Operator level hard-to-soft transition for $\beta$-ensembles

The soft and hard edge scaling limits of $\beta$-ensembles can be characterized as the spectra of certain random Sturm-Liouville operators. It has been shown that by tuning the parameter of the hard edge process one can obtain the soft edge process as a scaling limit. We prove that this limit can be realized on the level of the corresponding random operators. More precisely, the random operators can be coupled in a way so that the scaled versions of the hard edge operators converge to the soft edge operator a.s. in the norm resolvent sense.


Introduction
The size n Laguerre β-ensemble is a two-parameter family of distributions on R n + with density function p n,β,a (λ 1 , . . . , λ n ) = 1 Z n,β,a j<k |λ j − λ k | β n k=1 λ β 2 (a+1)−1 k The parameters satisfy β > 0 and a > −1, and Z n,β,a is an explicitly computable normalizing constant. This density corresponds to the Gibbs measure of n positively charged particles living on the positive half-line with a log-Gamma potential. For β = 1, 2 or 4 and a ∈ Z ≥0 , the density (1) is also the joint eigenvalue distribution for an n × n Wishart matrix with real, complex or quaternion ingredients, respectively. These are classical random matrix When n + a is of the same order as n, the macroscopic behavior of this ensemble is described by the famous Marchenko-Pastur limit law. Fix β > 0 and let a n > −1, n ≥ 1 be a sequence such that lim n→∞ n+an n = γ ∈ [1, ∞) exists. Denote by Λ n,β,an = (λ 1,n , . . . , λ n,n ) a size n Laguerre β-ensemble with parameter a n , and consider the scaled empirical spectral measure ν n := 1 n n k=1 δ λ k,n /n . The Marchenko-Pastur theorem ( [9], [7]) states that the sequence of random probability measures ν n , n ≥ 1 converges in distribution a.s. to a deterministic measure with density given by Note that in the case γ = 1, the density becomes The microscopic behavior of the Laguerre ensemble can be described by the large n limit of the point process c n (Λ n,β,an − d n ) where d n is the centering point and c n is the appropriate scaling parameter. In order to get a meaningful point process limit, the scaling parameter c n would need to be chosen so that it is roughly the inverse of the average spacing between the particles near d n . From now on, we will focus on the lower edge behavior i.e. the case d n := b − . (See [8] and [15] for the bulk and upper edge behavior.) The distribution of the limiting point process depends on the asymptotic behavior of the sequence a n . If a n = a > −1 does not depend on n, then Ramírez and Rider [12] showed that the scaling limit of nΛ n,β,a exists, and gave an explicit description of the limiting point process. This is called the hard edge scaling limit.
Theorem 1 (Hard edge limit of the Laguerre ensemble, [12]). Fix β > 0 and a > −1, and let Λ n,β,a be a size n Laguerre β-ensemble with parameter a. Then the sequence nΛ n,β,a converges in distribution to a point process Bessel β,a as n → ∞. The Bessel β,a process has the same distribution as the a.s. discrete spectrum of the random differential operator Here B a is a standard Brownian motion, and the operator G β,a is defined on a subset of L 2 (R + , m) with Dirichlet boundary condition at 0 and Neumann at infinity.
We will come back to the precise definition of G β,a in Section 2. Let us just mention that since the functions s, m are a.s. continuous, this differential operator fits into the framework of classical Sturm-Liouville operators.
If the sequence a n , n ≥ 1 goes to infinity with at least a constant speed then the Marchenko-Pastur theorem and the expression of the limiting measure (2) suggest a different scaling than the one seen in the hard edge case. This is called the soft edge scaling scaling limit. The description of the limiting point process follows from the work of [15].
The point process Airy β has the same distribution as the a.s. discrete spectrum of the random differential operator defined on a subset of L 2 (R + ) with Dirichlet boundary conditions at 0. Here B is the standard white noise on R + .
The precise definition of the operator A β will be discussed in Section 2. Note that a priori it is not even clear that the operator A β is well-defined, due to the irregularity of the white noise term in the potential.
It is natural to conjecture that the condition lim inf n→∞ a n /n > 0 in Theorem 2 could be relaxed to lim n→∞ a n = ∞, but the tools developed in [15] do not seem to be sufficient to prove this. (See however [4] for the treatment of the case β = 2, a n = c √ n, where the appropriate limit is proved using the determinantal structure present at β = 2.) This conjecture, together with a diagonal argument, would imply the following point process level transition from the Bessel β,a process to Airy β : See [18] for a similar diagonal argument for the transition between the soft edge and the bulk limiting processes.
The process level limit (6) is called hard to soft edge transition. It can be analyzed without considering the finite n ensembles, working directly with the limiting point processes appearing in the statement. This transition was first proved in [3] for β = 2 using again the determinantal structure present in this case. For general β > 0, Ramírez and Rider [12] proved the scaling limit for the first point of the respective point processes. This result was extended in [14] to a full process level limit.
In light of Theorems 1 and 2, the statement of (6) can be rewritten using the operators G β,2a and A β as where spec(Q) denotes the spectrum of the operator Q. It is natural to ask whether it is possible to prove the corresponding limit on the level of the operators. This is the main result of our paper. Theorem 3 below shows that one can realize the operator level limit as an a.s. limit with an appropriate coupling between the Brownian motion B a of the Bessel operator (3) and the white noise B of the Airy operator (5).
To describe our coupling, we introduce a simple transformation of G β,2a . For a > 0 let θ a be the 'stretching' transformation defined via and define the following transform of the hard-edge operator corresponding to 2a: As we will see in Section 2, G β,2a is a self-adjoint operator with the same spectrum as G β,2a , and the operators A −1 β and (G β,2a − a 2 ) −1 are Hilbert-Schmidt integral operators acting on the same space of L 2 (R + ) functions.
Theorem 3 (Operator level hard-to-soft transition). Let B be white noise on R + and let B be a Brownian motion defined as B(x) := Consider A β defined as (5) using the white noise B , and G β,2a defined with the Brownian motion B 2a via (3) and (8) for a > 0. Then We expect that with a more careful application of our methods one could also get estimates on the speed of convergence in our coupling. See Remark 23 in Section 6.
The theorem implies that a −4/3 (G β,2a − a 2 ) → A β a.s. in norm resolvent sense from which the process level transition a −4/3 (spec(G β,2a ) − a 2 ) ⇒ spec(A β ), and therefore the limit (6) follow. The coupling of the operators produces a coupling of the point processes in a way that almost surely the points in the scaled hard edge processes converge to the points in the soft edge point process. More precisely, a version of the Hoffman-Wielandt inequality (see e.g. [1]) shows that if we denote the ordered points in the scaled hard edge 4 process a −4/3 (Bessel β,2a −a 2 ) by λ k,2a , k ≥ 0, and the ones in the soft edge process Airy β by λ k , k ≥ 0, then in the coupling of Theorem 3 we have a.s.
Moreover, as the spectrum of the operators are discrete, and each eigenvalue has multiplicity 1, the a.s. norm resolvent convergence also implies the a.s. convergence of the respective normalized eigenfunctions in L 2 .
The structure of the rest of the paper is as follows. In Section 2 we show how one can describe the appearing differential operators using the generalized Sturm-Liouville theory, show that A −1 β and (G β,2a − a 2 ) −1 are Hilbert-Schmidt integral operators, and describe their kernels in terms of certain diffusions. Section 3 outlines the main steps of the proof of the main Theorem 3. Our proof uses the approximation of the integral operators by their truncated version. We state the convergences of the truncated operators towards their full operator as well as the convergence of the truncated hard edge integral operators to the truncated soft edge integral operator in several lemmas whose proofs are postponed to later sections. Section 4 estimates the truncation error of the soft edge integral operator. Section 5 shows that the truncated hard edge integral operators converge to the truncated soft edge integral operator by proving that the integral kernels converge uniformly on compacts with probability one. Section 6 describes the asymptotic behavior of the diffusions connected to the operator G β,2a and provides the results needed to estimate the truncation error for the hard edge integral operators. Finally, the final section gathers the proof of some technical lemmas needed for the results of Sections 4 and 6.
2 The operators A β and G β,2a as generalized Sturm-

Liouville operators
This section briefly introduces the background for the differential operators appearing in this work, and shows how it can be used to describe the random differential operators G β,2a , G β,2a , A β and their inverses. We use the classical theory discussed in [19] and Chapter 9 of [17]. 5

Generalized Sturm-Liouville operators
We consider generalized Sturm-Liouville (S-L) operators of the form where u is a real valued function on [0, L] for some L > 0 or on R + (which we consider to be the L = ∞ case in the following). We assume that the real functions p 0 , p 1 , q 0 , r are continuous on [0, ∞) and r(x), p 1 (x) > 0 for x ≥ 0.
The operation τ u is well-defined if both u and p 1 u − q 0 u are absolutely continuous on From the standard theory of differential equations we have that for any λ ∈ C the differential equation τ u = λu has a unique differentiable solution on [0, L] with initial conditions u(0) = c 0 , u (0) = c 1 . We note that if f 1 , f 2 are both solutions of τ f = λf then integration by parts shows that the Wronskian We consider differential operators satisfying the following three assumptions: (A2) There is a unique solution u ∞ of the equation τ u ∞ = 0, with initial condition u ∞ (0) = 1 that is in L 2 (R + , r).
Under these assumptions, the operator τ can be made self-adjoint on an appropriate subset of L 2 ([0, L], r) or L 2 (R + , r). We introduce The continuity of the functions p 0 , p 1 , q 0 and r implies that the operator τ is regular at 0 and at any finite L and therefore is limit circle at those points. The condition (A1) implies that the operator τ is limit point at +∞ thanks to the Weyl's alternative theorem.
Conditions (A2) and (A3) ensure that the inverse and the resolvent are Hilbert Schmidt operators.
The following propositions summarize the basic properties of generalized Sturm-Liouville differential operators satisfying conditions (A1)-(A3).
Proposition 4 (Self-adjoint version of τ ). Assume that τ is of the form (9) and that it satisfies the condition (A1-A3), and let L ∈ (0, ∞]. Then there is a self-adjoint version of the operator on [0, L] with Dirichlet boundary conditions on the domain where the end condition u(L) = 0 is dropped in the case L = ∞. We denote this self-adjoint operator by τ L .
Proposition 5 (Inverse as an integral operator). Consider the operator τ L from Proposition 4. If L is finite then assume that u d (L) = 0 (i.e. that 0 is not an eigenvalue of τ L ). Then Here u d is defined in (A1). If L = ∞ then u L is u ∞ from (A2), and in the case L < ∞ the function u L is defined as the solution of τ u L = 0 with u L (0) = 1, u L (L) = 0. The inverse operator τ −1 L is a Hilbert-Schmidt operator in L 2 ([0, L], r), and it has a bounded pure point spectrum.
Proposition 6 (Resolvent as an integral operator). Consider τ L from Proposition 4, and assume that a given λ ∈ R is not an eigenvalue of τ L . Then the resolvent (τ L − λ) −1 is a Hilbert-Schmidt integral operator of the same form as K (L) from (10), where now u d , u L are the appropriate solutions of τ u = λu with the respective boundary conditions. For L = ∞ the function u L = u ∞ is the unique solution of τ u ∞ = λu ∞ with u ∞ (0) = 1 and u ∞ ∈ L 2 (R + , r).
The proofs of these propositions follow from the theory of Sturm-Liouville operators.
Again, we refer to the monograph [19]. Note that the classical theory (when q 0 = 0) is treated in a self-contained way in Chapter 9 of [17] (see in particular Theorems 9.6 and 9.7).

Bessel and Airy operators as generalized S-L operators
The operators G β,a , G β,2a , and A β can be represented as a generalized Sturm-Liouville operators for which Assumptions (A1-A3) are satisfied, and hence the appropriate resolvents are a.s. Hilbert-Schmidt integral operators. We summarize the relevant results in the propositions below.
is the unique strong solution of the stochastic differential equation with the corresponding initial conditions.
Proof. The fact that G β,2a is a Sturm-Liouville operator is contained in the statement of Theorem 1, the statement about the solution of the eigenvalue equation can be checked with Itô's formula (see [12]). As explained in [13], the Neumann boundary condition for G β,2a at ∞ for a > 0 can be dropped. The SDE (11) satisfies the usual conditions for existence and uniqueness, so (φ, φ ) is a well-defined process for all times.
We only need to check that the conditions (A1-A3) are satisfied for a > 1/2. This can be done directly using the a.s. sublinear growth of the Brownian motion by noting that be the unique strong solution of (11) with λ = a 2 and initial conditions φ(0) = 0, φ (0) = 1. Let E a be the event that a 2 is not an eigenvalue of G β,2a . Denote by φ On the event E a the operator a 4/3 (G β,2a − a 2 ) −1 has a bounded pure point spectrum that is the same as the spectrum of a 4/3 (G β,2a − a 2 ) −1 .
Proof. By Proposition 6, the function φ (2a) ∞ is well-defined on E a , and the operator ( Recalling the definition of G β,2a from (8) we get that a 4/3 (G β,2a − a 2 ) −1 is a Hilbert-Schmidt integral operator on L 2 (R + ) with kernel from which the proposition follows.
Note that for any fixed a > 1/2, the event E a has a probability 1, see Remark 28. Later, in Corollary 21 in Section 6 we show that in our coupling if a is large enough then a 2 is not an eigenvalue for G β,2a .
Proposition 9 (The operator A β as a generalized S-L operator). The operator A β is a generalized Sturm-Liouville operator of the form (9) The operator satisfies the conditions (A1-A3) with probability one.

If ψ solves the equation
which is well defined for all times, and satisfies A.s. 0 is not an eigenvalue of A β , and the operator A −1 β is a Hilbert-Schmidt integral operator with kernel Here Figure 1).
Proof. The fact that the soft-edge operator Airy β can be represented as a generalized Sturm-Liouville operator of the form (9) with the listed coefficients was shown in [2] (see also [10]).
The SDE representation of the solutions of A β ψ = 0 with a deterministic initial condition is shown in [15]. Since the SDE (13) satisfies the usual conditions of existence and uniqueness for SDEs, the solution is well defined for all times. The asymptotics (14) was stated without proof in [15], we include a proof of this statement in Proposition 14 in Section 7.1 below for completeness.
To check that the conditions (A1)-(A3) are satisfied we first observe that if ψ d is the solution of A β ψ = 0 with Dirichlet initial condition then by (14) for any fixed ε > 0 we have hence ψ d is not in L 2 (R + ). This means that a.s. there can be at most one L 2 (R + ) solution of A β ψ = 0 with initial condition ψ(0) = 1. We will construct such a function using ψ d .
Denote by z 0 the largest zero of ψ d on R + , and let z 0 = 0 if such a zero does not exists.
Motivated by the Wronskian identity we introduce the function which is well defined for x > z 0 . One can check that ψ ∞ satisfies A β ψ ∞ = 0 and the Wronskian identity for x > z 0 . Then, the function ψ ∞ can be uniquely extended to R + as a solution of A β ψ = 0.
Using (17) we see that for x > z 0 we have and from (14) we get the bounds for some random C < ∞. Together with the bound (16) this is now sufficient to show that By Propositions 5 and 9 it follows immediately that A −1 β is almost surely a Hilbert-Schmidt integral operator with kernel given in (15).
Remark 10. Using the identity (17) and the limit (14) one can show that x a.s. as x → ∞, and that ψ ∞ (x) ≤ e −(2/3−ε)x 3/2 for x large enough. This behavior was also noted in [15]. See Figure 1 for an illustration for the behavior of ψ d , ψ ∞ . Figure 1: Representation of the log-derivatives of ψ d and ψ ∞ .
We record here the Wronskian identities for the appropriate operators: where we dropped the a-dependence in φ ∞ to alleviate the notation. From the second equation of (20) one can obtain the following analogue of the identity (17) for the hard edge diffusions: if x is larger than the largest zero of φ d .
Note that the functions ψ d , φ d are diffusions with respect to the natural filtrations of the Brownian motions B, B 2a . This is not the case for the functions ψ ∞ and φ ∞ , as the starting values of these processes depend on the σ-field generated by the whole Brownian motion B(t), t ≥ 0. In particular, those functions are not Markovian.

Proof of Theorem 3
Proof of Theorem 3. In order to prove the theorem, we first need to show that in our coupling with probability one a 2 is not an eigenvalue of the operator G β,2a if a is large enough. This will be the content of Corollary 21 in Section 6: we will show that there is an a.s. finite random variable C ev such that the operator G β,2a − a 2 is invertible for all a > C ev . In particular, this means that on the event {a > C ev } the operator (G β,2a − a 2 ) −1 is a well-defined integral operator with kernel given in Proposition 8.
By the results of Section 2, to prove Theorem 3 we need to show that we have We do this by approximating K A and K G,2a with the resolvent kernels of the appropriate differential operators restricted to [0, L], with L > 0. We denote these operators by K where (Note that φ andφ depend on a as well, which we do not denote.) By the triangle inequality we have We will show that all three terms on the right will vanish in the limit if we let a → ∞ and then L → ∞ along a particular sequence, this is the content of the Lemmas 11, 12 and 13 below. From these three lemmas, we deduce the convergence (22), and hence Theorem 3 follows.
More precisely, we will prove the following three lemmas.
Lemma 11 (Truncation of the Airy operator). We prove Lemma 11 in Section 4 using the the asymptotics (16). The proof of Lemma 12 is given in Section 5, we will show that for a fixed L < ∞ the kernel K (L) G,2a converges uniformly to K (L) A on [0, L] 2 as a → ∞. Finally, the proof of Lemma 13 will be given in Section 6, and it will rely on a careful analysis of the asymptotic behavior of φ

Truncation of the Airy operator
We analyze the solutions of the SDE (13) via the Riccati transform ψ (t) ψ(t) . Suppose that ψ, ψ is the strong solution of the SDE (13) with deterministic initial conditions ψ(0) = c 0 , ψ(t) , by Itô's formula X satisfies the SDE with initial condition X(0) = c 1 /c 0 . The initial condition is ∞ if c 0 = 0, c 1 = 0. Note that the diffusion blows up to −∞ at the zeros of ψ, and it restarts at ∞ instantaneously whenever this happens.
The drift in (25) vanishes on the parabola x 2 = t, it is positive for |x| < √ t, and negative for |x| > √ t. This suggests that the asymptotic behavior of X(t) should be √ t (since the branch x = − √ t is unstable), as stated in (14). The proposition below proves this statement by providing quantitative bounds on |X(t) − √ t|. See Figure 2 for an illustration of the asymptotic behavior of X. Note that less precise asymptotic bounds on X were also proved in [6] for the study of the small β limit.
Proposition 14. Let ψ, ψ be the strong solution of (13) with deterministic initial conditions Then there is an a.s. finite random time T such that Our upper bound in (26) is not optimal. In fact by evaluating the error terms in the proof given below it can be shown that t −1/4 ln t can be replaced with t −1/4 √ ln t g(t) for any positive function g(t) satisfying lim t→∞ g(t) = ∞.
The proof of Proposition 14 relies on the following two technical lemmas, whose proofs are postponed to Section 7.1.
Lemma 15. Let X be a strong solution of the SDE (25). For a given s ≥ 10 set Then σ s is a.s. finite. with initial condition X(t 0 ) = x 0 , and denote by P t 0 ,x 0 its distribution. Then Lemma 15 shows that for any solution X of the SDE (25) and any s ≥ 10 the process X(t) − √ t will get close enough to 0 after time s. Lemma 16 shows that if X(t) − √ t is close to 0 for a given large t = t 0 then with a high probability it will stay close to 0 for all t ≥ t 0 .
Proof of Proposition 14. Let f (t) = t −1/4 ln t. By Lemma 15 for any fixed s ≥ 10 there is an a.s. finite stopping time σ s with σ s ≥ s so that |X(σ s ) − √ σ s | ≤ 1 2 f (σ s ) with probability one. Lemma 16 shows that if the diffusion is close to √ t then with a high probability it will stay close forever.
More precisely, for a given ε > 0 one can choose s ≥ 10 so that The strong Markov property and Lemma 15 now imply that the inequality (26) holds with T = σ s with probability at least 1 − ε. This shows that the random time is finite with probability at least 1 − ε, hence it is a.s. finite. Therefore (26) holds with probability one with T = T 0 .
We can now prove Lemma 11.
Proof of Lemma 11. By Proposition 9 with probability one the operator A −1 β is a Hilbert-Schmidt integral operator with kernel K A . From (14) and the estimate (16) it follows that ψ d has a largest zero (if it has one), hence if L is larger than that, the linearity of the equation Hence the truncated operator K we get By Proposition 9, with probability one we have K A We now estimate the first term on the right hand side of (30). By symmetry we have From (29), for L large enough, and 0 ≤ x ≤ y ≤ L, we get From the bounds of (19) we get that the expression in (31) is bounded by a random constant times L −2 , and thus it converges to zero a.s. as L → ∞. This concludes the proof of Lemma 11.

Convergence of the truncated operators
Recall the definition ofφ L , ψ L from Section 3. Lemma 12 will follow from the following statement: Proof. To ease notation, we drop the dependence on a in u a ,û a . By Proposition 7 the process (u(t), u (t)) satisfies the SDE The initial conditions forû arê hence by the conditions of the proposition we see that (û(0),û (0)) → (η 0 , η 1 ). Note that u (x) = −a 1/3û (x) + e −a 1/3 x u (a −2/3 x), by Itô's formula and (32) we have that This means thatû,û satisfies where ε = a −1/3 and With a bit of abuse of notation we will useû ε ,û ε to denote the dependence on ε ∈ (0, 1]. The functions F 1 , F 2 can be continuously extended to ε = 0 by setting F i (0, x, p, q) = 0.
. Note that ψ d (L) = 0 with probability one for a fixed L, so ψ L is a.s. well-defined. This also implies that for a fixed L the random variableφ d (L) is not zero if a is larger than a random constant, and in this caseφ L is also well-defined.
The function ψ L satisfies A β ψ L = 0 with ψ L (0) = 1, ψ L (L) = 0. By our previous and we can check (by plugging in x = 0 and But this now implies thatφ L → ψ L uniformly on [0, L] with probability one, completing the proof.

Truncation of the Bessel operator
In order to control K G,2a − K Itô's formula together with (11) implies that p(t) satisfies the diffusion Proposition 19 (Behavior of the Bessel diffusion). Let d 1 , d 2 > 0. For a given L > 0 and a 1 ≥ 1, define C L,a 1 to be the event where the following inequalities hold for all a ≥ a 1 : Proposition 20. Define There are absolute constants c, c so that for all a 1 ≥ c L , the following inequalities hold on the event C L,a 1 (as defined in Proposition 19): for all a ≥ a 1 , Proof. We first prove the case when t ≥ s ≥ t 0 in (41). From this point on we will work on the event C L,a 1 with a 1 ≥ c L , allowing us to assume the inequalities (36)-(38). Let us define On the event C L,a 1 , and for t ≥ t 0 , q(t) is well defined as p(t) > 0. By Itô's formula the process q satisfies the following differential equation: with the initial condition q(t 0 ) = ln(p(t 0 )/a) > 0. Note that the drift of the diffusion q will be close to a(2 − e q ) for large t. The corresponding diffusion converges to a stationary distribution supported on R (which can be computed explicitly).
This suggests that q behaves like the stationary solution ofq, and hence we cannot expect to get a uniform constant bound on a(e q(t) − 1) = p(t) − a in (40). Because of this we instead look for a bound on the integral term in (40).
We start with the following identity: for all t ≥ s ≥ t 0 , we have Using the lower bound from (37) and the fact that −a −1/6 ln t ≥ −t + t 0 for all t ≥ t 0 , we get and thus Using the inequality ln t ≤ ln s + t −1 0 (t − s) for t ≥ s ≥ t 0 , the bounds (37), (38), and by our choice of c L , we get that there exist positive constants c 1 , c 1 such that for all t ≥ s ≥ t 0 , we have I(s, t) ≤ −c 1 a (t − s) + 5a −1/6 ln s + c 1 .
This completes the proof of (41) in the case t ≥ s ≥ t 0 .
As a −2/3 L ≤ s ≤ t 0 and a ≥ a 1 ≥ c L , we get that there exists a constant c I such that: For t ≥ t 0 ≥ s ≥ a −2/3 L, note that I(s, t) = I(s, t 0 ) + I(t 0 , t). Therefore, we get We choose c = min{c 1 , c 2 } and c = max{c 1 , c I , c I } to conclude the proof of (41).
As a consequence of Proposition 19, we can also show that a 2 is not an eigenvalue of G β,2a if a is large enough.
Corollary 21. Let a 1 ≥ c L . On the event C L,a 1 defined in Proposition 19, a 2 is not an eigenvalue of G β,2a for all a ≥ a 1 . As a consequence, there exists an a.s. finite random variable C ev > 0 such that a 2 is not an eigenvalue of G β,2a on the event {a ≥ C ev }.
Proof. The value a 2 is not an eigenvalue of G β,2a exactly if the function φ Using the above lower bound on the integral of ae q(t) , and the bounds (37) and (38), we get where c(t 0 ) is an a.s. finite random constant. Choosing a ≥ a 1 ≥ c L ≥ (1 − e −t 0 ) −2 , we get that ∞ 0 φ d (t) 2 m 2a (t)dt is infinite, proving the statement. Now set If a ≥ C ev then a 2 is not an eigenvalue of G β,2a . By the limit (39), the random variable C ev is a.s. finite, which completes the proof.
Proposition 22. Recall the definition of the event C L,a 1 from Proposition 19. On this event a 2 is not an eigenvalue of G β,2a (or G β,2a ) if a ≥ a 1 ≥ c L by Corollary 21, henceφ ∞ is welldefined. There exist deterministic constants c 1 , c > 0 such that for all L ≥ 10 and a 1 ≥ c L , the following inequalities hold on C L,a 1 : for all a ≥ a 1 , and Moreover, under the same conditions, we also get the following inequality for all y ≥ x ≥ L: Proof. Recall the definition ofφ d ,φ ∞ from (12). On C L,a 1 , the diffusion p(t) does not explode on [a −2/3 L, ∞), which also implies the largest zero of φ (2a) d is smaller than a −2/3 L. By the Wronskian identity (21), for all x ≥ L we havẽ where For the productφ d (y) −2φ d (x) 2 for y ≥ x ≥ L, we havẽ For a 1 ≥ c L , (45) follows from (41) directly. Integrating the exponential of (40) and using the upper bounds (41), we get (44) and the statement of the proposition. The inequality (46) follows by comparing the upper bounds in (44) and (45).
We now turn to the proof of Lemma 13. We will use the following identity, that follows from the linearity of the equation G β,2a φ = a 2 φ: By Propositions 19 and 20, we have thatφ d (L) = 0 andφ ∞ is well-defined for all a ≥ a 1 on the event C L,a 1 .
Proof of Lemma 13. For L ≥ 10 define the event The family of events C The family C L,a 1 is non-decreasing in a 1 for fixed L and the events ∪ a 1 C We now prove inequalities on the event C (2) L,a 1 for all a 1 ≥ c L . In the following, c is a constant that may change from line to line. We start with the following identity: The first term 2φ d (L) −2 L 0φ d (x) 2 dx is bounded from above by 6 L −1/2 . For the second term, we split the integral, and apply Proposition 22 to get the following upper bound: At last, on R 2 we have We use (44) and (45) to bound the first integral, For the second integral, we use (44) and (46), Recall that the family of events C L,a 1 is non-decreasing in a 1 for fixed L, and the events C By choosing L = L a to be dependent on a with L a → ∞ at some rate, one could potentially obtain a bound on the rate of convergence in (22). This would require the extension of the result of Lemma 17 to increasing intervals [0, L a ]. We do not explore this path in this paper, but we want to present a hand-waving argument to show that our methods are not expected to give better than logarithmic convergence.
In the proof of Proposition 18, we viewed the process (û,û ) as a stochastic flow depending on two variables ε = a −1/3 and x. It is reasonable to expect that if the statement of Lemma 17 holds on the interval [0, L a ] then sup x≤La |û ε (x) −û 0 (x)| should vanish as a → ∞. This quantity should be of the same order as ε sup x≤La |v(x)| where v(x) = ∂ εûε (x)| ε=0 . One can check that v satisfies the stochastic differential equation, with initial values v(0) = 0 and v (0) = 0. If we assume that v grows at least as fast as the contribution of the 2 βû 0 (x)dx term then we would get that v grows at least as fast as e Proof of Lemma 15. We will prove that This means that with higher and higher probability we will hit the region |X(t)− √ t| ≤ 1 2 f (t) within a small time interval, which implies that σ s < ∞ with probability one.
To prove (49) we consider X with initial condition X(t 0 ) = x 0 with t 0 ≥ 10, x 0 ∈ R, and Therefore, we obtain that For a large enough deterministic which completes the proof of (52).

Bounds for the hard edge diffusion
We start this section with a lemma controlling the fluctuations of Brownian motion. Although the bounds in the lemma are not optimal they are sufficient for our purposes.
Lemma 24. Let B be a standard Brownian motion. Then there is a random finite positive C so that a.s. we have the following inequality: This implies in particular the following simple bounds: with a random constant C 1 .
Proof. First set h = 2 n , s = m2 n , for n ∈ Z and m ∈ N. We have which is summable for n ∈ Z, m ∈ N. Hence by the Borel-Cantelli Lemma, there is a random for all s = m2 n , h = 2 n . For general s > 0, h > 0, there exist n ∈ Z, m ∈ N such that 2 n < h ≤ 2 n+1 and m2 n < s ≤ (m + 1)2 n . Using (55) and the triangle inequality, we get which proves the first part of the lemma with C = 8C. L,a 1 be the event that Then lim L→∞ lim a 1 →∞ P (A L,a 1 ) = 1.
Fix L large and define the event: Note that the family A L is non-decreasing in L.
L,a 1 . By Itô's formula, for t ≥ a −2/3 L we have The diffusion q blows-up when p reaches 0, so q may not be well-defined on the whole interval [a −2/3 L, +∞).
The next proposition controls the growth of q from small times starting at a −2/3 L until a positive deterministic time. In this time-interval, q is small and therefore p is close to a(1 + q). Analyzing the drift of the q diffusion for small t and q, we see that one can compare the behavior of q with the diffusion X defined in (25). This allows us to bound q with constant multiples of the square root function with large probability.
Next we estimate the growth of q(t) in the time interval t ∈ [t 0 , ∞). As we will see, q will have a different behavior for large times: it oscillates near the value ln 2 with possibly making large excursions away from this value. We will prove bounds on those fluctuations using a comparison with a non-exploding, stationary version of the diffusion q.
Then, there exists a constant c > 0 such that lim L→∞ lim a 1 →∞ P A L,a 1 = 1.
Proof. For each a, we bound q(t) using two stationary diffusions q 1 (t) = q (t), and we show that the growth of q 1 , q 2 is at most logarithmic with a large probability.
Proof of Proposition 19. The statement follows from Propositions 26 and 27, and the inequality (60).
Remark 28. A more careful analysis of the diffusion φ (2a) d (using ideas described in the proofs of Lemma 28 and Lemma 26) can provide a logarithmic bound on the diffusion q for a fixed a > 0. More precisely, it can be shown that for a fixed a > 1/2 with probability one the diffusion q satisfies |q(t)| ≤ 2(32) 2 β a ln t for all large t. In particular, this result implies that φ d := φ (2a) d is a.s. not in L 2 (R + , m 2a ) for a > 1/2 thanks to the identities (42) and