The hyperbolic Brownian plane Thomas Budzinski ENS Paris July 7th, 2016 Thomas Budzinski The hyperbolic Brownian plane
Planar maps Definitions A planar map is a locally finite, connected graph embedded in the plane in such a way that : no two edges cross, except at a common endpoint, every compact subset of the plane intersects finitely many vertices and edges, considered up to orientation-preserving homeomorphism. The faces of the map are the connected components of its complementary. The degree of a face is the number of half-edges adjacent to this face. Thomas Budzinski The hyperbolic Brownian plane
Planar maps Definitions A planar map is a locally finite, connected graph embedded in the plane in such a way that : no two edges cross, except at a common endpoint, every compact subset of the plane intersects finitely many vertices and edges, considered up to orientation-preserving homeomorphism. The faces of the map are the connected components of its complementary. The degree of a face is the number of half-edges adjacent to this face. = � = Thomas Budzinski The hyperbolic Brownian plane
Triangulations Definition A triangulation of the plane is an infinite planar map in which all the faces have degree 3. It may contain loops and multiple edges. A triangulation with a hole of perimeter p is a finite map in which all the faces have degree 3 except the external face, which has degree p . A rooted triangulation is a triangulation with a distinguished oriented edge. From now on, all the triangulations will be rooted. Examples : a rooted triangulation with a hole of perimeter 6. Thomas Budzinski The hyperbolic Brownian plane
t ⊂ T Definition If t is a triangulation of a p -gon and T a triangulation of the plane, we write t ⊂ T if T may be obtained by "filling" the hole of t with an infinite triangulation. ⊂ Thomas Budzinski The hyperbolic Brownian plane
The UIPT Theorem ( ≈ Angel-Schramm, 2003) There is a random triangulation of the plane T , called the UIPT (Uniform Infinite Planar Triangulation), such that for any triangulation t with a hole of perimeter p , we have = C p λ | t | � � P t ⊂ T c , 1 where | t | is the number of vertices of t and we have λ c = 3 and √ 12 √ 3 p ( 2 p )! p ! 2 3 p . C p = 2 Thomas Budzinski The hyperbolic Brownian plane
Picture by N. Curien. Thomas Budzinski The hyperbolic Brownian plane
Spatial Markov property Condition on t ⊂ T , and let e be an edge of ∂ t : e t
Spatial Markov property Condition on t ⊂ T , and let e be an edge of ∂ t : e f t Case I � � = C p + 1 λ | t | + 1 t + f ⊂ T P = C p + 1 � � Then P Case I = c C p λ c . � � C p λ | t | P t ⊂ T c
Spatial Markov property Condition on t ⊂ T , and let e be an edge of ∂ t : f f e f e e t t t Case II i (here i = 2) Case III i (here i = 3) Case I � � = C p + 1 λ | t | + 1 t + f ⊂ T P = C p + 1 � � Then P Case I = c C p λ c . � � C p λ | t | P t ⊂ T c � � � � P Case II i and P Case III i are also explicitely known, and depend only on p . Thomas Budzinski The hyperbolic Brownian plane
Peeling process and consequences Allows to discover T , almost "face by face", in a Markovian way. Very flexible : the choice of e may be adapted to the information we are looking for : growth in r 4 [Angel], critical probabilities for percolation [Angel, Angel-Curien, Richier], subdiffusivity of the random walk [Benjamini-Curien] Thomas Budzinski The hyperbolic Brownian plane
λ -Markovian triangulations Definition A random triangulation of the plane T is λ -Markovian if there are constants ( C p ) p ≥ 1 such that for any triangulation t with a hole of perimeter p we have = C p ( λ ) λ | t | . � � P t ⊂ T Thomas Budzinski The hyperbolic Brownian plane
λ -Markovian triangulations Definition A random triangulation of the plane T is λ -Markovian if there are constants ( C p ) p ≥ 1 such that for any triangulation t with a hole of perimeter p we have = C p ( λ ) λ | t | . � � P t ⊂ T Proposition (Curien 2014, B. 2016) If λ > λ c then there is no λ -Markovian triangulation. If 0 < λ ≤ λ c then there is a unique one (in distribution), that we write T λ . Besides we have � p − 1 p − 1 � 2 q � C p ( λ ) = 1 8 + 1 � � h q , λ h q q = 0 where h ∈ ( 0 , 1 h 4 ] is such that λ = ( 1 + 8 h ) 3 / 2 . Thomas Budzinski The hyperbolic Brownian plane
Hyperbolic behaviour Exponential volume growth [Curien] Anchored expansion : if A is a finite, connected set of vertices containing the root, then | ∂ A | ≥ c | A | [Curien]. The simple random walk has positive speed [Curien, Angel-Nachmias-Ray]. Thomas Budzinski The hyperbolic Brownian plane
Scaling limit of T A planar map can be seen as a (discrete) metric space, equipped with its graph distance and the counting measure on its vertices. The set of all (classes of) locally compact measured metric spaces can be equipped with the local Gromov-Hausdorff-Prokhorov distance. Thomas Budzinski The hyperbolic Brownian plane
Scaling limit of T A planar map can be seen as a (discrete) metric space, equipped with its graph distance and the counting measure on its vertices. The set of all (classes of) locally compact measured metric spaces can be equipped with the local Gromov-Hausdorff-Prokhorov distance. Theorem (Curien-Le Gall 14, B. 16) Let µ T be the counting measure on the set of vertices of T . We have the following convergence in distribution for the local Gromov-Hausdorff-Prokhorov distance : � 1 n T , 1 � ( d ) n 4 µ T a → + ∞ P − → where P is a random (pointed) measured metric space homeomorphic to the plane called the Brownian plane . Thomas Budzinski The hyperbolic Brownian plane
Scaling limit of T λ ? For λ < λ c fixed 1 n T λ cannot converge because T λ "grows too quickly". Thomas Budzinski The hyperbolic Brownian plane
Scaling limit of T λ ? For λ < λ c fixed 1 n T λ cannot converge because T λ "grows too quickly". We look for ( λ n ) → λ c such that 1 n T λ n converges. Thomas Budzinski The hyperbolic Brownian plane
Scaling limit of T λ ? For λ < λ c fixed 1 n T λ cannot converge because T λ "grows too quickly". We look for ( λ n ) → λ c such that 1 n T λ n converges. Theorem (B. 16) Let ( λ n ) n ≥ 0 be a sequence of numbers in ( 0 , λ c ] such that � 1 2 � � � λ n = λ c 1 − + o . 3 n 4 n 4 Then � 1 n T λ n , 1 � ( d ) n → + ∞ P h n 4 µ T λ n − → where P h is a random (pointed) measured metric space homeomorphic to the plane that we call the hyperbolic Brownian plane . Thomas Budzinski The hyperbolic Brownian plane
✶ Hull process of P For r ≥ 0 we write B r ( P ) for the hull of radius r of P , that is, the reunion of its ball of radius r and all the bounded connected components of its complementary. Thomas Budzinski The hyperbolic Brownian plane
✶ Hull process of P For r ≥ 0 we write B r ( P ) for the hull of radius r of P , that is, the reunion of its ball of radius r and all the bounded connected components of its complementary. Theorem (Curien-Le Gall 14) There is a natural notion of "perimeter" of B r ( P ) , that we � � write P r ( P ) , and P r ( P ) r ≥ 0 is a time-reversed stable branching process (in particular it is càdlàg with only negative jumps). Thomas Budzinski The hyperbolic Brownian plane
Hull process of P For r ≥ 0 we write B r ( P ) for the hull of radius r of P , that is, the reunion of its ball of radius r and all the bounded connected components of its complementary. Theorem (Curien-Le Gall 14) There is a natural notion of "perimeter" of B r ( P ) , that we � � write P r ( P ) , and P r ( P ) r ≥ 0 is a time-reversed stable branching process (in particular it is càdlàg with only negative jumps). If V r ( P ) is the volume of B r ( P ) , then � � ξ i | ∆ P r ( P ) | 2 � � � V r ( P ) r ≥ 0 = r ≥ 0 , t i ≤ r where ( t i ) is a measurable enumeration of the jumps of r ≥ 0 , and the ξ i are i.i.d. with density e − 1 / 2 x � � P r ( P ) 2 π x 5 ✶ x > 0 . √ Thomas Budzinski The hyperbolic Brownian plane
Description of P h Theorem For all r ≥ 0, the random variable B r ( P h ) has density � 1 e − 3 P 2 r ( P ) x 2 d x e − 2 V 2 r ( P ) e P 2 r ( P ) 0 with respect to B r ( P ) . Thomas Budzinski The hyperbolic Brownian plane
Sketch of proof We use the convergence of T to P and the absolute continuity relations between T and T λ : � λ � � B r ( T λ ) = t P � = C p ( λ ) � | t | � C p ( λ c ) λ c P B r ( T ) = t Thomas Budzinski The hyperbolic Brownian plane
Sketch of proof We use the convergence of T to P and the absolute continuity relations between T and T λ : � λ � � B r ( T λ ) = t P � = C p ( λ ) � | t | � C p ( λ c ) λ c P B r ( T ) = t Two main tools : precise asymptotics for the C p ( λ ) , a reinforcement of the convergence of T to P . Thomas Budzinski The hyperbolic Brownian plane
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