The two matrix model with quartic potential Arno Kuijlaars - PowerPoint PPT Presentation

The two matrix model with quartic potential Arno Kuijlaars Katholieke Universiteit Leuven, Belgium joint work with Maurice Duits (CalTech) Man Yue Mo (Bristol) to appear in Memoirs Amer. Math. Soc. Universidad Carlos III de Madrid, Spain, 26 January 2011

Unitary ensembles Probability measure on n × n Hermitian matrices 1 e − n Tr V ( M ) dM ˜ Z n V ( M ) = 1 2 M 2 This is GUE for Explicit formula for joint density of eigenvalues n 1 � � ( x k − x j ) 2 e − nV ( x j ) Z n 1 ≤ j < k ≤ n j =1

Global eigenvalue behavior As n → ∞ , there is a limiting mean eigenvalue density ρ V ( x ) . ✞ ☎ The probability measure d µ V ( x ) = ρ V ( x ) dx ✝ ✆ minimizes �� � 1 log | x − y | d µ ( x ) d µ ( y ) + V ( x ) d µ ( x ) Typical behavior for polynomial V : density ρ V is positive and real analytic on each interval and vanishes as a square root at endpoints.

Orthogonal polynomials Average characteristic polynomial P n , n ( x ) = E [det( xI n − M )] is n th degree orthogonal polynomial with respect to e − nV ( x ) on real line Orthogonality with respect to varying weight P k , n ( x ) = x k + · · · Monic OPs � ∞ P k , n ( x ) x j e − nV ( x ) dx = h k , n δ j , k , j = 0 , . . . , k . −∞

Determinantal correlation functions Eigenvalues are determinantal point process with correlation kernel n − 1 √ e − nV ( x ) √ P k , n ( x ) P k , n ( y ) � K n ( x , y ) = e − nV ( y ) h k , n k =0 This means that the k point eigenvalue correlation function (which is proportional to marginal density) is given by k × k determinant det [ K n ( x i , x j )] k i , j =1 Global eigenvalue behavior 1 lim nK n ( x , x ) = ρ V ( x ) n →∞

Local eigenvalue behavior Local eigenvalue statistics have universal behavior as n → ∞ . Sine kernel in the bulk: if c = ρ V ( x ∗ ) > 0 then 1 � � = sin π ( x − y ) x ∗ + x cn , x ∗ + y lim cnK n π ( x − y ) cn n →∞ Pastur, Shcherbina (1997), Bleher, Its (1999) Deift, Kriecherbauer, McLaughlin, Venakides, Zhou (1999) McLaughlin, Miller (2008), Lubinsky (2009) Airy kernel at the spectral edge (if ρ V vanishes as a square root at x ∗ ) 1 � � x y x ∗ + cn 2 / 3 , x ∗ + lim cn 2 / 3 K n cn 2 / 3 n →∞ = Ai( x ) Ai ′ ( y ) − Ai ′ ( x ) Ai( y ) x − y

Singular behavior Other limiting kernels at special points Painlev´ e II kernels at interior points where density vanishes. Bleher, Its (2003), Claeys, K (2006) Claeys, K, Vanlessen (2008), Shcherbina (2008) Painlev´ e I 2 kernels at edge points where density vanishes at higher order. Claeys, Vanlessen (2007) Claeys, Its, Krasovsky (2010)

Riemann-Hilbert problem Powerful tool for asymptotic analysis in case of real analytic V is the Riemann-Hilbert problem for OPs Fokas, Its, Kitaev (1992) (1) Y : C \ R → C 2 × 2 is analytic (2) Y has limiting values Y ± on R , satisfying � � e − nV ( x ) 1 Y + ( x ) = Y − ( x ) for x ∈ R , 0 1 � z n � 0 (3) Y ( z ) = ( I + O (1 / z )) as z → ∞ . z − n 0 Correlation kernel is √ e − nV ( x ) √ � 1 � e − nV ( y ) � � Y − 1 K n ( x , y ) = 0 1 + ( y ) Y + ( x ) 0 2 π i ( x − y )

Steepest descent analysis Asymptotics of orthogonal polynomials can be proved by means of a steepest descent analysis of the RH problem Essential role is played by minimizer of the energy functional �� � 1 log | x − y | d µ ( x ) d µ ( y ) + V ( x ) d µ ( x ) and associated g -function � g ( z ) = log( z − s ) ρ V ( s ) ds that satisfies a number of (in)equalities due to Euler-Lagrange variational conditions associated with the minimization problem.

Long term project Extend these results to other matrix ensembles where eigenvalues have determinantal structure Random matrices with external source 1 e − n Tr( V ( M ) − AM ) dM Z n Coupled random matrices (two matrix model) 1 e − n Tr( V ( M 1 )+ W ( M 2 ) − τ M 1 M 2 ) dM 1 dM 2 Z n Normal matrix model (for complex matrices M ) 1 e − n Tr( MM ∗ − V ( M ) − V ( M ∗ )) dM Z n We need extensions / analogues of Orthogonal polynomials Riemann-Hilbert problem Equilibrium problem

Two matrix model The Hermitian two matrix model 1 e − n Tr( V ( M 1 )+ W ( M 2 ) − τ M 1 M 2 ) dM 1 dM 2 Z n is a probability measure on pairs ( M 1 , M 2 ) of n × n Hermitian matrices. V and W are polynomial potentials τ � = 0 is a coupling constant

Biorthogonal polynomials Average characteristic polynomials P n , n ( x ) = E [det( xI n − M 1 )] Q n , n ( y ) = E [det( yI n − M 2 )] are biorthogonal polynomials Mehta,Shukla (1994), Eynard, Mehta (1998) � ∞ � ∞ P n , n ( x ) y j e − n ( V ( x )+ W ( y ) − τ xy ) dxdy = 0 , −∞ −∞ � ∞ � ∞ x j Q n , n ( y ) e − n ( V ( x )+ W ( y ) − τ xy ) dxdy = 0 , −∞ −∞ for j = 0 , . . . , n − 1 .

Multiple orthogonality Suppose W is a polynomial of degree r + 1 Then the biorthogonality condition for P n , n can be rewritten as MOP conditions with weights � ∞ y j e − n ( W ( y ) − τ xy ) dy , w j , n ( x ) = e − nV ( x ) j = 0 , . . . , r − 1 , −∞ and multi-index (assume n is a multiple of r ) � n = ( n / r , . . . , n / r ) � ∞ P n , n ( x ) x k w j , n ( x ) dx = 0 , −∞ for k = 0 , . . . , n r − 1 and j = 0 , . . . , r − 1 . K, McLaughlin (2005)

Riemann-Hilbert problem Multiple orthogonality leads to RH problem of size ( r + 1) × ( r + 1) . Van Assche, Geronimo, K (2001) In case deg W = 4 , (i.e., r = 3 ) (1) Y : C \ R → C 4 × 4 is analytic (2) Y has limiting values Y ± on R , satisfying for x ∈ R   1 w 0 , n ( x ) w 1 , n ( x ) w 2 , n ( x ) 0 1 0 0   Y + ( x ) = Y − ( x )   0 0 1 0   0 0 0 1 (3) as z → ∞   z n 0 0 0 z − n / 3 0 0 0   Y ( z ) = ( I + O (1 / z ))   z − n / 3 0 0 0   z − n / 3 0 0 0

Correlation kernel RH problem has a unique solution and Y 11 ( z ) = P n , n ( z ) Eigenvalues of M 1 are determinantal point process (multiple orthogonal polynomial ensemble) with correlation kernel K n ( x , y ) = 1 � � 0 w 0 , n ( y ) w 1 , n ( y ) w 2 , n ( y ) 2 π i ( x − y )   1 0   × Y − 1 + ( y ) Y + ( x )   0   0 This is based on the Christoffel-Darboux formula for multiple orthogonal polynomials. Daems, K (2004)

Quartic potential Two matrix model 1 e − n Tr( V ( M 1 )+ W ( M 2 ) − τ M 1 M 2 ) dM 1 dM 2 Z n with V even and W a quartic polynomial W ( y ) = 1 4 y 4 + α 2 y 2 There is a vector equilibrium problem for three measures that describes the limiting mean density for the eigenvalues of M 1 .

Quartic potential Two matrix model 1 e − n Tr( V ( M 1 )+ W ( M 2 ) − τ M 1 M 2 ) dM 1 dM 2 Z n with V even and W a quartic polynomial W ( y ) = 1 4 y 4 + α 2 y 2 There is a vector equilibrium problem for three measures that describes the limiting mean density for the eigenvalues of M 1 . Earlier work for α = 0 Duits, K (2009), Mo (2009) Very different description of limiting eigenvalue distributions is due to Guionnet (2004)

Notation �� 1 I ( µ ) = log | x − y | d µ ( x ) d µ ( y ) �� 1 I ( µ, ν ) = log | x − y | d µ ( x ) d ν ( y )

Vector equilibrium problem Energy functional E ( µ 1 , µ 2 , µ 3 ) = I ( µ 1 ) + I ( µ 2 ) + I ( µ 3 ) − I ( µ 1 , µ 2 ) − I ( µ 2 , µ 3 ) � � + V 1 ( x ) d µ 1 ( x ) + V 3 ( x ) d µ 3 ( x ) Minimize E ( µ 1 , µ 2 , µ 3 ) among µ 1 , µ 2 , µ 3 such that (a) µ 1 is a measure on R with µ 1 ( R ) = 1 , (b) µ 2 is a measure on i R with µ 2 ( R ) = 2 / 3 , (c) µ 3 is a measure on R with µ 3 ( R ) = 1 / 3 , and µ 2 ≤ σ 2 , where σ 2 is certain given measure on i R

External field V 1 acting on first measure Definition V 1 ( x ) = V ( x ) + min s ∈ R ( W ( s ) − τ xs ) 4 s 4 + α W ( s ) = 1 2 s 2 By Laplace’s method � ∞ 1 e − n ( V ( x )+ W ( s ) − τ xs ) ds V 1 ( x ) = − lim n log n →∞ −∞ W ( s ) − τ xs has a minimum for s ∈ R at s 1 ( x ) V 1 ( x ) = V ( x ) + W ( s 1 ( x )) − τ xs 1 ( x )

External field V 3 acting on the third measure W ( s ) − τ xs has more local extrema on the real line, say at s 2 ( x ) and s 3 ( x ) , if | x | < x ∗ ( α ) where � − α � 3 / 2 x ∗ ( α ) = 2 x ∗ ( α ) = 0 if α ≥ 0 , if α < 0 , τ 3 Definition if | x | < x ∗ ( α ) then V 3 ( x ) = ( W ( s 3 ( x )) − τ xs 3 ( x )) − ( W ( s 2 ( x )) − τ xs 2 ( x )) > 0 � �� � � �� � other local minimum local maximum if | x | ≥ x ∗ ( α ) then V 3 ( x ) = 0

Illustration For x close to 0 , the function W ( s ) − τ sx has a global minimum and one local non-global minimum and one local maximum. Then V 3 ( x ) is the difference between the value at the local non-global local minimum and the local maximum. For large x , there is only a global minimum and then V 3 ( x ) = 0 .

Upper constraint σ 2 acting on the second measure W ( s ) − τ zs with z ∈ i R has a global minimum on the imaginary axis � � 3 / 2 , � α 2 α > 0 , If z = iy with | y | > y ∗ ( α ) = τ 3 0 α ≤ 0 , then there no other local extrema Definition σ 2 is the measure on ( − i ∞ , − iy ∗ ( α )] ∪ [ iy ∗ ( α ) , i ∞ ) with density d σ 2 ( z ) = τ π Re s ( z ) | dz | where s ( z ) is the solution of W ′ ( s ) = τ z with Re s ( z ) > 0.