Frozen percolation on the triangular lattice
Demeter Kiss
Centrum Wiskunde & Informatica, Amsterdam, The Netherlands
9 August, 2013
Motivation
Polymerization model:
Motivation
Polymerization model:
I vertices ↔ atoms
Motivation
Polymerization model:
I vertices ↔ atoms
I clusters ↔ molecules
Motivation
Polymerization model:
I vertices ↔ atoms
I clusters ↔ molecules
I Dynamics: small molecules merge, while big ones do not
interact with other particles
The model:
N -parameter
I
frozen percolation
G = (V , E ) graph, N ∈ N
The model:
N -parameter
I
frozen percolation
G = (V , E ) graph, N ∈ N
The model:
N -parameter
I
frozen percolation
G = (V , E ) graph, N ∈ N
The model:
N -parameter
frozen percolation
I
G = (V , E ) graph, N ∈ N
I
∀v ∈ V sample τv ∼ Unif [0, 1]
independently from each other
The model:
N -parameter
frozen percolation
I
G = (V , E ) graph, N ∈ N
I
∀v ∈ V sample τv ∼ Unif [0, 1]
I
independently from each other
at t = 0 every vertex is closed (white)
The model:
N -parameter
frozen percolation
I
G = (V , E ) graph, N ∈ N
I
∀v ∈ V sample τv ∼ Unif [0, 1]
I
I
independently from each other
at t = 0 every vertex is closed (white)
at t = τv vertex v opens (red/blue) i the
diameter of the open clusters of the
neighbors is less than N
Example
N =1
I
G = (V , E ) graph, N ∈ N
I
∀v ∈ V sample τv ∼ Unif [0, 1]
I
I
independently from each other
at t = 0 every vertex is closed (white)
at t = τv vertex v opens (red/blue) i the
diameter of the open clusters of the
neighbors is less than N
Example
N =1
I
G = (V , E ) graph, N ∈ N
I
∀v ∈ V sample τv ∼ Unif [0, 1]
I
I
independently from each other
at t = 0 every vertex is closed (white)
at t = τv vertex v opens (red/blue) i the
diameter of the open clusters of the
neighbors is less than N
Example
N =1
I
G = (V , E ) graph, N ∈ N
I
∀v ∈ V sample τv ∼ Unif [0, 1]
I
I
independently from each other
at t = 0 every vertex is closed (white)
at t = τv vertex v opens (red/blue) i the
diameter of the open clusters of the
neighbors is less than N
Example
N =1
I
G = (V , E ) graph, N ∈ N
I
∀v ∈ V sample τv ∼ Unif [0, 1]
I
I
independently from each other
at t = 0 every vertex is closed (white)
at t = τv vertex v opens (red/blue) i the
diameter of the open clusters of the
neighbors is less than N
Example
N =1
I
G = (V , E ) graph, N ∈ N
I
∀v ∈ V sample τv ∼ Unif [0, 1]
I
I
independently from each other
at t = 0 every vertex is closed (white)
at t = τv vertex v opens (red/blue) i the
diameter of the open clusters of the
neighbors is less than N
Example
N =1
I
G = (V , E ) graph, N ∈ N
I
∀v ∈ V sample τv ∼ Unif [0, 1]
I
I
independently from each other
at t = 0 every vertex is closed (white)
at t = τv vertex v opens (red/blue) i the
diameter of the open clusters of the
neighbors is less than N
Example
N =1
I
G = (V , E ) graph, N ∈ N
I
∀v ∈ V sample τv ∼ Unif [0, 1]
I
I
independently from each other
at t = 0 every vertex is closed (white)
at t = τv vertex v opens (red/blue) i the
diameter of the open clusters of the
neighbors is less than N
−→ a cluster with diameter at least N
stops growing, i.e freezes
Example
N =1
I
G = (V , E ) graph, N ∈ N
I
∀v ∈ V sample τv ∼ Unif [0, 1]
I
I
independently from each other
at t = 0 every vertex is closed (white)
at t = τv vertex v opens (red/blue) i the
diameter of the open clusters of the
neighbors is less than N
−→ a cluster with diameter at least N
stops growing, i.e freezes
∞-parameter
frozen percolation
Our case: N large,
∞-parameter
frozen percolation
Our case: N large, G is the triangular lattice
∞-parameter
frozen percolation
Our case: N large, G is the triangular lattice
−→ Why not take N = ∞?
∞-parameter
frozen percolation
Our case: N large, G is the triangular lattice
−→ Why not take N = ∞? Does this make sense?
∞-parameter
frozen percolation
Our case: N large, G is the triangular lattice
−→ Why not take N = ∞? Does this make sense?
I
Aldous(1999): when G is the binary tree, it does.
∞-parameter
frozen percolation
Our case: N large, G is the triangular lattice
−→ Why not take N = ∞? Does this make sense?
I
I
Aldous(1999): when G is the binary tree, it does.
Benjamini and Schramm (1999): for the triangular lattice it
does not
∞-parameter
frozen percolation
Our case: N large, G is the triangular lattice
−→ Why not take N = ∞? Does this make sense?
I
I
Aldous(1999): when G is the binary tree, it does.
Benjamini and Schramm (1999): for the triangular lattice it
does not
−→ this is the reason why van den Berg, de Lima and Nolin
introduced the N -parameter models.
∞-parameter
frozen percolation
Our case: N large, G is the triangular lattice
−→ Why not take N = ∞? Does this make sense?
I
I
Aldous(1999): when G is the binary tree, it does.
Benjamini and Schramm (1999): for the triangular lattice it
does not
−→ this is the reason why van den Berg, de Lima and Nolin
introduced the N -parameter models.
Is there a limit N → ∞?
∞-parameter
frozen percolation
Our case: N large, G is the triangular lattice
−→ Why not take N = ∞? Does this make sense?
I
I
Aldous(1999): when G is the binary tree, it does.
Benjamini and Schramm (1999): for the triangular lattice it
does not
−→ this is the reason why van den Berg, de Lima and Nolin
introduced the N -parameter models.
Is there a limit N → ∞?
I Van den Berg, K., Nolin (2012): the N -parameter process on
the binary tree `converges' to the ∞-parameter process.
∞-parameter
frozen percolation
Our case: N large, G is the triangular lattice
−→ Why not take N = ∞? Does this make sense?
I
I
Aldous(1999): when G is the binary tree, it does.
Benjamini and Schramm (1999): for the triangular lattice it
does not
−→ this is the reason why van den Berg, de Lima and Nolin
introduced the N -parameter models.
Is there a limit N → ∞?
I Van den Berg, K., Nolin (2012): the N -parameter process on
the binary tree `converges' to the ∞-parameter process.
I What about the triangular lattice?
∞-parameter
frozen percolation
Our case: N large, G is the triangular lattice
−→ Why not take N = ∞? Does this make sense?
I
I
Aldous(1999): when G is the binary tree, it does.
Benjamini and Schramm (1999): for the triangular lattice it
does not
−→ this is the reason why van den Berg, de Lima and Nolin
introduced the N -parameter models.
Is there a limit N → ∞?
I Van den Berg, K., Nolin (2012): the N -parameter process on
the binary tree `converges' to the ∞-parameter process.
I What about the triangular lattice?
−→ simulation
Percolation process
Forget about freezing: v is open at time p with probability p, and
closed with probability 1 − p.
Percolation process
Forget about freezing: v is open at time p with probability p, and
closed with probability 1 − p.
p = 0.3
Percolation process
Forget about freezing: v is open at time p with probability p, and
closed with probability 1 − p.
p = 0.3
p = 0.7
Phase transition
With pc = 1/2 we have
Phase transition
With pc = 1/2 we have
I for p ≤ pc all open clusters are nite,
Phase transition
With pc = 1/2 we have
I for p ≤ pc all open clusters are nite,
I for p > pc there is a unique innite open cluster.
Phase transition
With pc = 1/2 we have
I for p ≤ pc all open clusters are nite,
I for p > pc there is a unique innite open cluster.
Corollary
For all
ε > 0, v ∈ V
PN (v
freezes before time
1/2 − ε) → 0 as N → ∞.
Phase transition
With pc = 1/2 we have
I for p ≤ pc all open clusters are nite,
I for p > pc there is a unique innite open cluster.
Corollary
For all
ε > 0, v ∈ V
PN (v
freezes before time
1/2 − ε) → 0 as N → ∞.
Theorem (K. 2013)
For all
ε > 0, v ∈ V
PN (v
freezes after
1/2 + ε) → 0 as N → ∞.
Correlation length
p = 0.3
L (p) is the `typical' size of
the big open clusters
Correlation length
p = 0.3
L (p) is the `typical' size of
the big open clusters
p = 0.7
L (p) `typical' size of
the big closed clusters
Near critical scaling
Set p such that L (p) N :
Near critical scaling
Set p such that L (p) N :
Take λ ∈ R and set
pλ (N) =
with r (N) ≈ N −3/4 .
1
+ λr (N)
2
Near critical scaling
Set p such that L (p) N :
Take λ ∈ R and set
pλ (N) =
1
+ λr (N)
2
with r (N) ≈ N −3/4 . Then
L (pλ (N)) |λ|−4/3 N.
Main result - strong version
Theorem (K. 2013)
For all
k >0
and
ε>0
there is
λ∈R
such that
lim sup PN (a cluster freezes in B (kN)
N→∞
after time
pλ (N)) < ε.
Main result - strong version
Theorem (K. 2013)
For all
k >0
and
ε>0
there is
λ∈R
such that
lim sup PN (a cluster freezes in B (kN)
after time
pλ (N)) < ε.
N→∞
Conjecture
Scale the space by
the
N -parameter
N
and the time close to
1/2 appropriately.
frozen percolation processes converge to a
(continuum) limiting process.
Then
Main result - strong version
Theorem (K. 2013)
For all
k >0
and
ε>0
there is
λ∈R
such that
lim sup PN (a cluster freezes in B (kN)
after time
pλ (N)) < ε.
N→∞
Conjecture
Scale the space by
the
N -parameter
N
and the time close to
1/2 appropriately.
frozen percolation processes converge to a
(continuum) limiting process.
Moreover, the limiting process completely describes the frozen
clusters.
Then
Thank you for your attention!