From Sine kernel to Poisson statistics

El e c t ro nic J

o f

Pr

ob a bi l i t y

Electron. J. Probab.19(2014), no. 114, 1–25.

ISSN:1083-6489 DOI:10.1214/EJP.v19-3742

From Sine kernel to Poisson statistics

Romain Allez

Laure Dumaz

Abstract

We study the Sineβprocess introduced in [B. Valkó and B. Virág. Invent. math.177 463-508 (2009)] when the inverse temperatureβtends to0. This point process has been shown to be the scaling limit of the eigenvalues point process in the bulk of β-ensembles and its law is characterized in terms of the winding numbers of the Brownian carrousel at different angular speeds. After a careful analysis of this family of coupled diffusion processes, we prove that the Sineβ point process converges weakly to a Poisson point process onR. Thus, the Sineβ point processes establish a smooth crossover between the rigid clock (or picket fence) process (corresponding to β=∞) and the Poisson process.

Keywords:Random matrices ; Diffusions ; Poisson point process ; Exit time problem.

AMS MSC 2010:60G55 ; 60G17 ; 60B20 ; 60J60.

Submitted to EJP on August 14, 2014, final version accepted on December 11, 2014.

1 Introduction and main result

Although random matrices were originally introduced by John Wishart [24] in 1928 as a tool to study population dynamics in biology through principal component analysis, they became very popular much later in 1951 when Wigner [23] postulated that the fluctuations in positions of the energy levels of heavy nuclei are well described (in terms of statistical properties) by the eigenvalues of a very large Hermitian random matrix. Random matrix theory (RMT) is now an active research area in mathematics and theoretical physics with applications in statistics, biology, financial mathematics, engineering and telecommunications, number theory etc. (see [6, 7, 18, 13, 1] for a state of the art).

The classical models of Hermitian random matrices are the Gaussian orthogonal, unitary and symplectic ensembles. It is well known that the joint law of the eigenvalues of the matrices in those Gaussian ensembles is the Boltzmann-Gibbs equilibrium measure of a one-dimensional repulsive Coulomb gas confined in a harmonic well. More precisely, this joint law has a probability densityP^{β on}R^{N (}N is the dimension of the square matrices) given by

P^β(λ1,· · · , λN) = 1 ZN

Y

i<j

|λi−λj|^βexp(−N β 4

N

X

i=1

λ²_i) (1.1)

*Weierstrass Institute, Berlin, Germany. E-mail:romain.allez@gmail.com

†Statistical Laboratory, Centre for Math. Sciences, Cambridge, UK. E-mail:L.dumaz@statslab.cam.ac.uk

where the inverse temperatureβ = 1for the Gaussian orthogonal ensemble, respectively β= 2,4for the unitary and symplectic ensembles. The linear statistics of the point pro- cesses with joint probability density functions (jpdf)P^β, β= 1,2,4have been extensively studied in the literature with different methods [6, 18].

In 2002, Dumitriu and Edelman [11] came up with a new explicit ensemble of random tri-diagonal matrices whose eigenvalues are distributed according to the jpdfP^{β for} anyβ > 0 (see also [5, 2] where invariant ensembles associated to general β were constructed).

Those tri-diagonal matrices have been very useful in the last decade, leading to important progress on the understanding of the local eigenvalues statistics in the limit of large dimensionN for generalβ >0. At the edge of the spectrum, it was first proved [20] that the largest eigenvalues converge jointly (when zooming in the edge-scaling region of widthN^−2/3around2) to the low lying eigenvalues of a random Schrödinger operator called the stochastic Airy operator (see also [12]). Similar results were proved for the bulk in [22] by Valkó and Virág. Forλbelonging to the Wigner sea(−2,2), the authors of [22] consider the point process

ΛN := (2πN ρ(λ) (λi−λ))_i=1,···,N (1.2) where(λ1,· · · , λN)is distributed according toP^βandρ(λ) = _2π¹ √

4−λ² is the Wigner semi-circle density. Indeed, the mean level spacing around levelλfor the points with lawP^βis approximately1/(N ρ(λ))whenN1. The mean point spacing ofΛN defined in (1.2) is therefore of order2πand in this scaling, one can now investigate the limiting statistics of this point process whenN → ∞. The authors of [22] precisely answer this question proving that the point processΛN converges in law¹to a point process Sineβ

onRfirst introduced in [22] and characterized in terms of a family(αλ)_λ∈Rof coupled one-dimensional diffusion processes, thestochastic sine equations. As expected for the eigenvalues statistics in the bulk, the point process Sineβ is translation-invariant in law. The family of diffusions(α_λ)_λ∈Rcan be interpreted as the hyperbolic angle of the Brownian carousel with parameterλand its law is characterized as follows: Given a (driving) complex Brownian motion(Zt)_t≥0, the diffusionsαλ, λ∈R^satisfy

dαλ=λβ

4e^−β4tdt+ Re((e^−iαλ−1)dZt), αλ(0) = 0. (1.3) Note that all the diffusionsα_λ, λ∈Rare adapted to the filtration of the Brownian motion (Z_t). This coupling induces a strong interaction between the diffusions which makes the joint law difficult to analyse, as we shall see. A key feature shared by the processes αλ, λ∈Ris that they all converge almost surely ast→ ∞^{to a limit}αλ(∞)which is an integer multiple of2π. The characterization of the law of the Sineβpoint process²can now be enunciated as follows

(Sineβ([λ, λ⁰]))_λ<λ0

(d)=

α_λ0(∞)−α_λ(∞) 2π

λ<λ⁰

. (1.4)

In this paper, we are interested in the limiting law of the Sineβ process when the inverse temperatureβgoes to0. Theorem 1.1 is the main result of this paper and gives the convergence asβ →0of the Sine_βprocess towards a Poisson point process onR^{. This} convergence at the continuous (N=∞) level seems rather natural since takingβ→0 amounts to decreasing the electrostatic repulsion (and hence the correlation) at the

1The convergence is with respect to vague topology for the counting measure of the point process.

2IfAis a Borel set ofR, Sineβ(A)insideAdenotes the number of points insideA. In other words, Sineβis the counting measure of the point process.

0 20 40 60 80 100

01234

Figure 1: (Color online). Sample paths of two diffusionsαλ/(2π)(blue curve) andαλ⁰/(2π) (red curve) forλ < λ⁰ and for a small value ofβ. We see that wheneverbαλ/(2π)c^jumps, bαλ⁰/(2π)cjumps at the same time in agreement with Lemma 3.4.

discrete level, i.e. between theN points distributed according toP^β. If one takesβ= 0 abruptly for fixedN, the probability densityP⁰corresponds (up to a rescaling depending onβ) to the joint law of independent Gaussian variables and it is then straightforward to check the convergence (asN → ∞) of the point process with lawP⁰towards a Poisson process. Theorem 1.1 exchanges the order of the limitsβ→0andN → ∞, describing the statistics whenN → ∞^{first and then}β →0. It also gives the precise rate of the convergence.

Theorem 1.1.As β → 0, the Sineβ point process converges weakly in the space of Radon measure (equipped with the topology of vague convergence [15]) to a Poisson point process onRwith intensity ^dλ_2π. In particular, we have, for anyk∈N^andλ < λ⁰,

P[Sineβ[λ, λ⁰] =k]→^β→0exp

λ−λ⁰ 2π

(λ⁰−λ)^k (2π)^kk! ,

and the numbers of points of Sineβ inside two disjoint intervals are asymptotically independent.

Let us briefly discuss some implications of Theorem 1.1 and mention a few related questions on the spectral statistics of random matrices and random Schrödinger opera- tors.

In [16], the authors have shown that the circularβ-ensemble, which was later shown to be Sine_βin [19], interpolates between Poisson and clock distributions on the circle (point process with rigid spacings like the numerals on a clock) by considering random CMV matrices. Theorem 1.1 provides a more precise description of this interpolating process on the Poisson process side.

In our study, we are led to examine a time homogeneous family of diffusions(θλ)λ∈R

defined as

dθλ=λβ

4dt+ Re((e^−iθλ−1)dZt), θλ(0) = 0. (1.5) This family of coupled diffusions also appears in [17] to describe the law of the limiting point process of a certain critical random discrete Schrödinger operator. Our result can

be extended in this context to prove that this critical Schrödinger operator continuously interpolates between the extended (clock/picket fence) and localized (Poisson) regimes.

More precisely, one could prove using our ideas that the random spectrum has Poissonian statistics in the limit of large temperature.

Let us also compare the results stated in Theorem 1.1 with those of a previous work [4] where we consider the stochastic Airy ensemble, Airyβ, obtained in the scaling limit ofβ-ensembles at the edge of the spectrum. In this context, we proved that the number of points Airy_β(]− ∞, λ])inside the interval]− ∞, λ]displays Poisson statistics in the smallβlimit. This permitted us to obtain the limiting distributions asβ→0of each of the lowest eigenvalues (individually) of the Airyβ ensemble. In particular, we obtained the weak convergence of theT W(β)distribution towards the Gumbel distribution. Although the Sineβ and Airyβ characterizations in law (in terms of a family of coupled diffusions) look very similar, the analysis of the limiting marginal statistics of the number of points inside a finite closed interval Airy_β[λ, λ⁰]forλ < λ⁰ and the asymptotic independence whenβ → 0 of the respective numbers of points of the Airy_β point process into two disjoint intervals remain open even after our study [4]. In this aspect, Theorem 1.1 gives a much more powerful and complete description of the Sineβ process in the smallβlimit.

In this case, we are able to prove the asymptotic independence between the number of points of the Sineβprocess in two disjoint intervals. This part of the proof requires new ideas in order to obtain estimates on the relative positions between two coupled diffusionsα_λandα_λ⁰. The nice feature of the Sine_βprocess is its translation invariance in law. This property makes the analysis of Sineβeasier than the one of the Airyβprocess.

The non-homogeneous intensity of the Airyβ process is governed by the edge-scaling crossover spectral density ofβ-ensembles computed explicitly in [8, 14] forβ= 2(see also [4]).

Other spectral statistics of random matrices at high temperature, i.e. whenβ→0, have been investigated in [2, 3]. In [2], the authors study the empirical eigenvalue density in the limit of large dimensionN forβ-ensembles whenβtends to0withN as β= 2c/N wherec >0is a constant. The authors compute the limiting spectral density ρc(λ)explicitly in terms of parabolic cylinder function and establish a Gauss-Wigner crossover, in the sense that the familyρc interpolate between the Gaussian probability distribution (c= 0) and the Wigner semi-circle (c→+∞). The case of Gaussian Wishart matrices has also been studied in [3].

It would be interesting to have a description of the crossover statistics of the β- ensembles obtained in the double scaling limits whenβ tends to0withN → ∞^(this question is briefly discussed in [4] for the statistics at the edge of the spectrum).

Organization of the paper. We start in section 2 by looking at the limiting marginal distributions of the random variables Sineβ[λ, λ⁰]forλ < λ⁰. We first study a classical problem on the exit time of a diffusion trapped in the well of a stationary potential (obtained by neglecting the slow evolution with time). Then, we prove that the jump process ofbα_λ/(2π)cconverges weakly to an (inhomogeneous) Poisson point process by first approximatingα_λwith diffusions processes with piecewise constant drifts on a subdivision of small intervals and then by using the convergence of the exit time of the stationary well established previously. This requires estimates on the sample paths of a single diffusion, in a spirit similar to [4, 10]. In Section 3, we investigate the asymptotic independence of the numbers of points of Sineβ in two disjoint intervals. We prove a crucial estimate regarding the typical relative positions of two diffusionsα_λ andα⁰_λfor λ < λ⁰. Loosely speaking, the main point is to use this estimate to prove that, in the limitβ→0, the jumps of the processbαλ/(2π)cimmediately follow those ofbαλ⁰/(2π)c (see Fig. 1) while the processesbαλ/(2π)c^andb(αλ⁰−αλ)/(2π)cnever jump at the same time (see Fig. 4). The asymptotic independence follows essentially from the fact that

two Poisson point processes adapted to the same filtration are independent if and only if they never jump simultaneously.

We gather in the next paragraph important properties of the family (αλ) already established in [Section 2.2, [22]] that we will use throughout the paper.

First properties of the coupling of the diffusionsα_λ:

(i) For allλ < λ⁰,α_λ⁰−α_λhas the same distribution asα_λ⁰_−λ; (ii) “Increasing property”:α_λ(t)is increasing inλ;

(iii) bαλ(t)/(2π)cis non-decreasing int; (iv) E[αλ(t)] =λ^β₄Rt

0e^−βs/4ds;

(v) lim_t→∞αλ(t)/(2π)exists and is an integer a.s.

We will also use the following notation:

{x}^2π=x−2πj x 2π

k .

Acknowledgments. Special thanks are addressed to Chris Janjigian and Benedek Valkó. We have benefited from insightful and precise comments from them. Their detailed feedback has helped us to improve the second version of this manuscript, especially the proofs of Lemmas 3.2 and 3.4. We are also grateful to them for pointing out references [16, 19, 17] and the connections with our work.

We thank Stéphane Benoist and Antoine Dahlqvist for useful comments and discus- sions.

R. A. received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement nr.

258237 and thanks the Statslab in DPMMS, Cambridge for its hospitality. The work of L. D. was supported by the Engineering and Physical Sciences Research Council under grant EP/103372X/1 and L.D. thanks the hospitality of the maths department of TU and the Weierstrass institute in Berlin.

2 Limiting marginal distributions

We are first interested in the limiting law of the random variable Sineβ[0, λ]when β→0for a single fixedλ. In this case, we re-write the diffusionαλin a more convenient way:

dαλ=λβ

4e^−β4tdt+ 2 sin(α_λ

2 )dBt, (2.1)

whereB is a real standard Brownian motion (which depends onλ). Let us introduce the change of variableRλ:= log(tan(αλ/4)). A straightforward computation (see [22]) shows that:

dRλ= 1 2

λβ

4e^−β4tcosh(Rλ) + tanh(Rλ)

dt+dBt, Rλ(0) =−∞. (2.2) 2.1 Trapping in the stationary potential

In this subsection, we study an exit time problem for a Langevin diffusionSλevolving in a stationary potentialVβ defined forr∈R^as

V_β(r) :=−1 2

λβ

4 sinh(r) + log cosh(r)

.

This problem is relevant to our study thanks to the slow variation with time of the non-stationary potential asβ →0in which the diffusionR_λevolves. The diffusionS_λ satisfies the following stochastic differential equation

dSλ=−V_β⁰(Sλ)dt+dBt, Sλ(0) =−∞. (2.3) In the whole paper, the diffusionsR_λ andS_λdefined respectively in (2.2) and (2.3) are coupled, driven by the same Brownian motionB. Under this coupling, we have almost surely

Rλ(t)≤Sλ(t) for allt≤ζwhereζis the first explosion (stopping) time

ζ:= inf{t>0 :Sλ(t) = +∞}.

' log(1/β)/2 ' log(1/β )

(0, 0)

aβ

b_β

Figure 2: The potentialV_β(r)as a function ofr.

In this paragraph we investigate the limiting law of the stopping timeζthrough its Laplace transform defined forξ >0as

gξ(r) :=E^r[e^−ξζ], whereris the initial position of the diffusionS_λ.

We know from classical diffusion theory (see also [4]) thatgξ satisfies 1

2g_ξ⁰⁰(r)−V_β⁰(r)g_ξ⁰(r) =ξg_ξ(r), withg_ξ(r)→1asr→+∞. (2.4)

Let us first examine the expectation of the first explosion time of S_λ. Due to the strong barrier separating the local minimum and the local maximum (see Fig. 2), it is natural to expect the asymptotic of this mean exit time not to depend on the starting point of the diffusion, whenβ → 0, as long as it is located in the well (“memory-loss property”). This is the purpose of Proposition 2.1. We then show that this first exit time properly rescaled by its mean value converges in law to an exponential distribution (Proposition 2.2) when the starting point is in the well.

Proposition 2.1.Suppose that the diffusionSλstarts fromr:=rβ such thatrβ→ −∞^. Then its expected exit time denotedtβ(r) :=E^r[ζ]has the following equivalent when β→0:

tβ(rβ)∼ 8π βλ.

Proof of Proposition 2.1. From (2.4), it is easy to see that the expected exit timetβ(r) satisfies the boundary value problem

1

2t⁰⁰_β(r)−V_β⁰(r)t⁰_β(r) =−1, witht_β(r)→0asr→+∞. (2.5) Solving (2.5) explicitly we obtain the integral form

t_β(r) = 2 Z +∞

r

dx Z x

−∞

exp (2 [V_β(x)−V_β(y)])dy . (2.6) By extracting carefully the asymptotic behavior of this integral in the limitβ →0(see Appendix A.1), we obtained the desired result.

The Laplace transform of the exit time satisfies the fixed point equation gξ(r) = 1−2ξ

Z +∞

r

dx Z x

−∞

exp(2[Vβ(x)−Vβ(y)])gξ(y)dy.

With classical arguments similar to those of [4], we prove the following proposition in Appendix A.1.

Proposition 2.2.Conditionally onSλ(0) =r:=rβsuch thatrβ → −∞^whenβ→0, the first explosion timeζof the diffusion(S_λ(t))_t>0converges weakly when rescaled by the expected exit time8π/(βλ)towards an exponential distribution with mean1.

The processθλsuch thatSλ:= log tan(θλ/4)satisfies dθλ=λβ

4dt+ 2 sin(θ_λ 2 )dBt.

Proposition (2.2) translates into a convergence of the stopping timeζ2π := inf{t >0 : θλ(t)>2π}^.

Proposition 2.3.Conditionally onθλ(0) =θ0:=θ^β₀ such thatθ₀^β→0+ whenβ →0, the stopping time ^βλ_8πζ2π converges weakly whenβ →0towards an exponential distribution with mean1.

2.2 Convergence of the jump process

We consider the diffusionαλdefined in (1.5) or equivalently for a singleλ, in (2.1).

Fork∈N^{, let}

ζk:= inf{t>0 :αλ(t)>2kπ}. (2.7) Note that those stopping times correspond to the jumps of the processbαλ/(2π)c^{. We} denote byF^tthe filtration associated to the diffusion processαλ. In this paragraph, we

will sometimes omit the subscript λinα_λ to simplify the notations. We consider the rescaled empirical measureµ^β_λ of the ^β₄ζ_kdefined onR+

µ^β_λ[0, t] =

+∞

X

k=1

δ_ζk

0,8πt β

. (2.8)

We divide the time intervalR⁺into random small intervalsIk := [^S_n^k8π_β,^Sk+1_n ^8π_β], k∈N^, independent of the diffusionαwhereS₀= 0and:

Sk :=

k

X

i=1

τi,

where theτ_iare i.i.d. random variables with mean1uniformly distributed on[1/2,3/2]. For eachk∈N, conditionally onF^Sk

n 8π

β

, we define the two diffusion processes α⁺_n andα⁻_n such that, fort∈Ik,

dα⁺_n =λβ

4e^−Skn2πdt+ 2 sin(α⁺_n

2 )dBt, α⁺_n(S_k n

8π

β ) =α(S_k n

8π β ), dα⁻_n =λβ

4e^{−Sk+1n 2π}dt+ 2 sin(α⁻_n

2 )dBt, α⁻_n(Sk

n 8π

β ) =α(Sk

n 8π

β ). By the increasing property, it follows that for anyk∈N^andt∈I_k, we have

α⁻_n(t)≤α(t)≤α⁺_n(t).

Theorem 2.4.Asβ →0, the empirical measureµ^β_λ converges weakly in the space of Radon measures (equipped with the topology of vague convergence [15]) to a Poisson point process onR⁺ with inhomogeneous intensityλ e^−2πtdt. In particular, we have, for anyk∈N^,

P[µ^β_λ[0, t] =k]→^β→0exp(−λ

2π(1−e^−2πt))(_2π^λ)^k(1−e^−2πt)^k

k! . (2.9)

The proof of Theorem 2.4 is done in the next paragraph 2.3. It uses a careful analysis of the behaviour of a single diffusionαλin the smallβ limit.

Remark 2.5.Theorem 2.4 establishes the convergence of the rescaled empirical mea- sure of the jumps of the processbαλ/2πcunder the initial conditionαλ(0) = 0. It is easy to generalize this result to other starting timess0and positionsαλ(s0)6= 0such thats0

scales with1/βandαλ(s0)is close enough to0modulo2π.

More precisely, fix a time s > 0 and denote by µ^β_λ,s,x the rescaled counting mea- sure of the jumps ofbα_λ/2πcon the time interval[8πs/β,+∞)with the initial condition αλ(8πs/β) =x. Then, for any family of starting points(xβ)such that{xβ}^2π64 arctan(β^1/4), the measureµ^β_λ,s,x

β converges weakly in the space of Radon measures towards a Poisson point process on[s,+∞)with inhomogeneous intensityλ e^−2πtdt.

Thanks to the equality in law

µ^β_λ(R⁺)^(d)= αλ(∞) 2π

(d)= Sineβ[0, λ],

we easily deduce from Theorem 2.4 the convergence of the marginals of the Sineβpoint process.

Corollary 2.6.Letλ < λ⁰. The random variableSineβ[λ, λ⁰]converges weakly asβ→0 to a Poisson law with parameter ^λ0_2π^−λ.

2.3 Estimates for a single diffusion and proof of Theorem 2.4

We analyse in this paragraph the diffusion αλ and the jumps ofbαλ/(2π)c ^{in the} limitβ→0. The results we obtain are derived using the diffusionR_λ:= log(tan(α_λ/4)) satisfying the SDE (2.2). We defer the proof of those technical lemmas in Appendix A.2.

In the following, for allt >0andx∈R⁺, the law of the diffusion processαλstarting fromxat timetis denotedP^{x,t. When}t= 0, we use the short cutsP^xforP^{x,0and and} P^forP^0,0.

In the following Lemma, we first prove that if the diffusionαλstarts just below2π modulo2π, thenbαλ/(2π)cwill jump in a time of order log_β¹ much shorter than the typical time between two jumps (of order1/β) with probability going to1.

Lemma 2.7.Let0< ε <1ands >0. Denotesˆ:= 8πs/β. We define the first reaching time

ζ_2π^λ := inf{t≥sˆ:α_λ(t) = 2π+ 2πbα_λ(ˆs)/(2π)c}. (2.10) Then, there exists a constantc >0(depending only ons,εandλ) such that for allβ >0 small enough,

P

ζ_2π^λ <9 log 1 β

{αλ(ˆs)}^2π= 2π−4 arctan(β^ε)

>1−β^c.

Lemma 2.7 permits us to show that the time spent byαλaway from0modulo2πis negligible compared to the time scale1/β. This is the content of Lemma 2.8.

Lemma 2.8.Lett > s≥0,x∈R+. Let us define Ξβ(s, t, x) :=Ex,8πs/β

"

Z ^8π_βt

8π βs

1_{{{αλ(u)}2π}≥4 arctan(β^1/4)}du

#

. (2.11)

Then, for allt≥s >0, there existsC >0(depending only ontandλ) such that for all β >0andx∈R^+,

Ξ_β(s, t, x)6 C

√β log1 β . We can now prove Theorem 2.4 using the previous estimates:

Proof of Theorem 2.4.

The proof follows the same lines as the proof of [Theorem 4.1 in [4]]. The idea is to approximate the number of jumps of the diffusionαby those of stationary diffusions and to use the increasing property. To this end, we will use the subdivision ofR+introduced above and the diffusionsα_n⁺andα⁻_n.

From Kallenberg’s theorem [15], we just need to see that, for any finite unionIof disjoint and bounded intervals, we have whenβ→0,

E[µ^β_λ(I)]−→λ Z

I

e^−2πtdt , (2.12)

P[µ^β_λ(I) = 0]−→exp(−λ Z

I

e^−2πtdt). (2.13)

Denote by[t₁;t₂] the right most interval ofI, byJ the union of disjoint and bounded intervals such thatI=J∪[t₁;t₂]and byt₀the supremum ofJ.

Let us use the random subdivision introduced above. It is crucial to control the position of the diffusion at the starting point of all the sub-intervals. As a consequence of

Lemma 2.8, with large probability, the diffusionαis close to0modulo2πin the beginning of each of the sub-intervals overlapping[0, t₂]³.

More precisely, denote byC^k :={{α((8π/β)Sk/n)}^2π 64 arctan(β^1/4)}and consider:

C:=

b2nt2c+1

\

k=0

C^k.

Every sub-interval intersecting [0, t2] is taken care of in this event as by definition S_k∈[k/2,3k/2]. Its probability is bounded from above by

P[C^c]6

b2nt2c+1

X

k=0

P[{α((8π/β)S_k/n)}^2π>4 arctan(β^1/4)]

62E[

Z 3nt2+3 0

1{{α((8π/β)u/n)}2π>4 arctan(β^1/4)}du]

6 n

4πβΞ_β(0,3t₂+ 1,0)→^β→00

where we used the notation and result of Lemma 2.8 for the last line.

We now turn to the proof of 2.12. Note that thanks to the linearity of the expectation, we simply need to prove (2.12) for intervalsIof the formI= [0, t₂]. The upper bound simply follows from the SDE form:

E[α(t)] =λβ 4

Z t 0

e^−βs/4ds . We immediately derive:

E[µβ[0, t2]] =E[bα((8π/β)t2)/(2π)c]6λ Z t₂

0

e^−2πsds .

For the lower bound, denote byN_k⁺,N_k⁻andN_kthe number of jumps ofα⁺_n,α⁻_n and αin the sub-interval[(8π/β)S_k/n,(8π/β)S_k+1/n].

E[µ_β[0, t₂]]>

b2nt2c+1

X

k=0

E[N_k⁻1_{(Sk/n)<t2}|C^k]−

b2nt2c+1

X

k=0

E[N_k⁻1_{(Sk/n)<t2}|C^k]P[Ck^c]. The second sum of the RHS can be simply bounded from above by:

E[number of jumps ofθλin[0,3t2(8π)/β]]× sup

k∈{0,···,b2nt2c+1}

P[Ck^c] P[C^k].

The first expectation is bounded from above by a number independent of β and the second term (bounded byP[C^c]/(1−P[C^c])) tends to0asβ →0. Moreover, thanks to Proposition 2.3, we have

E N_k⁺

C^k

→^β→0E[λ(τk+1/n) exp(−2πSk/n)].

Using the convergence of the Riemann sum whenn → ∞, it gives the desired lower bound.

Let us now examine the convergence 2.13 for a single interval[t1, t2]first. By the Markov property:

P[µ_β[t₁, t₂] = 0]6EhY^∞

k=0

P

N_k⁺= 0

C^k,(τ_i)_i

1_{{Sk+1/n>t1, Sk/n6t2}}i

+P[C^c].

3For the generalization of Remark 2.5, the initial position modulo2πat times0belongs to the interval [0,4 arctan(β^1/4))so that the eventC0occurs almost surely by definition.

Using the convergence in each of the sub-intervals (given by Proposition 2.3) P

N_k⁺= 0 C^k

→^β→0E[exp(−λ(τk+1/n) exp(−2πSk/n)))], we obtain:

lim sup

β→0 P[µβ[t1, t2] = 0]6E[

∞

Y

k=0

exp(−λ(τk+1/n) exp(−2πSk/n)))1_{Sk+1/n>t₁, S_k/n6t₂}]. Thanks to the convergence of the Riemann sum when n → ∞, we deduce the upper bound. The lower bound can be done using the same techniques as above. To generalize the result to any finite union of intervalsI, we use the simple Markov property which gives

P[µ^β_λ(I) = 0] =E[1_{µβ(J)=0}P[µ_β([t₁, t₂]) = 0|α(8πt₀/β)]].

To conclude, let us introduce an independent random time U sampled uniformly in [t0, t1]. With probability going to1,α(8πU/β)belongs to an interval of the type[2π,2π+ 4 arctan(β^1/4)). We can then iterate the previous arguments to obtain the result.

3 Asymptotic spatial independence

3.1 Ordering of two diffusions

Forλ < λ⁰, we now control the expected time spent by the process{α_λ⁰}^{2πbelow the} process{αλ}^2πwhereαλandαλ⁰ are coupled according to (1.5). The following Lemma is a crucial step towards the proof of the asymptotic independence of the limiting point process on disjoint intervals and will be used for the proof of Lemmas 3.4 and 3.5.

Lemma 3.1.Lett >0, λ < λ⁰and Θ_β(t) :=E

"

Z ^8π_βt 0

1_{{α

λ0(u)}2π≤{αλ(u)}2π}du

# .

Then, there exists two constantsc, C >0(depending only ontandλ⁰) such that for all β >0,

Θβ(t)6 C β^1−c. Proof. We set

E^u:={{αλ⁰(u)}^2π<{αλ(u)}^2π} .

Before evaluating the probability of the eventE^u, we need to introduce foru >0the last jump time of the processb^α2π^λ0cbefore timeu, i.e.

ζ^(u):= sup{ζ_k^λ0 : k∈N, ζ_k^λ0 ≤u}.

The main idea is to prove that any timeusuch thatE^u occurs is associated to a jump time ofαλ⁰ right beforeu.

We setu0:=u−9 log_β¹ and bound from above the probability of the eventE^uby P[E^u]≤Ph

E^u∩n

ζ^(u)∈[u₀, u]oi +Ph

E^u∩n

ζ^(u)< u₀oi

≤Ph E^u∩n

ζ^(u)∈[u₀, u]oi +P





\

s∈[u0,u]

E^s



 (3.1)

u 2π

0

ζ ^(u)

{ α _λ } 2π

{ α _λ

} 2π

u

Figure 3: Representation of the eventE^u∩ {ζ^(u)< u0}^. where we have noticed the inclusionE^u∩{ζ^(u)< u0} ⊆T

s∈[u0,u]E^sfor the second line (see Fig. 3). Indeed the definition ofζ^(u)implies that the processb^α2π^λ0chas not jumped during the time interval[ζ^(u), u]so that the relative ordering (modulo2π){α_λ⁰(t)}^2π<{α_λ(t)}^2π at timet=uhas to be preserved for allt∈[ζ^(u), u).

We now tackle the second probability of (3.1) using the fact that αλ is close to0 modulo2πwith large probability:

P





\

s∈[u0,u]

E^s



≤P





\

s∈[u0,u]

E^s∩n

{αλ(u0)}^2π≤4 arctan(β^1/4)o





+Ph

{α_λ(u₀)}^2π≥4 arctan(β^1/4)i . We introduce the stopping time

ζ>(u0) := inf{t≥u0:{αλ⁰(t)}^2π>{αλ(t)}^2π} and notice that, conditionally on the eventA^u0 :=E^u0 ∩

{α_λ(u₀)}^2π≤4 arctan(β^1/4) , we have almost surely for allt∈[u₀, ζ_>(u₀)],

{αλ⁰(t)−αλ(t)}^2π= 2π−({αλ(t)}^2π− {αλ⁰(t)}^2π).

We define a diffusion processαe_λ0−λand its associated first reaching time of2πsuch that dαeλ⁰−λ(t) = (λ⁰−λ)β

4e^−β4tdt+ Re((e^{−iαeλ0 −λ(t)}−1)dZt), t≥u0

withαe_λ⁰_−λ(u₀) = 2π−({α_λ(u₀)}^2π− {α_λ⁰(u₀)}^2π), andζe2π := inf{t≥u0:αeλ⁰−λ(t) = 2π}.

Endowed with those definitions, we have the following upper-bound

P





\

s∈[u₀,u]

E^s∩ A^u0



=P[{ζ_>(u₀)≥u} ∩ A^u0]

≤P

eζ2π≥9 log1 β

≤β^c

where we have used the equality in law(αeλ⁰−λ(t), t≥u0) = (αλ⁰(t)−αλ(t), t≥u0)for the second line and Lemma 2.7 (as well as the increasing property) to obtain the last

upper-bound. Gathering the above inequalities, we obtain P[E^u]≤Ph

ζ^(u)∈[u₀, u]i

+β^c+Ph

{α_λ(u₀)}^2π≥4 arctan(β^1/4)i .

Now, to conclude, we just have to integrate this latter inequality with respect tou. First notice that we have almost surely

Z ^8π_βt 9 log(1/β)

1{^ζ(u)∈[^{u−9 log}_β^1,u]}du≤9 log1

β µ^β_λ0[0,8π β t].

Integrating the other terms as well with respect tou∈[0,^8π_βt]and taking the expectation, we get

Θβ(t)≤9 log1 β

1 +E

µ^β_λ0[0,8π β t]

+ Ξβ(0, t,0)

≤9 log1 β

1 + λ

2π

+ 8π t β^c−1+ Ξ_β(0, t,0)

whereΞ_β(0, t,0)is defined in (2.11). The conclusion now follows from Lemma (2.8).

3.2 Limiting coupled Poisson processes

Lemma 3.1 shall be an important tool to prove the asymptotic independence between αλ(∞)andαλ⁰(∞)−αλ(∞)forλ < λ⁰.

Theorem 2.4 gives the weak convergence of the random measuresµ^β_λandµ^β_λ0 in the space of measures onR+ equipped with the topology of vague convergence denoted M⁺(R+). Due to the equality in law

αλ⁰−αλ

(d)= αλ⁰−λ, (3.2)

Theorem 2.4 also implies the weak convergence of the (positive) random measureµ^β_λ0−λ

such that for allt≥0,

µ^β_λ0−λ[0, t] :=

αλ⁰(t)−αλ(t) 2π

towards a Poisson measureP^λ0−λwith intensity(λ⁰−λ)e^−2πtdt.

We now fixλ < λ⁰< λ⁰⁰and work with the three diffusionsαλ,αλ⁰ andαλ⁰⁰coupled according to (1.5). We are interested in the limiting joint distribution of the triplet of random measures(µ^β_λ, µ^β_λ0, µ^β_λ00−λ⁰)according to this coupling.

From the above convergences, it is straightforward to deduce the relative-compactness of the family of the triplets of (random) measures

{ µ^β_λ, µ^β_λ0, µ^β_λ00−λ⁰

, β >0} ^(3.3)

for the weak topology over(M⁺(R⁺))³equipped with the product topology of vague convergence.

Let us take a sequenceβk→0whenk→ ∞such that the processes µ^β_λ^k, µ^β_λ^k0, µ^β_λ^k00−λ⁰

(3.4)

converge jointly weakly in the product space whenk→ ∞to a triplet (Pλ^(βk),Pλ^(β0k),Pλ^(β00k−λ^{) 0})

of point measures onR+whose marginal distributions are given respectively by the law of the Poisson measuresP^{λ and}P^{λ0 and}P^λ00−λ0 and whose joint law dependsa priori on the chosen sub-sequence(βk). In the following, we study this triplet and we drop the superscript(βk)to ease the notations. We shall in fact see later that all the possible limit points have the same law. Therefore the law of the triplet does not depend on the subsequence(βk)and the weak convergence of (3.3) holds (see Remark 3.6).

In the next Lemma, we regard the point measures P^λ,P^{λ0 and}P^{λ00−λ0 as Poisson} point processes and prove that they are indeed jointly Poisson processes on a common filtration. This is an important step for our needs.

Lemma 3.2.Letλ < λ⁰< λ⁰⁰andF:= (F^t)_t≥0be the natural filtration associated to the process(P^λ,P^λ0,P^λ00−λ0)i.e. such that

F^t:=σ(P^λ(s),P^λ0(s),P^λ00−λ0(s),0≤s≤t).

Then, the marginal processesP^{λ and}P^{λ0 and}P^{λ00−λ0 are}(F^t)-Poisson point processes with respective intensitiesλe^−2πtdtandλ⁰e^−2πtdtand(λ⁰⁰−λ⁰)e^−2πtdt.

The main point used to prove this Lemma is that all the diffusions (αλ, λ ≥0 are measurable with respect to the same driving (complex) Brownian motion(Zt)as they are strong solutions of the SDEs (1.5). A proof can be found in Appendix A.3.

Lemma 3.2 can easily be generalized to finitely manyλ’s:

Lemma 3.3.Let us fix an integerk≥3andλ0< λ1<· · ·< λk. For alli, j ∈ {1,· · · , k} such thatj > i, define the point measuresP^λi andP^λj−λ_i as above. LetF:= (F^t)t≥0be the filtration:

F^t:=σ(P^λi(s),P^λj−λi(s),0≤s≤t, i, j ∈ {1,· · · , k}^{such that}j > i).

Then, for all i, j ∈ {1,· · ·, k} ^with j > i, the marginal processes P^λi and P^λj−λi are (F^t)-Poisson point processes with respective intensitiesλie^−2πtdtand(λj−λi)e^−2πtdt. To fix notations, we will denote by(ξ^λ_i)_i∈N resp.(ξ^λ_i⁰)_i∈N,(ξ_i^λ0−λ)_i∈N the points associ- ated to the Poisson processesP^λandP^{λ0 and}P^{λ0−λsuch that}

P^λ[0, t] =X

i

δ_ξλ

i[0, t], P^λ0[0, t] =X

i

δ_ξλ0

i [0, t], P^λ0−λ[0, t] =X

i

δ_ξλ0 −λ i

[0, t].

Forβ >0, we also recall the notations µ^β_λ[0, t] =X

i

δ_ζλ i[0,8πt

β ], µ^β_λ0[0, t] =X

i

δ_ζλ0 i [0,8πt

β ], µ^β_λ0−λ[0, t] =X

i

δ_ζλ0 −λ i

[0,8πt β ]. Lemma 3.4.Letλ < λ⁰. Then, we have almost surely

P^λ⊆ P^λ0 i.e. for alli∈N, there existsj≥isuch thatξ^λ_i =ξ^λ_j⁰.

Proof. We have to prove that for anyt >0, P

∃i:ξ_i^λ< t, ξ_i^λ6∈ P^λ0

= 0. (3.5)

The probability (3.5) is the increasing limit of the sequence(pn)n∈Ndefined as p_n:=P

∃i:ξ^λ_i < t,∀j ≥i,|ξ_i^λ−ξ_j^λ0|> 1 2n

. (3.6)

It suffices to prove thatp_n= 0for anyn. Let us introduce the probability p^β_n:=P

∃i: β

8πζ_i^λ< t,∀j≥i, β

8π|ζ_i^λ−ζ_j^λ0|> 1 2n

. (3.7)

Denote byM^p(R+)the space of point measures and recall that it is closed in the space M⁺(R+)for the vague convergence topology. Notice that the set

{(µ, ν)∈(M^p(R⁺))² : µ=

∞

X

i=1

δx_i, ν =

∞

X

i=1

δy_i,∃isuch thatyi< tand∀j, |yi−xj|> 1 2n} is open in(M^p(R⁺))²equipped with the product topology of the vague convergence on the space of point measures. It comes from the straightforward fact that ifµk ∈ M^p(R⁺) converges towardsµ∈ M^p(R+)for the vague topology, the points ofµ_k belonging to [0, t]for all but finitely manykconverge to points ofµin[0, t].

If n is fixed, we therefore havepn ≤ lim inf_β→0p^β_n using the joint convergence of (µ^β_λ, µ^β_λ0) in the space(M^p(R⁺))² (along the subsequence (βk)) and the Portmanteau

theorem. It suffices to check that

lim sup

β→0

p^β_n = 0.

We need to work with a random subdivision of the interval[0,8πt/β]. As before, we consider a sequence(τ_k)_k∈Nof i.i.d. positive random variables distributed uniformly in [¹₂,³₂]and form the sumSk =Pk

i=1τi.

Noting that for all x, y such that |x−y| ≤ 1/(2n), there exists k ∈ N ^{such that} x, y∈[Sk/n, Sk/n+ 2/n], we obtain

p^β_n ≤P

∃k≤ b2ntc+ 1 :bα_λ

2πcjumps on the interval 8π β

S_k n,(S_k

n + 2 n)

but notbα_λ⁰ 2πc

. Due to the increasing property, the event inside the probability can not happen if the process{αλ}^2πstarts below{αλ⁰}^2πat the beginning of the interval. Therefore,

p^β_n≤

b2ntc+1

X

k=1

P

{α_λ⁰(8π β

S_k

n)}^2π≤ {α_λ(8π β

S_k n )}^2π

. which can in turn be upper-bounded as follows

b2ntc+1

X

k=1

P

{αλ⁰(8π β

Sk

n )}^2π≤ {αλ(8π β

Sk

n )}^2π

≤ βn 4πE

"

Z ^8π_β(3t+1) 0

1_{{{αλ0(u)}2π≤{αλ(u)}2π}}du

#

≤ βn

4πΘβ(3t+ 1) ≤β^c (3.8) where we have used Lemma 3.1 to obtain the last inequality which holds forβ small enough (cis a constant which does not depend onβ).

Lemma 3.5.Let λ < λ⁰. Then the (F^t)-Poisson point processes P^{λ and} P^{λ0−λ are} independent.

Proof of Lemma 3.5. From a classical result (see Proposition (1.7) Chapter XII, §1, p.473 in [21]) on Poisson processes, we know that it suffices to prove that the two (F^t)-Poisson processesP^λandP^λ0−λ do not jump simultaneously, i.e. that fort >0,

Ph

∃i, j∈N:ξ_i^λ< t, ξ_j^λ0−λ< t, ξ_i^λ=ξ_j^λ0−λi

= 0. (3.9)

0 20 40 60 80 100

0.00.51.01.52.02.53.03.5

Figure 4: (Color online). Sample paths of the diffusionsα_λ/(2π)(blue curve) andα_λ⁰/(2π) (red curve) together with(α_λ⁰−α_λ)/(2π)(green curve) forλ < λ⁰and for a small value of β. Again we see thatbαλ/(2π)c^and b(αλ⁰ −αλ)/(2π)cnever jump simultaneously whilebαλ⁰/(2π)c^andb(αλ⁰−αλ)/(2π)calways jump at the same times in agreement with Lemma 3.5, Lemma 3.4 and Remark 3.6.

Forn∈N, we consider the probability

p^β_n :=P

∃i, j∈N: β

8πζ_i^λ< t, β

8πζ_j^λ0−λ< t, β

8π|ζ_i^λ−ζ_j^λ0−λ|< 1 2n

. (3.10)

Ifnis fixed, then we have the convergence (the studied set is open as in the proof Lemma 3.4)

lim inf

β→0 p^β_n≥p_n :=P

∃i, j∈N:ξ^λ_i < t, ξ_j^λ0−λ< t, |ξ_i^λ−ξ^λ_j^0−λ|< 1 2n

.

To prove (3.9), it therefore suffices to prove that

lim sup

n→∞

lim sup

β→0

p^β_n = 0. (3.11)

For this, we need to work with a random subdivision of the interval [0,8πt/β]. As before, we consider a sequence(τk, k∈N)of i.i.d. positive random variables uniformly distributed in[1/2,3/2]and independent of the processes and form the sumSk=Pk

i=1τi

(S0:= 0).

2π

8π β

S_k n

α _λ

α _λ + 2π α _λ 0

2π`

2π(` + 1) 2π(` + 2)

Figure 5: Illustration for inclusion (3.15).

The probability (3.10) can be upper-bounded as follows P

∃i, j∈N, k≤ b2ntc+ 1 : 8π β

Sk

n ≤ζ_i^λ, ζ_j^λ0−λ≤8π β (Sk

n +2 n)]

=P

∃k≤ b2ntc+ 1 :bαλ⁰−αλ

2π c^andbαλ

2πcboth jump on the interval 8π β

Sk

n ,(Sk

n +2 n)

≤

b2ntc+1

X

k=1

P

{α_λ⁰(8π β

S_k

n)}^2π≤ {α_λ(8π β

S_k n )}^2π

(3.12)

+

b2ntc+1

X

k=1

P

bαλ⁰

2πcjumps two times during the interval 8π β

Sk

n,(Sk

n + 2 n)

. (3.13)

For this bound, we have used the fact that, conditionally on {αλ(8π

β Sk

n )}^2π≤ {αλ⁰(8π β

Sk

n )}^2π, (3.14)

the increasing property and the equality in law (3.2) impose

bαλ⁰−αλ

2π c^andbαλ

2πcboth jump on the interval 8π β

Sk

n,(Sk

n + 2 n)

(3.15)

⊆

bαλ⁰

2πcjumps two times during the interval8π β

Sk

n,(Sk

n + 2 n)

.

Indeed, the equality in law (3.2) implies thatb^{αλ0(t)−α}2π ^λ(t)cis increasing with respect tot (once the differenceα_λ⁰ −α_λ has reached the value2kπwherek ∈N, it remains forever above this value). This fact and the increasing property imply that under the event (3.15), the processb^α2π^λ0chas to jump two times on the interval ^8π_β[^S_n^k,(^S_n^k +_n²)]

(see Fig. 5). We now estimate the two sums in (3.12). The first one was already tackled in (3.8).