. . Ginibre point process and its Palm measures: absolute continuity and singularity . . . . . Tomoyuki SHIRAI 1 2 Kyushu University Dec. 7, 2011 1 Joint work with Hirofumi Osada (Kyushu University) 2 ”10th workshop on stochastic analysis on large scale interacting systems” at Kochi University, Dec. 5–7, 2011. . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 1 / 26
Contents . .. Ginibre point process and determinantal point process 1 . .. Main results 2 . .. Absolute continuity 3 . .. Singularity 4 . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 2 / 26
Ginibre matrix ensemble and complex eigenvalues M N : the space of N × N complex matrices ∼ = C N 2 P N ( dX ) = Z − 1 N exp( − T rX ∗ X ) dX , . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 3 / 26
Ginibre matrix ensemble and complex eigenvalues M N : the space of N × N complex matrices ∼ = C N 2 P N ( dX ) = Z − 1 N exp( − T rX ∗ X ) dX , or equivalently, Random matrix whose entries are all i.i.d. standard complex Gaussian. It is called Ginibre matrix ensemble of size N . . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 3 / 26
Ginibre matrix ensemble and complex eigenvalues M N : the space of N × N complex matrices ∼ = C N 2 P N ( dX ) = Z − 1 N exp( − T rX ∗ X ) dX , or equivalently, Random matrix whose entries are all i.i.d. standard complex Gaussian. It is called Ginibre matrix ensemble of size N . Probability density of complex eigenvalues was computed by Ginibre(1965) as follows: 1 p ( N ) ( z 1 , . . . , z N ) = ∏ | z i − z j | 2 ∏ N k =1 k ! 1 ≤ i < j ≤ N 1 det( z j − 1 ) N = i , j =1 i ∏ N k =1 k ! with respect to the standard complex Gaussian measure λ ⊗ N ( dz 1 . . . dz N ) with λ ( dz ) = π − 1 e −| z | 2 dz . . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 3 / 26
Random complex eigenvalues Ginibre N � 100 Ginibre N � 400 Ginibre N � 900 30 30 30 20 20 20 10 10 10 � 30 � 20 � 10 10 20 30 � 30 � 20 � 10 10 20 30 � 30 � 20 � 10 10 20 30 � 10 � 10 � 10 � 20 � 20 � 20 � 30 � 30 � 30 Figure: N = 100 , 400 , 900 . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 4 / 26
Random complex eigenvalues Ginibre N � 100 Ginibre N � 400 Ginibre N � 900 30 30 30 20 20 20 10 10 10 � 30 � 20 � 10 10 20 30 � 30 � 20 � 10 10 20 30 � 30 � 20 � 10 10 20 30 � 10 � 10 � 10 � 20 � 20 � 20 � 30 � 30 � 30 Figure: N = 100 , 400 , 900 w Bai showed that 1 ∑ N √ i =1 δ z i / → Uniform ( D 1 ) almost surely N N . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 4 / 26
Ginibre point process as determinantal point process . Definition (Determinantal point process (DPP)) . . . DPP is a point process having deteminantal correlation functions ρ n ( z 1 , z 2 , . . . , z n ) = det( K ( z i , z j ) n i , j =1 ) for some K ( z , w ) relative to a Radon measure λ ( dz ) = g ( z ) dz. . . . . . . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 5 / 26
Ginibre point process as determinantal point process . Definition (Determinantal point process (DPP)) . . . DPP is a point process having deteminantal correlation functions ρ n ( z 1 , z 2 , . . . , z n ) = det( K ( z i , z j ) n i , j =1 ) for some K ( z , w ) relative to a Radon measure λ ( dz ) = g ( z ) dz. . . . . . The N -particle Ginibre point process on C is rotation invariant DPP on C whose kernel relative to λ ( dz ) = π − 1 e −| z | 2 dz is given by N − 1 ( zw ) k ∑ K ( N ) ( z , w ) = k ! k =0 . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 5 / 26
Ginibre point process as determinantal point process . Definition (Determinantal point process (DPP)) . . . DPP is a point process having deteminantal correlation functions ρ n ( z 1 , z 2 , . . . , z n ) = det( K ( z i , z j ) n i , j =1 ) for some K ( z , w ) relative to a Radon measure λ ( dz ) = g ( z ) dz. . . . . . The N -particle Ginibre point process on C is rotation invariant DPP on C whose kernel relative to λ ( dz ) = π − 1 e −| z | 2 dz is given by N − 1 ( zw ) k e zw =: K ( z , w ) N →∞ ∑ K ( N ) ( z , w ) = → k ! k =0 . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 5 / 26
Ginibre point process as determinantal point process . Definition (Determinantal point process (DPP)) . . . DPP is a point process having deteminantal correlation functions ρ n ( z 1 , z 2 , . . . , z n ) = det( K ( z i , z j ) n i , j =1 ) for some K ( z , w ) relative to a Radon measure λ ( dz ) = g ( z ) dz. . . . . . The N -particle Ginibre point process on C is rotation invariant DPP on C whose kernel relative to λ ( dz ) = π − 1 e −| z | 2 dz is given by N − 1 ( zw ) k e zw =: K ( z , w ) N →∞ ∑ K ( N ) ( z , w ) = → k ! k =0 When correlation functions converge uniformly on any compacts, corresponding point processes converge weakly to a limit. . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 5 / 26
Ginibre point process Ginibre point process on C is defined as DPP with a kernel λ ( dz ) = π − 1 e −| z | 2 dz K ( z , w ) = e zw , . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 6 / 26
Ginibre point process Ginibre point process on C is defined as DPP with a kernel λ ( dz ) = π − 1 e −| z | 2 dz K ( z , w ) = e zw , In particular, ρ 1 ( z ) = g ( z ) K ( z , z ) = π − 1 ρ 2 ( z , w ) ≤ ρ 1 ( z ) ρ 1 ( w ) · · · negative correlation . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 6 / 26
Ginibre point process Ginibre point process on C is defined as DPP with a kernel λ ( dz ) = π − 1 e −| z | 2 dz K ( z , w ) = e zw , In particular, ρ 1 ( z ) = g ( z ) K ( z , z ) = π − 1 ρ 2 ( z , w ) ≤ ρ 1 ( z ) ρ 1 ( w ) · · · negative correlation Ginibre p.p. on C is invariant under translations and rotations. . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 6 / 26
Poisson and Ginibre Poisson Ginibre 10 10 5 5 0 0 � 5 � 5 � 10 � 10 � 10 � 5 0 5 10 � 10 � 5 0 5 10 Figure: Poisson(left) and Ginibre(right) . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 7 / 26
. . . Variance and large deviations for the number of points ξ ( D r ) is the number of points inside the disk of radius r . . . . 1 Variance Poisson case: Var ( ξ ( D r )) = r 2 Ginibre case: r Var ( ξ ( D r )) ∼ √ π . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 8 / 26
Variance and large deviations for the number of points ξ ( D r ) is the number of points inside the disk of radius r . . . . 1 Variance Poisson case: Var ( ξ ( D r )) = r 2 Ginibre case: r Var ( ξ ( D r )) ∼ √ π . . . 2 Large deviations Poisson case: P ( r − 2 ξ ( D r ) ≈ a ) ∼ exp( − I ( a ) r 2 ) Ginibre case: P ( r − 2 ξ ( D r ) ≈ a ) ∼ exp( − J ( a ) r 4 ) . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 8 / 26
Ginibre point process is considered as a Gibbs measure? Formal expression: i | z i | 2 ∞ µ = Z − 1 ∏ | z i − z j | 2 e − ∑ ∏ dz i i < j i =1 ∞ = Z − 1 exp | z i | 2 + 2 ∑ ∏ ∏ − log | z i − z j | dz i i i < j i =1 . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 9 / 26
Ginibre point process is considered as a Gibbs measure? Formal expression: i | z i | 2 ∞ µ = Z − 1 ∏ | z i − z j | 2 e − ∑ ∏ dz i i < j i =1 ∞ = Z − 1 exp | z i | 2 + 2 ∑ ∏ ∏ − log | z i − z j | dz i i i < j i =1 2-body potential Φ( z , w ) = − 2 log | z − w | is not even bounded. . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 9 / 26
Palm measure We focus on the (reduced) Palm measure of a simple point process µ defined as follows: for a = ( a 1 , a 2 , . . . , a n ) ∈ R n n µ a ( · ) := µ ( · − ∑ δ a i | ξ ( { a i } ) ≥ 1 , ∀ i = 1 , 2 , . . . n ) i =1 . . . . . . Tomoyuki SHIRAI (Kyushu University) Ginibre point process and its Palm measures Dec. 7, 2011 10 / 26
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