Gap Probabilities in Two-Matrix
Models
CRM Montréal 26/08/2008
Gernot Akemann
work with:
+ P. Damgaard: arXiv:0803.1171 (JHEP) & in preparation
+ J. Osborn + K. Splittorff: hep-th/0609059v2 (Nucl.Phys.B)
Outline
1. Two Matrix Model (2MM)
2. Physics Motivation: QCD
3. Objects:
densities, gaps, individual eigenvalues
4. Solution for densities
5. Expansion of gaps & eigenvalues in densities
6. Exact results for gaps & eigenvalues
7. Large-N & Examples
8. Remarks and open problems
Chiral Two-Matrix Model
Z=
Z
1
/
dΦdΨ ΠN
f =1 det[iD1
+
2
/2
mf ] ΠN
g=1 det[iD
+ ng ] e
−Tr(ΦΦ†+Ψ† Ψ)
• we are interested in real non-neg. eigenvalues of
0
Φ+µ1,2Ψ
/ 1,2 ≡
D
0
Φ†+µ1,2Ψ†
Φ, Ψ are matrices of sizes N × (N + ν) ∈ C
• change of variables Φj ≡ Φ+µj Ψ, j = 1, 2 couples them:
Z=
R
1
/
dΦ1dΦ2 ΠN
f =1 det[D1
+
2
/2
mf ] ΠN
g=1 det[D
+ ng ] e
• Unitary dof Φ1 = U1XV1 can be eliminated using a
group integral (HCIZ for non-chiral)
• Φi Hermitean N × N for non-chiral [Ercolani,McLaughlin01]
†
†
†
†
−Tr(c1 Φ1 Φ1+c2 Φ2 Φ2−d(Φ1Φ2 +Φ1Φ2 ))
Physics Motivation
• low energy QCD −→ chiral Perturbation Theory (χPT)
−→ const. Pion fields = Matrix Model (N → ∞)
[Shuryak, Verbaarschot 93;. . . ]
ZεχP T =
R
U (Nf
ν
dU
det[U
]
e
0
0
)
1 V F 2 µ2
4
†
[Γ,U0][Γ,U0 ]+ 12 ΣV
Tr
Tr
• effective χPT has 2 LEC’s: F Pion decay constant &
Σ chiral condensate
• determine by comparing analytical MM predictions to
Lattice QCD −→ need source µ to couple to F
• easiest: imaginary isospin chemical potential iµ
/
−→ keeps D(±iµ)
spectra real!
[Damgaard, Heller, Splittorff, Svetitzky, Toublan 05/06]
• Individual eigenvalues: num. easiest [G.A.,Damgaard 08]
†
M (U0+U0 )
Illustration Density vs. ’1st’ Eigenvalue on C
ρ(0,y)
0.4
ν=0
0.3
0.2
ν=1
0.1
0
0
1
2
3
4 y
• cut through density (left) vs. Lattice data [Bloch,Wettig’06]
• 1st eigenvalue (right) radially integrated vs same data
[+ G.A., Shifrin 07]
Eigenvalue Representation
• two sets of real eigenvalues: {x}, {y}
diagonalise Φ1 = U1XV1, Φ2 = U2Y V2 + use integral:
[Guhr,Wettig; Jackson,Sener,Verbaarschot 96]
Z
dU dV exp [N dℜeTr(V XU Y )] = QN
det1≤i,j≤N [Iν (N dxiyj )]
i=1 (xi yi )
ν∆
N ({x
• chiral 2MM:
N1
N2
N Z ∞
Y
Y
Y
dxidyi(xiyi)ν+1 (x2i + m2f ) (yi2 + n2g )
Z=
i=1
0
f =1
2
2
× ∆N (x )∆N (y ) det [Iν (2dN xk yl )] e
k,l
• non-Gaussian!! (non-chiral: Iν → e)
g=1
−N
PN
2
2
i (c1 xi +c2 yi )
/ operators = ave characteristic polynomials
• inserted D
2})∆
N ({y
2 })
Objects of Study
• Define jpdf: Z ≡
QN R ∞
i=1 0
dxidyi Pjpdf
• densities:
R∞
R∞
QN
QN
Rk,l (x1, . . . , xk ; y1, . . . , yl ) ∼ i=k+1 0 dxi j=l+1 0 dyj Pjpdf
• gap probabilities:
R∞
Rt
R∞
Rs
Ek,l(s, t) ∼ 0 dx1 . . . dxk s dxk+1 . . . dxN 0 dx1 . . . dxl t dyl+1 . . . dyN Pjpdf
• individual eigenvalues: xk = s & yl = t
Z ∞
Z s
dxk+1 . . . dxN
dx1 . . . dxk−1
pk,l(s, t) ∼
s
0
×
Z
t
dy1 . . . dyl−1
0
Z
t
∞
dyl+1 . . . dyN Pjpdf (. . . , xk = s, . . . , yl = t, . . .)
• not independent: e.g. ∂s∂tE0,0(s, t) = p1,1(s, t) for
smallest
Solution for all Densities
• bi-orthogonal polynomials δnk =
R∞
0
dxdy w (N1,N2)(x, y)Pn(x2)Qk (y 2)
N1
N2
Y
Y
(N1,N2)
ν+1
2
2
−N (c1 x2+c2 y 2 )
2
2
weight w
(x, y) = (xy)
(x +mf ) (y +mg )Iν (2dN xy) e
g
f =1
• in terms of 4 kernels: using
KN =
PN −1
k
Pk Qk , HN =
[Eynard, Mehta 98]
PN −1
k
χ̂k Pk , ĤN =
containing integral trafo χk (x) =
R∞
0
PN −1
k
χk Qk , MN =
PN −1
k
χ̂k χk
dy w(x, y)Pk (y) & ditto for Qk
⇒ all density correlation functions known explicitly
Rn,k ({x}n; {y}k ) =
det
1≤i1 ,i2≤n; 1≤j1 ,j2 ≤k
(N1,N2)
(xi1 , yj2 )
HN (xi1 , xi2 ) MN (xi1 , yj2 ) − w
KN (yj1 , xi2 )
ĤN (yj1 , yj2 )
Example Density: 2MM vs. 1MM
2
example quenched (N1 = N2 = 0) : Pk , Qk ∼ Lνk and χk , χ̂k ∼ e−x Lνk Laguerre
0.4
8
0.4
0.2
6
0
4
0.3
0.2
2
2
4
0.1
6
8
0
2
4
6
8
• 1MM N → ∞: Bessel ρ1(x) = x2 (Jν (x)2−Jν+1(x)Jν−1(x))
• 2MM N → ∞: generalised Bessel α2 = 2N δµ2
2
2
2
ρ1,1(x, y) = ρ1(x)ρ1(y)−xy K+(x, y) K−(x, y) − e−(x +y )/α Iν (xy/α2 )
±
K (x, y) ≡
R1
0
dt e
±t2 α2
Jν (tx)Jν (ty)
Densities – Gaps – Eigenvalues: Expansion
• all densities −→ all gap prob’s −→ all indiv. ev.
Ek,l (s, t) =
N
−k X
N −l
X
i=0 j=0
i+j
(−)
i!j!
Z
s
dx1 . . . dxk+i
0
Z
t
dy1 . . . dyl+j Rk+i,l+j ({x}k+i, {y}l+j )
0
• example p1,1 = ∂∂E0,0:
p1,1(s, t) = R1,1(s, t) −
Z
s
0
dx R2,1(x, s, t) −
Z
t
dy R1,2(s, t, y) + . . .
0
• same expansion for 1MM:
R
1st eigenvalue = density – 2-pt density +. . .
Expansion in 1MM
• exact result:
0.5
ρ1(x) Bessel
0.4
0.3
0.2
p1(x) = x/2 e−x
2 /4
0.1
2
4
6
8
x
• expansion to 3rd order
0.5
0.4
0.3
0.2
0.1
2
-0.1
-0.2
4
6
8
10
Expansion in 2MM
0.4
0.4
8
0.2
8
0.2
6
0
6
0
4
2
4
2
2
4
6
2
4
6
8
8
ρ1,1 generalised Bessel vs. p1,1 expanded to 2nd order
• Can we find an exact result as in the 1MM?
Reminder: Gaps in 1MM
N1
N Z ∞
Y
Y
1
dxi xνi
(xi + m2f ) e−N c1xi ∆N (x)2
E0(s) =
Z i=1 s2
f =1
• after changing variables x2 → x
Q
• Vandermonde i>j (xi − xj ) invariant under shift
⇒
2
E0(s) ∼ e−s × Z Gap = new Z with
– shifted masses m2f → m2f + s2,
– ν extra masses s2
– “effective ν = 0” [Damgaard, Nishigaki,Wettig 98]
−s2
Result E0(s) ∼ e
(ν = 0)
detf,k [LN +k (m2f + s2)]/∆({m2 + s2})
• same strategy for pk : shift jpdf of k-th eigenvalue
[Damgaard, Nishigaki 01]
Exact Results for 2MM
• partial results for objects totally symmetric in {y}
Q
−→ can replace det [Iν (xiyj )] by diagonal i Iν (xiyi)
⇒ Gaps of x only, k-th individual x-eigenvalues
/ of y’s
nontrivial due to presence of N2 D’s
• example 1st gap:
E0,0(s, t = 0) ∼
N Z
Y
• Shift easy as for 1MM?
i=1
s
∞
dxi
Z
0
∞
dyi Pjpdf
How to do 1st Gap in 2MM
E0,0(s, t = 0) ∼
1.)
R∞
0
Z
∞
dx exp mass1 ∆(x)
Z
∞
0
s
dy symmetric: replace det Iν (xy) →
2.) replace:
Iν (xy) →
P∞
j
dy exp mass2 ∆(y) det Iν (xy)
Q
Iν (xy)
aj Lj (x)Lj (y) and mass2 ∆(y) → det
orthogonality: ⇒ E0,0(s, t = 0) ∼
3.) det identity & shift & 2b.):
E0,0(s, t = 0) ∼
Z
∞
0
dx exp det
N +N1
QN
R∞
i=1 s
′
Lj (m )
Lj (x)
Lj (n)
Lj (y)
Lj (n)
dx exp mass1 ∆(x) det
aj Lj (x)
det
N +N2
• compute like average of characteristic polynomials
qj (n)
Lj (x)
First Gap in 2MM

K(m′1 , n1) . . .
det

E0,0(s, 0) ∼
′
∆(m )∆(n) K(m′
for N1 ≥ N2
K(m′1 , nN2 )
LN +N2 (m′2
1 )...
LN +N1−1(m′2
1)

..
..
..
..

′
′2
′2
N1 , n1 ) . . . K(mN1 , nN2 ) LN +N2 (mN1 ) . . . LN +N1 −1(mN1 )
• shifted mass m′ 2 = m2 + s2
PN +N2−1
′
Lj (m2 + s2)qj (n)
• Kernel K(m , n) = j=0
Pj
−ν−1 2
1
ν 2
L
(n
)L
• generalised shifted Laguerre qj (n) = l=0 (1−τ
l
j−l (s )
) l
1MM limµ1→µ2 τ (µ1 , µ2) = 0 :
qj (n) = Lj (n2 + s2)
• for N2 > N1 replace Lj → qj in right half of det
Averages of Characteristic Polynomials
*
N1
Y
/ 1 + mf ]
det[iD
f =1
N2
Y
/ 2 + ng ]
det[iD
g=1
+
[G.A., Vernizzi 03]
N1
N2
N Z
Y
Y
Y
1
dxidyiw(xi, yi) (x2i +m2f )
(yi2+n2g ) ∆N (x2)∆N (y 2)
=
Z i=1
g=1
f =1

PN +N2 (m21) . . .
PN +N1−1(m21)

K(m1 , n1) . . . K(m1 , nN2 )
const.
..
..
..
..


det
=
2
2
∆N1 (m )∆N2 (n )
K(mN1 , n1) . . . K(mN1 , nN2 ) PN +N2 (m2N1 ) . . . PN +N1−1(m2N1 )
for N1 ≥ N2
The k-th x-eigenvalue in 2MM
• definition
pk (x) =
Z
x
dx1
0
Z
x
dx2 . . .
x1
Z
x
dxk−1 Ωjpdf (x1, . . . , xk )
xk−2
• partial jpdf of the k-th ev
Ωjpdf (x1, . . . , xk ) ∼
• Result:
Ωjpdf ∼∼
N Z
Y
i=k+1
∞
0
N Z
Y
i=k+1
∞
N Z
Y
∞
L1j (m′)
L1j (x)
dxi
xk
dxi exp det
j=1
0
dyj Pjpdf
det
qj1(n)
L1j (x)
The Large-N Limit: Hard Edge
• rescale N xj , N yj , N m, N n,
and N δµ2 ’weak non-Hermitecity’
• building blocks:
2
√
2
* LN (N (m + s )) → I0( m̂2 + ŝ2 ) Laguerre
N
X
1
2
2
−1
(s
) (ν = 0)
L
(n
)L
l
N −l
l
(1 − τ )
l=0
Z
p
√
1 δ̂ 2
1 r δ̂ 2
1
1 1
2
ŝI1( (1 − r) ŝ) + e 2 I0(n̂)
dr e I0( r n̂) √
→
2 0
1−r
* qN (n) =
: L−1
0 = 1
√
(check: lim δ̂ → 0: → I0( n̂2 + ŝ2) Sonine identity)
• tricky:
P
→
R
⇒ Kernel asymptotic =
P
j
Lj qj & Gap
Examples
0.5
1
0.4
0.8
0.3
0.6
0.2
0.4
0.1
0.2
2
4
6
8
10
1
2
3
4
5
6
• left: ρ1(x) and p1(x) ∼ ∂q(n̂) for N1 = 0, N2 = 1
(δ̂ → 0: 1MM with 1 mass, δ̂ → ∞: 1MM quenched)
• right: gap 1 − E0,0(s, 0) comparing
N1 = 0, N2 = 2 and N1 = 1 = N2 at fixed mass and δ̂
7
Remarks
• for N2 = 0 integration over all {y}:
reduces 2MM to 1MM
• Factorisation into 1MM densities & indiv ev.:
for N1 = N2 = 0 and parameter µj NOT scaled
with N one of the kernels converges to the weight:
M (x, y) →
NX
=∞
χj (x)χ̂j (y) = w(x, y)
j=0
∞
X
(1 − τ )j Lj (x2)Lj (y 2) ∼ I0(xy)
j=0
= Mehler formula for Hermite or Laguerre
R1
2 2
2
(compare: for scaled N µ : 0 dt e−t α Jν (tx)Jν (ty) )
Open Problems
◦ asymptotic large-N rigorous and universality proof
◦ complete the solution of 2MM:
mixed individual eigenvalues of x- and y-type
◦ 2MM with complex eigenvalues:
exact individual eigenvalues, partial quenching?