Bulk scaling limit of the Laguerre ensemble

We consider the beta-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 ensembles exists for all beta>0 for a general family of parameters and it is the same as the bulk scaling limit of the corresponding beta-Hermite ensemble.


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 A is an n × (m + 1) matrix with i.i.d. standard normal entries. One can also define versions with i.i.d. complex or real quaternion standard normal random variables. Since we are only interested in the eigenvalues, we can assume m + 1 ≥ n.
Then the joint eigenvalue density on R n + exists and it is given by the following formula: where β = 1, 2 and 4 correspond to the real, complex and quaternion cases respectively and Z β n,m+1 is an explicitly computable constant. The density (1) defines a distribution on R n + for any β > 0, n ∈ N and m > n. The resulting family of distributions is called the β-Laguerre ensemble. Note that we intentionally shifted the parameter m by one as this will result in slightly cleaner expressions later on.
Another important family of distributions in random matrix theory is the Hermite (or Gaussian) β-ensemble. It is described by the density function (2) on R n . For β = 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 by √ n then the empirical spectral density converges to the Wigner semicircle distribution 1 In [13] the authors derive the bulk scaling limit of the β-Hermite ensemble, i.e. the point process limit of the spectrum it is scaled around a sequence of points away from the edges.
Note that the condition on µ n means that we are in the bulk of the spectrum, not too close to the edge. The limiting point process Sine β 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 2.4 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. If m/n → γ ∈ [1, ∞) then with probability one the scaled empirical spectral measures ν n = 1 n n k=1 δ λ k /n converge weakly to the Marchenko-Pastur distribution which is a deterministic measure with densitỹ This can be proved by the moment method or using Stieltjes-transform. (See [7] for the original proof and [5] for the general β case).
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 8 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 general β. 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 random variables one uses more general distributions in the construction described at the beginning of this section. The recent papers of Tao and Vu [12] and Erdős et al. [3] provide the bulk scaling limit in these cases.
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 where Airy β is a discrete simple point process given by the eigenvalues of the stochastic Airy operator 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: if lim inf m/n > 1 then 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 inf m/n > 1 is not optimal, the statement is expected to hold with m − n → ∞. This has been known for the classical cases β = 1, 2, 4 [8].
If m − 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 where Θ β,a is a simple point process that can be described as the sequence of eigenvalues of a certain random 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.

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. Let A n,m be the following n × n bidiagonal matrix: where χ βa ,χ βb are independent chi-distributed random variables with the appropriate pa- . Then the eigenvalues of the tridiagonal matrix A n,m A T n,m are distributed according to the density (1). If we want to find the bulk scaling limit of the eigenvalues of A n,m A T n,m then it is sufficient to understand the scaling limit of the singular values of A n,m .The following simple lemma will be a useful tool for this. Lemma 6. Suppose that B is an n × n bidiagonal matrix with a 1 , a 2 , . . . , a n in the diagonal and b 1 , b 2 , . . . , b n−1 below the diagonal. Consider the 2n × 2n symmetric tridiagonal matrix M which has zeros in the main diagonal and a 1 , b 1 , a 2 , b 2 , . . . , a n in the off-diagonal. If the singular values of B are λ 1 , λ 2 , . . . , λ n then the eigenvalues of M are ±λ i , i = 1 . . . n.
We learned about this trick from [2], we reproduce the simple proof for the sake of the reader.
Because of the previous lemma it is enough to study the eigenvalues of the (2n) × (2n) tridiagonal matrix The main advantage of this representation, as opposed to studying the tridiagonal matrix A n,m A T n,m , is that here the entries are independent modulo symmetry.
Remark 7. Assume that [u 1 , v 1 , u 2 , v 2 , . . . , u n , v n ] T is an eigenvector forÃ n,m with eigenvalue λ. Then [u 1 , u 2 , . . . , u n ] T is and eigenvector for A T n,m A n,m with eigenvalue λ 2 and [v 1 , v 2 , . . . , v n ] T is an eigenvector for A n,m A T n,m with eigenvalue λ 2 .

Bulk limit of the singular values
We can compute the asymptotic spectral density ofÃ n,m from the Marchenko-Pastur distribution. If m/n → γ ∈ [1, ∞) then the asymptotic density (when scaled with √ n) is This means that the spectrum ofÃ n,m in R + is asymptotically concentrated on the interval in a way that it is not too close to the edges. Near µ n the asymptotic eigenvalue density should be close to σ m/n (µ n / √ n) which explains the choice of the scaling parameters in the following theorem.
Assume that as n → ∞ we have and lim inf n→∞ m/n > 1 or lim n→∞ m/n = 1 and lim inf µ n / √ n > 0. Then The extra 1/2 in the definition of n 0 is introduced to make some of the forthcoming formulas nicer. We also note that the following identities hold: Note that we did not assume that m/n converges to a constant or that µ n = c √ n. By the discussions at the beginning of this section (Λ n ∩ R + ) 2 is distributed according to the Laguerre ensemble. If we assume that m/n → γ and µ n = √ c √ n with c ∈ (a(γ) 2 , b(γ) 2 ) then both (9) and (10) are satisfied. Since in this case n 0 n −1 →σ γ (c) the result of Theorem 8 implies Theorem 2.
Remark 9. We want prove that the weak limit of 4 √ n 0 (Λ n −µ n ) is Sine β , thus it is sufficient to prove that for any subsequence of n there is a further subsequence so that the limit in distribution holds. Because of this by taking an appropriate subsequence we may assume These assumptions imply that for m 1 = m − n + n 1 we have One only needs to check this in the m/n → 1 case, when from (13) and the definition of n 1 we get n 1 /n → c > 0.
Remark 10. The conditions of Theorem 8 are optimal if lim inf m/n > 1 and the theorem provides a complete description of the possible point process scaling limits of Λ L n . To see this first note that using Λ L n = (Λ n ∩ R + ) 2 we can translate the edge scaling limit of Theorem 3 to get If lim inf m/n > 1 then by the previous remark we may assume lim m/n = γ ∈ (1, ∞]. Then the previous statement can be transformed into n 1/6 (Λ n − ( √ m ± √ n)) d =⇒ Ξ where Ξ is a a linear transformation of Airy β . From this it is easy to check that if n 1/3 1 n −1 0 → c ∈ (0, ∞] then we need to scale Λ n − µ n with n 1/6 to get a meaningful limit (and the limit is a linear transformation of Airy β ) and if n 1/3 1 n −1 0 → 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. Here the statement should be true with the following condition instead of (10):

Counting eigenvalues of tridiagonal matrices
Assume that the tridiagonal k × k matrix M has positive off-diagonal entries.
If u = [u 1 , . . . , u k ] T is an eigenvector corresponding to λ then we have where we can we set u 0 = u k+1 = 0 (with c 0 , b k defined arbitrarily). This gives a single term recursion on R ∪ {∞} for the ratios r ℓ = u ℓ+1 u ℓ : This recursion can be solved for any parameter λ, and λ is an eigenvalue if and only if r k = r k,λ = 0.
We do not need to fully solve the recursion (17) in order to count eigenvalues. If we consider the reversed version of (17) started from index k with initial condition 0: then λ is an eigenvalue if and only if r ℓ,λ = r ⊙ k−ℓ,λ . Moreover, if we turn r ⊙ ℓ,λ into an angle φ ⊙ ℓ,λ (similarly as before for r and φ) we can also count eigenvalues in the interval [λ 0 , λ 1 ] by the formula In our case, by analyzing the scaling limit of a certain version of the phase function φ ℓ,λ we can identify the limiting point process. This method was used in [13] for the bulk scaling limit of the β Hermite 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.

The Sine β process
The distribution of the point process Sine β 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 for t ∈ [0, 1), λ ∈ R : It was proved in [13] that for any given λ the limit N(λ) = 1 2π lim t→1 α λ (t) exists, it is integer valued a.s. and N(λ) has the same distribution as the counting function of the point process Sine β evaluated at λ. Moreover, this is true for the joint distribution of (N(λ i ), i = 1, . . . , d) for any fixed vector (λ i , i = 1, . . . , d). Recall that the counting function at λ > 0 gives the number of points in the interval (0, λ], and negative the number of points in (λ, 0] for λ < 0.

The main steps of the proof of Theorem 8
The proof will be similar to one given for Theorem 1 in [13]. The basic idea is simple to explain: we will define a version of the phase function and the target phase function for the rescaled eigenvalue equation and consider (19) with a certain ℓ = ℓ(n). We will then show that the length of the interval in the left hand side of the equation converges to 2π(N(λ 1 ) − N(λ 0 )) while the left endpoint of that interval becomes uniform modulo 2π. This shows that the scaling limit of the eigenvalue process is given by Sine β .
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 8. Recall that Λ n denotes the multi-set of eigenvalues for the matrixÃ n,m which is defined in (6). We denote by N n (λ) the counting function of the scaled random multi-sets 4n where N(λ) = 1 2π lim t→1 α λ (t) as defined using the SDE (20). We will use the ideas described in Subsection 2.3 to analyze the eigenvalue equatioñ A n,m x = Λx, where x ∈ R 2n . Following the scaling given in (11) we set In Section 4 we will define the phase function ϕ ℓ,λ and the target phase function ϕ ⊙ ℓ,λ for ℓ ∈ [0, n 0 ). These will be independent of each other for a fixed ℓ (as functions in λ) and satisfy the following identity for λ < λ ′ : The phase function ϕ will be a regularized version of the phase function obtained from the ratio of the consecutive elements of the eigenvector. The regularization is needed in order to have a phase function which is asymptotically continuous. Indeed, in Proposition 16 of Section 5 we will show that for any 0 < ε < 1 the rescaled version of the phase function converges to a one-parameter family of stochastic differential equations.
Moreover we will prove that in the same region the relative phase function α ℓ,λ = ϕ ℓ,λ − ϕ ℓ,0 will converge to the solution α λ of the SDE (20) in the sense of finite dimensional distributions in λ. This will be the content of Corollary 17.
Next we will describe the asymptotic behavior of the phase functions ϕ ℓ,λ , α ℓ,λ and ϕ ⊙ (The constants ε, K will be determined later.) We will show that if the relative phase function is already close to an integer multiple of 2π at ⌊n 0 (1 − ε)⌋ then it will not change too much To be more precise, in Proposition 18 of Section 6 we will prove that there exists a constant c = c(λ, β) so that we have We will also show that if K → ∞ and K(n 1/3 1 ∨ 1)n −1 0 → 0 then the random angle ϕ n 2 ,0 becomes uniformly distributed module 2π as n → ∞ (see Proposition 22).
Next we will prove that the target phase function will loose its dependence on λ: for every λ ∈ R and K > 0 we have This will be the content of Proposition 23 in Section 7.
The proof now can be finished exactly the same way as in [13]. We can choose ε = ε(n) → 0 and K = K(n) → ∞ so that the following limits all hold simultaneously: This means that if we apply the identity (22) with λ = 0, λ ′ = λ i and ℓ = n 2 then the length of the random intervals converge to 2πN(λ i ) in distribution (jointly), while the common left endpoint of these intervals becomes uniform modulo 2π. (Since ϕ n 2 ,0 and ϕ ⊙ n 2 ,0 are independent and ϕ n 2 ,0 converges to a uniform distribution mod 2π.) This means that #{2kπ ∈ I i : k ∈ Z} converges to N(λ i ) which proves (21) and Theorem 8.

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

The eigenvalue equations
The first couple of moments of these random variables are explicitly computable using the moment generating function of the χ 2 -distribution and we get the following asymptotics: where the constants in the error terms only depend on β.
We consider the eigenvalue equation forÃ D n,m with a given Λ ∈ R and denote a nontrivial solution of the first 2n − 1 components by u 1 , v 1 , u 2 , v 2 , . . . , u n , v n . Then we have where we set v 0 = 0 and we can assume u 1 = 1 by linearity. We set r ℓ = r ℓ, with initial condition r 0 = ∞. We can set Y n = 0 and define r n via (29) with ℓ = n − 1, then Λ is an eigenvalue if and only if r n = 0.

The hyperbolic point of view
We use the hyperbolic geometric approach of [13] to study the evolution of r andr. 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 preserving H by PSL(2, R). The recursions for both r andr evolve by elements of this group of the form via the conformal bijection U(z) = i−z 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 to R, the universal cover of ∂U, each element T in PSL(2, R) becomes an R → R function. The lifted versions are uniquely determined up to shifts by 2π and will also form a group which we denote by UPSL(2, R). For any T ∈ UPSL(2, R) we can look at T as a function acting on ∂H , ∂U or R. We will denote these actions by: For every T ∈ UPSL(2, R) the function x → f (x) = x * T is monotone, analytic and quasiperiodic modulo 2π: f (x + 2π) = f (x) + 2π. It is clear from the definitions that e ix •T = e if (x) and (2 tan(x)).T = 2 tan f (x). Now we will introduce a couple of simple elements of UPSL(2, R). For a given α ∈ R we 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 by A(a, b) the affine map z → a(z + b) in H . This is an element of PSL(2, R) which fixes ∞ in H and −1 in ∂U. We specify its lifted version in UPSL(2, R) by making it fix π, this will uniquely determines it as a R → R function.
Given T ∈ UPSL(2, R), x, y ∈ R we define the angular shift which gives the change in the signed distance of x, y under T. This only depends on v = e ix , w = e iy and the effect of T on ∂U, so we can also view ash(T, ·, ·) as a function on ∂U × ∂U and the following identity holds: The following lemma appeared as Lemma 16 in [13], it provides a useful estimate for the angular shift. Then where for d = 1, 2, 3 and an absolute constant c we have If v = −1 then the previous bounds hold even in the case |z| > 1 3 .

Regularized phase functions
Because of the scaling in (11) we will set We introduce the following operators Then (28) and (29) can be rewritten as (We suppressed the λ dependence in r and the operators M,M.) Lifting these recursions from ∂H to R we get the evolution of the corresponding phase angle which we denote by Solving the recursion from the other end, with end condition 0 we get the target phase It is clear that φ ℓ,λ and φ ⊙ ℓ,λ are independent for a fixed ℓ (as functions in λ), they are monotone and analytic in λ and we can count eigenvalues using the formula (19).
Note that J ℓ andĴ ℓ do not depend on λ and they are not infinitesimal. The main part of the evolution is J ℓĴℓ . This is a rotation in the hyperbolic plane if it only has one fixed point in H. The fixed point equation ρ ℓ = ρ ℓ .J ℓĴℓ can be rewritten as This can be solved explicitly, and one gets the following unique solution in the upper half plane if ℓ < n 0 : (One also needs to use the identity p 2 ℓ − s 2 ℓ = m − n.) This shows that if ℓ < n 0 then J ℓĴℓ is a rotation in the hyperbolic plane. We can move the center of rotation to 0 in U by conjugating it with an appropriate affine transformation: In order to regularize the evolution of the phase function we introduce where Q ℓ = Q(2 arg(ρ 0ρ0 )) . . . Q(2 arg(ρ ℓρℓ )) and Q −1 is the identity. It is easy to check that the initial condition remains ϕ 0,λ = π. Then Note that the evolution operator is now infinitesimal: M ℓ ,M ℓ and T −1 ℓ T ℓ+1 are all asymptotically small, and the various conjugations will not change this.
We can also introduce the corresponding target phase function The new, regularized phase functions ϕ ℓ,λ and ϕ ⊙ ℓ,λ have the same properties as φ, φ ⊙ , i.e.: they are independent for a fixed ℓ (as functions in λ), they are monotone and analytic in λ and we can count eigenvalues using the formula (22).
We will further simplify the evolution using the following identities: From this we getĴ This allows us to write whereQ ℓ = Q ℓ Q(−2 arg(ρ ℓ )). Thus We will introduce the following operators to break up the evolution into smaller pieces: Then

SDE limit for the phase function
Let F ℓ denote the σ-field generated by ϕ j,λ , j ≤ ℓ. Then ϕ ℓ,λ is a Markov chain in ℓ with respect to F ℓ . We will show that this Markov chain converges to a diffusion limit after proper normalization. In order to do this we will estimate E [ϕ ℓ+1,λ − ϕ ℓ,λ |F ℓ ] and using the angular shift lemma, Lemma 11.

Single step estimates
Throughout the rest of the proof we will use the notation k = n 0 − ℓ. We will need to rescale the discrete time n 0 in order to get a limit, we will use t = ℓ/n 0 and also introducê We start with the identity Note that this means that where the sign in ℜρ ℓ is positive if µ n > √ m − n and negative otherwise.
For the angular shift estimates we need to consider We have the following estimates for the deterministic part (by Taylor expansion): where p(t) = p (n) (t) = m/n 0 − t and ρ(t) = ρ (n) (t),ρ(t) =ρ n (t) are defined by equations (38) and (39) with ℓ = n 0 t. For the random terms from (27) we get where the constants in the error term only depend on β and We introduce the notations Remark 12. We would like to note that the 'half-step' evolution rules ϕ ℓ,λ → ϕ ℓ+ 1 /2 ,λ , ϕ ℓ+ 1 /2 ,λ → ϕ ℓ+1,λ are very similar to the one-step evolution of the phase function ϕ in [13].
Besides the fact that the ℓ → ℓ + 1 /2 and ℓ + 1 /2 → ℓ + 1 steps are quite different from each other the other big difference between our case and [13] is that here the oscillating terms Q ℓ ,Q ℓ are more complicated.
The following proposition is the analogue of Proposition 22 in [13].
Proposition 13. For ℓ ≤ n 0 we have The oscillatory terms are Proof. We start with the identity Here we used the definition of the angular shift with the fact that S ℓ,λ (and any affine transformation) will preserve ∞ ∈ H which corresponds to −1 in U. A similar identity can be proved for ∆1 /2 ϕ ℓ+ 1 /2 ,λ .
The proof now follows exactly the same as in [13], it is a straightforward application of Lemma 11 using the estimates on v ℓ,λ ,v ℓ,λ , V ℓ ,V ℓ .

The continuum limit
In this section we will prove that ϕ (n) (t, λ) = ϕ ⌊tn 0 ⌋,λ converges to the solution of a oneparameter 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 14.
Fix T > 0, and for each n ≥ 1 consider a Markov chain X n ℓ ∈ R d with ℓ = 1 . . . ⌊nT ⌋. Let Y n ℓ (x) be distributed as the increment X n ℓ+1 − x given X n ℓ = x. We define Suppose that as n → ∞ we have and that there are functions a, b from R × [0, T ] to R d 2 , R d respectively with bounded first and second derivatives so that Assume also that the initial conditions converge weakly, X n 0 d =⇒ X 0 .
Then (X n ⌊nt⌋ , 0 ≤ t ≤ T ) converges in law to the unique solution of the SDE where B is a d-dimensional standard Brownian motion and σ : R d × [0, T ] is a square root of the matrix valued function a, i.e. a(t, x) = σ(t, x) σ(t, x) T .
We will apply this proposition to ϕ ℓ,λ with ℓ ≤ n 0 (1 − ε) and ℓ ∈ Z/2, so the single steps of the proposition correspond to half steps in our setup.
The following lemma shows that the oscillatory terms in the estimates of Proposition 13 average out in the 'long run'. Its proof relies on Proposition 13 and Lemma 25 of the Appendix.
Lemma 15. Let |λ|, |λ ′ | ≤λ and ε > 0. Then for any where t = ℓ/n 0 , the functions b λ , a are defined as and the implicit constants in O depend only on ε, β,λ. The indices in the summation ∼ run through half integers.
Proof of Lemma 15. We will only prove the first statement, the second one being similar.
λ (t). Summing the first and third estimates in Proposition 13 we get (45) with an error term 1 n 0 where the first two terms will be denoted ζ 1 , ζ 2 . Here where for this proof c denotes varying constants depending on ε. Using the fact that v λ ,v λ , q (1) , q (2) and their first derivatives are continuous on [0, 1 − ε] we get Applying Lemma 25 of the Appendix to the first sum in (47): Since ℓ 1 ≤ n 0 (1 − ε) we have |F (Recall that k = n 0 − ℓ.) For the estimate of ζ 2 we first note that We will use Lemma 25 if |ρ 2 ℓ ρ 2 ℓ + 1| is 'big', and a direct bound with (49) if it is small. To be more precise: we divide the sum into three pieces, we cut it at indices ℓ * 1 and ℓ * 2 so that Note that one or two of the resulting partial sums may be empty. We can always find such The term |ζ 2,3 | can be bounded exactly the same way, so we only need to deal with ζ 2,2 . Here we use (49) to get a direct estimate: Collecting all our estimates the statement follows.
Now we have the ingredients to prove the continuum limit.
Let B and W be independent real and complex standard Brownian motions, and for each λ ∈ R consider the strong solution of Then we have where the convergence is in the sense of finite dimensional distributions for λ and in path- Proof. The proof is very similar to the proof of Theorem 25 in [13]. One needs to check that for any fixed vector (λ 1 , . . . , λ d ) the Markov chain (ϕ ℓ, satisfies the conditions of Proposition 14 and to identify the variance matrix of the limiting diffusion. Note that because our Markov chain lives on the half integers one needs to slightly rephrase the proposition, but this is straightforward. The Lipshitz condition (42) and the moment condition (43) are easy to check from Proposition 13. The averaging condition (44) is satisfied because of Lemma 15, using the fact that because of the conditions of the proposition, the functions b λ (t), a(t, x, y) converge. This proves that the rescaled version of (ϕ ℓ,λ j , 1 ≤ j ≤ d) converges in distribution to an SDE in R d where the drift term is given by the limit of (b λ j , j = 1 . . . d) and the diffusion matrix is given by a(t, x) j,k = 2 The only step left is to verify that the limiting SDE can be rewritten in the form (51).
This follows easily using the fact that if Z is a complex Gaussian with i.i.d. standard real and imaginary parts and ω 1 , ω 2 ∈ C then The following corollary describes the scaling limit of the relative phase function α ℓ,λ . Proof. We just need to show that for any subsequence of n we can choose a further subsequence so that the convergence holds. By choosing an appropriate subsequence we can assume that m/n 0 , n/n 0 both converge and that µ n − √ m − n is always positive or nonnegative. Then the conditions of Proposition 16 are satisfied and α λ = ϕ λ − ϕ 0 will satisfy the SDE (20) with a complex Brownian motion Z t := t 0 e iϕ 0 (t) dW t . From this the statement of the corollary follows.

The relative phase function
The objective of this subsection is to show that the relative phase function α ℓ,λ does not change much in the middle stretch.
Indeed, the probability that (53) does not hold is at most c(n 0 − n 2 ) −1 ≤ cK −1 which can be absorbed in the error term of (52).
We first provide the one-step estimates for the evolution of the relative phase function.
By choosing c(β,λ) large enough we can assumeλ 4 √ n 0 k ≤ 1 10 for ℓ ≤ n 2 ≤ n − K. Using this with the cutoff (53) the random variables Z ℓ,λ ,Ẑ ℓ,λ defined in (40) are both less than 1/3 in absolute value. This means that we are allowed to use Lemma 11 in the general case for each operator appearing in (57) (i.e. the condition |z| ≤ 1/3 is always satisfied). From this point the proof is similar to the proof of Proposition 29 in [13]. We first write ∆1 /2 α ℓ,λ = ash(LT ℓ ℓ , −1, e iϕ ℓ,λη ℓρ we get the analogue of (54) for ∆1 /2 α ℓ,λ : We can prove the analogues of (55) and (56) and similar bounds for ∆1 /2 α ℓ+ 1 /2 ,λ the same way. We can also prove this is the analogue of Lemma 32 from [13] and it can be proved exactly the same way.
To get (54) we write where the last line follows from (60) and the just proved half step estimates. Now applying (59) and the corresponding estimate for ∆1 /2 α ℓ+ 1 /2 ,λ we get (54). The other to estimates follow similarly.
The next lemma provides a Gronwall-type estimate for the relative phase function. This will be the main ingredient in the proof of Proposition 18. The proof is based on the single step estimates of Proposition 19 and the oscillation estimates of Lemma 25, the latter will be proved in the Appendix.
Lemma 20. There exist constants c 0 , c 1 , c 2 depending onλ, β and a finite set J depending on n, n 1 , m 1 so that with y = n −1/2 0 (n Proof. Recall that k = n 0 − ℓ, k i = n 0 − ℓ i . We denote x ℓ = E[α ℓ |F ℓ 1 ] and set From Proposition 19 we can write k −1/2 n −1/2 0 whose terms we denote ζ 1 , ζ 2 , ζ 3 and ζ 4 respectively. Clearly, ζ 3 is of the right form and so we only need to bound the first two terms.
We will use which is the 'one-step' version of (60) and can be proved the same way as Lemma 32 in [13].
From this we get the estimates Then by Lemma 25 we have Collecting the estimates and using k 2 ≥ K(n In order to bound ζ 2 we use a similar strategy to the one applied in the proof of Lemma 15. We divide the index set [ℓ 1 , ℓ 2 ] into finitely many intervals I 1 , I 2 , . . . , I a so that for each 1 ≤ j ≤ a one of the following three statements holds: for each ℓ ∈ I j we have k ≥ √ n 1 m 1 and |ρ 2 for each ℓ ∈ I j we have k ≤ √ n 1 m 1 and |ρ 2 for each ℓ ∈ I j we have |ρ 2 It is clear that if we divide [ℓ 1 , ℓ 2 ] into parts at the √ n 1 m 1 and the solutions of |ρ 2 ℓ ρ 2 ℓ + 1| = k −1/2 then the resulting partition will satisfy the previous conditions. Since |ρ 2 ℓ ρ 2 ℓ +1| = k −1/2 has at most three roots (it is a cubic equation, see the proof of Lemma 25 for details) we can always get a suitable partition with at most five intervals. Moreover the endpoints of these intervals (apart from ℓ 1 and ℓ 2 ) will be the elements of a set of size at most four with elements only depending on n, m 1 , n 1 .
We will estimate the sums corresponding to the various intervals I j separately. If I j satisfies condition (63) then we use satisfies condition (61) then we use Lemma 25 to get We can bound the first term as using k ≥ max(n Collecting our estimates, noting that ℓ * 2 − 1 is the endpoint of one of the intervals I j and letting K be large enough we get the statement of the lemma.
The proof of Proposition 18 relies on the single step estimates of Proposition 19 and the following Gronwall-type lemma which was proved in [13].
Lemma 21. Suppose that for positive numbers x ℓ , b ℓ , c, integers ℓ 1 < ℓ ≤ ℓ 2 we have Now we are ready to prove Proposition 18.
Proof of Proposition 18. We will adapt the proof of Proposition 28 from [13]. Let a = α ℓ 1 ,λ and define a ♦ , a ♦ ∈ 2πZ so that [a ♦ , a ♦ ) is an interval of length 2π containing a. We can assume that λ ≥ 0, the other case being very similar. We will drop the index λ from α and we will write E(.) = E(.|F ℓ 1 ).
We will show that there exists c 0 so that if K > c 0 , then ifã = a ♦ or a ♦ then The claim of the proposition follows from this by an application of the triangle inequality, the additional condition κ > c 0 is treated via the error term 1/κ.
In order to prove (68) forã = a ♦ we follow the steps described in Proposition 28 from [13]. Using the exact same argument we only need to prove that for the coefficients b ℓ in Lemma 20 are bounded by a constant depending only onλ, β and that α never goes below an integer multiple of 2π that it passes.
In order to deal with theã = a ♦ case in (68) we define T ∈ Z/2 the first time when α T ≥ a ♦ . Note that α can only pass an integer multiple of 2π in the ℓ → ℓ + 1 /4 or ℓ + 1 /2 → ℓ + 3 /4 steps, and ϕ evolves deterministically in these steps. This means that T − 1 /2 is a stopping time with respect to F j , j ∈ Z/2.
For large enough K we haveλ 4 √ n 0 k ≤ 1 10 . Then by Lemma 11 we get the uniform bound By the strong Markov property and the bound (54) we get Using this together with the first part of the proof and the strong Markov property again we get Lemma 20 gives Then by (69) and the identity |a| = −a + 2a + we get Sinceα ℓ ≤ |α ℓ − a ♦ |, the Gronwall-type estimate in Lemma 21 implies (68) withã = a ♦ .

Last stretch
The purpose of this section is to prove that on the interval [n 2 , n] the relative target phase Proposition 23. For any fixed λ ∈ R and K > 0 we have α ⊙ The length of the interval [n 2 , n] is equal to n 1 + K(n 1/3 1 ∨ 1), up to an error of order 1. By taking an appropriate subsequence of n (see Remark 9) we may assume that n 1 has a finite or infinite limit. We will consider these two cases separately.
Proof of Proposition 23 in the lim n 1 < ∞ case. By (12) we have that |m − n − µ 2 n |/µ n converges, by taking an appropriate subsequence we can assume that the limit also exists without the absolute values. Note that condition (10) implies that lim m/n > 1 and thus m−n → ∞.
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 of ν is an eigenvalue for a family of recursions parameterized by ν 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 uniformly in ν with r 0 := w 1,ν /w 0,ν . Here the factor (mn) 1/3 ( √ m+ √ n) 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 Ξ ζ is also a.s. simple 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 m/n > 1 with (r n,ν + 1) −1 in (70). (In this case one first multiplies the off-diagonal entries of A n,m A T n,m by −1 before applying the arguments of [9], this will not change the eigenvalues.) If m/n → γ ∈ [1, ∞) then we can rewrite (70) jointly for the upper and lower soft edge as γ −1/3 ( √ γ ± 1) −2/3 n −1/3 (r 0,ν ∓ 1) −1 P −→ ζ, .
The multiplier is 1 in the case γ = ∞ and it is always a finite nonzero value unless γ = 1 and we are at the lower edge.
Proof of Proposition 23 if lim n 1 = ∞. By taking an appropriate subsequence, we may assume that µ n − √ m − n is always positive or always negative. According to the proof of Lemma 34 in [13] we need to consider the family of recursions r ℓ+1,ν = r ℓ,ν .J ℓ M ℓĴℓMℓ , n 2 ≤ ℓ ≤ n with initial condition r n 2 ,ν = x.T −1 n 2 = ℑ(ρ n 2 )x + ℜ(ρ n 2 ) where Λ = µ n + λ (We used that if γ * = 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.

Appendix
In this section we provide the needed oscillation estimates.
Proof. The first inequality is the same as Lemma 36 from [13] and the second inequality is straightforward.