Localization of the continuous Anderson Hamiltonian in 1-D

LOCALIZATION OF THE CONTINUOUS

ANDERSON HAMILTONIAN IN 1-D

LAURE DUMAZ AND CYRIL LABBÉ

ABSTRACT. We study the bottom of the spectrum of the Anderson Hamiltonian HL := −∂x2 + ξ on

[0, L] driven by a white noise ξ and endowed with either Dirichlet or Neumann boundary conditions. We show that, as L → ∞, the point process of the (appropriately shifted and rescaled) eigenvalues converges to a Poisson point process on R with intensity ex_{dx, and that the (appropriately rescaled) eigenfunctions}

converge to Dirac masses located at independent and uniformly distributed points. Furthermore, we show that the asymptotic shape of each eigenfunction, recentered around its maximum, is given by an explicit, deterministic function that does not depend on the rank of the corresponding eigenvalue. Finally, we show that the eigenvalues/eigenfunctions in the Dirichlet and Neumann cases are very close to each other and converge to the same limits.

AMS 2010 subject classifications: Primary 60H25, 60J60; Secondary 35P20.

Keywords: Anderson Hamiltonian; Hill’s operator; localization; Riccati transform; Diffusion.

CONTENTS

1. Introduction 1

2. Convergence of the eigenvalues towards the Poisson point process 11

3. Localization of the eigenfunctions 16

4. Proofs of some first estimates on the diffusion Xa 29

5. Crossing of the well 35

6. Neumann boundary conditions 48

References 49

1. INTRODUCTION

Consider the Anderson Hamiltonian

HL= −∂x2+ ξ , x ∈ (0, L) , (1)

endowed with either Dirichlet or Neumann boundary conditions. Here, the potential ξ is taken to be a real white noise on (0, L), that is, a mean zero, delta-correlated Gaussian field on (0, L). The operator HLis a random Schrödinger operator, sometimes called Hill’s operator, that models disordered solids in

physics.

There exist many different versions of the Anderson Hamiltonian according to whether the underlying space is discrete or continuous, and according to the potential ξ that one considers. The study of the spectrum of such operators has given rise to a vast literature. Notice that there is a competition between the two terms appearing in the operator: while the eigenvalues of the Laplacian are spread out over the whole box, the multiplication-by-ξ operator tends to concentrate the mass of the eigenvalues in very

Date: November 15, 2017.

1

small regions - a phenomenon usually referred to as localization; see for instance [Kir08] for an extended survey on Anderson localization and related topics.

In the present paper, we establish a localization phenomenon at the bottom of the spectrum of HL

when L → ∞.

This operator was first studied by the physicist Halperin [Hal65], see also the work of Frisch and Lloyd [FL60]. They were interested in the macroscopic picture of the eigenvalues in the large L limit. If N(λ) denotes the number of eigenvalues smaller than λ, the density of states is defined as the large L limit of the derivative of N(λ) divided by L. They found that the density of states of the operator HL

admits an explicit integral formula, see Equation (11) or Figure 1.

-2 0 2 4 6 8 10 0.00 0.05 0.10 0.15 0.20 0.25

FIGURE1. Density of states of the operator HL(plain line) and of u 7→ −∂x2u (dashed line).

Later on, Fukushima and Nakao [FN77_{] gave a precise formulation of the eigenvalue problem H}Lϕ =

λϕ. They proved that for any fixed L, almost surely HLis a self-adjoint operator on L2(0, L), bounded

below, that admits a pure point spectrum (λk)k≥1, with λ1 < λ2 < . . ., and that the associated

eigen-functions (ϕk)k≥1form an orthonormal basis of L2(0, L) and are Hölder 3/2−(that is, Hölder 3/2 − ε

for all ε > 0). They also rigorously derived the density of states.

Subsequently, McKean [McK94] established that the first eigenvalue, appropriately shifted and rescaled:

−4√aL(λ1+ aL) , (2)

converges as L → ∞ to a Gumbel distribution and therefore falls in one of the three famous extreme-value distributions (see also [Tex00] for the aymptotics of the k-th eigenextreme-value). The precise definition of aLis given right above Equation (12), let us simply mention that as L → ∞ we have

aL∼

3 8ln L

2/3 .

Let us also cite the works [CM99,CRR06] where the authors obtained an exact formula for the density distribution of the first eigenvalue in terms of an integral over the circular Brownian motion, and a precise asymptotic for its left tail. In these works, L is fixed and the operator is endowed with periodic boundary conditions.

In all the aforementioned studies, the starting point is the Riccati transform which translates the second order linear differential equation HLu = λ u into a first order non-linear one, see below. We will also

use this tool in the present paper.

Statement of our results: We are interested in the large L limit behavior of the smallest eigenvalues of HL and of their associated eigenfunctions. We will consider the operator HL endowed with either

homogeneous Dirichlet boundary conditions: ϕ(0) = ϕ(L) = 0, or homogeneous Neumann boundary conditions: ϕ′_{(0) = ϕ}′_{(L) = 0. In the sequel, if no mention is made of the boundary conditions, then}

they are taken to be homogeneous Dirichlet.

Let us define the point process of shifted and rescaled eigenvalues: QL:=

X

k≥1

δ_{4√aL(λk+aL)}. (3)

This object is a random variable in the space M of Radon measures on R endowed with the vague topology. We also introduce the rescaled eigenfunctions:

mk(dt) = Lϕ2k(tL)dt , t ∈ (0, 1) , (4)

that belong to the space P of probability measures on [0, 1] endowed with the topology of weak con-vergence. Setting mL:= (mk)k≥1, we then view QL, mL
as a random variable in the product space

M × PN

, endowed with the product topology and the associated Borel σ-field.

Our first result shows that asymptotically in L, the eigenvalues form a Poisson point process on R while the eigenfunctions converge to Dirac masses whose locations are uniformly distributed over [0, L] and independent from the eigenvalues.

Theorem 1. The sequence of r.v. _QL, mL
converges in law to Q∞, m∞
where Q∞ is a Poisson

point process on R with intensityexdx, m∞ = (δU∞

k )k≥1 and(U

∞

k )k≥1is a sequence of i.i.d. uniform

r.v. on_{[0, 1], independent of Q∞}.

We also obtain a precise description of the eigenfunctions near their maxima. We let Uk ∈ [0, 1] be

the (first, if many) point at which ϕ2

k(L ·) achieves its maximum, and we set

hk(t) := √ 2 a1/4_L   ϕk  UkL + t √_a L    , t ∈ R ,

where we implicitly set the value of this function to 0 whenever the argument of the function on the right does not belong to [0, L]. We also define

h(t) := 1 cosh(t) , t ∈ R . L 0 U3L U1L U4L U5L U2L 1 (ln L)1/3

FIGURE 2. A very schematic plot of the fifth eigenfunction ϕ5. The main peak occurs

at U5L, while near UiL, for every i < 5, peaks of smaller order occur. Observe that the

height of the peak near UiL decays with the distance |UiL − U5L|. 3

Theorem 2. As_{L → ∞, for every k ≥ 1 the location of the maximum U}kconverges toUk∞in

distribu-tion, and the processhkconverges toh uniformly over compact subsets of R in probability.

As we will see in the proof, the k-th eigenfunction admits local maxima nearby the points UiL for

i ∈ {1, . . . , k − 1}, see Figure 2, and the exponential decay between the localization center UkL and the

location UiL of a smaller peak is roughly exp(−(√aL+ o(1))|UiL − UkL|).

We now consider the case where HLis endowed with Neumann boundary conditions: we denote by

ϕ(N )_k and λ(N )_k the corresponding eigenfunctions and eigenvalues. Notice that ϕ(N )_k , λ(N )_k and ϕk, λk

are all defined on a same probability space. Our next result shows that the limiting behavior of the eigenvalues/eigenfunctions in the Neumann case is the same as in the Dirichlet case. Furthermore, the convergence can be taken jointly for the two types of boundary conditions and the limits are the same. Theorem 3. For Neumann boundary conditions, the eigenvalues and eigenfunctions satisfy the same results as those stated in Theorems 1 and 2. Furthermore, for every_{k ≥ 1, the r.v.}

(λ(N )_{k − λ}k)√aL, and (U (N )

k − Uk)L

√ a_L,

converge to_{0 in probability as L → ∞ and consequently the limiting r.v. satisfy a.s. Q}(N )_{∞ = Q∞}and U_k(N ),∞ = U∞

k .

These theorems show that, asymptotically in L, the first eigenfunctions of HLare strongly localized:

most of their masses are supported in regions of size 1/(ln L)1/3_{. As we will see in the proofs, in}

these regions the behavior of the white noise ξ is exceptional. Since in addition ξ is delta-correlated, the independence of the r.v. (Ui)i≥1(which correspond to the centers of these regions) is intuitive.

Fi-nally, Theorem 3 shows that the choice of Dirichlet/Neumann boundary conditions does not affect the asymptotic behavior of the eigenvalues/eigenfunctions.

Discussion: Before presenting the outline of the proof, let us give some motivations for studying this operator. The parabolic Anderson model (PAM) is the following Cauchy problem:

∂tu(t, x) = ∂x2u(t, x) − ξ(x)u(t, x) , u(0, x) = δ0(x) , x ∈ R . (5)

One expects its solution at time t to be well approximated by the solution of the same equation restricted to a segment [−L/2, L/2], with L = L(t) properly chosen, and endowed with Dirichlet boundary con-ditions. Thus, the spectral decomposition of HLwould yield

u(t, x) ≈

∞

X

k=1

exp(−λkt)ϕk(x)hϕk, δ0iL2 . (6)

The bottom eigenvalues and eigenfunctions of HLin the large L limit should therefore give the

asymp-totic behavior of the solution (6) when t → ∞: in particular, the solution should concentrate on a few islands corresponding to the “supports” of the first eigenfunctions. We will provide rigorous arguments on this heuristic discussion in a future work.

If the Anderson operator is multiplied by the imaginary unit on the right hand side of (5), the equation becomes the famous Anderson Schrödinger equation which is of fundamental importance in quantum mechanics. In this case, one needs to study the whole spectrum (not only its bottom part). This will be the object of a forthcoming work.

The discretization of HLand (5) has been investigated in many papers for a general dimension d. In

this case, the Laplacian is discrete on a grid, for instance Zd_{, and the white noise is replaced by i.i.d}

random variables. The discrete operator is first defined on a finite box B, say B = [0, L]d_{∩ Z}d_{, and}

taken with Dirichlet boundary conditions. We refer to the book of König [K¨16] for a state of the art on

the subject. Analogous results to our Theorem 1 are known for some distributions of potentials, see for instance Biskup and König [BK16].

When d = 1, much more precise results are known. Let us introduce the discrete Anderson Hamil-tonian

u 7→ −∆xu + ξ u,

acting on u : [0, L] ∩ Z → R with Dirichlet boundary conditions, and where ∆xdenotes the discrete

Laplacian and (ξ(x), x ∈ Z) are i.i.d random variables.

If the variance of ξ(0) does not depend on L, the eigenfunctions are localized [CKM87, KS81, GMP77], and the local statistics of the eigenvalues are Poissonian [Min96, Mol81, KN07]. On the other hand, if ξ ≡ 0 then the eigenfunctions are spread out and the eigenvalues are deterministic with locally regular spacings (clock-points).

The critical regime appears when the variance of ξ is of the order 1/L. It has been considered by Kritchevski, Valkó and Virág in [KVV12]. They proved that delocalization holds in this case and that the eigenvalues near a fixed bulk energy E have a point process limit depending on only one parameter τ (which is a simple function of the variance and energy E) they called Schrτ. Moreover, they showed

that this point process exhibits strong eigenvalue repulsion. Rifkind and Virág [RV16] also studied the associated eigenfunctions. They found that the shape of the eigenfunctions near their maxima is given by the exponential of a Brownian motion plus a linear drift, and is independent of the eigenvalue. Note that heuristically, this regime corresponds to the high eigenvalues and eigenfunctions of HL.

Another famous family of one-dimensional discrete random Schrödinger operators is given by the tridiagonal matrices called β-ensembles. These operators seen from the edge converge, in the large dimensional limit, to the continuous Airy Hamiltonian Aβ formally defined as:

Aβ : u 7→ −∂x2u + (x +

2 √

βξ) u .

In the paper [AD14b], the analogue of the result of McKean was proved, namely that the first eigenvalue of the Airy operator properly rescaled converges to a Gumbel distribution in the small β limit. The density of states was also derived. Using similar techniques to the present paper, one should be able to prove the convergence of the point process of the first eigenvalues (properly rescaled) to a Poisson point process of intensity ex_{dx. In the bulk, the large dimensional limit of the eigenvalues of the β-ensembles}

converges towards the Sineβprocess [VV09]. In the small β limit, the Sineβ process was also shown to

converge towards a (homogeneous) Poisson point process [AD14a] using its characterization via coupled diffusions. Closely related discrete models are Jacobi matrices with random decaying potential [KVV12] and CMV matrices [KS09].

The analogue of HL in dimension d greater than 1 can be considered. However, due to the

irregu-larity of the white noise, the eigenvalue problem becomes singular already in dimension 2 and one has to renormalize the operator by infinite constants. This has been carried out by Allez and Chouk [AC15] in dimension 2 by means of the recently introduced paracontrolled calculus of Gubinelli, Imkeller and Perkowski [GIP15]; a similar construction should be possible in dimension 3 using the theory of regu-larity structures [Hai14].

Another generalization would be to consider the multivariate Anderson Hamiltonian of the form −∂2

x+ W′, operating on the vector-valued function space L2([0, L], Rr) and where W′ is the

deriva-tive of a matrix valued Brownian motion. This study has been done in the case of the stochastic Airy operator by Bloemendal and Virág [BV16]. The eigenvalues of the multivariate Anderson Hamiltonian are characterized by a family of coupled SDEs studied by Allez and one of the authors in [AD15].

Main ideas of the proof: We now present the main arguments of the proof. For the sake of clarity, we restrict ourselves to the case of Dirichlet boundary conditions.

The Riccati transform. The main tool for proving our result is the Riccati transform f 7→ f′/f that reduces the eigenvalue problem to the study of a family of diffusions, indexed by a parameter a ∈ R. More precisely, we introduce the potential (see Figure 8)

Va(x) =

x3

3 − ax , for all x ∈ R , a ∈ R , (7)

and we consider the family of diffusions:

dXa(t) = −Va′(Xa(t))dt + dB(t) ,

= (a − Xa(t)2)dt + dB(t) , (8)

starting from Xa(0) = +∞. Here B(t) := hξ, 1[0,t]i is a one-dimensional Brownian motion. Let us

mention that this diffusion hits −∞ in finite time almost surely: we restart the diffusion back from +∞ every time it hits −∞.

A typical realization of the diffusion Xafor a fixed large a > 0 does the following. It comes down

from +∞ very quickly and oscillates around √a (bottom of the well of Va) for a long time. It makes

many attempts to get out of the well, and from time to time, it makes an exceptional excursion to −√a, spends a very short time near that point, and either goes back to √a or explodes to −∞ very quickly.

0 100 200 300 400 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 100 200 300 400 -4 -2 0 2 4

FIGURE3. (color online) On the left, simulation of the first five eigenfunctions

(respec-tively ordered in black, blue, purple, red, green) and on the right the Riccati transform associated to the first eigenfunction with the positions of the maxima of the first five eigenfunctions represented with their color.

From the identity HLϕk = λkϕk, it is possible to show that the process

χ

k:= ϕ′k/ϕksolves the same

differential equation as Xa for a = −λk, see for instance [McK94, RRV11]. Regarding the boundary

conditions, the assumption ϕk(0) = ϕk(L) = 0, together with the fact that ϕ′k(0), ϕ′k(L) are non-zero

(otherwise ϕkwould be identically zero), yields the following:

lim x↓0 ϕ′_k(x) ϕk(x) = +∞ , lim x↑L ϕ′_k(x) ϕk(x) = −∞ . 6

Consequently,

χ

_{k(0) = +∞ and}

χ

_{k(L) = −∞, and almost surely}

χ

_kcoincides with Xafor a = −λk.

Actually, as in the Sturm-Liouville theory, the process

χ

_kblows up to −∞ exactly k times on (0, L].

The aforementioned result of McKean shows that, as L becomes large, the first eigenvalue is negative with a huge probability and goes to −∞. The same is true for the k first eigenvalues for any given k ≥ 1, so that one should think of the λk’s as negative numbers.

Since the eigenvalues (λk, k ∈ N) depend on the realization of the underlying noise ξ, the process

χ

k

is not a diffusion and actually doesn’t look like a “typical” solution of (8). As we will see in the proofs of the theorems,

χ

_kspends a macroscopic time near −p|λk|, see Figures 3 and 5. However, it is possible

to extract information out of the collection of diffusions (Xa, a ∈ R). Indeed, we have the following

monotonicity property: a.s. for all a < a′_{and t}

0≥ 0, if Xa(t0) ≤ Xa′(t₀) then X_a(t₀+ ·) ≤ X_a′(t₀+ ·)

up to the next explosion time of Xa(see Figure 4). Consequently, the number of explosions #Xaof Xa

on [0, L] is larger than or equal to the number of explosions #Xa′ of X_a′ on [0, L]. The eigenvalues of

HLthen coincide with (the opposite of) the jump locations of the map a 7→ #Xa: almost surely −λ1 is

the largest a for which Xa explodes once on [0, L], −λ2 is the largest a for which Xaexplodes twice,

and so on. 0 1000 2000 3000 4000 5000 6000 -1 0 -5 0 5 10

FIGURE 4. (color online) Ordering of the diffusions X_auntil their first explosion time

for a = 1, 3 (black), a = 0, 845 (blue), a = 0, 7 (purple), a = 0, 55 (red).

Convergence of the eigenvalues. Let us now present the main steps of the proof of our theorems. We start with the convergence of the point process of the rescaled and recentered eigenvalues QLof Theorem

1, which is the shortest part of the paper. While tightness is easy to get, we identify the limit by showing that for all fixed r1 < . . . < rk the random variable QL((r1, r2]), . . . , QL((rk−1, rk])
converges to a

vector of independent Poisson r.v. with parameters eri_{− e}ri−1. To that end, we proceed as follows. First

we observe that QL((ri−1, ri]) = #Xai− #Xai−1where ai =

√_a

L− ri/(4√aL). We subdivide (0, L]

into the 2n _{disjoint subintervals (t}n

j, tnj+1] with tnj = j2−nL, and we introduce the diffusion X tn

j

ai that

starts from +∞ at time tn

j and follows the SDE (8) with parameter ai. Then, we show that with large

probability for all i and j:

• Xai explodes at most one time on (t

n j, tnj+1], • Xai explodes on (t n j, tnj+1] if and only if X tn j ai explodes on (tnj, tnj+1]. 7

This holds because the potential Vai possesses a large well and the diffusion goes quickly down from

+∞: its starting point quickly becomes irrelevant. Therefore, it suffices to deal with the diffusions Xtnj

ai restricted to (tnj, tnj+1]: these diffusions are

independent for different values of j, and are monotone in i, so that a very simple computation, see Lemma 2.7, allows to get the aforementioned convergence.

An important remark is that we only need the monotonicity of the coupled diffusions for this part of the proof: we use no finer information about this coupling. That is why the arguments in the proof could be applied to other situations where the number of explosions of coupled diffusions counts the eigenvalues (e.g. Airy_β [RRV11], Sine_β[VV09] or the Stochastic Bessel Operator [RR09]).

The first eigenfunction. Let us now concentrate on the asymptotic behavior of the k first eigenfunctions: for simplicity, we start with k = 1. According to the convergence (2), we consider a discretization ML

of a small neighborhood of aLof mesh much smaller than 1/√aLand we obtain precise estimates on

the behavior of Xa, simultaneously for all a’s in ML, with large probability. Then, we show that with

large probability, −λ1 falls within this neighborhood of aL and that there exist two points α, α′ in the

discretization such that −λ2 < α ≤ −λ1 < α′. By definition, Xαexplodes one time and Xα′ does not

explode on the time interval [0, L].

From the monotonicity property, we have Xα ≤

χ

1 < Xα′ up to the first explosion time of X_α. With

large probability, Xαand Xα′ are “typical”: in particular, they remain close to √α and√α′respectively

most of the time. If the mesh of the discretization MLhas been chosen small enough, we deduce that

χ

₁is squeezed in between those two typical diffusions that are close to each other with large probability,

up to the first explosion time of Xα. However, this does not provide any good control on

χ

1 after this

explosion time (it gives no lower bound). In particular, it does not say whether, for instance,

χ

₁remains

around −√α, or goes back to√α. To push the analysis further, we rely on a symmetry argument.

0 20 40 60 80 100 -4 -2 0 2 4

FIGURE 5. (color online)

Simulation of

χ

₁ (in black)

and of two diffusions Xa(in

red) and Xa′ (in blue) with

a < −λ1< a′. 0 20 40 60 80 100 -4 -2 0 2 4

FIGURE 6. (color online)

Simulation of

χ

₁ (in black)

with the two diffusions Xa

(in red) and ˆXa(in blue)

un-til their first explosion, with a < −λ1.

The Riccati transform can be applied not only forward in time from time 0 but also backward in time from time L. This yields another set of diffusions ( ˆXa, a ∈ R), called the time-reversed diffusions, which

is equal in law to (−Xa, a ∈ R). Namely, we have:

d ˆXa(t) = Va′( ˆXa(t))dt + d ˆB(t) ,

starting from ˆXa(0) = −∞ and where ˆB(t) := B(L − t) − B(L). A coupling argument shows that the

number of explosions of Xa and ˆXa coincide. This allows to say that

χ

1, run backward in time from

time L, is squeezed in between ˆXα and ˆXα′ up to the explosion time of ˆX_α. Then, it remains to show

that the explosion times of Xαand ˆXα are very close so that the intervals on which we bound the first

eigenfunction overlap (see Figure 6).

Since Xα and Xα′ spend most of their time near √α and √α′ and since α, α′ are of the same

or-der, we deduce that the eigenfunction grows exponentially fast up to the explosion time of Xa. Then,

they follow the time-reversed diffusions which spend most of their time near −√α, −√α′ _{so that the}

first eigenfunction decreases exponentially fast from the explosion time until time L. This proves the localization of the first eigenfunction near the explosion-time of its Riccati transform.

We push this analysis further to obtain a much more precise result on the localization, namely the shape of the first eigenfunction of Theorem 2. The explosion of Xα occurs right after an exceptional

descent from +√_{α to −}√_{α whose trajectory is concentrated around the deterministic function t 7→} −√α tanh(√α t) (appropriately shifted in time). This is a simple consequence of the large deviations principle satisfied by the diffusion, that ensures that the trajectory on this exceptional descent is roughly given by the solution of the ODE

dx(t) = V_α′(x(t))dt .

We will obtain a precise statement about this fact thanks to the Cameron-Martin-Girsanov Theorem. We then show that

χ

₁ remains very close to Xα upon this deterministic descent. A careful argument,

involving the time-reversed diffusion, shows that the maximum of ϕ1 is located at one of the zeros

of

χ

₁ achieved during this deterministic descent: since all these zeros lie in a tiny region, we get a

precise enough control on the location of the maximum. Applying the inverse of the Riccati transform to −√α tanh(√α t), one gets the deterministic shape h(t) of the statement of the theorem.

Up to this point, we have not proved yet that the localization center is asymptotically uniform on [0, L]. To that end, we rely on another symmetry property that the operator HLenjoys. The law of the

white noise is invariant under the action of any piecewise continuous bijection σ of [0, L] that preserves the Lebesgue measure. As a consequence, the operator ˜_HLassociated with the image of ξ under σ−1

has the same statistics as the original one. We show that the maxima of the eigenfunctions for ˜_HL are

essentially located at the image through σ of the same quantities for HL. Since the uniform distribution

is the only distribution which is invariant under such Lebesgue preserving bijections, this is enough to ensure that the maximum is asymptotically located at a uniform point.

The next eigenfunctions. For the k-th eigenfunction with k > 1, the situation is slightly more involved though the strategy is the same. We show that with large probability, −λk falls in the

aforemen-tioned neighborhood of aL and that there exist α, α′ in the discretization of that neighborhood such

that −λk+1 < α ≤ −λk < α′ ≤ −λk−1. The typical diffusions Xαand Xα′ explode k and k − 1 times

respectively. We show that they remain close to each other up to the additional explosion time of Xα,

thus providing a good control on

χ

_kup to this time. Then, we rely on the time-reversed diffusions to

complete the picture, as in the case of the first eigenfunction. We refer to Figure 7 for an illustration.

0 50 100 150 200 -4 -2 0 2 4 0 50 100 150 200 -4 -2 0 2 4 0 50 100 150 200 -4 -2 0 2 4 0 50 100 150 200 -4 -2 0 2 4

FIGURE 7. Simulation of the Riccati transform of the first 4 eigenfunctions (in order

black, blue, purple, red). On the first picture, the position of the maxima of the first four eigenfunctions are represented with their color. Notice that the number of explosions equals the numbering of the eigenfunction. The explosions occur near the maxima of the previous eigenfunctions.

Organization of the paper: In Section 2, we collect some first estimates on the diffusion Xa and we

prove the convergence of the point process of eigenvalues towards the Poisson point process with inten-sity ex_{dx. The proofs of the first estimates are postponed to Section 4. In Section 3, we prove Theorems 1}

and 2. The proofs are based on more precise estimates on the diffusion Xaon its exceptional excursions

to −√a, which are obtained in Section 5. Finally, in Section 6 we prove Theorem 3 about Neumann boundary conditions.

Notations. The first hitting time by a continuous function f of a point ℓ ∈ R ∪ {−∞} is defined as follows

τℓ(f ) := inf{t ≥ 0 : f(t) = ℓ} ,

Moreover, for any integrable function f we define its average on [s, t] by setting

t s f (x)dx := 1 t − s ˆ t s f (x)dx . When s = t, we set this quantity to f(s).

Acknowledgements. LD thanks Romain Allez and Benedek Valkó for useful discussions. The work of LD is supported by the project MALIN ANR-16-CE93-0003. CL is grateful to Julien Reygner for several useful comments on a preliminary version of this paper. The work of CL is supported by the project SINGULAR ANR-16-CE40-0020-01.

2. CONVERGENCE OF THE EIGENVALUES TOWARDS THEPOISSON POINT PROCESS

The goal of this section is to prove that the point process QL defined in (3) converges in law to a

Poisson point process of intensity ex_{dx. This is the first step in the proof of Theorem 1. After some}

technical preliminaries on the well-definiteness of our diffusions, we collect some first estimates on the diffusion Xa, and then, we prove the convergence.

2.1. Technical preliminaries. For any given continuous function b starting from 0, we consider the ODE:

(

x(t) = ₋´t

0Va′(x(s))ds + b(t) , t > 0 ,

x(0) = _{+∞ .}

This ODE admits a unique solution that leaves +∞ at time 0+, and restarts from +∞ whenever it hits −∞. Let B be a standard Brownian motion on the probability space (Ω, F, P). Without loss of generality, we can assume that B(ω) is continuous for all ω ∈ Ω. We apply deterministically the solution map associated to the ODE above for every realization of the Brownian motion. This yields a well-defined process (Xa(t), t ≥ 0, a ∈ R).

Remark 2.1. Actually, the process can be initialized not only at time0 but also at some arbitrary time t0 ≥ 0. The above construction still applies, and yields a family (Xat0(t), t ≥ t0, a ∈ R, t0≥ 0). In any

case,Xt0

a (t0) = +∞.

The monotonicity of the solution map associated to the ODE yields the following property: if a < a′

and Xa(t) ≤ Xa′(t) then X_a(t + ·) ≤ X_a′(t + ·) up to the next explosion time of X_a.

2.2. First estimates on the diffusion Xa. We now study the process just defined above:

dXa(t) = (a − Xa(t)2)dt + dB(t), Xa(0) = +∞.

This diffusion evolves in the potential Va(x) = −ax + x3/3. It blows-up to −∞ in finite time a.s.

and immediately restarts from +∞. From now on, we let 0 = ζa(0) < ζa(1) < ζa(2) < . . . be the

successive explosion times of the diffusion Xa. When a > 0, the potential Vahas a well centered at √a.

The diffusion has to cross the barrier from √a to −√a, which is of size ∆Va = (4/3)a3/2, in order to

explode to −∞ (see Figure 8).

∆_{V =}4

3a 3_/2

−√a 0 √a

FIGURE8. The potential Vawhen a > 0. 11

The expectation of the first explosion time ζa(1) admits the following expression, see [McK94]: m(a) := E[ζa(1)] = √ 2π ˆ +∞ 0 dv √ v exp  2av −1₆v3  (9) = √π aexp( 8 3a 3/2_{)(1 + o(1)), when a → +∞ . (10)}

We see that the explosion time is of order exp(2∆Va), which is in line with Kramers’ law.

Remark 2.2. Recall that the number of explosions ofXain[0, L] is equal to the number of eigenvalues

below _{−a. By the law of large numbers, it readily implies that the integrated density of states N(λ)} mentioned in the introduction is precisely given by

N (λ) = 1

m(−λ) for allλ ∈ R . (11)

The following result is due to McKean, we refer to [McK94,AD14b] for a proof.

Proposition 2.3. As _{a → ∞, the r.v. ζ}a(1)/m(a) converges in distribution to an exponential law of

parameter1.

As a direct corollary, we get the following result.

Proposition 2.4. The point process of the explosion times ofXarescaled bym(a) converges in law, for

the vague topology, to a Poisson point process on R+with intensity1.

It is then natural to introduce L 7→ aLas the reciprocal of a 7→ m(a). A simple computation yields

the following asymptotics aL= 3 8ln L 2/3 + 3−1/3_{· 2}−1(ln L)−1/3ln(3 1/3 2π ln L) + o(1)  . (12)

The diffusion Xa, with a close to aL, typically explodes a finite number of times on the time interval

[0, L]. More precisely, for all r ∈ R, using (10), we have as L → ∞: m aL+ r √_a L
= π √_a L exp(8 3a 3/2 L + 4r)(1 + o(1)) ,

so that the order of magnitude of the number of explosions of Xaon [0, L] for a = aL+ r/(4√aL) is

given by L/m(aL+ r/(4√aL)) = exp(−r)(1 + o(1)).

Until the end of this subsection, we drop the subscript a from Xaand from the explosion times ζa(k),

k ≥ 0 to alleviate the notations. Let us analyse the diffusion X until its first explosion time ζ := ζ(1). We set

ta=

ln a √

a . (13)

A typical realization of the process X behaves as follows:

1- Entrance: The diffusion is close to a deterministic path from +∞ until it gets close to the bottom of the well of the potential Va. Let ℓ :=√a + ln a/a1/4. We have

τℓ(X) =

3

8ta(1 + o(1)), together with the bound

sup t∈(0,38ta]  X(t) − √ a coth(√at)≤ 1 . 12

2- Explosion: Let ℓ := −√_{a − ln a/a}1/4. We have τ_{−∞(X) − τ}ℓ(X) =

3

8ta(1 + o(1)). together with the bound

sup t∈[τℓ(X),ζ)  X(t) +√a coth(√_{a(ζ − t))} ≤ 1 . We denote by DN

a the event on which the N first trajectories (X(t), t ∈ [ζ(k), ζ(k + 1))) for

k ∈ {0, · · · , N − 1} satisfy the estimates above.

Proposition 2.5. Fix_{N ∈ N. For all a large enough, we have} P_[DN_{a] ≥ 1 −} 1

ln a .

The proof of Proposition 2.5 relies on a comparison with an ordinary differential equation (in the same way as in [DV13]). The proof can be found in Section 4.

For t0> 0, let Xt0be the diffusion following the same SDE as X but starting from +∞ at time t0. The

following synchronization estimate ensures that the diffusion Xt0gets very close to the original diffusion

X after a short time of order O(ta), and that they stay together until their first explosion. In that result,

we assume that a and L are related to each other by the following condition: aL−1 < a < aL+1. Notice

that this condition is somehow arbitrary. Eventually, we will only consider values a in a window of size negligible compared to 1 and centered at aL, so that this condition is not restrictive for our purpose.

We also set τk

−2√a(X) := inf{t ≥ ζa(k − 1) : X(t) = −2

√ a}.

Proposition 2.6. Fix_{N ≥ 1 and t}0 = t0(a) ∈ (0, L). There exists C > 0 such that the following holds

true with a probability at least1 − O(1/ ln a) as a → ∞ and L → ∞ with aL− 1 < a < aL+ 1. If

there exists_{k ∈ {1, . . . , N} such that t}0∈ [ζa(k − 1), ζa(k)) then we have the bounds

 X(t) − Xt0(t)  ≤ ln(a) a1/4 , ∀t ∈ [t0+ 3 8ta, τ k −2√a(X) ∧ L] ,  τ_−∞(Xt0) ∧ L − ζ_a(k) ∧ L  < C √ a .

The proof of Proposition 2.6 is based on a coupling with a stationary diffusion. Since the arguments are rather classical, we postpone its proof to Section 4.

2.3. Proof of the convergence of the point process of the eigenvalues. By [Kal02, Lemma 14.15], the family (QL)L>1is tight if for every r > 0 and every ε > 0, there exists c > 0 such that

sup

L>1

PQ_{L([−r, r]) > c} < ε . (14) Since QL((−∞, r]) is equal to the number of explosions of XaL−r/(4√aL) on the time-interval [0, L],

we deduce from Proposition 2.4 that there exists c′_{> 0 such that}

lim

L∈N,L→∞

PQ_{L([−r, r]) > c}′ < ε , and therefore, that there exist c > 0 such that

sup

L∈N

P_{QL([−r, r]) > c < ε .}

Since aL= a_⌊L⌋+o(1/√a_⌊L⌋), we can bound the number of explosions of XaL−r/(4√aL)by the number

of explosions of Xa_⌊L⌋−r′_/(4√a

⌊L⌋)for some appropriately chosen r

′_{, thus ensuring that the latter bound} 13

extends to all L ∈ (1, ∞). We deduce that (14) holds true, thus ensuring the tightness of the family (QL)L>1.

It remains to prove that any limit Q∞is distributed as a Poisson point process with intensity exdx. By

classical arguments, it suffices to show that for every k ≥ 1, and every −∞ = r0 < r1 < r2< . . . < rk,

if we set

QL(i) := QL((ri−1, ri]) , i = 1, . . . , k ,

then (QL(1), . . . , QL(k)) converges in law to a vector of independent Poisson r.v. with parameters pi

where

pi:= eri− eri−1 , i = 1, . . . , k .

Let us fix from now on r1 < r2 < . . . < rk, and notice that if we denote by #Xai the number of

explosions of Xai on [0, L], where ai= aL− ri/(4√aL), then we have almost surely

QL(i) = #Xai− #Xai−1 , i = 1, . . . , k .

To identify the limiting law of (QL(1), . . . , QL(k)), we follow an indirect approach. First, we

intro-duce some simpler r.v. ˜QL(i) which are shown to converge to the right limits, and then we prove that

they are actually close in probability to the original ones.

We discretize the time-interval [0, L] in such a way that the studied diffusions will typically explode at most once on each sub-interval with large probability. Let n ≥ 1 be given. We set tn

j := j2−nL for all

j ∈ {0, . . . , 2n_{} and we let X}j a := X

tn j

a be the diffusion starting from +∞ at time tnj. The main idea of

the proof is to approximate the number of explosions of Xai on (t

n

j, tnj+1] by the number of explosions

of Xj

ai on the same interval. Such an approximation is justified by Proposition 2.6. The advantage of

considering the diffusions Xj

ai restricted to (tnj, tnj+1] is twofold: first, for every j, they are ordered in i

by the monotonicity of the diffusions; second, for different values j they are independent. For every j ∈ {0, . . . , 2n_{− 1}, and for every i ∈ {1, . . . , k}, we set U}

j(i) = 1 if Xaji explodes on

(tn

j, tnj+1], and Uj(i) = 0 otherwise. Recall that a1 > a2 > . . . > ak. Note that on each time-interval

(tn_j, tn_j+1], we have the ordering Xajk ≤ · · · ≤ X

j

a1 until the first explosion of X

j ak. Then, we set ˜ Q(n)_L (i) := 2n₋₁ X j=0  Uj(i) − Uj(i − 1)  , i = 1, . . . , k .

Lemma 2.7. The vector( ˜Q(n)_L (i))i=1,...,k converges in distribution, asL → ∞ and then n → ∞, to a

vector of independent Poisson r.v. with parameterspi.

Proof. First of all, the collection (indexed by j = 0, . . . , 2n_{−1) of the k-dimensional vectors (U}j(i), i =

1, . . . , k) is i.i.d., and for every j we have almost surely Uj(1) ≤ . . . ≤ Uj(k) since a1 > . . . > ak.

Henceforth we have for i ∈ {1, · · · , k − 1},

P[Uj(1) = . . . = Uj(i) = 0, Uj(i + 1) = . . . = Uj(k) = 1] = P[Uj(i) = 0, Uj(i + 1) = 1]

= P[Uj(i + 1) = 1) − P(Uj(i) = 1] ,

as well as

P_{[Uj(1) = . . . = Uj(k) = 0] = P[Uj(k) = 0] ,} and

P_{[Uj(1) = . . . = Uj(k) = 1] = P[Uj(1) = 1] .}

Consequently, the law of the vector Uj is completely characterized by its one-dimensional marginals.

Since m(aL)/m(ai) → erias L → ∞, Proposition 2.4 gives as L → ∞

P_{[Uj(i) = 1] → 1 − exp(−2}−n_eri_{) ,}

for any j. Moreover, the random variables Uj, j ∈ {0, . . . , 2n − 1} are independent. We have all

the elements at hand to compute the limiting distributions of the ˜Q(n)_L (i)’s. Fix ℓ1, . . . , ℓk such that

ℓ =Pk 1ℓi ≤ 2n. Then, we find: lim L→∞ Ph k \ i=1 { ˜Q(n)_L (i) = ℓi} i = lim L→∞  2n ℓ1, . . . , ℓk, 2n− ℓ  Yk i=1 P_{[U1(1) = . . . = U1(i − 1) = 0, U1(i) = . . . = U1(k) = 1)}ℓii × P[U1(1) = . . . = U1(k) = 0]2 n_−ℓ =  2n ℓ1, . . . , ℓk, 2n− ℓ  Yk i=1

1 − exp(−2−neri_{) − 1 + exp(−2}−n_eri−1₎
ℓi

exp(−2−nerk₍₂n_{− ℓ))} = k Y i=1 eri_{− e}ri−1
ℓi ℓi!

e−(eri−eri−1)_{(1 + O(2}−n)) ,

so that the limiting distribution as n → ∞ is the product of Poisson distributions with parameter pi. 

We now introduce an event on which ˜Q(n)_L and QLcoincide. Let FL,nbe the event on which for every

i ∈ {1, . . . , k} and j ∈ {0, · · · , 2n− 1}:

(i) Xai never explodes twice in any interval (tnj, tnj+1],

(ii) Xai explodes on (t

n

j, tnj+1] if and only if X j

ai explodes on (tnj, tnj+1].

On the event FL,n, the auxiliary random variables ˜Q(n)_L (i) indeed coincide with QL(i).

If we prove that the probability of FL,ntends to 1 when L → ∞ and then n → ∞, then by Lemma

2.7 we get: P_{[QL(1) = ℓ1, . . . , QL(k) = ℓk] =} k Y i=1 pℓi i ℓi! e−pi_{+ ε} L,n ,

where εL,n goes to 0 as L and then n go to infinity. This would ensure that the vector QLconverges in

distribution to a vector of independent Poisson random variables with parameters pi, and would conclude

the proof of this section.

Therefore, it remains to show that P[FL,n] goes to 1 as L → ∞ and then n → ∞. By Proposition 2.4

and Proposition 2.6, the probability that for all i ∈ {1, . . . , k}: (a) Xai never explodes twice in any interval (t

n j, tnj+1],

(b) Xai never explodes in any interval (tnj+1− taL, t

n j+1],

(c) the next explosion times of Xaiand X

j

aiafter time tnj lie within a distance of order 1/√aLfrom

each other,

goes to 1 as L → ∞ and then n → ∞. Property (a) ensures (i). The monotonicity of the diffusions ensures the following assertion: if Xj

ai explodes on (tnj, tnj+1], then Xai explodes on (tnj, tnj+1]. The

converse assertion is implied by (b) and (c), so that (ii) follows.

3. LOCALIZATION OF THE EIGENFUNCTIONS

In this section, we first introduce the time-reversed diffusions associated to the eigenvalue problem. Then, we collect some fine estimates on the diffusion during its exceptional excursion that leads it from √

a to −√a. Finally, we provide the proof of Theorems 1 and 2. 3.1. Time reversal. We define the time-reversed operator

ˆ

HL= −∂x2+ ˆξ , x ∈ (0, L) ,

with homogeneous Dirichlet boundary conditions, where ˆ_{ξ(·) := ξ(L − ·) in the distributional sense.} Let ( ˆϕk, ˆλk)k≥1 be the corresponding eigenfunctions/eigenvalues and observe that almost surely for all

k ≥ 1

ˆ

ϕk(·) = ϕk(L − ·) , λˆk= λk.

We can therefore study the Riccati transforms of the reversed operator and introduce the associated family of diffusions. For convenience, we take the opposite of the Riccati transform:

ˆ

χ

_{k(t) := −}ϕˆ ′ k(t) ˆ ϕk(t) , _{t ∈ (0, L) ,}

with boundary conditions ˆ

χ

_{k(0) = −∞ and ˆ}

χ

_{k(L) = +∞. We then have the identity:}

ˆ

χ

_{k(L − t) =}

χ

_k(t) .

It is natural to introduce the collection of diffusions ( ˆXa(t), t ≥ 0), a ∈ R, as solutions of:

d ˆXa(t) = Va′( ˆXa(t))dt + d ˆB(t) , (15)

starting from ˆXa(0) = −∞ and where ˆB(t) := B(L − t) − B(L). We then let ˆζa(k) be the k-th

explosion time of ˆXa. We also set ˆζa(0) = ζa(0) = 0.

Remark 3.1. Time0 for the diffusion ˆXacorresponds to timeL for the diffusion Xa.

Lemma 3.2. Almost surely, for every_{k ≥ 1 if X}aexplodes to−∞ exactly k times on [0, L] then

(1) ˆXaexplodes exactlyk times to +∞ on [0, L],

(2) for every i ∈ {1, . . . , k}, we have ζa(i − 1) ≤ L − ˆζa(k − i + 1) ≤ ζa(i).

Proof. (1) is immediate as the numbers of explosions of Xaand ˆXaare related to the number of

eigen-values below −a of HLand ˆHL, and these two operators share the same eigenvalues.

To prove the second property, note that almost surely, for all t0∈ Q ∩ (0, L) the operators

H0,t0 := −∂ 2 x+ ξ , x ∈ (0, t0) , and ˆ HL−t0,L:= −∂ 2 x+ ˆξ , x ∈ (L − t0, L) ,

share the same eigenvalues. Assume that for some i, L− ˆζa(k −i+1) > ζa(i) and pick a rational number

t0in between these two values. Since the diffusion Xaexplodes at least i times on the interval [0, t0], the

operator H0,t0 has at least i eigenvalues below −a. On the other hand, the diffusion ˆXaexplodes at most

i − 1 times on [L − t0, L] so that the diffusion ˆXaL−t0 that starts at −∞ at time L − t0 cannot explode

more than i − 1 times either. Consequently, ˆHL−t0,Lhas at most i − 1 eigenvalues below −a. This raises

a contradiction. The inequality ζa(i − 1) ≤ L − ˆζa(k − i + 1) follows by symmetry, thus concluding the

proof. 

3.2. Fine estimates on the exceptional excursions. In this subsection, we collect some precise esti-mates on the behavior of the diffusion Xaduring its first exceptional excursion to −√a.

We let θa:= τ₋√a(Xa) be the first hitting time of −√a, υabe the last hitting time of 0 before θaand

ιathe last hitting time of √a before θa. We take similar definitions for the time-reversed diffusion ˆXa:

ˆ

θais the first hitting time of +√a, ˆυais the last hitting time of 0 before ˆθaand ˆιais the last hitting time

of −√a before ˆθa.

We call excursion of Xa a portion of the trajectory that starts at √a, reaches −√a while staying

(strictly) below √a and then comes back to √a (either after an explosion and a restart from +∞ or without explosion). Note that the diffusion Xastarts its first excursion at time ιa.

In the next proposition, we control the behavior of Xaduring its first excursion. We refer to Figure 9

for an illustration of the path Xa.

(3/8)ta (3/4) ta (3/8)ta (3/8)ta

υa θa

√a

−√a

−√a + 1 Oscillations around√a

√a− 1

ιa

O(1/a1/4₎

O(m(a))

FIGURE9. Schematic path of Xaand its first excursion to −√a in the case of an explosion

Proposition 3.3(Typical diffusion on its first excursion). There exists some constant C > 0 such that for all_{a large enough, with a probability at least 1 − O(1/ ln ln a), the following holds:}

(i) Oscillations around √a: For all_{t ∈ [}3 8ta, τ−2√a(Xa) ∧ L), t 3 8ta Xa(s)ds ∈√a −ln(a) a1/4 , √ a +ln(a) a1/4 .

(ii) Crossing: For all t ∈ [ιa, θa],

 X_a(t) − √ a tanh(−√a(t − υa))  ≤ C √ a ln a , and  θ_a− υ_a− (3/8)t_a  ≤ C ln ln a √ a .

(iii) Explosion and/or return to √a: If after time θa the diffusion Xa explodes before coming back

to+√a, then Xa stays below −√a + 1 in [θa, ζa(1)) and explodes within a time (3/8) ta+

2(ln ln a)2_/√_a.

If it does not explode,Xareturns to√a within a time (3/4) ta+ C(ln ln a)2/√a. 17

Remark 3.4.

• We do not control in this proposition the time it takes to reach 0 after time ιa(which is possibly

very long) but it is not necessary for our purposes.

• In (iii), the time necessary to explode (resp. return to √a) is in fact (3/8)ta+ O(ln ln a/√a)

(resp.(3/4)ta+O(ln ln a/√a)) but we do not have a good enough control on the corresponding

probability.

The next proposition controls the difference between two diffusions Xaand Xa+ε, for ε not too large,

during the first excursion of Xa. Again, the bounds are not optimal but they suffice for our purposes.

Proposition 3.5(Coupling during the excursion). Let ε ∈ (a−2/3, 1). There exists some constant C > 0 such that with a probability at least_{1 − O(1/ ln a), the following holds for all a large enough:}

For all_{t ∈ [υ}a−₁₆1 ta, υa+₁₆1ta],



Xa+ε(t) − Xa(t)

 ≤ 1 , For all_{t ∈ [υ}a+₁₆1 ta, θa−₁₆1 ta], Xa+ε(t) ≤ −√a + C a3/7,

For allt ∈ [υa+₁₆1 ta, θa], Xa+ε(t) ≤√a − 1.

As we will see later on, the above estimates provide a good control on the first eigenfunction over the time-interval [ιa, θa] (for some well chosen a). We will control the eigenfunction after time θaby using

the time-reversed diffusions. The idea is that after the stopping time θa, the Brownian motion makes no

exceptional event with high probability so that ˆXa(L − ·) should oscillate around −√a.

Proposition 3.6 (Control after time θa). There exists ρ > 0 such that for all a large enough, with

probability greater than_{1 − (ln a)}−ρ, if ˆXadoes not explode before time(L − θa− 10 ta) ∨ 0, then it

does not explode before time_{(L − θ}a) ∨ 0 and for all t ∈ [θa, L] we have the bounds:

ˆ Xa(L − θa) ≤ −√a + ln(a) a1/4 , t θa ˆ Xa(L − s)ds ≤ −√a + ln(a) a1/4 .

For the sake of readability, we postpone the proofs of those technical propositions to Section 5 and proceed to the proof of the main results of this paper.

3.3. Preliminaries for the proof of Theorems 1 and 2. The collection (indexed by L) of random variables

QL, (Ui)i≥1, (mi)i≥1
,

is tight since the first coordinate converges in law, see Section 2, and since the remaining coordinates belong to the product space [0, 1]N_{× P}N_{which is compact. Let Q}

∞, (Ui∞)i≥1, (m∞i )i≥1
be the limit

of a converging subsequence. We already know that Q∞ is a Poisson point process of intensity exdx.

Our goal is to show that for any k ≥ 1, (U∞

1 , . . . , Uk∞) is i.i.d. uniform over [0, 1] and independent of

Q∞, that m∞i = δU∞

i for every i = 1 . . . k and that the shape around the maximum is given by the

inverse of a cosh. From now on, k is fixed.

As we explained in the introduction, we do not deal directly with the Riccati transforms of the eigen-functions, but rather control many typical diffusions that will be close enough to those Riccati transforms. Let ǫL be a function going to 0 at a small speed (for instance, ǫL = 1/ ln ln ln ln(L) will do). We

consider the window WL:=aL−_ǫL√1_aL, aL+_ǫL√1_aL



together with the following discretization: ML:=  a ∈ WL: ∃n ∈ Z, a = aL+ nǫL √_a L  . The number of points in this grid is of order ǫ−2

L . By the convergence of the eigenvalues towards the

Poisson point process that we established in Section 2, the probability that −λ1, . . . , −λkfall into distinct 18

subintervals of the grid MLgoes to 1 as L → ∞. On this event, we introduce the random variables αi

and α′

i which are the largest and smallest elements of the grid MLthat satisfy

αi ≤ −λi < α′i , ∀i ∈ {1, . . . , k} ,

(and we set arbitrary values to αiand α′i on the complementary event).

We then have αk≤ −λk< α′k≤ αk−1≤ . . . < α′2 ≤ α1 ≤ −λ1< α′1 . −λ1 −λ2 −λ3 −λ4 α′ 1 α1 α′ 2 α3 α4 α′4 no explosion 1 explosion 2 explosions 3 explosions

More than 4 explosions

ǫL/√aL

α′ 3= α2

FIGURE 10. Locations of the (opposite of the) eigenvalues −λ_k, together with their

approximations αkand α′k. Black dots correspond to points of ML.

Notice that Xaand ˆXa(thanks to Lemma 3.2) explode exactly i times up to time L, whenever a = αi

or a = α′ i+1.

From now on, we will use the notation tL := taL = ln(aL)/√aL. Notice that, for all a ∈ ML, the

difference tL− ta= O ln(aL)/(ǫLa2L)

is negligible compared to 1/√aLso that the estimates already

obtained carry through without modifications upon replacing taby tL.

As in Subsection 2.3, we divide [0, L] into 2n_{macroscopic sub-intervals [t}n

j, tnj+1] where tnj := j2−nL

and denote by Xj a:= X

tn j

a the diffusion starting from +∞ at time tnj and by ˆXaj+1:= ˆXL−t

n j+1

a the

time-reversed diffusion starting from −∞ at time L − tn

j+1. The underlying idea of the proof is that, on every

time-interval [tn

j, tnj+1], the diffusion X j

ais a faithful approximation of Xauntil their first explosions and

the content of Propositions 3.3, 3.5 and 3.6 can be applied to Xj

a and ˆXaj+1 (on an interval of length

2−nL instead of L).

Remark 3.7. The aforementioned approximation is supported by the following heuristics: the restriction of every eigenfunction of_HLto a small interval surrounding its maximum coincides, up to small errors,

to the principal eigenfunction of the operator_HLrestricted to that interval.

In the next proposition, we gather the estimates of the previous sections that we need:

Proposition 3.8(Typical diffusions). The following holds with probability greater than 1−O(ǫ−2_L (ln ln aL)−1).

For all_{a ∈ M}L, andj ∈ {0, · · · , 2n− 1}:

(i) Synchronization of Xaand Xaj: if tn_j ∈ [ζa(l − 1), ζa(l)) for some l ∈ {1, · · · , k} then

 Xa(t) − Xaj(t)  ≤ ln(a) a1/4 , ∀t ∈ [t n j + 3 8tL, τ l −2√a(Xa) ∧ tnj+1] ,  τ_−∞(X_aj_{) ∧ t}n_{j+1− ζ}a(l) ∧ tnj+1  < C √_a , whereτ₋₂l √_a(Xa) := inf{t ≥ ζa(l − 1) : Xa= −2√a}.

(ii) Typical behavior of Xj

aand Xa: The diffusions Xaj follow the behavior described in Proposition

3.3 and satisfy the conditions of the event_Da1, and the diffusionsXasatisfy the conditions of the

eventDk

aas well.

(iii) Coupling: Xaj andX_a+ǫj _L/√aLfollow the behavior described in Proposition 3.5. 19

(iv) Time-reversal: ˆXaj+1follows the behavior described in Proposition 3.6.

and similar statements for ˆXa(with its time reversalXa) up to obvious modifications in the statements

due to the fact that ˆXahas the same law as−Xa.

The next proposition contains several technical results which allow to approximate the diffusion Xa

by the collection of diffusions Xj

a, and which rule out some undesired boundary effects due to our

discretization of [0, L]. It is not difficult to prove that the stated properties hold with large probability for a given value a. But as we want to control simultaneously the diffusions for all parameters a ∈ ML, the

proof becomes more involved and relies on coupling arguments for different parameters a’s. Note that we use only (i) and (ii) for the localization of the first eigenvalue proved in paragraph 3.4 and the reader may skip the rest of the proposition at first reading.

Proposition 3.9(Control of the excursions). Fix δ > 0. There exists a0 = a0(δ) ∈ MLsuch that for

all_{n ≥ 1 large enough and all L large enough, the following holds with probability at least 1 − δ. The} diffusion Xa0 explodes at leastk times in [0, L]. Furthermore, for all a ∈ ML with a ≥ a0 and all

j ∈ {0, . . . , 2n− 1}:

(i) None of the explosion times of Xaand ˆXafall at distance less than2−2nL from any tnj.

(ii) Over the time-interval [tn

j, tnj+1], the diffusion Xaj (resp. ˆXaj+1(L − ·)) starts at most one

excur-sion to₋√a (resp.√_{a) that hits −}√a (resp.√a) before time tn_j+1(resp.tn_j). Over the first and last time-intervals (i.e.j = 0 and j = 2n_{− 1), the diffusion X}aj (resp. ˆXaj+1) does not hit−√a

(resp.√a).

(iii) If Xaj(resp. ˆXaj+1) explodes on[tn_j, tn_j+1], then neither Xaj−1(resp. ˆXaj) norXaj+1(resp. ˆXaj+2)

explodes on[tn

j−1, tnj] and [tnj+1, tnj+2].

(iv) If Xj

a (resp. ˆXaj+1) explodes on[tn_j, tn_j+1], then so does Xaj0 (resp. ˆX

j+1

a0 ) and their explosion

times lie at a distanceo(tL) from each other.

(v) The total number of explosions of Xa(resp. ˆXa) equals the sum overj of the number of

explo-sions ofXaj (resp. ˆXaj+1) on[tnj, tnj+1].

The proof of this proposition is postponed to Section 5 as well. We define E(n, δ) as the event on which:

(a) the eigenvalues −λ1, . . . , −λkfall into distinct subintervals of ML,

(b) Control of the excursions: the content of Proposition 3.9 is satisfied, (c) Typical diffusions: the content of Proposition 3.8 is satisfied.

By the convergence towards the Poisson point process for (a), Proposition 3.9 for (b) and Proposition 3.8 for (c), the probability of the event E(n, δ) is larger than 1 − 2δ for all n and L large enough.

In the next subsection, we show that m∞ 1 = δU∞

1 and that the shape of ϕ1around U1is asymptotically

given by the inverse of a hyperbolic cosine. In the subsequent subsection, we prove the same result but for the i-th eigenfunction with i ∈ {2, . . . , k}. In the last subsection, we prove that the r.v. (U∞

1 , . . . , Uk∞)

are i.i.d. uniform over [0, 1] and independent of Q∞. These three subsections therefore conclude the

proof of Theorems 1 and 2.

Remark 3.10. At a technical level, the symmetries that we rely on are similar in spirit to those appearing in[OWW14] where the invariant measure of the stochastic Allen-Cahn equation is studied.

3.4. The first eigenfunction ϕ1. Recall that we denote the Riccati transform of ϕ1by

χ

₁ = X−λ1. Let

α := α1 and α′ := α′1 be the approximations of −λ1. We work deterministically on the event E(n, δ)

for some arbitrary δ > 0. Recall that Xαand ˆXαexplode exactly one time on [0, L] while Xα′ and ˆX_α′

do not explode. Moreover, the diffusions Xαand Xα′are “typical” in the sense of (b) and (c) of E(n, δ).

There is an interval [tn

j, tnj+1) that contains the (unique) explosion time of Xα which occurs in [tnj +

2−2n_{L, t}n

j+1 − 2−2nL] thanks to part (i) of (b) of E(n, δ). Using part (i) of (c), we know that Xαj and

Xα synchronize after time tn_j + (3/8)tL and that Xαj explodes as well (only one time, by (ii) of (b)).

Moreover, using Lemma 3.2 on the interval [tn

j, tnj+1], we deduce that ˆX j+1

α (L − ·) explodes one time in

[tn_j, tn_j+1] and by (b) and (c) again, that the time-interval [t_jn+ 2−2nL, tn_{j+1− 2}−2nL] contains the unique explosion time of ˆXα(L − ·) as well.

The diffusion Xj

αmakes only one excursion to −√α on the time interval [tnj, tnj+1) which therefore

explodes to −∞ before coming back to√α: let us denote by θ its first hitting time of −√α after time tn

j, by υ its last hitting time of 0 before time θ and by ι its last hitting time of√α. The same holds for

ˆ

Xαj+1and we use the notations ˆθ, ˆυ and ˆι. See Figure 11 for an illustration of the above diffusions.

υ θ √ α −√α tn j tnj+1

X

_αj

X

α

No passage time to −√αfor Xα Xαand Xαjsynchronized No explosion of Xα 2−2n_L τ_−∞(Xj α) ι θ √α −√α tn j tnj+1 υ X_αj ˆ X_αj+1

FIGURE11. Typical behavior of Xα, Xαj and ˆXαj+1on the time-interval [tnj, tnj+1]

The next proposition provides some estimates on the first eigenfunction ϕ1 over the time interval

[υ − (1/16)tL, L]. Note that thanks to the time reversal, we obtain a similar description on the time

interval [0, L − ˆυ + (1/16)tL]. Proposition 3.12 will show that those two intervals overlap and this will

permit us to control the first eigenfunction on the whole interval [0, L]. See Figure 12 for an illustration. Proposition 3.11(Control after time υ − (1/16)tL). On the event E(n, δ), the following holds:

(1) Hyperbolic cosine around υ: ∀t ∈ [υ − (1/16)tL, υ + (1/16)tL], ϕ1(t) ϕ1(υ) = 1 cosh(√aL(t − υ))  1 + O(|t − υ| √_a L ln aL ). (2) Control over [υ + (1/16)tL, θ]: The eigenfunction |ϕ1| stays below |ϕ1(υ + (1/16)tL)| on the

time interval[υ + (1/16)tL, θ].

(3) Exponential decay after time θ:

∀t ∈ [θ, L − 3₈tL], |ϕ1(t)| ≤ |ϕ1(θ)|e− √_a

L(t−θ)(1+o(1))_,

and the eigenfunction_|ϕ1| stays below |ϕ1(L −3₈tL)| on the time-interval [L −3₈tL, L].

υ

_θ

1 16tL Control shape ϕ1 with Xαand Xα′ ϕ1stays below ϕ1(υ +161tL) (thanks to Xα′) ϕ1exponentially decreases (thanks toXˆα) L − ˆθ 3 8tL L − ˆυ Control of Proposition 3.11

FIGURE12. Schematic outline of the proof.

Proof. For (1): We have Xα(t) ≤

χ

1(t) ≤ Xα′(t) for all t ≤ ζ_α(1). As ζ_α(1) > tn_j + 2−2nL, we get

by monotonicity τ_−∞(Xαj) > tnj + 2−2nL. Moreover, τ−∞(Xαj) − υ ≤ (3/4)tL+ o(tL) (by (ii) of (c))

which implies that υ − (1/16)tL> tnj + (3/8)tLfor L large enough. Notice also that Xα′(t) ≤ X_αj′(t)

for all t ∈ [tn j, L].

Therefore, using (i) and (iii) of (c), for all t ∈ [υ−(1/16)tL, υ+(1/16)tL], we have |

χ

1(t)−X j α(t)| ≤ |Xα(t) − Xαj(t)| + |X_αj′(t) − Xαj(t)| ≤ 2 and: ϕ1(t) ϕ1(υ) = exp( ˆ t υ X_αj_{(s)ds + O(|t − υ|))} = exp( ˆ t υ √ α tanh(−√α(s − υ))ds + O(|t − υ| √ a_L ln aL )) = 1 cosh(√aL(t − υ))  1 + O(|t − υ| √ a_L ln aL ), where we used (ii) of (c) at the second line.

For (2): By (iii) of (c), the diffusion X_αj′ is below −

√

α(1 + o(1)) on the interval [υ + (1/16)tL, θ −

(1/16)tL]. Therefore, |ϕ1| decreases on this time interval and we have the bound

|ϕ1(θ − (1/16)tL)| ≤ |ϕ1(υ + 1 16tL)| exp  −√α (1 + o(1))tL 1 4+ o(1)
 . Since Xj

α′ stays below √α on [θ − (1/16)tL, θ], we deduce that |ϕ1| remains below |ϕ1(υ + ₁₆1 tL)| on

this time interval, thus yielding (2).

For (3): By (ii) of (c), Xαj explodes at time θ + (3/8 + o(1))tL. Applying Lemma 3.2, we deduce that

ˆ

Xαj+1(L − ·) does not explode on [θ + 10tL, tnj+1] so that (iv) of (c) yields the bound t θ ˆ X_αj+1_{(L − s)ds ≤ −}√α +ln(α) α1/4 , t ∈ [θ, t n j+1] . (16)

We thus get the following bound for all t ∈ [θ, L − (3/8)tL] and all L large enough:

ˆ t

θ

ˆ

Xα(L − s)ds ≤ −(t − θ)√aL(1 + o(1)) .

Indeed, for t ≤ tn

j+1−38tLthis is a direct consequence of (16) and of the synchronization estimate. For

t > tn_j+1− 3₈tL, it comes from (16), the synchronization estimate, the typical behavior of ˆXα, the fact

that tn

j+1− θ > 2−2nL and the simple calculation:

ˆ t θ ˆ Xα(L − s)ds = ˆ tn_j+1−3₈tL θ ˆ Xα(L − s)ds + ˆ L−3₈tL tn j+1−38tL ˆ Xα(L − s)ds − ˆ L−3₈tL t ˆ Xα(L − s)ds ≤ −(t − θ)(√a_{L− ln a}L) ,

for all L large enough. Since

χ

₁(t) ≤ ˆXα(L − t) for all t ∈ [θ, L), we deduce the first inequality.

Since in addition ˆXα(L − t) remains negative on [L − (3/8)taL, L] (as ˆXαsatisfies the analogue of

D1

α), the eigenfunction |ϕ1| stays below |ϕ1(L − 3₈taL)| on this time-interval. 

Proposition 3.12. On the event_{E(n, δ), we have}

L − ˆυ > υ −√ 2C α ln(α).

Remark 3.13. From the arguments presented in Step 1 below, one deduces that_{υ and L − ˆυ actually lie} at a distance of order1/(√α ln α) from each other.

Proof. Note that in the time-interval [L − ˆτ_+∞( ˆXαj+1), τ−∞(Xαj)], the diffusion ˆXαj+1(L −·) is bounded

from below by Xαj(·) as the Riccati transform of the first eigenfunction of the operator

−∂x2+ ξ , x ∈ (tnj, tnj+1) ,

stays in between those two diffusions. Using that the diffusion Xj

α is bounded from below by t 7→

√

α tanh(−√α(t − υ)) − C√α/ ln(α) on the time interval [ι, υ] and given the behavior of ˆXαj+1on its

excursion to +√α as prescribed by (ii) of (c), we deduce the desired lower bound.  We now prove the statements of Theorems 1 and 2 regarding the first eigenfunction (except the state-ment on the law of U∞

1 , which is presented in Subsection 3.6) in three steps. We work on the event

E(n, δ) whose probability is at least 1 − 2δ.

Step 1: Identifying the maximum.Let us show that υ is at a distance o(1/√a_L) from the location of the maximum of |ϕ1| that we called U1L. Necessarily, a local maximum of |ϕ1| corresponds to a zero of

χ

₁. By the arguments presented at the beginning of the proof of Proposition 3.11, we know that on the

time interval [υ − (1/16)tL, υ + (1/16)tL], all the zeros of

χ

1lie at a distance of order 1/(ln aL√aL)

from υ: therefore a neighborhood of order 1/(ln aL√a_L) around υ contains the local maximum of |ϕ1|

on [υ − (1/16)tL, υ + (1/16)tL]. By (2) and (3) of Proposition 3.11, this local maximum is actually a

maximum on the interval [υ − (1/16)tL, L]. Using the time reversal and Proposition 3.12, we deduce

that it is actually the maximum on the whole interval [0, L]. Consequently, U1L is at a distance of order

at most 1/(ln aL√aL) from υ.

Step 2: Localization aroundU1. By (2) and (3) of Proposition 3.11, we get

m1([θ/L, 1]) ≤ ϕ1(θ)2 ˆ L−(3/8)tL θ e−√aL(t−θ)(1+o(1))_{dt + (3/8)t} Le− √_a L(L−(3/8)tL−θ)(1+o(1)) ≤ ϕ1(υ + (1/16)tL)2tL.

By (ii) of (c) we know that θ − υ − (1/16)tLis of order (5/16)tL, so that using (2) of Proposition 3.11

again we obtain

m1([υ/L + (1/16)tL/L, θ/L]) ≤ ϕ1(υ + (1/16)tL)2O(tL) .

Integrating the identity (1) of Proposition 3.11, we obtain

m1([υ/L − (1/16)tL/L, υ/L + (1/16)tL/L]) = ϕ1(υ)2

2 √

a_L(1 + o(1)) . By (1) of Proposition 3.11 again, we deduce that

ϕ1(υ + (1/16)tL)2 = ϕ1(υ)2O(1/ cosh(√aLtL/16)2) = ϕ1(υ)2O(1/a1/8_L ) .

Consequently,

m1([υ/L + (1/16)tL/L, 1]) = o(m1([υ/L − (1/16)tL/L, υ/L + (1/16)tL/L])) .

Similarly, using the time-reversed version of Proposition 3.11, we find

m1([0, (L − ˆυ)/L − (1/16)tL/L]) = o(m1([(L − ˆυ)/L − (1/16)tL/L, (L − ˆυ)/L + (1/16)tL/L])) .

Actually, (1) of the time-reversed version of Proposition 3.11 allows to get the further bound

m1([0, (L − ˆυ)/L − (1/17)tL/L]) = o(m1([(L − ˆυ)/L − (1/17)tL/L, (L − ˆυ)/L + (1/16)tL/L])) .

By Proposition 3.12, we know that L − ˆυ − (1/17)tL > υ − (1/16)tL for L large enough. Recall

that m1is a probability measure. Combining all the above estimates, we deduce that the interval [υ/L −

(1/16)tL/L, υ/L+(1/16)tL/L] contains asymptotically all the mass of m1. Since U1L−υ is negligible

compared to tL, we thus deduce that any macroscopic neighborhood of U1carries asymptotically all the

mass of m1. Consequently, m∞1 is a Dirac mass at U1∞with probability at least 1 − 2δ, for δ arbitrarily

small.

Step 3: Shape of the eigenfunction. Since m1 is a probability measure, we readily deduce from the

estimates of the previous step that ϕ2

1(υ) = (

√

a_L/2)(1 + o(1)). Combining this with (1) of Proposition 3.11 and Step 1, we deduce the statement on the shape.

This concludes the proof of Theorems 1 and 2 for the first eigenfunction (except for the law of U∞ 1 ,

see Subsection 3.6).

√aL

−√aL

0 L

4

close to the forward diffusion

close to the time-reversed diffusion

argmax ϕ4

FIGURE13. Schematic path of

χ

₄.

3.5. The general case. We now turn to the i-th eigenfunction, for some i ∈ {2, . . . , k}. Let α = αiand

α′ _{= α}′

i. Again, we work deterministically on the event E(n, δ).

The main idea is that the trajectory

χ

_ifollows the forward diffusions Xαi and Xα′_iuntil the additional

explosion of Xαi. It then follows the time-reversed diffusions ˆXαi and ˆXα′_iup to time L (see Figure 13).

There exist j1 < j2 < . . . < jisuch that the i explosion times of Xα lie in the intervals [tnjℓ, t

n jℓ+1),

ℓ = 1, . . . , i. We then call υℓthe location of the last 0 of Xαjℓbefore its explosion and θℓ the location of

its first hitting time of −√α. There exists a unique i_{∗∈ {1, . . . , i} such that, for j∗} = ji∗, X

j∗

α explodes

but Xj∗

α′ does not. Without loss of generality, we can suppose that i∗ ≥ 2: indeed, the case i∗ = 1

corresponds to the case i_∗ = i upon reversing time.

Let us now study ϕi. We denote by 0 = ζ(0) < ζ(1) < . . . < ζ(i) = L the zeros of ϕi, or equivalently

the explosion times of

χ

_i. We also let Mℓbe the maximum of |ϕi| on the interval [ζ(ℓ − 1), ζ(ℓ)].

The following lemma describes the behavior of ϕion an interval where both Xαand Xα′ explode.

Lemma 3.14. Letℓ < i_∗. On the event_{E(n, δ), the maximum of |ϕ}i| on [ζ(ℓ − 1), ζ(ℓ)] is achieved at a

distance negligible compared toL from ζ(ℓ) and we have Mℓ= |ϕ′i(ζ(ℓ))|e √ a_Lo(L)_{, and M} ℓ= |ϕ′i(ζ(ℓ − 1))|e √ a_L(ζ(ℓ)−ζ(ℓ−1))(1+o(1))_.

Proof. We treat in detail the case ℓ = 1. The other cases follow from the same arguments, one simply has to notice that the diffusions Xαand Xα′ have a delay at the starting time of the interval, but since

this delay is negligible compared to tLthe proof carries through.

Set j = j1. Since Xαj and Xαremain close on [tnj + (3/8)taL, τ₋₂√aL(Xα)], the following holds true

(thanks to (i) and (ii) of (c) of E(n, δ)):

(I) For all t ∈ [(3/8)tL, θ1], bothffl_(3/8)tt _LXα(s)ds andffl_(3/8)tt _LXα′(s)ds belong to the interval

h√ a_L−ln aL a1/4_L , √ a_L+ln aL a1/4_L i ,

(II) After time θ1, both Xαand Xα′remain below −√a_L+ 2 and they hit −∞ within a time of order

(3/8)tL(1 + o(1)).