Heavy-tailed random matrices and the Poisson Weighted In fi nite Tree - PowerPoint PPT Presentation

Heavy-tailed random matrices and the Poisson Weighted In fi nite Tree Charles Bordenave CNRS & University of Toulouse Joint work with Pietro Caputo (Roma III) and Djalil Chafa¨ ı (Paris XI). 1

PART I : RANDOM MATRICES

SPECTRAL MEASURE Let X = ( X ij ) 1 ≤ i,j ≤ n be a n × n complex matrix. Let λ 1 , · · · , λ n be its eigenvalues, the spectral measure of X is n µ X = 1 � δ λ k . n k =1 - (random hermitian) : the array ( X ij ) i ≥ j ≥ 1 is i.i.d., X ij = ¯ X ji , - (random non-hermitian) : the array ( X ij ) i,j ≥ 1 is i.i.d.. = ⇒ As n goes to in fi nity, does the spectral measure converge ?

WIGNER’S SEMI-CIRCULAR LAW Theorem 1. If E X 11 = 0 , E | X 11 | 2 = 1 and A n = X/ √ n, then, almost surely, µ A n = ⇒ µ sc , √ 1 4 − x 2 dx . where µ sc ( dx ) = 2 π

GIRKO’S CIRCULAR LAW Theorem 2 (Tao & Vu (2008)) . If E X 11 = 0 , E | X 11 | 2 = 1 and A n = X/ √ n, then, almost surely, µ A n = ⇒ Unif ( D ) . where Unif ( D ) is the uniform distribution on the unit complex disc. = ⇒ Found by Girko, earlier versions due to Edelman, Bai, Pan & Zhou, G¨ otze & Tikhomirov ...

HEAVY-TAILED ENTRIES We now assume that P ( | X 11 | > t ) ∼ t − α for some 0 < α < 2 . De fi ne A n = X/n 1 /α , = ⇒ In the hermitian and non-hermitian cases, does µ A n converge to a measure µ ?

HERMITIAN CASE Theorem 3 (Ben Arous & Guionnet, 2008) . There exists a probability measure µ bc depending only on α such that, with the above assumptions, almost surely, µ A n = ⇒ µ bc . = ⇒ Found non-rigourously by Bouchaud-Cizeau (1994).

PROPERTIES OF THE LIMIT MEASURE Theorem 4. For all 0 < α< 2 , the probability measure µ bc (i) is symmetric and has a bounded density f bc on R , � 1 � � Γ ( 1 − α 2 ) α (ii) f bc (0) = 1 1 + 2 � π Γ , α Γ ( 1+ α 2 ) (iii) f bc ( t ) ∼ t →∞ α 2 t − α − 1 . = ⇒ Summarizes properties obtained by Ben Arous & Guionnet, Belinschi, Dembo & Guionnet, Bordenave, Caputo & Chafa¨ ı .

NON-HERMITIAN CASE Theorem 5. Assume that X 11 has a bounded density on R or C , and is asymptotically radial : � X 11 � � lim | X 11 | ∈ · � | X 11 | ≥ t = θ, t →∞ P � for some probability distribution θ on S 1 . Then, there exists a probability measure µ depending only on α such that, with the above assumptions, almost surely, µ A n = ⇒ µ.

PROPERTIES OF THE LIMIT SPECTRAL MEASURE Theorem 6. The measure µ has radial bounded density µ ( dz ) = f ( | z | ) dz , where 2 Γ(1 + 2 α ) 2 Γ(1 + α 2 ) f (0) = 1 α , 2 π Γ(1 − α 2 ) α and as r → ∞ f ( r ) ∼ c r 2( α − 1) e − α 2 r α .

PART II : OBJECTIVE METHOD AND LOCAL OPERATOR CONVERGENCE

HERMITIAN CASE : SPECTRAL MEASURE AT A VECTOR There exists a probability measure on R such that, for all t integers, � x t dµ ( k ) ( e k , A t n e k ) = A n , n n µ A n = 1 µ ( k ) µ ( k ) � � | ( e k , u i ) | 2 δ λ i . A n = and A n n k =1 i =1 The spectral measure at a vector is well de fi ned for all self-adjoint operators.

HERMITIAN CASE : REDUCTION TO LOCAL CONVERGENCE = ⇒ By exchangeability, we get E µ A n = E µ (1) A n = ⇒ From basic concentration inequality, µ A n − E µ A n converges a.s. to 0 . ⇒ It is enough to get the convergence of µ (1) = A n .

HERMITIAN CASE : LOCAL OPERATOR CONVERGENCE We look for a random operator A de fi ned in L 2 ( V ) for some countable set V such that there exists a sequence of bijections σ n : V → N , ø ∈ V , σ n (ø) = 1 and, for all φ ∈ L 2 ( V ) with compact support, weakly, σ − 1 n A n σ n φ → Aφ. If A is self-adjoint, then it would imply that µ (1) A n → µ (ø) A .

NON-HERMITIAN CASE The above strategy does not work. The spectral measure at a vector does not exist for non-normal matrices. The local operator convergence is not suf fi cient.     0 1 0 0 · · · 0 1 0 0 · · ·     0 0 1 0 · · · 0 0 1 0 · · ·         .   vs       · · · · · ·             0 · · · 0 0 0 1 · · · 0 0 0 An additional ingredient is needed. (For the moment, we skip this very important part).

PART III : CONVERGENCE TO POISSON WEIGHTED INFINITE TREE

ALDOUS’ PWIT Let V = ∪ k ∈ N N k with N 0 = ø . Consider the in fi nite tree on V : ø · · · 2 4 1 3 · · · · · · 1 , 2 1 , 3 1 , 1 · · · · · · 1 , 1 , 3 1 , 1 , 4 1 , 1 , 1 1 , 1 , 2

ALDOUS’ PWIT Let ( Z v ) v ∈ V be iid Poisson processes of intensity λ on R + , Z v = { 0 ≤ ζ v 1 ≤ ζ v 2 ≤ · · · } ø · · · ζ 1 ζ 2 ζ 3 ζ 4 2 4 1 3 · · · ζ 21 ζ 11 · · · ζ 12 ζ 22 ζ 13 ζ 23 1 , 2 1 , 3 1 , 1 · · · ζ 111 ζ 112 · · · ζ 113 ζ 114 1 , 1 , 3 1 , 1 , 4 1 , 1 , 1 1 , 1 , 2

GRAPHIC REPRESENTATION OF A MATRIX We think of the matrix A n as an oriented weighted graph on n vertices. A 11 A 22 � A 12 � 2 1 A 21 � A 14 � � A 23 � A 41 A 32 � A 24 � A 13 � � A 42 A 31 � A 34 � 3 4 A 43 A 44 A 33

ORDERED STATISTICS �� A 11 �� � A 12 � A 1 n � � , , · · · , The vector is reordered non-increasingly in A 11 A 21 A n 1 �� A 1 σ (1) � � A 1 σ 1 (2) � � A 1 σ ( n ) �� , , · · · , A σ (1)1 A σ (2)1 A σ ( n )1 with � A 1 σ (1) � � A 1 σ (2) � � � 1 ≥ � � 1 ≥ · · · . A σ (1)1 A σ (2)1 = ⇒ We restrict ourselves to non-hermetian case and non-negative random variables.

CONVERGENCE OF ORDERED STATISTICS �� A 1 σ (1) � � A 1 σ (2) � � A 1 σ ( n ) �� , , · · · , A σ (1)1 A σ (2)1 A σ ( n )1 converges to �� � � � � ε 1 ε 2 − 1 − 1 ζ , ζ , · · · , α α 1 2 1 − ε 1 1 − ε 2 where ( ζ k ) k ≥ 1 , ζ 1 ≤ ζ 2 ≤ · · · , is a Poisson point process of intensity Λ( dx ) = 21 I x> 0 dx and ( ε k ) iid Ber(1 / 2) random variables.

CONVERGENCE OF ORDERED STATISTICS For fi xed i , the vector �� A jσ ( i ) �� A σ ( i ) j j � =1 is reordered non-increasingly. It converges again to �� � � � � ε i 1 ε i 2 − 1 − 1 ζ i 1 , ζ i 2 , · · · , α α 1 − ε i 1 1 − ε i 2 where ( ζ ik ) k ≥ 1 are independent Poisson processes of intensity Λ and ( ε ik ) i,k iid Ber(1 / 2) r.v.

LOCAL CONVERGENCE TO ALDOUS’ PWIT ø 1 → ø · · · � ε 1 � ε 2 − 1 − 1 � ε 3 � ε 4 − 1 − 1 � � σ ( i ) → i ζ ζ � � α α ζ ζ α α 1 2 1 − ε 1 1 − ε 2 3 4 1 − ε 3 1 − ε 4 · · · 2 4 1 3 � ε 21 − 1 · · · � ε 11 � − 1 ζ α � ζ 21 � ε 22 α 1 − ε 21 · · · 11 � ε 12 1 − ε 11 − 1 − 1 � � ζ ζ α α 12 � ε 13 � ε 23 22 1 − ε 12 − 1 1 − ε 22 − 1 � � ζ ζ α α 13 23 1 − ε 13 1 − ε 23 A 11 A 22 � A 12 � 2 1 , 2 1 , 3 1 1 , 1 A 21 · · · � ε 111 − 1 � ζ � A 14 � α � A 23 � 111 1 − ε 111 A 41 A 32 � ε 112 − 1 � ζ α · · · � A 24 � A 13 � � 112 � ε 113 1 − ε 112 − 1 A 42 A 31 � ε 114 � − 1 ζ α � ζ 113 1 − ε 113 α 114 1 − ε 114 − → � A 34 3 � 1 , 1 , 3 1 , 1 , 4 4 1 , 1 , 1 1 , 1 , 2 A 43 A 44 A 33 n →∞

OPERATOR ON THE PWIT De fi ne the operator on compactly supported function of L 2 ( V ) , − 1 − 1 � Aδ v = (1 − ε vk ) ζ vk δ vk + ε v ζ α α δ a ( v ) , v k ≥ 1 where a ( v ) is the ancestor of v � = ø . = ⇒ There exists a sequence of bijections σ n : V → N , ø ∈ V , σ n (ø) = 1 and, for all φ ∈ L 2 ( V ) with compact support, weakly, σ − 1 n A n σ n φ → Aφ.

HERMITIAN CASE : OPERATOR ON THE PWIT In the hermitian case, the operator is de fi ned similarly, we simply forget about, the ε � v s : − 1 − 1 � Aδ v = ζ vk δ vk + ζ δ a ( v ) . α α v k ≥ 1 → Again, for all φ ∈ L 2 ( V ) with compact support, weakly, σ − 1 − n A n σ n φ → Aφ. Theorem 7. With probability one, the operator A is (essentially) self-adjoint. ⇒ As a corollary, we obtain the convergence of µ A n to µ bc := E µ (ø) = A .

HERMITIAN CASE : RECURSIVE DISTRIBUTIONAL EQUATION The resolvent formula and the recursive structure of the PWIT implies a RDE for, z ∈ C + = { z ∈ C : � ( z ) > 0 } , g ø ( z ) := � δ ø , ( A − z ) − 1 δ ø � � − 1 � d � g ø = − z + ξ k g k , k ∈ N where g ø , ( g k ) k ∈ N are i.i.d. independent of { ξ k } k ∈ N , a independent Poisson point 2 x − α process of R + with intensity α 2 − 1 dx .

RECURSIVE DISTRIBUTIONAL EQUATION If S is a positive α/ 2 -stable random variable, α d 2 � α S. ξ k g k = E [ g ø ] 2 k ∈ N α 2 = ⇒ The RDE can be solved in terms of a scalar fi xed point equation for E [ g ø ] 2 α . = ⇒ Since g is the Cauchy-Stieltjes transform of µ ø , we deduce the properties of µ bc = E µ (ø) A .

PART IV : CONVERGENCE IN THE NON-HERMITIAN CASE