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Hypergeometric SLE and Convergence of Critical Planar Ising - - PowerPoint PPT Presentation

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Hypergeometric SLE and Convergence of Critical Planar Ising Interfaces Hao Wu Yau Mathematical Sciences Center, Tsinghua University, China Hao Wu (THU) Hypergeometric SLE 1 / 28 Outline Ising Model and Percolation 1 SLE 2 Hypergeometric

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Hypergeometric SLE and Convergence of Critical Planar Ising Interfaces

Hao Wu

Yau Mathematical Sciences Center, Tsinghua University, China

Hao Wu (THU) Hypergeometric SLE 1 / 28

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SLIDE 2

Outline

1

Ising Model and Percolation

2

SLE

3

Hypergeometric SLE

Hao Wu (THU) Hypergeometric SLE 2 / 28

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SLIDE 3

Ising Model and Percolation

Table of contents

1

Ising Model and Percolation

2

SLE

3

Hypergeometric SLE

Hao Wu (THU) Hypergeometric SLE 3 / 28

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SLIDE 4

Ising Model and Percolation

Ising Model

Curie temperature [Pierre Curie, 1895] Ferromagnet exhibits a phase transition by losing its magnetization when heated above a critical temperature. Ising Model [Lenz, 1920] A model for ferromagnet, to understand the critical temperature G = (V, E) is a finite graph σ ∈ {⊕, ⊖}V The Hamiltonian H(σ) = −

  • x∼y

σxσy

⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊖ ⊕ ⊖ ⊕ ⊕ ⊕ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊖ a b

Hao Wu (THU) Hypergeometric SLE 4 / 28

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SLIDE 5

Ising Model and Percolation

Ising Model

Ising model is the probability measure of inverse temperature β > 0 : µβ,G[σ] ∝ exp(−βH(σ))

⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊖ ⊕ ⊖ ⊕ ⊕ ⊕ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊖ a b

Hao Wu (THU) Hypergeometric SLE 5 / 28

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SLIDE 6

Ising Model and Percolation

Ising Model

Ising model is the probability measure of inverse temperature β > 0 : µβ,G[σ] ∝ exp(−βH(σ))

⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊖ ⊕ ⊖ ⊕ ⊕ ⊕ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊖ a b

Kramers-Wannier, Onsager-Kaufman, 1940 Ising model on Z2 : βc = 1

2 log(1 +

√ 2).

Hao Wu (THU) Hypergeometric SLE 5 / 28

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SLIDE 7

Ising Model and Percolation

Ising Model

Ising model is the probability measure of inverse temperature β > 0 : µβ,G[σ] ∝ exp(−βH(σ))

⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊖ ⊕ ⊖ ⊕ ⊕ ⊕ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊖ a b

Kramers-Wannier, Onsager-Kaufman, 1940 Ising model on Z2 : βc = 1

2 log(1 +

√ 2). Interface Conformal invariance + Domain Markov Property

Hao Wu (THU) Hypergeometric SLE 5 / 28

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SLIDE 8

Ising Model and Percolation

Percolation

Site percolation on triangular lattice : each site is chosen independently to be black or white with probability p or 1 − p. When p < 1/2, white sites dominate. When p > 1/2, black sites dominate. When p = 1/2, critical, the system converges to something nontrivial.

Hao Wu (THU) Hypergeometric SLE 6 / 28

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SLIDE 9

Ising Model and Percolation

Percolation

Site percolation on triangular lattice : each site is chosen independently to be black or white with probability p or 1 − p. When p < 1/2, white sites dominate. When p > 1/2, black sites dominate. When p = 1/2, critical, the system converges to something nontrivial. Interface Conformal invariance + Domain Markov Property

Hao Wu (THU) Hypergeometric SLE 6 / 28

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SLE

Table of contents

1

Ising Model and Percolation

2

SLE

3

Hypergeometric SLE

Hao Wu (THU) Hypergeometric SLE 7 / 28

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SLE

SLE (Schramm Loewner Evolution)

Random fractal curves in D ⊂ C from a to b. Candidates for the scaling limit of discrete Statistical Physics models.

D ϕ(D) a b ϕ(b) ϕ(a) γ ϕ(γ) ϕ

Conformal invariance : If γ is in D from a to b, and ϕ : D → ϕ(D) conformal map, then ϕ(γ) d ∼ the one in ϕ(D) from ϕ(a) to ϕ(b).

D a b γ[0, t] γ[t, ∞) γ(t)

Domain Markov Property : the conditional law of γ[t, ∞) given γ[0, t] d ∼ the one in D \ γ[0, t] from γ(t) to b.

Hao Wu (THU) Hypergeometric SLE 8 / 28

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SLE

Examples of SLE

Lemma [Schramm 1999] There exists a one-parameter family of random curves that satisfies Conformal Invariance and Domain Markov Property : SLEκ for κ ≥ 0. Simple, κ ∈ [0, 4]; Self-touching, κ ∈ (4, 8); Space-filling, κ ≥ 8.

Courtesy to Tom Kennedy.

κ = 2 : LERW κ = 8 : UST (Lawler, Schramm, Werner) κ = 3 : Critical Ising κ = 16/3 : FK-Ising (Chelkak, Duminil-Copin, Hongler, Kempainen, Smirnov) κ = 6 : Percolation (Camia, Newman, Smirnov)

Hao Wu (THU) Hypergeometric SLE 9 / 28

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SLE

Percolation and Critical Ising

Thm [Smirnov 2000] The interface of critical site percolation on triangular lattice converges to SLE(6).

Hao Wu (THU) Hypergeometric SLE 10 / 28

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SLE

Percolation and Critical Ising

Thm [Smirnov 2000] The interface of critical site percolation on triangular lattice converges to SLE(6). Thm [Chelkak, Duminil-Copin, Hongler, Kempainen, Smirnov 2010] The interface of critical Ising model on Z2 with Dobrushin boundary condition converges to SLE(3).

Hao Wu (THU) Hypergeometric SLE 10 / 28

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SLE

Percolation and Critical Ising

Thm [Smirnov 2000] The interface of critical site percolation on triangular lattice converges to SLE(6). Thm [Chelkak, Duminil-Copin, Hongler, Kempainen, Smirnov 2010] The interface of critical Ising model on Z2 with Dobrushin boundary condition converges to SLE(3). Their Strategy Tightness : RSW Identify the scaling limit : Holomorphic observable

Hao Wu (THU) Hypergeometric SLE 10 / 28

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SLE

What does the convergence tell us about the model?

Application : arm exponents. Boundary arm exponents p+

n(r, R) = P

≈ R−α+

n ,

R → ∞ Interior arm exponents pn(r, R) = P ≈ R−αn, R → ∞

Hao Wu (THU) Hypergeometric SLE 11 / 28

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SLE

What does the convergence tell us about the model?

Application : arm exponents. Boundary arm exponents p+

n(r, R) = P

≈ R−α+

n ,

R → ∞ Interior arm exponents pn(r, R) = P ≈ R−αn, R → ∞ Q : How to calculate these exponents?

Hao Wu (THU) Hypergeometric SLE 11 / 28

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SLE

Percolation [Lawler & Schramm & Werner, Smirnov & Werner 2000] Interior arm exponents : αn = (n2 − 1)/12. Boundary arm exponents : α+

n = n(n + 1)/6.

Ising Model [W. 2016] Interior arm exponents : α2n = (16n2 − 1)/24. Boundary arm exponents : 6 patterns b.c. (⊖⊕)

η

⊕ ⊕ ⊖ ⊖ ⊕

η

⊖ ⊕ ⊕ ⊖ ⊖ ⊕

α+

n ≈ n2/3.

b.c. (⊖free)

η

⊖ free ⊖ ⊕ ⊕ ⊖

η

⊖ free ⊖ ⊕ ⊕ ⊖ ⊕

β+

n = n(n + 1)/3.

b.c.(freefree)

η

free free ⊖ ⊖ ⊕

η

free free ⊖ ⊖ ⊕ ⊕

γ+

n = n(2n − 1)/6.

Hao Wu (THU) Hypergeometric SLE 12 / 28

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SLE

Other results on the convergence?

Thm [Chelkak, Duminil-Copin, Hongler, Kempainen, Smirnov 2010] The interface of critical Ising model on Z2 with Dobrushin boundary condition converges to SLE(3).

Hao Wu (THU) Hypergeometric SLE 13 / 28

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SLE

Other results on the convergence?

Thm [Chelkak, Duminil-Copin, Hongler, Kempainen, Smirnov 2010] The interface of critical Ising model on Z2 with Dobrushin boundary condition converges to SLE(3). Different Models? Different lattices? Different Boundary Conditions?

Hao Wu (THU) Hypergeometric SLE 13 / 28

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SLE

Other results on the convergence?

Thm [Chelkak, Duminil-Copin, Hongler, Kempainen, Smirnov 2010] The interface of critical Ising model on Z2 with Dobrushin boundary condition converges to SLE(3). Different Models? Different lattices? Different Boundary Conditions? Many conjectures.

Hao Wu (THU) Hypergeometric SLE 13 / 28

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SLE

Other results on the convergence?

Thm [Chelkak, Duminil-Copin, Hongler, Kempainen, Smirnov 2010] The interface of critical Ising model on Z2 with Dobrushin boundary condition converges to SLE(3). Different Models? Different lattices? Different Boundary Conditions? Many conjectures. Universality : open.

Hao Wu (THU) Hypergeometric SLE 13 / 28

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SLE

Other results on the convergence?

Thm [Chelkak, Duminil-Copin, Hongler, Kempainen, Smirnov 2010] The interface of critical Ising model on Z2 with Dobrushin boundary condition converges to SLE(3). Different Models? Different lattices? Different Boundary Conditions? Many conjectures. Universality : open. Some results.

Hao Wu (THU) Hypergeometric SLE 13 / 28

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SLE

Open Question : Other Models

Conjecture For q ≤ 4, the interface of critical Random Cluster Model converges to SLE(κ) where κ = 4π/arccos(−√q/2).

Hao Wu (THU) Hypergeometric SLE 14 / 28

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SLE

Open Question : Other Models

Conjecture For q ≤ 4, the interface of critical Random Cluster Model converges to SLE(κ) where κ = 4π/arccos(−√q/2). Conjecture The interface of Double Dimer Model converges to SLE(4).

Hao Wu (THU) Hypergeometric SLE 14 / 28

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SLE

Open Question : Universality

Thm [Smirnov 2000] The interface of critical site percolation on triangular lattice converges to SLE(6).

Hao Wu (THU) Hypergeometric SLE 15 / 28

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SLE

Open Question : Universality

Thm [Smirnov 2000] The interface of critical site percolation on triangular lattice converges to SLE(6). Conjecture The interface of critical bond percolation on square lattice converges to SLE(6).

Hao Wu (THU) Hypergeometric SLE 15 / 28

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Hypergeometric SLE

Table of contents

1

Ising Model and Percolation

2

SLE

3

Hypergeometric SLE

Hao Wu (THU) Hypergeometric SLE 16 / 28

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SLIDE 29

Hypergeometric SLE

General Boundary Conditions

Thm [Izyurov 2014, W. 2017] The interface of critical Ising model on Z2 with alternating boundary condition converges to Hypergeometric SLE3, denoted by hSLE3.

Hao Wu (THU) Hypergeometric SLE 17 / 28

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Hypergeometric SLE

General Boundary Conditions

Thm [Izyurov 2014, W. 2017] The interface of critical Ising model on Z2 with alternating boundary condition converges to Hypergeometric SLE3, denoted by hSLE3. Q1 : What is Hypergeometric SLE? Q2 : Why are they the limit? Q3 : How do we prove the convergence?

Hao Wu (THU) Hypergeometric SLE 17 / 28

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Hypergeometric SLE

General Boundary Conditions

Thm [Izyurov 2014, W. 2017] The interface of critical Ising model on Z2 with alternating boundary condition converges to Hypergeometric SLE3, denoted by hSLE3. Q1 : What is Hypergeometric SLE? Q2 : Why are they the limit? Q3 : How do we prove the convergence? Answer to Q1 :

random fractal curves in quad q = (Ω; x1, x2, x3, x4) hSLEκ(ν) for κ ∈ (0, 8) and ν ∈ R. driving function : dWt = √κdBt + κ∂x1 log Zκ,ν(Wt, V 2

t , V 3 t , V 4 t )dt.

Hao Wu (THU) Hypergeometric SLE 17 / 28

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SLIDE 32

Hypergeometric SLE

General Boundary Conditions

Thm [Izyurov 2014, W. 2017] The interface of critical Ising model on Z2 with alternating boundary condition converges to Hypergeometric SLE3, denoted by hSLE3. Q1 : What is Hypergeometric SLE? Q2 : Why they are the limit? Q3 : How to prove the convergence? Answer to Q1 :

when ν = −2, it equals SLEκ

Hao Wu (THU) Hypergeometric SLE 18 / 28

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SLIDE 33

Hypergeometric SLE

General Boundary Conditions

Thm [Izyurov 2014, W. 2017] The interface of critical Ising model on Z2 with alternating boundary condition converges to Hypergeometric SLE3, denoted by hSLE3. Q1 : What is Hypergeometric SLE? Q2 : Why they are the limit? Q3 : How to prove the convergence? Answer to Q1 :

when ν = −2, it equals SLEκ when κ ∈ (4, 8), SLEκ in Ω from x1 to x4 conditioned to avoid (x2, x3) is hSLEκ(κ − 6)

Hao Wu (THU) Hypergeometric SLE 18 / 28

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SLIDE 34

Hypergeometric SLE

General Boundary Conditions

Thm [Izyurov 2014, W. 2017] The interface of critical Ising model on Z2 with alternating boundary condition converges to Hypergeometric SLE3, denoted by hSLE3. Q1 : What is Hypergeometric SLE? Q2 : Why they are the limit? Q3 : How to prove the convergence? Answer to Q1 :

when ν = −2, it equals SLEκ when κ ∈ (4, 8), SLEκ in Ω from x1 to x4 conditioned to avoid (x2, x3) is hSLEκ(κ − 6) reversibility : the time-reversal has the same law. proved for ν ≥ κ/2 − 4;

Hao Wu (THU) Hypergeometric SLE 18 / 28

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SLIDE 35

Hypergeometric SLE

General Boundary Conditions

Thm [Izyurov 2014, W. 2017] The interface of critical Ising model on Z2 with alternating boundary condition converges to Hypergeometric SLE3, denoted by hSLE3. Q1 : What is Hypergeometric SLE? Q2 : Why they are the limit? Q3 : How to prove the convergence? Answer to Q1 :

when ν = −2, it equals SLEκ when κ ∈ (4, 8), SLEκ in Ω from x1 to x4 conditioned to avoid (x2, x3) is hSLEκ(κ − 6) reversibility : the time-reversal has the same law. proved for ν ≥ κ/2 − 4;? should be true for ν > −4 ∨ (κ/2 − 6).

Hao Wu (THU) Hypergeometric SLE 18 / 28

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Hypergeometric SLE

Q2 : Why they are the limit?

Recall : Conformal Invariance + Domain Markov Property → SLE(κ). Assume the scaling limit exists, then the limit should satisfy (CI) Conformal Invariance (DMP) Domain Markov Property (SYM) Symmetry

Hao Wu (THU) Hypergeometric SLE 19 / 28

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SLIDE 37

Hypergeometric SLE

Q2 : Why they are the limit?

Recall : Conformal Invariance + Domain Markov Property → SLE(κ). Assume the scaling limit exists, then the limit should satisfy (CI) Conformal Invariance (DMP) Domain Markov Property (SYM) Symmetry Thm [W.2017] Suppose (Pq, q ∈ Q) is a collection of proba, measures on pairs of simple curves that satisfies CI, DMP , and SYM. Then there exist κ ∈ (0, 4] and ν < κ − 6 such that Pq ∼ hSLEκ(ν).

Hao Wu (THU) Hypergeometric SLE 19 / 28

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Hypergeometric SLE

Q3 : How to prove the convergence?

⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊕ ⊖ ⊕ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊕ ⊕ ⊖ xL xR ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ yL yR

(ηL; ηR) : any subseq. limit L(ηL | ηR) = SLE3 L(ηR | ηL) = SLE3

Hao Wu (THU) Hypergeometric SLE 20 / 28

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SLIDE 39

Hypergeometric SLE

Q3 : How to prove the convergence?

⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊕ ⊖ ⊕ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊕ ⊕ ⊖ xL xR ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ yL yR

(ηL; ηR) : any subseq. limit L(ηL | ηR) = SLE3 L(ηR | ηL) = SLE3 Proposition Fix κ ∈ (0, 4]. There exists a unique probability measure on (ηL; ηR) such that L(ηL | ηR) = SLEκ L(ηR | ηL) = SLEκ The marginal of ηR is hSLEκ from xR to yR.

Hao Wu (THU) Hypergeometric SLE 20 / 28

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SLIDE 40

Hypergeometric SLE

Q3 : How to prove the convergence?

⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊕ ⊖ ⊕ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊕ ⊕ ⊖ xL xR ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ yL yR

(ηL; ηR) : any subseq. limit L(ηL | ηR) = SLE3 L(ηR | ηL) = SLE3 Proposition Fix κ ∈ (0, 4]. There exists a unique probability measure on (ηL; ηR) such that L(ηL | ηR) = SLEκ L(ηR | ηL) = SLEκ The marginal of ηR is hSLEκ from xR to yR. Conclusion ηR : hSLE3 from xR to yR.

Hao Wu (THU) Hypergeometric SLE 20 / 28

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SLIDE 41

Hypergeometric SLE

Convergence of Ising Interface to hSLE3

⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊕ ⊖ ⊕ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊕ ⊕ ⊖ xL xR ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ yL yR

Dobrushin b.c. : Interface→SLE3 RSW = ⇒ tightness Holomorphic observable Alternating b.c. : Interface→hSLE3

Hao Wu (THU) Hypergeometric SLE 21 / 28

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SLIDE 42

Hypergeometric SLE

Convergence of Ising Interface to hSLE3

⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊕ ⊖ ⊕ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊕ ⊖ ⊖ ⊕ ⊕ ⊖ ⊖ ⊕ ⊕ ⊕ ⊖ xL xR ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ yL yR

Dobrushin b.c. : Interface→SLE3 RSW = ⇒ tightness Holomorphic observable Alternating b.c. : Interface→hSLE3 First approach [Izyurov] RSW = ⇒ tightness New holomorphic observable Second approach [W.] RSW = ⇒ tightness Cvg with Dobrushin b.c. Advantage : more general b.c. Advantage : more general b.c. and for all κ ≤ 4.

Hao Wu (THU) Hypergeometric SLE 21 / 28

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SLIDE 43

Hypergeometric SLE

What about more complicate b.c.?

courtesy to E. Peltola

Global Multiple SLEs A collection of N disjoint simple curves (η1, . . . , ηN) ∈ X α(Ω; x1, . . . , x2N) such that ∀j, L(ηj | η1, . . . , ηj−1, ηj+1, . . . , ηN) = SLEκ

Hao Wu (THU) Hypergeometric SLE 22 / 28

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SLIDE 44

Hypergeometric SLE

What about more complicate b.c.?

courtesy to E. Peltola

Global Multiple SLEs A collection of N disjoint simple curves (η1, . . . , ηN) ∈ X α(Ω; x1, . . . , x2N) such that ∀j, L(ηj | η1, . . . , ηj−1, ηj+1, . . . , ηN) = SLEκ Thm [Korzdon & Lawler, Beffara & Peltola & W.] Fix κ ∈ (0, 4] and link pattern α ∈ LPN. There exists a unique global multiple SLEκ associated to α.

Hao Wu (THU) Hypergeometric SLE 22 / 28

slide-45
SLIDE 45

Hypergeometric SLE

What about more complicate b.c.?

courtesy to E. Peltola

Global Multiple SLEs A collection of N disjoint simple curves (η1, . . . , ηN) ∈ X α(Ω; x1, . . . , x2N) such that ∀j, L(ηj | η1, . . . , ηj−1, ηj+1, . . . , ηN) = SLEκ Thm [Korzdon & Lawler, Beffara & Peltola & W.] Fix κ ∈ (0, 4] and link pattern α ∈ LPN. There exists a unique global multiple SLEκ associated to α. Corollary The collection of Ising interfaces with alternating boundary conditions converges to global multiple SLE3.

Hao Wu (THU) Hypergeometric SLE 22 / 28

slide-46
SLIDE 46

Hypergeometric SLE

What about more complicate b.c.?

courtesy to E. Peltola

Global Multiple SLEs A collection of N disjoint simple curves (η1, . . . , ηN) ∈ X α(Ω; x1, . . . , x2N) such that ∀j, L(ηj | η1, . . . , ηj−1, ηj+1, . . . , ηN) = SLEκ Thm [Korzdon & Lawler, Beffara & Peltola & W.] Fix κ ∈ (0, 4] and link pattern α ∈ LPN. There exists a unique global multiple SLEκ associated to α. Corollary The collection of Ising interfaces with alternating boundary conditions converges to global multiple SLE3. Q : what is the marginal law?

Hao Wu (THU) Hypergeometric SLE 22 / 28

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SLIDE 47

Hypergeometric SLE

What is the marginal law?

For Ω = H and x1 < · · · < x2N, dWt = √κdBt + κ∂x1 log Zα(Wt, V 2

t , . . . , V 2N t

)dt, where Zα is the so-called pure partition functions. Pure Partition Functions {Zα : α ∈ LP} is a collection of smooth functions satisfying PDE, COV, ASY, and POS.

PDE :

  • κ

2 ∂2 i + j=i

  • 2

xj −xi ∂j − (6−κ)/κ (xj −xi )2

  • Z(x1, . . . , x2N) = 0.

COV : Z(x1, . . . , x2N) = 2N

i=1 ϕ′(xi )h × Z(ϕ(x1), . . . , ϕ(x2N)).

ASY : limxj ,xj+1→ξ

Zα(x1,...,x2N ) (xj+1−xj )−2h

= Z ˆ

α(x1, . . . , xj−1, xj+2, . . . , x2N)

Hao Wu (THU) Hypergeometric SLE 23 / 28

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SLIDE 48

Hypergeometric SLE

What is the marginal law?

For Ω = H and x1 < · · · < x2N, dWt = √κdBt + κ∂x1 log Zα(Wt, V 2

t , . . . , V 2N t

)dt, where Zα is the so-called pure partition functions. Pure Partition Functions {Zα : α ∈ LP} is a collection of smooth functions satisfying PDE, COV, ASY, and POS.

PDE :

  • κ

2 ∂2 i + j=i

  • 2

xj −xi ∂j − (6−κ)/κ (xj −xi )2

  • Z(x1, . . . , x2N) = 0.

COV : Z(x1, . . . , x2N) = 2N

i=1 ϕ′(xi )h × Z(ϕ(x1), . . . , ϕ(x2N)).

ASY : limxj ,xj+1→ξ

Zα(x1,...,x2N ) (xj+1−xj )−2h

= Z ˆ

α(x1, . . . , xj−1, xj+2, . . . , x2N)

Q1 : Do they exist? Q2 : Are they unique? Q3 : Are they explicit?

Hao Wu (THU) Hypergeometric SLE 23 / 28

slide-49
SLIDE 49

Hypergeometric SLE

Pure Partition Functions

Uniqueness [Flores & Kleban 2015] If there exist collections of smooth functions satisfying PDE, COV and ASY, they are (essentially) unique. Existence [Kytöla & Peltola 2016] For κ ∈ (0, 8) \ Q, there exists {Zα : α ∈ LP} satisfying PDE, COV, ASY. missing POS!

Hao Wu (THU) Hypergeometric SLE 24 / 28

slide-50
SLIDE 50

Hypergeometric SLE

Pure Partition Functions

Uniqueness [Flores & Kleban 2015] If there exist collections of smooth functions satisfying PDE, COV and ASY, they are (essentially) unique. Existence [Kytöla & Peltola 2016] For κ ∈ (0, 8) \ Q, there exists {Zα : α ∈ LP} satisfying PDE, COV, ASY. missing POS! Existence [Peltola & W. 2017] For κ ∈ (0, 4], there exists {Zα : α ∈ LP} satisfying PDE, COV, ASY and POS.

Hao Wu (THU) Hypergeometric SLE 24 / 28

slide-51
SLIDE 51

Hypergeometric SLE

Pure Partition Functions

Uniqueness [Flores & Kleban 2015] If there exist collections of smooth functions satisfying PDE, COV and ASY, they are (essentially) unique. Existence [Kytöla & Peltola 2016] For κ ∈ (0, 8) \ Q, there exists {Zα : α ∈ LP} satisfying PDE, COV, ASY. missing POS! Existence [Peltola & W. 2017] For κ ∈ (0, 4], there exists {Zα : α ∈ LP} satisfying PDE, COV, ASY and POS. Corollary The Ising Interface with alternating b.c. converges to the Lowener chain associated to Zα with κ = 3.

Hao Wu (THU) Hypergeometric SLE 24 / 28

slide-52
SLIDE 52

Hypergeometric SLE

Pure Partition Functions

Existence [Kytöla & Peltola 2016] For κ ∈ (0, 8) \ Q, there exists {Zα : α ∈ LP} satisfying PDE, COV, ASY. missing POS! Existence [Peltola & W. 2017] For κ ∈ (0, 4], there exists {Zα : α ∈ LP} satisfying PDE, COV, ASY and POS.

Hao Wu (THU) Hypergeometric SLE 25 / 28

slide-53
SLIDE 53

Hypergeometric SLE

Pure Partition Functions

Existence [Kytöla & Peltola 2016] For κ ∈ (0, 8) \ Q, there exists {Zα : α ∈ LP} satisfying PDE, COV, ASY. missing POS! Existence [Peltola & W. 2017] For κ ∈ (0, 4], there exists {Zα : α ∈ LP} satisfying PDE, COV, ASY and POS. κ ∈ (0, 8) \ Q Coulomb gas techniques PDE, COV, ASY : clear POS : missing κ ∈ (0, 4] global multiple SLEs COV, ASY, POS : clear Smoothness/PDE :

  • J. Dubédat, Hypoellipticity

Hao Wu (THU) Hypergeometric SLE 25 / 28

slide-54
SLIDE 54

Hypergeometric SLE

Pure Partition Functions

Existence [Kytöla & Peltola 2016] For κ ∈ (0, 8) \ Q, there exists {Zα : α ∈ LP} satisfying PDE, COV, ASY. missing POS! Existence [Peltola & W. 2017] For κ ∈ (0, 4], there exists {Zα : α ∈ LP} satisfying PDE, COV, ASY and POS. κ ∈ (0, 8) \ Q Coulomb gas techniques PDE, COV, ASY : clear POS : missing κ ∈ (0, 4] global multiple SLEs COV, ASY, POS : clear Smoothness/PDE :

  • J. Dubédat, Hypoellipticity

The optimal upper bound [Peltola & W. 2017] 0 < Zα(x1, . . . , x2N) ≤

N

  • j=1

(xbj − xaj)−2h.

Hao Wu (THU) Hypergeometric SLE 25 / 28

slide-55
SLIDE 55

Hypergeometric SLE

Crossing Probabilities of Ising Interfaces

Courtesy to E. Peltola

Conjecture : In progress The connection of Ising interfaces forms a planar link pattern Aδ. lim

δ→0 P[Aδ = α] =

Zα(Ω; x1, . . . , x2N) ZIsing(Ω; x1, . . . , x2N).

Hao Wu (THU) Hypergeometric SLE 26 / 28

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SLIDE 56

Hypergeometric SLE

Known Connection Probabilities

LERW [Kozdron & Lawler, Kenyon & Wilson, Karrila & Kytölä & Peltola] The connection probability of branches in Uniform Spanning Tree : lim

δ→0 δ−2NP[Aδ = α] = Zα(Ω; x1, . . . , x2N).

The collection {Zα : α ∈ LP} is explicit for κ = 2. Level Lines of GFF [Peltola & W.] The connection probability of level lines of Gaussian Free Field : P[A = α] = Zα(Ω; x1, . . . , x2N) ZGFF(Ω; x1, . . . , x2N). The collection {Zα : α ∈ LP} is explicit for κ = 4. (Wilson & Kenyon)

Hao Wu (THU) Hypergeometric SLE 27 / 28

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SLIDE 57

Hypergeometric SLE

Thanks! References

Percolation Critical Percolation in the Plane

  • S. Smirnov (C.R. Acad. Sci. Paris Sér. I Math. 2001)

Critical exponents for 2D percolation

  • S. Smirnov, W, Werner (Math. Res. Lett. 2001)

Ising and FK-Ising model Convergence of Ising Interfaces to Schramm’s SLE Curves

  • D. Chelkak, H. Duminil-Copin, C. Hongler, A. Kemppainen, S. Smirnov (Comptes Rendus Mathematique, 2014.)

Ising Interfaces and Free Boundary Conditions

  • C. Hongler, K. Kytola (Journal of the American Mathematical Society, 2013.)

Smirnov’s observable for free boundary conditions, interfaces, and crossing probabilities

  • K. Izyurov (Comm. Math. Phys., 2015.)

Alternating arm exponents for the critical planar Ising model

  • H. Wu (To appear in The Annals of Probability)

Hypergeometric SLE Hypergeometric SLE and Convergence of the Critical Planar Ising Interfaces

  • H. Wu (arxiv :1703.02022)

Pure Partition Functions A Solution Space for a System of Null-state Partial Differential Equations : Part 1—Part 4

  • S. Flores, P

. Kleban (Comm. Math. Phys., 2015.) Pure partition functions of multiple SLEs

  • K. Kytölä, E. Peltola. (Comm. Math. Phys., 2016.)

Global and Local Multiple SLE with κ ≤ 4 and Connection Probabilities of Level Lines of GFF

  • E. Peltola, H. Wu (arXiv : 1703.00898)

Hao Wu (THU) Hypergeometric SLE 28 / 28