getdoc9b2f. 332KB Jun 04 2011 12:04:44 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y} Vol. 16 (2011), Paper no. 11, pages 314–346.

Journal URL

http://www.math.washington.edu/~ejpecp/

Bulk scaling limit of the Laguerre ensemble

Stéphanie Jacquot

∗

Benedek Valkó

†

Abstract

We consider theβ-Laguerre ensemble, a family of distributions generalizing the joint eigenvalue distribution of the Wishart random matrices. We show that the bulk scaling limit of these ensem-bles exists for allβ >0 for a general family of parameters and it is the same as the bulk scaling limit of the correspondingβ-Hermite ensemble.

Key words:Random matrices, eigenvalues, Laguerre ensemble, Wishart ensemble, bulk scaling limit.

AMS 2000 Subject Classification:Primary 60B20; Secondary: 60G55, 60H10. Submitted to EJP on June 7, 2010, final version accepted January 2, 2011.

∗_{University of Cambridge, Statistical Laboratory, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3}

0WB, UK. S.M.Jacquot@statslab.cam.ac.uk

†_{Department of Mathematics, University of Wisconsin - Madison, WI 53705, USA. valko@math.wisc.edu B.Valkó was} partially supported by the NSF Grant DMS-09-05820.

1

Introduction

The Wishart ensemble is one of the first studied random matrix models, introduced by Wishart in 1928 [15]. It describes the joint eigenvalue distribution of the n_×n random symmetric matrix M = AA∗ where Ais an n_×(m₋1) matrix with i.i.d. standard normal entries. We can use real, complex or real quaternion standard normal random variables as ingredients. Since we are only interested in the eigenvalues, we can assume m₋1_≥n. Then the joint eigenvalue density onRn

+

exists and it is given by the following formula for all three versions: 1

Z_n,mβ ₊₁ Y

j<k

|λj−λk|β n Y

k=1

λ

β

2(m−n)−1

k e−

β

2λk_{. (1)}

Hereβ=1, 2 and 4 correspond to the real, complex and quaternion cases respectively and Z_n,mβ ₊₁ is an explicitly computable constant.

The density (1) defines a distribution onRn₊for anyβ >0,n_∈Nandm>nwith a suitableZ_n,mβ ₊₁. The resulting family of distributions is called theβ-Laguerre ensemble. Note that we intentionally shifted the parametermby one as this will result in slightly cleaner expressions later on.

Another important family of distributions in random matrix theory is theβ-Hermite (or Gaussian) ensemble. It is described by the density function

1 ˜ Znβ

Y

1≤j<k≤n

|λj−λk|β n Y

k=1

e−β4λ 2

k_{. (2)}

onRn. Forβ=1, 2 and 4 this gives the joint eigenvalue density of the Gaussian orthogonal, unitary and symplectic ensembles. It is known that if we rescale the ensemble bypnthen the empirical spectral density converges to the Wigner semicircle distribution ₂1_π

p

4−x2₁

[−2,2](x). In [13]the

authors derive the bulk scaling limit of theβ-Hermite ensemble, i.e. the point process limit of the spectrum it is scaled around a sequence of points away from the edges.

Theorem 1(Valkó and Virág[13]). Ifµnis a sequence of real numbers satisfying n1/6(2pn− |µn|)→ ∞as n→ ∞andΛH

n is a sequence of random vectors with density (2) then Æ

4n−µ2 n(Λ

H

n −µn)⇒Sineβ (3)

whereSineβ is a discrete point process with density(2π)−1.

Note that the condition onµn means that we are in the bulk of the spectrum, not too close to the

edge. The limiting point process Sineβ can be described as a functional of the Brownian motion in

the hyperbolic plane or equivalently via a system of stochastic differential equations (see Subsection 2.3 for details).

The main result of the present paper provides the point process limit of the Laguerre ensemble in the bulk. In order to understand the order of the scaling parameters, we first recall the classical results about the limit of the empirical spectral measure for the Wishart matrices. Ifm/n_→γ∈[1,_∞)then with probability one the scaled empirical spectral measuresνn= 1_n

Pn

k=1δλk/n converge weakly to

the Marchenko-Pastur distribution which is a deterministic measure with density ˜

σγ(x) = p

(x−a2)(b2−x)

2πx 1[a2,b2](x), a=a(γ) =γ

1/2

This can be proved by the moment method or using Stieltjes-transform. (See [7] for the original proof and[5]for the generalβ case).

Now we are ready to state our main theorem:

Theorem 2(Bulk limit of the Laguerre ensemble). Fixβ >0, assume that m/n_→γ∈[1,_∞)and let c_∈(a2,b2)for a=a(γ),b= b(γ)defined in (4). LetΛ_nLdenote the point process given by (1). Then

2πσ˜γ(c)€Λ_nL₋cnŠ_⇒Sine_β (5)

whereSineβ is the bulk scaling limit of theβ-Hermite ensemble andσ˜γ is defined in (4).

We will actually prove a more general version of this theorem: we will also allow the cases when m/n _{→ ∞} or when the center of the scaling gets close to the spectral edge. See Theorem 9 in Subsection 2.2 for the details.

Although this statement has been known for the classical cases (β=1, 2 and 4)[8], this is the first proof for generalβ. Our approach relies on the tridiagonal matrix representation of the Laguerre ensemble introduced by Dumitriu and Edelman[1]and the techniques introduced in[13].

There are various other ways one can generalize the classical Wishart ensembles. One possibility is that instead of normal distribution one uses more general real or complex distributions in the construction described at the beginning of this section. It has been conjectured that the bulk scaling limit for these generalized Wishart matrices would be the same as in theβ=1 and 2 cases for the Laguerre ensemble. The recent papers of Tao and Vu[12]and Erd˝os et al.[3]prove this conjecture for a wide range of distributions (see[12]and[3]for the exact conditions).

Our theorem completes the picture about the point process scaling limits of the Laguerre ensemble. The scaling limit at the soft edge has been proved in [9], where the edge limit of the Hermite ensemble was also treated.

Theorem 3(Ramírez, Rider and Virág[9]). If m>n_{→ ∞}then

(mn)1/6 (pm+pn)4/3(Λ

L n−(

p

n+pm)2)_⇒Airy_β

whereAiry_β is a discrete simple point process given by the eigenvalues of the stochastic Airy operator

Hβ =−

d2

d x2 +x+

2

p

β b′_x.

Here b′_x is white noise and the eigenvalue problem is set up on the positive half line with initial conditions f(0) =0,f′(0) =1. A similar limit holds at the lower edge: iflim infm/n>1then

(mn)1/6 (pm₋pn)4/3((

p

m₋pn)2₋Λ_nL)_⇒Airy_β.

Remark 4. The lower edge result is not stated explicitly in [9], but it follows by a straightforward modification of the proof of the upper edge statement. Note that the condition lim infm/n>1 is not optimal, the statement is expected to hold withm−n→ ∞. This has been known for the classical casesβ=1, 2, 4[8].

Ifm₋n_→a_∈(0,_∞)then the lower edge of the spectrum is pushed to 0 and it becomes a ‘hard’ edge. The scaling limit in this case was proved in[10].

Theorem 5(Ramírez and Rider[10]). If m₋n_→a_∈(0,_∞)then nΛ_nL_⇒Θβ,a

where Θβ,a is a simple point process that can be described as the sequence of eigenvalues of a certain

random operator (the Bessel operator).

In the next section we discuss the tridiagonal representation of the Laguerre ensemble, recall how to count eigenvalues of a tridiagonal matrix and state a more general version of our theorem. Section 3 will contain the outline of the proof while the rest of the paper deals with the details of the proof.

2

Preparatory steps

2.1

Tridiagonal representation

In [1] Dumitriu and Edelman proved that the β-Laguerre ensemble can be represented as joint eigenvalue distributions for certain random tridiagonal matrices.

Lemma 6(Dumitriu, Edelman[1]). Let An,mbe the following n×n bidiagonal matrix:

A_n,m= _p1

β



      

˜ χβ(m−1)

χβ(n−1) χ˜β(m−2)

... ...

χβ·2 χ˜β(m−n+1)

χβ χ˜β(m−n)



      

.

whereχβa, ˜χβbare independent chi-distributed random variables with the appropriate parameters (1≤

a≤n−1,m−1_≤b≤m−n). Then the eigenvalues of the tridiagonal matrix An,mATn,m are distributed

according to the density (1).

If we want to find the bulk scaling limit of the eigenvalues ofA_n,mAT_n,mthen it is sufficient to under-stand the scaling limit of the singular values of An,m.The following simple lemma will be a useful

tool for this.

Lemma 7. Suppose that B is an n× n bidiagonal matrix with a1,a2, . . . ,an in the diagonal and

b₁,b₂, . . . ,b_n−1 below the diagonal. Consider the2n_×2n symmetric tridiagonal matrix M which has zeros in the main diagonal and a₁,b₁,a₂,b₂, . . . ,a_n in the off-diagonal. If the singular values of B are λ1,λ2, . . . ,λn then the eigenvalues of M are±λi,i=1 . . .n.

We learned about this trick from[2], we reproduce the simple proof for the sake of the reader.

Proof. Consider the matrix ˜B = –

0 BT

B 0

™

. If Bu = λiv and BTv = λiu then [u,±v]T is

an eigenvector of ˜B with eigenvalue _±λi. Let C be the permutation matrix corresponding to (2, 4, . . . , 2n, 1, 3, . . . , 2n−1). ThenCTBC˜ is exactly the tridiagonal matrix described in the lemma and its eigenvalues are exactly±λi,i=1 . . .n.

Because of the previous lemma it is enough to study the eigenvalues of the(2n)_×(2n)tridiagonal matrix

˜ An,m=

1

p

β



         

0 χ˜β(m−1)

˜

χβ(m−1) 0 χβ(n−1)

χβ(n−1) 0 χ˜β(m−2)

... ... ...

˜

χβ(m−n+1) 0 χβ

χβ 0 χ˜β(m−n)

˜

χβ(m−n) 0



         

(6)

The main advantage of this representation, as opposed to studying the tridiagonal matrixAn,mATn,m,

is that here the entries are independent modulo symmetry.

Remark 8. Assume that [u1,v1,u2,v2, . . . ,un,vn]T is an eigenvector for ˜An,m with eigenvalue λ.

Then[u₁,u₂, . . . ,u_n]T is and eigenvector forAT_n,mA_n,mwith eigenvalueλ2 and[v₁,v₂, . . . ,v_n]T is an eigenvector forA_n,mAT_n,mwith eigenvalueλ2.

2.2

Bulk limit of the singular values

We can compute the asymptotic spectral density of ˜A_n,mfrom the Marchenko-Pastur distribution. If m/n_→γ∈[1,_∞)then the asymptotic density (when scaled withpn) is

σγ(x) = 2_|x_|σ˜γ(x2) = p

(x2_−a2_)(b2_−x2₎

π|x_| 1[a,b](|x|)

= p

(x₋a)(x+a)(b₋x)(b+x)

π|x_| 1[a,b](|x|). (7) This means that the spectrum of ˜A_n,m in R+ _{is asymptotically concentrated on the interval[}p_m

− p_n,p_m+p_{n]. We will scale around}

µn ∈(pm−pn,pm+pn) where µn is chosen in a way

that it is not too close to the edges. Nearµn the asymptotic eigenvalue density should be close to

σm/n(µn/ p

n)which explains the choice of the scaling parameters in the following theorem.

Theorem 9. Fixβ >0and suppose that m=m(n)>n. LetΛ_n denote the set of eigenvalues ofA˜_n,m and set

n₀ = π

2

4 nσ

m/n€

µnn−1/2 Š2

−1

2, n1=n− π2

4 nσ

m/n€

µnn−1/2 Š2

. (8)

Assume that as n_{→ ∞}we have

n1₁/3n−₀1_→0 (9)

and

lim inf

n→∞ m/n>1 or nlim→∞m/n=1 and lim infn→∞ µn/

p

n>0. (10)

Then

The extra 1/2 in the definition ofn₀is introduced to make some of the forthcoming formulas nicer. We also note that the following identities hold:

n0+

1 2 =

2(m+n)µ2_n−(m₋n)2₋µ4_n 4µ2

n

, n1= €

m₋n₋µ2_nŠ2 4µ2

n

. (12)

Note that we did not assume thatm/nconverges to a constant or thatµn=pcpn. By the

discus-sions at the beginning of this section(Λ_n∩R+)2 _{is distributed according to the Laguerre ensemble.}

If we assume that m/n_→ γ andµn = p

cpn with c _∈(a(γ)2,b(γ)2) then both (9) and (10) are satisfied. Since in this casen0n−1→σ˜γ(c)the result of Theorem 9 implies Theorem 2.

Remark 10. We want prove that the weak limit of 4pn₀(Λ_n−µn)is Sineβ, thus it is sufficient to

prove that for any subsequence ofnthere is a further subsequence so that the limit in distribution holds. Because of this by taking an appropriate subsequence we may assume that

m/n→γ∈[1,∞], and if m/n→1 then µn/ p

n→c∈(0, 2]. (13) These assumptions imply that form1=m−n+n1we have

lim inf

n→∞ m1/n>0. (14)

One only needs to check this in them/n_→1 case, when from (13) and the definition ofn₁ we get n1/n→c>0.

Remark 11. The conditions of Theorem 9 are optimal if lim infm/n>1 and the theorem provides a complete description of the possible point process scaling limits ofΛ_nL. To see this first note that usingΛL

n = (Λn∩R+)2 we can translate the edge scaling limit of Theorem 3 to get

2(mn)1/6

(pm±pn)1/3(Λn−( p

m_±pn))_{⇒ ±}Airy_β. (15)

If lim infm/n> 1 then by the previous remark we may assume limm/n = γ∈(1,_∞]. Then the previous statement can be transformed into n1/6(Λ_n−(pm_±pn)) =d_⇒ Ξ where Ξ is a a linear transformation of Airy_β. From this it is easy to check that ifn1₁/3n₀−1→c∈(0,_∞]then we need to scaleΛ_n−µn withn1/6 to get a meaningful limit (and the limit is a linear transformation of Airyβ)

and ifn1₁/3n−₀1→0 then we get the bulk case.

If m/n_→1 then the condition (10) is suboptimal, this is partly due to the fact that the lower soft edge limit in this case is not available. Assuming lim infm−n>0 the statement should be true with the following condition instead of (10):

µn p

n(m₋n)−1/3₋1

2(m−n)

2/3

→ ∞. (16)

It is easy to check that this condition is necessary for the bulk scaling limit. By choosing an appro-priate subsequence we may assume thatm−n→a>0 orm−n→ ∞. Then if (16) does not hold then we can use Theorem 5 (ifm₋n_→a>0) or (15) (ifm₋n_{→ ∞}) to show that an appropriately scaled version ofΛ_n−µn converges to a shifted copy of the hard edge or soft edge limiting point

2.3

The

Sine

β

process

The distribution of the point process Sineβ from Theorem 1 was described in[13]as a functional of

the Brownian motion in the hyperbolic plane (the Brownian carousel) or equivalently via a system of stochastic differential equations. We review the latter description here. Let Z be a complex Brownian motion with i.i.d. standard real and imaginary parts. Consider the strong solution of the following one parameter system of stochastic differential equations fort_∈[0, 1),λ∈R:

dαλ=

λ

2p1−td t+

p

2

p

β(1−t)ℜ ”

(e−iαλ_{−1)d Z}—_{, α}

λ(0) =0. (17)

It was proved in [13] that for any given λ the limit N(λ) = 1

2πlimt→1αλ(t) exists, it is integer

valued a.s. andN(λ)has the same distribution as the counting function of the point process Sineβ

evaluated atλ. Moreover, this is true for the joint distribution of(N(λi),i=1, . . . ,d)for any fixed

vector(λi,i=1, . . . ,d). Recall that the counting function atλ >0 gives the number of points in the

interval(0,λ], and negative the number of points in(λ, 0]forλ <0.

2.4

Counting eigenvalues of tridiagonal matrices

Assume that the tridiagonalk_×kmatrixM has positive off-diagonal entries.

M=



      

a₁ b₁ c₁ a₂ b₂

... ...

ck−2 ak−1 bk−1

ck−1 ak 

      

, b_i >0,c_i>0.

Ifu=

u₁, . . . ,u_kT

is an eigenvector corresponding toλthen we have

cℓ−1uℓ−1+aℓuℓ+bℓuℓ+1=λuℓ, ℓ=1, . . .k (18)

where we can we set u₀ = u_k+1 = 0 (with c₀,b_k defined arbitrarily). This gives a single term recursion onR_{∪ {∞}}for the ratios rℓ=

uℓ+1 uℓ :

r0=∞, rℓ=

1 b_ℓ

−cℓ−1

r_ℓ−1 +λ−aℓ

, ℓ=1, . . .k. (19)

This recursion can be solved for any parameterλ, andλis an eigenvalue if and only ifrk=rk,λ=0.

Induction shows that for a fixedℓ >0 the functionλ→ rℓ,λ is just a rational function inλ which

is analytic and increasing between its blow-ups. (In fact, it can be shown that r_ℓ is a constant multiple of pℓ(λ)/pℓ−1(λ) where pℓ(·) is the characteristic polynomial of the top left ℓ×ℓminor

of M.) From this it follows easily that for each 0_≤ ℓ≤ k we can define a continuous monotone increasing functionλ →φℓ,λ which satisfies tan(φℓ,λ/2) =rℓ,λ. The functionϕℓ,· is unique up to translation by integer multiples of 2π. Clearly the eigenvalues ofMare identified by the solutions of φk,λ=0 mod 2π. Sinceφℓ,·is continuous and monotone this provides a way to identify the number of eigenvalues in(λ0,λ1]from the valuesφk,λ0 andφk,λ1:

#¦(φk,λ0,φk,λ1]∩2πZ ©

This is basically a discrete version of the Sturm-Liouville oscillation theory. (Note that if we shift φk,· by 2πthen the expression on the right stays the same, so it does not matter which realization ofφk,· we take.)

We do not need to fully solve the recursion (19) in order to count eigenvalues. If we consider the reversed version of (19) started from indexkwith initial condition 0:

r_k⊙=0, r_ℓ⊙₋₁=₋c_ℓ€a_ℓ−λ+b_ℓr_ℓ⊙Š−1, ℓ=1, . . .k. (20) thenλis an eigenvalue if and only ifrℓ,λ=r_ℓ⊙_,λ. Moreover, we can turnr_ℓ⊙_,λinto an angleφ_ℓ⊙_,λwhich

will be continuous and monotone decreasing inλ(similarly as before forrandφ) which transforms the previous condition toφℓ,λ−φ⊙ℓ,λ =0 mod 2π. This means that we can also count eigenvalues

in the interval(λ0,λ1]by the formula

#n(φℓ,λ0−φ

⊙

ℓ,λ0,φℓ,λ1−φ

⊙

ℓ,λ1]∩2πZ o

=#_{eigenvalues in(λ0,λ1]} (21)

Ifh:R_→Ris a monotone increasing continuous function withh(x+2π) =h(x)then the solutions ofφℓ,λ=φℓ⊙,λ mod 2πwill be the same as that ofh(φℓ,λ) =h(φℓ⊙,λ)mod 2π. Sinceh(φℓ,λ)−h(φℓ⊙,λ)

is also continuous and increasing we get #

n

(h(φℓ,λ0)−h(φ

⊙

ℓ,λ0),h(φℓ,λ1)−h(φ

⊙

ℓ,λ1)]∩2πZ o

=#_{eigenvalues in(λ0,λ1]}. (22)

In our case, by analyzing the scaling limit ofh(φℓ,·)andh(φℓ⊙,·)for a certainℓandhwe can identify the limiting point process. This method was used in[13]for the bulk scaling limit of theβHermite ensemble. An equivalent approach (via transfer matrices) was used in[6]and[14]to analyze the asymptotic behavior of the spectrum for certain discrete random Schrödinger operators.

3

The main steps of the proof

The proof will be similar to one given for Theorem 1 in[13]. The basic idea is simple to explain: we will use (22) with a certainℓ=ℓ(n)andh. Then we will show that the length of the interval on the left hand side of the equation converges to 2π(N(λ1)−N(λ0)) while the left endpoint of that

interval becomes uniform modulo 2π. SinceN(λ1)−N(λ0)is a.s. integer the number of eigenvalues

in(λ0,λ1]converges toN(λ1)−N(λ0)which shows that the scaling limit of the eigenvalue process

is given by Sineβ.

The actual proof will require several steps. In order to limit the size of this paper and not to make it overly technical, we will recycle some parts of the proof in[13]. Our aim is to give full details whenever there is a major difference between the two proofs and to provide an outline of the proof if one can adapt parts of[13]easily.

Proof of Theorem 9. Recall thatΛ_n denotes the multi-set of eigenvalues for the matrix ˜A_n,m which is defined in (6). We denote byN_n(λ)the counting function of the scaled random multi-sets 4n1₀/2(Λ_n−

µn), we will prove that for any(λ1,· · ·,λd)∈Rd we have

N_n(λ1),· · ·,Nn(λd)

d

=_⇒ N(λ1),· · ·,N(λd)

whereN(λ) = ₂1_πlim_t→1αλ(t)as defined using the SDE (17).

We will use the ideas described in Subsection 2.4 to analyze the eigenvalue equation ˜An,mx = Λx,

where x∈R2n_{. Following the scaling given in (11) we set} Λ =µn+

λ 4pn₀.

In Section 4 we will define the regularized phase function ϕℓ,λ and target phase functionϕ_ℓ⊙_,λ for

ℓ∈[0,n₀). These will be independent of each other for a fixedℓ(as functions inλ) and satisfy the following identity forλ < λ′:

#

n

(ϕℓ,λ−ϕ_ℓ⊙_,λ,ϕℓ,λ′−ϕ_ℓ⊙_,λ′]∩2πZ

o

=N_n(λ′)₋N_n(λ). (24) The functions ϕℓ,λ and ϕ⊙ℓ,λ will be transformed versions of the phase function and target phase

function φℓ,λ and φℓ⊙,λ so (24) will be just an application of (22). The regularization is needed

in order to have a version of the phase function which is asymptotically continuous. Indeed, in Proposition 17 of Section 5 we will show that for any 0< ǫ <1 the rescaled version of the phase functionϕℓ,λ in

0,n0(1−ǫ)

converges to a one-parameter family of stochastic differential equa-tions. Moreover we will prove that in the same region the relative phase functionαℓ,λ=ϕℓ,λ−ϕℓ,0

will converge to the solutionαλ of the SDE (17)

α_⌊n0(1−ǫ)⌋,λ

d

=_⇒αλ(1−ǫ), asn→ ∞ (25)

in the sense of finite dimensional distributions inλ. This will be the content of Corollary 18. Next we will describe the asymptotic behavior of the phase functionsϕℓ,λ,αℓ,λandϕℓ⊙,λin the stretch

ℓ∈[_⌊n₀(1₋ǫ)_⌋,n₂]where

n₂=_⌊n_{0− K}(n₁1/3_∨1)_⌋. (26)

(The constants ǫ,K will be determined later.) We will show that if the relative phase function is already close to an integer multiple of 2π at _⌊n₀(1₋ǫ)_⌋ then it will not change too much in the interval[_⌊n0(1−ǫ)⌋,n2]. To be more precise, in Proposition 19 of Section 6 we will prove that there

exists a constantc=c(λ,¯ β)so that we have

E”_|α_⌊n0(1−ǫ)⌋,λ−αn2,λ| ∧1 —

≤c h

dist(α_⌊n0(1−ǫ)⌋,λ, 2πZ) + p

ε+n−₀1/2(n₁1/6_∨logn₀) +_K−1 i

(27) for allK >0,ε∈(0, 1),λ≤ |λ¯|. Note that we already know thatα_⌊n0(1−ǫ)⌋converges toαλ(1−ǫ)

in distribution (asn_{→ ∞}) and thatαλ(1−ǫ)converges a.s. to an integer multiple of 2π(asǫ→0).

By the conditions onn₀,n₁the term n₀−1/2(n1₁/6_∨logn₀)converges to 0.

We will also show that ifK → ∞andKn−₀1(n₁1/3∨1)_→0 then the random angleϕ_n2,0becomes

uniformly distributed modulo 2πasn_{→ ∞}(see Proposition 23 of Section 7).

Next we will prove that the target phase function will loose its dependence onλ: for everyλ∈R

and_K >0 we have

α⊙_n2,λ=ϕ⊙_n2,λ−ϕ_n⊙

2,0 P

−→0, asn→ ∞. (28)

The proof now can be finished exactly the same way as in[13]. Using the previous statements and a standard diagonal argument we can chooseǫ=ǫ(n)_→0 and_K =_K(n)_{→ ∞}so that the following limits all hold simultaneously:

(α_⌊n0(1−ǫ)⌋,λi,i=1, . . . ,d) d

=_⇒ (2πN(λi),i=1, . . . ,d),

ϕn2,0 P

−→ Uniform[0, 2π] modulo 2π, α_⌊n0(1−ǫ)⌋,λi −αn2,λi

P

−→ 0, i=1, . . . ,d, α⊙_n

2,λi P

−→ 0, i=1, . . . ,d.

This means that if we apply the identity (24) withλ=0,λ′=λi andℓ=n2 then the length of the

random intervals

Ii = (ϕn2,0−ϕ

⊙

n2,0,ϕn2,λi−ϕ

⊙

n2,λi]

converge to 2πN(λi) in distribution (jointly), while the common left endpoint of these intervals

becomes uniform modulo 2π. (Since ϕn2,0 and ϕ

⊙

n2,0 are independent and ϕn2,0 converges to a

uniform distribution mod 2π.) This means that #{2kπ ∈ I_i : k _∈Z_} converges to N(λi) which

proves (23) and Theorem 9.

The following figure gives an overview of the main components of the proof.

n₀,n₁: defined in (8), phase functions: ϕℓ,λ,ϕ_ℓ⊙_,λ,αℓ,λ,α⊙_ℓ,λ, (defined in Section 4) ✓

✒

✏ ✑

SDE limit, α_⌊n0(1−ǫ)⌋,λ⇒αλ(1−ǫ)

(Section 5)

✓ ✒

✏ ✑

αℓ,λdoes not change much for

ℓ∈[n₀(1₋ǫ),n₂]

(Section 6)

☛ ✡

✟ ✠

α⊙_n

2,λ converges to 0

(Section 7)

❄ ❄ ❄

✻

ℓ 1

‘first stretch’ ‘middle stretch’ ‘last stretch’

⌊n₀(1₋ǫ)_⌋ n₂=_⌊n_{0− K}(n1₁/3_∨1)_⌋ n

☛ ✡

✟ ✠

ϕn2,0becomes uniform mod 2π

(Section 6.2)

4

Phase functions

In this section we introduce the phase functions used to count the eigenvalues.

4.1

The eigenvalue equations

Lets_j=pn− j−1/2 and p_j =pm−j−1/2. Conjugating the matrix ˜An,m (6) with a(2n)×(2n)

diagonal matrixD=D(n)with diagonal elements

D_1,1=1, D_2i,2i= χ˜β(_pm−i−1)

βp_i

i−1 Y

ℓ=1

˜

χβ(m−ℓ)χβ(n−ℓ)

βpℓsℓ

, D_2i+1,2i+1= i Y

ℓ=1

˜

χβ(m−ℓ)χβ(n−ℓ)

βpℓsℓ

we get the tridiagonal matrix ˜AD_n,m=D−1A˜_n,mD:

˜ AD_n,m=



         

0 p₀+X₀

p₁ 0 s₀+Y₀

s1 0 p1+X1

... ... ...

pn−1 0 sn−2+Yn−2

s_n−1 0 p_n−1+X_n−1

p_n 0



         

(29)

where

Xℓ=

˜

χ_β(2 _m−ℓ−1)

βp_ℓ+1 −pℓ, 0≤ℓ≤n−1, Yℓ=

χ_β(2 _n−ℓ−1)

βs_ℓ+1 −sℓ, 0≤ℓ≤n−2.

The first couple of moments of these random variables are explicitly computable using the moment generating function of theχ2-distribution and we get the following asymptotics:

EX_ℓ=_O((m₋ℓ)−3/2), EX_ℓ2=2/β+_O((m₋ℓ)−1), EX_ℓ4=_O(1),

EY_ℓ=_O((n₋ℓ)−3/2), EY_ℓ2=2/β+_O((n₋ℓ)−1), EY_ℓ4=_O(1), (30) where the constants in the error terms only depend onβ.

We consider the eigenvalue equation for ˜AD_n,m with a givenΛ_∈Rand denote a nontrivial solution of the first 2n−1 components byu1,v1,u2,v2, . . . ,un,vn. Then we have

sℓvℓ+ (pℓ+Xℓ)vℓ+1 = Λuℓ+1, 0≤ℓ≤n−1,

pℓ+1uℓ+1+ (sℓ+Yℓ)uℓ+2 = Λvℓ+1, 0≤ℓ≤n−2,

where we setv₀=0 and we can assumeu₁=1 by linearity. We setr_ℓ=r_ℓ,Λ=u_ℓ+1/v_ℓ, 0_≤ℓ≤n₋1 andˆrℓ= ˆrℓ,Λ=vℓ/uℓ, 1≤ℓ≤n. These are elements ofR∪ {∞}satisfying the recursion

ˆrℓ+1 =

−1

rℓ ·

sℓ

pℓ

+ Λ

pℓ

1+ Xℓ

pℓ

−1

, 0_≤ℓ≤n−1 (31)

r_ℓ+1 =

−_ˆ1

rℓ+1 ·pℓ+1

sℓ

+ Λ

sℓ

1+ Yℓ

sℓ

−1

, 0_≤ℓ≤n₋2, (32)

with initial conditionr0=∞. We can setYn=0 and definernvia (32) withℓ=n−1, thenΛis an

4.2

The hyperbolic point of view

We use the hyperbolic geometric approach of[13]to study the evolution of r andˆr. We will view

R_{∪ {∞}}as the boundary of the hyperbolic plane H=_{ℑz>0 :z∈C_}in the Poincaré half-plane model. We denote the group of linear fractional transformations preservingH by PSL(2,R). The recursions for bothr andˆr evolve by elements of this group of the form x7→b−a/x witha>0. The Poincaré half-plane model is equivalent to the Poincaré disk modelU=_{|z_|< 1_} via the con-formal bijectionU(z) = iz+1

z+i which is also a bijection between the boundaries∂H=R∪ {∞}and

∂U = _{|z_| = 1,z _∈C_}. Thus elements of PSL(2,R) also act naturally on the unit circle ∂U. By lifting these maps toR, the universal cover of∂U, each elementT in PSL(2,R)becomes anR_→R

function. The lifted versions are uniquely determined up to shifts by 2πand will also form a group which we denote by UPSL(2,R). For anyT _∈UPSL(2,R) we can look atTas a function acting on ∂H,∂UorR. We will denote these actions by:

∂H_→∂H :z_7→z.T, ∂U_→∂U :z_7→z◦T, ∂R→∂R :z7→z∗T.

For everyT _∈UPSL(2,R) the function x _7→ f(x) = x∗T is monotone, analytic and quasiperiodic modulo 2π: f(x +2π) = f(x) +2π. It is clear from the definitions that ei x◦T = ei f(x) and

(2 tan(x)).T=2 tanf(x).

Now we will introduce a couple of simple elements of UPSL(2,R). For a givenα∈Rwe will denote by Q(α) the rotation by α in U about 0. More precisely, ϕ∗Q(α) = ϕ+α. For a > 0,b ∈R we denote byA(a,b)the affine mapz_→a(z+b)inH. This is an element of PSL(2,R)which fixes_∞ inH and₋1 in∂U. We specify its lifted version in UPSL(2,R)by making it fixπ, this will uniquely determines it as aR_→Rfunction.

GivenT∈UPSL(2,R), x,y∈Rwe define the angular shift

ash(T,x,y) = (y∗T−x∗T)−(y−x)

which gives the change in the signed distance ofx,yunderT. This only depends onv=ei x,w=ei y and the effect ofTon∂U, so we can also view ash(T,·,·)as a function on∂U×∂U and the following identity holds:

ash(T,v,w) =arg_[0,2π)(w◦T/v◦T)−arg[0,2π)(w/v).

The following lemma appeared as Lemma 16 in[13], it provides a useful estimate for the angular shift.

Lemma 12. Suppose that for aT_∈UPSL(2,R)we have(i+z).T=i with_|z_{| ≤}1

3. Then

ash(T,v,w) = _ℜh(w¯−¯v)₋z− i(2+₄¯v+w¯)z2 i

+ǫ3 = _−ℜ[(w¯₋¯v)z] +ǫ2=ǫ1,

(33)

where for d=1, 2, 3and an absolute constant c we have

|ǫ_d| ≤c|w−v||z|d≤2c|z|d. (34) If v=₋1then the previous bounds hold even in the case|z|> 1₃.

4.3

Regularized phase functions

Because of the scaling in (11) we will set

Λ =µn+

λ 4n1₀/2

. We introduce the following operators

J_ℓ=Q(π)A(s_ℓ/p_ℓ,µn/sℓ), Mℓ=A((1+Xℓ/pℓ)−1,λ/(4n

1/2 0 pℓ))A(

pℓ

pℓ+1

, 0),

ˆ

Jℓ=Q(π)A(pℓ/sℓ,µn/pℓ), Mˆℓ=A((1+Yℓ/sℓ)−1,λ/(4n1 /2 0 sℓ)).

Then (31) and (32) can be rewritten as

r_ℓ+1=r_ℓ.J_ℓM_ℓˆJ_ℓMˆ_ℓ, r₀=_∞.

(We suppressed theλdependence in r and the operatorsM,Mˆ.) Lifting these recursions from ∂H

toRwe get the evolution of the corresponding phase angle which we denote byφℓ=φℓ,λ.

φℓ+1=φℓ∗JℓMℓJˆℓMˆℓ, φ0=−π. (35)

Solving the recursion from the other end, with end condition 0 we get the target phase function φ⊙_ℓ,λ:

φ_ℓ⊙=φ_ℓ+⊙₁∗Mˆ−_ℓ1ˆJ_ℓ−1M−_ℓ1J−_ℓ1, φ⊙_n =0. (36) It is clear thatφℓ,λ andφℓ⊙,λ are independent for a fixedℓ(as functions inλ), they are monotone

and analytic inλand we can count eigenvalues using the formula (21).

In our case bothMℓ andMˆℓ will be small perturbations of the identity soJℓˆJℓ will be the main part

of the evolution. This is a rotation in the hyperbolic plane if it only has one fixed point inH. The fixed point equationρℓ=ρℓ.JℓˆJℓ can be rewritten as

ρℓ=

p_ℓ s_ℓ



 

µn

p_ℓ − p_{ℓ µ}

n sℓ −

1

ρ_ℓ



 =

ρℓ(µ2n−p 2

ℓ)−µnsℓ

ρℓµnsℓ−s2ℓ

.

This can be solved explicitly, and one gets the following unique solution in the upper half plane if ℓ <n0+1/2:

ρℓ=

µ2_n−m+n 2µ_nsℓ

+i s

1₋ (µ

2

n−m+n)2

4µ2 ns2ℓ

. (37)

One also needs to use the identity p2_ℓ −s2_ℓ =m−nand (12). This shows that ifℓ <n0 thenJℓˆJℓ is

a rotation in the hyperbolic plane. We can move the center of rotation to 0 inU by conjugating it with an appropriate affine transformation:

J_ℓˆJ_ℓ=Q(₋2 arg(ρℓρˆℓ))T

−1

ℓ .

HereT_ℓ=A(_ℑ(ρℓ)−1,−ℜρℓ),XY=Y−1XYand

ˆ

ρℓ=

µ2

n+m−n

2µnpℓ

+i s

1₋ (µ

2

n+m−n)2

4µ2 npℓ2

In order to regularize the evolution of the phase function we introduce ϕℓ,λ:=φℓ,λ∗TℓQℓ−1, 0≤ℓ <n0

where Qℓ =Q(2 arg(ρ0ρˆ0))· · ·Q(2 arg(ρℓρˆℓ)) andQ−1 is the identity. It is easy to check that the

initial condition remainsϕ0,λ=π. Then

ϕℓ+1 = ϕℓ∗Q−_ℓ−1₁T_ℓ−1JℓMℓˆJℓMˆℓTℓ+1Qℓ

= ϕℓ∗Q−_ℓ−1₁T_ℓ−1Q(−2 arg(ρℓ))T

−1

ℓ MˆJℓ

ℓ MˆℓTℓT−

1

ℓ Tℓ+1Qℓ

= ϕℓ∗



MˆJℓ

ℓ

T_ℓ‹Q_ℓ €

( ˆM_ℓ)TℓŠQℓ€_T−1

ℓ Tℓ+1 ŠQ_ℓ

Note that the evolution operator is now infinitesimal:M_ℓ,Mˆ_ℓ andT−_ℓ1T_ℓ+1 will all be asymptotically small, and the various conjugations will not change this.

We can also introduce the corresponding target phase function

ϕ_ℓ⊙_,λ:=φ_ℓ⊙_,λ∗TℓQℓ−1, 0≤ℓ <n0. (39)

The new, regularized phase functions ϕℓ,λ and ϕ_ℓ⊙_,λ have the same properties asφ,φ⊙, i.e.: they

are independent for a fixedℓ(as functions inλ), they are monotone and analytic inλand we can count eigenvalues using the formula (24). (See (22) and the discussion before it.)

We will further simplify the evolution using the following identities:

−a_r +b= ‚

b2+1 a r−b

Œ

Q

arg

_b

−i b+i

, r.ˆJℓTℓ=−

1 r

p_ℓ s_ℓℑρℓ

+ µn

s_ℓℑρℓ−

ℜρℓ

ℑρℓ

. From this we get

ˆ

J_ℓT_ℓ= ˆT_ℓQ_ℓ(₋2 arg( ˆρℓ))

where

r.ˆTℓ =

‚ s_ℓ

ℑρℓpℓ

−2ℜρℓ

ℑρℓ

µ_n+ µ 2 n ℑρℓpℓsℓ

Œ

r− µn

s_ℓℑρℓ

+ℜρℓ ℑρℓ

= 1 ℑρˆℓ

r₋ℜρˆℓ

ℑρˆℓ

. This allows us to write



MˆJℓ

ℓ

T_ℓ‹Qℓ

= (MTˆℓ

ℓ )

Q(−2 arg( ˆρℓ))Qℓ_{= (M}Tˆℓ

ℓ ) ˆ

Qℓ_{. (40)}

where

ˆ

Q_ℓ=Q_ℓQ(₋2 arg( ˆρℓ)).

Thus

ϕℓ+1 = ϕℓ∗

MˆTℓ

ℓ

Qˆ_ℓ ˆ MTℓ

ℓ

Q_ℓ€

We will introduce the following operators to break up the evolution into smaller pieces:

Lℓ,λ=A(1,λ/(4n1 /2

0 pℓ)), Lˆℓ,λ=A(1,λ/(4n1 /2 0 sℓ)),

Sℓ,λ=L ˆ

T_ℓ

ℓ,λ

ˆ

T−_ℓ1A( pℓ

pℓ+1

(1+Xℓ/pℓ)−1, 0) ˆTℓ

, (41)

ˆ

S_ℓ,λ= ˆLTℓ

ℓ,λ

€

T_ℓ−1A((1+Y_ℓ/s_ℓ)−1, 0)T_ℓ+1Š. Then

ϕℓ+1 = ϕℓ∗

LTˆℓ

ℓ

Qˆ_ℓ€

S_ℓ,0ŠQˆℓ

ˆ LTℓ

ℓ

Q_ℓ€ ˆ

S_ℓ,0ŠQℓ =ϕℓ∗

€

S_ℓ,λŠQˆℓ€ˆS_ℓ,λŠQℓ. (42) We also introduce the relative (regularized) phase function and target phase function:

αℓ,λ:=ϕℓ,λ−ϕℓ,0, αℓ⊙,λ:=ϕℓ⊙,λ−ϕ⊙ℓ,0. (43)

5

SDE limit for the phase function

Let Fℓ denote the σ-field generated by ϕj,λ,j ≤ ℓ−1. Then ϕℓ,λ is a Markov chain in ℓ with

respect to _Fℓ. Indeed, the relation (42) shows that ϕℓ+1,λ = hℓ,λ(ϕℓ+1,λ,Xℓ,Yℓ) where hℓ,λ is a

deterministic function depending on ℓ andλ. Since X_ℓ,Y_ℓ are independent of _Fℓ it follows that

E”ϕℓ+1,λ|Fℓ

—

= E”ϕℓ+1,λ|ϕℓ,λ

—

. We will show that this Markov chain converges to a diffusion limit after proper normalization. In order to do this we will estimate E”ϕℓ+1,λ−ϕℓ,λ|Fℓ

—

and

E”(ϕℓ+1,λ−ϕℓ,λ)(ϕℓ+1,λ′−ϕ_ℓ,λ′)|F_ℓ

—

using the angular shift lemma, Lemma 12.

To simplify the computations we introduce ‘intermediate’ values for the processϕℓ,λby breaking the

evolution operator in (42) into two parts: ϕℓ+1_/2,λ=ϕ_ℓ∗

€

S_ℓ,λŠQˆℓ, _Fℓ+1_/2=σ(Fℓ∪ {ϕℓ+1_/2,λ}).

Note thatϕℓ,λis still a Markov chain if we consider it as a process on the half integers.

Remark 13. We would like to note that the ‘half-step’ evolution rules ϕℓ,λ →ϕℓ+1/2,λ, ϕℓ+1/2,λ →

ϕℓ+1,λ arevery similar to the one-step evolution of the phase functionϕ in [13]. Indeed, in[13],

the evolution ofϕis of the typeϕℓ+1=ϕℓ∗

€

˜

S_ℓ,λŠQ˜ℓ where ˜Sis an affine transformation and ˜Qis a rotation similar to ourS,ˆSandQ,Qˆ. In our case the evolution ofϕℓ+1,λ is the composition of two

transformations with similar structure. The main difficulties in our computations are caused by the fact thatQandQˆ are rather different which makes the oscillating terms more complicated.

5.1

Single step estimates

Throughout the rest of the proof we will use the notation k= n₀₋ℓ. We will need to rescale the discrete time by n0 in order to get a limit, we will uset =ℓ/n0 and also introduceˆs(t) =

p

We start with the identity p_ℓℑρˆℓ=sℓℑρℓ =

È s2_{ℓ −}(µ

2

n−m+n)2

4µ2 n

= È

n₋(µ

2

n−m+n)2

4µ2 n

−ℓ−1

2

= pn0−ℓ= p

k=pn0ˆs(t).

Note that this means that ρℓ = ±

r

n−n0−1/2

n₋ℓ−1/2 +i

r

n0−ℓ

n₋ℓ−1/2=±

r n1

n₁+k+i È

k

n₁+k, (44)

ˆ

ρℓ =

r

m₋n₀₋1/2 m₋ℓ−1/2 +i

r

n₀₋ℓ m₋ℓ−1/2 =

r m₁ m₁+k+i

È k

m₁+k (45)

where the sign in_ℜρℓ is positive ifµn> p

m−nand negative otherwise. For the angular shift estimates we need to consider

Zℓ,λ := i.S−ℓ,1λ−i=

ˆ

ρℓXℓ

p_n 0ˆs(t)

· pℓ+1

p_ℓ +

− λ

4n₀ˆs(t)+ ˆ

ρℓ(pℓ+1−pℓ)

p_ℓℑρˆℓ

=:Vℓ+vℓ,

ˆ

Zℓ,λ := i.Sˆ−ℓ,1λ−i=

ρℓYℓ

p_n 0ˆs(t)

+

− λ

4n₀ˆs(t)+

ρℓ+1−ρℓ

ℑρℓ

=:Vˆℓ+ ˆvℓ. (46)

HereV_ℓ,Vˆ_ℓ are the random andv_ℓ,ˆv_ℓ are the deterministic parts of the appropriate expressions. We have the following estimates for the deterministic parts (by Taylor expansion):

vℓ,λ =

v_λ(t)

n0

+O(k−2), vλ(t) =−

λ 4ˆs(t)−

ˆ

ρ(t)

2p(t)ˆs(t), |vℓ,λ| ≤

c k,

ˆ

vℓ,λ =

ˆv_λ(t)

n₀ +O(k

−2_{), ˆv}

λ(t) =−

λ 4ˆs(t)+

d d tρ(t)

ℑρ(t), |ˆvℓ,λ| ≤

c k,

wherep(t) =p(n)(t) =pm/n₀₋t andρ(t) =ρ(n)(t),ρˆ(t) = ˆρn(t)are defined by equations (44) and (45) withℓ=n0t. For the random terms from (30) we get

EVℓ = EVˆℓ=O(k−1/2(n−ℓ)−3/2),

EV_ℓ2 = 1

n₀q

(1)_{(t) +}

O(k−1(n−ℓ)−1), EVˆ_ℓ2= 1

n₀q

(2)_{(t) +}

O(k−1(n−ℓ)−1),

E_|V_ℓ2| = E_|Vˆ_ℓ2|= 1

n₀q

(3)_{(t) +}

O(k−1(n−ℓ)−1), E_|V_ℓd|,E_|Vˆ_ℓ2|=_O(k−d/2),d=3, 4, where the constants in the error term only depend onβand

q(1)(t) =2ρˆ(t) 2

βˆs(t)2, q

(2)_{(t) =}2ρ(t)

2

βˆs(t)2, q

(3)_{(t) =} 2

βˆs(t)2. (47)

We introduce the notations

∆1_/2fx,λ= fx+1/2,λ−fx,λ, ∆fx,λ= fx+1,λ− fx,λ

and we also set forℓ∈Z₊

ηℓ=ρ02ρˆ 2 0ρ

2 1ρˆ

2 1. . .ρ

2

ℓρˆ2ℓ.

Proposition 14. Forℓ≤n₀we have

E”∆1/2ϕℓ,λ

ϕ_ℓ,λ=x —

= 1

n0

b(_λ1)(t) + 1

n0

osc(1)+_O(k−3/2) =_O(k−1)

E”∆1_/2ϕℓ,λ∆1/2ϕℓ,λ′

ϕ_ℓ,λ= x,ϕ_ℓ,λ′= y

— = 1

n₀a

(1)_{(t,x,y) +} 1

n₀osc

(2)₊

O(k−3/2)

E”∆1/2ϕℓ+1/2,λ

ϕ_ℓ+1/

2,λ=x

— = 1

n₀b

(2) λ (t) +

1 n₀osc

(3)₊

O(k−3/2) =_O(k−1)

E”∆1_/2ϕℓ+1_/2,λ∆1/2ϕℓ+1_/2,λ′

ϕ_ℓ+1_/2,λ=x,ϕℓ+1_/2,λ′= y

— = 1

n0

a(2)(t,x,y) + 1

n0

osc(4)+_O(k−3/2),

E”_|∆1_/2ϕℓ,λ|d

ϕ_ℓ,λ=x —

,E”_|∆1_/2ϕℓ+1_/2,λ|d

ϕ_ℓ,λ=x —

=_O(k−d/2), d=2, 3 where t=ℓ/n₀,

b(_λ1)= ₄λ_ˆs+_2pℜρˆ_ˆs+₂ℑ_βˆρˆ_s22, b

(2)

λ =

λ 4ˆs−

ℜ_{d t}d ρ

ℑρ +

ℑρ2 2βˆs2,

a(1)=_β1_ˆ s2ℜ

”

ei(x−y)—+1+_βˆℜρˆ2

s2 , a

(2)₌ 1

βˆs2ℜ ”

ei(x−y)—+1+ℜρ 2

βˆs2 .

The oscillatory terms are

osc(1) = _ℜ((₋vℓ−iq(1)/2)e−i xρˆ_ℓ−2ηℓ) +ℜ(ie−2i xρˆ−_ℓ4η2ℓq (1)₎

/4,

osc(2) = q(3)_ℜ(e−i xρˆ_ℓ−2ηℓ+e−i yρˆℓ−2ηℓ)/2+ℜ(q(1)(e−i xρˆℓ−2ηℓ+e−i yρˆℓ−2ηℓ+e−i(x+y)ρˆℓ−4η

2

ℓ))/2,

osc(3) = _ℜ((₋ˆvℓ−iq(2)/2)e−i xηℓ) +ℜ(ie−2i xη2ℓq (2)_)/4,

osc(4) = q(3)ℜ(e−i xηℓ+e−i yηℓ)/2+ℜ(q(2)(e−i xηℓ+e−i yηℓ+e−i(x+y)η2ℓ))/2.

Proof. We start with the identity

ϕℓ+1/2,λ−ϕℓ,λ=ϕℓ+1,λ∗Qˆ_ℓ−1−ϕℓ,λ∗Qˆ_ℓ−1=ϕℓ,λ∗Qˆ_ℓ−1Sℓ,λ−ϕℓ,λ∗Qˆ−_ℓ1=ash(Sℓ,λ,eiϕℓ,λη¯ℓρˆ−_ℓ2,−1).

Here we used the definition of the angular shift with the fact that Sℓ,λ (and any affine

transfor-mation) will preserve_{∞ ∈} Hwhich corresponds to ₋1 in U. A similar identity can be proved for

∆1_/2ϕℓ+1_/2,λ.

The proof now follows exactly the same as in[13], it is a straightforward application of Lemma 12 using the estimates onv_ℓ,λ,ˆv_ℓ,λ,V_ℓ,Vˆ_ℓ.

5.2

The continuum limit

In this section we will prove thatϕ(n)_{(t,λ) =ϕ}

⌊t n0⌋,λ converges to the solution of a one-parameter

family of stochastic differential equations on t_∈[0, 1). The main tool is the following proposition, proved in[13](based on[11]and[4]).

Proposition 15. Fix T >0, and for each n≥1consider a Markov chain X_ℓn∈Rdwithℓ=1, . . . ,⌊nT⌋. Let Y_ℓn(x)be distributed as the increment X_ℓ+n ₁₋x given X_ℓn=x. We define

whereR_ℓ,λ= ˆM−_ℓ1ˆJ_ℓ−1M−_ℓ1J−_ℓ1. From (31) and (32) we get

r.R_n−j,λ=

rs2_n−j+rs_n−jY_n−j−s_n−jΛ

p_n−jp_n−j+1−Λ2+pn−j+1Xn−j+rsn−jΛ +r Yn−jΛ

whereΛ =µn+λ/(4n

1/2

0 ). Using (12),sn−j= p

j₋1/2,p_n−j=pm₋n+ j₋1/2 andm₋n_{→ ∞}

we get thatr.Rn−j,λ−r.Rn−j,0

P

−→0 for any fixed j. This leads to α⊙_n

2,λ=ϕ ⊙

n−ξ,λ∗Q−1n−ξ−1−ϕn⊙−ξ,0∗Q−1n−ξ−1→0 asn→ ∞.

If lim_n→n1=∞then we will need the edge scaling results proved in[9]which are summarized in Theorem 3 of the introduction. The initial condition f(0) =0,f′(0) =1 for the operatorHβ in the theorem comes from the fact that the discrete eigenvalue equation forA_n,mAT_n,m with an eigenvalue Λ′= (pn+pm)2+(p₍m+pn)4/3

mn)1/6 ν is equivalent to a three-term recursion for the vector entries wℓ,Λ

(c.f. (18)) with the initial conditionw_0,ν=0 andw1,ν6=0.

By [9], Remark 3.8, the results of[9] extend to solutions of the same three-term recursion with more general initial conditions. We say that a value ofν is an eigenvalue for a family of recursions parametrized byν if the corresponding recursion reaches 0 in its last step. Suppose that for given ζ∈[_−∞,_∞]the initial condition for the three-term recursion equation satisfies

(mn)−1/3 (pm+pn)−2/3

w0,ν

(w1,ν−w0,ν)

= (mn) −1/3

(pm+pn)−2/3(r0,ν−1) −1 P

−→ζ, (76)

uniformly inν withr₀:=w_1,ν/w_0,ν. Here the factor (mn)1/3

(pm+pn)2/3 is the spatial scaling for the problem ([9], Section 5). Then the eigenvalues of this family of recursions converge to those of the stochastic Airy operator with initial condition f(0)/f′(0) = ζ. The corresponding point process Ξζ is also

a.s. simple (i.e. there are no multiple points) and it will satisfy the following non-atomic property: for any x ∈ R we have P(x ∈ Ξ_ζ) = 0 (see [9], Remark 3.8). Similar statement holds at the lower edge if lim inf_n→∞m/n> 1 with (r_n,ν +1)−1 in (76). (In this case one first multiplies the

off-diagonal entries of A_n,mAT_n,m by₋1 before applying the arguments of [9], this will not change the eigenvalues.)

Ifm/n→γ∈[1,_∞)then we can rewrite (76) jointly for the upper and lower soft edge as γ−1/3

(pγ±1)−2/3n −1/3_(r

0,ν∓1)−1 P

−→ζ, . (77)

The multiplier is 1 in the caseγ=_∞and it is always a finite nonzero value unlessγ=1 and we are at the lower edge.

Proof of Proposition 24 iflim_n→∞n₁=_∞. By taking an appropriate subsequence, we may assume thatµn−

p

m−nis always positive or always negative. According to the proof of Lemma 34 in[13] we need to consider the family of recursions

with initial condition

rn2,ν =x.T −1

n2 =ℑ(ρn2)x+ℜ(ρn2)

for a givenx ∈Rand show that the probability of having an eigenvalue in[0,λ]converges to 0 as

n_{→ ∞}. We introduce

n∗=n−n2, m∗=m−n2.

Note that the recursion is determined by the bottom(2n∗)_×(2n∗)submatrix of ˜AD_n,(n)_m in (29) which has the exact same distribution as the matrix ˜AD_n∗(n∗)_,m∗. Thus we can consider the eigenvalue equation

for ˜AD_n∗(n_,∗_m)∗ with generalized initial condition

r_0,∗_λ=_ℑ(ρn2)x+ℜ(ρn2).

(We will use_∗to denote that we are working with the smaller matrix.) We will transform this back to an eigenvalue equation for ˜An∗_,m∗ and thenA_n∗_,m∗AT_n∗,m∗ with a generalized initial condition.

Recall the recursion (31) and thatr_ℓ∗=u∗_ℓ+1/v_ℓ∗,ˆr_ℓ∗=v_ℓ∗/u∗_ℓ. From this we get

v₁∗

v₀∗ = r

∗ 0ˆr1∗=r0∗

‚ −_r1_∗

0

· s

∗ 0

p∗₀ +

Λ

p∗₀ Œ ‚

1+ X ∗ 0

p∗₀ Œ−1

=

‚ −s

∗ 0

p∗₀ + (ℑ(ρn2)x+ℜ(ρn2)) Λ

p₀∗ Œ

βp∗₁p∗₀

˜ χ_β2_(m∗₋₁₎

(78) whereΛ =µn+₄pλ_n

0. Denote the solution of the generalized eigenvalue equation for ˜An∗,m∗ corre-sponding toΛwith ˜u∗₁, ˜v₁∗, . . . . If we remove the conjugation with D(n∗)from ˜AD_n∗(n∗)_,m∗ then from (78)

we get

˜

v₁∗

˜

v₀∗ = ‚

−s

∗ 0

p∗₀ + (ℑ(ρn2)x+ℜ(ρn2)) Λ

p₀∗ Œ

χβ(n∗₋₁₎p₀∗

˜

χβ(m∗−1)s1∗

. (79)

By Remark 8 this is exactly the initial condition for the generalized eigenvalue equation for

A_n∗_,m∗AT_n∗_,m∗ with eigenvalueΛ2=µ2_n+

µnλ

2pn0+

λ2 16n0.

The spectrum of the matrixA_n∗_,m∗AT_n∗_,m∗ is concentrated asymptotically to[(

p

m∗₋p_n∗₎2_,(p_m∗+

p

n∗)2_{]. A direct computation shows that}p_m

1±pn1=µn, where we have+ifµn− p

m−n>0 and

−otherwise. This means that ifµn< p

m₋nthen our original bulk scaling aroundµn corresponds

to the lower edge scaling of An∗_,m∗A_nT∗_,m∗ and ifµn > p

m−nthen we get the upper edge scaling. (Note that because of our assumptions ifµn<

p

m₋nthan lim infm/n>1.)

Again, by taking an appropriate subsequence, we may assume that m∗/n∗_→γ∗∈[1,_∞]. We first check that the initial condition (79) satisfies (77), this is equivalent to showing that(n∗)1/3



˜

v₁∗

˜

v₀∗ ∓1 ‹

converges in probability to a constant. Sincen_{1→ ∞}we have

m∗>n∗→ ∞, ρn2=±1+i

p

K(n∗)−1/3+o((n∗)−1/3),

n₁=n∗_{− K}(n∗)1/3, m₁=m∗_{− K}(n∗)1/3, µn=

p

m∗_±p_n∗₋K 2 (n

∗₎1/3€

Sinceℓ−1/2χℓ P

−→1 asℓ→ ∞we get that in probability lim(n∗)1/3

‚

˜

v₁∗

˜

v₀∗∓1 Œ

=p_Kx€1_±(γ∗)−1/2Š

and the convergence is uniform if we assume thatλis bounded. This means that we may apply the edge scaling result, we just need to convert the scaling from the bulk to the edge. In the bulk we were interested in the interval I = [µ2_n,(µn+λn−

1/2 0 /4)

2_{]. If we apply the edge scaling, then this} interval becomes (m∗n∗)1/6

(pm∗_±p_n∗₎4/3(I−(

p

m∗_±p_n∗₎2_{). Using our asymptotics forµ}

n we get that for the

end point of the interval (m∗n∗)1/6 (pm∗±pn∗)4/3(µ

2

n−( p

m∗±pn∗)2)_{→ −}K

2 (1±(γ

∗₎−1/2₎1/6_, and for the length we get

lim

n→∞

(µn+λn−

1/2 0 /4)

2

−µ2_n (m ∗_n∗₎1/6 (pm∗_±p_n∗₎4/3 =

(γ∗)1/6

(pγ∗_±1)1/3nlim→∞ λ

p_n

0 =0.

(We used that ifγ∗=1 then we must have+in the±.) This means that after the edge rescaling the interval shrinks to a point, meaning that the probability that our original recursion has an eigenvalue in[0,λ]converges to the probability that the limiting edge point process has a point at a given value which is equal to 0.

8

Appendix

In this section we provide the needed oscillation estimates.

Lemma 25. Suppose that2π > θ1> θ2>. . .> θm>0and let sℓ= Pℓ

j=1θj. Then

| m X

j=1

eisj_{| ≤c(θ}−1

m + (2π−θ1)−1)≤c′(|eiθm/2−1|−1+|eiθ1/2+1|−1).

Proof. The first inequality is the same as Lemma 36 from[13]and the second inequality is straight-forward.

Lemma 26. Let1_≤ℓ1< ℓ2≤n0and F

(i)

ℓ1,ℓ2=

Pℓ2

j=ℓ1η

i

j for i=1, 2and gℓ∈C. Then for i=1, 2:

ℜ ℓ2

X

ℓ=ℓ1

g_ℓηi_ℓ

≤ |F_ℓ(i)

1,ℓ2||gℓ2|+

ℓ2−1

X

ℓ=ℓ1

|F_ℓ(i)

1,ℓ||gℓ+1−gℓ|, (80)

ℜ ℓ2

X

ℓ=ℓ1

gℓηiℓ

≤ |F_ℓ(i)

1,ℓ2|gℓ1||+

ℓ2−1

X

ℓ=ℓ1

|F_ℓ(₊i)_1,ℓ

We also have the following estimates: |F_ℓ(1)

1,ℓ2| ≤ c(1+n 1/2 1 k

−1/2

2 ) (82)

|F_ℓ(2)

1,ℓ2| ≤





 c

|ρ_ℓ2

1ρˆ 2

ℓ1+1|

−1_+1+n1/2 1 k−

1/2 2

if (⋆) or (⋆⋆),

c

|ρ_ℓ2 2ρˆ

2

ℓ2+1|

−1_+1+n1/2 1 k

−1/2 1

if (⋆ ⋆ ⋆)

(83)

where the conditions are given by

(⋆) : µn≥ p

m₋nwithk₀,k_1≥pm₁n₁ or k₀,k_1≤pm₁n₁, (⋆⋆) : µn<

p

m−nwithk0,k1≥pm1n1, (84)

(⋆ ⋆ ⋆) : µn< p

m−nwithk0,k1≤pm1n1.

Proof. The bounds (80) and (81) follow from partial summation. In order to prove the bounds on

F_ℓ(i)

1,ℓ2 we will apply Lemma 25, but we need to consider various cases. Note that the constant c might change from line to line.

Case 1:µn≥ p

m₋n,_ℜρℓ≥0.

We have the bounds

k1/2(n₁+k)−1/2_≤argρℓ, n

1/2

1 (n1+k)−1/2≤π/2−argρℓ, k1/2(m₁+k)−1/2_≤argρˆℓ, m11/2(m1+k)−1/2≤π/2−argρˆℓ.

and arg(ρℓρˆℓ) is decreasing. The sequence arg( ˆρ2ℓρℓ2) satisfies the conditions of Lemma 25 and

usingm₁>n₁ andm₁>cn>cn₀ we get the bound

| ℓ2

X

ℓ=ℓ1 ηℓ| ≤c

(n₁+k₂)1/2k−₂1/2+ (k₁+m₁)1/2m−₁1/2

≤c(n1₁/2k₂−1/2+1). We have

ρℓρˆℓ=

p_m

1n1−k

p k+m1

p k+n1

+ i

p

k pm₁+pn₁ p

k+m1

p k+n1

which means that ifk1,k2≥pm1n1ork1,k2≤pm1n1then we can use Lemma 25 for the sequence arg( ˆρ_ℓ4ρ_ℓ4). (In the first caseπ/2≤arg( ˆρℓρℓ)< πwhile in the second case 0<arg( ˆρℓρℓ)≤π/2.)

From the lemma we get

| ℓ2

X

ℓ=ℓ1

η2_ℓ| ≤c

|ρ2_ℓ 1ρˆ

2

ℓ1+1| −1₊

|ρ_ℓ2 2ρˆ

2

ℓ2−1| −1

and explicit computation together withm₁>cn>ck gives

|ρ_ℓ2ρˆ_ℓ2−1|−1=

p k+m1

p k+n1 2pk pm1+pn1

≤c(1+n

1/2 1 k−

This finishes the proof of the lemma in this case. Case 2:µn<

p

m−n,_ℜρℓ<0.

Now we have

ρℓρˆℓ=−

k+pm1pn1

p

k+m₁pk+n₁

+ i

p

k pm1−pn1

p

k+m₁pk+n₁

. (85)

meaningπ/2<arg( ˆρℓρℓ)< π. Differentiation of the real part shows that argρℓρˆℓ decreases ifk> p_m

1n1and then it increases. This means that we can apply Lemma 25 for the sequences arg( ˆρ2ℓρℓ2)

or arg( ˆρ_ℓ4ρ4_ℓ) if k₁,k_{2 ≥} pn₁m₁ and the reversed versions of these sequences if k₁,k_{2 ≤} pn₁m₁. From (85) we get

π≤argρ_ℓ2ρˆ2_ℓ, 2π−argρ_ℓ2ρˆ2_ℓ ≥ 2 p

k pm₁₋pn₁ p

k+m₁pk+n₁ ≥c

(k−1/2n1₁/2+1)−1 (86)

where we used pm1−n1 ≥ cpm1 which follows from m1 > cn > cn1. Applying Lemma 25 to arg( ˆρ_ℓ2ρ_ℓ2)(or the reversed sequence) we get

| ℓ2

X

ℓ=ℓ1 ηℓ| ≤

¨

c(n1₁/2k−₁1/2+1) ifk₁,k_2≥pn₁m₁,

c(n1₁/2k−₂1/2+1) ifk₁,k_2≤pn₁m₁.

Noting thatk−1₁ /2 ≤k₂−1/2 this proves (82) in Case 2 ifk1,k2 ≤pn1m1 or k1,k2≥pn1m1. Ifk1 >

p_n

1m1 andk2 <pn1m1 then we can cut the sum in to parts atpn1m1 and since fork≥pm1n1 we haven1₁/2k−1/2_≤1 we have (82) in this case as well.

To prove (83) we apply Lemma 25 to arg( ˆρ_ℓ4ρ4_ℓ)(or its reversed) to get

| ℓ2

X

ℓ=ℓ1 η2_ℓ| ≤





 c

|ρ2_ℓ

1ρˆ 2

ℓ1+1| −1₊

|ρ_ℓ2 2ρˆ

2

ℓ2−1|

−1 _ifk

1,k2≥pn1m1,

c

|ρ2_ℓ 2ρˆ

2

ℓ2+1| −1₊

|ρ_ℓ2 1ρˆ

2

ℓ1−1|

−1 _ifk

1,k2≤pn1m1. From this (83) follows by noting that

|ρ_ℓ2ρˆ_ℓ2−1_|−1=

p

k+m₁pk+n₁

2pk pm1−pn1

≤c(n

1/2 1 k−

1/2_+1). where the first equality is explicit computation and the inequality is from (86).

Acknowledgments. The authors wish to thank the anonymous referees for valuable comments and suggestions.

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