Asymptotics of orthogonal polynomials in normal matrix ensemble - PowerPoint PPT Presentation

Asymptotics of orthogonal polynomials in normal matrix ensemble Seung-Yeop Lee (University of South Florida) Cincinnati, September 20th 2014 1 / 32

Joint work with Roman Riser. Many discussions with Marco Bertola, Robert Buckingham, Maurice Duits, Kenneth McLaughlin, ... 2 / 32

Main actors: ◮ Orthogonal polynomials ◮ Two dimensional Coulomb gas ◮ Hele-Shaw flow 3 / 32

Orthogonal polynomials on C Orthogonal polynomials : p n ( z ) = z n + ... � p j ( z ) p k ( z ) e − NQ ( z ) d A ( z ) = h j δ jk . C Q : C → R is the external field; N is a positive parameter. Examples: – When Q ( z ) = | z | 2 , p n ( z ) = z n . – When Q ( z ) = (1 − t )( Re z ) 2 + (1 + t )( Im z ) 2 , � √ � � 2 n z t n p n ( z ) ∝ H n ; F 0 = 2 (1 − t 2 ) N . F 0 4 / 32

2D Coulomb gas (Eigenvalues) Using the same Q , probablity density function of n point particles, { z 1 , · · · , z n } ⊂ C , are given by � � �� n � � PDF ( { z j ’s } ) = 1 Q ( z j ) + 2 1 exp − N log . Z n N | z j − z k | j =1 j < k � �� � 2D Coulomb energy For Q ( z ) = | z | 2 − c log | z − a | . 5 / 32

Droplet K (compact set in C ) – Support of the equilibrium measure. – Throughout this talk, we assume that ∆ Q = const. – For logarhthmic/rational Hele-Shaw potential, the exterior of K c is a quadrature domain . – As T := n / N grows, K grows monotonically in T : We call T := n / N the total charge or (Hele-Shaw) time . The deformation of K under T follows Hele-Shaw flow . 6 / 32

Exterior conformal map of K For simplicity, we assume that K is simply connected so that we can define the unique riemann mapping f : K c → D c such that f ( z ) = z ρ + O (1) , ρ > 0 , as | z | → ∞ . Geometry of K is encoded in f . For example, the curvature of the boundary of K is given by � � 1 − f ′′ f | f ′ | κ = Re ( f ′ ) 2 where the prime ′ stands for the complex derivative. 7 / 32

Similar cases: – Bergman orthogonal polynomials: � p n ( z ) p m ( z ) dA ( z ) = h n δ nm . D p n ( z ) = ρ n +1 f ′ ( z ) f ( z ) n (1 + (corrections)) . – Szeg¨ o orthogonal polynomials: � p n ( z ) p m ( z ) | dz | = h n δ nm . Γ p n ( z ) = ρ n � ρ f ′ ( z ) f ( z ) n (1 + (corrections)) . In both cases, if the relevant geometry has a smooth boundary , the correction term is exponentially small in n . 8 / 32

Conjecture (If the potential Q is such that K has real analytic boundary,) the strong asymptotics of p n ( z ) as n → ∞ and N → ∞ while T := n / N is finite, is given by � � 1 �� � 1 + 1 ρ f ′ ( z ) e ng ( z ) p n ( z ) = N Ψ( z ) + O , z / ∈ K . N 2 The function g (called g -function) is the complex logarithmic potential generated by the measure 1 K : � 1 g ( z ) = log( z − ζ ) dA ( ζ ) . π T K The function Ψ is in the next page. 9 / 32

The function Ψ is given by � Ψ( z ) = i Φ( ζ ) df ( ζ ) � , � 2 π f ( ζ ) f ( z ) f ( ζ ) − 1 ∂ K where � f ′′′ f 2 � Φ := κ 2 f ′′ 2 f 2 12 + 1 2 κ ( | f ′ | − κ ) + 1 − 1 | f ′ | 2 + i 2 ∂ � | f ′ | . 4 Re f ′ 2 f ′ 4 2 Remark. The method (that we will explain) can generate the corrections in the arbitrary order in 1 / N . 10 / 32

Known examples of strong asymptotics: Q ( z ) = | z | 2 : K is a disk Q ( z ) = | z | 2 + a Re z 2 : K is ellipse (Felder-Riser ’13) Q ( z ) = | z | 2 + a Re z 3 : K is a hypotrochoid (Bleher-Kuijlaars ’12) Q ( z ) = | z | 2 + a Re z p : (Kuijlaars - Lopez-Garcia) Q ( z ) = | z | 2 − c log | z − a | : K is a Joukowsky airfoil (Balogh-Bertola-Lee-McLaughlin ’13) *** The correction term is checked explicitly only for the first two cases. 11 / 32

Restating the conjecture... Claim. If the following (WKB) expansion � � 1 �� n g ( z ) + Ψ 0 ( z ) + 1 (A1) p n ( z ) = exp N Ψ 1 ( z ) + O , N 2 holds (in some region around the boundary), and if the kernel satisfies certain asymptotic behavior such that the density is given by � 1 � ρ ( z ) = 1 (A2) π + O , N 2 (uniformly) inside (a compact subset of) K , then the conjecture is true. 12 / 32

Relation between OP and CG: Several fundamental facts: – OP = Average of characteristic polynomial: � � n � p n ( z ) = E ( z − z j ) . j =1 – Density of the CG = Sum of the absolute square of OPs: n − 1 � ρ ( z ) = 1 | p j ( z ) | 2 e − NQ ( z ) . N j =0 � � � � n − 1 � K n ( z , w ) = 1 p j ( z ) p j ( w ) e − N Q ( z )+ Q ( w ) . 2 N j =0 13 / 32

Hele-Shaw potential The density of the Coulomb gas is given by � n � dA ( z j ) → ∆ Q ρ ( z ) := PDF ( z , z 2 , · · · , z n ) 4 π when z ∈ K . j =2 Q ( z ) = | z | 2 − t Re ( z 2 ) Q ( z ) = | z | 2 14 / 32

Quantum Hele-Shaw flow The plot of | p n ( z ) | 2 e − NQ ( z ) : (Left: single; Right: several consecutive) Gaussian peak along the boundary is from � � e − N Q ( z ) − Tg ( z ) − Tg ( z ) . 15 / 32

D-bar approach From the orthogonality we have � � � p n ( w ) e − NQ ( w ) 1 1 d A ( w ) = O . z n +1 π z − w C Again by the orthogonality, we have � � p n ( w ) e − NQ ( w ) p n ( w ) p n ( w ) e − NQ ( w ) 1 d A ( w ) = 1 1 d A ( w ) . π z − w π p n ( z ) z − w C C The numerator in RHS has the following property. Theorem (Ameur-Hedenmalm-Makarov) | p n ( z ) | 2 e − NQ ( z ) dA ( z ) → Harmonic measure on K c 16 / 32

1 / N -expansion of Cauchy transform For a smooth test function f , � � � f ( ζ ) e − N Q ( ζ ) − g ( ζ ) − g ( ζ )+ ℓ dA ( ζ ) C � π � � � κ 2 � f ( ζ ) + 1 12 f ( ζ ) + 3 κ 8 ∂ n f ( ζ ) + 1 8 ∂ 2 = n f ( ζ ) 2 N N ∂ K � 1 �� + O | d ζ | . N 2 (This is obtained by using Schwarz function.) We take p n ( ζ ) | 2 f ( ζ ) = | � ζ − z where � p n is all the subleading parts of p n : � � 1 �� 1 + 1 p n ( z ) := p n ( z ) e − ng ( z ) = e Ψ 0 � N Ψ 1 + O . N 2 17 / 32

1 / N -expansion of Cauchy transform (cont.) One obtains the following. � π � � | � p n ( w ) | 2 1 � C n ( z ) = � p n ( z ) 2 N z − w � κ 2 � | � � 1 �� p n ( w ) | 2 + 1 12 + 3 κ 8 ∂ n + 1 8 ∂ 2 + O | dw | . n N 2 N z − w – Note that this is the “electric force” from the measure | p n | 2 e − NQ dA . – By using the “convergence to harmonic measure” the leading term of � C ( z ) must vanish inside K . 18 / 32

Therefore, in the leading order, p n ( w ) | 2 ≈ | e 2Ψ 0 | ∝ | f ′ | . | � And this leads to � e Ψ 0 ( z ) = ρ f ′ ( z ) . (This is not the main point.) To calculate the next order, we claim that � C n vanishes even at the second order. This is not proven in general, however it follows from certain asymptotics of the kernel (which is also not proven in general). 19 / 32

Kernel → Cauchy transform Recall � ρ (1) n ( z ) := PDF n ( { z , z 2 , · · · , z n } ) dA ( z 2 ) · · · dA ( z n ) . = 1 N K n ( z , z ) . � ρ (2) n ( z , w ) := PDF n ( { z , w , z 3 , · · · , z n } ) dA ( z 3 ) · · · dA ( z n ) . � K n ( z , z ) K n ( w , w ) − | K n ( z , w ) | 2 � 1 = . N ( n − 1) 20 / 32

Taking ∂ z on the first equation: � � � n n � � 1 ∂ρ (1) − NQ ′ ( z ) + n ( z ) = ρ n ( { z , z 2 , · · · , z n } ) dA ( z j ) z − z j j =2 j =2 � dA ( w ) n � = − NQ ′ ( z ) ρ (1) z − w ρ (2) n ( z ) + ( n − 1) n ( { z , w , z 3 , · · · , z n } ) dA ( z j ) j =3 � dA ( w ) � K n ( z , z ) K n ( w , w ) − | K n ( z , w ) | 2 � n ( z ) + 1 = − NQ ′ ( z ) ρ (1) . N z − w Divide the whole equation by ρ (1) n ( z ) = 1 N K n ( z , z ). Obtain the same equation for ρ (1) n +1 and take the difference of the two. We obtain � | p n ( w ) | 2 e − NQ ( w ) dA ( w ) � | K n ( z , w ) | 2 dA ( w ) 1 = z − w K n ( z , z ) z − w +(terms with ∂ρ (1) n ( z )) 21 / 32

Asymptotics of kernel Theorem [Riser] For ellipse case, Q ( z ) = | z | 2 − t Re ( z 2 ), | K n ( z , w ) | 2 = N π e − N | z − w | 2 (1 + O ( N −∞ )) , when z and w are both inside the ellipse and sufficiently close to each other. Proof) Based on the Christoffel-Darboux identity: � 2 ( | z | 2 + | w | 2 − 2 zw ) � 1 N N ∂ w K n ( z , w ) e � n t p n ( z ) p n − 1 ( w ) − p n − 1 ( z ) p n ( w ) N 2 ( − 2 zw + t Re ( z 2 )+ t Re ( w 2 )) . = � √ e N 1 − t 2 h n h n − 1 When z and w are inside the bulk (and close to each other), the polynomials in the right hand side peak on the boundary. 22 / 32

QUESTION : For real analytic potential of the type: Q ( z ) = | z | 2 + (harmonic) the kernel inside the bulk is asymptotically given by | K n ( z , w ) | 2 = N π e − N | z − w | 2 (1 + O ( N −∞ )) . This observation shows that the term � | K n ( z , w ) | 2 dA ( w ) 1 K n ( z , z ) z − w and ∂ρ (1) are both exponentially small in N inside the bulk of n the ellipse. 23 / 32

Let us come back to � C n and use the expansion with: � � � 1 + 1 ρψ ′ ( z ) � p n ( z ) = N Ψ( z ) + ... and we define Φ( z ) such that � � κ 2 � ρ | ψ ′ ( w ) | ρ | ψ ′ ( w ) | Φ( w ) 12 + 3 κ 8 ∂ n + 1 8 ∂ 2 dA ( w ) := z − w dA ( w ) . n z − w ∂ K Using Plemelj-Sokhotski relation, we get, at the second order in 1 / N , the following identity: � � � � 2 π 3 ρ ψ ′ ( z ) � � C n ( z ) | in − � � C n ( z ) | out = − Ψ( z ) + Φ( z ) . ψ ( z ) N 1 / N 24 / 32