Supercritical causal maps: geodesics and simple random walk
Thomas Budzinski
ENS Paris et Université Paris Saclay
Journée Cartes, Orsay 11 Avril 2018
Supercritical causal maps
Lett be an infinite plane tree. We define thecausal mapC(t)and thecausal slice S(t) associated tot as follows :
t ρ
C(t) ρ
S(t) ρ
Goal : studyC=C(T), where T is asupercritical Galton–Watson tree conditionned to survive.
Supercritical causal maps
Lett be an infinite plane tree. We define thecausal mapC(t)and thecausal slice S(t) associated tot as follows :
t ρ
C(t) ρ
S(t) ρ
Goal : studyC=C(T), where T is asupercritical Galton–Watson tree conditionned to survive.
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 ?
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 ?
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 ?
A nice picture
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.
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.
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.
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.
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.
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.
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.
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.
Setting
We fix a supercritical distributionµ onN, i.e. P
i≥0iµ(i)>1.
Let T be a Galton–Watson tree with offspring distributionµ, conditionned to be infinite. Let ρ be its root.
If G is a graph, we let dG be the graph distance onG. Theheight h(v) of a vertex v is its distance to the root in T, and also inC.
If x ∈ C has infinitely many descendants, letS[x]be the map formed by the descendants of x. It has the same distribution as S(T).
"Usual" Gromov-hyperbolicity
Definition
We say that a graphG isGromov-hyperbolicif there is a constant k≥0 such that for every verticesx,y andz ofG and every geodesicsγxy,γyz andγzx fromx toy,y toz andz 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 !
"Usual" Gromov-hyperbolicity
Definition
We say that a graphG isGromov-hyperbolicif there is a constant k≥0 such that for every verticesx,y andz ofG and every geodesicsγxy,γyz andγzx fromx toy,y toz andz 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 !
Weak anchored hyperbolicity
Definition
We say that a planar mapM is weakly anchored hyperbolicif there is a constantk ≥0 such that for every verticesx,y andz ofM and every geodesicsγxy,γyz andγzx fromx toy,y to z andz to x such that the triangle they form surrounds ρ, we have
dM(ρ, γxy∪γyz ∪γzx)≤k.
x y
z
ρ
≤k
Bi-infinite geodesics
Definition
Abi-infinite geodesicin a graph G is a family of vertices(γ(i))i∈
such that for everyi,j ∈Z, Z
dG (γ(i), γ(j)) =|i−j|.
Such geodesics exist inZd, 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 mapC is weakly anchored hyperbolic and admits bi-infinite geodesics.
Bi-infinite geodesics
Definition
Abi-infinite geodesicin a graph G is a family of vertices(γ(i))i∈
such that for everyi,j ∈Z, Z
dG (γ(i), γ(j)) =|i−j|.
Such geodesics exist inZd, 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 mapC is weakly anchored hyperbolic and admits bi-infinite geodesics.
Bi-infinite geodesics
Definition
Abi-infinite geodesicin a graph G is a family of vertices(γ(i))i∈
such that for everyi,j ∈Z, Z
dG (γ(i), γ(j)) =|i−j|.
Such geodesics exist inZd, 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 mapC is weakly anchored hyperbolic and admits
Our main tool
Letγ` (resp.γr) be its left (resp. right) boundaries ofS =S(T).
Proposition
There is a (random)K ≥0 such that any geodesic in S from a vertex ofγ`to a vertex onγr contains a vertex of height at mostK. Proof :
Let γ be a geodesic in S fromγ`(i)to γr(j), and let h be the minimal height on γ.
The path γ`(i)→ρ→γr(j) has lengthi+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.
Our main tool (proof)
ρ
Zh=5
γ` γr
h
Let Zh be the number of vertices at heighth with infinitely many descendants.
γ stays at height ≥h, so it must crossZh slices, which requires at leastZh−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.
Our main tool (proof)
ρ
Zh=5
γ` γr
h
Let Zh be the number of vertices at heighth with infinitely many descendants.
γ stays at height ≥h, so it must crossZh slices, which requires at leastZh−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.
Our main tool (proof)
ρ
Zh=5
γ` γr
h
Let Zh be the number of vertices at heighth with infinitely many descendants.
γ stays at height ≥h, so it must crossZh slices, which requires at leastZh−1 horizontal steps.
Proof of weak anchored hyperbolicity
a1 ρ
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 toy must crossS[a1], so it contains a vertex v with d(ρ,v)≤d(ρ,a1) +K1.
Proof of weak anchored hyperbolicity
a1 ρ
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 toy must crossS[a1], so it contains a vertex v with d(ρ,v)≤d(ρ,a1) +K1.
Proof of weak anchored hyperbolicity
a1 ρ
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 toy must crossS[a1], so it contains
Existence of bi-infinite geodesics
ρ a1 a2
S[a1] S[a2]
γ` γr
γ`(i) γr(j)
γ`(i0) γr(j0)
γ
Let ai,j =i+j−dC(γ`(i), γr(j)). By the triangular inequality, (ai,j) is non-decreasing in i andj.
A geodesic from γ`(i) toγr(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 γ`(i0) toγr(j0) withγ` andγr.
Existence of bi-infinite geodesics
ρ a1 a2
S[a1] S[a2]
γ` γr
γ`(i) γr(j)
γ`(i0) γr(j0)
γ
Let ai,j =i+j−dC(γ`(i), γr(j)). By the triangular inequality, (ai,j) is non-decreasing in i andj.
A geodesic from γ`(i) toγr(j) must crossS[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 γ`(i0) toγr(j0) withγ` andγr.
Existence of bi-infinite geodesics
ρ a1 a2
S[a1] S[a2]
γ` γr
γ`(i) γr(j)
γ`(i0) γr(j0)
γ
Let ai,j =i+j−dC(γ`(i), γr(j)). By the triangular inequality, (ai,j) is non-decreasing in i andj.
A geodesic from γ`(i) toγr(j) must crossS[a1]or S[a2], so it visits a bounded height, so the ai,j are bounded.
Take (i ,j ) such that a =max{a |i,j ≥0}, and
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 ofC.
To prove the Proposition, two obstacles :
some steps may be both "horizontal" and "vertical", the branchesγr andγ` 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].
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 ofC.
To prove the Proposition, two obstacles :
some steps may be both "horizontal" and "vertical", the branchesγr andγ` 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].
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 ofC.
To prove the Proposition, two obstacles :
some steps may be both "horizontal" and "vertical", the branchesγr andγ` 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
Positive speed for the simple random walk
Let(Xn)be the simple random walk onC started from ρ.
Theorem (B., 18)
Assumeµ(0) =0. Then there is a constantv >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 ofCguarantees that we do not discover
"very bad" points,
a regeneration argument gives the positive speed.
Positive speed for the simple random walk
Let(Xn)be the simple random walk onC started from ρ.
Theorem (B., 18)
Assumeµ(0) =0. Then there is a constantv >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 ofCguarantees that we do not discover
"very bad" points,
a regeneration argument gives the positive speed.
Half-plane model
To gain stationarity, we work in the following half-plane modelH, where(Ti)i∈Z are i.i.d. Galton–Watson trees :
T−2 T−1T0 T1 T2
. . . .
ρ
We will prove positive speed away from the boundary on H.
To pass fromH toC, show that the SRW onHstays in the same tree eventually.
Bad vertices
The drift at a vertex withi children is i−1i+3 ≥0, so we need (Xn) to spend a lot of time at vertices with≥2 children.
We define ak-bad vertex as follows :
k
k k
Exemple :P(ρ is k-bad) =µ(1)k(2k+1)≈e−k2. Lemma
There is a constantc such that almost surely, for n large enough, none of the verticesX0,X1, . . . ,Xn iscp
logn-bad.
Bad vertices
The drift at a vertex withi children is i−1i+3 ≥0, so we need (Xn) to spend a lot of time at vertices with≥2 children.
We define ak-bad vertex as follows :
k
k k
Exemple :P(ρ is k-bad) =µ(1)k(2k+1)≈e−k2. Lemma
Proof of the "bad vertex lemma"
v v v
We "explore" Halong the walkXn. At timen, we discover Xn
and all its ancestors.
Let k =cp
logn. When we discover a new vertexv, if itsk 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.
Proof of the "bad vertex lemma"
v v v
We "explore" Halong the walkXn. At timen, we discover Xn
and all its ancestors.
Let k =cp
logn. When we discover a new vertexv, if itsk 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.
Proof of the "bad vertex lemma"
v
v v
We "explore" Halong the walkXn. At timen, we discover Xn
and all its ancestors.
Let k=cp
logn. When we discover a new vertexv, if itsk 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.
Proof of the "bad vertex lemma"
v
v
v
We "explore" Halong the walkXn. At timen, we discover Xn
and all its ancestors.
Let k=cp
logn. When we discover a new vertexv, if itsk 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.
Proof of the "bad vertex lemma"
v v
v
We "explore" Halong the walkXn. At timen, we discover Xn
and all its ancestors.
Let k=cp
logn. When we discover a new vertexv, if itsk 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.
Proof of the "bad vertex lemma"
v v
v
We "explore" Halong the walkXn. At timen, we discover Xn
and all its ancestors.
Let k=cp
logn. When we discover a new vertexv, if itsk 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.
Quasi-positive speed
A vertex is goodif it has≥2 children.
For every 0≤i ≤n, the vertexXi is at distance at most cp
logn from a good vertex.
Hence, the probability to reach a good vertex from Xi in cp
logn steps is at least 4−c
√
logn=n−o(1).
Hence, the number of good vertices visited between time 0 andn is n1−o(1), so the drift accumulated isn1−o(1). We obtainh(Xn) =n1−o(1) almost surely.
More careful computations : h(Xn) is "almost increasing", with explicit, subpolynomial tail bounds.
Regeneration times
We say thatn >0 is aregeneration time if
∀k <n,h(Xk)<h(Xn) and∀k ≥n,h(Xk)≥h(Xn).
We list these times as 0< τ1 < τ2 < . . ..
X0
Regeneration times
We say thatn >0 is aregeneration time if
∀k <n,h(Xk)<h(Xn) and∀k ≥n,h(Xk)≥h(Xn).
We list these times as 0< τ1 < τ2 < . . ..
X0
X0
Regeneration times
We say thatn >0 is aregeneration time if
∀k <n,h(Xk)<h(Xn) and∀k ≥n,h(Xk)≥h(Xn).
We list these times as 0< τ1 < τ2 < . . ..
X0
Regeneration times
We say thatn >0 is aregeneration time if
∀k <n,h(Xk)<h(Xn) and∀k ≥n,h(Xk)≥h(Xn).
We list these times as 0< τ1 < τ2 < . . ..
X0
X0
Regeneration times
We know thath(Xn)→+∞a.s., so every new height has a positive probability to be a regeneration time, so all the τi are finite.
The blocs betweenτi andτi+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 often and, when they do, we know quickly if they are
regeneration times. We obtain thatτ2−τ1 has subpolynomial
Robustness and other questions
Highly non-robust proof : relies heavily on the fact that the local drift is nonnegative at everyvertex.
With different ideas, results about the Poisson boundary : even ifµ(0)>0, description of the Poisson boundary, ifT 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" ?