Supercritical causal maps: geodesics and simple random walk Thomas - - PowerPoint PPT Presentation

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 : t ρ C(t) ρ S(t) ρ

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 : t ρ C(t) ρ S(t) ρ 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 : pc < pu 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 : pc < pu 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 : pc < pu 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 : pc < pu 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 iµ(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 dG 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 γxy, γyz and γzx from x to y, y to z and z to x, we have ∀v ∈ γxy, dG (v, γyz ∪ γzx) ≤ k. x y z v ≤ k

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 γxy, γyz and γzx from x to y, y to z and z to x, we have ∀v ∈ γxy, dG (v, γyz ∪ γzx) ≤ k. x y z v ≤ k Problem : if e.g. µ(1) > 0, then C contains arbitrarily large portions of the square lattice, which is not hyperbolic. We need an "anchored" version !

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 γxy, γyz and γzx from x to y, y to z and z to x such that the triangle they form surrounds ρ, we have dM (ρ, γxy ∪ γyz ∪ γzx) ≤ k. x y z ρ ≤ k

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, dG (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, dG (γ(i), γ(j)) = |i − j|. Such geodesics exist in Zd, 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, dG (γ(i), γ(j)) = |i − j|. Such geodesics exist in Zd, 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 γℓ (resp. γr) 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 γℓ to a vertex on γr contains a vertex of height at most K. Proof : Let γ be a geodesic in S from γℓ(i) to γr(j), and let h be the minimal height on γ. The path γℓ(i) → ρ → γr(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 − 2h.

Thomas Budzinski Supercritical causal maps

Our main tool (proof)

ρ Zh = 5 γℓ γr h Let Zh be the number of vertices at height h with infinitely many descendants.

Thomas Budzinski Supercritical causal maps

Our main tool (proof)

ρ Zh = 5 γℓ γr h Let Zh be the number of vertices at height h with infinitely many descendants. γ stays at height ≥ h, so it must cross Zh slices, which requires at least Zh − 1 horizontal steps.

Thomas Budzinski Supercritical causal maps

Our main tool (proof)

ρ Zh = 5 γℓ γr h Let Zh be the number of vertices at height h with infinitely many descendants. γ stays at height ≥ h, so it must cross Zh slices, which requires at least Zh − 1 horizontal steps. We obtain i + j ≥ |γ| ≥ i + j − 2h + Zh − 1, so Zh ≤ 2h + 1, which is only true for finitely many h by exponential growth.

Thomas Budzinski Supercritical causal maps

Proof of weak anchored hyperbolicity

ρ a1 a2 a3 a4 S[a1] S[a2] S[a3] S[a4] Let (ai)1≤i≤4 be four vertices with infinitely many descendants, neither of which is an ancestor of another.

Thomas Budzinski Supercritical causal maps

Proof of weak anchored hyperbolicity

ρ a1 a2 a3 a4 S[a1] S[a2] S[a3] S[a4] x y z Let (ai)1≤i≤4 be four vertices with infinitely many descendants, neither of which is an ancestor of another. If x, y, z form a geodesic triangle, assume none of them is in S[a1].

Thomas Budzinski Supercritical causal maps

Proof of weak anchored hyperbolicity

ρ a1 a2 a3 a4 S[a1] S[a2] S[a3] S[a4] x y z v Let (ai)1≤i≤4 be four vertices with infinitely many descendants, neither of which is an ancestor of another. If x, y, z form a geodesic triangle, assume none of them is in S[a1]. Then the geodesic from x to y must cross S[a1], so it contains a vertex v with d(ρ, v) ≤ d(ρ, a1) + K1.

Thomas Budzinski Supercritical causal maps

Existence of bi-infinite geodesics

ρ a1 a2 S[a1] S[a2] γℓ γr Let ai,j = i + j − dC (γℓ(i), γr(j)). By the triangular inequality, (ai,j) is non-decreasing in i and j.

Thomas Budzinski Supercritical causal maps

Existence of bi-infinite geodesics

ρ a1 a2 S[a1] S[a2] γℓ γr γℓ(i) γr(j) Let ai,j = i + j − dC (γℓ(i), γr(j)). By the triangular inequality, (ai,j) is non-decreasing in i and j. A geodesic from γℓ(i) to γr(j) must cross S[a1] or S[a2], so it visits a bounded height, so the ai,j are bounded.

Thomas Budzinski Supercritical causal maps

Existence of bi-infinite geodesics

ρ a1 a2 S[a1] S[a2] γℓ γr γℓ(i0) γr(j0) γ Let ai,j = i + j − dC (γℓ(i), γr(j)). By the triangular inequality, (ai,j) is non-decreasing in i and j. A geodesic from γℓ(i) to γr(j) must cross S[a1] or S[a2], so it visits a bounded height, so the ai,j are bounded. Take (i0, j0) such that ai0,j0 = max{ai,j|i, j ≥ 0}, and concatenate a geodesic from γℓ(i0) to γr(j0) with γℓ and γr.

Thomas Budzinski Supercritical causal maps

Robustness

Theorem (B., 18) Any planar map containing (an injective embedding of) a supercritical Galton–Watson tree conditionned to survive is weakly anchored hyperbolic and admits bi-infinite geodesics. The conclusion from the Proposition is almost the same as in the particular case of C.

Thomas Budzinski Supercritical causal maps

Robustness

Theorem (B., 18) Any planar map containing (an injective embedding of) a supercritical Galton–Watson tree conditionned to survive is weakly anchored hyperbolic and admits bi-infinite geodesics. The conclusion from the Proposition is almost the same as in the particular case of C. To prove the Proposition, two obstacles :

some steps may be both "horizontal" and "vertical", the branches γr and γℓ are no longer geodesics of the tree.

The proof of the Proposition relies on the fact that a supercritical Galton–Watson process survives even if a reasonable number of individuals are killed at each generation.

Thomas Budzinski Supercritical causal maps

Robustness

Theorem (B., 18) Any planar map containing (an injective embedding of) a supercritical Galton–Watson tree conditionned to survive is weakly anchored hyperbolic and admits bi-infinite geodesics. The conclusion from the Proposition is almost the same as in the particular case of C. To prove the Proposition, two obstacles :

some steps may be both "horizontal" and "vertical", the branches γr and γℓ are no longer geodesics of the tree.

The proof of the Proposition relies on the fact that a supercritical Galton–Watson process survives even if a reasonable number of individuals are killed at each generation. This general setting includes the PSHIT, which are hyperbolic variants of the UIPT [B., 18].

Thomas Budzinski Supercritical causal maps

Positive speed for the simple random walk

Let (Xn) be the simple random walk on C started from ρ. Theorem (B., 18) Assume µ(0) = 0. Then there is a constant v > 0 such that dC (ρ, Xn) n

a.s.

− − − − →

n→+∞ v.

Thomas Budzinski Supercritical causal maps

Positive speed for the simple random walk

Let (Xn) be the simple random walk on C started from ρ. Theorem (B., 18) Assume µ(0) = 0. Then there is a constant v > 0 such that dC (ρ, Xn) n

a.s.

− − − − →

n→+∞ v.

Proved in 1995 for Galton–Watson trees, by finding a stationary environment [Lyons, Pemantle, Peres]. All the similar proofs make heavy use of the tree structure. Two main tools in our proof :

an exploration method of C guarantees that we do not discover "very bad" points, a regeneration argument gives the positive speed.

Thomas Budzinski Supercritical causal maps

Half-plane model

To gain stationarity, we work in the following half-plane model H, where (Ti)i∈Z are i.i.d. Galton–Watson trees : T−2 T−1 T0 T1 T2 . . . . . . ρ We will prove positive speed away from the boundary on H. To pass from H to C, show that the SRW on H stays in the same tree eventually.

Thomas Budzinski Supercritical causal maps

Bad vertices

The drift at a vertex with i children is i−1

i+3 ≥ 0, so we need

(Xn) to spend a lot of time at vertices with ≥ 2 children. We define a k-bad vertex as follows : k k k

Thomas Budzinski Supercritical causal maps

Bad vertices

The drift at a vertex with i children is i−1

i+3 ≥ 0, so we need

(Xn) to spend a lot of time at vertices with ≥ 2 children. We define a k-bad vertex as follows : k k k Exemple : P (ρ is k-bad) = µ(1)k(2k+1) ≈ e−k2. Lemma There is a constant c such that almost surely, for n large enough, none of the vertices X0, X1, . . . , Xn is c

• log n-bad.

Thomas Budzinski Supercritical causal maps

Proof of the "bad vertex lemma"

We "explore" H along the walk Xn. At time n, we discover Xn and all its ancestors.

Thomas Budzinski Supercritical causal maps

Proof of the "bad vertex lemma"

We "explore" H along the walk Xn. At time n, we discover Xn and all its ancestors.

Thomas Budzinski Supercritical causal maps

Proof of the "bad vertex lemma"

v We "explore" H along the walk Xn. At time n, we discover Xn and all its ancestors. Let k = c

• log n. When we discover a new vertex v, if its k

neighbours on the left (or on the right) are undiscovered, then P (v is k-bad) ≤ µ(1)k2.

Thomas Budzinski Supercritical causal maps

Proof of the "bad vertex lemma"

v We "explore" H along the walk Xn. At time n, we discover Xn and all its ancestors. Let k = c

• log n. When we discover a new vertex v, if its k

neighbours on the left (or on the right) are undiscovered, then P (v is k-bad) ≤ µ(1)k2.

Thomas Budzinski Supercritical causal maps

Proof of the "bad vertex lemma"

v We "explore" H along the walk Xn. At time n, we discover Xn and all its ancestors. Let k = c

• log n. When we discover a new vertex v, if its k

neighbours on the left (or on the right) are undiscovered, then P (v is k-bad) ≤ µ(1)k2. To ensure this is the case, everytime a "narrow pit" of width ≤ 2k is created, we discover its interior completely.

Thomas Budzinski Supercritical causal maps

Proof of the "bad vertex lemma"

v We "explore" H along the walk Xn. At time n, we discover Xn and all its ancestors. Let k = c

• log n. When we discover a new vertex v, if its k

neighbours on the left (or on the right) are undiscovered, then P (v is k-bad) ≤ µ(1)k2. To ensure this is the case, everytime a "narrow pit" of width ≤ 2k is created, we discover its interior completely.

Thomas Budzinski Supercritical causal maps

Quasi-positive speed

A vertex is good if it has ≥ 2 children. For every 0 ≤ i ≤ n, the vertex Xi is at distance at most c

• log n from a good vertex.

Hence, the probability to reach a good vertex from Xi in c

• log n steps is at least

4−c√

log n = n−o(1).

Hence, the number of good vertices visited between time 0 and n is n1−o(1), so the drift accumulated is n1−o(1). We obtain h(Xn) = n1−o(1) almost surely. More careful computations : h(Xn) is "almost increasing", with explicit, subpolynomial tail bounds.

Thomas Budzinski Supercritical causal maps

Regeneration times

We say that n > 0 is a regeneration time if ∀k < n, h(Xk) < h(Xn) and ∀k ≥ n, h(Xk) ≥ h(Xn). We list these times as 0 < τ1 < τ2 < . . . . X0

Thomas Budzinski Supercritical causal maps

Regeneration times

We say that n > 0 is a regeneration time if ∀k < n, h(Xk) < h(Xn) and ∀k ≥ n, h(Xk) ≥ h(Xn). We list these times as 0 < τ1 < τ2 < . . . . X0

Thomas Budzinski Supercritical causal maps

Regeneration times

We say that n > 0 is a regeneration time if ∀k < n, h(Xk) < h(Xn) and ∀k ≥ n, h(Xk) ≥ h(Xn). We list these times as 0 < τ1 < τ2 < . . . . X0

Thomas Budzinski Supercritical causal maps

Regeneration times

We say that n > 0 is a regeneration time if ∀k < n, h(Xk) < h(Xn) and ∀k ≥ n, h(Xk) ≥ h(Xn). We list these times as 0 < τ1 < τ2 < . . . . X0

Thomas Budzinski Supercritical causal maps

Regeneration times

We know that h(Xn) → +∞ a.s., so every new height has a positive probability to be a regeneration time, so all the τi are finite. The blocs between τi and τi+1 are i.i.d. ! In particular, the variables (τi+1 − τi) and

• h(Xτi+1) − h(Xτi)
• are i.i.d..

By the law of large numbers, we obtain Xn n → E [h(Xτ2) − h(Xτ1)] E [τ2 − τ1] , so it is enough to prove E [τ2 − τ1] < +∞. By the "quasi-positive speed" estimates, "fresh" times occur

• ften and, when they do, we know quickly if they are

regeneration times. We obtain that τ2 − τ1 has subpolynomial tail.

Thomas Budzinski Supercritical causal maps

Robustness and other questions

Highly non-robust proof : relies heavily on the fact that the local drift is nonnegative at every vertex. With different ideas, results about the Poisson boundary :

even if µ(0) > 0, description of the Poisson boundary, if T is filled with i.i.d. slices, the map we obtain is non-Liouville.

Heat kernel decay ? Our proof of positive speed gives P (Xn = ρ) = o(n−β) for every β, we expect exp(−n1/3). λ-biased random walk ? No regime where the positive drift is "too strong" ?

Thomas Budzinski Supercritical causal maps

THANK YOU !

Thomas Budzinski Supercritical causal maps