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Random matrices, differential operators andcarousels
Benedek Valko(University of Wisconsin – Madison)
joint with B. Virag (Toronto)
March 24, 2016
Basic question of RMT:
What can we say about the spectrum of a large random matrix?
-60 -40 -20 0 20 40 60
5
10
15
20
25
30
35
Ln
b HLn - aL
global local
In this talk: local picture (point process limits)
Basic question of RMT:
What can we say about the spectrum of a large random matrix?
-60 -40 -20 0 20 40 60
5
10
15
20
25
30
35
Ln
b HLn - aL
global local
In this talk: local picture (point process limits)
Basic question of RMT:
What can we say about the spectrum of a large random matrix?
-60 -40 -20 0 20 40 60
5
10
15
20
25
30
35
Ln
b HLn - aL
global local
In this talk: local picture (point process limits)
A classical example: Gaussian Unitary Ensemble
M = A+A∗√2
, A is n × n with iid complex std normal.
Global picture: Wigner semicircle law -60 -40 -20 0 20 40 60
5
10
15
20
25
30
35
Local picture: point process limit in the bulk and near the edge
(Bulk: Dyson-Gaudin-Mehta, edge: Tracy-Widom)
The limit processes are characterized by their joint intensity functions.
Roughly: what is the probability of finding points near x1, . . . , xn
A classical example: Gaussian Unitary Ensemble
M = A+A∗√2
, A is n × n with iid complex std normal.
Global picture: Wigner semicircle law
-60 -40 -20 0 20 40 60
5
10
15
20
25
30
35
Local picture: point process limit in the bulk and near the edge
(Bulk: Dyson-Gaudin-Mehta, edge: Tracy-Widom)
The limit processes are characterized by their joint intensity functions.
Roughly: what is the probability of finding points near x1, . . . , xn
A classical example: Gaussian Unitary Ensemble
M = A+A∗√2
, A is n × n with iid complex std normal.
Global picture: Wigner semicircle law -60 -40 -20 0 20 40 60
5
10
15
20
25
30
35
Local picture: point process limit in the bulk and near the edge
(Bulk: Dyson-Gaudin-Mehta, edge: Tracy-Widom)
The limit processes are characterized by their joint intensity functions.
Roughly: what is the probability of finding points near x1, . . . , xn
A classical example: Gaussian Unitary Ensemble
M = A+A∗√2
, A is n × n with iid complex std normal.
Global picture: Wigner semicircle law -60 -40 -20 0 20 40 60
5
10
15
20
25
30
35
Local picture: point process limit in the bulk and near the edge
(Bulk: Dyson-Gaudin-Mehta, edge: Tracy-Widom)
The limit processes are characterized by their joint intensity functions.
Roughly: what is the probability of finding points near x1, . . . , xn
A classical example: Gaussian Unitary Ensemble
M = A+A∗√2
, A is n × n with iid complex std normal.
Global picture: Wigner semicircle law -60 -40 -20 0 20 40 60
5
10
15
20
25
30
35
Local picture: point process limit in the bulk and near the edge
(Bulk: Dyson-Gaudin-Mehta, edge: Tracy-Widom)
The limit processes are characterized by their joint intensity functions.
Roughly: what is the probability of finding points near x1, . . . , xn
Point process limit
Ln
b HLn - aL
Finite n: spectrum of a random Hermitian matrix
Limit point process: spectrum of ??
Point process limit
Ln
b HLn - aL
Finite n: spectrum of a random Hermitian matrix
Limit point process: spectrum of ??
Detour to number theory
Riemann zeta function: ζ(s) =∞∑n=1
1ns , for Re s > 1.
(Analytic continuation to C \ {1})
Riemann hypothesis: the non-trivial zeros are on the line Re s = 12 .
Dyson-Montgomery conjecture:
After some scaling:
non-trivial zeros of ζ(1
2+ i s) ∼ bulk limit process of GUE
(Sine2 process)
I Strong numerical evidence: Odlyzko
I Certain weaker versions are proved(Montgomery, Rudnick-Sarnak)
Detour to number theory
Riemann zeta function: ζ(s) =∞∑n=1
1ns , for Re s > 1.
(Analytic continuation to C \ {1})
Riemann hypothesis: the non-trivial zeros are on the line Re s = 12 .
Dyson-Montgomery conjecture:
After some scaling:
non-trivial zeros of ζ(1
2+ i s) ∼ bulk limit process of GUE
(Sine2 process)
I Strong numerical evidence: Odlyzko
I Certain weaker versions are proved(Montgomery, Rudnick-Sarnak)
Detour to number theory
Riemann zeta function: ζ(s) =∞∑n=1
1ns , for Re s > 1.
(Analytic continuation to C \ {1})
Riemann hypothesis: the non-trivial zeros are on the line Re s = 12 .
Dyson-Montgomery conjecture:
After some scaling:
non-trivial zeros of ζ(1
2+ i s) ∼ bulk limit process of GUE
(Sine2 process)
I Strong numerical evidence: Odlyzko
I Certain weaker versions are proved(Montgomery, Rudnick-Sarnak)
Hilbert-Polya conjecture: the Riemann hypotheses is true because
non-trivial zeros of ζ(1
2+ i s)
= ev’s of an unbounded self-adjoint operator
A famous attempt to make this approach rigorous: de Branges
(based on the theory of Hilbert spaces of entire functions)
This approach would produce a self-adjoint differential operatorwith the appropriate spectrum.
Hilbert-Polya conjecture: the Riemann hypotheses is true because
non-trivial zeros of ζ(1
2+ i s)
= ev’s of an unbounded self-adjoint operator
A famous attempt to make this approach rigorous: de Branges
(based on the theory of Hilbert spaces of entire functions)
This approach would produce a self-adjoint differential operatorwith the appropriate spectrum.
Hilbert-Polya conjecture: the Riemann hypotheses is true because
non-trivial zeros of ζ(1
2+ i s)
= ev’s of an unbounded self-adjoint operator
A famous attempt to make this approach rigorous: de Branges
(based on the theory of Hilbert spaces of entire functions)
This approach would produce a self-adjoint differential operatorwith the appropriate spectrum.
Natural question:
Is there a self-adjoint differential operator with a spectrum givenby the bulk limit of GUE?
Disclaimer: A positive answer would not get us closer to any of the conjecturesor the Riemann hypothesis (unfortunately...)
Borodin-Olshanski, Maples-Najnudel-Nikeghbali:
‘operator-like object’ with generalized eigenvalues distributed as Sine2
Natural question:
Is there a self-adjoint differential operator with a spectrum givenby the bulk limit of GUE?
Disclaimer: A positive answer would not get us closer to any of the conjecturesor the Riemann hypothesis (unfortunately...)
Borodin-Olshanski, Maples-Najnudel-Nikeghbali:
‘operator-like object’ with generalized eigenvalues distributed as Sine2
Natural question:
Is there a self-adjoint differential operator with a spectrum givenby the bulk limit of GUE?
Disclaimer: A positive answer would not get us closer to any of the conjecturesor the Riemann hypothesis (unfortunately...)
Borodin-Olshanski, Maples-Najnudel-Nikeghbali:
‘operator-like object’ with generalized eigenvalues distributed as Sine2
Starting point for deriving the Sine2 process:
Joint eigenvalue density of GUE:
1
Zn
∏i<j≤n
|λj − λi |2n∏
i=1
e−12λ2i
Many of the classical random matrix ensembles have jointeigenvalue densities of the form
1
Zn,f ,β
∏i<j≤n
|λj − λi |βn∏
i=1
f (λi )
with β = 1, 2 or 4 and f a specific reference density.
Starting point for deriving the Sine2 process:
Joint eigenvalue density of GUE:
1
Zn
∏i<j≤n
|λj − λi |2n∏
i=1
e−12λ2i
Many of the classical random matrix ensembles have jointeigenvalue densities of the form
1
Zn,f ,β
∏i<j≤n
|λj − λi |βn∏
i=1
f (λi )
with β = 1, 2 or 4 and f a specific reference density.
Starting point for deriving the Sine2 process:
Joint eigenvalue density of GUE:
1
Zn
∏i<j≤n
|λj − λi |2n∏
i=1
e−12λ2i
Many of the classical random matrix ensembles have jointeigenvalue densities of the form
1
Zn,f ,β
∏i<j≤n
|λj − λi |βn∏
i=1
f (λi )
with β = 1, 2 or 4 and f a specific reference density.
β-ensemble: finite point process with joint density
1
Zn,f ,β
∏i<j≤n
|λj − λi |βn∏
i=1
f (λi )
f (·): reference density
Examples:
I Hermite or Gaussian: normal density
I Laguerre or Wishart: gamma density
I Jacobi or MANOVA: beta density
I circular: uniform on the unit circle
β = 1, 2, 4: classical random matrix models
Scaling limits - global picture
Hermite β-ensemble semicircle lawLaguerre β-ensemble Marchenko-Pastur law
-2 2 1 2 3 4
↑ ↑ ↗ ↑ ↑ ↑soft edge bulk s. e. hard edge bulk s. e.
Local limits
Soft edge: Rider-Ramırez-Virag (Hermite, Laguerre)Airyβ process
Hard edge: Rider-Ramırez (Laguerre)Besselβ,a processes
Bulk: Killip-Stoiciu, V.-Virag (circular, Hermite)CβE and Sineβ processes
Instead of joint intensities, the limit processes are described viatheir counting functions using coupled systems of SDEs.
sign(λ) · (# of points in [0, λ])
Local limits
Soft edge: Rider-Ramırez-Virag (Hermite, Laguerre)Airyβ process
Hard edge: Rider-Ramırez (Laguerre)Besselβ,a processes
Bulk: Killip-Stoiciu, V.-Virag (circular, Hermite)CβE and Sineβ processes
Instead of joint intensities, the limit processes are described viatheir counting functions using coupled systems of SDEs.
sign(λ) · (# of points in [0, λ])
Operators at the edge
Soft edge: Airyβ is the spectrum of
Aβ = − d2
dx2+ x +
2√βdB
dB: white noise
Hard edge: Besselβ,a is the spectrum of
Bβ,a = −e(a+1)x+ 2√βB(x) d
dx
{e−ax− 2√
βB(x) d
dx
}B: standard Brownian motion
Random second order self-adjoint differential operators on [0,∞).
Edelman-Sutton: non-rigorous versions of these operators
What about the bulk? Is there an operator for CβE or Sineβ?
Operators at the edge
Soft edge: Airyβ is the spectrum of
Aβ = − d2
dx2+ x +
2√βdB
dB: white noise
Hard edge: Besselβ,a is the spectrum of
Bβ,a = −e(a+1)x+ 2√βB(x) d
dx
{e−ax− 2√
βB(x) d
dx
}B: standard Brownian motion
Random second order self-adjoint differential operators on [0,∞).
Edelman-Sutton: non-rigorous versions of these operators
What about the bulk? Is there an operator for CβE or Sineβ?
Operators at the edge
Soft edge: Airyβ is the spectrum of
Aβ = − d2
dx2+ x +
2√βdB
dB: white noise
Hard edge: Besselβ,a is the spectrum of
Bβ,a = −e(a+1)x+ 2√βB(x) d
dx
{e−ax− 2√
βB(x) d
dx
}B: standard Brownian motion
Random second order self-adjoint differential operators on [0,∞).
Edelman-Sutton: non-rigorous versions of these operators
What about the bulk? Is there an operator for CβE or Sineβ?
The Sineβ operator
Thm (V-Virag):There is a self-adjoint differential operator (Dirac-operator)
f → 2R−1t
[0 −11 0
]f ′(t), f : [0, 1)→ R2.
where Rt is a random 2× 2 positive definite matrix valued functionso that the spectrum is the Sineβ process.
Rt is given a simple function of a hyperbolic Brownian motion.
This is a first order differential operator.
The Sineβ operator
Thm (V-Virag):There is a self-adjoint differential operator (Dirac-operator)
f → 2R−1t
[0 −11 0
]f ′(t), f : [0, 1)→ R2.
where Rt is a random 2× 2 positive definite matrix valued functionso that the spectrum is the Sineβ process.
Rt is given a simple function of a hyperbolic Brownian motion.
This is a first order differential operator.
The Sineβ operator
Thm (V-Virag):There is a self-adjoint differential operator (Dirac-operator)
f → 2R−1t
[0 −11 0
]f ′(t), f : [0, 1)→ R2.
where Rt is a random 2× 2 positive definite matrix valued functionso that the spectrum is the Sineβ process.
Rt is given a simple function of a hyperbolic Brownian motion.
This is a first order differential operator.
Digression: the hyperbolic plane H
Disk model
Halfplane model
A geometric description of Sineβ
Hyperbolic carousel: (η0, η∞, γ) point process
η0, η∞: points on the boundary of the hyperbolic plane H
γ : [0, 1)→ H: a path in the hyperbolic plane
η0
γ(t)
η∞ zλ(t)
For each λ ∈ R we start a point zλ from η0 and rotate itcontinuously around γ(t) with rate λ. (This is just an ODE!)
N(λ): # of times zλ hits η∞. This is the counting function of thepoint process.
A geometric description of Sineβ
Hyperbolic carousel: (η0, η∞, γ) point process
η0, η∞: points on the boundary of the hyperbolic plane H
γ : [0, 1)→ H: a path in the hyperbolic plane
η0
γ(t)
η∞ zλ(t)
For each λ ∈ R we start a point zλ from η0 and rotate itcontinuously around γ(t) with rate λ. (This is just an ODE!)
N(λ): # of times zλ hits η∞. This is the counting function of thepoint process.
A geometric description of Sineβ
Hyperbolic carousel: (η0, η∞, γ) point process
η0, η∞: points on the boundary of the hyperbolic plane H
γ : [0, 1)→ H: a path in the hyperbolic plane
η0
γ(t)
η∞ zλ(t)
For each λ ∈ R we start a point zλ from η0 and rotate itcontinuously around γ(t) with rate λ. (This is just an ODE!)
N(λ): # of times zλ hits η∞. This is the counting function of thepoint process.
A geometric description of SineβV.-Virag (’07): if γ is a time changed hyperbolic Brownian motion,η∞ is its limit point and η0 is a fixed boundary point then
(η0, η∞, γ) Sineβ
(β only appears in the time change: t → − 4β log(1− t))
Carousel ∼ Dirac operator
Suppose that γ(t) = xt + iyt in the half-plane coordinates.
From the carousel (η0, η∞, γ) we can build the self-adjoint Dirac operator
τ : f → 2(UTU)−1[
0 −11 0
]f ′(t), U =
1√yt
[1 −xt0 yy
](η0, η∞ boundary conditions)
point process produced by (η0, η∞, γ)= spectrum of τ
Main idea of the proof: Sturm-Liouville oscillation theory
τv(t, λ) = λv(t, λ), v1(t, λ) + iv2(t, λ) = r(t, λ)e iθ(t,λ)
The spectrum can be identified from θ(·, ·) which basically evolvesaccording to a carousel.
Carousel ∼ Dirac operator
Suppose that γ(t) = xt + iyt in the half-plane coordinates.
From the carousel (η0, η∞, γ) we can build the self-adjoint Dirac operator
τ : f → 2(UTU)−1[
0 −11 0
]f ′(t), U =
1√yt
[1 −xt0 yy
](η0, η∞ boundary conditions)
point process produced by (η0, η∞, γ)= spectrum of τ
Main idea of the proof: Sturm-Liouville oscillation theory
τv(t, λ) = λv(t, λ), v1(t, λ) + iv2(t, λ) = r(t, λ)e iθ(t,λ)
The spectrum can be identified from θ(·, ·) which basically evolvesaccording to a carousel.
Carousel ∼ Dirac operator
Suppose that γ(t) = xt + iyt in the half-plane coordinates.
From the carousel (η0, η∞, γ) we can build the self-adjoint Dirac operator
τ : f → 2(UTU)−1[
0 −11 0
]f ′(t), U =
1√yt
[1 −xt0 yy
](η0, η∞ boundary conditions)
point process produced by (η0, η∞, γ)= spectrum of τ
Main idea of the proof: Sturm-Liouville oscillation theory
τv(t, λ) = λv(t, λ), v1(t, λ) + iv2(t, λ) = r(t, λ)e iθ(t,λ)
The spectrum can be identified from θ(·, ·) which basically evolvesaccording to a carousel.
Carousel ∼ Dirac operator
Suppose that γ(t) = xt + iyt in the half-plane coordinates.
From the carousel (η0, η∞, γ) we can build the self-adjoint Dirac operator
τ : f → 2(UTU)−1[
0 −11 0
]f ′(t), U =
1√yt
[1 −xt0 yy
](η0, η∞ boundary conditions)
point process produced by (η0, η∞, γ)= spectrum of τ
Main idea of the proof: Sturm-Liouville oscillation theory
τv(t, λ) = λv(t, λ), v1(t, λ) + iv2(t, λ) = r(t, λ)e iθ(t,λ)
The spectrum can be identified from θ(·, ·) which basically evolvesaccording to a carousel.
Carousel ∼ Dirac operator
(η0, η∞, γ) τ : f → 2R−1t
[0 −11 0
]f ′(t),
Under mild conditions: τ−1 is a Hilbert-Schmidt integral operatorwith kernel
K(x , y) =(u0u
T1 1(x < y) + u1u
T0 1(x ≥ y)
)R(y)
u0, u1: boundary conditions in τ
Nice property: if the path γ lives on [0,T ) then the operator canbe approximated using the path restricted to [0,T − ε).
Carousel ∼ Dirac operator
(η0, η∞, γ) τ : f → 2R−1t
[0 −11 0
]f ′(t),
Under mild conditions: τ−1 is a Hilbert-Schmidt integral operatorwith kernel
K(x , y) =(u0u
T1 1(x < y) + u1u
T0 1(x ≥ y)
)R(y)
u0, u1: boundary conditions in τ
Nice property: if the path γ lives on [0,T ) then the operator canbe approximated using the path restricted to [0,T − ε).
Carousel ∼ Dirac operator
(η0, η∞, γ) τ : f → 2R−1t
[0 −11 0
]f ′(t),
Under mild conditions: τ−1 is a Hilbert-Schmidt integral operatorwith kernel
K(x , y) =(u0u
T1 1(x < y) + u1u
T0 1(x ≥ y)
)R(y)
u0, u1: boundary conditions in τ
Nice property: if the path γ lives on [0,T ) then the operator canbe approximated using the path restricted to [0,T − ε).
Carousel ∼ Dirac operator
τ : f → 2(UTU)−1[
0 −11 0
]f ′(t), U =
1√yt
[1 −xt0 yy
]
Brownian carousel representation of Sineβ
⇓random differential operator for Sineβ
xt + iyt : time-changed hyperbolic Brownian motion
Carousel ∼ Dirac operator
τ : f → 2(UTU)−1[
0 −11 0
]f ′(t), U =
1√yt
[1 −xt0 yy
]
Brownian carousel representation of Sineβ
⇓random differential operator for Sineβ
xt + iyt : time-changed hyperbolic Brownian motion
Additional results
I CβEd= Sineβ
Using the hyperbolic carousel ∼ Dirac operator connection
(Nakano: independent proof)
I Dirac operator description for deterministic unitary matrices
I Random Dirac-operator description for other classical models
driving paths: ‘affine’ hyperbolic Brownian motions
I Soft edge limit: representation as a canonical system[0 −11 0
]f ′(t) = λRt f (t)
(rank(Rt) = 1)
I Bulk convergence via the operators
Additional results
I CβEd= Sineβ
Using the hyperbolic carousel ∼ Dirac operator connection
(Nakano: independent proof)
I Dirac operator description for deterministic unitary matrices
I Random Dirac-operator description for other classical models
driving paths: ‘affine’ hyperbolic Brownian motions
I Soft edge limit: representation as a canonical system[0 −11 0
]f ′(t) = λRt f (t)
(rank(Rt) = 1)
I Bulk convergence via the operators
Additional results
I CβEd= Sineβ
Using the hyperbolic carousel ∼ Dirac operator connection
(Nakano: independent proof)
I Dirac operator description for deterministic unitary matrices
I Random Dirac-operator description for other classical models
driving paths: ‘affine’ hyperbolic Brownian motions
I Soft edge limit: representation as a canonical system[0 −11 0
]f ′(t) = λRt f (t)
(rank(Rt) = 1)
I Bulk convergence via the operators
Additional results
I CβEd= Sineβ
Using the hyperbolic carousel ∼ Dirac operator connection
(Nakano: independent proof)
I Dirac operator description for deterministic unitary matrices
I Random Dirac-operator description for other classical models
driving paths: ‘affine’ hyperbolic Brownian motions
I Soft edge limit: representation as a canonical system[0 −11 0
]f ′(t) = λRt f (t)
(rank(Rt) = 1)
I Bulk convergence via the operators
Additional results
I CβEd= Sineβ
Using the hyperbolic carousel ∼ Dirac operator connection
(Nakano: independent proof)
I Dirac operator description for deterministic unitary matrices
I Random Dirac-operator description for other classical models
driving paths: ‘affine’ hyperbolic Brownian motions
I Soft edge limit: representation as a canonical system[0 −11 0
]f ′(t) = λRt f (t)
(rank(Rt) = 1)
I Bulk convergence via the operators
CβEd= Sineβ
Circular β-ensemble: n points on the unit circle with joint density
1
Zn,β
∏i<j≤n
|e iλi − e iλj |β
Killip-Stoiciu: {nλi , 1 ≤ i ≤ n} converges to a point process CβE
Description: coupled SDE system counting function
Similar to the Sineβ description, but time is reversed, different end cond.
One can reformulate the Killip-Stoiciu description as a hyperboliccarousel driven by a time-reversed version of the Sineβ carousel.
Reversing time in the carousel one can show that CβEd= Sineβ
CβEd= Sineβ
Circular β-ensemble: n points on the unit circle with joint density
1
Zn,β
∏i<j≤n
|e iλi − e iλj |β
Killip-Stoiciu: {nλi , 1 ≤ i ≤ n} converges to a point process CβE
Description: coupled SDE system counting function
Similar to the Sineβ description, but time is reversed, different end cond.
One can reformulate the Killip-Stoiciu description as a hyperboliccarousel driven by a time-reversed version of the Sineβ carousel.
Reversing time in the carousel one can show that CβEd= Sineβ
CβEd= Sineβ
Circular β-ensemble: n points on the unit circle with joint density
1
Zn,β
∏i<j≤n
|e iλi − e iλj |β
Killip-Stoiciu: {nλi , 1 ≤ i ≤ n} converges to a point process CβE
Description: coupled SDE system counting function
Similar to the Sineβ description, but time is reversed, different end cond.
One can reformulate the Killip-Stoiciu description as a hyperboliccarousel driven by a time-reversed version of the Sineβ carousel.
Reversing time in the carousel one can show that CβEd= Sineβ
CβEd= Sineβ
Circular β-ensemble: n points on the unit circle with joint density
1
Zn,β
∏i<j≤n
|e iλi − e iλj |β
Killip-Stoiciu: {nλi , 1 ≤ i ≤ n} converges to a point process CβE
Description: coupled SDE system counting function
Similar to the Sineβ description, but time is reversed, different end cond.
One can reformulate the Killip-Stoiciu description as a hyperboliccarousel driven by a time-reversed version of the Sineβ carousel.
Reversing time in the carousel one can show that CβEd= Sineβ
CβEd= Sineβ
Circular β-ensemble: n points on the unit circle with joint density
1
Zn,β
∏i<j≤n
|e iλi − e iλj |β
Killip-Stoiciu: {nλi , 1 ≤ i ≤ n} converges to a point process CβE
Description: coupled SDE system counting function
Similar to the Sineβ description, but time is reversed, different end cond.
One can reformulate the Killip-Stoiciu description as a hyperboliccarousel driven by a time-reversed version of the Sineβ carousel.
Reversing time in the carousel one can show that CβEd= Sineβ
CβEd= Sineβ
Circular β-ensemble: n points on the unit circle with joint density
1
Zn,β
∏i<j≤n
|e iλi − e iλj |β
Killip-Stoiciu: {nλi , 1 ≤ i ≤ n} converges to a point process CβE
Description: coupled SDE system counting function
Similar to the Sineβ description, but time is reversed, different end cond.
One can reformulate the Killip-Stoiciu description as a hyperboliccarousel driven by a time-reversed version of the Sineβ carousel.
Reversing time in the carousel one can show that CβEd= Sineβ
Dirac operators for unitary matrices
V : n × n unitary matrix with distinct eigenvaluese: a cyclic unit vector
Apply G-S to e,Ve, . . . ,V n−1e Szego recursion for OPUC[Φk+1(z)Φ∗k+1(z)
]=
[1 −αk
−αk 1
] [z 00 1
] [Φk(z)Φ∗k(z)
],
[Φ∗0(z)Φ0(z)
]=
[11
]αk : Verblunsky coefficients, |αk | ≤ 1
z is an e.v. ⇔[z 00 1
] [Φn−1(z)Φ∗n−1(z)
]‖[αn−1
1
]
Dirac operators for unitary matrices
V : n × n unitary matrix with distinct eigenvaluese: a cyclic unit vector
Apply G-S to e,Ve, . . . ,V n−1e Szego recursion for OPUC
[Φk+1(z)Φ∗k+1(z)
]=
[1 −αk
−αk 1
] [z 00 1
] [Φk(z)Φ∗k(z)
],
[Φ∗0(z)Φ0(z)
]=
[11
]αk : Verblunsky coefficients, |αk | ≤ 1
z is an e.v. ⇔[z 00 1
] [Φn−1(z)Φ∗n−1(z)
]‖[αn−1
1
]
Dirac operators for unitary matrices
V : n × n unitary matrix with distinct eigenvaluese: a cyclic unit vector
Apply G-S to e,Ve, . . . ,V n−1e Szego recursion for OPUC[Φk+1(z)Φ∗k+1(z)
]=
[1 −αk
−αk 1
] [z 00 1
] [Φk(z)Φ∗k(z)
],
[Φ∗0(z)Φ0(z)
]=
[11
]αk : Verblunsky coefficients, |αk | ≤ 1
z is an e.v. ⇔[z 00 1
] [Φn−1(z)Φ∗n−1(z)
]‖[αn−1
1
]
Dirac operators for unitary matrices
V : n × n unitary matrix with distinct eigenvaluese: a cyclic unit vector
Apply G-S to e,Ve, . . . ,V n−1e Szego recursion for OPUC[Φk+1(z)Φ∗k+1(z)
]=
[1 −αk
−αk 1
] [z 00 1
] [Φk(z)Φ∗k(z)
],
[Φ∗0(z)Φ0(z)
]=
[11
]αk : Verblunsky coefficients, |αk | ≤ 1
z is an e.v. ⇔[z 00 1
] [Φn−1(z)Φ∗n−1(z)
]‖[αn−1
1
]
Dirac operators for unitary matrices
[Φk+1(z)Φ∗k+1(z)
]=
[1 −αk
−αk 1
] [z 00 1
] [Φk(z)Φ∗k(z)
],
[Φ∗0(z)Φ0(z)
]=
[11
]
We can introduce a transformed version of
[Φk(z)Φ∗k(z)
]satisfying
gk+1 = M−1k
[e i
µ2n 0
0 e−iµ2n
]Mkgk , z = e i
µn
Mk : product of
[1 −αj
−αj 1
]matrices
This gives an actual Dirac operator with piecewise continuous Rt .
The path γ: a discrete walk in H.
Dirac operators for unitary matrices
[Φk+1(z)Φ∗k+1(z)
]=
[1 −αk
−αk 1
] [z 00 1
] [Φk(z)Φ∗k(z)
],
[Φ∗0(z)Φ0(z)
]=
[11
]
We can introduce a transformed version of
[Φk(z)Φ∗k(z)
]satisfying
gk+1 = M−1k
[e i
µ2n 0
0 e−iµ2n
]Mkgk , z = e i
µn
Mk : product of
[1 −αj
−αj 1
]matrices
This gives an actual Dirac operator with piecewise continuous Rt .
The path γ: a discrete walk in H.
Dirac operators for unitary matrices
[Φk+1(z)Φ∗k+1(z)
]=
[1 −αk
−αk 1
] [z 00 1
] [Φk(z)Φ∗k(z)
],
[Φ∗0(z)Φ0(z)
]=
[11
]
We can introduce a transformed version of
[Φk(z)Φ∗k(z)
]satisfying
gk+1 = M−1k
[e i
µ2n 0
0 e−iµ2n
]Mkgk , z = e i
µn
Mk : product of
[1 −αj
−αj 1
]matrices
This gives an actual Dirac operator with piecewise continuous Rt .
The path γ: a discrete walk in H.
More β-ensembles
1
Zn,f ,β
∏i<j≤n
|λj − λi |βn∏
i=1
f (λi )
Dumitriu-Edelman: tridiagonal matrix models for Hermite andLaguerre β-ensembles
Killip-Nenciu: models for the circular β-ensembles, using theSzego-recursion and random Verblunsky coefficients
Edelman-Sutton: the rescaled tridiagonal models can be viewed asdiscrete versions of random differential operators
More β-ensembles
1
Zn,f ,β
∏i<j≤n
|λj − λi |βn∏
i=1
f (λi )
Dumitriu-Edelman: tridiagonal matrix models for Hermite andLaguerre β-ensembles
Killip-Nenciu: models for the circular β-ensembles, using theSzego-recursion and random Verblunsky coefficients
Edelman-Sutton: the rescaled tridiagonal models can be viewed asdiscrete versions of random differential operators
More β-ensembles
1
Zn,f ,β
∏i<j≤n
|λj − λi |βn∏
i=1
f (λi )
Dumitriu-Edelman: tridiagonal matrix models for Hermite andLaguerre β-ensembles
Killip-Nenciu: models for the circular β-ensembles, using theSzego-recursion and random Verblunsky coefficients
Edelman-Sutton: the rescaled tridiagonal models can be viewed asdiscrete versions of random differential operators
Operator convergence
One can find the discrete versions of the limit operators in thefinite tridiagonal models.
Soft edge: in the appropriate scaling the tridiagonal matrix can bewritten as a sum of a discrete Laplacian, a discrete white noisepotential and a potential approximating the function x
Hard edge: the inverse of the tridiagonal matrix (as a product oftwo bidiagonal matrices) can be written as an integral operatorapproximating the inverse of the Bβ,a operator
What about the bulk?
Operator convergence
One can find the discrete versions of the limit operators in thefinite tridiagonal models.
Soft edge: in the appropriate scaling the tridiagonal matrix can bewritten as a sum of a discrete Laplacian, a discrete white noisepotential and a potential approximating the function x
Hard edge: the inverse of the tridiagonal matrix (as a product oftwo bidiagonal matrices) can be written as an integral operatorapproximating the inverse of the Bβ,a operator
What about the bulk?
Operator convergence
One can find the discrete versions of the limit operators in thefinite tridiagonal models.
Soft edge: in the appropriate scaling the tridiagonal matrix can bewritten as a sum of a discrete Laplacian, a discrete white noisepotential and a potential approximating the function x
Hard edge: the inverse of the tridiagonal matrix (as a product oftwo bidiagonal matrices) can be written as an integral operatorapproximating the inverse of the Bβ,a operator
What about the bulk?
Operator level bulk limit
discrete model
↓discrete ‘differential operator’
↓discrete integral operator
↓limiting integral operator
The previous methods required the derivation of a one-parameterfamily of SDE system.
Here we need to understand the limit of the integral kernel: asingle SDE.
Operator level bulk limit
discrete model
↓discrete ‘differential operator’
↓discrete integral operator
↓limiting integral operator
The previous methods required the derivation of a one-parameterfamily of SDE system.
Here we need to understand the limit of the integral kernel: asingle SDE.
Dirac operators for other models
I finite circular β-ensemble and circular Jacobi ensembles
I limits of circular Jacobi ensembles
I hard edge limits
I certain one dimensional random Schrodinger operators
In each case the path γ is a random walk or diffusion on H.
I finite circular β-ensemble and circular Jacobi ensembles: γ isa random walk in H
I Hard edge: γ is a real BM with drift embedded in HI circular Jacobi: γ is a ‘hyperbolic BM with drift’
Dirac operators for other models
I finite circular β-ensemble and circular Jacobi ensembles
I limits of circular Jacobi ensembles
I hard edge limits
I certain one dimensional random Schrodinger operators
In each case the path γ is a random walk or diffusion on H.
I finite circular β-ensemble and circular Jacobi ensembles: γ isa random walk in H
I Hard edge: γ is a real BM with drift embedded in HI circular Jacobi: γ is a ‘hyperbolic BM with drift’
Dirac operators for other models
I finite circular β-ensemble and circular Jacobi ensembles
I limits of circular Jacobi ensembles
I hard edge limits
I certain one dimensional random Schrodinger operators
In each case the path γ is a random walk or diffusion on H.
I finite circular β-ensemble and circular Jacobi ensembles: γ isa random walk in H
I Hard edge: γ is a real BM with drift embedded in HI circular Jacobi: γ is a ‘hyperbolic BM with drift’
Dirac operators from tridiagonal matrices?
The eigenvalue equation is a three-term recursion
Mu = λu a`u`−1 + b`u` + a`u`+1 = λu`
This can be reformulated with transfer matrices:
T`
[u`−1u`
]−[
u`u`+1
]=
[0 0λ 0
] [u`u`+1
],
[u0u1
]=
[01
].
After conjugation and some averaging, one can recover theeigenvalue equation of a Dirac operator.
Dirac operators from tridiagonal matrices?
The eigenvalue equation is a three-term recursion
Mu = λu a`u`−1 + b`u` + a`u`+1 = λu`
This can be reformulated with transfer matrices:
T`
[u`−1u`
]−[
u`u`+1
]=
[0 0λ 0
] [u`u`+1
],
[u0u1
]=
[01
].
After conjugation and some averaging, one can recover theeigenvalue equation of a Dirac operator.
Dirac operators from tridiagonal matrices?
The eigenvalue equation is a three-term recursion
Mu = λu a`u`−1 + b`u` + a`u`+1 = λu`
This can be reformulated with transfer matrices:
T`
[u`−1u`
]−[
u`u`+1
]=
[0 0λ 0
] [u`u`+1
],
[u0u1
]=
[01
].
After conjugation and some averaging, one can recover theeigenvalue equation of a Dirac operator.
THANK YOU!