Local inhomogeneous circular law Johannes Alt Institute of Science - PowerPoint PPT Presentation

Local inhomogeneous circular law Johannes Alt Institute of Science and Technology Austria December 4, 2017 Mathematical Physics Seminar, Geneva University Joint work with László Erdős and Torben Krüger. Partially funded by ERC Advanced Grant RANMAT No. 338804.

Table of Contents Introduction 1 Local inhomogeneous circular law 2 Ideas of the proof 3 Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Introduction Random matrix: matrix-valued random variable. Introduced by Wishart [1928] and Wigner [1955]. Applications/Motivations: Mathematical statistics, Quantum physics, Wireless communication, Neural networks, Number theory, Free probability Some important questions in random matrix theory Do the eigenvalues exhibit any deterministic behaviour? Are there any universal characteristics in this deterministic behaviour? How universal are their fluctuations? Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Empirical spectral measure and density of states Empirical spectral measure of an n × n random matrix X : n µ X = 1 � δ z i n i =1 where z 1 , . . . , z n ∈ C are the eigenvalues of X . Goal: Study eigenvalue density µ X for large n . Typical situation : Empirical spectral measure becomes deterministic! There is a (deterministic) measure µ ( self-consistent density of states ) such that n � f d µ X = 1 � � f ( z i ) ≈ f d µ n C C i =1 for all f with compact support. Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Spectra of Hermitian and non-Hermitian random matrices Hermitian 0 . 3 X = X ∗ i.i.d. entries 0 . 2 Semicircular law: 0 . 1 µ X ≈ 1 � (4 − x 2 ) + d x 0 − 2 − 1 0 1 2 2 π 1 0 . 5 Non-Hermitian 0 X i.i.d. entries Circular law: − 0 . 5 µ X ≈ 1 − 1 π 1 D (0 , 1) d 2 z − 1 − 0 . 5 0 0 . 5 1 X = ( x ij ) n In this talk: i,j =1 with x ij centered, independent but non-identically distributed Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Example with non-identical variances Random matrix X = ( x ij ) n i,j =1 with independent entries . = E | x ij | 2 , . = ( s ij ) n s ij . S . E x ij = 0 , i,j =1 . 1 1 0 . 5 S = 1 0 . 2 0 − 0 . 5 − 1 − 1 0 1 (b) Variance profile S such that ρ ( S ) . . = max | spec( S ) | = 1 . (a) Eigenvalue locations Figure: Averaging 100 matrices of size n = 1000 with centered complex Gaussian entries and the variance profile S . Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Example with non-identical variances 0 . 6 0 . 4 0 . 2 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 | z | Figure: Eigenvalue histogram and self-consistent density of states. Averaging 100 matrices of size n = 1000 with centered complex Gaussian entries and the variance profile S . Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Behaviour of eigenvalues on different scales Different scales ( X normalized such that � X � = O (1) ): Global law µ X ( U ) → µ ( U ) diam( U ) = O (1) diam( U n ) = O ( n − a ) , a ∈ (0 , 1 / 2) Local law µ X ( U n ) → µ ( U n ) Local scales: rescale and shift test function: . = n 2 a f ( n a ( z − w )) f w,a ( z ) . for z, w ∈ C and a > 0 . Local law: For a ∈ (0 , 1 / 2) , we have n 1 � � f w,a ( z i ) ≈ f w,a ( z )d µ ( z ) . n C i =1 Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Related previous results for non-Hermitian case Global laws ( a = 0 ) Ginibre [1965]: i.i.d. entries with complex Gaussian distribution Girko [1984]: independent entries with variance profile Bai [1997]: i.i.d. entries with bounded density Tao, Vu [2010]: i.i.d. entries with second moments Cook [2016], Cook, Hachem, Najim, Renfrew [2016]: independent entries with irreducible variance profile; control on smallest singular value Local laws ( a ∈ (0 , 1 / 2) ) Bourgade, Yau, Yin [2014]; Yin [2014] Bulk and edge local law on optimal scale for independent entries with identical variances E | x ij | 2 = n − 1 Local statistics ( a = 1 / 2 ) Ginibre [1965]: k -point correlation function for complex Gaussians → expected universal statistics Tao, Vu [2014]: i.i.d. entries with four matching moments Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Local inhomogeneous circular law Assumptions The random matrix X = ( x ij ) n i,j =1 has independent centered entries. We assume There are some constants s ∗ , s ∗ > 0 such that n ≤ E | x ij | 2 ≤ s ∗ s ∗ for all i, j. n Uniformly bounded moments √ n | m ≤ c m E | x ij for all m ∈ N and i, j. √ n has an L p density for some p > 1 . The distribution of x ij Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Local inhomogeneous circular law We recall the matrix of variances . = ( s ij ) n s ij = E | x ij | 2 . S . i,j =1 , For z ∈ C and η > 0 , let ( v 1 , v 2 ) ∈ R 2 n + be the unique solution of | z | 2 | z | 2 1 1 = η + S t v 1 + = η + Sv 2 + , . η + S t v 1 v 1 v 2 η + Sv 2 Self-consistent density of states of X � ∞ − 1  if | z | 2 < ρ ( S ) , ∆ z � v 1 ( η ; z ) � d η,  σ ( z ) . 2 π . = 0 if | z | 2 ≥ ρ ( S ) . 0 ,  ( ρ ( S ) . . = max | Spec( S ) | spectral radius of S , � v 1 � average of v 1 ) Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Local inhomogeneous circular law With R . � . = ρ ( S ) , we recall � ∞ − 1  ∆ z � v 1 ( η ; z ) � d η, if | z | < R,  σ ( z ) . 2 π . = 0 0 , if | z | ≥ R.  Proposition (Properties of σ ) (A., Erdős, Krüger; 2016) (i) there are c 2 > c 1 > 0 such that for all z ∈ D (0 , R ) c 1 ≤ σ ( z ) ≤ c 2 . (ii) infinitely often differentiable in D (0 , R ) . (iii) supp σ = D (0 , R ) Cook et al. [2016]: σ is a continuous, rotationally symmetric probability density with supp σ ⊆ D (0 , R ) . Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Local inhomogeneous circular law Local scales: . = n 2 a f ( n a ( z − w )) f w,a ( z ) . Theorem (A., Erdős, Krüger; 2016) � ρ ( S ) and f ∈ C 2 (i) For any ε, D > 0 , a ∈ (0 , 1 / 2) , | w | < R . . = 0 ( C ) , we have �� n � � 1 � � ≤ � ∆ f � 1 ≥ 1 − C � � � f w,a ( z ) σ ( z )d 2 z P f w,a ( z i ) − � � n 1 − 2 a − ε n D n � � C � i =1 (ii) For any D > 0 and ε > 0 , we have ≥ 1 − C � � Spec( X ) ⊆ D (0 , R + ε ) P n D Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Ideas of the proof Girko’s Hermitization trick Idea: Transform problem to question for a Hermitian random matrix Instead of X study the family of Hermitian random 2 n × 2 n matrices � � 0 X − z 1 H z . . = X ∗ − ¯ z 1 0 for z ∈ C . Relation between X and H z : log | det( X − z 1 ) | = 1 2 log | det H z | . This connects the quantity we want to study with H z since n 1 1 � � ∆ f w,a ( z ) log | det( X − z 1 ) | d 2 z. f w,a ( z i ) = n 2 πn C i =1 Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Girko’s Hermitization trick � ∞ log | det H z | = − 2 n Im m z (i η )d η, 0 where m z is the Stieltjes transform of the empirical spectral measure of H z 2 n � x − ζ d µ H z ( x ) = 1 1 1 m z ( ζ ) = � λ i ( z ) − ζ , 2 n R i =1 with ζ ∈ C , Im ζ > 0 and λ i ( z ) eigenvalues of H z . Starting formula n 1 � � f w,a ( z ) σ ( z )d 2 z f w,a ( z i ) − n C i =1 � ∞ = − 1 � � � Im m z (i η ) − � v 1 ( η ; z ) � d η d 2 z. ∆ f w,a ( z ) 2 π C 0 Goal (Local law) | m z (i η ) − i � v 1 ( η ; z ) �| � 1 ( with high prob. ) nη Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Local laws for Hermitian random matrices For ζ ∈ C , Im ζ > 0 , we have m z ( ζ ) = 1 . = ( H z − ζ ) − 1 , G ( ζ ) . 2 n Tr G ( ζ ) . The local law follows from G ( ζ ) ≈ M z ( ζ ) , Im ζ ≥ n − 1+ γ . Here, M = M z is the unique solution of Matrix Dyson equation − M ( ζ ) − 1 = ζ − A + S [ M ( ζ )] with positive definite imaginary part Im M = 1 2i ( M − M ∗ ) and A . S [ R ] . . = E H , . = E [( H − A ) R ( H − A )] . Existence and uniqueness: Helton, Rashidi Far, Speicher [2007]. Relationship between v 1 and M : � v 1 ( η ; z ) � = 1 2 n Tr Im M z (i η ) . Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017

Heuristic derivation of the Matrix Dyson equation If Y is a centered real Gaussian random variable and f a differentiable function then Y 2 � f ′ ( Y ) � � � E [ Y f ( Y )] = E E . (1) With the notations A . Y . S [ R ] . D . . = E H , . = H − A , . = E [ Y RY ] , . = ( − Y − S [ G ]) G the definition of the resolvent G can be rewritten as − 1 = ( ζ − H ) G = ( ζ − A − Y ) G = ( ζ − A + S [ G ]) G + D . For real Gaussian H , using a multidimensional analogue of (1), we obtain E [ Y G ] = − E [ S [ G ] G ] ⇒ E D = 0 . Therefore, if G can be approximated by a deterministic matrix M then it is plausible that − 1 = ( ζ − A + S [ M ]) M . Johannes Alt (IST Austria) Local inhomogeneous circular law December 4, 2017