[PPT] - Percolation on isoradial graphs Ioan Manolescu Joint work with PowerPoint Presentation

SLIDE 1

Percolation on isoradial graphs

Ioan Manolescu Joint work with Geoffrey Grimmett

University of Geneva

15 August 2013

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 1 / 25

SLIDE 2

General Percolation

Percolation

Under Pp, an edge e is

pen with probability pe

closed with probability 1 − pe

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 2 / 25

SLIDE 3

General Percolation

Percolation

Under Pp, an edge e is

pen with probability pe

closed with probability 1 − pe

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 2 / 25

SLIDE 4

General Percolation

Percolation

Under Pp, an edge e is

pen with probability pe

closed with probability 1 − pe

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 2 / 25

SLIDE 5

General Percolation

Percolation

Under Pp, an edge e is

pen with probability pe

closed with probability 1 − pe

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 2 / 25

SLIDE 6

General Percolation

Percolation

Under Pp, an edge e is

pen with probability pe

closed with probability 1 − pe

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 2 / 25

SLIDE 7

General Percolation

Percolation

Under Pp, an edge e is

pen with probability pe

closed with probability 1 − pe

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 2 / 25

SLIDE 8

General Percolation

Percolation

Under Pp, an edge e is

pen with probability pe

closed with probability 1 − pe

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 2 / 25

SLIDE 9

General Percolation

Percolation

Under Pp, an edge e is

pen with probability pe

closed with probability 1 − pe

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 2 / 25

SLIDE 10

General Percolation

Homogeneous percolation on Z2: all edges have intensity p ∈ [0, 1]. Question: is there an infinite connected component?

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 3 / 25

SLIDE 11

General Percolation

Homogeneous percolation on Z2: all edges have intensity p ∈ [0, 1]. Question: is there an infinite connected component?

1 Supercriticality: Existence of infinite cluster Subcriticality: No infinite cluster

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 3 / 25

SLIDE 12

General Percolation

Homogeneous percolation on Z2: all edges have intensity p ∈ [0, 1]. Question: is there an infinite connected component?

1 Supercriticality: Existence of infinite cluster Subcriticality: No infinite cluster Exponential tail for cluster size Unique infinite cluster Exponential tail for dis- tance to infinite cluster. Trivial large scale behaviour! Trivial large scale behaviour!

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 3 / 25

SLIDE 13

General Percolation

Homogeneous percolation on Z2: all edges have intensity p ∈ [0, 1]. Question: is there an infinite connected component?

1 Supercriticality: Existence of infinite cluster Subcriticality: No infinite cluster Exponential tail for cluster size Unique infinite cluster Exponential tail for dis- tance to infinite cluster. Trivial large scale behaviour! Trivial large scale behaviour! Criticality Scale invariance Large scale limit. Universality. pc

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 3 / 25

SLIDE 14

Isoradial Percolation

Isoradial percolation

θe e

Each face of G is inscribed in a circle of radius 1. PG percolation with pe: pe 1 − pe = sin( π−θ(e)

3

) sin( θ(e)

3 )

.

1 π

1 2 π 2

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 4 / 25

SLIDE 15

Isoradial Percolation

Bond Percolation on Z2

p p

Isoradiality: p = 1

2

Theorem (Kesten 80) p ≤ 1

2, a.s. no infinite cluster;

p > 1

2, a.s. existence of an infinite cluster.

Method:

self-duality + RSW + sharp-threshold

( ( ( (

= 1

2

≥ c (0 ↔ ∞) > 0

⇒ ⇒ P 1

2+ǫ

P 1

2

P 1

2

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 5 / 25

SLIDE 16

Isoradial Percolation

Bond Percolation on Z2

p p

Isoradiality: p = 1

2

Theorem (Kesten 80) p ≤ 1

2, a.s. no infinite cluster;

p > 1

2, a.s. existence of an infinite cluster.

Method:

self-duality + RSW + sharp-threshold

( ( ( (

= 1

2

≥ c (0 ↔ ∞) > 0

⇒ ⇒ P 1

2+ǫ

P 1

2

P 1

2

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 5 / 25

SLIDE 17

Percolation - Details The phase transition

The box-crossing property (RSW)

A model satisfies the box-crossing property if for all rectangles ABCD there exists c(BC/AB) = c(ρ) > 0 s. t. for all N large enough:

∈ [c, 1 − c] A B C D N ρN

P Equivalent for the primal and dual model. Theorem If Pp satisfies the box-crossing property, then it is critical.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 6 / 25

SLIDE 18

Percolation - Details The phase transition

The box-crossing property (RSW)

A model satisfies the box-crossing property if for all rectangles ABCD there exists c(BC/AB) = c(ρ) > 0 s. t. for all N large enough:

∈ [c, 1 − c] A B C D N ρN

P Equivalent for the primal and dual model. Theorem If Pp satisfies the box-crossing property, then it is critical.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 6 / 25

SLIDE 19

Percolation - Details The phase transition

The box-crossing property (RSW)

A model satisfies the box-crossing property if for all rectangles ABCD there exists c(BC/AB) = c(ρ) > 0 s. t. for all N large enough:

∈ [c, 1 − c] A B C D N ρN

P Equivalent for the primal and dual model. Theorem If Pp satisfies the box-crossing property, then it is critical.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 6 / 25

SLIDE 20

Percolation - Details The phase transition

Results I: the box-crossing property

For an isoradial graph G with the percolation measure PG, subject to conditions: Theorem PG satisfies the box-crossing property. Corollary PG is critical. Pp(infinite cluster) = 0, Pp+ǫ(infinite cluster) = 1.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 7 / 25

SLIDE 21

Percolation - Details The phase transition

Results I: the box-crossing property

For an isoradial graph G with the percolation measure PG, subject to conditions: Theorem PG satisfies the box-crossing property. Corollary PG is critical. Pp(infinite cluster) = 0, Pp+ǫ(infinite cluster) = 1.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 7 / 25

SLIDE 22

Percolation - Details Critical exponents

Arm exponents

For a critical percolation measure P, as n → ∞, we expect:

ne-arm exponent

5 48:

P(rad(C0) ≥ n) = P(A1(n)) ≈ n−ρ1, 2j-alternating-arms exponents 4j2−1

12 :

P[A2j(n)] ≈ n−ρ2j. Moreover ρi does not depend on the underlying model.

O n

A4(n) Power-law bounds are given by the box-crossing property.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 8 / 25

SLIDE 23

Percolation - Details Critical exponents

Critical exponents

For Pp critical we expect:

Exponents at criticality. Volume exponent δ = 91

5 :

Pp(|C0| = n) ≈ n−1−1/δ. Connectivity exponent η =

5 24:

Pp(0 ↔ x) ≈ |x|−η. Radius exponent ρ = 48

5 :

Pp(rad(C0) = n) ≈ n−1−1/ρ. (ρ =

1 ρ1 )

Exponents near criticality. Percolation probability β =

5 36:

Pp+ǫ(|C0| = ∞) ≈ ǫβ as ǫ ↓ 0. Correlation length ν = 4

3:

ξ(p − ǫ) ≈ ǫ−ν as ǫ ↓ 0, were − 1

n log Pp−ǫ(rad(C0) ≥ n) →n→∞ 1 ξ(p−ǫ).

Mean cluster-size γ = 43

18:

Pp+ǫ(|C0|; |C0| < ∞) ≈ |ǫ|−γ as ǫ → 0. Gap exponent ∆ = 91

36: Pp+ǫ(|C0|k+1;|C0|<∞) Pp+ǫ(|C0|k;|C0|<∞) ≈ |ǫ|−∆. for k ≥ 1, as ǫ → 0.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 9 / 25

SLIDE 24

Percolation - Details Critical exponents

Results II: arm exponents

For an isoradial graph G with the percolation measure PG, subject to conditions Theorem For k ∈ {1, 2, 4, . . .} there exist constants c1, c2 > 0 such that: c1PZ2[Ak(n)] ≤ PG[Ak(n)] ≤ c2PZ2[Ak(n)], for n ∈ N.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 10 / 25

SLIDE 25

Percolation - Details Critical exponents

Results II: arm exponents

For an isoradial graph G with the percolation measure PG, subject to conditions Theorem For k ∈ {1, 2, 4, . . .} there exist constants c1, c2 > 0 such that: c1PZ2[Ak(n)] ≤ PG[Ak(n)] ≤ c2PZ2[Ak(n)], for n ∈ N. Corollary The one arm exponent and the 2j alternating arm exponents are universal for percolation on isoradial graphs.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 10 / 25

SLIDE 26

Isoradial graphs – details

Isoradial Graphs

G isoradial graph

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

SLIDE 27

Isoradial graphs – details

Isoradial Graphs

G isoradial graph

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

SLIDE 28

Isoradial graphs – details

Isoradial Graphs

G isoradial graph

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

SLIDE 29

Isoradial graphs – details

Isoradial Graphs

G isoradial graph G ∗ dual isoradial graph

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

SLIDE 30

Isoradial graphs – details

Isoradial Graphs

G isoradial graph G ∗ dual isoradial graph

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

SLIDE 31

Isoradial graphs – details

Isoradial Graphs

G isoradial graph G ∗ dual isoradial graph G diamond graph

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

SLIDE 32

Isoradial graphs – details

Isoradial Graphs

G isoradial graph G ∗ dual isoradial graph G diamond graph

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

SLIDE 33

Isoradial graphs – details

Isoradial Graphs

G isoradial graph G ∗ dual isoradial graph G diamond graph

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

SLIDE 34

Isoradial graphs – details

Isoradial Graphs

G isoradial graph G ∗ dual isoradial graph G diamond graph

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

SLIDE 35

Isoradial graphs – details

Isoradial Graphs

G isoradial graph G ∗ dual isoradial graph G diamond graph

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

SLIDE 36

Isoradial graphs – details

Isoradial Graphs

G isoradial graph G ∗ dual isoradial graph G diamond graph

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

SLIDE 37

Isoradial graphs – details

Isoradial Graphs

G isoradial graph G ∗ dual isoradial graph G diamond graph Track system

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

SLIDE 38

Isoradial graphs – details

Isoradial Graphs

G isoradial graph G ∗ dual isoradial graph G diamond graph Track system

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 11 / 25

SLIDE 39

Isoradial graphs – details Conditions

Conditions for isoradial graphs.

Bounded angles condition: There exist ǫ0 > 0 such that for any edge e, θe ∈ [ǫ0, π − ǫ0].

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 12 / 25

SLIDE 40

Isoradial graphs – details Conditions

Conditions for isoradial graphs.

Bounded angles condition: There exist ǫ0 > 0 such that for any edge e, θe ∈ [ǫ0, π − ǫ0]. Square grid property: Families of ”parallel” tracks (si)i∈Z and (tj)j∈Z. The number of intersections on si between tj and tj+1 is uniformly bounded by a con- stant I. (same for t). t−1 t0 t1 s0 s1 s2

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 12 / 25

SLIDE 41

Isoradial graphs – details Conditions

Examples: Penrose tilings and no square grid

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 13 / 25

SLIDE 42

Isoradial graphs – details Conditions

Examples: Penrose tilings and no square grid

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 13 / 25

SLIDE 43

Isoradial graphs – details Conditions

Examples: Penrose tilings and no square grid

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 13 / 25

SLIDE 44

Isoradial graphs – details Conditions

Examples: Penrose tilings and no square grid

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 13 / 25

SLIDE 45

Isoradial graphs – details Conditions

Examples: Penrose tilings and no square grid

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 13 / 25

SLIDE 46

Isoradial graphs – details Conditions

Examples: Penrose tilings and no square grid

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 13 / 25

SLIDE 47

Star–triangle transformation

A C B C A B O p0 p1 p2

κ△(p) = p0 + p1 + p2 − p0p1p2 = 1. Take ω, respectively ω′, according to the measure on the left, respectively right. The families of random variables

x

ω

← → y : x, y = A, B, C

,
x

ω′

← → y : x, y = A, B, C

,

have the same joint law.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 14 / 25

SLIDE 48

Star–triangle transformation

Coupling

and similarly for all pairs of edges

(1 − p0)p1p2 P p0p1p2 P p0(1 − p1)p2 P p0p1(1 − p2) P (1 − p0)p1p2 P p0p1p2 P p0(1 − p1)p2 P p0p1(1 − p2) P

and similarly for all single edges T T S S S T

where P = (1 − p0)(1 − p1)(1 − p2).

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 15 / 25

SLIDE 49

Star–triangle transformation

Path transformation

A B C A B C A B C A B C A B C O O A B C A B C O

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 16 / 25

SLIDE 50

Proof for box-crossing property From Z2 to isoradial square lattice.

Track exchange

Two parallel tracks s1 and s2 with no intersection between them. We may exchange s1 and s2 using star–triangle transformations.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

SLIDE 51

Proof for box-crossing property From Z2 to isoradial square lattice.

Track exchange

Two parallel tracks s1 and s2 with no intersection between them. We may exchange s1 and s2 using star–triangle transformations.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

SLIDE 52

Proof for box-crossing property From Z2 to isoradial square lattice.

Track exchange

Two parallel tracks s1 and s2 with no intersection between them. We may exchange s1 and s2 using star–triangle transformations.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

SLIDE 53

Proof for box-crossing property From Z2 to isoradial square lattice.

Track exchange

Two parallel tracks s1 and s2 with no intersection between them. We may exchange s1 and s2 using star–triangle transformations.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

SLIDE 54

Proof for box-crossing property From Z2 to isoradial square lattice.

Track exchange

Two parallel tracks s1 and s2 with no intersection between them. We may exchange s1 and s2 using star–triangle transformations.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

SLIDE 55

Proof for box-crossing property From Z2 to isoradial square lattice.

Track exchange

Two parallel tracks s1 and s2 with no intersection between them. We may exchange s1 and s2 using star–triangle transformations.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

SLIDE 56

Proof for box-crossing property From Z2 to isoradial square lattice.

Track exchange

Two parallel tracks s1 and s2 with no intersection between them. We may exchange s1 and s2 using star–triangle transformations.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

SLIDE 57

Proof for box-crossing property From Z2 to isoradial square lattice.

Track exchange

Two parallel tracks s1 and s2 with no intersection between them. We may exchange s1 and s2 using star–triangle transformations.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

SLIDE 58

Proof for box-crossing property From Z2 to isoradial square lattice.

Track exchange

Two parallel tracks s1 and s2 with no intersection between them. We may exchange s1 and s2 using star–triangle transformations.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

SLIDE 59

Proof for box-crossing property From Z2 to isoradial square lattice.

Track exchange

Two parallel tracks s1 and s2 with no intersection between them. We may exchange s1 and s2 using star–triangle transformations.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

SLIDE 60

Proof for box-crossing property From Z2 to isoradial square lattice.

Track exchange

Two parallel tracks s1 and s2 with no intersection between them. We may exchange s1 and s2 using star–triangle transformations.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

SLIDE 61

Proof for box-crossing property From Z2 to isoradial square lattice.

Track exchange

Two parallel tracks s1 and s2 with no intersection between them. We may exchange s1 and s2 using star–triangle transformations.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

SLIDE 62

Proof for box-crossing property From Z2 to isoradial square lattice.

Track exchange

Two parallel tracks s1 and s2 with no intersection between them. We may exchange s1 and s2 using star–triangle transformations.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

SLIDE 63

Proof for box-crossing property From Z2 to isoradial square lattice.

Track exchange

Two parallel tracks s1 and s2 with no intersection between them. We may exchange s1 and s2 using star–triangle transformations.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 17 / 25

SLIDE 64

Proof for box-crossing property From Z2 to isoradial square lattice.

Principal

utcome

Secondary

utcome

Probability

f secondary
utcome

pπ−θ1pθ2 pθ1pπ−θ2 pπ−θ1pπ−θ2+θ1 pθ1pθ2−θ1 pθ2pπ−θ2+θ1 pπ−θ2pθ2−θ1 pθ2pπ−θ2+θ1 pπ−θ2pθ2−θ1 pπ−θ1pπ−θ2+θ1 pθ1pθ2−θ1

Initial configuration

θ1 θ2 θ1 θ2 θ1 θ2 θ1 θ2 θ1 θ2

Open paths are preserved (unless the deleted edge was part of the path).

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 18 / 25

SLIDE 65

Proof for box-crossing property From Z2 to isoradial square lattice.

Strategy

Proposition If two isoradial square lattices have same transverse angles for the vertical/horizontal tracks, and one has the box-crossing property, then so does the

ther.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 19 / 25

SLIDE 66

Proof for box-crossing property From Z2 to isoradial square lattice.

Strategy

Proposition If two isoradial square lattices have same transverse angles for the vertical/horizontal tracks, and one has the box-crossing property, then so does the

ther.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 19 / 25

SLIDE 67

Proof for box-crossing property From Z2 to isoradial square lattice.

Transport of horizontal crossings

Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

SLIDE 68

Proof for box-crossing property From Z2 to isoradial square lattice.

Transport of horizontal crossings

Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

SLIDE 69

Proof for box-crossing property From Z2 to isoradial square lattice.

Transport of horizontal crossings

Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

SLIDE 70

Proof for box-crossing property From Z2 to isoradial square lattice.

Transport of horizontal crossings

Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

SLIDE 71

Proof for box-crossing property From Z2 to isoradial square lattice.

Transport of horizontal crossings

Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

SLIDE 72

Proof for box-crossing property From Z2 to isoradial square lattice.

Transport of horizontal crossings

Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

SLIDE 73

Proof for box-crossing property From Z2 to isoradial square lattice.

Transport of horizontal crossings

Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

SLIDE 74

Proof for box-crossing property From Z2 to isoradial square lattice.

Transport of horizontal crossings

Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

SLIDE 75

Proof for box-crossing property From Z2 to isoradial square lattice.

Transport of horizontal crossings

Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

SLIDE 76

Proof for box-crossing property From Z2 to isoradial square lattice.

Transport of horizontal crossings

Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

SLIDE 77

Proof for box-crossing property From Z2 to isoradial square lattice.

Transport of horizontal crossings

Construct a mixed isoradial square lattice: ”regular” in the gray part, ”irregular” in the rest.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 20 / 25

SLIDE 78

Proof for box-crossing property From square lattices to general graphs

Track stacking

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

SLIDE 79

Proof for box-crossing property From square lattices to general graphs

Track stacking

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

SLIDE 80

Proof for box-crossing property From square lattices to general graphs

Track stacking

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

SLIDE 81

Proof for box-crossing property From square lattices to general graphs

Track stacking

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

SLIDE 82

Proof for box-crossing property From square lattices to general graphs

Track stacking

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

SLIDE 83

Proof for box-crossing property From square lattices to general graphs

Track stacking

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

SLIDE 84

Proof for box-crossing property From square lattices to general graphs

Track stacking

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

SLIDE 85

Proof for box-crossing property From square lattices to general graphs

Track stacking

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

SLIDE 86

Proof for box-crossing property From square lattices to general graphs

Track stacking

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

SLIDE 87

Proof for box-crossing property From square lattices to general graphs

Track stacking

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

SLIDE 88

Proof for box-crossing property From square lattices to general graphs

Track stacking

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

SLIDE 89

Proof for box-crossing property From square lattices to general graphs

Track stacking

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

SLIDE 90

Proof for box-crossing property From square lattices to general graphs

Track stacking

Pgen(Ch[B(ρN, N)]) ≥ Psq(Ch[B(IρN, N)])Psq(Cv[B(N, N)])2

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 21 / 25

SLIDE 91

Proof for box-crossing property Arm exponents

Transport of the arm exponents . . .

. . . using the same strategy as for the box-crossing property.

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 22 / 25

SLIDE 92

Proof for box-crossing property Arm exponents

Square lattices

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 23 / 25

SLIDE 93

Proof for box-crossing property Arm exponents

Square lattices

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 23 / 25

SLIDE 94

Proof for box-crossing property Arm exponents

Square lattices

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 23 / 25

SLIDE 95

Proof for box-crossing property Arm exponents

Square lattices

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 23 / 25

SLIDE 96

Proof for box-crossing property Arm exponents

Square lattices

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 23 / 25

SLIDE 97

Proof for box-crossing property Arm exponents

Square lattices

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 23 / 25

SLIDE 98

Proof for box-crossing property Arm exponents

Square lattices

c1Preg(Ak(n)) ≤ Pirreg(Ak(n))

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 23 / 25

SLIDE 99

Proof for box-crossing property Arm exponents

Square lattices

c1Preg(Ak(n)) ≤ Pirreg(Ak(n))

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 23 / 25

SLIDE 100

Proof for box-crossing property Arm exponents

Square lattices

c1Preg(Ak(n)) ≤ Pirreg(Ak(n)) ≤ c2Preg(Ak(n)).

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 23 / 25

SLIDE 101

Proof for box-crossing property Arm exponents

From square lattices to general graphs

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 24 / 25

SLIDE 102

Proof for box-crossing property Arm exponents

From square lattices to general graphs

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 24 / 25

SLIDE 103

Proof for box-crossing property Arm exponents

From square lattices to general graphs

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 24 / 25

SLIDE 104

Proof for box-crossing property Arm exponents

From square lattices to general graphs

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 24 / 25

SLIDE 105

Proof for box-crossing property Arm exponents

From square lattices to general graphs

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 24 / 25

SLIDE 106

Proof for box-crossing property Arm exponents

From square lattices to general graphs

c1Psq(Ak(n)) ≤ Pgen(Ak(n)) ≤ c2Psq(Ak(n)).

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 24 / 25

SLIDE 107

Proof for box-crossing property Arm exponents

Thank you!

Ioan Manolescu (University of Geneva) Percolation on isoradial graphs 15 August 2013 25 / 25