Eigenvalues of random matrices in the general linear group

Eigenvalues of random matrices in the general lineargroup in the large-N limit

Brian C. HallUniversity of Notre Dame

April 13, 2019

Brian Hall Eigenvalues of random matrices April 13, 2019 1 / 57

Credits and references

Joint work with Bruce Driver and Todd Kemp

Web page: www.nd.edu/˜bhall/

Hall and Kemp, preprint arXiv:1810.00153 [math.FA]

Driver, Hall, and Kemp, preprint arXiv:1903.11015 [math.PR]

Results in physics literature:

E. Gudowska-Nowak, R. Janik, J. Jurkiewicz, and M. Nowak, Nucl.Phys. B 670 2003R. Lohmayer, H. Neuberger, and T. Wettig, JHEP (2008) 053


OUTLINE:

Part 1: Random matrix theory

Part 2: Brown measure and the large-N limit

Part 3: Random matrices in GL(N; C)

Part 4: The PDE and its solution

Part 5: Letting ε tend to zero


PART 1: RANDOM MATRIX THEORY


Random matrix theory: History

Introduced by Wigner in 1955 to model energy levels of large nuclei

Interested in large-N behavior

“Quantum chaos”: quantum mechanics of systems that are classically“chaotic”

Spacing between energy levels conjectured to behave like spacingbetween consecutive eigenvalues of certain Hermitian randommatrices


A basic example: Ginibre ensemble

Simple way to choose a random matrix (not necessarily Hermitian)

Choose N ×N matrix randomly with independent, identicallydistributed entries

Ginibre ensemble: entries are complex Gaussian with variance 1/N


Circular law

Circular law: For large N, eigenvalues will (with high probability) bemostly in unit disk and almost uniformly distributed

Bulk distribution of eigenvalues becomes nonrandom in limit

-1.0 -0.5 0.5 1.0

-1.0

-0.5

0.5

1.0


PART 2: BROWN MEASURE AND THE LARGE-NLIMIT


Notion of a large-N limit

“∗-moments” of matrix-valued random variable X : e.g.

E{tr[X 2(X ∗)3X (X ∗)7X 5]}

where tr is the normalized trace

If X depends on N, consider large-N limits

Seek operator algebra A with “trace” τ : A → C and x ∈ A with

τ[word in x and x∗] = limN→∞

E{tr[word in XN and (XN)∗]}


Brown measure

Note: x is a single operator in A, not a random variable

Want to define something like “eigenvalue distribution” of x

If x is normal (x∗x = xx∗) can use spectral theorem

If x is non-normal, use “Brown measure” (L. Brown, 1986)


Motivating computation

For an N ×N matrix A, set

s(λ) = log(|det(A− λ)|2/N), λ ∈ C

Or

s(λ) =2

N

N

∑j=1

log|λ− λj |

Then Laplacian ∆ of s is zero except at the eigenvalues


Motivating computation

Indeed

1

4π∆s(λ) =

1

N

N

∑j=1

δλj(λ)

(probability measure with mass 1/N at each eigenvalue)

Can also write s as

s(λ) =1

Ntrace{log[(A− λ)∗(A− λ)]}


Motivating example

Plot of −s(λ)


Brown measure: imitating the previous construction

Given x in (A, τ), define

S(λ, ε) := τ{log[(x − λ)∗(x − λ) + ε]}, λ ∈ C

Limit ε→ 0 exists as subharmonic function

s(λ) := limε→0+

S(λ, ε)

If x is N ×N matrix and τ = 1N trace, then s is as in previous slide

Define Brown measure of x :

µx =1

4π∆s(λ)


Properties of Brown measure

µx is a probability measure

Supported on spectrum σ(x) of x

Satisfies ∫σ(x)

λn dµx (λ) = τ(xn)

but this does not uniquely determine µx .


Brown measure of circular law

Let XN be distributed as N ×N Ginibre ensemble

Then XN converges to a “circular element” x

Brown measure of x is uniform prob. measure on unit disk


General strategy

Step 1: Identify large-N limit and find its Brown measure

Step 2: Prove that eigenvalue distribution converges to Brownmeasure

Strategy pioneered by Girko for general circular law (entriesindependent and identically distributed but not normal)

Strategy used by Tao and Vu in strongest version of circular law


PART 3: RANDOM MATRICES IN GL(N;C)


Brownian motion in GL(N;C)

General linear group = group of all N ×N invertible matrices

Consider Brownian motion bNt in GL(N; C), starting at I

Compute as

bNt ≈k

∏j=1

(I +

√t√k

Ginj

), k large

where {Ginj} are independent N ×N Ginibre-distributed matrices

Distribution of bNt is heat kernel measure


Comparison to Brownian motion in U(N)

Brownian motion uNt in the unitary group U(N)

Arises in 2D Yang–Mills theory on plane

Brown measure νt of large-N limit found by Philippe Biane

νt supported on proper subset of S1 until t = 4

-π -π

2

π

2π



In GL(N; C), problem has been mostly open until now

For small t, have bNt ≈ I +√t Gin

For small t and large N, eigenvalues approximately uniform on disk ofradius

√t centered at 1 (t = 0.12 and N = 2, 000)

0.6 0.8 1.0 1.2 1.4

-0.4

-0.2

0.0

0.2

0.4



For larger t, eigenvalues of bNt cluster as N → ∞ into domain Σt

Nonrigorous argument for this in the cited papers in physics literature

N = 2, 000 with t = 2 and t = 3.9

0 5 10

-5

0

5

0 5 10

-5

0

5



For larger t, eigenvalues of bNt cluster as N → ∞ into domain Σt

N = 2, 000 with t = 4 and t = 4.1

0 5 10

-5

0

5

0 5 10

-5

0

5


Large-N limit

bNt converges as N → ∞ to Biane’s free multiplicative Brownianmotion bt

Full convergence result proved by Kemp

Goal: Show that Brown measure of bt is supported on Σt andcompute the Brown measure


Description of Σt

Domains introduced by Philippe Biane in 1997

Also appear in cited papers in physics literature

Set

T (λ) = |λ− 1|2 log(|λ|2)|λ|2 − 1

DefineΣt = {λ ∈ C|T (λ) < t}


Properties of Σt

Simply connected for t ≤ 4, doubly connected for t > 4

Shown for t = 4:

0 5 10

-5

0

5

-1.5 -1.0 -0.5 0.0 0.5-1.0

-0.5

0.0

0.5

1.0


Properties of Σt

Plot of T (λ) for values from 0 to 5


Mystery

How does the Brown measure of bt know about Σt?

Or: how does T (λ) arise?


One answer

Use conformal map ft of Σct to (support(νt))c :

ft(λ) = λet21+λ1−λ

Use large-N “Segal–Bargmann transform” to show Brown measure iszero outside Σt [H–Kemp, 2018]


PART 4: THE PDE AND ITS SOLUTION


The goal

Following def. of Brown meaure, set

S(t, λ, ε) := τ[log((bt − λ)∗(bt − λ) + ε)]

andst(λ) := lim

ε→0+S(t, λ, ε).

Brown measure is then1

4π∆st(λ).


The PDE

Theorem (Driver–H–Kemp)

The function S(t, λ, ε) satisfies the following PDE:

∂S

∂t= ε

∂S

∂ε

(1 + (|λ|2 − ε)

∂S

∂ε− a

∂S

∂a− b

∂S

∂b

), λ = a+ ib,

subject to the initial condition

S(0, λ, ε) = log(|λ− 1|2 + ε).

Want S(t, λ, 0) but PDE involves derivatives with respect to ε

Not correct (in general) to say ∂S/∂t = 0 when ε = 0


Hamilton–Jacobi method

“Hamiltonian” H read off from PDE:

H(a, b, ε, pa, pb, pε) = −εpε

(1 + (|λ|2 − ε)pε − apa − bpb

)Consider Hamilton’s equations

da

dt=

∂H

∂pa;

dpadt

= −∂H

∂a

and similarly for b, pb, ε, pε.

Miracle: In our case, these ODE’s can be solved explicitly!


Hamilton–Jacobi method

Then

S(t, λ(t), ε(t)) = log(|λ0 − 1|2 + ε0)

+ ε0t

(1

|λ0 − 1|2 + ε0

)2

+ log |λ(t)| − log |λ0|

RHS mostly in terms of initial conditions λ0 and ε0.

Variant of “method of characteristics”: find special curves alongwhich the solution can be computed


PART 5: LETTING ε TEND TO ZERO


Strategy

Given an arbitrary t and λ, try to find ε0 and λ0 so that

ε(t) = 0; λ(t) = λ.

Then get st(λ) := S(t, λ, 0) = function of ε0 and λ0

Then take Laplacian in λ


First attempt: ground assault

Ground assault: To get ε(t) ≈ 0, try ε0 ≈ 0

Then λ(t) ≈ λ0

st(λ) = log |λ− 1|2

st(λ) is independent of t

Result: If this works, Brown measure is zero!


Ground assault

The catch: Only works if λ is outside Σt !

If ε0 ≈ 0 and λ0 = λ then ε(t) is small and positive until time T (λ),when ε(t) ceases to exist

T(λ)t

ϵ0

ϵ(t)


Ground assault

Same T (λ) as in definition of Σt !

Answer to our “mystery”:

T (λ) is the lifetime of a path with ε0 ≈ 0 and λ0 ≈ λ.


Ground assault

Region {λ|t > T (λ)} = Σt is inaccessible to this simple approach.

Reproduces result that Brown measure is zero outside Σt

Ground assault cannot get into Σt !


Second attempt: aerial assault

Aerial assult: Must really embrace ε > 0 !

Choose ε0 > 0 and try to “land” at ε = 0 at time t with λ(t) = λ


Aerial assault

For (t, λ) with λ ∈ Σt , exist (in principle) ε0 and λ0 with ε(t) = 0and λ(t) = λ


Aerial assault

Formula is for (t, λ) in terms of (ε0, λ0)

Have to invert this relationship to get S(t, λ, 0)

Still have to take Laplacian w.r.t. λ


Results

Theorem (Driver–H–Kemp)

For points in Σt , the density Wt of the Brown measure has the followingform in polar coordinates:

Wt(r , θ) =1

r2wt(θ)

for a certain function wt .

Amazingly simple dependence on r !


The formula for wt

wt is determined by the geometry of Σt as follows:

wt(θ) =1

4π

(2

t+

∂

∂θ

2rt(θ) sin θ

rt(θ)2 + 1− 2rt(θ) cos θ

)where rt(θ) is “outer radius” of Σt :

1 2 3 4

-2

-1

1

2

θ

rt(θ)


The formula for wt

Use implicit differentiation on T (r , θ) = t to get:

wt(θ) =1

2πtω(rt(θ), θ)

for some explicit function ω(r , θ)


The formula for wt

The formula for ω:

ω(r , θ) = 1 + h(r)α(r) + β(r) cos θ

β(r) + α(r) cos θ,

where

h(r) = rlog(r2)

r2 − 1;

α(r) = (r2 + 1)h(r)− 2r ;

β(r) = r2 + 1− 2rh(r).


Plots of Wt

t = 1


Plots of Wt

t = 4


Plots of Wt

t = 4 detail


Simulation of eigenvalues

t = 2, Brown measure and eigenvalues


Consequence of main result

Corollary

The push-forward of the Brown measure under the complex log map

λ 7→ log |λ|+ iθ

has density in log(Σt) given by

wt(θ)

independent of log |λ| .


More simulations of eigenvalues

Eigenvalues for N = 2, 000 and their logarithms for t = 4.1

0 5 10

-5

0

5

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3


Connection to Biane’s measure

Recall νt—limiting eigenvalue distribution for Br. motion in U(N)

Recall map ft(λ) := λet21+λ1−λ —takes boundary of Σt to support of νt

Define Ft : Σt → S1 as ft on boundary, constant along radialsegments



Theorem

The push-forward of the Brown measure of bt under Ft is νt !

Plot of νt versus histogram of {Ft(λj )}Nj=1 for t = 2

-π -π

20

π

2π



Brown measure of bt is actually the unique measure on Σt with:(1) this push-forward property,(2) a density of the form 1

r2g(θ).


Concluding remarks

THANK YOU FOR YOUR ATTENTION!


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Eigenvalues of random matrices in the general linear group