Why is electrostatics in the complex plane interesting ...baloghf/homepage/talks/ism_conf_2007.pdf · the complex plane interesting from a mathematical point of view? Ferenc Balogh

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



Ferenc Balogh

Overview

Motivation



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



Ferenc Balogh

Overview

Motivation



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).



Ferenc Balogh

Overview

Motivation



Random Matrices

References


∆f =∂2f

∂x2+

∂2f

∂y2=

1

r

∂

∂r

(r∂f

∂r

)+

∂2f

∂ϕ2.


F (r) = A log r + B


V (r) = q log1

r.


Uµ(z) :=

∫log

1

|z − t|dµ(t).



Ferenc Balogh

Overview

Motivation



Random Matrices

References


∆f =∂2f

∂x2+

∂2f

∂y2=

1

r

∂

∂r

(r∂f

∂r

)+

∂2f

∂ϕ2.


F (r) = A log r + B


V (r) = q log1

r.


Uµ(z) :=

∫log

1

|z − t|dµ(t).



Ferenc Balogh

Overview

Motivation



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



Ferenc Balogh

Overview

Motivation



Random Matrices

References





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 |




Ferenc Balogh

Overview

Motivation



Random Matrices

References





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 |




Ferenc Balogh

Overview

Motivation



Random Matrices

References


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.



Ferenc Balogh

Overview

Motivation



Random Matrices

References


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.



Ferenc Balogh

Overview

Motivation



Random Matrices

References


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?



Ferenc Balogh

Overview

Motivation



Random Matrices

References


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?



Ferenc Balogh

Overview

Motivation



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}.



Ferenc Balogh

Overview

Motivation



Random Matrices

References


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}.



Ferenc Balogh

Overview

Motivation



Random Matrices

References


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.



Ferenc Balogh

Overview

Motivation



Random Matrices

References


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.




Ferenc Balogh

Overview

Motivation



Random Matrices

References


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.




Ferenc Balogh

Overview

Motivation



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.



Ferenc Balogh

Overview

Motivation



Random Matrices

References


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:∫





Ferenc Balogh

Overview

Motivation



Random Matrices

References




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.



Ferenc Balogh

Overview

Motivation



Random Matrices

References







Ferenc Balogh

Overview

Motivation



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



Ferenc Balogh

Overview

Motivation



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.



Ferenc Balogh

Overview

Motivation



Random Matrices

References




∞∑n=0

Hn(x)




n!

dn

dxn

(e−xxn+α

),


(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.



Ferenc Balogh

Overview

Motivation



Random Matrices

References




∞∑n=0

Hn(x)




n!

dn

dxn

(e−xxn+α

),


(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.



Ferenc Balogh

Overview

Motivation



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.



Ferenc Balogh

Overview

Motivation



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.



Ferenc Balogh

Overview

Motivation



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 .



Ferenc Balogh

Overview

Motivation



Random Matrices

References

The Energy Problem


I(µ) :=

∫Uµ(z)dµ(z) =

∫∫log

1



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 .



Ferenc Balogh

Overview

Motivation



Random Matrices

References

The Energy Problem


I(µ) :=

∫Uµ(z)dµ(z) =

∫∫log

1



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 .



Ferenc Balogh

Overview

Motivation



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.



Ferenc Balogh

Overview

Motivation



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.



Ferenc Balogh

Overview

Motivation



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



Ferenc Balogh

Overview

Motivation



Random Matrices

References



Q : Σ → (−∞,∞]




I cap({z ∈ Σ | Q(z) < ∞}) > 0,

I Q(z)− log |z | → ∞ as |z | → ∞ in Σ.


IQ(µ) := I(µ) + 2

∫Qdµ



Ferenc Balogh

Overview

Motivation



Random Matrices

References



Q : Σ → (−∞,∞]




I cap({z ∈ Σ | Q(z) < ∞}) > 0,

I Q(z)− log |z | → ∞ as |z | → ∞ in Σ.


IQ(µ) := I(µ) + 2

∫Qdµ



Ferenc Balogh

Overview

Motivation



Random Matrices

References


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.



Ferenc Balogh

Overview

Motivation



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.



Ferenc Balogh

Overview

Motivation



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)



Ferenc Balogh

Overview

Motivation



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)



Ferenc Balogh

Overview

Motivation



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].



Ferenc Balogh

Overview

Motivation



Random Matrices

References



QN(x + iy) = x2 + Ny2.


EN =

(x + iy ∈ C

˛ x2

a2N

+y2

b2N

≤ 1

)

whereaN =

sN

N + 1→ 1 bN =

s1

N(N + 1)→ 0 (N →∞).





Ferenc Balogh

Overview

Motivation



Random Matrices

References



QN(x + iy) = x2 + Ny2.


EN =

(x + iy ∈ C

˛ x2

a2N

+y2

b2N

≤ 1

)

whereaN =

sN

N + 1→ 1 bN =

s1

N(N + 1)→ 0 (N →∞).





Ferenc Balogh

Overview

Motivation



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



Ferenc Balogh

Overview

Motivation



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



Ferenc Balogh

Overview

Motivation



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).



Ferenc Balogh

Overview

Motivation



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).



Ferenc Balogh

Overview

Motivation



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).



Ferenc Balogh

Overview

Motivation



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).



Ferenc Balogh

Overview

Motivation



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



Ferenc Balogh

Overview

Motivation



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

Documents

Why is electrostatics in the complex plane interesting ...baloghf/homepage/talks/ism_conf_2007.pdf · the complex plane interesting from a mathematical point of view? Ferenc Balogh