Extreme gap problems in random matrix theory Renjie Feng BIMCR, Peking University Renjie Feng (BICMR) 1 / 21
Joint density of eigenvalues of G β E G β E: Given n point λ 1 , · · · , λ n ( β > 0) with the joint density n 1 e − β n | λ j − λ i | β , 4 λ 2 � k � J ( λ 1 , · · · , λ n ) = Z β, n k =1 i < j here, Z β, n is a norming constant which can be computed by the Selberg integral, β = 1 is corresponding to GOE, β = 2 for GUE, β = 4 for GSE. Renjie Feng (BICMR) 2 / 21
Joint density of eigenvalues of C β E C β E: Given n points on the unit circle e i θ 1 , · · · , e i θ n with joint density 1 β � � � � e i θ j − e i θ i J ( θ 1 , · · · , θ n ) = , � � C β, n � i < j C β, n = (2 π ) n Γ(1+ β n / 2) (Γ(1+ β/ 2)) n , β = 1 is corresponding to COE, β = 2 for CUE, β = 4 for CSE. Renjie Feng (BICMR) 3 / 21
Extreme gaps I: smallest gaps for CUE Let e i θ 1 , · · · , e i θ n be n eigenvalues of CUE, consider the 2-dimensional process of spacing of eigenangles and its position, n � χ n = δ ( n 4 / 3 ( θ i +1 − θ i ) ,θ i ) . i =1 Theorem (Vinson, Soshinikov, Ben Arous-Bourgade) χ n tends to a Poisson point process χ with intensity � 1 � �� � � du u 2 du E χ ( A × I ) = . 24 π 2 π A I Let t n 1 < t n 2 · · · < t n k be the first k smallest eigenangles gaps, denote τ n k = (72 π ) − 1 / 3 t n k , then as a consequence, 3 ( k − 1)! x 3 k − 1 e − x 3 dx . n → + ∞ P ( τ n lim k ∈ [ x , x + dx ]) = Renjie Feng (BICMR) 4 / 21
Extreme gaps I: smallest gaps for CUE Let e i θ 1 , · · · , e i θ n be n eigenvalues of CUE, consider the 2-dimensional process of spacing of eigenangles and its position, n � χ n = δ ( n 4 / 3 ( θ i +1 − θ i ) ,θ i ) . i =1 Theorem (Vinson, Soshinikov, Ben Arous-Bourgade) χ n tends to a Poisson point process χ with intensity � 1 � �� � � du u 2 du E χ ( A × I ) = . 24 π 2 π A I Let t n 1 < t n 2 · · · < t n k be the first k smallest eigenangles gaps, denote τ n k = (72 π ) − 1 / 3 t n k , then as a consequence, 3 ( k − 1)! x 3 k − 1 e − x 3 dx . n → + ∞ P ( τ n lim k ∈ [ x , x + dx ]) = Renjie Feng (BICMR) 4 / 21
Extreme gaps I: smallest gaps for C β E When β is an positive integer, consider 2-dimensional process n � χ n = δ β +2 β +1 ( θ i +1 − θ i ) ,θ i ) ( n i =1 Theorem (F.-Wei) χ n tends to a Poisson point process χ with intensity E χ ( A × I ) = A β | I | � u β du , 2 π A where A β = (2 π ) − 1 ( β/ 2) β (Γ( β/ 2+1)) 3 Γ(3 β/ 2+1)Γ( β +1) . In particular, the result holds for COE, CUE and CSE with A 1 = 1 1 1 24 , A 2 = 24 π, A 4 = 270 π. Renjie Feng (BICMR) 5 / 21
Extreme gaps I: smallest gaps for C β E When β is an positive integer, consider 2-dimensional process n � χ n = δ β +2 β +1 ( θ i +1 − θ i ) ,θ i ) ( n i =1 Theorem (F.-Wei) χ n tends to a Poisson point process χ with intensity E χ ( A × I ) = A β | I | � u β du , 2 π A where A β = (2 π ) − 1 ( β/ 2) β (Γ( β/ 2+1)) 3 Γ(3 β/ 2+1)Γ( β +1) . In particular, the result holds for COE, CUE and CSE with A 1 = 1 1 1 24 , A 2 = 24 π, A 4 = 270 π. Renjie Feng (BICMR) 5 / 21
Extreme gaps I: smallest gaps for C β E Corollary Let’s denote t n k as the k -th smallest gap, and k = n ( β +2) / ( β +1) × ( A β / ( β + 1)) 1 / ( β +1) t n τ n k , then for any bounded interval A ⊂ R + , we have β + 1 ( k − 1)! x k ( β +1) − 1 e − x β +1 dx . n → + ∞ P ( τ n lim k ∈ [ x , x + dx ]) = No determinantal point process structure can be used as CUE (which is used by Soshinikov and Ben Arous-Bourgade, Figalli-Guionnet), we have to start from the Selberg integral Conjecture: The result must be true for any β > 0, but our method does not work other than integer β . Renjie Feng (BICMR) 6 / 21
Extreme gaps I: smallest gaps for C β E Corollary Let’s denote t n k as the k -th smallest gap, and k = n ( β +2) / ( β +1) × ( A β / ( β + 1)) 1 / ( β +1) t n τ n k , then for any bounded interval A ⊂ R + , we have β + 1 ( k − 1)! x k ( β +1) − 1 e − x β +1 dx . n → + ∞ P ( τ n lim k ∈ [ x , x + dx ]) = No determinantal point process structure can be used as CUE (which is used by Soshinikov and Ben Arous-Bourgade, Figalli-Guionnet), we have to start from the Selberg integral Conjecture: The result must be true for any β > 0, but our method does not work other than integer β . Renjie Feng (BICMR) 6 / 21
Extreme gaps I: smallest gaps for C β E Corollary Let’s denote t n k as the k -th smallest gap, and k = n ( β +2) / ( β +1) × ( A β / ( β + 1)) 1 / ( β +1) t n τ n k , then for any bounded interval A ⊂ R + , we have β + 1 ( k − 1)! x k ( β +1) − 1 e − x β +1 dx . n → + ∞ P ( τ n lim k ∈ [ x , x + dx ]) = No determinantal point process structure can be used as CUE (which is used by Soshinikov and Ben Arous-Bourgade, Figalli-Guionnet), we have to start from the Selberg integral Conjecture: The result must be true for any β > 0, but our method does not work other than integer β . Renjie Feng (BICMR) 6 / 21
Extreme gaps I: why such order heuristically? We have the gap probability P ( n ( θ j +1 − θ i ) < x ) ∼ x β +1 , thus for a single gap P ( s < x ) = P ( ns < nx ) ∼ ( nx ) β +1 if we treat the gaps ’independently’, we have E (# { gaps < x } ) ∼ n P ( s < x ) ∼ n ( nx ) β +1 , hence, we must have x ∼ n − β +2 β +1 to get some nontrivial result. Renjie Feng (BICMR) 7 / 21
Extreme gaps I: how we get A β ? The constant A β is very meaningful, it appears when one studied the k th factorial moment of χ n . To prove χ n (ignoring the position) tends to Poisson, we may consider the process with k -pair of smallest gaps, � ρ n = δ β +1 ( θ i 2 k − θ i 2 k − 1 ) . β +2 β +2 β +1 ( θ i 2 − θ i 1 ) , ··· , n n We proved that � E ρ n ( A k ) → ( A β u β du ) k , A where Z β, n − 2 k , k A k β = lim Z β, n n k β . n → + ∞ Renjie Feng (BICMR) 8 / 21
Extreme gaps I: how we get A β ? The constant A β is very meaningful, it appears when one studied the k th factorial moment of χ n . To prove χ n (ignoring the position) tends to Poisson, we may consider the process with k -pair of smallest gaps, � ρ n = δ β +1 ( θ i 2 k − θ i 2 k − 1 ) . β +2 β +2 β +1 ( θ i 2 − θ i 1 ) , ··· , n n We proved that � E ρ n ( A k ) → ( A β u β du ) k , A where Z β, n − 2 k , k A k β = lim Z β, n n k β . n → + ∞ Renjie Feng (BICMR) 8 / 21
Extreme gaps I: how we get A β ? For one-component log-gas of n particles with charge +1, � β � � � � e i θ j − e i θ i Z β, n = d θ 1 ... d θ n . � � � [0 , 2 π ] n 1 ≤ i < j ≤ n For two-component log-gas of n − 2 k particles of charge +1 and k particles of charge +2, � q i q j β � � � e i θ j − e i θ i � Z β, n − 2 k , k = d θ 1 ... d θ n − k � � � [0 , 2 π ] n − k 1 ≤ i < j ≤ n − k where q i = 1 for 1 ≤ i ≤ n − 2 k ; q i = 2 for n − 2 k + 1 ≤ i ≤ n − k . Renjie Feng (BICMR) 9 / 21
Extreme gaps I: how we get A β ? For one-component log-gas of n particles with charge +1, � β � � � � e i θ j − e i θ i Z β, n = d θ 1 ... d θ n . � � � [0 , 2 π ] n 1 ≤ i < j ≤ n For two-component log-gas of n − 2 k particles of charge +1 and k particles of charge +2, � q i q j β � � � e i θ j − e i θ i � Z β, n − 2 k , k = d θ 1 ... d θ n − k � � � [0 , 2 π ] n − k 1 ≤ i < j ≤ n − k where q i = 1 for 1 ≤ i ≤ n − 2 k ; q i = 2 for n − 2 k + 1 ≤ i ≤ n − k . Renjie Feng (BICMR) 9 / 21
Extreme gaps II: smallest gaps for GUE Consider the 2-dimensional process of (interior) eigenvalues of GUE n � χ n = δ ( n 3 ( λ i +1 − λ i ) ,λ i ) 1 | λ i | < 2 − η 4 i =1 Theorem (Vinson, Soshinikov, Ben Arous-Bourgade) χ n tends to a Poisson point process χ with intensity 1 � � u 2 du )( (4 − x 2 ) 2 dx ) , E χ ( A × I ) = ( 48 π 2 A I where A ⊂ R + and I ⊂ ( − 2 + η, 2 − η ). I (4 − x 2 ) 2 dx / 144 π 2 ) 1 / 3 t n The k -th smallest gaps τ n � k = ( k has the limiting ( k − 1)! x 3 k − 1 e − x 3 , same as CUE. 3 density Renjie Feng (BICMR) 10 / 21
Extreme gaps II: smallest gaps for GUE Consider the 2-dimensional process of (interior) eigenvalues of GUE n � χ n = δ ( n 3 ( λ i +1 − λ i ) ,λ i ) 1 | λ i | < 2 − η 4 i =1 Theorem (Vinson, Soshinikov, Ben Arous-Bourgade) χ n tends to a Poisson point process χ with intensity 1 � � u 2 du )( (4 − x 2 ) 2 dx ) , E χ ( A × I ) = ( 48 π 2 A I where A ⊂ R + and I ⊂ ( − 2 + η, 2 − η ). I (4 − x 2 ) 2 dx / 144 π 2 ) 1 / 3 t n The k -th smallest gaps τ n � k = ( k has the limiting ( k − 1)! x 3 k − 1 e − x 3 , same as CUE. 3 density Renjie Feng (BICMR) 10 / 21
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