Hadamards formula and couplings of SLE with GFF K. Izyurov and K. - - PowerPoint PPT Presentation

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Hadamards formula and couplings of SLE with GFF K. Izyurov and K. - - PowerPoint PPT Presentation

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Hadamards formula and couplings of SLE with GFF K. Izyurov and K. Kyt ol a Universit e de Gen` eve May 24, 2010 K. Izyurov and K. Kyt ol a Hadamards formula and couplings of SLE with GFF The Gaussian Free Field K.

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Hadamard’s formula and couplings of SLE with GFF

  • K. Izyurov and K. Kyt¨
  • l¨

a

Universit´ e de Gen` eve

May 24, 2010

  • K. Izyurov and K. Kyt¨
  • l¨

a Hadamard’s formula and couplings of SLE with GFF

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The Gaussian Free Field

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The Gaussian Free Field

A random (gaussian) field Φ : Ω → R in a planar domain

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The Gaussian Free Field

A random (gaussian) field Φ : Ω → R in a planar domain The mean of the field M(z) = EΦ(z) is a harmonic function

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The Gaussian Free Field

A random (gaussian) field Φ : Ω → R in a planar domain The mean of the field M(z) = EΦ(z) is a harmonic function (usually defined by boundary conditions: Dirichlet, Neumann, etc...)

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The Gaussian Free Field

A random (gaussian) field Φ : Ω → R in a planar domain The mean of the field M(z) = EΦ(z) is a harmonic function (usually defined by boundary conditions: Dirichlet, Neumann, etc...) The covariance of the field C(z1, z2) = G(z1, z2) is a Green’s function in Ω

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The Gaussian Free Field

A random (gaussian) field Φ : Ω → R in a planar domain The mean of the field M(z) = EΦ(z) is a harmonic function (usually defined by boundary conditions: Dirichlet, Neumann, etc...) The covariance of the field C(z1, z2) = G(z1, z2) is a Green’s function in Ω (with corresponding homogeneous boundary conditions)

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Relations to SLE: level lines

Schramm & Sheffield ’2006

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Relations to SLE: level lines

Schramm & Sheffield ’2006 Domains with two marked points x, x1, with Dirichlet boundary conditions ±λ = ± π

8 .

Dirichlet boundary valued Green’s function as covariance Discretize the field, take the mesh to zero

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Relations to SLE: level lines

Schramm & Sheffield ’2006 Domains with two marked points x, x1, with Dirichlet boundary conditions ±λ = ± π

8 .

Dirichlet boundary valued Green’s function as covariance Discretize the field, take the mesh to zero ⇒ level lines converge to SLE4

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Soft approach: coupling

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Soft approach: coupling

In the continuum: there exists a coupling of SLE4 and GFF, such that the curve behaves like a level line.

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Soft approach: coupling

In the continuum: there exists a coupling of SLE4 and GFF, such that the curve behaves like a level line. Namely: Conditionally on the curve γt, the law of the field is that of the GFF in Ω\γt, the jump has moved to the tip

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Constructive formulation: sample SLE4 curve up to time t; sample GFF in Ω\γt; forget the curve ⇒ obtain a new field ˜ Φ in Ω

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Constructive formulation: sample SLE4 curve up to time t; sample GFF in Ω\γt; forget the curve ⇒ obtain a new field ˜ Φ in Ω which appears to have the same law as Φ.

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Soft approach: coupling

  • K. Izyurov and K. Kyt¨
  • l¨

a Hadamard’s formula and couplings of SLE with GFF

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Soft approach: coupling

Other boundary conditions far away from the curve?

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Soft approach: coupling

Other boundary conditions far away from the curve? Doubly connected domains?

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: simply-connected case

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: simply-connected case

Three arcs, boundary values −λ, λ, Neumann: dipolar SLE4.

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: simply-connected case

Three arcs, boundary values −λ, λ, Neumann: dipolar SLE4. Three arcs, boundary values −λ, λ, 0: dipolar SLE4.

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: simply-connected case

Three arcs, boundary values −λ, λ, Neumann: dipolar SLE4. Three arcs, boundary values −λ, λ, 0: dipolar SLE4. Three arcs, boundary values −λ, λ, Riemann-Hilbert: ∂σM(z) = 0, σ = eiατ: SLE4(ρ) with ρ depending on α.

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: simply-connected case

More marked points, jump-Dirichlet boundary conditions: SLE4(ρ1, ρ2, . . . ) with ρ′s proportional to jumps (Schramm & Sheffield, Cardy, Dub´ edat).

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: simply-connected case

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: simply-connected case

More marked points, mixed boundary conditions: not SLE4(ρ)!

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: simply-connected case

More marked points, mixed boundary conditions: not SLE4(ρ)! But the drift still can be computed. Expression involves derivatives of M and its harmonic conjugate w.r.t marked points.

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: doubly-connected case

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: doubly-connected case

One marked point on the outer boundary with jump −2λ ⇒ multi-valued mean. Neumann boundary conditions on the inner boundary

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: doubly-connected case

One marked point on the outer boundary with jump −2λ ⇒ multi-valued mean. Neumann boundary conditions on the inner boundary Coupled with annular SLE4.

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: doubly-connected case

Two marked points on the outer boundary (Hagendorf, Bauer, Bernard’09 via partition function): some annulus analogs of SLE4(ρ). On the inner boundary: either Neumann or Dirichlet

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: doubly-connected case

Two marked points on the outer boundary (Hagendorf, Bauer, Bernard’09 via partition function): some annulus analogs of SLE4(ρ). On the inner boundary: either Neumann or Dirichlet Drifts are computed explicitly (in terms of Schwarz kernels in the annulus), and the existence of couplings is proven.

  • K. Izyurov and K. Kyt¨
  • l¨

a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: doubly-connected case

Two marked points on the outer boundary (Hagendorf, Bauer, Bernard’09 via partition function): some annulus analogs of SLE4(ρ). On the inner boundary: either Neumann or Dirichlet Drifts are computed explicitly (in terms of Schwarz kernels in the annulus), and the existence of couplings is proven. Easily generalizes to many marked points x1, x2, . . . on the

  • uter boundary (of total jump 2λ in Dirichlet case)
  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: doubly-connected case

  • K. Izyurov and K. Kyt¨
  • l¨

a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: doubly-connected case

One marked point on the inner boundary; Dirichlet boundary conditions.

  • K. Izyurov and K. Kyt¨
  • l¨

a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: doubly-connected case

One marked point on the inner boundary; Dirichlet boundary conditions. Still one integer parameter to fix: can add an integer multiple

  • f λ on the inner boundary.
  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: doubly-connected case

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: doubly-connected case

This leads to a curve with a prescribed winding.

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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The zoo of examples: doubly-connected case

This leads to a curve with a prescribed winding. Indeed; eventually the winding is as it’s supposed to be.

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Different κ?

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Different κ?

All results concerning Dirichlet boundary conditions generalize to different κ;

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a Hadamard’s formula and couplings of SLE with GFF

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Different κ?

All results concerning Dirichlet boundary conditions generalize to different κ; The conformal transformation rule of the field: ΦΩ(z) = Φϕ(Ω)(ϕ(z)) + β arg ϕ′(z);

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Different κ?

All results concerning Dirichlet boundary conditions generalize to different κ; The conformal transformation rule of the field: ΦΩ(z) = Φϕ(Ω)(ϕ(z)) + β arg ϕ′(z); Neumann boundary conditions do not generalize.

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Different κ?

All results concerning Dirichlet boundary conditions generalize to different κ; The conformal transformation rule of the field: ΦΩ(z) = Φϕ(Ω)(ϕ(z)) + β arg ϕ′(z); Neumann boundary conditions do not generalize. Question: what is the natural coupling of annulus GFF with SLEκ for κ = 4?.

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Multiply connected domains

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a Hadamard’s formula and couplings of SLE with GFF

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Multiply connected domains

Still possible to prove that there exists a coupling for a unique drift...

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Multiply connected domains

Still possible to prove that there exists a coupling for a unique drift... and compute that drift

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Multiply connected domains

Still possible to prove that there exists a coupling for a unique drift... and compute that drift in terms of derivatives of M and ˜ M etc w. r. t. marked points and conformal moduli parameters.

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Proof: simple case

Domain: half-plane H; two marked points x, x1

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Proof: simple case

Domain: half-plane H; two marked points x, x1 Fields ΦH, ΦH\γt coincide in distribution ⇒

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a Hadamard’s formula and couplings of SLE with GFF

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Proof: simple case

Domain: half-plane H; two marked points x, x1 Fields ΦH, ΦH\γt coincide in distribution ⇒ EΦ(z) = ESLEEΦΩt(z)=E(Φ ◦ gt)(z)

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Proof: simple case

Domain: half-plane H; two marked points x, x1 Fields ΦH, ΦH\γt coincide in distribution ⇒ EΦ(z) = ESLEEΦΩt(z)=E(Φ ◦ gt)(z) EΦ(z1)Φ(z2) = E(Φ ◦ gt)(z1))(Φ ◦ gt)(z2))

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Proof: simple case

Domain: half-plane H; two marked points x, x1 Fields ΦH, ΦH\γt coincide in distribution ⇒ EΦ(z) = ESLEEΦΩt(z)=E(Φ ◦ gt)(z) EΦ(z1)Φ(z2) = E(Φ ◦ gt)(z1))(Φ ◦ gt)(z2)) We actually prove:

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Proof: simple case

Domain: half-plane H; two marked points x, x1 Fields ΦH, ΦH\γt coincide in distribution ⇒ EΦ(z) = ESLEEΦΩt(z)=E(Φ ◦ gt)(z) EΦ(z1)Φ(z2) = E(Φ ◦ gt)(z1))(Φ ◦ gt)(z2)) We actually prove: M(Xt, gt(x1), gt(z)) is a martingale;

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Proof: simple case

Domain: half-plane H; two marked points x, x1 Fields ΦH, ΦH\γt coincide in distribution ⇒ EΦ(z) = ESLEEΦΩt(z)=E(Φ ◦ gt)(z) EΦ(z1)Φ(z2) = E(Φ ◦ gt)(z1))(Φ ◦ gt)(z2)) We actually prove: M(Xt, gt(x1), gt(z)) is a martingale; G(gt(z1), gt(z2)) + M(Xt, gt(x1), gt(z))M(Xt, gt(x1), gt(z)) is a martingale.

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Proof: one-point function is a martingale

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Proof: one-point function is a martingale

Let M = ℑF, F analytic in z; dXt := √κdBt + Dtdt

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a Hadamard’s formula and couplings of SLE with GFF

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Proof: one-point function is a martingale

Let M = ℑF, F analytic in z; dXt := √κdBt + Dtdt dM(Xt, gt(x1), gt(z)) = ℑ

  • κ

2∂xxF + 2 z−x ∂zF + 2 x1−x ∂x1F + Dt∂xF

  • dt +

ℑκ∂xFdBt|x,x1,z→Xt,gt(x1),gt(z)

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a Hadamard’s formula and couplings of SLE with GFF

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Proof: Two-point function is a martingale

Second equation: dG(gt(z1), gt(z2)) = −d[M(..., ..., gt(z1))M(..., ..., gt(z2))]

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Proof: Two-point function is a martingale

Second equation: dG(gt(z1), gt(z2)) = −d[M(..., ..., gt(z1))M(..., ..., gt(z2))] If the first equation holds, then M is a martingale, with dB part equal to = √κℑ∂xF(gt(z1))dB

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Proof: Two-point function is a martingale

Second equation: dG(gt(z1), gt(z2)) = −d[M(..., ..., gt(z1))M(..., ..., gt(z2))] If the first equation holds, then M is a martingale, with dB part equal to = √κℑ∂xF(gt(z1))dB= −√κPXT (gt(z1))dB

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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Proof: Two-point function is a martingale

Second equation: dG(gt(z1), gt(z2)) = −d[M(..., ..., gt(z1))M(..., ..., gt(z2))] If the first equation holds, then M is a martingale, with dB part equal to = √κℑ∂xF(gt(z1))dB= −√κPXT (gt(z1))dB The equation above is Hadamard’s formula; easily generalizes to other cases.

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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One-point function is a martingale

First equation: dM(Xt, gt(x1), gt(z)) = ℑ

  • κ

2∂xxF + 2 z−x ∂zF + 2 x1−x ∂x1F + Dt∂xF

  • = 0
  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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SLIDE 64

One-point function is a martingale

First equation: dM(Xt, gt(x1), gt(z)) = ℑ

  • κ

2∂xxF + 2 z−x ∂zF + 2 x1−x ∂x1F + Dt∂xF

  • = 0

LHS: Zero Dirichlet boundary condtions apart from x;

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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SLIDE 65

One-point function is a martingale

First equation: dM(Xt, gt(x1), gt(z)) = ℑ

  • κ

2∂xxF + 2 z−x ∂zF + 2 x1−x ∂x1F + Dt∂xF

  • = 0

LHS: Zero Dirichlet boundary condtions apart from x; Possible singularity at x

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF

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SLIDE 66

One-point function is a martingale

First equation: dM(Xt, gt(x1), gt(z)) = ℑ

  • κ

2∂xxF + 2 z−x ∂zF + 2 x1−x ∂x1F + Dt∂xF

  • = 0

LHS: Zero Dirichlet boundary condtions apart from x; Possible singularity at x There exists a unique Dt that cancels it out!

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a Hadamard’s formula and couplings of SLE with GFF

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Thank you!

  • K. Izyurov and K. Kyt¨
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a Hadamard’s formula and couplings of SLE with GFF