Supercritical causal maps: geodesics and simple random walk Thomas Budzinski ENS Paris et Université Paris Saclay Journée Cartes, Orsay 11 Avril 2018 Thomas Budzinski Supercritical causal maps
Supercritical causal maps Let t be an infinite plane tree. We define the causal map C ( t ) and the causal slice S ( t ) associated to t as follows : ρ ρ ρ C ( t ) S ( t ) t Thomas Budzinski Supercritical causal maps
Supercritical causal maps Let t be an infinite plane tree. We define the causal map C ( t ) and the causal slice S ( t ) associated to t as follows : ρ ρ ρ C ( t ) S ( t ) t Goal : study C = C ( T ) , where T is a supercritical Galton–Watson tree conditionned to survive. Thomas Budzinski Supercritical causal maps
Motivations As in the critical case, closely related models have been considered by theoretical physicists [Ambjørn, Loll...]. Thomas Budzinski Supercritical causal maps
Motivations As in the critical case, closely related models have been considered by theoretical physicists [Ambjørn, Loll...]. A toy-model and an "extremal" case of maps containing a Galton–Watson tree : some of our results can be generalized to more general maps containing a supercritical GW tree (ex : PSHIT), applications to the UIPT in the critical case [Curien, Ménard]. Thomas Budzinski Supercritical causal maps
Motivations As in the critical case, closely related models have been considered by theoretical physicists [Ambjørn, Loll...]. A toy-model and an "extremal" case of maps containing a Galton–Watson tree : some of our results can be generalized to more general maps containing a supercritical GW tree (ex : PSHIT), applications to the UIPT in the critical case [Curien, Ménard]. Better understanding of the properties of supercritical GW trees : when is the tree structure necessary ? Thomas Budzinski Supercritical causal maps
A nice picture Thomas Budzinski Supercritical causal maps
Hyperbolicity What does it mean for a graph to be hyperbolic ? Thomas Budzinski Supercritical causal maps
Hyperbolicity What does it mean for a graph to be hyperbolic ? Metric properties : exponential growth, Gromov-hyperbolicity, existence of "a lot" of infinite geodesics, bi-infinite geodesics... Thomas Budzinski Supercritical causal maps
Hyperbolicity What does it mean for a graph to be hyperbolic ? Metric properties : exponential growth, Gromov-hyperbolicity, existence of "a lot" of infinite geodesics, bi-infinite geodesics... Isoperimetric inequalities : nonamenability, anchored expansion... Thomas Budzinski Supercritical causal maps
Hyperbolicity What does it mean for a graph to be hyperbolic ? Metric properties : exponential growth, Gromov-hyperbolicity, existence of "a lot" of infinite geodesics, bi-infinite geodesics... Isoperimetric inequalities : nonamenability, anchored expansion... Simple random walk : transience, non-Liouville, positive speed, quick heat kernel decay... Thomas Budzinski Supercritical causal maps
Hyperbolicity What does it mean for a graph to be hyperbolic ? Metric properties : exponential growth, Gromov-hyperbolicity, existence of "a lot" of infinite geodesics, bi-infinite geodesics... Isoperimetric inequalities : nonamenability, anchored expansion... Simple random walk : transience, non-Liouville, positive speed, quick heat kernel decay... Other random processes : p c < p u for percolation, uniform spanning forest... Thomas Budzinski Supercritical causal maps
Hyperbolicity What does it mean for a graph to be hyperbolic ? Metric properties : exponential growth, Gromov-hyperbolicity, existence of "a lot" of infinite geodesics, bi-infinite geodesics... Isoperimetric inequalities : nonamenability, anchored expansion... Simple random walk : transience, non-Liouville, positive speed, quick heat kernel decay... Other random processes : p c < p u for percolation, uniform spanning forest... Contrast with the critical case : most of these properties are common to the supercritical GW tree T , and the map C ( T ) . Thomas Budzinski Supercritical causal maps
Hyperbolicity What does it mean for a graph to be hyperbolic ? Metric properties : exponential growth , Gromov-hyperbolicity, existence of "a lot" of infinite geodesics, bi-infinite geodesics... Isoperimetric inequalities : nonamenability, anchored expansion... Simple random walk : transience , non-Liouville, positive speed, quick heat kernel decay... Other random processes : p c < p u for percolation, uniform spanning forest... Contrast with the critical case : most of these properties are common to the supercritical GW tree T , and the map C ( T ) . For C ( T ) , some of these properties are easy , Thomas Budzinski Supercritical causal maps
Hyperbolicity What does it mean for a graph to be hyperbolic ? Metric properties : exponential growth , Gromov-hyperbolicity , existence of "a lot" of infinite geodesics, bi-infinite geodesics ... Isoperimetric inequalities : nonamenability, anchored expansion... Simple random walk : transience , non-Liouville, positive speed , quick heat kernel decay... Other random processes : p c < p u for percolation, uniform spanning forest... Contrast with the critical case : most of these properties are common to the supercritical GW tree T , and the map C ( T ) . For C ( T ) , some of these properties are easy , some others are the goal of this talk . Thomas Budzinski Supercritical causal maps
Setting We fix a supercritical distribution µ on N , i.e. � i ≥ 0 i µ ( i ) > 1. Let T be a Galton–Watson tree with offspring distribution µ , conditionned to be infinite. Let ρ be its root. If G is a graph, we let d G be the graph distance on G . The height h ( v ) of a vertex v is its distance to the root in T , and also in C . If x ∈ C has infinitely many descendants, let S [ x ] be the map formed by the descendants of x . It has the same distribution as S ( T ) . Thomas Budzinski Supercritical causal maps
"Usual" Gromov-hyperbolicity Definition We say that a graph G is Gromov-hyperbolic if there is a constant k ≥ 0 such that for every vertices x , y and z of G and every geodesics γ xy , γ yz and γ zx from x to y , y to z and z to x , we have ∀ v ∈ γ xy , d G ( v , γ yz ∪ γ zx ) ≤ k . z ≤ k y x v Thomas Budzinski Supercritical causal maps
"Usual" Gromov-hyperbolicity Definition We say that a graph G is Gromov-hyperbolic if there is a constant k ≥ 0 such that for every vertices x , y and z of G and every geodesics γ xy , γ yz and γ zx from x to y , y to z and z to x , we have ∀ v ∈ γ xy , d G ( v , γ yz ∪ γ zx ) ≤ k . z Problem : if e.g. µ ( 1 ) > 0, then C contains arbitrarily large portions of the square lattice, ≤ k which is not hyperbolic. y We need an "anchored" version ! x v Thomas Budzinski Supercritical causal maps
Weak anchored hyperbolicity Definition We say that a planar map M is weakly anchored hyperbolic if there is a constant k ≥ 0 such that for every vertices x , y and z of M and every geodesics γ xy , γ yz and γ zx from x to y , y to z and z to x such that the triangle they form surrounds ρ , we have d M ( ρ, γ xy ∪ γ yz ∪ γ zx ) ≤ k . z ≤ k ρ y x Thomas Budzinski Supercritical causal maps
Bi-infinite geodesics Definition A bi-infinite geodesic in a graph G is a family of vertices ( γ ( i )) i ∈ Z such that for every i , j ∈ Z , d G ( γ ( i ) , γ ( j )) = | i − j | . Thomas Budzinski Supercritical causal maps
Bi-infinite geodesics Definition A bi-infinite geodesic in a graph G is a family of vertices ( γ ( i )) i ∈ Z such that for every i , j ∈ Z , d G ( γ ( i ) , γ ( j )) = | i − j | . Such geodesics exist in Z d , but are expected to disappear after perturbations (first-passage percolation, UIPT). FPP on hyperbolic graphs admits bi-infinite geodesics [Benjamini–Tessera, 2016]. Thomas Budzinski Supercritical causal maps
Bi-infinite geodesics Definition A bi-infinite geodesic in a graph G is a family of vertices ( γ ( i )) i ∈ Z such that for every i , j ∈ Z , d G ( γ ( i ) , γ ( j )) = | i − j | . Such geodesics exist in Z d , but are expected to disappear after perturbations (first-passage percolation, UIPT). FPP on hyperbolic graphs admits bi-infinite geodesics [Benjamini–Tessera, 2016]. Theorem (B., 18) Almost surely, the map C is weakly anchored hyperbolic and admits bi-infinite geodesics. Thomas Budzinski Supercritical causal maps
Our main tool Let γ ℓ (resp. γ r ) be its left (resp. right) boundaries of S = S ( T ) . Proposition There is a (random) K ≥ 0 such that any geodesic in S from a vertex of γ ℓ to a vertex on γ r contains a vertex of height at most K . Proof : Let γ be a geodesic in S from γ ℓ ( i ) to γ r ( j ) , and let h be the minimal height on γ . The path γ ℓ ( i ) → ρ → γ r ( j ) has length i + j , so | γ | ≤ i + j . Every step of γ is either horizontal or vertical. Number of vertical steps ≥ ( i − h ) + ( j − h ) = i + j − 2 h . Thomas Budzinski Supercritical causal maps
Our main tool (proof) γ ℓ γ r Z h = 5 h ρ Let Z h be the number of vertices at height h with infinitely many descendants. Thomas Budzinski Supercritical causal maps
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