A result on precise asymptotics for largest eigenvalues of β ensembles

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R E S E A R C H

Open Access

A result on precise asymptotics for largest

eigenvalues of

β

ensembles

Junshan Xie

1*

_{and Jing Zhao}

2

*_{Correspondence:}

[email protected] 1_{College of Mathematics and}

Information, Henan University, Kaifeng, 475000, P.R. China Full list of author information is available at the end of the article

Abstract

The paper focuses on the precise asymptotics of the largest eigenvalues of

β-Hermite

ensemble and

β-Laguerre ensemble. In particular, we obtain a general law on the

precise moment convergence rates for a type of weighted series constructed by the ﬁrst-order conditional moment of the largest eigenvalues of

β

ensembles. The results are motivated by the complete convergence for partial sums of independent random variables. The proofs depend on the small deviations for largest eigenvalue of

β

ensembles and non-asymptotic tail probability inequalities of the general

β

Tracy-Widom law.

MSC: 60F15; 60G50; 60G70

Keywords:

β

ensembles; largest eigenvalue; moment convergence rate; general

β

Tracy-Widom law

1 Introduction and main results

In random matrix theory,β-Hermite ensemble andβ-Laguerre ensemble introduced by Baker and Forrester [] are two classical ensembles, and they have played an important role in lattice gas theory and statistical mechanics. The ﬁrst one, denoted byHβ, is a probability

measure related to a random point process onR, whose joint probability density function has the form

P(λ,λ, . . . ,λn) =



Zn,β

≤j<k≤n

|λj–λk|βe–nβ

n

k=λk_, _(.)

whereλi∈R,Zn,βis a normalizing constant, and for the parameterβcan be taken anyβ>

 and it can be interpreted as the inverse temperature of a Coulomb gas with logarithmic potential in lattice gas theory. In particular, whenβ= , , , the joint densities are shared by the eigenvalues of Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE), and Gaussian symplectic ensemble (GSE), respectively.

For the Gaussian ensembles, because of the facts that there exist corresponding explicit random matrix models and that some powerful analytical methods are available in the derivation of their exact or asymptotical properties, lots of limiting spectral properties, including the behavior of the extremal eigenvalue, have been discussed in depth. As one of the major achievements in random matrix theory, it has been shown by Tracy and Widom [–] that the distribution of the largest eigenvalue ofG(O/U/S)E, after being centered and scaled properly, converges to the Tracy-Widom type distribution, which is an important

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probability distribution and has aroused great interest of the mathematicians and statisti-cians. According to the results of Tracy and Widom, for the largest eigenvalueλmax(A) of

the random matrixA, whenever it comes fromGOE(β= ),GUE(β= ) orGSE(β= ), its distribution function

Fn,β(x) =P

λmax(A)≤x

, β= , , ,

satisﬁes the following Tracy-Widom limit law:

lim

n→∞Fn,β

x

n

+ 

=Fβ(x), β= , , ,

which is given by

F(x) =exp

–

∞

x

(y–x)q(y)dy

,

F(x) =exp

– 

∞

x

q(y)dyF(x)_,

F

x √



=cosh

 

∞

x

q(y)dyF(x)

 _.

Hereq(x) is the unique solution to thePainlevé IIequationq=xq+ q_{with the}

asymp-totics

q(x)∼  π

–__x–__exp

– x

 

asx→ ∞.

As for theβ-Hermite ensemble, where forβ>  can be taken any positive real value, a primary question is whether we can establish an internal relation with the random matrix model, whose eigenvalues obey the desired joint probability law. This problem was ﬁrst posed by Dumitriu and Edelman [], who successfully constructed a tridiagonal sparse random matrix model forHβ, and the eigenvalues of the matrix model obey the desired

joint density function. We usually also call the corresponding matrix model asHβ and

denote the largest eigenvalue of theβ-Hermite ensemble asλmax(Hβ). Making essential

use of the tridiagonal matrix model, Ramírezet al.[] proved thatλmax(Hβ), after being

centered and scaled, converges weakly to the generalβTracy-Widom lawTWβ(β> ),

that is,

n_λ_max₍_H_β_{) – } ⇒ _TW_β_, _β_{> .} _(.)

They deﬁne the generalβTracy-Widom law through a random variational principle:

TWβ

d

=sup

f∈L



√

β

∞



f(x)dB(x) –

∞



f(x)+xf(x)dx

,

wherex→B(x) is a standard Brownian motion,Lis the space of functionsf that vanish at the origin and satisfy_∞f₍_x₎_dx_{= ,}∞

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Ramírezet al.[] also obtained the tail properties of the distribution function ofTWβ.

LetFβ(x) =P(TWβ≤x) for anyβ>  and largea, and we have

Fβ(–a) =exp

–  βa

_{ +}_o_() _(.)

and

 –Fβ(a) =exp

– βa



_{ +}_o_()_. _(.)

In order to investigate the rates of the concentration for various eigenvalue statistics, some non-asymptotic results for certain random matrix ensembles have also been devel-oped. Ledoux and Rider [] presented some small deviation results related toλmax(Hβ),

speciﬁcally, for alln≥,  <ε≤, andβ≥, we have

Pλmax(Hβ)≥ +ε

≤Ce–

β

Cnε  

(.)

and

Pλmax(Hβ)≤ –ε

≤Ce–

β

Cn _ε_

, (.)

whereCandCare positive constants.

In this paper, we are interested in the precise asymptotics of the largest eigenvalues of βensembles. Now we will give some attention to the precise asymptotics of random vari-ables in classical probability theory. Let{X,Xn;n≥}be a sequence of i.i.d. random

vari-ables,Sn=

n

k=Xk. Letϕ(x) andf(x) be the positive functions deﬁned on [, +∞), which

satisfy∞_n_=ϕ(n) =∞,f(x)↑ ∞asx→ ∞. Since Hsu and Robbins [] introduced the con-cept of complete convergence, some researchers discussed the convergence of the series

∞

n=ϕ(n)P(|Sn| ≥εf(n)). Note that the sums tend to inﬁnity asε; it is interesting to

find the precise convergence rates when this occurs. This amounts to finding the appro-priate normalizations in terms of the functions ofε, which, multiplied by the series, has a non-trivial limit. The researchers call the discussion in this field ‘precise asymptotics’. Later, some researchers extended the discussion to the cases that the series is constructed by conditional moment related toSn, or the series is constructed by other variables, such

as the associated counting process, empirical process, self-normalized sums. For more de-tails on the topic of precise asymptotics of independent random variables, one can refer to Hsu and Robbins [], Baum and Katz [], Chow [], Gut and Spˇataru [], Zhanget al.

[], Chen and Zhang [] and Zang and Huang [].

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Letx+=max{x, }and [x] be the integer part of real numberx. The main results can be listed as follows.

Theorem . Assume that g(x)is diﬀerentiable on the interval[, +∞), which is non-negative and strictly increasing to∞,and the diﬀerentiable function g(x)is non-negative.

We further suppose that g(x)is monotone,and if g(x)is monotone non-decreasing,we have

limx→∞g ₍_x_+)

g(x) = .Then for any s> ,β≥,we have

lim

ε(ε) 

s ∞

n=

g(n)En_λ

max(Hβ) – 

–εgs(n)₊= 

∞



xs_{ –}_F_β₍_x₎_dx_.

Theorem . Under the assumptions of Theorem.,we also have

lim

ε

– logε

∞

n= g(n)

g(n)E n



_λ_max₍_H_β_{) – }_–

εg

s₍_n₎

+

=  s

∞



 –Fβ(x)

dx.

Remark . The assumptions on the functiong(x) in the above theorems are all rather mild conditions, it is easy to see thatg(x) =xa_,_g₍_x_{) = (}_log_x₎b_or_g₍_x_{) = (}_{log log}_x₎c_{with some}

suitable conditions ofa> ,b>  orc>  and some others all satisfy these assumptions. In particular, wheng(x) =x_,_s_{= , it can lead to a result which is consistent with that of}

Xie [].

Remark . As it is diﬃcult to get the explicit expression ofFβ(x), we do not present the

exact values of the right hand of the above equations. However, the approximate values of them could be computed by Bornemann [].

There is another probability measure, related to a random point process onRwhose joint probability density function is

Pλ_,λ_, . . . ,λ_n= 

Z_n_,β

≤j<k≤n

λ_j–λ_kβ

n

i=

λ(

β

)(m–n+)–

i e–

nβ



n i=λi_,

whereλ_i∈R+,m>n– , andβ> ;Z_n_,_βis also a normalizing constant. It is usually called theβ-Laguerre ensemble and denotedLβfor short. Dumitriu and Edelman [] also

con-structed a tridiagonal sparse random matrix model for Lβ, and we usually also call the

corresponding matrix modelLβand denote the largest eigenvalue of theβ-Hermite

en-semble asλmax(Lβ). Ramírezet al.[] also obtained, form+  >n→ ∞, m_n→γ≥,

m_n

(√m+√n)

λmax(Lβ) – ( +√γ)

⇒ TWβ.

At the same time, Ledoux and Rider [] proved that for allcn≤m≤cnwithc≥c≥, β≥, and  <ε< , there existC> ,C> , and

P

λmax(Lβ)≥

 +

m

n 

( +ε)

≤Ce–

β

C_ √

mnε(√

ε∧(

m n)

 )

≤Ce–Cβ_nε

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and

P

λmax(Lβ)≤

 +

m

n 

( –ε)

≤Ce–

β

C_mnε(√ε∧(

m n)

 )

≤Ce–

β

Cn _ε

.

With the help of the two important results, we could reach a similar result on the β-Laguerre ensemble.

Theorem . Under the assumptions of Theorem.,when m= [γn], ≤γ <∞andβ≥ ,for any s> ,we have

lim

εε 

s ∞

n=

g(n)E m  _n

(√m+√n)

λmax(Lβ) – ( +√γ)

–εgs₍_n₎

+

=

∞



xs_{ –}_F_β₍_x₎_dx_.

Theorem . Under the assumptions of Theorem.,we have

lim

ε

 –logε

∞

n= g(n)

g(n)E

m_n

(√m+√n)

λmax(Lβ) – ( +√γ)

–εgs(n)

+

=

s

∞



 –Fβ(x)

dx. (.)

The main proofs will be obtained in the next section, and they mainly depend on the non-asymptotic exponential inequalities about the generalβTracy-Widom law, weak con-vergence, and small deviation inequalities about the largest eigenvalues of theβ ensem-bles. Throughout this paper, letCdenote an absolutely positive constant whose value can be diﬀerent from one appearance to another.

2 The proofs

Proof of Theorem. SetA(ε) = [g–₍_M_ε–_s_{)], where}_g–₍_x_{) is the inverse function of}_g₍_x₎

andMis an arbitrary positive real number. The proof of Theorem . relies on the follow-ing four propositions.

Proposition . Under the assumptions of Theorem.,we have

lim

εε 

s ∞

n=

g(n)ETWβ– εgs(n)

+=  –s

∞



xs_{ –}_F_β₍_x₎_dx_.

Proof Observing the fact that, for any random variableξanda∈R,

EξI{ξ≥a}=aP(ξ≥a) +

∞

a

P(ξ≥x)dx, (.)

thus we have

ETWβ– εgs(n)

+=

∞

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Ifg(x) is monotone non-increasing, by the assumption of Theorem ., we can see that

g(x)_∞_ε_gs₍_x₎P(TWβ≥t)dtis also non-increasing, thus

∞

 g(y)

∞

εgs₍_y₎P(TWβ≥t)dt dy

≤

∞

n=

+

≤

∞

 g(y)

∞

εgs₍_y₎

P(TWβ≥t)dt dy. (.)

If g(x) is monotone non-decreasing, by the assumption thatlim_n_→∞g(n+)

g(n) = , for any

 <δ< , there exists a suﬃcient large numberN=N(δ), such that g

₍_n_+)

g(n) <  +δand

g(n+)

g(n) >  –δfor alln≥N. Thus we can get

  +δ

∞

N

g(y)

∞

εgs₍_y₎

P(TWβ≥t)dt dy

≤

∞

n=N

+

≤ 

 –δ

∞

N– g(y)

∞

εgs₍_y₎P(TWβ≥t)dt dy. (.) Making a change of variables and performing integration by parts, for anyθ≥, we can deduce that

lim

εε 

s

∞

θ g(y)

∞

εgs₍_y₎P(TWβ≥t)dt dy =lim

ε



s –_s

∞

εgs₍_θ₎x



s–

∞

x

P(TWβ≥t)dt dx

=lim

ε



s –_s

∞

εgs₍_θ₎P(TWβ≥t)

t

εgs₍_θ₎x



s–_{dx dt}

=lim

ε –s

∞

εgs₍_θ₎

ts _{– (ε)}s_g_(θ₎_{ –}_F_β₍_t₎_dt

= –s

∞



ts_{ –}_F_β₍_t₎_dt_. _(.)

By (.)-(.) and the fact that

lim

εε 

s ∞

n=

+=_εlim_ε 

s ∞

n=N

+,

lettingδ→, we can complete the proof.

Proposition . Under the assumptions of Theorem.,for any M> ,we have

lim

εε 

s

n≤A(ε)

g(n)En_λ_max₍_H_β_{) – }_–_ε_gs₍_n₎ +–E

TWβ– εgs(n)

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Proof Using (.), we have

n:=sup y∈R

Pn_λ_max₍_H_β_{) – }≥_y_–_P₍_TW_β≥_y₎→_ _as_n→ ∞_. _(.)

By (.), it is easy to see that

En_λ_max₍_H_β_{) – }_–_ε_gs₍_n₎ +–E

TWβ– εgs(n)

+

=

∞



Pn_λ_max₍_H_β_{) – }≥_x_{+ ε}_gs₍_n₎_dx_–

∞



PTWβ≥x+ εgs(n)

dx

≤

∞



Pn_λ

max(Hβ) – 

≥x+ εgs(n)–PTWβ≥x+ εgs(n)dx

= (n+n), (.)

where

n= – n



Pn_λ

max(Hβ) – 

≥x+ εgs(n)–PTWβ≥x+ εgs(n)dx

and

n=

∞

– n

Pn_λ_max₍_H_β_{) – }≥_x_{+ ε}_gs₍_n₎_–_P_TW_β≥_x_{+ ε}_gs₍_n₎_dx_.

For the termn, using the relation (.), we can get

n≤n

–_

n =

 

n → asn→ ∞. (.)

For the termn, by (.) and (.), for largen, we obtain

PTWβ≥x+ εgs(n)

≤Ce–β(x+εgs(n))  

and

Pn_λ_max₍_H_β_{) – }≥_x_{+ }_n_ε≤_Ce–

βn C_(n

– _

x+εn– gs(n)) ≤Ce–C_βx

 

.

Thus

n≤C

∞

– n

e–Cβx  

dx+C

∞

– n

e–β(x+εgs(n))  

dx→ asn→ ∞. (.)

Note that

εs

n≤A(ε)

g(n)≤Cεs

A(ε) 

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by the Toeplitz lemma (see Loève [], p.), we can ﬁnd that

εs

n≤A(ε)

g(n)(n+n)→ asε.

Then the proof is completed.

Proposition . Under the assumptions of Theorem.,for uniformlyε> ,we have

lim

M→∞ε



s

n>A(ε)

+= .

Proof Following the same line as of the proof of Proposition ., we can deduce that

εs

n>A(ε)

+

=εs

n>A(ε) g(n)

∞

εgs₍_n₎

P(TWβ≥t)dt

≤Cεs

_∞

A(ε) g(y)

_∞

εgs₍_y₎P(TWβ≥t)dt dy

≤C

∞

εgs₍_A₍_ε₎₎x



s–

∞

x

P(TWβ≥t)dt dx

≤C

∞

MsP(TWβ≥t)

t

Msx



s–_{dx dt}

≤C

∞

Mst

 _e–βt

 

dt→ asM→ ∞,

where (.) is used again in the last inequality. The proof is completed.

Proposition . Under the assumptions of Theorem.,for uniformlyε> ,we have

lim

M→∞

n>A(ε)

g(n)En_λ_max₍_H_β_{) – }_–_ε_gs₍_n₎ += .

Proof It follows by the same argument as in Proposition . that

εs

n>A(ε)

g(n)En_λ

max(Hβ) – 

–εgs(n)₊

= ε



s

n>A(ε) g(n)

_∞

εgs₍_n₎P

n_λ_max₍_H_β_{) – }≥_t_dt

≤C

∞

MsP

λmax(Hβ)≥ +

 n

–__t t Msx



s–_{dx dt}

≤C

∞

Mst

 _e–

β

Ct  

dt→ asM→ ∞,

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Lettingε and thenM→ ∞, a combination of Propositions .-. can complete the proof of Theorem . immediately.

Proof of Theorem. The proof of Theorem . is analogous to that of Theorem .. The only diﬀerence is that we will change the cut-oﬀ pointA(ε) = [g–(Mε–s_{)] to a new one,}

ˆ

A(ε) = [g–_(ε–v_)],_v_>

s. The following four propositions corresponding to Propositions

.-. are required, and the details are omitted.

Proposition . Under the assumptions of Theorem.,we have

lim

ε

– logε

∞

n= g(n)

g(n)E

TWβ–εgs(n)

+=



s

∞



 –Fβ(x)

dx.

Proposition . Under the assumptions of Theorem.,we have

lim

ε

– logε

n≤ ˆA(ε) g(n)

g(n)

E n_λ_max₍_H_β_{) – }_–

εg

s₍_n₎

+

–ETWβ–εgs(n)

+ = .

Proposition . Under the assumptions of Theorem.,we have

lim

ε

– logε

n>Aˆ(ε) g(n)

g(n)E

TWβ–εgs(n)

+= .

Proposition . Under the assumptions of Theorem.,we have

lim

ε

– logε

n>Aˆ(ε) g(n)

g(n)E n



_λ_max₍_H_β_{) – }_–

εg

s₍_n₎

+

= .

Proofs of Theorems.and. The proofs of Theorems . and . are essentially the same as those of Theorems . and ., respectively, and the details of the proofs will not

be presented.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

JX designed the work and drafted the manuscript, and JZ participated in the calculation of the work. All authors read and approved the ﬁnal manuscript.

Author details

1_{College of Mathematics and Information, Henan University, Kaifeng, 475000, P.R. China.}2_{Department of Mathematics,}

Kaifeng Institute of Education, Kaifeng, 475000, P.R. China.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Nos. 11326173 and 11401169) and Foundation of Henan Educational Committee (No. 13A110087).

Received: 30 April 2014 Accepted: 16 September 2014 Published:16 Oct 2014 References

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