Thick points for generalized Gaussian fields under different - - PowerPoint PPT Presentation

Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Thick points for generalized Gaussian fields under different cut-offs

Alessandra Cipriani1 Rajat Subhra Hazra2

1 Weierstrass Institute for Applied Analysis and Stochastics, Berlin 2 Indian Statistical Institute, Kolkata

October 24, 2014

Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Table of contents

Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Massive Gaussian Free Field

• For m ≥ 0 we consider S(Rd) and define on it the

differential operator Λm := (mI − ∆)d/2 .

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Massive Gaussian Free Field

• For m ≥ 0 we consider S(Rd) and define on it the

differential operator Λm := (mI − ∆)d/2 .

• It induces a norm φ :=

ΛφL2(Rd).

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Massive Gaussian Free Field

• For m ≥ 0 we consider S(Rd) and define on it the

differential operator Λm := (mI − ∆)d/2 .

• It induces a norm φ :=

ΛφL2(Rd).

• The closure is a Hilbert space (H, · ).

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Massive Gaussian Free Field

• For m ≥ 0 we consider S(Rd) and define on it the

differential operator Λm := (mI − ∆)d/2 .

• It induces a norm φ :=

ΛφL2(Rd).

• The closure is a Hilbert space (H, · ).
• Define

L(φ) = exp

• −1

2φ2

• .

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Massive Gaussian Free Field

• For m ≥ 0 we consider S(Rd) and define on it the

differential operator Λm := (mI − ∆)d/2 .

• It induces a norm φ :=

ΛφL2(Rd).

• The closure is a Hilbert space (H, · ).
• Define

L(φ) = exp

• −1

2φ2

• .
• Remark If m > 0, Λmφ = Gm ∗ φ, with

Gm(x) = +∞

1

k(ux) u du. (1)

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

MBS theorem

Theorem (Minlos, Bochner-Schwartz)

A characteristic functional L(·) on H such that

• i. L(0) = 1,
• ii. L is continuous in the induced Fréchet topology on H,
• iii. L is positive semi-definite on H∗

denotes uniquely a probability measure W on (H∗, H) by L(φ) =

• H∗ eix, φ W(dx).

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

MBS theorem

Theorem (Minlos, Bochner-Schwartz)

A characteristic functional L(·) on H such that

• i. L(0) = 1,
• ii. L is continuous in the induced Fréchet topology on H,
• iii. L is positive semi-definite on H∗

denotes uniquely a probability measure W on (H∗, H) by L(φ) =

• H∗ eix, φ W(dx).
• Moral of the story: H + · + L(·) ❀ (H∗, H, W) .

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

MBS theorem

Theorem (Minlos, Bochner-Schwartz)

A characteristic functional L(·) on H such that

• i. L(0) = 1,
• ii. L is continuous in the induced Fréchet topology on H,
• iii. L is positive semi-definite on H∗

denotes uniquely a probability measure W on (H∗, H) by L(φ) =

• H∗ eix, φ W(dx).
• Moral of the story: H + · + L(·) ❀ (H∗, H, W) .
• X ∼ W is a random distribution, not a function!

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Examples

• In the case of (mI − ∆)d/2 X is the massive Gaussian

Free Field.

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Examples

• In the case of (mI − ∆)d/2 X is the massive Gaussian

Free Field.

• In the case of D ⊆ R2 bounded domain, H = H1

0(D) and

Λ = −∆ X is the planar Gaussian Free Field with Dirichlet boundary conditions.

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Some remarks

• MBS yields a Gaussian structure.

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Some remarks

• MBS yields a Gaussian structure.
• Formally one can write X = (X(x)) a centered Gaussian

field with covariance kernel K(·, ·).

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Some remarks

• MBS yields a Gaussian structure.
• Formally one can write X = (X(x)) a centered Gaussian

field with covariance kernel K(·, ·).

• We look at fields X such that

E [X(x)X(y)] = K(x, y) ∼ − log x − y as x → y: generalized Gaussian fields.

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Why do we look at such fields?

• Physicists are interested in random measures of the form

µγ(dx) = exp

• γX(x) − γ2

2 E

• X 2(x)
• dx, γ > 0

where X is the planar GFF: quantum gravity.

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Why do we look at such fields?

• Physicists are interested in random measures of the form

µγ(dx) = exp

• γX(x) − γ2

2 E

• X 2(x)
• dx, γ > 0

where X is the planar GFF: quantum gravity.

• Being X a distribution µγ doesn’t make sense ❀ need for

approximation.

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Why do we look at such fields?

• Physicists are interested in random measures of the form

µγ(dx) = exp

• γX(x) − γ2

2 E

• X 2(x)
• dx, γ > 0

where X is the planar GFF: quantum gravity.

• Being X a distribution µγ doesn’t make sense ❀ need for

approximation.

• KPZ relation and quantum gravity:

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

The integral cut-offs

One can define an approximating field Xǫ by approximating the covariance kernel K.

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

The integral cut-offs

One can define an approximating field Xǫ by approximating the covariance kernel K. K(x, y) = ∞

1 k(ux−y) u

du

(1)

Kǫ(x, y) = ǫ−1

1 k(ux−y) u

du

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

The integral cut-offs

One can define an approximating field Xǫ by approximating the covariance kernel K. K(x, y) = ∞

1 k(ux−y) u

du

(1)

KD(x, y) = ∞

0 pD(t, x, y)dt

Kǫ(x, y) = ǫ−1

1 k(ux−y) u

du K (ǫ)

D =

∞

ǫ

pD(t, x, y)dt

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

The integral cut-offs

One can define an approximating field Xǫ by approximating the covariance kernel K. K(x, y) = ∞

1 k(ux−y) u

du

(1)

KD(x, y) = ∞

0 pD(t, x, y)dt

K(x, y) := +∞

k=1 pk(x, y)

Kǫ(x, y) = ǫ−1

1 k(ux−y) u

du K (ǫ)

D =

∞

ǫ

pD(t, x, y)dt Kn(x, y) :=

k≤n pk(x, y)

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

The integral cut-offs

One can define an approximating field Xǫ by approximating the covariance kernel K. K(x, y) = ∞

1 k(ux−y) u

du

(1)

KD(x, y) = ∞

0 pD(t, x, y)dt

K(x, y) := +∞

k=1 pk(x, y)

Kǫ(x, y) = ǫ−1

1 k(ux−y) u

du K (ǫ)

D =

∞

ǫ

pD(t, x, y)dt Kn(x, y) :=

k≤n pk(x, y)

Xǫ with covariance kernel Kǫ is a well-defined field.

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

The white noise cut-off

White noise representation Let W be a standard complex white noise. Then formally the massive GFF X on Rd can be represented as X(x) =

• Rd e−iπ(x, ξ)Rd
• K(ξ)W (dξ).

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

The white noise cut-off

White noise representation Let W be a standard complex white noise. Then formally the massive GFF X on Rd can be represented as X(x) =

• Rd e−iπ(x, ξ)Rd
• K(ξ)W (dξ).

White noise cut-off Xǫ(x) :=

• B(0, ǫ−1) e−iπ(x, ξ)Rd
• K(ξ)W (dξ).

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Hausdorff dimension of the thick points

The set of a-thick points is T(a, D) =

• x ∈ D : lim

ǫ→0

Xǫ(x) Var (Xǫ(x)) = a

• ,

a > 0.

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Hausdorff dimension of the thick points

The set of a-thick points is T(a, D) =

• x ∈ D : lim

ǫ→0

Xǫ(x) Var (Xǫ(x)) = a

• ,

a > 0.

Theorem (C. - Hazra ’14)

Let X be a generalized Gaussian field on D and Xǫ a cut-off. Under the assumptions (A)-(D) it holds that a. s. dimH(T(a, D)) = d − a2

2 for a ≤

√ 2d, and T(a, D) is empty for a > √ 2d.

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Support of quantum gravity

Figure: Quantum gravity measure on the square (M. Biskup, O. Louidor)

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Upper bound

Theorem

A centered Gaussian process (Xǫ(x))ǫ≥0, x∈D, d ≥ 2, satisfying (A) for all R > 0 and for all x, y ∈ D and ǫ, η ≥ 0 E

• (Xǫ(x) − Xη(y))2

≤ x − y + |η − ǫ| η ∧ ǫ ,

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Upper bound

Theorem

A centered Gaussian process (Xǫ(x))ǫ≥0, x∈D, d ≥ 2, satisfying (A) for all R > 0 and for all x, y ∈ D and ǫ, η ≥ 0 E

• (Xǫ(x) − Xη(y))2

≤ x − y + |η − ǫ| η ∧ ǫ , (B) E(Xǫ(x)2) ∼ǫ→0 − log ǫ,

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Upper bound

Theorem

A centered Gaussian process (Xǫ(x))ǫ≥0, x∈D, d ≥ 2, satisfying (A) for all R > 0 and for all x, y ∈ D and ǫ, η ≥ 0 E

• (Xǫ(x) − Xη(y))2

≤ x − y + |η − ǫ| η ∧ ǫ , (B) E(Xǫ(x)2) ∼ǫ→0 − log ǫ, is such that almost surely

• dimH(T(a, D)) ≤ d − a2

2 for a ≤

√ 2d,

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Upper bound

Theorem

A centered Gaussian process (Xǫ(x))ǫ≥0, x∈D, d ≥ 2, satisfying (A) for all R > 0 and for all x, y ∈ D and ǫ, η ≥ 0 E

• (Xǫ(x) − Xη(y))2

≤ x − y + |η − ǫ| η ∧ ǫ , (B) E(Xǫ(x)2) ∼ǫ→0 − log ǫ, is such that almost surely

• dimH(T(a, D)) ≤ d − a2

2 for a ≤

√ 2d,

• T(a, D) = ∅ for a >

√ 2d.

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Upper bound

Theorem

A centered Gaussian process (Xǫ(x))ǫ≥0, x∈D, d ≥ 2, satisfying (A) for all R > 0 and for all x, y ∈ D and ǫ, η ≥ 0 E

• (Xǫ(x) − Xη(y))2

≤ x − y + |η − ǫ| η ∧ ǫ , (B) E(Xǫ(x)2) ∼ǫ→0 − log ǫ, is such that almost surely

• dimH(T(a, D)) ≤ d − a2

2 for a ≤

√ 2d,

• T(a, D) = ∅ for a >

√ 2d. Main tool of the proof: Kolmogorov-Centsov theorem by (A).

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Lower bound

Theorem

A centered Gaussian process (Xn(x))n∈N, x∈[0,1]d with covariance kernels Kn(x, y) satisfying: (C) for x = y, Kn(x, y) ≤ − 1

2 log x − y + H(x, y) where

supx=y∈[0,1]d H(x, y) < C < ∞,

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Lower bound

Theorem

A centered Gaussian process (Xn(x))n∈N, x∈[0,1]d with covariance kernels Kn(x, y) satisfying: (C) for x = y, Kn(x, y) ≤ − 1

2 log x − y + H(x, y) where

supx=y∈[0,1]d H(x, y) < C < ∞, (D) there exists a sequence of positive definite covariance kernels Kn(x, y) such that Kn(x, y) =

k≤n

Kk(x, y), with Kk(x, x) = 1.

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Lower bound

Theorem

A centered Gaussian process (Xn(x))n∈N, x∈[0,1]d with covariance kernels Kn(x, y) satisfying: (C) for x = y, Kn(x, y) ≤ − 1

2 log x − y + H(x, y) where

supx=y∈[0,1]d H(x, y) < C < ∞, (D) there exists a sequence of positive definite covariance kernels Kn(x, y) such that Kn(x, y) =

k≤n

Kk(x, y), with Kk(x, x) = 1. is such that for 0 < a ≤ √ 2d dimH T(a) = dimH

• x ∈ [0, 1]d :

lim

n→∞

Xn n = a

• ≥ d−a2

2

• a. s.

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Sketch of proof

Kahane’s strategy: Peyrière’s or rooted measures.

• Construct the measures exp
• aXn(x) − a2

2 Kn(x, x)

• dx.

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Sketch of proof

Kahane’s strategy: Peyrière’s or rooted measures.

• Construct the measures exp
• aXn(x) − a2

2 Kn(x, x)

• dx.
• Break up Xn(x) =

k≤n

Xk(x) almost surely.

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Sketch of proof

Kahane’s strategy: Peyrière’s or rooted measures.

• Construct the measures exp
• aXn(x) − a2

2 Kn(x, x)

• dx.
• Break up Xn(x) =

k≤n

Xk(x) almost surely.

• Under the rooted measure,
• Xk
• k are i. i. d. LLN yields
• k≤n

Xk(x) n

→ a almost surely.

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Sketch of proof

Kahane’s strategy: Peyrière’s or rooted measures.

• Construct the measures exp
• aXn(x) − a2

2 Kn(x, x)

• dx.
• Break up Xn(x) =

k≤n

Xk(x) almost surely.

• Under the rooted measure,
• Xk
• k are i. i. d. LLN yields
• k≤n

Xk(x) n

→ a almost surely.

• Conclude by finite energy methods.

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Universality

Theorem (C. - Hazra ’14)

Let Xǫ and Yǫ be two cut-off families for the same GGF. Call Zǫ(x) := Xǫ(x) − Yǫ(x). Suppose there exist constants C, C ′ such that

• i. E [Zǫ(x)2] ≤ C,
• ii. E [(Zǫ(x) − Zǫ(y))2] ≤ C ′ |

|x−y| | ǫ

. Then the thick points of Xǫ and Yǫ have the same Hausdorff dimension almost surely.

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Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points

Thank you for your attention!

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