Asymptotics of beta-Hermite Ensembles

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Asymptotics of β-Hermite Ensembles

Department of Mathematics, Linköping University Filip Berglund

LiTH-MAT-EX–2020/10–SE

Credits: 16 hp Level: G2

Supervisor: Xiangfeng Yang,

Department of Mathematics, Linköping University Examiner: Martin Singull,

Department of Mathematics, Linköping University Linköping: November 2020

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Abstract

In this thesis we present results about some eigenvalue statistics of the β-Hermite ensembles, both in the classical cases corresponding to β = 1, 2, 4, that is the Gaussian orthogonal ensemble (consisting of real symmetric matrices), the Gaussian unitary ensemble (consisting of complex Hermitian matrices) and the Gaussian symplectic ensembles (consisting of quaternionic self-dual matri-ces) respectively. We also look at the less explored general β-Hermite ensembles (consisting of real tridiagonal symmetric matrices).

Specifically we look at the empirical distribution function and two different scalings of the largest eigenvalue. The results we present relating to these statis-tics are the convergence of the empirical distribution function to the semicircle law, the convergence of the scaled largest eigenvalue to the Tracy-Widom distri-butions, and with a different scaling, the convergence of the largest eigenvalue to 1. We also use simulations to illustrate these results.

For the Gaussian unitary ensemble, we present an expression for its level density. To aid in understanding the Gaussian symplectic ensemble we present properties of the eigenvalues of quaternionic matrices.

Finally, we prove a theorem about the symmetry of the order statistic of the eigenvalues of the β-Hermite ensembles.

Keywords:

β-Hermite ensembles, Gaussian ensembles, empirical distribution function, level density, largest eigenvalue, order statistic, Sturm sequence, Hermite polynomials

URL for electronic version:

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-171096

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Sammanfattning

I denna kandidatuppsats presenterar vi resultat om några olika egenvärdens-statistikor från β-Hermite ensemblerna, först i de klassiska fallen då β = 1, 2, 4, det vill säga den gaussiska ortogonala ensemblen (bestående av reella sym-metriska matriser), den gaussiska unitära ensemblen (bestående av komplexa hermitiska matriser) och den gaussiska symplektiska ensemblen (bestående av kvaternioniska själv-duala matriser). Vi tittar även på de mindre undersökta generella β-Hermite ensemblerna (bestående av reella symmetriska tridiagonala matriser).

Specifikt tittar vi på den empiriska fördelningsfunktionen och två olika nor-meringar av det största egenvärdet. De resultat vi presenterar för dessa statisti-kor är den empiriska fördelningsfunktionens konvergens mot halvcirkel-fördelningen, det normerade största egenvärdets konvergens mot Tracy-Widom fördelningen, och, med en annan normering, största egenvärdets konvergens mot 1. Vi illu-strerar även dessa resultat med hjälp av simuleringar.

För den gaussiska unitära ensemblen presenterar vi ett uttryck för dess ni-våtäthet. För att underlätta förståelsen av den gaussiska symplektiska ensemb-len presenterar vi egenskaper hos egenvärdena av kvaternioniska matriser.

Slutligen bevisar vi en sats om symmetrin hos ordningsstatistikan av egen-värdena av β-Hermite ensemblerna.

Nyckelord:

β-Hermite ensemblerna, gaussiska ensemblerna, empiriska fördelningsfunk-tionen, nivåtäthet, största egenvärdet, ordningsstatiska, Sturmföljder, Her-mitepolynom

URL för elektronisk version:

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-171096

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Acknowledgements

I would like to thank my supervisor Xiangfeng, for helping me understand this rather difficult topic. All the teachers whose courses I have taken, thank you for the efforts you take in making the content comprehensible and engaging. My classmates, thank you for showing me the joy of exploring ideas with other people. My mother, my father and my brother, thank you, foremost for being amazing but also for showing up and helping me with practical things. Cecilia, thank you for your endless support and encouragement.

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Introduction

The main objective of this thesis is to survey the asymptotic behavior of the eigenvalues of β-Hermite ensembles. There are many different statistics of the eigenvalues one may study, and in this thesis we limit ourselves to studying the distribution of the largest eigenvalue and the empirical distribution function.

Though the subject we study in this thesis is interesting in its own right there are also some interesting connections to other areas of mathematics and physics. For instance, the β-Hermite ensembles are useful in the study of the Coulomb gas model, where 1/β is proportional to the temperature [1, Chapter 9]. The distribution of the largest eigenvalue shows up in many places see e.g. [2, 3], three examples of which are; the length of the longest increasing subsequence of random permutations [4], the distribution of the number of piles needed for patience sorting [2] and random tilings [3].

1.1 Aims

The aim of this thesis is to present a concise and clear overview of the asymptotic behavior of the eigenvalues of the β-Hermite ensembles. Using simulations we also aim to aid the reader in developing an intuition for these results.

1.2 Content

Chapter 1: Presents the definitions of the β-Hermite ensembles and gives def-initions of the relevant modes of convergence of random variables.

Chapter 2: Gives the definitions of the empirical distribution function, the level density and the largest eigenvalue. Chapter 2 also gives the results of the

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2 Chapter 1. Introduction

asymptotics of the empirical distribution function and the largest eigenvalue for the Gaussian orthogonal ensemble.

Chapter 3: Gives the results of the asymptotics of the empirical distribution function and the largest eigenvalue for the Gaussian unitary ensemble. This chapter also contains a presentation of the analytical level density.

Chapter 4: Starts with a presentation of the theory of eigenvalues of quater-nionic matrices, then goes on to give the results of the asymptotics of the empir-ical distribution function and the largest eigenvalue for the Gaussian symplectic ensemble.

Chapter 5: Gives the results of the asymptotics of the empirical distribution function and the largest eigenvalue for the general β-Hermite ensembles. This chapter also contains a proof of the symmetry of the order statistic of the β-Hermite ensembles.

Chapter 6: Is a short conclusion of what we have seen in the thesis.

1.3 Definitions of the β-Hermite Ensembles

Different authors use different definitions of the β-Hermite ensembles. These definitions do not differ substantially, but still sufficiently that we think it would be good to clarify how we, and the papers we cite, define the relevant matrix ensembles.

Definitions in [5]

In [5] the n×n Gaussian orthogonal ensemble is defined as the symmetric matrix A1= a1ij

1≤i,j≤n with

a1_ii∼ N (0, 1), a1_ij ∼ N (0, 1/2)for j > i

and all being independent. The n × n Gaussian unitary ensemble is defined as the hermitian matrix A2= a2ij

1≤i,j≤n with

a2_ii∼ N (0, 1), a2_ij ∼ N (0, 1/2) + iN (0, 1/2), for j > i

and all being independent. The n × n Gaussian symplectic ensemble is defined as the self-dual matrix A4= a4ij

1≤i,j≤n with

a4_ii∼ N (0, 1), a4

ij ∼ N (0, 1/2) + iN (0, 1/2) + jN (0, 1/2) + kN (0, 1/2), for j > i and all being independent. These ensembles are labeled by the indexes β = 1, 2, 4, respectively, indicating how many real components the elements of the

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1.3. Definitions of the β-Hermite Ensembles 3

matrices have. The names of these ensembles comes from the fact that the GOE, GUE and GSE are invariant under orthogonal, unitary and symplectic conjugations, respectively. These can be generalized to the β-Hermite ensembles (with β > 0), which are defined as the n × n real symmetric tridiagonal matrix

Aβ= 1 √ 2        N (0, 2) χ(n−1)β χ(n−1)β N (0, 2) χ(n−2)β ... ... ... χ2β N (0, 2) χβ χβ N (0, 2)        ,

where χ is the distribution of the square root of the χ2 _{distribution [5]. Let} ΛA_i , 1 ≤ i ≤ n be the eigenvalues of Aβ, then the joint probability density function of these eigenvalues is

fβ λA1, . . . , λ A n = C A nβexp ( − n X i=1 λA_i 2/2 ) Y 1≤i<j≤n λA_i − λA j β , where CA nβ is a normalization constant. Definitions in [6]

The β-Hermite ensembles Bβ are defined in [6] as Bβ=

r 2 βAβ. Let ΛB

i , 1 ≤ i ≤ n be the eigenvalues of Bβ, then ΛBi = q

2

βΛ

A

i and the joint p.d.f. of these eigenvalues is fβ λB1, . . . , λ B n = C B nβexp ( − n X i=1 β λB_i 2 /4 ) Y 1≤i<j≤n λB_i − λB j β . Definitions in [7]

This is the scaling which is used in this thesis. The β-Hermite ensembles Hβ are defined in [7] as Hβ= 1 √ 2Bβ= 1 √ βAβ.

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4 Chapter 1. Introduction

Let ΛH

i , 1 ≤ i ≤ n be the eigenvalues of Hβ, then ΛHi = 1 √ 2Λ B i = 1 √ βΛ A i and

the joint p.d.f. of these eigenvalues is fβ λH1 , . . . , λ H n = C H nβexp ( − n X i=1 β λH_i 2 /2 ) Y 1≤i<j≤n λH_i − λH j β .

For the eigenvalues ΛH

i and the normalization constants CnβH we will from here on simply write Λi and Cnβ.

1.4 Types of convergence

A note on a few different kinds of convergence. Let Xn, n ∈ N, be a sequence of random variables.

Definition 1(Convergence in distribution). We say that Xn converges in dis-tribution to the random variable X if

lim

n→∞P (Xn≤ t) = P (X ≤ t).

We write this as Xn d → X.

This is the weakest type of convergence, in the sense that the following types imply convergence in distribution. Therefore it is sometimes called weak convergence.

Definition 2(Convergence in probability). We say that Xn converges in prob-ability to the random variable X if ∀ε > 0,

P (|Xn− X| > ε) → 0, as n → ∞. We write this as Xn

p → X.

Definition 3 (Almost sure convergence). We say that Xn converges almost surely to the random variable X if

P ({ω : Xn(ω) → X(ω) as n → ∞}) = 1. We write this as Xn

a.s. → X.

These different types of convergence relate to each other as Xn a.s. → X ⇒ Xn p → X ⇒ Xn d → X.

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Chapter 2

The Gaussian Orthogonal

Ensemble

The simplest Gaussian ensemble is the Gaussian orthogonal ensemble. It is a symmetric random matrix with real gaussian entries. In this chapter we present results about some of the GOE’s eigenvalue statistics.

We recall the result about the joint probability density function of the eigen-values.

Theorem 1. The joint p.d.f. of the eigenvalues of H1 is given by

f1(λ1, . . . , λn) = Cn1exp ( − n X i=1 λ2_i/2 ) Y 1≤i<j≤n |λi− λj| .

2.1 The Empirical Distribution

The first statistic of the eigenvalues we will consider is the empirical distribution function.

Definition 4. Let Λ1, . . . , Λn be the eigenvalues of a matrix H then the empir-ical distribution function of the eigenvalues of H is defined as

1 n n X i=1 I[Λi,∞)(t),

where I is the indicator function.

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6 Chapter 2. The Gaussian Orthogonal Ensemble

Note that the empirical distribution function is a random function.

Theorem 2(The semicircle law [5]). Let Fn(1)(t) denote the empirical distribu-tion funcdistribu-tion of the eigenvalues of a matrix drawn from the Gaussian orthogonal ensemble. Then

F_n(1)(t√2n)−→ S(t), as n → ∞,a.s. where _dtdS(t) = _π2√1 − t2_I

[−1,1](t). That is, the p.d.f. of S is given by a

“semi-circle”.

An alternative view of this theorem is that the empirical distribution function of H1/

√

2n tends to the semicircle law. Furthermore, we note that Fn(1) con-verges to a degenerate random variable. So if we sample F(1)

n for large n we should, with high probability, get a function that is close to S(t). For small n however we may look at the expected empirical distribution function to get an idea of its appearance. This leads us to another definition.

Definition 5. The level density of the ensemble is defined as

ρn(t) = d dtE 1 n n X i=1 I[Λi,∞)(t √ 2n) ! . Since Fn(t √

2n) → S(t)and S(t) is deterministic we have E(Fn(t

√

2n)) → E(S(t)) = S(t).

This means that the level density converges to the p.d.f. of the semicircle law. We can verify that the empirical distribution function converges to the semicircle distribution by simulations in Matlab. The result is shown in Figure 2.1. Note that the number of eigenvalues in each plot is equal to n × #samples. (The purpose of the magenta curve in the figure will be made clear later.)

2.2 The Distribution of the Largest Eigenvalue

Another statistic of the eigenvalue distribution of interest is the largest eigen-value, specifically the asymptotics of the largest eigenvalue. We define

Λmax(n) := max {Λ1, . . . , Λn} .

For {Λmax(n)}∞n=1to converge as n → ∞ we need appropriate scaling, which is given in the following theorem by Craig A. Tracy and Harold Widom.

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2.2. The Distribution of the Largest Eigenvalue 7 -1 -0.5 0 0.5 1 t 0 0.2 0.4 0.6 Probability -1 -0.5 0 0.5 1 t 0 0.2 0.4 0.6 Probability -1 -0.5 0 0.5 1 t 0 0.2 0.4 0.6 Probability

Figure 2.1: Histogram for the derivative of the empirical distribution function of the GOE with different matrix and sample sizes. The upper and middle plots correspond to simulations of the level density whereas the bottom plot is a sample of the derivative of the empirical distribution function. The exact semicircle distribution in red and 1

n d

dtT W1 tn

−2/3_{/2 + 1}_{in magenta, which is} too small to be seen when n = 107_.

Theorem 3 (Tracy–Widom [7]). The asymptotic distribution of the scaled largest eigenvalue of the GOE is given by

T W1(t) := lim n→∞P Λmax(n) − √ 2n 2−1/2_n−1/6 ≤ t ! ,

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for β = 1. It is given explicitly by T W1(t) = exp −1 2 Z ∞ t q(x) + (x − t) q (x)2dx , where q is the solution to

(_d2_q

dt2 = tq + 2q

3_,

q(t) ∼ Ai(t) as t → ∞,

where Ai(x) = _π1R₀∞cos 1₃t3+ xt dt is the airy function and “q(t) ∼ Ai(t) as t → ∞” means that limt→∞q(t)/Ai(t) = 1.

We verify this theorem with simulations in Matlab. The result is shown in Figure 2.2.

2.3 The Law of Large Numbers

In this section we wish to study the distribution of the largest eigenvalue with the same scaling as in the result of the semicircle law, i.e., we wish to find the distribution of Λmax/

√

2n as n → ∞. From the semicircle law we can see that asymptotically almost every eigenvalue of H1/

√

2nis in the interval [−1, 1], where we expect the largest eigenvalue to be close to one. An attempt at finding the asymptotic distribution might start with the following rough estimation of the distribution of Λmax/

√ 2n. We observe that P Λmax√ (n) 2n ≤ t = P Λmax(n) − √ 2n 2−1/2_n−1/6 ≤ (t − 1)2n 2/3 ! ,

which can be loosely approximated by T W1 (t − 1)2n2/3

. Thus the contribu-tion of Λmax/

√

2nto the empirical distribution is about 1 nT W1 (t − 1)2n2/3. In Figure 2.1 we see 1 n d dtT W1 (t − 1)2n2/3

in magenta. From the figure it is reasonable to expect that Λmax/ √

2n → 1as n → ∞in some mode of convergence.

Theorem 4(LLN for the largest eigenvalue). Λmax(n)/ √

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2.3. The Law of Large Numbers 9 -6 -5 -4 -3 -2 -1 0 1 2 3 4 t 0 0.2 Probability -6 -5 -4 -3 -2 -1 0 1 2 3 4 t 0 0.2 Probability -6 -5 -4 -3 -2 -1 0 1 2 3 4 t 0 0.2 Probability -6 -5 -4 -3 -2 -1 0 1 2 3 4 t 0 0.2 Probability

Figure 2.2: The simulated distribution of the scaled largest eigenvalue of the GOE with different matrix sizes n and 106 _{samples. The Tracy–Widom} distri-bution in red.

Proof. We want to show that ∀ε, δ > 0 ∃N(ε) s.t. ∀n > N P Λmax(n) √ 2n − 1 > δ < ε.

Step 1. From the fact that T W1 is a probability distribution we have that ∃N1(ε)s.t. ∀n > N1(ε) PT W1> p 2N1δN 1/6 1 + PT W1< − p 2N1δN 1/6 1 < ε/2. (i) Step 2. From the convergence in law of the maximum eigenvalue we have that

lim n→∞P n1/6Λmax(n) − √ 2n≤ x= P (T W1≤ x)

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for any fixed x. Now take x =√2N1δN 1/6 1 , then ∃N2(ε)s.t. ∀n > N2(ε) Pn1/6Λmax(n) − √ 2n> x≤ P (T W1> x) + ε 4, (ii) and ∃N3(ε)s.t. ∀n > N3(ε) Pn1/6Λmax(n) − √ 2n≤ −x≤ P (T W1≤ −x) + ε 4. (iii) Step 3. Take N(ε) = max {N1(ε), N2(ε), N3(ε)}Then for any n > N(ε) P Λmax(n) √ 2n − 1 > δ = P Λmax√ (n) 2n > 1 + δ + P Λmax√ (n) 2n < 1 − δ = P Λmax(n) − √ 2n n−1/6 > √ 2nδn1/6 ! + P Λmax(n) − √ 2n n−1/6 < − √ 2nδn1/6 ! ≤ P Λmax(n) − √ 2n n−1/6 > √ 2N δN1/6 ! + P Λmax(n) − √ 2n n−1/6 < − √ 2N δN1/6 ! (ii)+(iii) ≤ PT W1> √ 2N δN1/6+ε 4+ P T W1< − √ 2N δN1/6+ε 4 (i) < ε. We illustrate this result in Figure 2.3, where it is now clearer to what degree T W1 (t − 1)2n2/3fits the distribution of Λmax/

√ 2n.

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2.3. The Law of Large Numbers 11 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 t 0 5 10 Probability 0.9988 0.999 0.9992 0.9994 0.9996 0.9998 1 1.0002 1.0004 1.0006 1.0008 t 0 500 1000 Probability 0.999997 0.999998 0.999999 1 1.000001 t 0 5 Probability 105 0.9999999997 0.9999999998 0.9999999999 1 1.0000000001 1.0000000002 t 0 5 Probability 109

Figure 2.3: The simulated distribution of Λmax(n)/ √

2n with different matrix sizes and 106_samples. d

dtT W1 (t − 1)2n

2/3

in red, which loosely approximates the distribution of Λmax(n)/

√ 2n.

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Chapter 3

The Gaussian Unitary

Ensemble

The complex case is in some ways simpler than both the real (GOE) and quater-nion (GSE) cases. However, since we do not derive the results presented in this chapter it will not be clear that this is so, apart from one extra result that we are able to present relating to the level density.

f2(λ1, . . . , λn) = Cn2exp ( − n X i=1 λ2_i ) Y 1≤i<j≤n |λi− λj|2.

3.1 The Empirical Distribution

As for the GOE we look at the asymptotics of the empirical distribution function. Theorem 6(The semicircle law [5]). Let Fn(2)(t) denote the empirical distribu-tion funcdistribu-tion of the eigenvalues of a matrix drawn from the Gaussian unitary ensemble. Then

F_n(2)(t√2n)a.s.→ S(t), as n → ∞, where _dtdS(t) =_π2√1 − t2_I

[−1,1](t).

As we saw in Figure 2.1 (and will also see in figures 3.1, 4.1 and 5.1) the semicircle law is a good fit for the empirical distribution function for large n.

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14 Chapter 3. The Gaussian Unitary Ensemble

For small n we may also want to have an expression for the level density. In fact, expressions for the level density of the GOE, GUE and GSE are known [1, p. 118-119:148-152:175-178]. However, the expressions for the level density of the GOE and GSE are rather complicated compared to that of the GUE, wherefore we do not look at these cases.

To give the expression of the level density of the GUE we need to define a few terms.

Definition 6. The Hermite polynomials are defined as Hn(x) = (−1)nex

2_/2 dn dxne

−x2_/2

, n = 0, 1, . . . . The first few of which are given by

H0(x) = 1, H1(x) = x, H2(x) = x2− 1, H3(x) = x3− 3x.

Definition 7. The oscillator wave functions are defined as ϕn(t) = (2π)−1/4(n!)−1/2e−x

2_/4 Hn(t). Theorem 7. The level density of H2(n)/

√ 2n is given by ρn(t) = 2 √ n N −1 X n=1 ϕn(2 √ nt)2.

We can verify that the empirical distribution function converges to the semicircle distribution by simulations in Matlab. We also plot ρn(t) for n = 10, 100. The result is shown in Figure 3.1.

3.2 The Distribution of the Largest Eigenvalue

The asymptotic distribution of the largest eigenvalue of the GUE is also known. Theorem 8 (Tracy–Widom [7]). The asymptotic distribution of the scaled largest eigenvalue of the GUE is given by

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3.2. The Distribution of the Largest Eigenvalue 15 -1 -0.5 0 0.5 1 t 0 0.2 0.4 0.6 Probability -1 -0.5 0 0.5 1 t 0 0.2 0.4 0.6 Probability -1 -0.5 0 0.5 1 t 0 0.2 0.4 0.6 Probability

Figure 3.1: Histogram for the derivative of the empirical distribution function of the GUE with different matrix and sample sizes. The upper and middle plots correspond to simulations of the level density whereas the bottom plot is a sample of the derivative of the empirical distribution function. For n = 10, 100 we have the exact level density in magenta. For n = 107 _{we have the} exact semicircle distribution in red.

this convergence is in distribution, and T W2 is the Tracy–Widom distribution for β = 2. It is given explicitly by

T W2(t) = exp − Z ∞ t (x − t) q (x)2dx , where q is the solution to

(_d2_q

dt2 = tq + 2q

3_,

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16 Chapter 3. The Gaussian Unitary Ensemble

We verify this theorem with simulations in Matlab. The result is shown in Figure 3.2. -6 -5 -4 -3 -2 -1 0 1 2 3 4 t 0 0.2 0.4 Probability -6 -5 -4 -3 -2 -1 0 1 2 3 4 t 0 0.2 0.4 Probability -6 -5 -4 -3 -2 -1 0 1 2 3 4 t 0 0.2 0.4 Probability -6 -5 -4 -3 -2 -1 0 1 2 3 4 t 0 0.2 0.4 Probability

Figure 3.2: The simulated distribution of the scaled largest eigenvalue of the GUE with different matrix sizes and 106 _{samples. The Tracy–Widom} distribu-tion in red.

3.3 The Law of Large Numbers

For the GUE we obtain the same result for the convergence of Λmax/ √

2nas in the case of the GOE, with the same proof.

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3.3. The Law of Large Numbers 17

Simulation results for theorem 9 are found in Figure 3.3. Here we can see that T W2 (t − 1)2n2/3 is a better approximation for Λ

(GU E) max (n)/ √ 2n than T W1 (t − 1)2n2/3 is for Λ(GOE) max / √ 2n. 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 t 0 10 Probability 0.9988 0.999 0.9992 0.9994 0.9996 0.9998 1 1.0002 1.0004 1.0006 1.0008 t 0 1000 Probability 0.999997 0.999998 0.999999 1 1.000001 t 0 5 Probability 105 0.9999999997 0.9999999998 0.9999999999 1 1.0000000001 1.0000000002 t 0 5 Probability 109

dtT W2 (t − 1)2n2/3

_{in red, which loosely approximates} the distribution of Λmax(n)/

√ 2n.

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Chapter 4

The Gaussian Symplectic

Ensemble

The quaternion case is a bit trickier than the real (GOE) and complex (GUE) cases. The reader might wonder what the eigenvalues of quaternion matrices could be and if there are always n of them. We begin this chapter by exploring eigenvalues of quaternionic matrices before moving on to the results about said eigenvalues.

4.1 Eigenvalues of Quaternionic Matrices

The following about quaternions and eigenvalues of quaternionic matrices is taken from [8, 9, 10]. Let us begin by defining what a quaternion is.

Definition 8 (Quaternion). Let H = R4 be a 4 dimensional vector space and let {1, i, j, k} be the basis elements. Define multiplication in H by

1i = i1 = i, 1j = j1 = j, 1k = k1 = k, i2_{= j}2_{= k}2_{= −1,} ₁2_{= 1,}

ij = −ji = k, jk = −kj = i, ki = −ik = j. Furthermore, if x, y, z ∈ H and λ, µ ∈ R then

x(y + z) = xy + xz, (x + y)z = xz + yz, (µx)(λy) = (µλ)(xy). An element x = x0+ x1i + x2j + x3k of H is called a quaternion.

Definition 9. Let x = x0+ x1i + x2j + x3k, y = y0+ y1i + y2j + y3k ∈ H. Then we define the following:

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20 Chapter 4. The Gaussian Symplectic Ensemble 1. Re(x) = x0, Im(x) = x1i + x2j + x3k, 2. |x| =px2 0+ x21+ x22+ x23∈ R, 3. ¯x = x0− x1i − x2j − x3k is the conjugate of x, 4. if x 6= 0 then x−1= ¯x/|x|2,

5. x is similar to y if x = u−1_{yu for some u ∈ H\{0}, notation x ∼ y,} 6. the equivalence class containing x is denoted by [x] = {y ∈ H : x ∼ y}. Theorem 10. x ∼ y if and only if Re(x) = Re(y) and |Im(x)| = |Im(y)|. Remark 1. If x ∈ H is such that Im(x) = 0 then [x] = {x}.

A first observation about eigenvalues of quaternionic matrices is that since quaternions do not commute, the left (right) eigenvalues defined as Ax = λx (Ax = xλ) are in general not the same.

Theorem 11. If λ ∈ H is a right eigenvalue of A ∈ Hn×n _{then so is every} λ0∈ [λ].

Theorem 11 implies some quaternionic matrices have infinitely many right eigenvalues. For example the right eigenvalues of

A =0 i j 0

are all quaternions λ ∈ [1/√2 + i/√2], see [9, p. 90-91]. In particular λ = 1/√2 ± i/√2 are the complex eigenvalues of A.

We note that a quaternionic matrix H can be written as a unique sum of two complex matrices, i.e., H = C1+ C2j. The right eigenvalues of H are equivalent to the eigenvalues of the complex matrix

CH :=

C1 C2 −C2 C1

in the following way; if λi ∈ C, i = 1, . . . , 2n are the eigenvalues of CH then S

i[λi]are the right eigenvalues of H.

Since a matrix H4 from the GSE is self-dual, i.e., H4 = H4∗ we have that C1= C1∗ and C2 = −C2T, where .∗ is the conjugate transpose and .

T _{is simply} the transpose. Therefore CH4 is Hermitian and has 2n real eigenvalues, which can be shown to be doubly degenerate. Thus H4 has n real right eigenvalues. Furthermore, since quaternions commute with real numbers the left and right eigenvalues coincide for the GSE.

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4.2. The Empirical Distribution 21

Note that CH4is a 2n×2n matrix. This is why the GSE is sometimes defined as these matrices, see e.g. [7].

Now that we have established that the eigenvalues are real we recall the result about the joint eigenvalue density.

f4(λ1, . . . , λn) = Cn4exp ( − n X i=1 2λ2_i ) Y 1≤i<j≤n |λi− λj|4.

4.2 The Empirical Distribution

As we did for the previous ensembles we also look at the asymptotics of the empirical distribution function for the GSE.

Theorem 13(The semicircle law [5]). Let Fn(4)(t) denote the empirical distribu-tion funcdistribu-tion of the eigenvalues of a matrix drawn from the Gaussian symplectic ensemble. Then Fn(4)(t √ 2n)a.s.→ S(t), as n → ∞, where _dtdS(t) = 2 π √ 1 − t2_I [−1,1](t).

We can verify that the empirical distribution function converges to the semi-circle distribution by simulations in Matlab. The result is shown in Figure 4.1.

4.3 The Distribution of the Largest Eigenvalue

Likewise for the GSE the distribution of the scaled largest eigenvalue converges when n → ∞.

Theorem 14 (Tracy–Widom [7]). The asymptotic distribution of the scaled largest eigenvalue of the GSE is given by

this convergence is in distribution, and T W4 is the Tracy–Widom distribution for β = 4. It is given explicitly by

T W4 t √ 2 =pT W2(t) cosh −1 2 Z ∞ t q(x)dx ,

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22 Chapter 4. The Gaussian Symplectic Ensemble -1 -0.5 0 0.5 1 t 0 0.2 0.4 0.6 Probability -1 -0.5 0 0.5 1 t 0 0.2 0.4 0.6 Probability -1 -0.5 0 0.5 1 t 0 0.2 0.4 0.6 Probability

Figure 4.1: Histogram for the derivative of the empirical distribution function of the GSE with different matrix and sample sizes. The upper and middle plots correspond to simulations of the level density whereas the bottom plot is a sample of the derivative of the empirical distribution function. The exact semicircle distribution in red.

where q is the solution to (_d2_q

dt2 = tq + 2q

3_,

q(t) ∼ Ai(t) as t → ∞.

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4.4. The Law of Large Numbers 23 -6 -5 -4 -3 -2 -1 0 1 2 3 4 t 0 0.5 Probability -6 -5 -4 -3 -2 -1 0 1 2 3 4 t 0 0.5 Probability -6 -5 -4 -3 -2 -1 0 1 2 3 4 t 0 0.5 Probability -6 -5 -4 -3 -2 -1 0 1 2 3 4 t 0 0.5 Probability

Figure 4.2: The simulated distribution of the scaled largest eigenvalue of the GSE with different matrix sizes and 106 _{samples. The Tracy–Widom} distribu-tion in red.

4.4 The Law of Large Numbers

For the GSE we obtain the same result for the convergence of Λmax/ √

2nas in the case of the previous ensembles, with the same proof.

2n → 1 in probabil-ity.

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24 Chapter 4. The Gaussian Symplectic Ensemble 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 t 0 10 20 Probability 0.9988 0.999 0.9992 0.9994 0.9996 0.9998 1 1.0002 1.0004 1.0006 1.0008 t 0 1000 2000 Probability 0.999997 0.999998 0.999999 1 1.000001 t 0 5 10 Probability 105 0.9999999997 0.9999999998 0.9999999999 1 1.0000000001 1.0000000002 t 0 5 10 Probability 109

dtT W4 (t − 1)2n

2/3

in red, which loosely approximates the distribution of Λmax(n)/

√ 2n.

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Chapter 5

The β−Hermite Ensembles

As stated in Chapter 1 the β−Hermite ensembles is a generalization of the Gaussian orthogonal, unitary and symplectic ensembles. It has the same joint eigenvalue distribution as the G(O/U/S)E when β = 1, 2, 4 respectively [11]. The β−Hermite ensemble representation of the the aforementioned ensembles is what allows us to not only store but also to compute the eigenvalues of large matrices. Specifically, it allows us to utilize the theory of Sturm sequences, see e.g. [12]. Not as much is known about the general β-Hermite ensembles as for the β = 1, 2, 4 cases, but some of the same results are known.

Theorem 16. The joint p.d.f. of the eigenvalues of Hβ is given by

fβ(λ1, . . . , λn) = Cn,βexp ( − n X i=1 βλ2_i/2 ) Y 1≤i<j≤n |λi− λj| β .

5.1 The Empirical Distribution

The semicircle law is one result that remains.

Theorem 17(The semicircle law [5]). Let Fn(β)(t) denote the empirical distribu-tion funcdistribu-tion of the eigenvalues of a matrix drawn from the β-Hermite ensemble. Then F_n(β)(t√2n)a.s.→ S(t), as n → ∞, where _dtdS(t) = 2 π √ 1 − t2_I [−1,1](t). Berglund, 2020. 25

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26 Chapter 5. The β−Hermite Ensembles

Likewise for the β-Hermite ensembles we can verify that the empirical dis-tribution function converges to the semicircle disdis-tribution by simulations in Matlab. However, we need to choose a specific value of β. Since β = 1, 2, 4 corresponds to the ensembles we looked at in previous chapters we pick an ar-bitrary value different from 1, 2, 4, say β = π2_/6_{, which will be used for our} simulations in this chapter. The result is shown in Figure 5.1.

-1 -0.5 0 0.5 1 t 0 0.2 0.4 0.6 Probability -1 -0.5 0 0.5 1 t 0 0.2 0.4 0.6 Probability -1 -0.5 0 0.5 1 t 0 0.2 0.4 0.6 Probability

Figure 5.1: Histogram for the derivative of the empirical distribution function of the β-Hermite ensemble with β = π2_/6_{, and different matrix and sample} sizes. The upper and middle plots correspond to simulations of the level density whereas the bottom plot is a sample of the derivative of the empirical distribu-tion funcdistribu-tion. The exact semicircle distribudistribu-tion in red.

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5.2. The Distribution of the Largest Eigenvalue 27

5.2 The Distribution of the Largest Eigenvalue

The scaled largest eigenvalue does converge and with the same scaling as for the G(O/U/S)E.

Theorem 18(Convergence of the largest eigenvalue [6]). The asymptotic distri-bution of the scaled largest eigenvalue of the β-Hermite ensembles exists ∀β > 0 and is given by T Wβ(t) := lim n→∞P Λmax(n) − √ 2n 2−1/2_n−1/6 ≤ t ! .

5.3 The Law of Large Numbers

We can also look at the convergence of Λmax(n)/ √

2n for the β−Hermite en-sembles, which display the same behavior as that of the G(O/U/S)E. The same proof as in Chapter 2 works here too.

2n → 1 in probabil-ity.

Here we try to give a rough approximation of the T Wpi2_/6 distribution. By linear interpolation between T W1 and T W2 we may obtain a distribution that hints at the real T Wπ2_/6distribution. This method works about as well as you would expect, that is, not at all for general β. However, for β = π2_/6 _this method is sufficient for our illustrative purposes. In Figure 5.3 we have plotted

d dtµT W1 (t − 1)2n2/3+ d dt(1 − µ)T W2 (t − 1)2n2/3, where µ = 2 − π2_/6_{, i.e., the solution to µ1 + (1 − µ)2 = π}2_/6_.

5.4 The Order Statistic of the β-Hermite

Ensem-bles

Thus far, the only order statistic we have looked at is the largest eigenvalue. Because X ∼ N(0, σ2₎_{is even (i.e., −X}_{= X}d _{) it is reasonable to expect that for} the G(O/U/S)E we might have that

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28 Chapter 5. The β−Hermite Ensembles -6 -5 -4 -3 -2 -1 0 1 2 3 4 t 0 0.2 Probability -6 -5 -4 -3 -2 -1 0 1 2 3 4 t 0 0.2 0.4 Probability -6 -5 -4 -3 -2 -1 0 1 2 3 4 t 0 0.2 0.4 Probability -6 -5 -4 -3 -2 -1 0 1 2 3 4 t 0 0.2 0.4 Probability

Figure 5.2: The simulated distribution of the scaled largest eigenvalue of the β-Hermite Ensemble with β = π2_/6_{, different matrix sizes and 10}6 _samples. where Λmin:= min {Λ1, . . . , Λn}. For the β-Hermite ensembles it is not obvious that this should hold, but the symmetry of the semicircle law is a hint that this may also hold for the β-Hermite ensembles. In fact, a more general statement than this holds.

Theorem 20. Let Λ(1:n)≤ · · · ≤ Λ(n:n) be the order statistic of the eigenvalues of Hβ. Then for β > 0 we have that

P (Λ(r:n)< t) = P (−Λ(n−r+1:n)< t).

Proof. First we show that

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5.4. The Order Statistic of the β-Hermite Ensembles 29 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 t 0 10 Probability 0.9988 0.999 0.9992 0.9994 0.9996 0.9998 1 1.0002 1.0004 1.0006 1.0008 t 0 1000 Probability 0.999997 0.999998 0.999999 1 1.000001 t 0 5 Probability 105 0.9999999997 0.9999999998 0.9999999999 1 1.0000000001 1.0000000002 t 0 5 Probability 109

Figure 5.3: Law of Large Numbers for the β-Hermite ensemble with β = π2_/6_, different matrix sizes and 106 _{samples. The approximated and scaled T W}

β distribution that loosely approximates the distribution of Λmax(n)/

√

2n in red.

where F(r:m)(t)is the distribution of the r:th order statistic of a sample of size m obtained by dropping n − m eigenvalues from the full spectrum. Note that which eigenvalues are dropped does not matter since the eigenvalues are equally distributed. The general order statistic F(r:m)(t) is difficult to express as an integral but the first order statistic is simple, and is given by

F(1:j)(t) = 1 − P (Λ1> t, . . . , Λj > t) = 1 − Z ∞ t · · · Z ∞ t | {z } j Z ∞ −∞ · · · Z ∞ −∞ | {z } n−j f (λ1, . . . , λn) dλ1dλ2· · · dλn.

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30 Chapter 5. The β−Hermite Ensembles e λi= −λi, i = 1, . . . , n, we get F(1:j)(t) = 1 − Z −∞ −t · · · Z −∞ −t | {z } j Z −∞ ∞ · · · Z −∞ ∞ | {z } n−j f (eλ1, . . . , eλn)(−1)ndeλ1· · · deλn.

Renaming ˜λ with λ we get

F(1:j)(t) = 1 − Z −t −∞ · · · Z −t −∞ Z ∞ −∞ · · · Z ∞ −∞ f (λ1, . . . , λn)dλ1· · · dλn = 1 − P (λ1< −t, . . . , λj< −t) = 1 − F(j:j)(−t). From [13] we have that

P (Λ(r:n)< t) = n X j=r (−1)j+rj − 1 r − 1 n j F(j:j)(t) (ii) = n X j=n−r+1 (−1)j+n−r+1 j − 1 n − r n j F(1:j)(t).

And from (i) we thus get

P (Λ(r:n)< t) = n X j=n−r+1 (−1)j+n−r+1 j − 1 n − r n j 1 − F(j:j)(−t) .

By the change of variableset = −tand_er = n − r + 1we get

P (Λ(r:n)< t) = n X j=_er (−1)j+er j − 1 e r − 1 n j 1 − F(j:j)(et) (ii) = n X j=_er (−1)j+er j − 1 e r − 1 n j − P (Λ(er:n) <et).

Finally, since P (Λ(r:n) < t) + P (Λ(n−r+1:n) < −t) → 1 when t → ∞ we have

that _n X j=r_e (−1)j+er j − 1 e r − 1 n j = 1. Thus P (Λ(r:n)< t) = P (−Λ(n−r+1:n)< t).

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5.4. The Order Statistic of the β-Hermite Ensembles 31

Analogous results to those of the maximum eigenvalue can now be derived. Corollary 1. The asymptotic distribution of the scaled smallest eigenvalue of the β-Hermite ensemble exists ∀β > 0 and is given by

SEβ(t) := lim n→∞P Λmin(n) + √ 2n 2−1/2_n−1/6 ≤ t ! .

Corollary 2. For β = 1, 2, 4 we have that the smallest eigenvalue distributions are SEβ(t) = 1 − T Wβ(−t).

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Chapter 6

Conclusions and Final

Observations

In this thesis we have seen that the empirical distribution function of the β-Hermite ensembles converges almost surely to the semicircle law. The closely related level density of the GUE, which we defined as the expected empirical density function, can be computed using Hermite polynomials and from the simulations we could observe that the levels (i.e., Λ(i:n)) give rise to peaks in the level density. These peaks are more pronounced for larger values of β. For a study of the order statistic for large values of β see e.g. [5, Chapter 7].

Furthermore, we presented and illustrated the fact that the largest eigenvalue of the β-Hermite ensembles scaled as

Λmax(n) − √

2n 2−1/2_n−1/6

converges for all β > 0, and, as we have seen, in the cases when β = 1, 2, 4 this limiting distribution is known. Moreover, the largest eigenvalue of the β-Hermite ensembles scaled as

Λmax(n)/ √

2n converges to 1 in probability.

Finally we proved that the order statistic of the eigenvalues of the β-Hermite ensembles is symmetric, i.e.,

Λ(r:n)

d

= −Λ(n−r+1:n).

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34 Chapter 6. Conclusions and Final Observations

The proof of this only relies on the fact that the joint eigenvalue density is symmetric and thus holds for all random vectors X = (X1, . . . , Xn)where the components Xi are equally distributed, no independence is imposed on the components of X, and X d

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Bibliography

[1] M. L. Mehta. Random Matrices, volume 142 of Pure and applied mathe-matics. Elsevier, 3rd edition, 2004.

[2] P. Deift. Universality for mathematical and physical systems. 2006. URL https://arxiv.org/abs/math-ph/0603038v2.

[3] C. A. Tracy and H. Widom. The distributions of random matrix theory and their applications. In V. Sidoravičius, editor, New Trends in Mathematical Physics, pages 753–765, Dordrecht, 2009. Springer Netherlands. ISBN 978-90-481-2810-5.

[4] J. Baik, P. Deift, and K. Johansson. On the distribution of the length of the longest increasing subsequence of random permutations. Journal of the American Mathematical Society, 12(4):1119–1178, 1999. doi: 10.1090/ S0894-0347-99-00307-0.

[5] I. Dumitriu. Eigenvalue Statistics for Beta-Ensembles. PhD thesis, Mas-sachusetts Institute of Technology, 2003. URL http://hdl.handle.net/ 1721.1/29347.

[6] J. A. Ramírez, B. Rider, and B. Virág. Beta ensembles, stochastic airy spectrum, and a diffusion. Journal of the American Mathematical Society, 24(4):919–944, 2011. doi: 10.1090/s0894-0347-2011-00703-0.

[7] C. A. Tracy and H. Widom. The distribution of the largest eigenvalue in the gaussian ensembles: β = 1, 2, 4. In J. F. van Diejen and L. Vinet, editors, Calogero—Moser—Sutherland Models, pages 461–472. Springer, New York, NY, 2000.

[8] F. Zhang. Quaternions and matrices of quaternions. Linear Algebra and its Applications, 251:21–57, 1997. doi: 10.1016/0024-3795(95)00543-9.

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36 Bibliography

[9] R. Leiba. Topics in Quaternion Linear Algebra. Princeton Series in Applied Mathematics. Princeton University Press, 1st edition, 2014.

[10] F. Benaych-Georges and F. Chapon. Random right eigenvalues of gaussian quaternionic matrices. URL https://arxiv.org/pdf/1104.4455.pdf. [11] I. Dumitriu and A. Edelman. Matrix models for beta ensembles. Journal

of Mathematical Physics, 43(11):5830–5847, 2002. doi: 10.1063/1.1507823. [12] J. T. Albrecht, C. P. Chan, and A. Edelman. Sturm sequences and random eigenvalue distributions. Foundations of Computational Mathematics, 9(4): 461–483, 2009. doi: 10.1007/s10208-008-9037-x.

[13] N. Balakrishnan, S. M. Bendre, and H. J. Malik. General relations and identities for order statistics from non-independent non-identical variables. Annals of the Institute of Statistical Mathematics, 44(1):177–183, 1992. doi: 10.1007/BF00048680.

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Asymptotics of beta-Hermite Ensembles

Asymptotics of β-Hermite Ensembles

Abstract

Sammanfattning

Acknowledgements

Contents

Chapter 1

Introduction

1.1

Aims

1.2

Content

1.3

Definitions of the β-Hermite Ensembles

1.4

Types of convergence

Chapter 2

The Gaussian Orthogonal

Ensemble

2.1

The Empirical Distribution

2.2

The Distribution of the Largest Eigenvalue

2.3

The Law of Large Numbers

Chapter 3

The Gaussian Unitary

Ensemble

3.1

The Empirical Distribution

3.2

The Distribution of the Largest Eigenvalue

3.3

The Law of Large Numbers

Chapter 4

The Gaussian Symplectic

Ensemble

4.1

Eigenvalues of Quaternionic Matrices

4.2

The Empirical Distribution

4.3

The Distribution of the Largest Eigenvalue

4.4

The Law of Large Numbers

Chapter 5

The β−Hermite Ensembles

5.1

The Empirical Distribution

5.2

The Distribution of the Largest Eigenvalue

5.3

The Law of Large Numbers

5.4

The Order Statistic of the β-Hermite

Ensem-bles

Chapter 6

Conclusions and Final

Observations

Bibliography

Linköping University Electronic Press

Copyright

Upphovsrätt