Cardy embedding of random planar maps
Nina Holden
ETH Zürich, Institute for Theoretical Studies

Collaboration with Xin Sun. Based on our joint works with Bernardi,
Garban, Gwynne, Lawler, Li, and Sepúlveda.
August 1, 2019

Holden (ETH Zürich)

August 1, 20191 / 20Two random surfaces

random planar map (RPM)

Holden (ETH Zürich)

Liouville quantum gravity (LQG)

August 1, 20192 / 20Planar maps
A planar map M is a finite connected graph drawn in the sphere,
viewed up to continuous deformations.

Holden (ETH Zürich)

August 1, 20193 / 20Planar maps
A planar map M is a finite connected graph drawn in the sphere,
viewed up to continuous deformations.

=
Holden (ETH Zürich)

August 1, 20193 / 20Planar maps
A planar map M is a finite connected graph drawn in the sphere,
viewed up to continuous deformations.

6=
Holden (ETH Zürich)

August 1, 20193 / 20Planar maps
A planar map M is a finite connected graph drawn in the sphere,
viewed up to continuous deformations.
A triangulation is a planar map where all faces have three edges.

6=
Holden (ETH Zürich)

August 1, 20193 / 20Planar maps
A planar map M is a finite connected graph drawn in the sphere,
viewed up to continuous deformations.
A triangulation is a planar map where all faces have three edges.
Given n ∈ N let M be a uniformly chosen triangulation with nvertices.

6=
Holden (ETH Zürich)

August 1, 20193 / 20Planar maps
A planar map M is a finite connected graph drawn in the sphere,
viewed up to continuous deformations.
A triangulation is a planar map where all faces have three edges.
Given n ∈ N let M be a uniformly chosen triangulation with nvertices.
Enumeration results by Tutte and Mullin in 60’s.

6=
Holden (ETH Zürich)

August 1, 20193 / 20

The Gaussian free field (GFF)
Hamiltonian H(f ) quantifies how much f deviates from being harmonicH(f ) =

1X
(f (x) − f (y ))2 ,
2 x∼y

f :

1 2
Z ∩ [0, 1]2 → R.
n

1
1
n

1
Holden (ETH Zürich)

August 1, 20194 / 20

The Gaussian free field (GFF)
Hamiltonian H(f ) quantifies how much f deviates from being harmonic
1X
1
H(f ) =(f (x) − f (y ))2 ,f : Z2 ∩ [0, 1]2 → R.
2 x∼yn
Discrete Gaussian free field (GFF) hn : n1 Z2 ∩ [0, 1]2 → R is a random function
with hn |∂[0,1]2 = 0 and probability density rel. to Lebesgue measure proportional toexp(−H(hn )).

n = 20, n = 100

Holden (ETH Zürich)

August 1, 20194 / 20

The Gaussian free field (GFF)
Hamiltonian H(f ) quantifies how much f deviates from being harmonic
1X
1
H(f ) =(f (x) − f (y ))2 ,f : Z2 ∩ [0, 1]2 → R.
2 x∼yn
Discrete Gaussian free field (GFF) hn : n1 Z2 ∩ [0, 1]2 → R is a random function
with hn |∂[0,1]2 = 0 and probability density rel. to Lebesgue measure proportional toexp(−H(hn )).
hn (z) ∼ N (0, π2 log n + O(1)) and Cov(hn (z), hn (w )) = − π2 log |z − w | + O(1).

n = 20, n = 100

Holden (ETH Zürich)

August 1, 20194 / 20

The Gaussian free field (GFF)
Hamiltonian H(f ) quantifies how much f deviates from being harmonic
1X
1
H(f ) =(f (x) − f (y ))2 ,f : Z2 ∩ [0, 1]2 → R.
2 x∼yn
Discrete Gaussian free field (GFF) hn : n1 Z2 ∩ [0, 1]2 → R is a random function
with hn |∂[0,1]2 = 0 and probability density rel. to Lebesgue measure proportional toexp(−H(hn )).
hn (z) ∼ N (0, π2 log n + O(1)) and Cov(hn (z), hn (w )) = − π2 log |z − w | + O(1).
The Gaussian free field h is the limit of hn when n → ∞.

n = 20, n = 100
Holden (ETH Zürich)

August 1, 20194 / 20

The Gaussian free field (GFF)
Hamiltonian H(f ) quantifies how much f deviates from being harmonic
1X
1
H(f ) =(f (x) − f (y ))2 ,f : Z2 ∩ [0, 1]2 → R.
2 x∼yn
Discrete Gaussian free field (GFF) hn : n1 Z2 ∩ [0, 1]2 → R is a random function
with hn |∂[0,1]2 = 0 and probability density rel. to Lebesgue measure proportional toexp(−H(hn )).
hn (z) ∼ N (0, π2 log n + O(1)) and Cov(hn (z), hn (w )) = − π2 log |z − w | + O(1).
The Gaussian free field h is the limit of hn when n → ∞.
The GFF is a random distribution (i.e., random generalized function).

n = 20, n = 100
Holden (ETH Zürich)

August 1, 20194 / 20Liouville quantum gravity (LQG)
If h : [0, 1]2 → R smooth and γ ∈ (0, 2), then e γh (dx 2 + dy 2 ) defines the metrictensor of a Riemannian manifold.

Holden (ETH Zürich)

August 1, 20195 / 20Liouville quantum gravity (LQG)
If h : [0, 1]2 → R smooth and γ ∈ (0, 2), then e γh (dx 2 + dy 2 ) defines the metrictensor of a Riemannian manifold.
γ-Liouville quantum gravity (LQG): h is the Gaussian free field.

Holden (ETH Zürich)

August 1, 20195 / 20Liouville quantum gravity (LQG)
If h : [0, 1]2 → R smooth and γ ∈ (0, 2), then e γh (dx 2 + dy 2 ) defines the metrictensor of a Riemannian manifold.
γ-Liouville quantum gravity (LQG): h is the Gaussian free field.
The definition does not make literal sense, since h is not a function.

discrete
GFF

Holden (ETH Zürich)

August 1, 20195 / 20Liouville quantum gravity (LQG)
If h : [0, 1]2 → R smooth and γ ∈ (0, 2), then e γh (dx 2 + dy 2 ) defines the metrictensor of a Riemannian manifold.
γ-Liouville quantum gravity (LQG): h is the Gaussian free field.
The definition does not make literal sense, since h is not a function.
Area measure e γh d 2 z and metric defined via regularized versions h of h:Z
2µ(U) = lim γ /2e γh (z) d 2 z, U ⊂ [0, 1]2 ,→0UZd(z, w ) = lim c infe γh (z)/d dz, z, w ∈ [0, 1]2(2019).
→0

P:z→w

P

discrete
GFF

Holden (ETH Zürich)

LQG
area

August 1, 20195 / 20Liouville quantum gravity (LQG)
If h : [0, 1]2 → R smooth and γ ∈ (0, 2), then e γh (dx 2 + dy 2 ) defines the metrictensor of a Riemannian manifold.
γ-Liouville quantum gravity (LQG): h is the Gaussian free field.
The definition does not make literal sense, since h is not a function.
Area measure e γh d 2 z and metric defined via regularized versions h of h:Z
2µ(U) = lim γ /2e γh (z) d 2 z, U ⊂ [0, 1]2 ,→0UZd(z, w ) = lim c infe γh (z)/d dz, z, w ∈ [0, 1]2(2019).
→0

P:z→w

P

discrete
GFF

Holden (ETH Zürich)

LQG
area

August 1, 20195 / 20Liouville quantum gravity (LQG)
If h : [0, 1]2 → R smooth and γ ∈ (0, 2), then e γh (dx 2 + dy 2 ) defines the metrictensor of a Riemannian manifold.
γ-Liouville quantum gravity (LQG): h is the Gaussian free field.
The definition does not make literal sense, since h is not a function.
Area measure e γh d 2 z and metric defined via regularized versions h of h:Z
2µ(U) = lim γ /2e γh (z) d 2 z, U ⊂ [0, 1]2 ,→0UZd(z, w ) = lim c infe γh (z)/d dz, z, w ∈ [0, 1]2(2019).
→0

P:z→w

P

The area measure is non-atomic and any open set has positive mass a.s., but themeasure is a.s. singular with respect to Lebesgue measure.

discrete
GFF

Holden (ETH Zürich)

LQG
area

August 1, 20195 / 20Random planar maps converge to LQG
Two models for random surfaces:
Random planar maps (RPM)
Liouville quantum gravity (LQG).

Holden (ETH Zürich)

August 1, 20196 / 20Random planar maps converge to LQG
Two models for random surfaces:
Random planar maps (RPM)
Liouville quantum gravity (LQG).
Conjectural relationship used by physicists to predict/calculate thedimension of random fractals and exponents of statistical physics models.

1nv

P[4 arms from v ] ≈ n−5/4
Holden (ETH Zürich)

August 1, 20196 / 20Random planar maps converge to LQG
Two models for random surfaces:
Random planar maps (RPM)
Liouville quantum gravity (LQG).
Conjectural relationship used by physicists to predict/calculate thedimension of random fractals and exponents of statistical physics models.
.
What does it mean for a RPM to converge?
Metric structure (Le Gall’13, Miermont’13)
Conformal structure (H.-Sun’19)
Statistical physics observables (Duplantier-Miller-Sheffield’14, ...)

Holden (ETH Zürich)

August 1, 20196 / 20Conformally embedded RPM converge top

8/3-LQG

AT
Cardy
embedding φ

random planar map (RPM) Mn

scaling limit

I⇒

B

C
embedded random planar map

p

8/3-LQG h√
Uniform triangulation Mn with n vertices, boundary length d ne.

Holden (ETH Zürich)

August 1, 20197 / 20Conformally embedded RPM converge top

8/3-LQG

AT
Cardy
embedding φ

random planar map (RPM) Mn

scaling limit

I⇒

B

C
embedded random planar map

p

8/3-LQG h√
Uniform triangulation Mn with n vertices, boundary length d ne.
Cardy embedding: uses properties of percolation on the RPM.

Holden (ETH Zürich)

August 1, 20197 / 20Conformally embedded RPM converge top

8/3-LQG

AT
Cardy
embedding φ

random planar map (RPM) Mn

scaling limit

I⇒

B

C
embedded random planar map

p

8/3-LQG h√
Uniform triangulation Mn with n vertices, boundary length d ne.
Cardy embedding: uses properties of percolation on the RPM.
Let µn be renormalized counting measure on the vertices in T.

Holden (ETH Zürich)

August 1, 20197 / 20Conformally embedded RPM converge top

8/3-LQG

AT
Cardy
embedding φ

random planar map (RPM) Mn

scaling limit

I⇒

B

C
embedded random planar map

p

8/3-LQG h√
Uniform triangulation Mn with n vertices, boundary length d ne.
Cardy embedding: uses properties of percolation on the RPM.
Let µn be renormalized counting measure on the vertices in T.
Let dn be a metric (distance function) on T prop. to graph distances.

Holden (ETH Zürich)

August 1, 20197 / 20Conformally embedded RPM converge top

8/3-LQG

AT
Cardy
embedding φ

random planar map (RPM) Mn

scaling limit

I⇒

B

C
embedded random planar map

p

8/3-LQG h√
Uniform triangulation Mn with n vertices, boundary length d ne.
Cardy embedding: uses properties of percolation on the RPM.
Let µn be renormalized counting measure on the vertices in T.
Let dn be a metric (distance function) on T prop. to graph distances.
p
Let µ be 8/3-LQG area measure in T, and d the associated metric.
Holden (ETH Zürich)

August 1, 20197 / 20Conformally embedded RPM converge top

8/3-LQG

AT
Cardy
embedding φ

random planar map (RPM) Mn

scaling limit

I⇒

B

C
embedded random planar map

p

8/3-LQG h√
Uniform triangulation Mn with n vertices, boundary length d ne.
Cardy embedding: uses properties of percolation on the RPM.
Let µn be renormalized counting measure on the vertices in T.
Let dn bep
a metric (distance function) on T prop. to graph distances.
Let µ be 8/3-LQG area measure in T, and d the associated metric.

Theorem (H.-Sun’19)

In the above setting, (µn , dn ) ⇒ (µ, d).
Holden (ETH Zürich)

August 1, 20197 / 20Conformally embedded RPM converge top

8/3-LQG

AT
Cardy
embedding φ

random planar map (RPM) Mn

scaling limit

I⇒

B

C
embedded random planar map

p

8/3-LQG h

Theorem (H.-Sun’19)
In the above setting, (µn , dn ) ⇒ (µ, d).
More precisely, ∃ coupling of Mn and h s.t. with probability 1, as n → ∞,RR
fdµn → fdµ ∀ continuous f : T → [0, 1] (measure convergence)dn (z, w ) → d(z, w ), uniformly in z, w ∈ T (metric convergence)

Holden (ETH Zürich)

August 1, 20197 / 20

The Schramm-Loewner evolution (SLE)
One-parameter family of random fractal curves indexed by κ ≥ 0,
which describe the scaling limit of statistical physics models
loop-erased random walk, κ = 2
Ising, κ = 3, and FK-Ising, κ = 16/3
percolation, κ = 6
discrete Gaussian free field level line, κ = 4
uniform spanning tree, κ = 8

SLE0
Holden (ETH Zürich)

SLE2

SLE4
August 1, 20198 / 20

The Schramm-Loewner evolution (SLE)
One-parameter family of random fractal curves indexed by κ ≥ 0,
which describe the scaling limit of statistical physics models
loop-erased random walk, κ = 2
Ising, κ = 3, and FK-Ising, κ = 16/3
percolation, κ = 6
discrete Gaussian free field level line, κ = 4
uniform spanning tree, κ = 8

Holden (ETH Zürich)

August 1, 20198 / 20

The Schramm-Loewner evolution (SLE)
One-parameter family of random fractal curves indexed by κ ≥ 0,
which describe the scaling limit of statistical physics models
loop-erased random walk, κ = 2
Ising, κ = 3, and FK-Ising, κ = 16/3
percolation, κ = 6
discrete Gaussian free field level line, κ = 4
uniform spanning tree, κ = 8

Holden (ETH Zürich)

August 1, 20198 / 20

The Schramm-Loewner evolution (SLE)
One-parameter family of random fractal curves indexed by κ ≥ 0,
which describe the scaling limit of statistical physics models
loop-erased random walk, κ = 2
Ising, κ = 3, and FK-Ising, κ = 16/3
percolation, κ = 6
discrete Gaussian free field level line, κ = 4
uniform spanning tree, κ = 8

Introduced by Schramm’99: SLE uniquely characterized by conformal
invariance and domain Markov property.

Holden (ETH Zürich)

August 1, 20198 / 20Percolation on uniform triangulations ⇒ SLE6
B

B

ηn

C

ηn

A

A

C

Smirnov’01, Camia-Newman’06: ηn ⇒ SLE6 on triangular lattice.

H.-Sun’19: ηn ⇒ SLE6 on Cardy embedded triangulation in aquenched sense.

Holden (ETH Zürich)

August 1, 20199 / 20Percolation on uniform triangulations ⇒ CLE6
B

B

Γn

C

Γn

A

A

C

The conformal loop ensemble (CLE6 ) is the loop version of SLE6 .
Smirnov’01, Camia-Newman’06: Γn ⇒ CLE6 on triangular lattice.
H.-Sun’19: Γn ⇒ CLE6 on Cardy embedded triangulation.
Holden (ETH Zürich)

August 1, 201910 / 20

Convergence of percolation crossing probability√
Let Mn be a uniformly chosen triangulation with n (resp. d ne) inner(resp. boundary) vertices.

Holden (ETH Zürich)

August 1, 201911 / 20

Convergence of percolation crossing probability√
Let Mn be a uniformly chosen triangulation with n (resp. d ne) inner(resp. boundary) vertices.
Pick edges an , bn , cn , dn uniformly at random from ∂Mn .

cn

dn

bn
an
Holden (ETH Zürich)

August 1, 201911 / 20

Convergence of percolation crossing probability√
Let Mn be a uniformly chosen triangulation with n (resp. d ne) inner(resp. boundary) vertices.
Pick edges an , bn , cn , dn uniformly at random from ∂Mn .
Let Pn = Pn (Mn , an , bn , cn , dn ) ∈ [0, 1] denote the probability of ablue crossing from an bn to cn dn .

cn

dn

cn

dn

Pn =
bn
an
Holden (ETH Zürich)bnanAugust 1, 201911 / 20

Convergence of percolation crossing probability√
Let Mn be a uniformly chosen triangulation with n (resp. d ne) inner(resp. boundary) vertices.
Pick edges an , bn , cn , dn uniformly at random from ∂Mn .
Let Pn = Pn (Mn , an , bn , cn , dn ) ∈ [0, 1] denote the probability of ablue crossing from an bn to cn dn .
The random variable Pn converges in law as n → ∞.
cn

dn

cn

dn

Pn =
bn
an
Holden (ETH Zürich)bnanAugust 1, 201911 / 20

Convergence of percolation crossing probability√
Let Mn be a uniformly chosen triangulation with n (resp. d ne) inner(resp. boundary) vertices.
Pick edges an , bn , cn , dn uniformly at random from ∂Mn .
Let Pn = Pn (Mn , an , bn , cn , dn ) ∈ [0, 1] denote the probability of ablue crossing from an bn to cn dn .
The random variable Pn converges in law as n → ∞.
Pn gives some notion of extremal distance between an bn and cn dn .
cn

dn

cn

dn

Pn =
bn
an
Holden (ETH Zürich)bnanAugust 1, 201911 / 20

Cardy embedding: percolation-based embedding

A
T
Cardy embedding φ

random planar map

BC
embedded random planar map

Holden (ETH Zürich)

August 1, 201912 / 20

Cardy embedding: percolation-based embedding

A
Tb
Cardy embedding φacrandom planar map

BC
embedded random planar map

Holden (ETH Zürich)

August 1, 201912 / 20

Cardy embedding: percolation-based embedding
What is the “correct” position of v in T?

b

v

a

c
Holden (ETH Zürich)

August 1, 201912 / 20

Cardy embedding: percolation-based embedding
What is the “correct” position of v in T?

B
bluecrossingx
C
Holden (ETH Zürich)A
August 1, 201912 / 20

Cardy embedding: percolation-based embedding
What is the “correct” position of v in T?

B
bluecrossing

BpA (x) = Pxx
C
Holden (ETH Zürich)

A

CA
August 1, 201912 / 20

Cardy embedding: percolation-based embedding
What is the “correct” position of v in T?
Map v ∈ V (M) to x ∈ T such that(pA (x), pB (x), pC (x)) = (bpa (v ), pbb (v ), pbc (v )).

b

Bpba (v) = P

pA (x) = P
xC

Holden (ETH Zürich)

A

v

a

c
August 1, 201912 / 20

RPM ⇒ LQG under conformal embedding
A

T
Cardy
embedding φ

random planar map (RPM) Mn

scaling limit

I⇒

B

C
embedded random planar map

p

8/3-LQG h

Our result is for uniform triangulations and the Cardy embedding, but
is also believed to hold for other1

conformal embeddings,

2

local map constraints, and3

universality classes of random planar maps.
Holden (ETH Zürich)

August 1, 201913 / 20

Discrete conformal embeddings
Circle packing
Riemann uniformization
Tutte embedding
Cardy embedding

circle packing (sphere topology)
Holden (ETH Zürich)circle packing (disk topology)
August 1, 201914 / 20

Discrete conformal embeddings
Circle packing
Riemann uniformization
Tutte embedding
Cardy embedding

Random planar map

Holden (ETH Zürich)Riemannian manifold

August 1, 201914 / 20

Discrete conformal embeddings
Circle packing
Riemann uniformization
Tutte embedding
Cardy embedding.
Uniformization theorem: For any simply connected Riemann surface M
there is a conformal map φ from M to either D, C or S2 .

φ

Holden (ETH Zürich)

August 1, 201914 / 20

Discrete conformal embeddings
Circle packing
Riemann uniformization
Tutte embedding
Cardy embedding

Tutte embedding
Holden (ETH Zürich)

August 1, 201914 / 20

Conformally embedded RPM converge top

8/3-LQG

AT
Cardy
embedding φ

random planar map (RPM) Mn

Holden (ETH Zürich)scaling limit

I⇒

B

C
embedded random planar map

p

8/3-LQG h

August 1, 201915 / 20

Conformally embedded RPM converge to
The proof is based on multiple works, including:
Percolation on triangulations: a bijective path to
Liouville quantum gravity (Bernardi-H.-Sun)p

8/3-LQG

Minkowski content of Brownian cut points(Lawler-Li-H.-Sun)Natural parametrization of percolation interface
and pivotal points (Li-H.-Sun)
Uniform triangulations with simple boundary
converge to the Brownian disk (Albenque-H.-Sun)
Joint scaling limit of site percolation on random
triangulations in the metric and peanosphere sense(Gwynne-H.-Sun)
Liouville dynamical percolation(Garban-H.-Sepúlveda-Sun)
Convergence of uniform triangulations under the
Cardy embedding (H.-Sun)

Holden (ETH Zürich)

August 1, 201916 / 20

Convergence as metric measure space√
Mn is a uniform triangulation with n vertices and bdy length d ne.

Holden (ETH Zürich)

August 1, 201917 / 20

Convergence as metric measure space√
Mn is a uniform triangulation with n vertices and bdy length d ne.
Mn is a random metric measure space.

n−1/4

Holden (ETH Zürich)

August 1, 201917 / 20

Convergence as metric measure space√
Mn is a uniform triangulation with n vertices and bdy length d ne.
Mn is a random metric measure space.

Gromov-Hausdorff-Prokhorov (GHP) topology on the space of
metric measure spaces.

n−1/4

Holden (ETH Zürich)

August 1, 201917 / 20

Convergence as metric measure space√
Mn is a uniform triangulation with n vertices and bdy length d ne.
Mn is a random metric measure space.
Gromov-Hausdorff-Prokhorov (GHP) topology on the space of
metric measure spaces.

Theorem (Albenque-H.-Sun’19)
Mn ⇒ M in the GHP topology, where M isBrownian disk).

p
8/3-LQG (equivalently, theBuilding on Le Gall’13, Miermont’13, Bettinelli–Miermont’17, Poulalhon–Schaeffer’06,
Addario-Berry–Albenque’17, Addario-Berry–Wen’17

n−1/4

Holden (ETH Zürich)

August 1, 201917 / 20

Convergence as metric measure space with loops
Theorem (Albenque-H.-Sun’19)
Mn ⇒ M in the GHP topology, where M is

Holden (ETH Zürich)p
8/3-LQG (the Brownian disk).

August 1, 201918 / 20

Convergence as metric measure space with loops
Theorem (Albenque-H.-Sun’19)
Mn ⇒ M in the GHP topology, where M isp
8/3-LQG (the Brownian disk).

Let Pn be a uniform percolation on Mn .

Holden (ETH Zürich)

August 1, 201918 / 20

Convergence as metric measure space with loops
Theorem (Albenque-H.-Sun’19)
Mn ⇒ M in the GHP topology, where M isp
8/3-LQG (the Brownian disk).

Let Pn be a uniform percolation on Mn .
Gromov-Hausdorff-Prokhorov-uniform (GHPU) topology on the space of metricmeasure spaces with a collection of loops.

Holden (ETH Zürich)

August 1, 201918 / 20

Convergence as metric measure space with loops
Theorem (Albenque-H.-Sun’19)
Mn ⇒ M in the GHP topology, where M isp
8/3-LQG (the Brownian disk).

Let Pn be a uniform percolation on Mn .
Gromov-Hausdorff-Prokhorov-uniform (GHPU) topology on the space of metricmeasure spaces with a collection of loops.

Theorem (Gwynne-H.-Sun’19)(Mn , Pn ) ⇒ (M, Γ) in the GHPU topology, where Γ is the conformal loop ensemble CLE6 .
Building on Gwynne-Miller’17, Bernardi-H.-Sun’18

Holden (ETH Zürich)

August 1, 201918 / 20

Liouville dynamical percolation
Dynamical percolation (Pt )t≥0 on M: Each vertex has an exponential clock andits color is resampled when its clock rings.

Holden (ETH Zürich)

August 1, 201919 / 20

Liouville dynamical percolation
Dynamical percolation (Pt )t≥0 on M: Each vertex has an exponential clock andits color is resampled when its clock rings.

Holden (ETH Zürich)

August 1, 201919 / 20

Liouville dynamical percolation
Dynamical percolation (Pt )t≥0 on M: Each vertex has an exponential clock andits color is resampled when its clock rings.

Holden (ETH Zürich)

August 1, 201919 / 20

Liouville dynamical percolation
Dynamical percolation (Pt )t≥0 on M: Each vertex has an exponential clock andits color is resampled when its clock rings.

Holden (ETH Zürich)

August 1, 201919 / 20

Liouville dynamical percolation
Dynamical percolation (Pt )t≥0 on M: Each vertex has an exponential clock andits color is resampled when its clock rings.

Holden (ETH Zürich)

August 1, 201919 / 20

Liouville dynamical percolation
Dynamical percolation (Pt )t≥0 on M: Each vertex has an exponential clock andits color is resampled when its clock rings.

Holden (ETH Zürich)

August 1, 201919 / 20

Liouville dynamical percolation
Dynamical percolation (Pt )t≥0 on M: Each vertex has an exponential clock andits color is resampled when its clock rings.
(Pn−1/4 t )t≥0 ⇒ (Γt )t≥0 , for (Γt )t≥0 Liouville dynamical percolation.
Γt is a CLE6 for each t ≥ 0.

Holden (ETH Zürich)

August 1, 201919 / 20

Liouville dynamical percolation
Dynamical percolation (Pt )t≥0 on M: Each vertex has an exponential clock andits color is resampled when its clock rings.
(Pn−1/4 t )t≥0 ⇒ (Γt )t≥0 , for (Γt )t≥0 Liouville dynamical percolation.
Γt is a CLE6 for each t ≥ 0.

Holden (ETH Zürich)

August 1, 201919 / 20

Liouville dynamical percolation
Dynamical percolation (Pt )t≥0 on M: Each vertex has an exponential clock andits color is resampled when its clock rings.
(Pn−1/4 t )t≥0 ⇒ (Γt )t≥0 , for (Γt )t≥0 Liouville dynamical percolation.
Γt is a CLE6 for each t ≥ 0.

Holden (ETH Zürich)

August 1, 201919 / 20

Liouville dynamical percolation
Dynamical percolation (Pt )t≥0 on M: Each vertex has an exponential clock andits color is resampled when its clock rings.
(Pn−1/4 t )t≥0 ⇒ (Γt )t≥0 , for (Γt )t≥0 Liouville dynamical percolation.
Γt is a CLE6 for each t ≥ 0.

Holden (ETH Zürich)

August 1, 201919 / 20

Liouville dynamical percolation
Dynamical percolation (Pt )t≥0 on M: Each vertex has an exponential clock andits color is resampled when its clock rings.
(Pn−1/4 t )t≥0 ⇒ (Γt )t≥0 , for (Γt )t≥0 Liouville dynamical percolation.
Γt is a CLE6 for each t ≥ 0.

Holden (ETH Zürich)

August 1, 201919 / 20

Liouville dynamical percolation
Dynamical percolation (Pt )t≥0 on M: Each vertex has an exponential clock andits color is resampled when its clock rings.
(Pn−1/4 t )t≥0 ⇒ (Γt )t≥0 , for (Γt )t≥0 Liouville dynamical percolation.
Γt is a CLE6 for each t ≥ 0.

Holden (ETH Zürich)

August 1, 201919 / 20

Liouville dynamical percolation
Dynamical percolation (Pt )t≥0 on M: Each vertex has an exponential clock andits color is resampled when its clock rings.
(Pn−1/4 t )t≥0 ⇒ (Γt )t≥0 , for (Γt )t≥0 Liouville dynamical percolation.
Γt is a CLE6 for each t ≥ 0.

Holden (ETH Zürich)

August 1, 201919 / 20

Liouville dynamical percolation
Dynamical percolation (Pt )t≥0 on M: Each vertex has an exponential clock andits color is resampled when its clock rings.
(Pn−1/4 t )t≥0 ⇒ (Γt )t≥0 , for (Γt )t≥0 Liouville dynamical percolation.
Γt is a CLE6 for each t ≥ 0.

Holden (ETH Zürich)

August 1, 201919 / 20

Liouville dynamical percolation
Dynamical percolation (Pt )t≥0 on M: Each vertex has an exponential clock andits color is resampled when its clock rings.
(Pn−1/4 t )t≥0 ⇒ (Γt )t≥0 , for (Γt )t≥0 Liouville dynamical percolation.
Γt is a CLE6 for each t ≥ 0.
(Γt )t≥0 is mixing (in particular, ergodic): Γt is asymptotically indep. of Γ0 .
limt→∞ Cov(E1 (Γ0 ), E2 (Γt )) = 0 for all events E1 , E2 .

Holden (ETH Zürich)

August 1, 201919 / 20

Liouville dynamical percolation
Dynamical percolation (Pt )t≥0 on M: Each vertex has an exponential clock andits color is resampled when its clock rings.
(Pn−1/4 t )t≥0 ⇒ (Γt )t≥0 , for (Γt )t≥0 Liouville dynamical percolation.
Γt is a CLE6 for each t ≥ 0.
(Γt )t≥0 is mixing (in particular, ergodic): Γt is asymptotically indep. of Γ0 .
limt→∞ Cov(E1 (Γ0 ), E2 (Γt )) = 0 for all events E1 , E2 .
Noise sensitivity: If a fraction Cn−1/4 of the vertices are resampled for C  1, we
get an essentially independent limiting CLE6 .

Holden (ETH Zürich)

August 1, 201919 / 20

Liouville dynamical percolation
Dynamical percolation (Pt )t≥0 on M: Each vertex has an exponential clock andits color is resampled when its clock rings.
(Pn−1/4 t )t≥0 ⇒ (Γt )t≥0 , for (Γt )t≥0 Liouville dynamical percolation.
Γt is a CLE6 for each t ≥ 0.
(Γt )t≥0 is mixing (in particular, ergodic): Γt is asymptotically indep. of Γ0 .
limt→∞ Cov(E1 (Γ0 ), E2 (Γt )) = 0 for all events E1 , E2 .
Noise sensitivity: If a fraction Cn−1/4 of the vertices are resampled for C  1, we
get an essentially independent limiting CLE6 .
Corollary: k indep. percolations on a map M give k indep. CLE6 ’s in the scaling
limit ⇒ convergence of Cardy embedding of M via LLN argument.

(M, P )
Holden (ETH Zürich)(M, Pe)August 1, 201919 / 20

Liouville dynamical percolation
Dynamical percolation (Pt )t≥0 on M: Each vertex has an exponential clock andits color is resampled when its clock rings.
(Pn−1/4 t )t≥0 ⇒ (Γt )t≥0 , for (Γt )t≥0 Liouville dynamical percolation.
Γt is a CLE6 for each t ≥ 0.
(Γt )t≥0 is mixing (in particular, ergodic): Γt is asymptotically indep. of Γ0 .
limt→∞ Cov(E1 (Γ0 ), E2 (Γt )) = 0 for all events E1 , E2 .
Noise sensitivity: If a fraction Cn−1/4 of the vertices are resampled for C  1, we
get an essentially independent limiting CLE6 .
Corollary: k indep. percolations on a map M give k indep. CLE6 ’s in the scaling
limit ⇒ convergence of Cardy embedding of M via LLN argument.

(M, P )
Holden (ETH Zürich)(M, Pe)August 1, 201919 / 20.

Thanks!

Holden (ETH Zürich)

August 1, 201920 / 20