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 ˜ Zn e−n Tr V (M) dM

This is GUE for V (M) = 1

2M2

Explicit formula for joint density of eigenvalues 1 Zn

• 1≤j<k≤n

(xk − xj)2

n

• j=1

e−nV (xj)

Global eigenvalue behavior

As n → ∞, there is a limiting mean eigenvalue density ρV(x). The probability measure

✞ ✝ ☎ ✆

dµV (x) = ρV(x)dx minimizes

• log

1 |x − y|dµ(x)dµ(y) +

• V (x)dµ(x)

Typical behavior for polynomial V : density ρV is positive and real analytic on each interval and vanishes as a square root at endpoints.

Orthogonal polynomials

Average characteristic polynomial Pn,n(x) = E [det(xIn − M)] is nth degree orthogonal polynomial with respect to e−nV (x) on real line

Orthogonality with respect to varying weight

Monic OPs Pk,n(x) = xk + · · · ∞

−∞

Pk,n(x) xj e−nV (x) dx = hk,nδj,k, j = 0, . . . , k.

Determinantal correlation functions

Eigenvalues are determinantal point process with correlation kernel Kn(x, y) = √ e−nV (x)√ e−nV (y)

n−1

• k=0

Pk,n(x)Pk,n(y) hk,n

This means that the k point eigenvalue correlation function (which is proportional to marginal density) is given by k × k determinant det [Kn(xi, xj)]k

i,j=1

Global eigenvalue behavior lim

n→∞

1 nKn(x, x) = ρV (x)

Local eigenvalue behavior

Local eigenvalue statistics have universal behavior as n → ∞.

Sine kernel in the bulk: if c = ρV (x∗) > 0 then lim

n→∞

1 cnKn

• x∗ + x

cn, x∗ + y cn

• = sin π(x − y)

π(x − y)

Pastur, Shcherbina (1997), Bleher, Its (1999) Deift, Kriecherbauer, McLaughlin, Venakides, Zhou (1999) McLaughlin, Miller (2008), Lubinsky (2009)

Airy kernel at the spectral edge (if ρV vanishes as a square root at x∗) lim

n→∞

1 cn2/3 Kn

• x∗ +

x cn2/3 , x∗ + y cn2/3

• = 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 I2 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 → C2×2 is analytic (2) Y has limiting values Y± on R, satisfying Y+(x) = Y−(x)

• 1

e−nV (x) 1

• for x ∈ R,

(3) Y (z) = (I + O(1/z)) zn z−n

• as z → ∞.

Correlation kernel is Kn(x, y) = √ e−nV (x)√ e−nV (y) 2πi(x − y)

• 1
• Y −1

+ (y)Y+(x)

1

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

• log

1 |x − y|dµ(x)dµ(y) +

• V (x)dµ(x)

and associated g-function g(z) =

• log(z − s)ρV(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 Zn e−n Tr(V (M)−AM)dM Coupled random matrices (two matrix model) 1 Zn e−n Tr(V (M1)+W (M2)−τM1M2)dM1dM2 Normal matrix model (for complex matrices M) 1 Zn e−n Tr(MM∗−V (M)−V (M∗))dM

We need extensions / analogues of

Orthogonal polynomials Riemann-Hilbert problem Equilibrium problem

Two matrix model

The Hermitian two matrix model 1 Zn e−n Tr(V (M1)+W (M2)−τM1M2) dM1dM2 is a probability measure on pairs (M1, M2) of n × n Hermitian matrices. V and W are polynomial potentials τ = 0 is a coupling constant

Biorthogonal polynomials

Average characteristic polynomials Pn,n(x) = E [det(xIn − M1)] Qn,n(y) = E [det(yIn − M2)] are biorthogonal polynomials Mehta,Shukla (1994), Eynard, Mehta (1998) ∞

−∞

∞

−∞

Pn,n(x)y je−n(V (x)+W (y)−τxy) dxdy = 0, ∞

−∞

∞

−∞

xjQn,n(y)e−n(V (x)+W (y)−τxy) dxdy = 0, for j = 0, . . . , n − 1.

Multiple orthogonality

Suppose W is a polynomial of degree r + 1 Then the biorthogonality condition for Pn,n can be rewritten as MOP conditions with weights wj,n(x) = e−nV (x) ∞

−∞

y je−n(W (y)−τxy) dy, j = 0, . . . , r−1, and multi-index (assume n is a multiple of r)

• n = (n/r, . . . , n/r)

∞

−∞

Pn,n(x)xkwj,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 → C4×4 is analytic (2) Y has limiting values Y± on R, satisfying for x ∈ R Y+(x) = Y−(x)     1 w0,n(x) w1,n(x) w2,n(x) 1 1 1     (3) as z → ∞ Y (z) = (I + O(1/z))     zn z−n/3 z−n/3 z−n/3    

Correlation kernel

RH problem has a unique solution and Y11(z) = Pn,n(z) Eigenvalues of M1 are determinantal point process (multiple orthogonal polynomial ensemble) with correlation kernel Kn(x, y) = 1 2πi(x − y)

• w0,n(y)

w1,n(y) w2,n(y)

• × Y −1

+ (y)Y+(x)

    1     This is based on the Christoffel-Darboux formula for multiple orthogonal polynomials. Daems, K (2004)

Quartic potential

Two matrix model 1 Zn e−n Tr(V (M1)+W (M2)−τM1M2) dM1 dM2 with V even and W a quartic polynomial W (y) = 1 4y 4 + α 2 y 2 There is a vector equilibrium problem for three measures that describes the limiting mean density for the eigenvalues of M1.

Quartic potential

Two matrix model 1 Zn e−n Tr(V (M1)+W (M2)−τM1M2) dM1 dM2 with V even and W a quartic polynomial W (y) = 1 4y 4 + α 2 y 2 There is a vector equilibrium problem for three measures that describes the limiting mean density for the eigenvalues of M1. Earlier work for α = 0 Duits, K (2009), Mo (2009) Very different description of limiting eigenvalue distributions is due to Guionnet (2004)

Notation

I(µ) =

• log

1 |x − y|dµ(x)dµ(y) I(µ, ν) =

• log

1 |x − y|dµ(x)dν(y)

Vector equilibrium problem

Energy functional E(µ1, µ2, µ3) = I(µ1) + I(µ2) + I(µ3) − I(µ1, µ2) − I(µ2, µ3) +

• V1(x)dµ1(x) +
• V3(x)dµ3(x)

Minimize E(µ1, µ2, µ3) among µ1, µ2, µ3 such that (a) µ1 is a measure on R with µ1(R) = 1, (b) µ2 is a measure on iR with µ2(R) = 2/3, (c) µ3 is a measure on R with µ3(R) = 1/3, and µ2 ≤ σ2, where σ2 is certain given measure on iR

External field V1 acting on first measure

Definition V1(x) = V (x) + min

s∈R(W (s) − τxs)

W (s) = 1

4s4 + α 2 s2

By Laplace’s method V1(x) = − lim

n→∞

1 n log ∞

−∞

e−n(V (x)+W (s)−τxs)ds W (s) − τxs has a minimum for s ∈ R at s1(x) V1(x) = V (x) + W (s1(x)) − τxs1(x)

External field V3 acting on the third measure

W (s) − τxs has more local extrema on the real line, say at s2(x) and s3(x), if |x| < x∗(α) where x∗(α) = 2 τ −α 3 3/2 if α < 0, x∗(α) = 0 if α ≥ 0, Definition if |x| < x∗(α) then V3(x) = (W (s3(x)) − τxs3(x))

• local maximum

− (W (s2(x)) − τxs2(x))

• ther local minimum

> 0 if |x| ≥ x∗(α) then V3(x) = 0

Illustration

For x close to 0, the function W (s) − τsx has a global minimum and one local non-global minimum and one local maximum. Then V3(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 V3(x) = 0.

Upper constraint σ2 acting on the second measure

W (s) − τzs with z ∈ iR has a global minimum on the imaginary axis If z = iy with |y| > y ∗(α) =

• 2

τ

α

3

3/2 , α > 0, α ≤ 0, then there no other local extrema Definition σ2 is the measure on (−i∞, −iy ∗(α)] ∪ [iy ∗(α), i∞) with density dσ2(z) |dz| = τ π Re s(z) where s(z) is the solution of W ′(s) = τz with Re s(z) > 0.

Vector equilibrium problem

Minimize I(µ1) + I(µ2) + I(µ3) − I(µ1, µ2) − I(µ2, µ3) +

• V1(x)dµ1(x) +
• V3(x)dµ3(x)

among measures µ1, µ2, µ3 such that (a) µ1 is a measure on R with µ1(R) = 1, (b) µ2 is a measure on iR with µ2(R) = 2/3, (c) µ3 is a measure on R with µ3(R) = 1/3, and µ2 ≤ σ2

Results on equilibrium measures

Theorem (Duits, K, Mo) There is a unique minimizer (µ1, µ2, µ3) of the vector equilibrium problem.

Structure of minimizer

The support of µ1 is a finite union of intervals S(µ1) =

N

• j=1

[aj, bj] The support of µ2 is equal to the support of σ2, and the constraint σ2 is active on a symmetric interval around 0, possibly empty, S(σ2 − µ2) = iR \ (−ic2, ic2), c2 ≥ 0. The support of µ3 has at most one gap S(µ3) = R \ (−c3, c3), c3 ≥ 0

Riemann surface

The supports of the measures S(µ1), S(σ2 − µ2), S(µ3) are the cuts for a four-sheeted Riemann surface Four sheets R1 = C \ S(µ1) R2 = C \ (S(µ1) ∪ S(σ2 − µ2)) R3 = C \ (S(σ2 − µ2) ∪ S(µ3)) R4 = C \ S(µ3)

Riemann surface (Case I)

Case I: 0 ∈ S(µ1), 0 ∈ S(σ2 − µ2), 0 ∈ S(µ3)

• Cut along S(µ1)
• Cut along S(σ2 − µ2)
• Cut along S(µ3)

Riemann surface (Case II)

Case II: 0 ∈ S(µ1), 0 ∈ S(σ2 − µ2), 0 ∈ S(µ3)

• Cut along S(µ1)
• Cut along S(σ2 − µ2)
• Cut along S(µ3)
• Cases I and II, i.e. S(µ3) = R, are the only cases

that can happen if α ≥ 0.

Riemann surface (Case III)

Case III: 0 ∈ S(µ1), 0 ∈ S(σ2 − µ2), 0 ∈ S(µ3)

• Cut along S(µ1)
• Cut along S(σ2 − µ2)
• Cut along S(µ3)

Riemann surface (Case IV)

Case IV: 0 ∈ S(µ1), 0 ∈ S(σ2 − µ2), 0 ∈ S(µ3)

• Cut along S(µ1)
• Cut along S(σ2 − µ2)
• Cut along S(µ3)

Riemann surface (Case V)

Case V: 0 ∈ S(µ1), 0 ∈ S(σ2 − µ2), 0 ∈ S(µ3)

• Cut along S(µ1)
• Cut along S(σ2 − µ2)
• Cut along S(µ3)

Meromorphic function

Proposition The function ξ1(z) = V ′(z) −

• 1

z − x dµ1(x), z ∈ R1, extends to a meromorphic function on the Riemann surface pole of order deg V at infinity on first sheet simple pole at the other point at infinity Consequence: ξ1 is solution of a quartic equation (spectral curve), which in case V (x) = 1

2x2 is:

ξ4 − zξ3 + (1 + ατ 2)ξ2 − (ατ 2 + τ 4)zξ + τ 4z2 + C = 0

Phase diagram for V (x) = 1

2x2 All transitions between cases can be calculated for V (x) = 1

2x2

Duits, Geudens, K (preprint) τ α τ = √α + 2 ατ 2 = −1

1 −1 −2 √ 2

Case I Case IV Case III Case II

Main result

Theorem (Duits, K, Mo) The first component µ1 of the minimizer is equal to the limiting mean eigenvalue density of the matrix M1 in the two-matrix model We also have the usual local scaling limits from random matrix theory: sine kernel in the bulk and Airy kernel at typical edge points. We see new critical phenomena that are not possible in the one-matrix model: Pearcey kernels and more...

Phase diagram for V (x) = 1

2x2 Singular behavior at 0 corresponds to change in cases I-V, that are typically described by Painlev´ e II kernels or Pearcey kernels τ α Painlev´ e II transition Pearcey transition

1 −1 −2 √ 2

Case I Case IV Case III Case II

Special point for α = −1, τ = 1

One very special point in phase diagram τ α Painlev´ e II transition Pearcey transition ??

1 −1 −2 √ 2

Case I Case IV Case III Case II

Special point for α = −1, τ = 1

Theorem (Duits, Geudens (in preparation)) Local eigenvalue correlations around 0 for the special values α = −1, τ = 1 are given by the same correlation kernels as for the non-intersecting Brownian motions at a tacnode

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −2 −1.5 −1 −0.5 0.5 1 1.5 2