Gaussian complex zeros and eigenvalues: Rare events and the emergence of the ‘forbidden’ region Les Diablerets, 13/02/2018 Alon Nishry - Tel Aviv University Joint work with Subhroshekhar Ghosh (NUS)
Random point configurations Poisson Point Process Ginibre ensemble Gaussian zeros
Random point configurations Poisson Point Process Ginibre ensemble Gaussian zeros
Random point configurations Poisson Point Process Ginibre ensemble Gaussian zeros
Random point configurations Poisson Point Process Ginibre ensemble Gaussian zeros
Point processes - Invariance ◮ X = { z j } j ∈ I - random point configuration ◮ n ( K ) is the number of points in a compact set K ◮ T : C → C a transformation (automorphism) ◮ The distribution of the point process X is invariant with respect to T if X d ∼ T ( X ) or ( n ( K 1 ) ,..., n ( K n )) d ∼ ( n ( T ( K 1 )) ,..., n ( T ( K n )))
Invariant point processes All the examples we consider: ◮ Homogeneous Poisson Point Process ◮ Infinite Ginibre ensemble ◮ Zeros of the Gaussian Entire Function are invariant with respect to: ◮ Translations ◮ Rotations ◮ Reflections
(Homogeneous) Poisson Point Process ◮ The distribution is given by � 1 � n ( K ) ∼ Poisson π Area ( K ) . ◮ No “correlations”: K 1 , K 2 ,..., K N ⊂ C compact sets K 1 , K 2 ,..., K N disjoint = ⇒ n ( K 1 ) , n ( K 2 ) ,..., n ( K N ) independent ◮ Can be thought of as “independent” points uniformly distributed in the plane. ◮ “Gas” with no particle-particle interactions.
Ginibre ensemble (random eigenvalues) Finite Ginibre ◮ Complex eigenvalues of non-Hermitian N × N matrix ◮ Entries are independent standard complex Gaussian π e −| w | 2 , w ∈ C . ◮ Standard complex Gaussian: density 1 Infinite Ginibre - limit of finite Ginibre as N → ∞ ◮ Determinantal point process ◮ Probabilities are governed by eigenvalues of some integral operators ◮ Gas with particle-particle interactions (repulsion) embedded in uniform background
Gaussian Entire Function (GEF) ◮ { ξ n } - sequence of independent standard complex Gaussians. ◮ The GEF is given by the Gaussian Taylor series: ∞ z n ∑ F ( z ) = ξ n √ z ∈ C . , n ! n = 0 ◮ Infinite radius of convergence (almost surely). ◮ Zero set: Z ( F ) = F − 1 ( 0 ) is a discrete set in C .
Gaussian Entire Function - invariance ◮ F is a Gaussian function, distribution determined by covariance kernel: ∞ ( zw ) n � � ∑ = e zw . K ( z , w ) = E F ( z ) F ( w ) = n ! n = 0 ◮ Rotation: follows from rotation invariance of complex Gaussians. ◮ Reflection: F ( z ) and F ( z ) have the same distribution. ◮ Translation: For a ∈ C easy to check that 2 | a | 2 have the same distribution F ( z + a ) and F ( z ) e za − 1 2 | a | 2 � = 0 so zeros are invariant with respect to translations. ◮ e za − 1
Gaussian Entire Function - zero set ◮ Counting measure of zeros: n F = ∑ δ z z ∈ Z ( F ) ◮ Logarithm - fundamental solution to Laplacian: δ z = 1 ⇒ d n F ( w ) = 1 2 π ∆ log |·− z | = 2 π ∆ log | F ( w ) | ◮ Not difficult to check: K ( w , w )+ C = 1 2 | w | 2 + C � E [ log | F ( w ) | ] = log ◮ Using Fubini: E [# { Z ( F ) ∩ D } ] = 1 π Area ( D ) .
Gaussian Entire Function - distribution of zeros ◮ GEF is unique (up to scaling) among Gaussian analytic functions in the plane with invariant zero set. ◮ Statistics of zero set are well understood. Some examples: ◮ Edelman - Kostlan ’95 ◮ Forrester - Honner ’99 (Variance asymptotics) ◮ Sodin - Tsirelson ’04 (Central limit theorem)
Point processes - rare events ◮ Write n ( r ) = n ( {| z | ≤ r } ) ◮ number of points in a (large) disk ◮ For all three models we normalize E [ n ( r )] = r 2 ◮ Consider rare events of the type: � � pr 2 �� n ( r ) = p � = 1 , ◮ p = 0 - ‘hole’ (no points) ◮ p < 1 - deficiency ◮ p > 1 - overcrowding
Rare events and conditional distribution ◮ We find the asymptotic rate of decay of � � pr 2 �� P n ( r ) = as r → ∞ . , ◮ (very) rare events ◮ We also find the distribution of the points conditioned on this rare event.
Poisson Point Process ◮ n ( r ) number of points in {| z | ≤ r } has � r 2 � Poisson distribution. ◮ In particular, not difficult to calculate these probabilities: ◮ log P � � pr 2 �� is of order − C p r 2 n ( r ) =
Poisson Point Process ◮ By the definition of the process: ◮ For disjoint sets the distribution of the points in each set is independent .
Infinite Ginibre ensemble (random ‘eigenvalues’) ◮ Determinantal point process ◮ The number of points n ( r ) can be written as the sum of independent Bernoulli random variables. ◮ Moreover: Set of radii � � | z 1 | 2 , | z 2 | 2 ,... has same distribution as the set of independent random variables { Γ( 1 , 1 ) , Γ( 2 , 1 ) ,... } .
Infinite Ginibre ensemble (random ‘eigenvalues’) ◮ It is possible to find the asymptotic decay of rare events’ probabilities. ◮ Shirai (’06) ◮ of order − C p r 4 ◮ Because the radii are independent not too difficult to find the conditional distribution. ◮ Jancovici, Lebowitz, Manificat (’93) ◮ Unlike Poisson, the number of points is ‘conserved’.
Gaussian Entire Function (random zeros) ◮ Zeros of the GEF ∞ z n ∑ F ( z ) = ξ n √ n ! n = 0 Z ( F ) = F − 1 ( 0 ) ⊂ C
Gaussian Entire Function (random zeros) ◮ Zeros of the GEF ∞ z n ∑ F ( z ) = ξ n √ n ! n = 0 Z ( F ) = F − 1 ( 0 ) ⊂ C ◮ Not a determinantal point process.
Gaussian Entire Function (random zeros) ◮ Not a determinantal point process. ◮ Sodin and Tsirelson (’05): Found decay rates are qualitatively like Ginibre ensemble.
Gaussian Entire Function (random zeros) ◮ Not a determinantal point process. ◮ Sodin and Tsirelson (’05): Found decay rates are qualitatively like Ginibre ensemble. ◮ F. Nazarov and M. Sodin asked what is the conditional distribution for the GEF. ◮ In particular, is there a gap in the distribution?
Ginibre vs. GEF zeros - deficiency n ( r ) = 1 4 r 2 Ginibre ensemble: GEF zeros:
Ginibre vs. GEF zeros - deficiency n ( r ) = 1 4 r 2 Ginibre ensemble: GEF zeros:
Ginibre vs. GEF zeros - deficiency n ( r ) = 1 4 r 2 Ginibre ensemble: GEF zeros:
Ginibre vs. GEF zeros - overcrowding n ( r ) = 2 r 2 Ginibre ensemble: GEF zeros:
Ginibre vs. GEF zeros - overcrowding n ( r ) = 2 r 2 Ginibre ensemble: GEF zeros:
Ginibre vs. GEF zeros - overcrowding n ( r ) = 2 r 2 Ginibre ensemble: GEF zeros:
GEF zeros - hole event Consider just the ‘hole’ event: Hole ( r ) = { Z ( F ) ∩{| z | ≤ r } = / 0 } Limiting measure µ H : Zeros on Hole ( r ) :
GEF zeros - hole event Consider just the ‘hole’ event: Hole ( r ) = { Z ( F ) ∩{| z | ≤ r } = / 0 } Limiting measure µ H : Zeros on Hole ( r ) :
GEF zeros - hole event Consider just the ‘hole’ event: Hole ( r ) = { Z ( F ) ∩{| z | ≤ r } = / 0 } Zeros on Hole ( r ) : Limiting measure µ H : d m ( w ) Limiting measure: d µ H ( w ) = e · δ {| w | = 1 } + 1 { | w |≥√ e } π
Main results [Ghosh, N. ’16] ◮ ϕ - a smooth test function with compact support. The random variable (called linear statistics ) � z � n ( ϕ ; r ) = ∑ ϕ r z ∈ Z ( F ) Theorem C ϕ ( w ) d µ H ( w ) · r 2 + o � � r 2 � E [ n ( ϕ ; r ) | Hole ( r )] = r → ∞ . ,
Main results [Ghosh, N. ’16] ◮ ϕ - a smooth test function with compact support. The random variable (called linear statistics ) � z � n ( ϕ ; r ) = ∑ ϕ r z ∈ Z ( F ) Theorem C ϕ ( w ) d µ H ( w ) · r 2 + o � � r 2 � E [ n ( ϕ ; r ) | Hole ( r )] = r → ∞ . , ◮ What is the actual number of zeros in the annulus? r ( 1 + ε ) ≤ | w | ≤ √ er ( 1 − ε ) ◮ N ε , r = # � � �� Z ( F ) ∩
Main results [Ghosh, N. ’16] ◮ ϕ - a smooth test function with compact support. The random variable (called linear statistics ) � z � n ( ϕ ; r ) = ∑ ϕ r z ∈ Z ( F ) Theorem C ϕ ( w ) d µ H ( w ) · r 2 + o � � r 2 � E [ n ( ϕ ; r ) | Hole ( r )] = r → ∞ . , ◮ What is the actual number of zeros in the annulus? r ( 1 + ε ) ≤ | w | ≤ √ er ( 1 − ε ) ◮ N ε , r = # � � �� Z ( F ) ∩ Theorem N ε , r > r 1 + ε | Hole ( r ) − C ε r 2 ( 1 + ε ) � � � � P ≤ exp r → ∞ . ,
Some ideas - Joint distribution ◮ Approximate the Taylor series with polynomials N z n ∑ ξ n √ P ( z ) = n ! n = 0
Some ideas - Joint distribution ◮ Approximate the Taylor series with polynomials N z n N = ξ n ( z − z j ) =: ξ n ∑ ∏ ξ n √ √ √ P ( z ) = Q N ( z ) . n ! n ! n ! n = 0 j = 1 ◮ Joint density of the zeros { z 1 ,..., z N } is known, but complicated: 1 | z j − z k | 2 A N ∏ j < k
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