why is electrostatics in the complex plane interesting...
TRANSCRIPT
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Why is electrostatics in the complex planeinteresting from a mathematical point of
view?
Ferenc Balogh
Concordia University
12 May, 2007
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Overview
I Motivation Electrostatics in the Plane, NumericalQuadrature, Orthogonal Polynomials, CotesNumbers as Charge Locations
I Classical Potential Theory Energy Problem,Frostman’s Theorem, Equilibrium Measures,Capacity
I Logarithmic Potentials with External FieldsAdmissible Background Potentials, EquilibriumMeasures, Examples
I Application to Random Matrices Scaling limit,Density of States, Wigner’s Semicircle Law
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Electrostatics in the PlaneLaplacian on the plane:
∆f =∂2f
∂x2+
∂2f
∂y2=
1
r
∂
∂r
(r∂f
∂r
)+
∂2f
∂ϕ2.
Search for ∆f = 0 with f = F (r) in the punctured plane:
F (r) = A log r + B
Electrostatic potential of a point charge q at 0 in theplane:
V (r) = q log1
r.
µ is a compactly supported finite positive measure on CLogarithmic potential of µ
Uµ(z) :=
∫log
1
|z − t|dµ(t).
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Electrostatics in the PlaneLaplacian on the plane:
∆f =∂2f
∂x2+
∂2f
∂y2=
1
r
∂
∂r
(r∂f
∂r
)+
∂2f
∂ϕ2.
Search for ∆f = 0 with f = F (r) in the punctured plane:
F (r) = A log r + B
Electrostatic potential of a point charge q at 0 in theplane:
V (r) = q log1
r.
µ is a compactly supported finite positive measure on CLogarithmic potential of µ
Uµ(z) :=
∫log
1
|z − t|dµ(t).
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Electrostatics in the PlaneLaplacian on the plane:
∆f =∂2f
∂x2+
∂2f
∂y2=
1
r
∂
∂r
(r∂f
∂r
)+
∂2f
∂ϕ2.
Search for ∆f = 0 with f = F (r) in the punctured plane:
F (r) = A log r + B
Electrostatic potential of a point charge q at 0 in theplane:
V (r) = q log1
r.
µ is a compactly supported finite positive measure on CLogarithmic potential of µ
Uµ(z) :=
∫log
1
|z − t|dµ(t).
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
An Equilibrium Problem on [−1, 1]
The physical setting:
I fixed positive charges p and q at 1 and −1
I n particles of unit charge confined to [−1, 1]
Problem: find an equilibrium configuration of thelocation x1, x2, . . . , xn of the movable charges, i. e.minimize the functional∑1≤k<l≤n
log1
|xk − xl |+
n∑k=1
[p log
1
|1− xk |+ q log
1
|1 + xk |
]self-energy + background terms
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
An Equilibrium Problem on [−1, 1]
The physical setting:
I fixed positive charges p and q at 1 and −1
I n particles of unit charge confined to [−1, 1]
Problem: find an equilibrium configuration of thelocation x1, x2, . . . , xn of the movable charges
, i. e.minimize the functional∑1≤k<l≤n
log1
|xk − xl |+
n∑k=1
[p log
1
|1− xk |+ q log
1
|1 + xk |
]self-energy + background terms
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
An Equilibrium Problem on [−1, 1]
The physical setting:
I fixed positive charges p and q at 1 and −1
I n particles of unit charge confined to [−1, 1]
Problem: find an equilibrium configuration of thelocation x1, x2, . . . , xn of the movable charges, i. e.minimize the functional∑1≤k<l≤n
log1
|xk − xl |+
n∑k=1
[p log
1
|1− xk |+ q log
1
|1 + xk |
]self-energy + background terms
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
An Equilibrium Problem on [−1, 1]
There exists a unique minimizing configuration.
Taking derivatives, the following system of equations hasto be solved:
∑1≤l≤nl 6=k
1
xk − xl+
p
xk − 1+
q
xk + 1= 0 k = 1, 2, . . . , n.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
An Equilibrium Problem on [−1, 1]
There exists a unique minimizing configuration.
Taking derivatives, the following system of equations hasto be solved:
∑1≤l≤nl 6=k
1
xk − xl+
p
xk − 1+
q
xk + 1= 0 k = 1, 2, . . . , n.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
An Equilibrium Problem on [−1, 1]
Stieltjes (1885):Using f (x) := (x − x1)(x − x2) · · · · · (x − xn) it reads as
1
2
f ′′(xk)
f ′(xk)+
p
xk − 1+
q
xk + 1= 0 k = 1, 2, . . . , n.
Hence f is a monic polynomial of degree n and solves thesecond-order differential equation
(1− x2)d2F
dx2+ 2(q − p − (q + p)x)
dF
dx+ cnF = 0.
For what cn are there polynomial solutions of thisequation?
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
An Equilibrium Problem on [−1, 1]
Stieltjes (1885):Using f (x) := (x − x1)(x − x2) · · · · · (x − xn) it reads as
1
2
f ′′(xk)
f ′(xk)+
p
xk − 1+
q
xk + 1= 0 k = 1, 2, . . . , n.
Hence f is a monic polynomial of degree n and solves thesecond-order differential equation
(1− x2)d2F
dx2+ 2(q − p − (q + p)x)
dF
dx+ cnF = 0.
For what cn are there polynomial solutions of thisequation?
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Numerical Quadrature
Let µ be a fixed compactly supported finite positivemeasure on R. The numerical quadrature is a method toapproximate the integral
I (f ) =
∫f (x)dµ(x)
of a function f by a finite sum
In(f ) =n∑
k=1
f (xj ,n)βj ,n.
using n distinct nodes x1,n, x2,n, . . . , xn,n and quadratureconstants β1,n, β2,n, . . . , βn,n (independent of f ).
Degree of accuracy of In:
max{d ∈ N | In(p) = I (p),∀p ∈ R[x ], deg p = d}.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Numerical Quadrature
Let µ be a fixed compactly supported finite positivemeasure on R. The numerical quadrature is a method toapproximate the integral
I (f ) =
∫f (x)dµ(x)
of a function f by a finite sum
In(f ) =n∑
k=1
f (xj ,n)βj ,n.
using n distinct nodes x1,n, x2,n, . . . , xn,n and quadratureconstants β1,n, β2,n, . . . , βn,n (independent of f ).Degree of accuracy of In:
max{d ∈ N | In(p) = I (p),∀p ∈ R[x ], deg p = d}.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Numerical Quadrature
Problem. Choose the nodes and quadrature coefficientsto have maximal degree of accuracy.
Remark. For all choice of n distinct nodes, there are β’ssuch that the degree of accuracy is at least n − 1.
Theorem (Gauss-Jacobi-Christoffel)There exist a choice of n nodes (the so-called Cotesnumbers) and corresponding quadrature coefficients suchthat the degree of accuracy is at least 2n − 1 (!).The nodes are the zeroes of the n-th orthogonalpolynomial Pn(x) with respect to the measure µ.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Numerical Quadrature
Problem. Choose the nodes and quadrature coefficientsto have maximal degree of accuracy.Remark. For all choice of n distinct nodes, there are β’ssuch that the degree of accuracy is at least n − 1.
Theorem (Gauss-Jacobi-Christoffel)There exist a choice of n nodes (the so-called Cotesnumbers) and corresponding quadrature coefficients suchthat the degree of accuracy is at least 2n − 1 (!).The nodes are the zeroes of the n-th orthogonalpolynomial Pn(x) with respect to the measure µ.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Numerical Quadrature
Problem. Choose the nodes and quadrature coefficientsto have maximal degree of accuracy.Remark. For all choice of n distinct nodes, there are β’ssuch that the degree of accuracy is at least n − 1.
Theorem (Gauss-Jacobi-Christoffel)There exist a choice of n nodes (the so-called Cotesnumbers) and corresponding quadrature coefficients suchthat the degree of accuracy is at least 2n − 1 (!).The nodes are the zeroes of the n-th orthogonalpolynomial Pn(x) with respect to the measure µ.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Orthogonal Polynomials on the Real Line
Let µ be a positive Borel measure on R.Assume that µ has infinite support and finite momentsand that {xn}∞n=1 is complete in L2(µ).
The Gram-Schmidt ortogonalization procedure gives theorthogonal polynomials Pn with respect to µ:∫
RPm(x)Pn(x)dµ(x) = δmn, m, n ∈ N0.
Problems in Approximation Theory:Problem. Find the various kinds of asymptotics of OP’sPn in the large n limit.Problem. Find the asymptotics of the zero distributionof Pn in the large n limit.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Orthogonal Polynomials on the Real Line
Let µ be a positive Borel measure on R.Assume that µ has infinite support and finite momentsand that {xn}∞n=1 is complete in L2(µ).The Gram-Schmidt ortogonalization procedure gives theorthogonal polynomials Pn with respect to µ:∫
RPm(x)Pn(x)dµ(x) = δmn, m, n ∈ N0.
Problems in Approximation Theory:Problem. Find the various kinds of asymptotics of OP’sPn in the large n limit.Problem. Find the asymptotics of the zero distributionof Pn in the large n limit.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Orthogonal Polynomials on the Real Line
Let µ be a positive Borel measure on R.Assume that µ has infinite support and finite momentsand that {xn}∞n=1 is complete in L2(µ).The Gram-Schmidt ortogonalization procedure gives theorthogonal polynomials Pn with respect to µ:∫
RPm(x)Pn(x)dµ(x) = δmn, m, n ∈ N0.
Problems in Approximation Theory:
Problem. Find the various kinds of asymptotics of OP’sPn in the large n limit.Problem. Find the asymptotics of the zero distributionof Pn in the large n limit.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Orthogonal Polynomials on the Real Line
Let µ be a positive Borel measure on R.Assume that µ has infinite support and finite momentsand that {xn}∞n=1 is complete in L2(µ).The Gram-Schmidt ortogonalization procedure gives theorthogonal polynomials Pn with respect to µ:∫
RPm(x)Pn(x)dµ(x) = δmn, m, n ∈ N0.
Problems in Approximation Theory:Problem. Find the various kinds of asymptotics of OP’sPn in the large n limit.Problem. Find the asymptotics of the zero distributionof Pn in the large n limit.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Classical Orthogonal Polynomials
Hermite Laguerre Legendrew(x) = e−x2
w(x) = I (x ≥ 0)e−x w(x) = I (−1 ≤ x ≤ 1)
Chebyshev I Chebyshev II Jacobiw(x) = 1q
1−x2w(x) =
p1− x2 w(x) = (1 + x)α(1− x)β
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Special Properties of the Classical OPs
These classical orthogonal polynomials have
I generating functions
∞∑n=0
Hn(x)
n!wn = exp(2xw − w2),
I Rodrigues-type formula
Lαn (x) = exx−α 1
n!
dn
dxn
(e−xxn+α
),
I second-order differential equation
(1− x2)d2P
(α,β)n
dx2+ (β − α− (α + β + 2)x)
dP(α,β)n
dx
+n(n + 1 + α + β)P(α,β)n (x) = 0.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Special Properties of the Classical OPs
These classical orthogonal polynomials have
I generating functions
∞∑n=0
Hn(x)
n!wn = exp(2xw − w2),
I Rodrigues-type formula
Lαn (x) = exx−α 1
n!
dn
dxn
(e−xxn+α
),
I second-order differential equation
(1− x2)d2P
(α,β)n
dx2+ (β − α− (α + β + 2)x)
dP(α,β)n
dx
+n(n + 1 + α + β)P(α,β)n (x) = 0.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Special Properties of the Classical OPs
These classical orthogonal polynomials have
I generating functions
∞∑n=0
Hn(x)
n!wn = exp(2xw − w2),
I Rodrigues-type formula
Lαn (x) = exx−α 1
n!
dn
dxn
(e−xxn+α
),
I second-order differential equation
(1− x2)d2P
(α,β)n
dx2+ (β − α− (α + β + 2)x)
dP(α,β)n
dx
+n(n + 1 + α + β)P(α,β)n (x) = 0.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Approximation Theory and Electrostatics
The solution of the equilibrium problem:The optimal locations x1,n, x2,n, . . . xn,n are given by the
zeroes of the Jacobi OPs P(α,β)n (x), where α = 2p − 1,
β = 2q − 1.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Approximation Theory and Electrostatics
The solution of the equilibrium problem:The optimal locations x1,n, x2,n, . . . xn,n are given by the
zeroes of the Jacobi OPs P(α,β)n (x), where α = 2p − 1,
β = 2q − 1.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
The Energy Problem
Let µ be a compactly supported finite positive measureon C. The energy stored in the charge configuration:
I(µ) :=
∫Uµ(z)dµ(z) =
∫∫log
1
|z − t|dµ(t)dµ(z).
Given a compact set K ⊂ C, M(K ) denotes the set of allprobability measures supported on K .
VK := infµ∈M(K)
I(µ).
Frostman’s Theorem. If VK is finite then there exists aunique measure µK ∈M(K ) such that I(µK ) = VK .This µK is called the equilibrium measure of K .
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
The Energy Problem
Let µ be a compactly supported finite positive measureon C. The energy stored in the charge configuration:
I(µ) :=
∫Uµ(z)dµ(z) =
∫∫log
1
|z − t|dµ(t)dµ(z).
Given a compact set K ⊂ C, M(K ) denotes the set of allprobability measures supported on K .
VK := infµ∈M(K)
I(µ).
Frostman’s Theorem. If VK is finite then there exists aunique measure µK ∈M(K ) such that I(µK ) = VK .
This µK is called the equilibrium measure of K .
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
The Energy Problem
Let µ be a compactly supported finite positive measureon C. The energy stored in the charge configuration:
I(µ) :=
∫Uµ(z)dµ(z) =
∫∫log
1
|z − t|dµ(t)dµ(z).
Given a compact set K ⊂ C, M(K ) denotes the set of allprobability measures supported on K .
VK := infµ∈M(K)
I(µ).
Frostman’s Theorem. If VK is finite then there exists aunique measure µK ∈M(K ) such that I(µK ) = VK .This µK is called the equilibrium measure of K .
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
The Energy Problem
Examples
I D closed disk of radius r centered at a:µD is the normalized arclength measure on thecircumference of D
I [−a/2, a/2] closed interval on R
dµ[−a/2,a/2](x) =1
π
dx√a2/4− x2
, x ∈ [−a/2, a/2].
For K compact, its capacity is defined as
cap(K ) := e−VK .
cap(D(a, r)) = r , cap([−a/2, a/2]) = a/4.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
The Energy Problem
Examples
I D closed disk of radius r centered at a:µD is the normalized arclength measure on thecircumference of D
I [−a/2, a/2] closed interval on R
dµ[−a/2,a/2](x) =1
π
dx√a2/4− x2
, x ∈ [−a/2, a/2].
For K compact, its capacity is defined as
cap(K ) := e−VK .
cap(D(a, r)) = r , cap([−a/2, a/2]) = a/4.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Energy Problem with a Background Potential
Idea. Replace the compactness assumption by thepresence of a background potential
Q : Σ → (−∞,∞]
on the conductor Σ. Q must be strong enough to confinethe charges to a finite region bounded away from ∞.
Q is said to be admissible if
I Q is lower-semicontinuous,
I cap({z ∈ Σ | Q(z) < ∞}) > 0,
I Q(z)− log |z | → ∞ as |z | → ∞ in Σ.
Weighted energy functional:
IQ(µ) := I(µ) + 2
∫Qdµ
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Energy Problem with a Background Potential
Idea. Replace the compactness assumption by thepresence of a background potential
Q : Σ → (−∞,∞]
on the conductor Σ. Q must be strong enough to confinethe charges to a finite region bounded away from ∞.
Q is said to be admissible if
I Q is lower-semicontinuous,
I cap({z ∈ Σ | Q(z) < ∞}) > 0,
I Q(z)− log |z | → ∞ as |z | → ∞ in Σ.
Weighted energy functional:
IQ(µ) := I(µ) + 2
∫Qdµ
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Energy Problem with a Background Potential
Idea. Replace the compactness assumption by thepresence of a background potential
Q : Σ → (−∞,∞]
on the conductor Σ. Q must be strong enough to confinethe charges to a finite region bounded away from ∞.
Q is said to be admissible if
I Q is lower-semicontinuous,
I cap({z ∈ Σ | Q(z) < ∞}) > 0,
I Q(z)− log |z | → ∞ as |z | → ∞ in Σ.
Weighted energy functional:
IQ(µ) := I(µ) + 2
∫Qdµ
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Energy Problem with a Background Potential
Mhaskar-Saff-Totik Theorem.If Q is an admissible potential then
I
VQ := infMΣ
IQ(µ) < ∞
I There exists a unique µQ ∈M(Σ) such thatIQ(µQ) = VQ with finite self-energy.
I The support of µQ is compact.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Gallery of Equilibrium Measures
Q(z) = α|z|2 Q(z) = α“|z|2 + tRe(z2)
”Q(z) = α
“|z|2 + tRe(z3)
”
Remark. The density is always uniform in these specialcases because, we know that, in general,
dµQ
dm(z) =
1
2π∆Q(z)
at points z where µQ is regular enough.
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Gallery of Equilibrium Measures II
Perturbation of the Gaussian using a point charge:
Q(z) = α|z |2 + β log1
|z − a|(α > 0, β > 0)
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Gallery of Equilibrium Measures II
Perturbation of the Gaussian using a point charge:
Q(z) = α|z |2 + β log1
|z − a|(α > 0, β > 0)
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Equilibrium measures restricted to R
What happens if Σ = R and Q(x) = x2 ?Physical intuition: Consider the N →∞ limit of:
QN(x + iy) = x2 + Ny2.
The equilibrium measure of QN is the normalizedLebesgue measure restricted to the ellipse
EN =
(x + iy ∈ C
˛ x2
a2N
+y2
b2N
≤ 1
)
whereaN =
sN
N + 1→ 1 bN =
s1
N(N + 1)→ 0 (N →∞).
µQN⇀ µW (N →∞)
where µW is the Wigner semicircle distribution on [−1, 1].
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Equilibrium measures restricted to R
What happens if Σ = R and Q(x) = x2 ?Physical intuition: Consider the N →∞ limit of:
QN(x + iy) = x2 + Ny2.
The equilibrium measure of QN is the normalizedLebesgue measure restricted to the ellipse
EN =
(x + iy ∈ C
˛ x2
a2N
+y2
b2N
≤ 1
)
whereaN =
sN
N + 1→ 1 bN =
s1
N(N + 1)→ 0 (N →∞).
µQN⇀ µW (N →∞)
where µW is the Wigner semicircle distribution on [−1, 1].
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Equilibrium measures restricted to R
What happens if Σ = R and Q(x) = x2 ?Physical intuition: Consider the N →∞ limit of:
QN(x + iy) = x2 + Ny2.
The equilibrium measure of QN is the normalizedLebesgue measure restricted to the ellipse
EN =
(x + iy ∈ C
˛ x2
a2N
+y2
b2N
≤ 1
)
whereaN =
sN
N + 1→ 1 bN =
s1
N(N + 1)→ 0 (N →∞).
µQN⇀ µW (N →∞)
where µW is the Wigner semicircle distribution on [−1, 1].
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Random Matrices
Unitary Ensembles of Hermitian MatricesHn: the space of n × n Hermitian matricesdM: Lebesgue measure on Hn
We are interested in densities pn(M) on Hn s.t. theprobability measure pn(M)dM is U(n)-invariant.Let V : R → R be a function increasing fast enough atinfinity.
pn,N(M) := cn,N exp(−NTr(V (M))).
The joint probability distribution on the eigenvalues:
fn,N(λ1, λ2, . . . , λn) =1
Zn,N
∏i<j
(λi − λj)2e−N
Pnj=1 V (λj ).
level repulsion of the λj ’s – strongly dependent variables
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Random Matrices
Unitary Ensembles of Hermitian MatricesHn: the space of n × n Hermitian matricesdM: Lebesgue measure on Hn
We are interested in densities pn(M) on Hn s.t. theprobability measure pn(M)dM is U(n)-invariant.Let V : R → R be a function increasing fast enough atinfinity.
pn,N(M) := cn,N exp(−NTr(V (M))).
The joint probability distribution on the eigenvalues:
fn,N(λ1, λ2, . . . , λn) =1
Zn,N
∏i<j
(λi − λj)2e−N
Pnj=1 V (λj ).
level repulsion of the λj ’s – strongly dependent variables
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Eigenvalues of Random Matrices
fn,N(λ1, λ2, . . . , λn) =
1
Zn,N
∏i<j
(λi − λj)2e−N
Pnj=1 V (λj ) =
1
Zn,Nexp
−∑i 6=j
log1
|λi − λj |− N
n∑j=1
V (λj)
=
1
Zn,Nexp
−n2
∑i 6=j
1
n2log
1
|λi − λj |− N
n
n∑j=1
1
nV (λj)
.
The eigenvalues are behaving like charged particles in thepresence of a background potential.
Scaling limit: N →∞, n →∞, N/n = γ. Thebackground potential is Q(x) = γ/2V (x).
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Eigenvalues of Random Matrices
fn,N(λ1, λ2, . . . , λn) =
1
Zn,N
∏i<j
(λi − λj)2e−N
Pnj=1 V (λj ) =
1
Zn,Nexp
−∑i 6=j
log1
|λi − λj |− N
n∑j=1
V (λj)
=
1
Zn,Nexp
−n2
∑i 6=j
1
n2log
1
|λi − λj |− N
n
n∑j=1
1
nV (λj)
.
The eigenvalues are behaving like charged particles in thepresence of a background potential.Scaling limit: N →∞, n →∞, N/n = γ. Thebackground potential is Q(x) = γ/2V (x).
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Density of States
Density of States: normalized random counting measureof the eigenvalues
Sn(H) :=1
n|σ(M) ∩ H| H ∈ B(R).
In the scaling limit, E(Sn) tends to a measure µ.(’Law of large numbers’ for Sn)
Important Fact. This measure is the same as theequilibrium measure on R of the background potentialQ(x) = γ
2V (x).
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Density of States
Density of States: normalized random counting measureof the eigenvalues
Sn(H) :=1
n|σ(M) ∩ H| H ∈ B(R).
In the scaling limit, E(Sn) tends to a measure µ.(’Law of large numbers’ for Sn)
Important Fact. This measure is the same as theequilibrium measure on R of the background potentialQ(x) = γ
2V (x).
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
Wigner’s Semicircle Law
Gaussian Unitary Ensemble (GUE)
V (x) = 2x2
The limiting density of states is the semicircle distributionon [−1, 1]:
dµW (x) =√
1− x2.
50× 50 GUE eigenvalues by Maple
Why is electrostatics inthe complex planeinteresting from a
mathematical point ofview?
Ferenc Balogh
Overview
Motivation
Classical PotentialTheory
Potential Theory withExternal Fields
Random Matrices
References
References
P. Deift,
Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach.Courant Lecture Notes, 2000.
T. J. Ransford,
Potential Theory in the Complex Plane.Cambridge University Press, 1995.
E. B. Saff, V. Totik,
Logarithmic Potentials with External Fields.Springer, 1997.
G. Szego,
Orthogonal Polynomials.AMS, 1959.
V. Totik,
Orthogonal Polynomials.Surveys in Approx. Theory, (1) 2005, pp. 70-125.
W. Van Assche,
Orthogonal Polynomials in the Complex Plane and on the Real Line.Fields Institute Communications