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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
Brian Hall Eigenvalues of random matrices April 13, 2019 2 / 57
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
Brian Hall Eigenvalues of random matrices April 13, 2019 3 / 57
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
Brian Hall Eigenvalues of random matrices April 13, 2019 5 / 57
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
Brian Hall Eigenvalues of random matrices April 13, 2019 6 / 57
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
Brian Hall Eigenvalues of random matrices April 13, 2019 7 / 57
PART 2: BROWN MEASURE AND THE LARGE-NLIMIT
Brian Hall Eigenvalues of random matrices April 13, 2019 8 / 57
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)∗]}
Brian Hall Eigenvalues of random matrices April 13, 2019 9 / 57
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)
Brian Hall Eigenvalues of random matrices April 13, 2019 10 / 57
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
Brian Hall Eigenvalues of random matrices April 13, 2019 11 / 57
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)]}
Brian Hall Eigenvalues of random matrices April 13, 2019 12 / 57
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)
Brian Hall Eigenvalues of random matrices April 13, 2019 14 / 57
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 .
Brian Hall Eigenvalues of random matrices April 13, 2019 15 / 57
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
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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
Brian Hall Eigenvalues of random matrices April 13, 2019 17 / 57
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
Brian Hall Eigenvalues of random matrices April 13, 2019 19 / 57
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
Brian Hall Eigenvalues of random matrices April 13, 2019 20 / 57
Brownian motion in GL(N;C)
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
Brian Hall Eigenvalues of random matrices April 13, 2019 21 / 57
Brownian motion in GL(N;C)
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
Brian Hall Eigenvalues of random matrices April 13, 2019 22 / 57
Brownian motion in GL(N;C)
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
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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
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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}
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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
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Properties of Σt
Plot of T (λ) for values from 0 to 5
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Mystery
How does the Brown measure of bt know about Σt?
Or: how does T (λ) arise?
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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]
Brian Hall Eigenvalues of random matrices April 13, 2019 29 / 57
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).
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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
Brian Hall Eigenvalues of random matrices April 13, 2019 32 / 57
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!
Brian Hall Eigenvalues of random matrices April 13, 2019 33 / 57
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
Brian Hall Eigenvalues of random matrices April 13, 2019 34 / 57
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
Brian Hall Eigenvalues of random matrices April 13, 2019 36 / 57
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!
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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)
Brian Hall Eigenvalues of random matrices April 13, 2019 38 / 57
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.
Brian Hall Eigenvalues of random matrices April 13, 2019 39 / 57
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 !
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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
Brian Hall Eigenvalues of random matrices April 13, 2019 41 / 57
Aerial assault
For (t, λ) with λ ∈ Σt , exist (in principle) ε0 and λ0 with ε(t) = 0and λ(t) = λ
Brian Hall Eigenvalues of random matrices April 13, 2019 42 / 57
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
Brian Hall Eigenvalues of random matrices April 13, 2019 43 / 57
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 !
Brian Hall Eigenvalues of random matrices April 13, 2019 44 / 57
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)
Brian Hall Eigenvalues of random matrices April 13, 2019 45 / 57
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)
Brian Hall Eigenvalues of random matrices April 13, 2019 46 / 57
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).
Brian Hall Eigenvalues of random matrices April 13, 2019 47 / 57
Simulation of eigenvalues
t = 2, Brown measure and eigenvalues
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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| .
Brian Hall Eigenvalues of random matrices April 13, 2019 52 / 57
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
Brian Hall Eigenvalues of random matrices April 13, 2019 53 / 57
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
Brian Hall Eigenvalues of random matrices April 13, 2019 54 / 57
Connection to Biane’s measure
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
Brian Hall Eigenvalues of random matrices April 13, 2019 55 / 57
Connection to Biane’s measure
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).
Brian Hall Eigenvalues of random matrices April 13, 2019 56 / 57
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