Propagation of singular behavior in
UE and GUE sums
Arno Kuijlaars (KU Leuven, Belgium)
with
Tom Claeys, Karl Liechty, Dong Wang
Random matrix theory and strongly correlated
systems
University of Warwick, UK, 21 March 2016
1. Unitary Ensemble plus GUE
Unitary ensemble (UE)
M is random n × n Hermitian matrix from UE
1 −n Tr V (M)
e
dM
Zn
Eigenvalues have joint p.d.f.
n
Y
1
2
∆n (x)
e −nV (xj ) ,
Z̃n
j=1
∆n (x) =
Y
(xj − xi )
1≤i<j≤n
Equilibrium measure µV is the minimizer of
ZZ
Z
1
log
dµ(s)dµ(t) + V (t)dµ(t)
|s − t|
µV is limiting eigenvalue distribution as n → ∞
GUE matrix
H is n × n (scaled) GUE matrix with distribution
1 − n Tr H 2
dH
e 2
Zn
Eigenvalues of
variance τ
dλτ (s) =
√
τ H follow semi-circle law with
1 √
4τ − s 2 ds,
2πτ
√ √
s ∈ [−2 τ , 2 τ ].
Sum of UE and GUE
We are interested in eigenvalues of
√
X = M + τH
with M from a unitary ensemble, H from a (scaled)
GUE, independent from M, and τ > 0.
Eigenvalues are determinantal point process
Interpretation as non-intersecting paths
Free probability:
µV λτ
is limiting distribution of eigenvalues as n → ∞
Determinantal point process
Brézin and Hikami (1998), Zinn-Justin (1998),
Johansson (2001)
If M is fixed with eigenvalues
a1 , . . . , an , then
√
eigenvalues of M + τ H have joint density
in
h n
1
− 2τ (xk −aj )2
∝
· ∆n (x) · det e
∆n (a)
j,k=1
If M is random from UE , then after averaging over
a1 , . . . , an ,
Z ∞
n
j−1 − nτ
(xk −a)2 −nV (a)
2
a e
e
da
∝ ∆n (x) · det
−∞
j,k=1
It is polynomial ensemble (special case of DPP)
Non-intersecting paths
Johansson (2001), Bleher and Kuijlaars (2004):
Non-intersecting Brownian bridges with starting
positions a1 , . . . , an at time t = 0 and ending
positions b1 , . . . , bn at time t = T
Joint density for particles at time t ∈ (0, T ):
∝
h
i
h n
i
n
2 n
1
2 n
·det e − 2(T −t) (xj −bk )
·det e − 2t (aj −xk )
∆n (a)∆n (b)
j,k=1
j,k=1
In limit when all bk → 0
n
in
h n
Y
n
2
1
− 2t (aj −xk )2
∝
· det e
· ∆n (x) ·
e − 2(T −t) xj
∆n (a)
j,k=1
j=1
Figure
1.5
1
0.5
0
−0.5
−1
−1.5
0
0.25
Picture if all aj → ±1.
0.5
0.75
1
Random starting points
If aj are random eigenvalues of matrix from UE
ensemble Z1n e −nV (M) dM then, after averaging over
a1 , . . . , an ,
Z
∞
∝ ∆n (x) · det
a
−∞
n
j−1 − 2t
(xk −a)2 −nV (a)
e
e
n
da
·
n
Y
n
2
e − 2(T −t) xj
j,k=1 j=1
For T →
√∞, this is exactly the same as eigenvalues
of M + tH.
2. Singular potential
Typical behavior of equilibrium measure
Suppose V is real analytic.
µV is supported on finite union of intervals with
density ψV
Generically ψV has square root vanishing at
endpoints, and is positive in the interior of each of
the intervals.
Singular behavior
Singular interior point
ψV vanishes at interior point x ∗
ψV (x) ∼ (x − x ∗ )κ ,
κ = 2k for integer k ≥ 1.
Singular edge point
ψV vanishes to higher order at edge point x ∗
ψV (x) ∼ |x − x ∗ |κ ,
κ = 2k +
1
2
for integer k ≥ 1.
Correlation kernels
Limiting correlation kernels
Sine kernel at regular interior point
Airy kernel at regular edge point
Painlevé kernels at singular points
3. Results
(a) Propagation of singular density for τ < τcr
(b) Propagation of correlation kernel
(c) Density at critical τcr
4 Propagation of singular density
Propagation of singular density
Suppose µV has density
ψV (s) = c0κ+1 |s − x ∗ |κ h(s),
h(x ∗ ) = 1,
where h is real analytic at x ∗ . Define
−1
Z
Z
dµV (s)
dµV (s)
∗
∗
and
x
=
x
−
τ
τ < τcr =
τ
(s − x ∗ )2
s − x∗
Propagation of singular density
Suppose µV has density
ψV (s) = c0κ+1 |s − x ∗ |κ h(s),
h(x ∗ ) = 1,
where h is real analytic at x ∗ . Define
−1
Z
Z
dµV (s)
dµV (s)
∗
∗
and
x
=
x
−
τ
τ < τcr =
τ
(s − x ∗ )2
s − x∗
Theorem
µτ = µV λτ has density ψτ satisfying
ψτ (s) = cτκ+1 |s − xτ∗ |κ hτ (s),
cτ =
where hτ is real analytic at xτ∗
τcr
c0 ,
τcr − τ
hτ (x ∗ ) = 1
Propagation of singular density
The singularity at an interior point
or edge point
√
propagates in the model M + τ H.
Interpretation in terms of non-intersecting
Brownian paths
Connection with two-matrix model
Duits (2014)
About the proof
Biane (1997) describes how to calculate the density of
µτ = µV λτ
The mapping
Z
w 7→ w + τ
dµV (s)
w −s
has an inverse z 7→ Fτ (z) that extends continuously and
injectively to C+ . Then
Z
1
dµV (s)
ψτ (x) = − Im
,
x ∈ R.
π
Fτ (x) − s
About the proof
1
ψτ (x) = − Im
π
Z
dµV (s)
Fτ (x) − s
,
x ∈ R.
R
Expand w + τ dµwV−s(s) around w = x ∗
Expand its inverse Fτ (z) around z = xτ∗
Combine this to find first non-zero term in
expansion of ψτ (x) around x = xτ∗ .
5 Propagation of correlation kernel
Propagation of correlation kernel
M is from unitary ensemble with eigenvalue
correlation kernel KnM (x, y ) and scaling limit
x
1
y
∗
M
∗
= Kcrit,κ (x, y )
lim
Kn x +
,x +
n→∞ c0 n γ
c0 n γ
c0 n γ
with γ = (κ + 1)−1
√
X = M + τ H has correlation kernel KnX (x, y )
Propagation of correlation kernel
M is from unitary ensemble with eigenvalue
correlation kernel KnM (x, y ) and scaling limit
x
1
y
∗
M
∗
= Kcrit,κ (x, y )
lim
Kn x +
,x +
n→∞ c0 n γ
c0 n γ
c0 n γ
with γ = (κ + 1)−1
√
X = M + τ H has correlation kernel KnX (x, y )
Theorem
Under these conditions
x
y
e −Hn (x)+Hn (y ) X
∗
∗
lim
Kn xτ +
,x +
= Kcrit,κ (x, y )
n→∞
cτ nγ
cτ nγ τ cτ nγ
for certain function Hn
About the proof
For X = M +
KnX (x, y )
√
τ H,
n
=
2πiτ
Z
x ∗ +i∞
Z
∞
dt KnM (s, t)
ds
x ∗ −i∞
−∞
n
n
2 −(t−y )2 )
× e 2 (V (s)−V (t)) e 2τ ((s−x)
Claeys-K-Wang (2015)
Proof is basically a steepest descent analysis of this
double integral.
Local part and rest
Separate
X
X
(x, y )
(x, y ) + Kn,rest
KnX (x, y ) = Kn,loc
with
X
Kn,loc
(x, y )
n
=
2πiτ
Z
x ∗ +iRn−γ
Z
x ∗ +Rn−γ
ds
x ∗ −iRn−γ
x ∗ −Rn−γ
n
dt KnM (s, t)
n
2 −(t−y )2 )
× e 2 (V (s)−V (t)) e 2τ ((s−x)
R is large, but fixed constant, independent of n,
but it will depend on x and y .
Analysis of local part
After change of variables
1
x
y
∗
X
∗
x +
K
,x +
cτ nγ n,loc τ cτ nγ τ cτ nγ
Z x ∗ +ic0 R Z x ∗ +c0 R
n1−2γ
=
ds
dt e Fn (s,x)−Fn (t,y )
2πic0 cτ x ∗ −ic0 R
∗
x −c0 R
s
t
1
M
∗
∗
x +
×
K
,x +
c0 n γ n
c0 nγ
c0 n γ
|
{z
}
→ Kcrit,κ (s, t)
with
n
Fn (s, x) = V
2
s
n
s
τ 0 ∗
x
∗
x +
+
− V (x ) −
c0 n γ
2τ c0 nγ
2
cτ nγ
Taylor expansion of Fn
Expand as n → ∞
Fn (s, x) =
τn
n1−γ 0 ∗
n1−2γ 00 ∗ 2
n
V (x ∗ ) + V 0 (x ∗ )2 +
V (x )x +
V (x )x
2
8
2cτ
4c0 cτ
n1−2γ +
(s − x)2 + O(n−γ )
2τ c0 cτ
No term sn−γ because of choice for xτ∗
Complete square (s − x)2 because of choice for cτ
∂Fn
Saddle point equation
= 0 gives the saddle
∂s
x + O(n−γ )
Analysis of rest
X
X
(x, y )
(x, y ) = KnX (x, y ) − Kn,loc
Kn,rest
Scaled version
e
−Hn (x)+Hn (y )
becomes O(e −cn
1−γ
X
Kn,rest
xτ∗
x
y
+
, xτ∗ +
γ
cτ n
cτ n γ
) as n → ∞ if R is large enough.
Proof uses deformation of t-contour into the
complex plane, and uses bounds on orthogonal
polynomials that come from steepest descent
analysis of Riemann-Hilbert problem
6 Density at critical τcr
Density at critical τcr
dµV (s)
What about the density for critical τcr =
(s − x ∗ )2
−1
?
Density at critical τcr
dµV (s)
What about the density for critical τcr =
(s − x ∗ )2
−1
We have results for
Singular interior point ψV (s) ∼ (s − x ∗ )2k as s → x ∗
κ = 2k = 2
Z
dµV (s)
=0
(s − x ∗ )3
Z
dµV (s)
κ = 2k ≥ 4 and
6= 0
(s − x ∗ )3
κ = 2k ≥ 4 and
1
Singular edge point ψV (s) ∼ (x ∗ − s)2k+ 2 as s → x ∗ −
?
Density at critical τcr
Singular interior point with exponent κ = 2k = 2
Theorem
1/2
ψτcr (s) = A± s − xτ∗cr + O(s − xτ∗cr ),
as s → xτ∗cr ±
where
A−
A+
!
=
cos θ
1
(πτcr c0
)3/2 (1
π
θ = + arctan
4
R
+ ( π1 −
h(s)
ds)2 )1/4
s−x ∗
Z
1
h(s)
−
ds
π s − x∗
sin θ
!
Density at critical τcr
Singular interior point with exponent κ = 2k ≥ 4.
Theorem
Z
Suppose
dµV (s)
ds = 0.
(s − x ∗ )3
Then
1/3
ψτcr (s) = A s − xτ∗cr + O(|s − xτ∗cr |2/3 ),
where
√
A=
4/3
2πτcr
Z
3
dµV (s)
ds
(s − x ∗ )4
1/3
as s → xτ∗cr
Density at critical τcr
Singular interior point with exponent κ = 2k ≥ 4.
Theorem
Z
Suppose
ψτcr (s) =
dµV (s)
ds > 0.
(s − x ∗ )3
Then
 1/2
A s − xτ∗cr + O(s − xτ∗cr ),
as s → xτ∗cr +
k−1/2
 B s − xτ∗cr + O((s − xτ∗cr )k )
as s → xτ∗cr −
where
1
A=
3/2
πτcr
R
dµV (s)
ds
(s−x ∗ )3
1/2 ,
c02k+1
B=
k+1/2
2τcr
R
dµV (s)
ds
(s−x ∗ )3
k+1/2
Density at critical τcr
Singular right edge point with exponent κ = 2k + 12 .
Theorem
Z
Suppose
dµV (s)
ds > 0.
(x ∗ − s)3
ψτcr (s) = A xτ∗cr − s
1/2
Then
+ O((s − xτ∗cr )3/4 )
as s → xτ∗cr −
where
1
A=
3/2 1/4
τcr c0
Z
dµV (s)
ds
(s − x ∗ )3
1/2
Speculations about correlation kernel
We have no results yet for the correlation kernels at
critical τcr
Case κ = 2k = 2 leads to exponent 1/2:
Tacnode kernel of Duits-Geudens (2013) !?
Speculations about correlation kernel
We have no results yet for the correlation kernels at
critical τcr
Case κ = 2k = 2 leads to exponent 1/2:
Tacnode kernel of Duits-Geudens (2013) !?
Z
dµV (s)
= 0 leads to
Case κ = 2k ≥ 4 with
(s − x ∗ )3
exponent 1/3:
Pearcey kernel !??
Speculations about correlation kernel
We have no results yet for the correlation kernels at
critical τcr
Case κ = 2k = 2 leads to exponent 1/2:
Tacnode kernel of Duits-Geudens (2013) !?
Z
dµV (s)
= 0 leads to
Case κ = 2k ≥ 4 with
(s − x ∗ )3
exponent 1/3:
Pearcey kernel !??
Z
dµV (s)
Case κ = 2k ≥ 4 with
≤ 0 leads to two
(s − x ∗ )3
exponents 1/2 and k − 1/2:
· · · kernel ???
Speculations about correlation kernel
We have no results yet for the correlation kernels at
critical τcr
Case κ = 2k = 2 leads to exponent 1/2:
Tacnode kernel of Duits-Geudens (2013) !?
Z
dµV (s)
= 0 leads to
Case κ = 2k ≥ 4 with
(s − x ∗ )3
exponent 1/3:
Pearcey kernel !??
Z
dµV (s)
Case κ = 2k ≥ 4 with
≤ 0 leads to two
(s − x ∗ )3
exponents 1/2 and k − 1/2:
· · · kernel ???
Case κ = 2k + 1/2 leads to exponent 1/2:
Airy kernel ??
Thank you for your attention