Fluctuations from the Semicircle Law Lecture 1
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1 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, / 30
2 1 Review 2 Fluctuations 3 Calculation of the Variance Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
3 Review Random (Real) Facts Distributions are generalized functions, better understood through the effect they have on functions Probability distributions define random variables via characteristic 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, / 30
4 Random (Real) Facts Review 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, / 30
5 The Moment Method Review A nice distribution is described by the collection of its moments, {E[X k ], k Z, k 0}. This leads to the moment method. More on the relationship between the distribution and its moments in today s Review Session. Convergence of moments is weak convergence, i.e., convergence in distribution. Stronger: in probability and almost surely. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
6 Review Wigner Matrices n n real (or complex, quaternion) matrices W; symmetric: W = W T (or Hermitian W = W, self-dual W = W D ); entries are independent up to symmetry (w ij = w ji ); entries are identically distributed up to symmetry (all w ij with i < j are equidistributed, resp. all w ii are equidistributed); Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
7 Review Wigner Matrices the distributions have moments of all orders; in particular, E(Z 4 ) is the 4th moment; all variables are centered (expectation 0) and all variances are 1 (could also consider variance σ 2 on the diagonal). Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
8 Wigner Matrices Review... actually, we consider the normalized Wigner matrices W n = 1 n W. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
9 The Semicircle Law Review n = 500; A = rand(n); A = (A + A )/ n; hist(eig(a)) semicircle Semicircle law with n= Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
10 Review Convergence to the Semicircle To understand convergence through experiments: Convergence in distribution: pick an n n random Wigner matrix W, pick one of its eigenvalues at random. Repeat many times. Plot histogram. Almost surely: pick a single matrix W, and plot a histogram of all its eigenvalues. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
11 LLN and CLT Review 1 D: let x 1, x 2,..., x n,... be independent samples from a distribution with mean µ and variance σ 2. n Law of Large Numbers: 1 n x i µ 0 as n. Central Limit Theorem: What about matrices? i=1 n x i nµ N(0, 1). nσ i=1 Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
12 LLN for Matrices Review The Semicircle Law is akin to a Law of Large Numbers. Showed: ( n ) 1 n E(tr(Wk )) = 1 { n E λ k 0, k odd, i C k/2, k even. i=1 On the right hand side are the moments of the semicircle distribution, with density s(x) = 1 2π 4 x 2. Convergence of moments means that the expected distribution of a random eigenvalue converges in distribution to the semicircle law. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
13 LLN for Matrices Review What this implies immediately is that for all reasonable functions f : [ 2, 2] R, ( n ) 1 2 n E f(λ i ) f(x)s(x)dx. i=1 2 Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
14 Review Exact expressions for the distribution The actual expected distributions can be computed, for the GOE/GUE/GSE, for any n. The expressions are not very complicated. 0.5 Distribution of one random eigenvalue, Wigner case, n= Figure: Distribution of one random eigenvalue, n = 1 Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
15 Review Exact expressions for the distribution The actual expected distributions can be computed, for the GOE/GUE/GSE, for any n. The expressions are not very complicated Distribution of one random eigenvalue, Wigner case, n= Figure: Distribution of one random eigenvalue, n = 2 Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
16 Review Exact expressions for the distribution The actual expected distributions can be computed, for the GOE/GUE/GSE, for any n. The expressions are not very complicated Distribution of one random eigenvalue, Wigner case, n= Figure: Distribution of one random eigenvalue, n = 6 Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
17 Review Exact expressions for the distribution The actual expected distributions can be computed, for the GOE/GUE/GSE, for any n. The expressions are not very complicated. 0.5 Distribution of one random eigenvalue, Wigner case, n= Figure: Distribution of one random eigenvalue, n = 100 Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
18 Fluctuations Bumps = Fluctuations The actual expected distributions can be computed, for the GOE/GUE/GSE, for any n. The expressions are not very complicated Distribution of one random eigenvalue, Wigner case, n=1,2,6,100 n=1 n=2 n=6 n= 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, / 30
19 Fluctuations Bumps = Fluctuations The actual expected distributions can be computed, for the GOE/GUE/GSE, for any n. The expressions are not very complicated Distribution of one random eigenvalue, Wigner case, n=1,2,6,100 n=1 n=2 n=6 n= 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, / 30
20 Format of CLT Fluctuations Recall that in 1 D, n X i nµ i=1 = nσ n i=1 ( n ) X i E X i i=1 N(0, 1) nσ For Wigner matrices we will have something similar: ( n n ) f (λ i ) E f (λ i ) i=1 i=1 N(0, 1), σ f provided that f is smooth enough. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
21 Format of CLT Fluctuations Recall that in 1 D, n X i nµ i=1 = nσ n i=1 ( n ) X i E X i i=1 N(0, 1) nσ For Wigner matrices we will have something similar: ( n n ) f (λ i ) E f (λ i ) i=1 i=1 N(0, 1), σ f provided that f is smooth enough. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
22 Format of CLT Fluctuations Start with the simplest smooth functions, f (x) = x k ; must then show that for a Wigner real matrix W n has the property that X n,k := tr(w k n) E(tr(W k n)), X n,k Var(Xn,k ) N(0, 1), where N(0, 1) is the standard normal variable, and convergence is in distribution. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
23 Calculation of the Variance Variance Need to show that all moments of X n,k / Var(X n,k ) converge to those of the standard normal variable. The first step is to calculate the variance ( ( ) ) 2 ( 2 Var(X n,k ) = E tr(wn) k E(tr(Wn))) k. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
24 Calculation of the Variance Expansion of the trace power Write tr(w k n) = I I w I, where I := {I = (i 1, i 2,..., i k ), 1 i 1, i 2,..., i k n}, that is, ordered k-tuples; we also use the notation w I = w i1 i 2 w i2 i 3... w ik i 1. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
25 Calculation of the Variance Expansion of the trace power Then ( ( ) ) 2 ( ) 2 Var(X n,k ) = E tr(wn) k E(tr(Wn)) k = E ( ) w I w J E(wI )E(w J ). I,J I To each I I there corresponds a graph G I, with vertex labels {i 1,..., i k }, with v vertices and e edges, having an edge between vertices i j and i l if they occur consecutively in I (loops are ok). Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
26 Calculation of the Variance Graphs and labels Consider the graph which is the union of the two graphs corresponding 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 w I, w J independent and E(w I w J ) E(w I )E(w J ) = 0. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
27 Calculation of the Variance Graphs and labels If any edge has multiplicity 1 in the union of the walks, E(w I w J ) = 0 = E(w I )E(w J ).... therefore only need to consider walks where all edges are repeated. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
28 Calculation of the Variance Graphs and labels So Total # 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 i 1, i 2,..., i k, j 1, j 2,..., j k, the total number of such graphs is independent of n. Asymptotics are given by those graphs for which v is as large as possible. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
29 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, / 30
30 Calculation of the Variance First Attempt: v = k + 1 Must have v = k + 1, e = v 1; so join graph is a tree on which each edge is walked on twice. Hence the two closed walks that form it are trees on which each edge is walked on twice. No edge overlap, so w I is independent from w J. Term contributes 0 to covariance. Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 1 May 20, / 30
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