The model and its background Main results Variations of the scheme The dual tree of a recursive triangulation of the disk Henning Sulzbach, INRIA Paris-Rocquencourt Journées Alea, Luminy, March 2014 joint work with Nicolas Broutin (INRIA)
The model and its background Main results Variations of the scheme Outline 1. The model and its background 2. Main results 3. Variations of the scheme
The model and its background Main results Variations of the scheme Outline 1. The model and its background 2. Main results 3. Variations of the scheme
The model and its background Main results Variations of the scheme Recursive laminations of the disk Curien and Le Gall 2011: In each step, connect two uniformly chosen points unless the chord intersects any previously inserted. 0
The model and its background Main results Variations of the scheme Recursive laminations of the disk Curien and Le Gall 2011: In each step, connect two uniformly chosen points unless the chord intersects any previously inserted. 0
The model and its background Main results Variations of the scheme Recursive laminations of the disk Curien and Le Gall 2011: In each step, connect two uniformly chosen points unless the chord intersects any previously inserted. 0
The model and its background Main results Variations of the scheme Recursive laminations of the disk Curien and Le Gall 2011: In each step, connect two uniformly chosen points unless the chord intersects any previously inserted. 0
The model and its background Main results Variations of the scheme Recursive laminations of the disk Curien and Le Gall 2011: In each step, connect two uniformly chosen points unless the chord intersects any previously inserted. X 0
The model and its background Main results Variations of the scheme Recursive laminations of the disk Curien and Le Gall 2011: In each step, connect two uniformly chosen points unless the chord intersects any previously inserted. 0
The model and its background Main results Variations of the scheme Recursive laminations of the disk Curien and Le Gall 2011: In each step, connect two uniformly chosen points unless the chord intersects any previously inserted. Number of inserted chords at time n is about √ π n . Lamination: L n = set of inserted chords at time n .
The model and its background Main results Variations of the scheme The limit triangulation Theorem (Curien, Le Gall) L ∞ := � n ≥ 1 L n is a triangulation, that is, its complement consists of triangles with vertices on the circle. Observe : Triangulations are maximal, that is, they cannot be increased by additional chords.
The model and its background Main results Variations of the scheme The dual tree C n ( s ) 0 s T n : dual tree, d gr : graph distance on T n . C n ( s ) = depth of node at s ∈ [ 0 , 1 ] in T n . Scaling limit of the dual tree T n ? Scaling limit of the contour process C n ( s ) ?
The model and its background Main results Variations of the scheme Trees encoded by excursions Let f : [ 0 , 1 ] → R + be a continuous excursion. [y] d (x,y) f [x] x y T f 0 1 T f = [ 0 , 1 ] / ∼ where s ∼ t with s ≤ t if d f ( s , t ) = 0 where d f ( s , t ) = f ( s ) + f ( t ) − 2 inf { f ( x ) : s ≤ x ≤ t } . ( T f , d f ) is a compact tree-like metric space (an R -tree).
The model and its background Main results Variations of the scheme Triangulations encoded by excursions Let f : [ 0 , 1 ] → R + be a continuous excursion with distinct local minima. L f contains chords connecting s ≤ t if and only if d f ( s , t ) = 0. 0 L f 0 1 Inner nodes of T f correspond to triangles in L f .
The model and its background Main results Variations of the scheme The Brownian world - Aldous ’94 Consider uniform triangulations of the n -gon P n : contour process (Dyck path) 0 4n-6
The model and its background Main results Variations of the scheme The Brownian world - Aldous ’94 Consider uniform triangulations of the n -gon P n : contour process (Dyck path) 0 4n-6 ↓ δ Haus ↓ d GH ↓ � · � ∞ 2.0 1.5 1.0 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 t
The model and its background Main results Variations of the scheme Outline 1. The model and its background 2. Main results 3. Variations of the scheme
The model and its background Main results Variations of the scheme The dual tree of the lamination C n ( s ) 0 s C n ( s ) = depth of node at s ∈ [ 0 , 1 ] in T n . Theorem (Broutin, S. ’14) There exists a random continuous process Z ( s ) , s ∈ [ 0 , 1 ] , such that, uniformly in s ∈ [ 0 , 1 ] , almost surely, √ C n ( s ) 17 − 3 n β/ 2 → Z ( s ) , β = = 0 . 561 . . . 2
The model and its background Main results Variations of the scheme The dual tree of the lamination Theorem (Broutin, S. ’14) There exists a random continuous process Z ( s ) , s ∈ [ 0 , 1 ] , such that, uniformly in s ∈ [ 0 , 1 ] , almost surely, √ C n ( s ) 17 − 3 n β/ 2 → Z ( s ) , β = = 0 . 561 . . . 2 Moreover, L ∞ = L Z (already proved by Curien and Le Gall). Almost surely, ( T n , n − β/ 2 d gr ) → ( T Z , d Z ) in the Gromov-Hausdorff topology on the space of (isometry classes of) compact metric spaces.
The model and its background Main results Variations of the scheme A simulation of the limit 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 E [ Z ( s )] ∼ ( s ( 1 − s )) β Optimal Hölder exponent: β = 0 . 561 . . . .
The model and its background Main results Variations of the scheme Recursive decomposition U Attempted insertions in (0) I n subfragments d I ( 0 ) = Bin ( n − 1 , ( 1 − ( V − U )) 2 ) 0 n (1) I n ∼ n ( 1 − ( V − U )) 2 d I ( 1 ) = Bin ( n − 1 , ( V − U ) 2 ) n ∼ n ( V − U ) 2 V
The model and its background Main results Variations of the scheme Recursive decomposition s U 0 V
The model and its background Main results Variations of the scheme Recursive decomposition s U 0 V � s � = 1 [ 0 , U ] ( s ) C ( 0 ) d C n ( s ) I ( n ) 1 − ( V − U ) 0
The model and its background Main results Variations of the scheme Recursive decomposition U 0 s V � s � � s − ( V − U ) � = 1 [ 0 , U ] ( s ) C ( 0 ) d + 1 ( V , 1 ] ( s ) C ( 0 ) C n ( s ) I ( n ) I ( n ) 1 − ( V − U ) 1 − ( V − U ) 0 0
The model and its background Main results Variations of the scheme Recursive decomposition U 0 s V � s � � s − ( V − U ) � = 1 [ 0 , U ] ( s ) C ( 0 ) d + 1 ( V , 1 ] ( s ) C ( 0 ) C n ( s ) I ( n ) I ( n ) 1 − ( V − U ) 1 − ( V − U ) 0 0
The model and its background Main results Variations of the scheme Recursive decomposition U 0 s V � s � � s − ( V − U ) � = 1 [ 0 , U ] ( s ) C ( 0 ) d + 1 ( V , 1 ] ( s ) C ( 0 ) C n ( s ) I ( n ) I ( n ) 1 − ( V − U ) 1 − ( V − U ) 0 0 � s − U � � U � �� 1 + C ( 0 ) + C ( 1 ) + 1 ( U , V ] ( s ) I ( n ) I ( n ) 1 − ( V − U ) V − U 0 1
The model and its background Main results Variations of the scheme Characterizing Z ( U , V ) min and max of two ind. uniforms, here U = 0 . 32 , V = 0 . 56 Z ( 0 ) Z ( 1 )
The model and its background Main results Variations of the scheme Characterizing Z ( U , V ) min and max of two ind. uniforms, here U = 0 . 32 , V = 0 . 56 Z ( 0 ) , ( 1 − ( V − U )) β Z ( 0 ) Z ( 1 ) , ( V − U ) β Z ( 1 )
The model and its background Main results Variations of the scheme Characterizing Z ( U , V ) min and max of two ind. uniforms, here U = 0 . 32 , V = 0 . 56 Z ( 0 ) , ( 1 − ( V − U )) β Z ( 0 ) Z ( 1 ) , ( V − U ) β Z ( 1 )
The model and its background Main results Variations of the scheme The fractal dimension Theorem (Broutin, S.) Almost surely, we have dim ( T Z ) = 1 β = 1 . 781 . . . both for Minkowski and Hausdorff dimension. Compare: dim ( T e ) = 2 for the CRT. Very roughly, dim ( T f ) = s means that, as r → 0, | B r ( x ) | ≈ r s with B r ( x ) = { y ∈ T f : d f ( x , y ) < r } .
The model and its background Main results Variations of the scheme Outline 1. The model and its background 2. Main results 3. Variations of the scheme
The model and its background Main results Variations of the scheme A homogeneous model In each step • choose one fragment uniformly at random • insert a chord uniformly at random Observe: I ( n ) is uniformly distributed (Polya urn!), hence 0 I ( n ) 0 → W , n → ∞ , n where W is uniform on [ 0 , 1 ] and independent of ( U , V ) .
The model and its background Main results Variations of the scheme A homogeneous model Theorem (Broutin, S. ’14) There exists a random continuous process H ( s ) , s ∈ [ 0 , 1 ] , such that, uniformly in s ∈ [ 0 , 1 ] , almost surely, C h n ( s ) n 1 / 3 → H ( s ) . Moreover, E [ H ( s )] ∼ ( s ( 1 − s )) 1 / 2 .
The model and its background Main results Variations of the scheme A simulation of H 1.0 0.5 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 √ Optimal Hölder exponent: 3 − 2 2 = 0 . 057 . . . 3
The model and its background Main results Variations of the scheme The characterization of H ( U , V ) : as before and W another independent uniform. H ( 0 ) H ( 1 )
Recommend
More recommend
