Isoperimetry in the infinite cluster
Julian Gold
UCLA
May 9th
Frontier Probability Days 2016
Outline
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Percolation
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Isoperimetric problems in the discrete setting
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A conjecture of Benjamini
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Attack via “shape theorem” in d = 2
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The Wulff construction in lattice models
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d ≥3
Percolation: setup
Bernoulli (bond) percolation:
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Zd with standard graph structure, d ≥ 2
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p ∈ [0, 1]
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(Xe )e∈E (Zd ) iid Bernoulli(p)
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e is open if Xe = 1 and is closed otherwise
Zd with only open edges:
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random subgraph of the original lattice
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connected components are open clusters
Percolation: the infinite cluster
Notation:
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C(0) is the open cluster containing 0 ∈ Zd
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θp (d) := Pp (|C(0)| = ∞)
d ≥ 2: There is a critical probability pc (d) ∈ (0, 1) so that
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p > pc (d) implies θp (d) > 0
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p < pc (d) implies θp (d) = 0
“supercritical”
p > pc (d): ∃! infinite (open) cluster Pp -a.s., denoted C∞ .
Goal: explore the geometry of C∞
Figure: Percolation on Z2 with p = .511
1
source: wikipedia.
The Cheeger constant: bottlenecks and robustness
Definition
Let G = (V , E ) be a finite, connected graph. The Cheeger
constant of G is
|∂H|
: H ⊂ G , 0 < |H| ≤ |G |/2
ΦG := min
|H|
where ∂H is the edge boundary of H in G .
The Cheeger constant of the giant component
Notation:
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Cn := C∞ ∩ [−n, n]d
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C̃n is the largest connected component of Cn
Thanks to the work of
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Benjamini-Mossel ’03
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Mathieu-Remy ’04
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Rau ’07
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Berger-Biskup-Hoffman-Kozma ’07
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Pete ’08
...it’s known that ΦC̃n n−1
A conjecture
Conjecture (Benjamini)
The limit
lim nΦC̃n
n→∞
exists Pp -almost surely.
Progress
Theorem (Procaccia, Rosenthal ’11)
Let d ≥ 2, and let p > pc (d).
var nΦC̃n ≤ n2−d
And in 2012...
...Benjamini’s conjecture was settled in d = 2 for a modification of
the Cheeger constant.
A natural modification of the Cheeger constant
Definition
In d = 2, the modified Cheeger constant of C̃n is
ω
|∂ H|
Φ̃n := min
: H ⊂ C̃n , 0 < |H| ≤ |C̃n |/2
|H|
where ∂ ω H is the open edge boundary of H in C∞ .
Figure: We treat subgraphs of C̃n as living in C∞ .
Asymptotics in d = 2 through a shape theorem
Gn :=
|∂ ω H|
H ⊂ C̃n :
= Φ̃n
|H|
Theorem (Biskup, Louidor, Procaccia, Rosenthal ’12)
Let d = 2, and let p > pc (2). There is Wp ⊂ [−1, 1]2 convex so
that Pp -a.s.,
max inf dH n−1 Gn , Wp + x −−−→ 0
Gn ∈Gn x∈R2
n→∞
Here dH is the `∞ -Hausdorff distance on compact sets.
Asymptotics in d = 2 through a shape theorem
Theorem (Biskup, Louidor, Procaccia, Rosenthal ’12)
Let d = 2, and let p > pc (d). There is a norm βp on R2 so that
Pp -a.s.,
lim nΦ̃n =
n→∞
lengthβp (∂Wp )
θp (2)L2 (Wp )
where Wp is the limit shape from the previous theorem.
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L2 is two-dimensional Lebesgue measure
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L2 (Wp ) = 22 /2
Scaling limit
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Wp is called the Wulff shape
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Wp is the optimizer of
minimize:
lengthβp (E )
L2 (E )
subject to: L2 (E ) ≤ 22 /2
Discrete, random isoperimetric problems scale to a continuous,
deterministic isoperimetric problem.
The Wulff construction
in isoperimetric problems
Given a norm τ on R2 , there is a standard way (due to Wulff) to
construct a solution to
minimize:
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lengthτ (E )
L2 (E )
subject to: L2 (E ) ≤ (const.)
Define:
cτ :=
W
\
{x ∈ R2 : n · x ≤ τ (n)}
n∈S 1
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cτ so that L2 (Wτ ) = (const.)
Let Wτ be a dilate of W
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Wτ is the solution, unique up to translations2
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This works in higher dimensions (see work of Taylor from the ’70s).
The Wulff construction
in isoperimetric problems
βp gives Wp
Q: How do we extract a norm from our lattice model?
A: There is an established theory (of analysis of equilibrium crystal
shapes), due to work of
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Dobrushin-Kotecký-Schlosman ’90
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Alexander-Chayes-Chayes ’90
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Ioffe-Schonmann ’98
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Cerf, Cerf-Pisztora ’98, ’00
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Bodineau ’99
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+ more
The Wulff construction
in lattice models
Setup: we have a material with two distinct “phases”
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open versus closed in percolation
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+ versus - in Ising
Goal: study the shape of a “droplet” of one phase immersed within
another
Postulate: “surface energy” between droplet phase and ambient
phase should be minimized
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surface energy gives rise to a norm by examining direction
dependence
The Wulff construction
in lattice models
In our context: Gn ∈ Gn is a droplet in C∞ \ Gn .
Q: What kind of surface energy is being minimized at the
boundary of this droplet?
The Wulff construction
in lattice models
In a given direction, the surface energy is the “cost” of forming an
interface between droplet and ambient region
At the microscopic scale, this minimal cost is a random variable
depending on the orientation and size of the box.
Extracting a surface energy
Consider a square Sn (v ) in R2 of side-length 2n, oriented so that
its “top” and “bottom” faces are orthogonal to v ∈ S 1 .
Xn (v ) := the (random) minimum size of a cutset
separating the top and bottom faces of Sn (v )
Subadditivity:
Ep Xn (v )
n→∞
n
βp (v ) := lim
This defines βp as a function on S 1 . Extend to a function on R2 .
βp is a norm
βp is well-studied (Kesten, Zhang, Cerf, Théret, Garet), and it is
known that βp is a norm on R2 .
I am lying:
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In [BLPR], βp is defined using paths, not cutsets
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Cutsets need to be “anchored” in order to use subadditivity
Upshot:
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This definition generalizes to d ≥ 3
Summary of d=2
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βp gives Wp through Wulff construction
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Must show that βp is the “correct” norm
One more modification
Definition
In d ≥ 3, the modified Cheeger constant of C̃n is
ω
|∂ H|
Φ̃n := min
: H ⊂ C̃n , 0 < |H| ≤ |C̃n |/d!
|H|
where ∂ ω H is the open edge boundary of H in C∞ .
d ≥3
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βp,d defined analogously
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Wp,d the associated Wulff shape
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Ld (Wp,d ) = 2d /d!
Theorem (G ’16)
Let d ≥ 3, and let p > pc (d). There is Wp,d ⊂ [−1, 1]d convex so
that Pp -a.s.,
max inf
Gn ∈Gn x∈Rd
||1Gn − 1n(Wp,d +x)∩Cn ||`1
nd
−−−→ 0
n→∞
d ≥3
The shape theorem gives Cheeger asymptotics.
Theorem (G ’16)
Let d ≥ 3, and let p > pc (d). There is a norm βp,d on Rd so that
Pp -a.s.,
lim nΦ̃n =
n→∞
perimeterβp,d (∂Wp,d )
θp (d)Ld (Wp,d )
where Wp,d is the limit shape from the previous theorem, and has
volume 2d /d!
Remarks
Figure: Why (a priori) we use `1 convergence
Summary
Challenges in d ≥ 3:
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finding a suitable norm
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we lose graph duality
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we lose theory of plane curves
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vacant percolation
Ingredients:
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concentration estimates
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(enhanced) renormalization argument of Zhang
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geometric measure theory
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+ more
Open questions
The Wulff shape:
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regularity?
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facets?
Boundary conditions:
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original conjecture?
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on a torus?
`∞ in d ≥ 3?
Figure: from Crystal Saga, Jean
Giraud 1989
Thank you!