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Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 1 / 30
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Page 1: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Fluctuations from the Semicircle LawLecture 1

Ioana Dumitriu

University of Washington

Women and Math, IAS 2014

May 20, 2014

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 1 / 30

Page 2: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

1 Review

2 Fluctuations

3 Calculation of the Variance

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 2 / 30

Page 3: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Review

Random (Real) Facts

Distributions are generalized functions, better understoodthrough the effect they have on functionsProbability distributions define random variables viacharacteristic functions:

X ∼ F if ∀[a, b] ∈ R , P[X ∈ [a, b]] = F([a, b]) .

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 3 / 30

Page 4: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Review

Random (Real) Facts

A probability distribution F may be given by a (positive) function(the probability density function) f with

∫R f (x)dx = 1. We say

dµ(x) = f (x)dx.

In this case P[X ∈ [a, b]] =∫ b

a f (x)dx.

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 4 / 30

Page 5: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Review

The Moment Method

A “nice” distribution is described by the collection of its moments,{E[Xk] , ∀k ∈ Z, k ≥ 0}. This leads to the “moment method”.

More on the relationship between the distribution and itsmoments in today’s Review Session.

Convergence of moments is weak convergence, i.e., convergencein distribution. Stronger: in probability and almost surely.

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 5 / 30

Page 6: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Review

Wigner Matrices

n× n real (or complex, quaternion) matrices W;symmetric: W = WT (or Hermitian W = W∗, self-dual W = WD);entries are independent up to symmetry (wij = wji);entries are identically distributed up to symmetry (all wij withi < j are equidistributed, resp. all wii are equidistributed);

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 6 / 30

Page 7: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Review

Wigner Matrices

the distributions have moments of all orders; in particular, E(Z4)is the 4th moment;all variables are centered (expectation 0) and all variances are 1(could also consider variance σ2 on the diagonal).

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 7 / 30

Page 8: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Review

Wigner Matrices

... actually, we consider the normalized Wigner matrices Wn = 1√n W.

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 8 / 30

Page 9: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Review

The Semicircle Law

n = 500; A = rand(n);A = (A + A′)/√

n;hist(eig(A))semicircle

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Semicircle law with n=500

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 9 / 30

Page 10: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Review

Convergence to the Semicircle

To understand convergence through experiments:Convergence in distribution: pick an n× n random Wignermatrix W, pick one of its eigenvalues at random. Repeat manytimes. Plot histogram.

Almost surely: pick a single matrix W, and plot a histogram of allits eigenvalues.

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 10 / 30

Page 11: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Review

LLN and CLT

1−D: let x1, x2, . . . , xn, . . . be independent samples from a distributionwith mean µ and variance σ2.

Law of Large Numbers: 1n

n∑i=1

xi − µ→ 0 as n→∞.

Central Limit Theorem:

n∑i=1

xi − nµ√

nσ∼ N(0, 1) .

What about matrices?

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 11 / 30

Page 12: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Review

LLN for Matrices

The Semicircle Law is akin to a Law of Large Numbers.Showed:

1nE(tr(Wk)) =

1nE

(n∑

i=1

λki

)→{

0, k odd,Ck/2, k even.

On the right hand side are the moments of the semicircle distribution,with density s(x) = 1

√4− x2.

Convergence of moments means that the expected distribution of arandom eigenvalue converges in distribution to the semicircle law.

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 12 / 30

Page 13: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Review

LLN for Matrices

What this implies immediately is that for all reasonable functionsf : [−2, 2]→ R,

1nE

(n∑

i=1

f(λi)

)→∫ 2

−2f(x)s(x)dx .

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 13 / 30

Page 14: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Review

Exact expressions for the distribution

The actual expected distributions can be computed, for theGOE/GUE/GSE, for any n. The expressions are not very complicated.

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Distribution of one random eigenvalue, Wigner case, n=1

Figure: Distribution of one random eigenvalue, n = 1

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 14 / 30

Page 15: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Review

Exact expressions for the distribution

The actual expected distributions can be computed, for theGOE/GUE/GSE, for any n. The expressions are not very complicated.

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Distribution of one random eigenvalue, Wigner case, n=2

Figure: Distribution of one random eigenvalue, n = 2

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 15 / 30

Page 16: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Review

Exact expressions for the distribution

The actual expected distributions can be computed, for theGOE/GUE/GSE, for any n. The expressions are not very complicated.

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Distribution of one random eigenvalue, Wigner case, n=6

Figure: Distribution of one random eigenvalue, n = 6

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 16 / 30

Page 17: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Review

Exact expressions for the distribution

The actual expected distributions can be computed, for theGOE/GUE/GSE, for any n. The expressions are not very complicated.

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Distribution of one random eigenvalue, Wigner case, n=100

Figure: Distribution of one random eigenvalue, n = 100

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 17 / 30

Page 18: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Fluctuations

Bumps = Fluctuations

The actual expected distributions can be computed, for theGOE/GUE/GSE, for any n. The expressions are not very complicated.

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Distribution of one random eigenvalue, Wigner case, n=1,2,6,100

n=1

n=2

n=6

n=100

Figure: Distribution of one random eigenvalue, n = 1, 2, 6, 100

How can we compute the fluctuation?Compute moments more carefully.

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 18 / 30

Page 19: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Fluctuations

Bumps = Fluctuations

The actual expected distributions can be computed, for theGOE/GUE/GSE, for any n. The expressions are not very complicated.

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Distribution of one random eigenvalue, Wigner case, n=1,2,6,100

n=1

n=2

n=6

n=100

Figure: Distribution of one random eigenvalue, n = 1, 2, 6, 100

How can we compute the fluctuation?Compute moments more carefully.

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 19 / 30

Page 20: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Fluctuations

Format of CLT

Recall that in 1−D,

n∑i=1

Xi − nµ√

nσ=

n∑i=1

Xi − E(

n∑i=1

Xi

)√

nσ→ N(0, 1)

• For Wigner matrices we will have something similar:

n∑i=1

f (λi)− E(

n∑i=1

f (λi)

)σf

→ N(0, 1) ,

provided that f is “smooth enough”.

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 20 / 30

Page 21: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Fluctuations

Format of CLT

Recall that in 1−D,

n∑i=1

Xi − nµ√

nσ=

n∑i=1

Xi − E(

n∑i=1

Xi

)√

nσ→ N(0, 1)

For Wigner matrices we will have something similar:

n∑i=1

f (λi)− E(

n∑i=1

f (λi)

)σf

→ N(0, 1) ,

provided that f is “smooth enough”.

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 21 / 30

Page 22: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Fluctuations

Format of CLT

Start with the simplest smooth functions, f (x) = xk; must then showthat for a Wigner real matrix Wn

Xn,k := tr(Wkn)− E(tr(Wk

n)) ,

has the property that

Xn,k√Var(Xn,k)

→ N(0, 1) ,

where N(0, 1) is the standard normal variable, and convergence is indistribution.

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 22 / 30

Page 23: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Calculation of the Variance

Variance

Need to show that all moments of Xn,k/√

Var(Xn,k) converge to thoseof the standard normal variable.

The first step is to calculate the variance

Var(Xn,k) = E((

tr(Wkn))2)−(E(tr(Wk

n)))2

.

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 23 / 30

Page 24: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Calculation of the Variance

Expansion of the trace power

Writetr(Wk

n) =∑I∈I

wI ,

whereI := {I = (i1, i2, . . . , ik), | 1 ≤ i1, i2, . . . , ik ≤ n} ,

that is, ordered k-tuples; we also use the notation

wI = wi1i2wi2i3 . . .wiki1 .

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 24 / 30

Page 25: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Calculation of the Variance

Expansion of the trace power

Then

Var(Xn,k) = E((

tr(Wkn))2)−(E(tr(Wk

n)))2

=∑I,J∈I

E(wIwJ

)− E(wI)E(wJ) .

To each I ∈ I there corresponds a graph GI, with vertex labels{i1, . . . , ik}, with v vertices and e edges, having an edge betweenvertices ij and il if they occur consecutively in I (loops are ok).

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 25 / 30

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Calculation of the Variance

Graphs and labels

Consider the graph which is the union of the two graphscorresponding to I and J (for a given I, J).

In the walks corresponding to I and J, edges may be repeated;loops are also possible.Total # of edges in the walks, with multiplicities, = 2k.Enough to consider the case when the graph is connected;otherwise wI, wJ independent and

E(wIwJ)− E(wI)E(wJ) = 0 .

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 26 / 30

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Calculation of the Variance

Graphs and labels

If any edge has multiplicity 1 in the union of the walks,E(wIwJ) = 0 = E(wI)E(wJ).... therefore only need to consider walks where all edges arerepeated.

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 27 / 30

Page 28: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Calculation of the Variance

Graphs and labels

SoTotal # of edges in the union of walks with multiplicities = 2k,every edge repeated.The graph is connected.So total # of actual edges e ≤ k and e ≥ v− 1, v ≤ k + 1.

Given i1, i2, . . . , ik, j1, j2, . . . , jk, the total number of such graphs isindependent of n.Asymptotics are given by those graphs for which v is as large aspossible.

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 28 / 30

Page 29: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Calculation of the Variance

First Attempt: v = k + 1

No such terms are relevant.

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 29 / 30

Page 30: Fluctuations from the Semicircle Law Lecture 1Fluctuations from the Semicircle Law Lecture 1 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 20, 2014 Ioana Dumitriu

Calculation of the Variance

First Attempt: v = k + 1

Must have v = k + 1, e = v− 1; so join graph is a tree on whicheach edge is walked on twice.Hence the two closed walks that form it are trees on which eachedge is walked on twice.No edge overlap, so wI is independent from wJ.Term contributes 0 to covariance.

Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, 2014 30 / 30