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Ergodicity of STIT tessellations
Raphael Lachieze-Rey
Laboratoire Paul Painleve (Universite Lille 1)
January 13th, 2011
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 1 / 33
1 Random tessellations
2 STIT tessellations
3 Direct construction of the model
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 2 / 33
1 Random tessellations
2 STIT tessellations
3 Direct construction of the model
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 3 / 33
Tessellations
Definition
Tessellation: Set C = {C1,C2, ...} of convex compact cells, locally finite(for all compact K , {i ; Ci ∩ K 6= ∅} is finite), such that
Rd = ∪iCi ,
int(Ci ) ∩ int(Cj) = ∅, i 6= j .
The corresponding closed set is M = ∪i∂Ci .
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 4 / 33
Some examples of real structures – Potential applications.
Craquelee on a ceramic (Photo: G. Weil)
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 4 / 33
cracking simulation (H.-J. Vogel)
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 4 / 33
Rat muscle tissue (I. Erzen)
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 4 / 33
Perla 2.jpgGranit joints (D. Nikolayev, S. Siegesmund, S. Mosch, A. Hoffmann)
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 4 / 33
Gris Perla.jpg
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 4 / 33
Poisson tessellations
Union of random lines.
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 5 / 33
Voronoi and Delaunay tessellation
Π: Point process on Rd .
x ∈ Π,Vx : Set of points of Rd for which x is the closest element ofΠ,
Vx = {y ∈ R2; ‖x − y‖ = infx ′∈Π‖x ′ − y‖}
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 6 / 33
3D tessellations
3D Poisson hyperplanes tesselation:
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 7 / 33
Mixed tessellations
( Schmidt, Voss 2010)
Grey tessellation: Low-level servers.
Black spots: High-level servers.
Black tessellation (High-level network): Voronoi tessellationcorresponding to black points.
Left: Low-level servers= Voronoi tessellation.Right: Low-level servers=Poisson line tessellation.
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 8 / 33
1 Random tessellations
2 STIT tessellations
3 Direct construction of the model
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 9 / 33
Parameters of the construction
Let H be the class of hyperplanes of Rd .
Intensity: a > 0.
Stationary measure ν on H, i.e. invariant under the action oftranslations, and locally finite.
W : Compact window of Rd .
[W ] = {H ∈ H : H ∩W 6= ∅}.
ν locally finite:ν([W ]) < +∞.
Renormalised restriction of ν to W :
νW (·) =1
ν([W ])ν([W ] ∩ ·).
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 10 / 33
Modelisation of cracking on compact window
Start from a bounded window W , and after a random exponential timewith rate ν([W ]), cut the window W by a random line drawn according toνW .
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 10 / 33
Modelisation of cracking on compact window
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Each sub-cell C created behaves independantly: it is divided after arandom time ∼ E(ν([C ])) by a random line drawn according to νC .
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 10 / 33
Modelisation of cracking on compact window
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Each sub-cell C created behaves independantly: it is divided after arandom time ∼ E(ν([C ])) by a random line drawn according to νC .
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 10 / 33
Modelisation of cracking on compact window
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Each sub-cell C created behaves independantly: it is divided after arandom time ∼ E(ν([C ])) by a random line drawn according to νC .
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 10 / 33
Modelisation of cracking on compact window
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Each sub-cell C created behaves independantly: it is divided after arandom time ∼ E(ν([C ])) by a random line drawn according to νC .
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 10 / 33
Modelisation of cracking on compact window
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Each sub-cell C created behaves independantly: it is divided after arandom time ∼ E(ν([C ])) by a random line drawn according to νC .
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 10 / 33
Modelisation of cracking on compact window
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Each sub-cell C created behaves independantly: it is divided after arandom time ∼ E(ν([C ])) by a random line drawn according to νC .
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 10 / 33
Modelisation of cracking on compact window
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Stop the process when time a is reached.
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 10 / 33
Nagel, Weiss, Mecke (Jena)
Model of cell division.
Time process without memory (birth and death process).
Call MW ,a,ν the obtained “tessellation”,under the form of the union of thecells boundaries.
MW ,a,ν =⋃
C cell existing at time a
∂C .
It is a random element with values in the class F(W ) of closed sets of W .F(W ) is endowed with the Fell topology, and the corresponding Borelσ-algebra B.
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 11 / 33
Examples(Simulations: J. Ohser)
Simulations of isotropic STIT tessellations. (ν is stationary and isotropic,i.e. invariant under the action of rotations).
In the isotropic case, the death rate of a cell ν([C ]) is proportionnal to itsperimeter.
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 12 / 33
Non-isotropic example
Examples where ν is stationary (but not isotropic).
Here, ν([C ]) is proportionnal to the perimeter of the smallest rectanglewith sides parallels to the axes containing C .
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 13 / 33
Binary tree
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 14 / 33
Tessellation on Rd .Consistency propertiesNagel and Weiss 2005:If W ⊆W ′, then
MW ,a,ν ∩ int(W )(d)= MW ′,a,ν ∩ int(W ).
Theorem
Let {Wi ; i ∈ N} be a family of compact windows such that
(i) Wi ↑ Rd ,
(ii) Wi ⊂ int(Wi+1).
If a family of random closed sets {FWi⊆Wi} satisfy
FWi∩ int(Wi )
(d)= FWj
∩ int(Wi ), j > i ,
then there exists a random closed set F of Rd such that
F ∩ int(Wi )(d)= FWi
, i ∈ N.Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 15 / 33
There exists a random tessellation Ma,ν ∈ F(Rd) such that
(Ma,ν ∩W ) ∪ ∂W(d)= Ma,ν,W
for all compact W . It is the STIT tessellation with parameters a and ν.We have
a = EHd−1(M ∩ [0, 1]d)
and a is the intensity.
There exists a direct construction of the tessellation with the help of apoint process on H× R+ (marked point process of hyperplanes).
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 16 / 33
Iteration
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 17 / 33
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 18 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 19 / 33
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 20 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 21 / 33
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 22 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 23 / 33
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 24 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 25 / 33
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 26 / 33
Rescaling
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 27 / 33
Rescaling
× 2⇒
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 28 / 33
Iteration
Let M,M ′ be two random tessellations.
C1,C2, . . . cells of M.
M ′1,M′2, . . . independent copies of M ′, independent of the Ci .
Define the iterate of M and M ′by
M � M ′ = 2 ∪i ∪Cj cell of M′i∂(Cj ∩ Ci ).
It is a definition in distribution.
The operation is not commutative.
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 29 / 33
STable under ITeration (Mecke, Nagel, Weiss)
Every STIT tessellation Ma,ν satisfies
Ma,ν � Ma,ν(d)= Ma,ν .
Furthermore, every random tessellation M that satisfies this property is aSTIT.
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 30 / 33
Attraction pool
Let M be a stationary tessellation. Define by induction{M1 = M,
Mn+1 = Mn � Mn.
ThenMn ⇒ Ma,ν ,
for a certain STIT tessellation Ma,ν .
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 31 / 33
Mixing properties
A stationary tessellation M is mixing if
P(M ∩ K = ∅,M ∩ (K ′ + h) = ∅)→‖h‖→∞ P(M ∩ K = ∅)P(M ∩ K ′ = ∅),
for all compacts K ,K ′.
Theorem (L.,2009)
Let M be a STIT tessellation. Then for all compacts K and K ′,
P(K ∩M = ∅, (K ′ + h) ∩M = ∅)− P(K ∩M = ∅)P((K ′ + h) ∩M = ∅)= O(1/‖h‖)
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 32 / 33
1 Random tessellations
2 STIT tessellations
3 Direct construction of the model
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
Poisson point processes,marked with birth times
space
time
-
6 R× (0,∞)
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
Poisson point processes,marked with birth times
space
time
-
6
r
r
r
r r
r
rr r
rR× (0,∞)
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
2. Poisson point processes,marked with birth times
space
time
-
6
r
r
r
r r
r
rr r
rR× (0,∞)
t
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
Poisson point processes,marked with birth times
space
time
-
6
r
r
r
r r
r
rr r
rR× (0,∞)
t
r r r r r? ? ? ? ?
Ψt
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
Now replace R byH ... the set of all lines in R2
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
Poisson line processes Γt ,marked with birth times
space
time
-
6
r
r
r
r r
r
rr r
rH× (0,∞)
t
r r r r r? ? ? ? ?
Γt
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
Poisson point process Γ on H× (0,∞)with intensity measure ν × `+
ν translation invariant on H`+ ... Lebesgue measure on (0,∞)For all t > 0
Γt = {h : (h, s) ∈ Γ : s < t}
is a spatially homogeneous Poisson line process in R2.
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
A preliminary constructionHow to start a division of the whole plane when all segments have to havea finite length?And all nodes are of T -type.
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The crucial idea (by Joseph Mecke):Consider the process (Zt)t>0 of the o-cells of (Γt)t>0.
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
For all t > 0 the Poisson line process Γt is a.s. not empty.Assume that the directional distribution R of the lines is not concentratedin a single point.Then Γt generates a tessellation with a compact convex polygon Zt thata.s. contains the origin o in its interior.=⇒ Stochastic process (Zt)t>0 of o-cells of (Γt)t>0.
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
The isotonyΓt1 ⊆ Γt2 for t1 < t2
impliesZt1 ⊇ Zt2 for t1 < t2.
The process (Zt)t>0 is piecewise constant.Z . . . the set of all integers.Monotonic sequence (σk)k∈Z of times where (Zt)t>0 changes its state.σk is the time when the interior int Zσk−1
is hit by a line from Γ.
. . . < σ−2 < σ−1 < σ0 < σ1 < 1 < σ2 < . . .
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
We obtainlim
k→−∞σk = 0 and lim
k→∞σk =∞
Crucial for the construction
Zσk↑ R2 a.s. if k ↓ −∞
AlsoZσk↓ {o} a.s. if k ↑ ∞
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
A preliminary tessellation of R2
For t > 0 we define a tessellation Ψt with the cells
Zt and Zσk−1\ Zσk
, σk < t.
All these cells are compact, convex and have a pairwise disjoint interior.Due to
Zσk↑ R2 a.s. if k ↓ −∞
the cells fill the plane, i.e. for all t > 0
Zt ∪⋃σk<t
Zσk−1\ Zσk
= R2
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
The random tessellation Ψt is non-homogeneous(spatially non-stationary).Intuitively, the older cells of Ψt ,Zσk−1
\ Zσkwith σk close to the time 0
(themoment of the ’Big Bang’)are very far from the origin o ∈ R2
and they tend to be larger than the younger ones.
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
The final steps of the construction –generating a spatially homogeneousrandom tessellation
For t > 0: non-homogeneous tessellation Ψt with the cells
Zt and Zσk−1\ Zσk
, σk < t.
A cellZσk−1
\ Zσk
is born at the time σk < t.During the time interval (σk , t) this bounded cell is divided by randomchords as described in the beginning.
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
Restricted to the cell Zσk−1\ Zσk
and starting at time σk
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
Restricted to the cell Zσk−1\ Zσk
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
Restricted to the cell Zσk−1\ Zσk
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
Restricted to the cell Zσk−1\ Zσk
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
Restricted to the cell Zσk−1\ Zσk
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Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
Restricted to the cell Zσk−1\ Zσk
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... and finishing at time t.
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
Thus the cells Zσk−1\ Zσk
are filled during the time interval (σk , t)such that the resulting tessellation Φt
is spatially homogeneous,
STIT, i.e. stable under iteration/nesting of tessellations.
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33
Mecke, J., Nagel, W., Weiß, V. (2008). A global construction ofhomogeneous random planar tessellations that are stable under iteration.Stochastics: An Int. J. of Prob. and Stoch. Proc. 80, 51-67.
Mecke, J., Nagel, W., Weiß, V. (2008). The iteration of randomtessellations and a construction of a homogeneous process of cell divisions.Advances in Applied Probability 40, 49-59.
Nagel, W., Weiß, V. (2005). Crack STIT tessellations – characterization ofthe stationary random tessellations which are stable with respect toiteration.Advances in Applied Probability 37, 859-883.
Lachieze-Rey R., Mixing properties of STIT tessellations.Advances in Applied Probability43.1 (march 2011)
Raphael Lachieze-Rey () Ergodicity of STIT tessellations January 13th, 2011 33 / 33