(M − 1)2∞∑n=1

rnDEV0

3⋂j=1

F j(n)

E4 := (M − 1)2

∞∑n=1

rnDEV0

4⋂j=1

F j(n)

1 2

1

4

1 2

3

11

The case d = 2


Z0(F ) =1− E1 − E2 + E3 − E4,

where

E1 := 2M(M − 1)∞∑n=1

rnDEV0(F 1(n) ∩ F 2(n))

E2 := 2(M − 1)2∞∑n=1

rnDp2n

E3 := 4(M − 1)2∞∑n=1

rnDp3n

E4 := (M − 1)2∞∑n=1

rnDEV0

4⋂j=1

F j(n)

1 2

1

4

1 2

3

11

The case d = 2


Z0(F ) =1− E1 − E2 + E3 − E4,

where

E1 := 2M(M − 1)∞∑n=1

rnDEV0(F 1(n) ∩ F 2(n))

E2 := 2(M − 1)2∞∑n=1

rnDp2n

E3 := 4(M − 1)2∞∑n=1

rnDp3n

E4 := (M − 1)2∞∑n=1

rnDp4n

1 2

1

4

1 2

3

11

Reducing the dimension

I Let K1, K2 be two independent (1-dim.) fractal percolationsdefined on I = [0, 1] (with the same M and p as F ).

I For n ∈ N, let Ki (n), i = 1, 2 denote the n-th steps of theirconstruction.

I Let Ki (n), i = 1, 2 be the random set, which equals Ki (n)with probability p and is empty otherwise.

Then for each n ∈ N, in distribution

F 1(n) ∩ F 2(n) = ψ(K1(n − 1) ∩ K2(n − 1)),

where ψ : I → R2 is the similarity mapping I to J1 ∩ J2.

F 1(n) F 2(n)

12



I For n ∈ N, let Ki (n), i = 1, 2 denote the n-th steps of theirconstruction.

I Let Ki (n), i = 1, 2 be the random set, which equals Ki (n)with probability p and is empty otherwise.

Then for each n ∈ N, in distribution

F 1(n) ∩ F 2(n) = ψ(K1(n − 1) ∩ K2(n − 1)),


In particular,

EVk(F 1(n) ∩ F 2(n)) = rkp2EVk(K1(n − 1) ∩ K2(n − 1)).

12

Expected Euler characteristic (d = 2)

Z0(F ) = 1− E1 + (M − 1)2(− 2p

M2 − p+

4p2

M2 − p2− p3

M2 − p3

)E1 = 2(M − 1)2

((3

M − 1− 4p

M − p+

p2

M − p2

)p

M − p

− 2Mp

(M − 1)(M2 − p)+

4Mp2

(M − p)(M2 − p2)− Mp3

(M − p2)(M2 − p3)

)

.

13


Z0(F ) = 1− E1 + (M − 1)2(− 2p

M2 − p+

4p2

M2 − p2− p3

M2 − p3

)E1 = 2(M − 1)2

((3

M − 1− 4p

M − p+

p2

M − p2

)p

M − p

− 2Mp

(M − 1)(M2 − p)+

4Mp2

(M − p)(M2 − p2)− Mp3

(M − p2)(M2 − p3)

)

0

0.5

1

0 0.2 0.4 0.6 0.8 1

Z0

p

M=2

M=3

M=4

13


Z0(F ) = 1− E1 + (M − 1)2(− 2p

M2 − p+

4p2

M2 − p2− p3

M2 − p3

)E1 = 2(M − 1)2

((3

M − 1− 4p

M − p+

p2

M − p2

)p

M − p

− 2Mp

(M − 1)(M2 − p)+

4Mp2

(M − p)(M2 − p2)− Mp3

(M − p2)(M2 − p3)

)

0

0.5

1

0 0.2 0.4 0.6 0.8 1

pcZ0

p

M=2

13


Z0(F ) = 1− E1 + (M − 1)2(− 2p

M2 − p+

4p2

M2 − p2− p3

M2 − p3

)E1 = 2(M − 1)2

((3

M − 1− 4p

M − p+

p2

M − p2

)p

M − p

− 2Mp

(M − 1)(M2 − p)+

4Mp2

(M − p)(M2 − p2)− Mp3

(M − p2)(M2 − p3)

)

0

0.5

1

0 0.2 0.4 0.6 0.8 1

pcZ0

p

M=3

13

A lower bound for pc?Is the zero of the rescaled limit of expected Euler characteristics alower bound for pc?

Does Z0(Fp,M) > 0 imply p < pc?

Z0(Fp,M) := limn→∞

rDnV0(Fp,M(n))

It is well known [Chayes, Chayes 89] that limM→∞

pc(M) = psite , where

psite ≈ 0.5927.. is the percolation threshhold of site percolation onthe lattice Z2. The zero of f (p) := lim

M→∞Z0(Fp,M) stays even well

below psite,8 ≈ 0.407...

0

0.5

1

0 0.2 0.4 0.6 0.8 1

psite,8

Z0

p

10

100

M

14

A lower bound for pc?Is the zero of the rescaled limit of expected Euler characteristics alower bound for pc? Does Z0(Fp,M) > 0 imply p < pc?


rDnV0(Fp,M(n))






0

0.5

1

0 0.2 0.4 0.6 0.8 1

psite,8

Z0

p

10

100

M

14



rDnV0(Fp,M(n))



psite ≈ 0.5927.. is the percolation threshhold of site percolation onthe lattice Z2.

The zero of f (p) := limM→∞

Z0(Fp,M) stays even well


0

0.5

1

0 0.2 0.4 0.6 0.8 1

psite,8

Z0

p

10

100

M

14



rDnV0(Fp,M(n))






0

0.5

1

0 0.2 0.4 0.6 0.8 1

psite,8

Z0

p

10

100

M

14

Another approximation of FInstead of the sequence F (n) consider the union G (n) of thoseboxes Q of level n, which have a nonempty intersection with F .Then

I G (n) ⊆ F (n),

I F =⋂

n G (n);

I G (n) are still polyconvex;

I similar formulas hold for the rescaled limits of Vk(G (n));

I the survival probability q := q(p) := Pp(F 6= ∅) appears

q = (1− p + pq)M2

F (2)

1M

15

Another approximation of FInstead of the sequence F (n) consider the union G (n) of thoseboxes Q of level n, which have a nonempty intersection with F .Then

I G (n) ⊆ F (n),I F =

⋂n G (n);

I G (n) are still polyconvex;I similar formulas hold for the rescaled limits of Vk(G (n));I the survival probability q := q(p) := Pp(F 6= ∅) appears

q = (1− p + pq)M2

Z0(F ) = ....1− E1 + (M − 1)2(− 2pq2

M2 − p+

4p2q3

M2 − p2− p3q4

M2 − p3

),

E1 = 2(M − 1)2q2((

3

M − 1− 4pq

M − p+

p2q2

M − p2

)p

M − p

− 2Mp

(M − 1)(M2 − p)+

4Mp2q

(M − p)(M2 − p2)− Mp3q2

(M − p2)(M2 − p3)

)15

Approximation by F (n) vs. G (n)

16

Yet another approximation of F

In F (n) diagonal connections are counted,which produce many extra holes.

But di-agonal connections do not survive in F .Consider the closed complements of F (n)in J = [0, 1]d :

C (n) = J \ F (n).

Then −V0(C (n)) corresponds to the Eu-ler characteristic of F (n) with a 4-neigh-borhood (no diagonal connections).

Yk(F ) := limn→∞

rn(D−k)EVk(C (n))

17


In F (n) diagonal connections are counted,which produce many extra holes. But di-agonal connections do not survive in F .

Consider the closed complements of F (n)in J = [0, 1]d :

C (n) = J \ F (n).



rn(D−k)EVk(C (n))

17


In F (n) diagonal connections are counted,which produce many extra holes. But di-agonal connections do not survive in F .Consider the closed complements of F (n)in J = [0, 1]d :

C (n) = J \ F (n).



rn(D−k)EVk(C (n))

17

Approximation by the complements


Let F be fractal percolation in Rd with parameters M ≥ 2 andp ∈ (1/Md , 1].

Then, for each k < D, the limit


rn(D−k)EVk(C (n))


cd ,k +∑

T⊂{1,...,Md},|T |≥2

(−1)|T |−1∞∑n=1


C j(n)),

where cd,k :=Md−k(1− p)

Md−kp − 1Vk([0, 1]d).

C j(n) is the union of the level-n cubescontained in Jj ∩ C (n).

J

1M

J1 J2 J3

J4 J5 J6

J9J8J7

18



Let F be fractal percolation in Rd with parameters M ≥ 2 andp ∈ (1/Md , 1]. Then, for each k < D, the limit


rn(D−k)EVk(C (n))

exists

and is given by

cd ,k +∑

T⊂{1,...,Md},|T |≥2

(−1)|T |−1∞∑n=1


C j(n)),


Md−kp − 1Vk([0, 1]d).

C j(n) is the union of the level-n cubescontained in Jj ∩ C (n).

J

1M

J1 J2 J3

J4 J5 J6

J9J8J7

18



Let F be fractal percolation in Rd with parameters M ≥ 2 andp ∈ (1/Md , 1]. Then, for each k < D, the limit


rn(D−k)EVk(C (n))


cd ,k +∑

T⊂{1,...,Md},|T |≥2

(−1)|T |−1∞∑n=1


C j(n)),


Md−kp − 1Vk([0, 1]d).

C j(n) is the union of the level-n cubescontained in Jj ∩ C (n). J

1M

J1 J2 J3

J4 J5 J6

J9J8J7

18

The case d = 2

In R2 (and for k = 0), the general formula reduces to

Y0(F ) =M2(1− p)

M2p − 1− E1 − E2 + E3 − E4,

where

E1 := 2M(M − 1)∞∑n=1

rnD

E2 := 2(M − 1)2∞∑n=1

rnD

E3 := 4(M − 1)2∞∑n=1

rnD

E4 := (M − 1)2∞∑n=1

rnD

1 2

1

4

1 2

3

19

The case d = 2


Y0(F ) =M2(1− p)

M2p − 1− E1 − E2 + E3 − E4,

where

E1 := 2M(M − 1)∞∑n=1

rnDEV0(C 1(n) ∩ C 2(n))

E2 := 2(M − 1)2∞∑n=1

rnD

E3 := 4(M − 1)2∞∑n=1

rnD

E4 := (M − 1)2∞∑n=1

rnD

1 2

1

4

1 2

3

19

The case d = 2


Y0(F ) =M2(1− p)

M2p − 1− E1 − E2 + E3 − E4,

where

E1 := 2M(M − 1)∞∑n=1

rnDEV0(C 1(n) ∩ C 2(n))

E2 := 2(M − 1)2∞∑n=1

rnDEV0(C 1(n) ∩ C 4(n))

E3 := 4(M − 1)2∞∑n=1

rnD

E4 := (M − 1)2∞∑n=1

rnD

1 2

1

4

1 2

3

19

The case d = 2


Y0(F ) =M2(1− p)

M2p − 1− E1 − E2 + E3 − E4,

where

E1 := 2M(M − 1)∞∑n=1

rnDEV0(C 1(n) ∩ C 2(n))

E2 := 2(M − 1)2∞∑n=1

rnDEV0(C 1(n) ∩ C 4(n))

E3 := 4(M − 1)2∞∑n=1

rnDEV0

3⋂j=1

C j(n)

E4 := (M − 1)2∞∑n=1

rnD

1 2

1

4

1 2

3

19

The case d = 2In R2 (and for k = 0), the general formula reduces to

Y0(F ) =M2(1− p)

M2p − 1− E1 − E2 + E3 − E4,

where

E1 := 2M(M − 1)∞∑n=1

rnDEV0(C 1(n) ∩ C 2(n))

E2 := 2(M − 1)2∞∑n=1

rnDEV0(C 1(n) ∩ C 4(n))

E3 := 4(M − 1)2∞∑n=1

rnDEV0

3⋂j=1

C j(n)

E4 := (M − 1)2

∞∑n=1

rnDEV0

4⋂j=1

C j(n)

1 2

1

4

1 2

3

19

The case d = 2In R2 (and for k = 0), the general formula reduces to

Y0(F ) =M2(1− p)

M2p − 1− E1 − E2 + E3 − E4,

where

E1 := 2M(M − 1)∞∑n=1

rnDEV0(C 1(n) ∩ C 2(n))

E2 := 2(M − 1)2∞∑n=1

rnD(1− pn)2

E3 := 4(M − 1)2∞∑n=1

rnDEV0

3⋂j=1

C j(n)

E4 := (M − 1)2

∞∑n=1

rnDEV0

4⋂j=1

C j(n)

1 2

1

4

1 2

3

19

The case d = 2


Y0(F ) =M2(1− p)

M2p − 1− E1 − E2 + E3 − E4,

where

E1 := 2M(M − 1)∞∑n=1

rnDEV0(C 1(n) ∩ C 2(n))

E2 := 2(M − 1)2∞∑n=1

rnD(1− pn)2

E3 := 4(M − 1)2∞∑n=1

rnD(1− pn)3

E4 := (M − 1)2∞∑n=1

rnDEV0

4⋂j=1

C j(n)

1 2

1

4

1 2

3

19

The case d = 2


Y0(F ) =M2(1− p)

M2p − 1− E1 − E2 + E3 − E4,

where

E1 := 2M(M − 1)∞∑n=1

rnDEV0(C 1(n) ∩ C 2(n))

E2 := 2(M − 1)2∞∑n=1

rnD(1− pn)2

E3 := 4(M − 1)2∞∑n=1

rnD(1− pn)3

E4 := (M − 1)2∞∑n=1

rnD(1− pn)4

1 2

1

4

1 2

3

19



I For n ∈ N, let Ki (n), i = 1, 2 denote the n-th steps of theirconstruction and set Di (n) := Ki (n)c .

I Then, for Ki (n) (the random set, which equals Ki (n) withprobability p and is empty otherwise), i = 1, 2, we had

F 1(n) ∩ F 2(n) = ψ(K1(n − 1) ∩ K2(n − 1)),


I This implies

F 1(n) F 2(n)

20



I For n ∈ N, let Ki (n), i = 1, 2 denote the n-th steps of theirconstruction and set Di (n) := Ki (n)c .

I Then, for Ki (n) (the random set, which equals Ki (n) withprobability p and is empty otherwise), i = 1, 2, we had

F 1(n) ∩ F 2(n) = ψ(K1(n − 1) ∩ K2(n − 1)),


I This implies

C 1(n) ∩ C 2(n) = ψ(D1(n − 1) ∩ D2(n − 1))

where Di (n) is the random set which equals Di (n) with probabilityp and I with probability 1− p.

20

Expected Euler characteristic (complements)

Y0(F ) =M2(1− p)

M2p − 1− E1

+ (M − 1)2(

1

M2p − 1− 4p

M2 − 1+

4p

M2 − p− p3

M2 − p3

),

E1 =2M(M − 1)

M − p

(2(1− p)

M − 1+

2(M − 1)p

M2 − p− p(1− p2)

M − p2

)− 2M(M − 1)2p3

(M − p2)(M2 − p3)+

2M(M − 1)

M2p − 1− 8M

M + 1+

4M(M − 1)p

M2 − p.

21


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

2× 2

pc

p

Z0

Y0

22


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

3× 3

pc

p

Z0

Y0

22


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

3× 3

χ

p

n = 4

6

8

22

A lower bound for pc?

I Does Y0(Fp,M) > 0 imply p < pc?

Y0(Fp,M) := limn→∞

rDnV0(Fp,M(n))

I Recall that limM→∞

pc(M) = psite , where psite ≈ 0.5927...

0

0.5

1

0 0.2 0.4 0.6 0.8 1

psite,8 psite

Z0

p

10

100

M

What is the red limit curve? limM→∞−Y0(Fp,M) = 1− 2p + p3

23




rDnV0(Fp,M(n))



0

0.5

1

0 0.2 0.4 0.6 0.8 1

psite,8 psite

Z0

p

10

100

M

What is the red limit curve?

limM→∞−Y0(Fp,M) = 1− 2p + p3

23




rDnV0(Fp,M(n))



0

0.5

1

0 0.2 0.4 0.6 0.8 1

psite,8 psite

Z0

p

10

100

M

What is the red limit curve? limM→∞−Y0(Fp,M) = 1− 2p + p3

23

Final remarks and open questions

I Is there a rigorous connection between of expected Eulercharacteristics and pc?

I What is the ‘right’ approximation of F?(F (n))n? (C (n))n? Parallel sets? ...?

I similar relations for other scale invariant models (Booleanmultiscale models, Brownian loop soup, ...) [Broman,Camia 10]

0

0.5

1

0 0.2 0.4 0.6 0.8 1

psite,8 psite

Z0

p

10

100

M

24

Final remarks and open questions

I Is there a rigorous connection between of expected Eulercharacteristics and pc?

I What is the ‘right’ approximation of F?(F (n))n? (C (n))n? Parallel sets? ...?

I similar relations for other scale invariant models (Booleanmultiscale models, Brownian loop soup, ...) [Broman,Camia 10]

0

0.5

1

0 0.2 0.4 0.6 0.8 1

psite,8 psite

Z0

p

10

100

M24

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