When J. Ginibre met E. Schr¨ odinger Thomas J. Bothner Department of Mathematics King’s College London Joint with Jinho Baik, arXiv:1808.02419 CIRM - Integrability and Randomness April 11th, 2019 Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 1 / 37
Did they actually meet? 2 1.5 1 0.5 0 -0.5 -1 F K G -1.5 Lf gingin z L this i L g n iz i -2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Figure 1: E.S.: 1887 - 1961 and J.G.: 1938 - ??? Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 2 / 37
GOE to the comparison Consider the Gaussian Orthogonal Ensemble (GOE), i.e. matrices X = 1 2( Y + Y T ) ∈ R n × n : iid ∼ N (0 , 1) . (Mehta 1960) Y jk Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 3 / 37
GOE to the comparison Consider the Gaussian Orthogonal Ensemble (GOE), i.e. matrices X = 1 2( Y + Y T ) ∈ R n × n : iid ∼ N (0 , 1) . (Mehta 1960) Y jk Equivalently think of this setup as a log-gas system λ 1 < λ 2 < . . . < λ n , λ j ∈ R , Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 3 / 37
GOE to the comparison Consider the Gaussian Orthogonal Ensemble (GOE), i.e. matrices X = 1 2( Y + Y T ) ∈ R n × n : iid ∼ N (0 , 1) . (Mehta 1960) Y jk Equivalently think of this setup as a log-gas system λ 1 < λ 2 < . . . < λ n , λ j ∈ R , with joint pdf for the particles’ locations equal to (Hsu 1939) � n � f ( λ 1 , . . . , λ n ) = 1 − 1 � � λ 2 | λ k − λ j | exp . j Z n 2 1 ≤ j < k ≤ n j =1 Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 3 / 37
GOE to the comparison Consider the Gaussian Orthogonal Ensemble (GOE), i.e. matrices X = 1 2( Y + Y T ) ∈ R n × n : iid ∼ N (0 , 1) . (Mehta 1960) Y jk Equivalently think of this setup as a log-gas system λ 1 < λ 2 < . . . < λ n , λ j ∈ R , with joint pdf for the particles’ locations equal to (Hsu 1939) � n � f ( λ 1 , . . . , λ n ) = 1 − 1 � � λ 2 | λ k − λ j | exp . j Z n 2 1 ≤ j < k ≤ n j =1 Objective: What can we say about the underlying limit laws? Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 3 / 37
The eigenvalues { λ j } n j =1 form a Pfaffian point process (Dyson 1970), n � n ! � � � k R k ( λ 1 , . . . , λ k ) := R n − k f ( λ 1 , . . . , λ n ) d λ j = Pf K n ( λ i , λ j ) i , j =1 , ( n − k )! j = k +1 with a Hilbert-Schmidt class 2 × 2 matrix-valued kernel K n . Now analyze R k asymptotically in different scaling regimes. Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 4 / 37
The eigenvalues { λ j } n j =1 form a Pfaffian point process (Dyson 1970), n � n ! � � � k R k ( λ 1 , . . . , λ k ) := R n − k f ( λ 1 , . . . , λ n ) d λ j = Pf K n ( λ i , λ j ) i , j =1 , ( n − k )! j = k +1 with a Hilbert-Schmidt class 2 × 2 matrix-valued kernel K n . Now analyze R k asymptotically in different scaling regimes. (A) The global eigenvalue regime: define the ESD µ X ( s ) = 1 n # { 1 ≤ j ≤ n , λ j ≤ s } , s ∈ R , Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 4 / 37
The eigenvalues { λ j } n j =1 form a Pfaffian point process (Dyson 1970), n � n ! � � � k R k ( λ 1 , . . . , λ k ) := R n − k f ( λ 1 , . . . , λ n ) d λ j = Pf K n ( λ i , λ j ) i , j =1 , ( n − k )! j = k +1 with a Hilbert-Schmidt class 2 × 2 matrix-valued kernel K n . Now analyze R k asymptotically in different scaling regimes. (A) The global eigenvalue regime: define the ESD µ X ( s ) = 1 n # { 1 ≤ j ≤ n , λ j ≤ s } , s ∈ R , then, as n → ∞ , the random measure µ X / √ n converges almost surely to the Wigner semi-circular distribution (Wigner 1955) ρ sc ( λ ) = 1 � 2 − λ 2+ d λ. π Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 4 / 37
0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 Figure 2: Wigner’s law for one (rescaled) 2000 × 2000 GOE matrix on the left, plotted is the rescaled histogram of the 2000 eigenvalues and the semicircular density ρ sc ( λ ). On the right we compare Wigner’s law to the exact eigenvalue density for n = 4 and the associated eigenvalue histogram (sampled 4000 times). Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 5 / 37
Universality I Wigner’s law is a universal limiting law (Arnold 1967, ...), it holds true for any (properly centered and scaled) symmetric or Hermitian j , k =1 with E | X jk | 2 < ∞ where X jk , j < k are Wigner matrix X = ( X jk ) n iid real or complex variables and X jj iid real variables independent of the upper triangular ones. Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 6 / 37
Universality I Wigner’s law is a universal limiting law (Arnold 1967, ...), it holds true for any (properly centered and scaled) symmetric or Hermitian j , k =1 with E | X jk | 2 < ∞ where X jk , j < k are Wigner matrix X = ( X jk ) n iid real or complex variables and X jj iid real variables independent of the upper triangular ones. (B) The local eigenvalue regime: We shall zoom in on the right edge √ point λ 0 = 2 n and let n be even (Forrester, Nagao, Honner 1999), � √ � √ 1 x y K n 2 n + , 2 n + → Q Ai ( x , y ) , √ √ √ 1 1 1 2 n 2 n 2 n 6 6 6 as n → ∞ uniformly in x , y ∈ R chosen from compact subsets. Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 6 / 37
Here, Q Ai is Hilbert-Schmidt on L 2 ( s 0 , ∞ ) with kernel entries � y Q 11 ( x , y ) = Q 22 ( y , x ) = K Ai ( x , y ) + 1 2 Ai( x ) Ai( t ) d t −∞ Q 12 ( x , y ) = − ∂ ∂ y K Ai ( x , y ) − 1 2 Ai( x )Ai( y ) , and � ∞ � y K Ai ( t , y ) d t − 1 Q 21 ( x , y ) = − Ai( t ) d t 2 x x � ∞ � ∞ + 1 Ai( t ) d t − 1 Ai( t ) d t · 2 sgn( x − y ) , 2 x y Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 7 / 37
Here, Q Ai is Hilbert-Schmidt on L 2 ( s 0 , ∞ ) with kernel entries � y Q 11 ( x , y ) = Q 22 ( y , x ) = K Ai ( x , y ) + 1 2 Ai( x ) Ai( t ) d t −∞ Q 12 ( x , y ) = − ∂ ∂ y K Ai ( x , y ) − 1 2 Ai( x )Ai( y ) , and � ∞ � y K Ai ( t , y ) d t − 1 Q 21 ( x , y ) = − Ai( t ) d t 2 x x � ∞ � ∞ + 1 Ai( t ) d t − 1 Ai( t ) d t · 2 sgn( x − y ) , 2 x y where we use the trace-class kernel (on L 2 ( s 0 , ∞ )) � ∞ K Ai ( x , y ) = Ai( x + s )Ai( y + s ) d s . 0 Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 7 / 37
Universality II The limiting kernel Q Ai ( x , y ) is once more universal (Soshnikov 1999), it governs the soft edge scaling limits of the k -point correlation functions for any (properly centered and scaled) real Wigner matrix X (modulo some decay constraints). Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 8 / 37
Universality II The limiting kernel Q Ai ( x , y ) is once more universal (Soshnikov 1999), it governs the soft edge scaling limits of the k -point correlation functions for any (properly centered and scaled) real Wigner matrix X (modulo some decay constraints). The above kernel allows us to formulate the following central limit theorem for the largest eigenvalue λ max in the GOE, as n → ∞ , √ 1 λ max ( X ) ⇒ 2 n + √ 6 F 1 1 2 n Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 8 / 37
Universality II The limiting kernel Q Ai ( x , y ) is once more universal (Soshnikov 1999), it governs the soft edge scaling limits of the k -point correlation functions for any (properly centered and scaled) real Wigner matrix X (modulo some decay constraints). The above kernel allows us to formulate the following central limit theorem for the largest eigenvalue λ max in the GOE, as n → ∞ , √ 1 λ max ( X ) ⇒ 2 n + √ 6 F 1 1 2 n where the cdf of F 1 equals (Tracy, Widom 2005) � 2 = det � 2 (1 − GQ Ai G − 1 ↾ L 2 ( s , ∞ ) ⊕ L 2 ( s , ∞ ) ) . P ( F 1 ≤ s ) √ with G = diag( g , g − 1 ) and g ( x ) = 1 + x 2 . Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 8 / 37
0.45 1 0.4 0.35 0.8 0.3 0.25 0.6 0.2 0.4 0.15 0.1 0.2 0.05 0 0 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 Figure 3: Tracy-Widom distribution F 1 (blue) versus N (0 , 1) (red). mean variance skewness kurtosis N (0 , 1) 0 1 0 0 F 1 -1.20653 1.60778 0.29346 0.16524 Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 9 / 37
There are other explicit formulæ for the cdf of F 1 : 1. Airy determinant and resolvent formula (Forrester 2006) � 2 = det(1 − ( K Ai + U ⊗ V ) ↾ L 2 ( s , ∞ ) ) � P ( F 1 ≤ s ) � x where ( U ⊗ V )( x , y ) = Ai( x ) A ( y ) and A ( x ) = −∞ Ai( y ) d y . Thomas J. Bothner (KCL) Ginibre and Schr¨ odinger April 11th, 2019 10 / 37
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