Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points Thick points for generalized Gaussian fields under different cut-offs Alessandra Cipriani 1 Rajat Subhra Hazra 2 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 2 / 16
Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points Massive Gaussian Free Field • For m ≥ 0 we consider S ( R d ) and define on it the differential operator Λ m := ( m I − ∆) d / 2 . 3 / 16
Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points Massive Gaussian Free Field • For m ≥ 0 we consider S ( R d ) and define on it the differential operator Λ m := ( m I − ∆) d / 2 . • It induces a norm � φ � := � � Λ φ � L 2 ( R d ) . 3 / 16
Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points Massive Gaussian Free Field • For m ≥ 0 we consider S ( R d ) and define on it the differential operator Λ m := ( m I − ∆) d / 2 . • It induces a norm � φ � := � � Λ φ � L 2 ( R d ) . • The closure is a Hilbert space ( H , � · � ) . 3 / 16
Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points Massive Gaussian Free Field • For m ≥ 0 we consider S ( R d ) and define on it the differential operator Λ m := ( m I − ∆) d / 2 . • It induces a norm � φ � := � � Λ φ � L 2 ( R d ) . • The closure is a Hilbert space ( H , � · � ) . • Define � � − 1 2 � φ � 2 L ( φ ) = exp . 3 / 16
Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points Massive Gaussian Free Field • For m ≥ 0 we consider S ( R d ) and define on it the differential operator Λ m := ( m I − ∆) d / 2 . • It induces a norm � φ � := � � Λ φ � L 2 ( R d ) . • The closure is a Hilbert space ( H , � · � ) . • Define � � − 1 2 � φ � 2 L ( φ ) = exp . • Remark If m > 0, Λ m φ = G m ∗ φ, with � + ∞ k ( u � x � ) G m ( x ) = (1) d u . u 1 3 / 16
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 � H ∗ e i � x , φ � W ( d x ) . L ( φ ) = 4 / 16
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 � H ∗ e i � x , φ � W ( d x ) . L ( φ ) = • Moral of the story: H + � · � + L ( · ) ❀ ( H ∗ , H , W ) . 4 / 16
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 � H ∗ e i � x , φ � W ( d x ) . L ( φ ) = • Moral of the story: H + � · � + L ( · ) ❀ ( H ∗ , H , W ) . • X ∼ W is a random distribution, not a function! 4 / 16
Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points Examples • In the case of ( m I − ∆) d / 2 X is the massive Gaussian Free Field. 5 / 16
Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points Examples • In the case of ( m I − ∆) d / 2 X is the massive Gaussian Free Field. • In the case of D ⊆ R 2 bounded domain, H = H 1 0 ( D ) and Λ = − ∆ X is the planar Gaussian Free Field with Dirichlet boundary conditions. 5 / 16
Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points Some remarks • MBS yields a Gaussian structure. 6 / 16
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 ( · , · ) . 6 / 16
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. 6 / 16
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 � �� γ X ( x ) − γ 2 � X 2 ( x ) µ γ ( d x ) = exp 2 E d x , γ > 0 where X is the planar GFF: quantum gravity. 7 / 16
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 � �� γ X ( x ) − γ 2 � X 2 ( x ) µ γ ( d x ) = exp 2 E d x , γ > 0 where X is the planar GFF: quantum gravity. • Being X a distribution µ γ doesn’t make sense ❀ need for approximation. 7 / 16
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 � �� γ X ( x ) − γ 2 � X 2 ( x ) µ γ ( d x ) = exp 2 E d x , γ > 0 where X is the planar GFF: quantum gravity. • Being X a distribution µ γ doesn’t make sense ❀ need for approximation. • KPZ relation and quantum gravity: 7 / 16
Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points The integral cut-offs One can define an approximating field X ǫ by approximating the covariance kernel K . 8 / 16
Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points The integral cut-offs One can define an approximating field X ǫ by approximating the covariance kernel K . � ∞ � ǫ − 1 k ( u � x − y � ) k ( u � x − y � ) K ( x , y ) = d u K ǫ ( x , y ) = d u 1 u 1 u (1) 8 / 16
Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points The integral cut-offs One can define an approximating field X ǫ by approximating the covariance kernel K . � ∞ � ǫ − 1 k ( u � x − y � ) k ( u � x − y � ) K ( x , y ) = d u K ǫ ( x , y ) = d u 1 u 1 u (1) � ∞ � ∞ K ( ǫ ) K D ( x , y ) = 0 p D ( t , x , y ) d t D = p D ( t , x , y ) d t ǫ 8 / 16
Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points The integral cut-offs One can define an approximating field X ǫ by approximating the covariance kernel K . � ∞ � ǫ − 1 k ( u � x − y � ) k ( u � x − y � ) K ( x , y ) = d u K ǫ ( x , y ) = d u 1 u 1 u (1) � ∞ � ∞ K ( ǫ ) K D ( x , y ) = 0 p D ( t , x , y ) d t D = p D ( t , x , y ) d t ǫ K ( x , y ) := � + ∞ K n ( x , y ) := � k = 1 p k ( x , y ) k ≤ n p k ( x , y ) 8 / 16
Generalized Gaussian fields Approximating the field Hausdorff dimension of the thick points The integral cut-offs One can define an approximating field X ǫ by approximating the covariance kernel K . � ∞ � ǫ − 1 k ( u � x − y � ) k ( u � x − y � ) K ( x , y ) = d u K ǫ ( x , y ) = d u 1 u 1 u (1) � ∞ � ∞ K ( ǫ ) K D ( x , y ) = 0 p D ( t , x , y ) d t D = p D ( t , x , y ) d t ǫ K ( x , y ) := � + ∞ K n ( x , y ) := � k = 1 p k ( x , y ) k ≤ n p k ( x , y ) X ǫ with covariance kernel K ǫ is a well-defined field. 8 / 16
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 R d can be represented as � � � R d e − i π ( x , ξ ) R d X ( x ) = K ( ξ ) W ( d ξ ) . 9 / 16
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 R d can be represented as � � � R d e − i π ( x , ξ ) R d X ( x ) = K ( ξ ) W ( d ξ ) . White noise cut-off � � � B ( 0 , ǫ − 1 ) e − i π ( x , ξ ) R d X ǫ ( x ) := K ( ξ ) W ( d ξ ) . 9 / 16
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 � � X ǫ ( x ) T ( a , D ) = x ∈ D : lim Var ( X ǫ ( x )) = a a > 0 . , ǫ → 0 10 / 16
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