Decompositions of log-correlated fields with applications Eero Saksman (University of Helsinki) Based on joint work with Janne Junnila (EPFL Lausanne) and Christian Webb (Aalto University) CONFERENCE IN HONOR OF JOHN AND DON AND JOHN Seattle 23/08/19 Saksman Decompositions of log-correlated fields Seattle 23/08/19 1 / 25
GFF A centered Gaussian field on a domain Ω ⊂ R d can be be thought of • as a random function X : Ω → C such that all the evaluation vectors vectors ( X ( z 1 ) , . . . , X ( z n )) are multivariate and centered Gaussians. Their statistical properties (for us) are determined by knowledge of the C X ( z , z ′ ) := EX ( z ) X ( z ′ ) . covariance function Saksman Decompositions of log-correlated fields Seattle 23/08/19 2 / 25
GFF A centered Gaussian field on a domain Ω ⊂ R d can be be thought of • as a random function X : Ω → C such that all the evaluation vectors vectors ( X ( z 1 ) , . . . , X ( z n )) are multivariate and centered Gaussians. Their statistical properties (for us) are determined by knowledge of the C X ( z , z ′ ) := EX ( z ) X ( z ′ ) . covariance function • Example: Consider the centered Gaussian field X on the torus T := {| z | = 1 } with the covariance structure 1 z , z ′ ∈ T . K X c i ( z , z ′ ) = ” E X c i ( z ) X c i ( z ′ )” = log � � for | z − z ′ | Such random functions ” T ∋ z �→ X ( z )” form the Gaussian Free Field (GFF) , restricted to T . Saksman Decompositions of log-correlated fields Seattle 23/08/19 2 / 25
GFF A centered Gaussian field on a domain Ω ⊂ R d can be be thought of • as a random function X : Ω → C such that all the evaluation vectors vectors ( X ( z 1 ) , . . . , X ( z n )) are multivariate and centered Gaussians. Their statistical properties (for us) are determined by knowledge of the C X ( z , z ′ ) := EX ( z ) X ( z ′ ) . covariance function • Example: Consider the centered Gaussian field X on the torus T := {| z | = 1 } with the covariance structure 1 z , z ′ ∈ T . K X c i ( z , z ′ ) = ” E X c i ( z ) X c i ( z ′ )” = log � � for | z − z ′ | Such random functions ” T ∋ z �→ X ( z )” form the Gaussian Free Field (GFF) , restricted to T . • X c i takes values in the generalized functions – need some care! Saksman Decompositions of log-correlated fields Seattle 23/08/19 2 / 25
GFF A centered Gaussian field on a domain Ω ⊂ R d can be be thought of • as a random function X : Ω → C such that all the evaluation vectors vectors ( X ( z 1 ) , . . . , X ( z n )) are multivariate and centered Gaussians. Their statistical properties (for us) are determined by knowledge of the C X ( z , z ′ ) := EX ( z ) X ( z ′ ) . covariance function • Example: Consider the centered Gaussian field X on the torus T := {| z | = 1 } with the covariance structure 1 z , z ′ ∈ T . K X c i ( z , z ′ ) = ” E X c i ( z ) X c i ( z ′ )” = log � � for | z − z ′ | Such random functions ” T ∋ z �→ X ( z )” form the Gaussian Free Field (GFF) , restricted to T . • X c i takes values in the generalized functions – need some care! ∞ 1 � � � √ n • Existence: Set GFF | T ( z ) := A n cos( n θ ) + B n sin( n θ ) , n =1 Saksman Decompositions of log-correlated fields Seattle 23/08/19 2 / 25
GFF The actual GFF is defined in a domain Ω ⊂ R 2 , and its covariance is • given by G Ω ( z , z ′ ) , where G Ω is the Green’s function of Ω ⇒ again logarithmic singularity in the covariance. Saksman Decompositions of log-correlated fields Seattle 23/08/19 3 / 25
GFF The actual GFF is defined in a domain Ω ⊂ R 2 , and its covariance is • given by G Ω ( z , z ′ ) , where G Ω is the Green’s function of Ω ⇒ again logarithmic singularity in the covariance. • Usually one considers Greens function w.r. to zero bry values. Then A n λ − 1 / 2 � GFF ( z ) = ϕ n ( z ) , n n ≥ 1 where λ n , ϕ n are the Dirichlet eigenvalues and eigenfunctions in Ω. Saksman Decompositions of log-correlated fields Seattle 23/08/19 3 / 25
GFF The actual GFF is defined in a domain Ω ⊂ R 2 , and its covariance is • given by G Ω ( z , z ′ ) , where G Ω is the Green’s function of Ω ⇒ again logarithmic singularity in the covariance. • Usually one considers Greens function w.r. to zero bry values. Then A n λ − 1 / 2 � GFF ( z ) = ϕ n ( z ) , n n ≥ 1 where λ n , ϕ n are the Dirichlet eigenvalues and eigenfunctions in Ω. • GFF appear in several connections. E.g. as scaling limits of fluctuations of several random models of statistical physics. Saksman Decompositions of log-correlated fields Seattle 23/08/19 3 / 25
Log-correlated fields • More generally, one sometimes needs to consider a centered Gaussian field X say on Ω ⊂ R 2 (or on a subdomain of R d ), such that 1 � � K X ( x , y ) := E X ( x ) X ( y ) = log + g ( x , y ) for x , y ∈ Q 0 , | x − y | where g is continuous (often smooth). Then X is called a log-correlated field. Realizations are (rather mild) generalised functions as before. Saksman Decompositions of log-correlated fields Seattle 23/08/19 4 / 25
Log-correlated fields • More generally, one sometimes needs to consider a centered Gaussian field X say on Ω ⊂ R 2 (or on a subdomain of R d ), such that 1 � � K X ( x , y ) := E X ( x ) X ( y ) = log + g ( x , y ) for x , y ∈ Q 0 , | x − y | where g is continuous (often smooth). Then X is called a log-correlated field. Realizations are (rather mild) generalised functions as before. • Given such a covariance K X existence not difficult to prove. Realizations are (rather mild) generalised functions as before. Saksman Decompositions of log-correlated fields Seattle 23/08/19 4 / 25
Multiplicative Gaussian chaos • The chaos obtained from from a log-correlated field X on U is (formally) the random measure on T with the exponential density ” e β X ( z ) d | z | ” . Here β > 0 is a constant (inverse temperature). Saksman Decompositions of log-correlated fields Seattle 23/08/19 5 / 25
Multiplicative Gaussian chaos • The chaos obtained from from a log-correlated field X on U is (formally) the random measure on T with the exponential density ” e β X ( z ) d | z | ” . Here β > 0 is a constant (inverse temperature). X = � ∞ k =1 X ′ k where the fields X ′ • k are nice, centered and independent Gaussian fields. Denote n � X ′ X n = . k k =1 Saksman Decompositions of log-correlated fields Seattle 23/08/19 5 / 25
Multiplicative Gaussian chaos • The chaos obtained from from a log-correlated field X on U is (formally) the random measure on T with the exponential density ” e β X ( z ) d | z | ” . Here β > 0 is a constant (inverse temperature). X = � ∞ k =1 X ′ k where the fields X ′ • k are nice, centered and independent Gaussian fields. Denote n � X ′ X n = . k k =1 Then the n :th martingale approximation of the chaos is given by β X n ( x ) − β 2 β X n ( x ) − 1 � 2 E ( β X n ( x )) 2 � � � d µ n := exp dx = exp 2 K X n ( x , x ) dx Saksman Decompositions of log-correlated fields Seattle 23/08/19 5 / 25
Existence of continuous chaos β X n ( x ) − β 2 β X n ( x ) − 1 � 2 E ( β X n ( x )) 2 � � � d µ n := exp dx = exp 2 K X n ( x , x ) dx √ THEOREM (Kahane) For 0 < β < 2 d there exists the limit w ∗ µ := lim n →∞ µ n ( a.s. limit ) . Saksman Decompositions of log-correlated fields Seattle 23/08/19 6 / 25
Existence of continuous chaos β X n ( x ) − β 2 β X n ( x ) − 1 � 2 E ( β X n ( x )) 2 � � � d µ n := exp dx = exp 2 K X n ( x , x ) dx √ THEOREM (Kahane) For 0 < β < 2 d there exists the limit w ∗ µ := lim n →∞ µ n ( a.s. limit ) . Saksman Decompositions of log-correlated fields Seattle 23/08/19 6 / 25
Existence of continuous chaos β X n ( x ) − β 2 β X n ( x ) − 1 � 2 E ( β X n ( x )) 2 � � � d µ n := exp dx = exp 2 K X n ( x , x ) dx √ THEOREM (Kahane) For 0 < β < 2 d there exists the limit w ∗ µ := lim n →∞ µ n ( a.s. limit ) . Kahane’s original proof is based on estimating moments in L 2 and on a • rather complicated iteration argument. There are now many different approaches to existence, and properties of chaos measures have been studied intensively. Saksman Decompositions of log-correlated fields Seattle 23/08/19 6 / 25
Existence of continuous chaos β X n ( x ) − β 2 β X n ( x ) − 1 � 2 E ( β X n ( x )) 2 � � � d µ n := exp dx = exp 2 K X n ( x , x ) dx √ THEOREM (Kahane) For 0 < β < 2 d there exists the limit w ∗ µ := lim n →∞ µ n ( a.s. limit ) . Kahane’s original proof is based on estimating moments in L 2 and on a • rather complicated iteration argument. There are now many different approaches to existence, and properties of chaos measures have been studied intensively. • a quite elegant proof of existence was given by Berestycki couple of years ago. Saksman Decompositions of log-correlated fields Seattle 23/08/19 6 / 25
Where do we meet mutiplicatice chaos? Multiplicative chaos appears e.g. as • as a scaling limit in random matrix theory Saksman Decompositions of log-correlated fields Seattle 23/08/19 7 / 25
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