Large deviations for near-extreme eigenvalues in the beta-ensembles

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Large deviations for near-extreme eigenvalues in the beta-ensembles

Catherine Donati-Martin, Alain Rouault

To cite this version:

Catherine Donati-Martin, Alain Rouault. Large deviations for near-extreme eigenvalues in the beta-

ensembles. 2015. �hal-01236334�

Large deviations for near-extreme eigenvalues in the beta-ensembles

Catherine Donati-Martin¹ and Alain Rouault¹

1Laboratoire de Math´ematiques de Versailles UVSQ, CNRS

Universit´e Paris-Saclay 78035-Versailles Cedex France

e-mail:catherine.donati-martin@uvsq.fr;alain.rouault@uvsq.fr; url:

http://lmv.math.cnrs.fr/annuaire/donati-martin-catherine/;http://rouault.perso.math.cnrs.fr/

Abstract: For beta ensembles with convex poynomial potentials, we prove a large deviation principle for the empirical spectral distribution seen from the rightmost particle. This modified spectral distribution was introduced by Perret and Schehr (J. Stat. Phys. 2014) to study the crowding near the maximal eigenvalue, in the case of the GUE. We prove also convergence of fluctuations.

AMS 2000 subject classifications:15B52, 60G57, 60F10.

Keywords and phrases:Beta ensembles, spectral distribution, top eigen- value, large deviations.

December 1, 2015 1. Introduction

In random matrix models, the most popular statistics is the empirical spectral distribution (ESD). For aN×N matrix with real eigenvalues (λ₁,· · · , λ_N)

µ^(N):= 1 N

N

X

k=1

δ_λk. (1)

The first step in asymptotic study is to prove the convergence ofµ^(N)and also of the so called integrated density of statesEµ^(N). The limiting distributionσ is more often compactly supported. A second step is to prove the convergence of the largest eigenvalueλ_(N)= max(λ1,· · · , λN) to the end of the support of σ. At a more precise level, it is sometimes possible to establish large deviations.

In the so-calledβ-models, the density of eigenvalues is P^NV,β(dλ1,· · · , dλN) = (Z_V,β^N )⁻¹|∆(λ)|^βexp−N β 2

N

X

1

V(λk)

N

Y

1

dλk (2) where ∆(λ) is the Vandermonde determinant. Under convenient assumptions on the potentialV, the ESD satisfy the Large deviation Principle (LDP) with speedβN²/2 and good rate function

I^V(µ) =−Σ(µ) + Z

V dµ−cV if Z

|V|dµ <∞ (3)

1

where Σ is the logarithmic entropy Σ(µ) =

Z Z

ln(|x−y|)dµ(x)dµ(y), (4)

and

cV = inf

ν −Σ(ν) + Z

V dν . (5)

MoreoverI^V achieves its minimum 0 at a unique probability measureµ_V which is compactly supported, and which is consequently the limit ofµ^(N).

The most famous example is the Gaussian Unitary Ensemble which corre- sponds to V(x) = x²/2 and β = 2. The limiting distribution µV is then the semicircle distribution

µSC(dx) = 1 2π

p4−x² 1_[−2,2](x)dx (6)

and the result of large deviations is due to [BAG97], with c_V = 3/4.

Moreover under appropriate conditions again, the support ofµ_V is an interval [a_V, b_V] and the maximal eigenvalueλ_(N)converges tob_V.

To analyze the ”crowding” phenomenon near the largest eigenvalue, Perret and Schehr proposed in [PS14] and [PS15] to study the empirical measure,

µ_N := 1 N−1

N−1

X

k=1

δ_{λ(N)−λ(k)}∈M₁(R⁺) (7) whereλ₍₁₎ < λ₍₂₎ <· · ·< λ_(N)are the eigenvalues of M_N ranked increasingly.

They considered the Gaussian case with the Dyson valuesβ= 1,2,4 and made a complete study of the density of states near the maximumEµN, in the limit N→ ∞both in the bulk and at the edge.

In the present paper, we consider more general potentialsV, actually convex polynomials of even degree. We first prove thatµN converges in probability to the pushforwardνV of ofµV by the mappingx7→bV−x. Then we prove that the family of distributions of (µN)N satisfies the LDP with speed N² and a “new”

rate function which we callI_V^DOS, referring to the name “Density of States near the maximum” given by Perret and Schehr toEµN. There are two striking facts.

The first one is that the LDP is obtained for a Wasserstein topology (and not for the usual weak topology). This ensures in particular that the rate function is lower semicontinuous. The second one is that the LDP is weak i.e. we do not have a large deviation upperbound for closed sets but only for compact sets.

This implies that we could not deduce the convergence to the limit from the LDP as usual. In the Gaussian case, we haveV(x) =x²/2 and

I_V^DOS(ν) =−Σ(ν) +1

2Varν−3

4. (8)

where forν∈M₁(R⁺) such thatR

xdν(x)<∞, we define Varν=

Z

x²dν(x)− Z

xdν(x) 2

∈[0,∞]. (9)

Section2 is devoted to LDPs : Proposition 2.7 and Corollary2.8 study the pair (λ^(N), µ_N) which prepare the main result, the LDP forµ_N in Theorem2.9.

The proofs are in Section3. To complete the description of the asymptotic be- havior ofµN we prove also the convergence of fluctuations in Section4. Finally, in the Appendix of Section 5, we gather some properties of the Wasserstein distance on probability measures.

2. Assumptions and main result

To begin with, let us recall the definition of the Wasserstein distance.

Definition 2.1 Let p∈[1,∞[, and M₁^p(R) ={ν ∈M₁(R),R

|x|^pdν(x)<∞}.

For two probabilitiesµandν inM₁^p(R), the Wasserstein distance of order pis defined by

d_Wp(µ, ν) =

inf

π∈Π(µ,ν)

Z

R

|x−y|^pdπ(x, y) ^1/p

(10) whereΠ(µ, ν)is the set of probabilities on R² with first marginalµand second marginalν.

Besides, we denote by d the usual distance for the weak topology, given by Lipschitz bounded functions. It is known that

d≤d_W1≤d_Wq forq≥1. (11) We assume that

Assumption 2.2 V is a convex polynomial of even degreep≥2.

This assumption guarantees thatµV is unique, with support [aV, bV]. Moreover we have

Theorem 2.3 The sequence of distributions of(µ^(N))_N satisfies a large devia- tion principle in(M₁(R), d), with speedβN²/2with good rate functionI_V given by (3).

This result is Th. 2.6.1 in [AGZ10]. As we will prove in the following section, it can be improved :

Corollary 2.4 The LDP still holds onM₁^q, endowed with the distancedWq for anyq < p.

Fot the largest eigenvalue, we have Proposition 2.5 Under assumption2.2,

1. λ(N) converges in probability tobV,

2. λ_(N) satisfies a large deviation principle with speed N, with a good rate function J_V⁺ satisfyingJ_V⁺(x) = +∞forx < b_V, i.e;

lim

N

2

βN lnPV,β(λ_(N)> x) =−J_V⁺(x) , x > bV (12) withJ⁺(b_V) = 0

3. λ_(N) satisfies a large deviation principle with speedN², with a rate func- tionJ_V⁻, on the left ofbV i.e.

limN

2

βN²lnPV,β(λ(N)≤x) =−J_V⁻(x) :=− inf

µ:Suppµ⊂(−∞,x]I^V(µ) , x < bV

(13) Point 1. and 2. are in [AGZ10] Prop. 2.6.6, but it is more readable in [BG13]

Prop. 2.1. For Point 3. , see [Joh00] Rem. 2.3, [F´er08] Sect. 4.2, [DM06] and [VMB07]).

We are now interested in the behavior of µ_N. First, we have the following convergence result:

Proposition 2.6 We denote byτcµthe probability defined by Z

f(x)τcµ(dx) = Z

f(c−x)µ(dx) Then,µ_N converges in probability to the probability measure

ν_V :=τ_bVµ_V . (14)

Our main result rules the large deviations of the pair (λ_(N), µ_N). We equip M₁^p(R⁺) withd_Wp and denote byB(µ;δ) the ball aroundµof radiusδ.

We define

I_V(c, ν) =I^V(τ_cν), (15) which, since Σ is invariant by the transformationτ_c, is also

I_V(c, ν) =−Σ(ν) + Z

V dτ_cν−c_V . (16)

Proposition 2.7 We have

1. For any c∈Randµ∈M₁^p(R⁺),

δ&0,δlim⁰&0lim inf

N→∞

2

βN²ln(P^NV,β(λ_(N)∈[c−δ⁰, c+δ⁰], µN ∈BW_p(µ, δ)))

≥ −I_V(c, µ).(17) 2. For any closed setF ⊂Randµ∈M₁^p(R⁺),

lim

δ&0lim sup

N→∞

2

βN²ln(P^NV,β(λ_(N)∈F, µ_N ∈B_Wp(µ, δ)))

≤ −inf

c∈FIV(c, µ). (18) Corollary 2.8 The sequence of distributions of(λ_(N), µN)N, underP^NV,β, sat- isfies a weak LDP onR×M₁^p(R⁺)equipped with the product topology, at speed βN²/2 with rate functionI_V.

From these results, we may deduce one the one hand a weak LDP for the random measureµ_N, and on the other hand a conditional LDP forµ_N, knowing λ_(N).

Theorem 2.9 The sequence of distributions of (µ_N)_N, under P^NV,β, satisfies a weak LDP in(M₁^p(R⁺), d_Wp)at speedβN²/2 with rate function

I_V^DOS(ν) := inf

c∈(−∞,∞)I_V(c, ν) =−Σ(ν) +G_V(ν)−c_V . (19) with

GV(ν) := inf

c∈(−∞,∞)

Z

V dτcν . (20)

The propertiesI_V^DOS andGV are ruled by the following lemma:

Lemma 2.10 Letp−1≤q≤p.

1. The infimum in (20) is reached at a unique point which we callκV(ν).

2. ν7→κV(ν)is continuous fordW_q. 3. ν7→GV(ν) =R

V(κV(ν)−x)dν(x)is lower semicontinuous for dW_q. 4. I_V^DOS is is well defined onM₁^q(R)with values in [0,+∞] and lower semi-

continuous for theWq topology.

5. For b≥bV,

I_V^DOS(τbµV) = 0.

It follows from the property 5 in Lemma2.10that I_V^DOS is not a good rate function since the level sets are not compact. Then, (µN) is not exponentially tight in scaleN² (see [DZ98, Lemma 1.2.18]) and we do not know if the large deviations upper bound in Theorem2.9is true for closed sets. Nevertheless, we can prove exponential tightness in a weaker topology, conditionally that λ_(N) remains bounded, which leads to :

Proposition 2.11 Letp−1≤q < p.

For any closed setF ofM₁^q(R⁺) andC∈(0,∞) large enough, lim sup

N→∞

2

βN²ln(P^NV,β(µN ∈F|λ_(N)∈[−C, C]))≤ − inf

µ∈FI_V^DOS(µ). (21) The conditional large deviations are ruled by the following theorem.

Theorem 2.12 Letp−1≤q < p. LetFbe closed andGbe open in M₁^q(R⁺).

1. If c > b_V, we have lim sup

N

2

βN²lnP^NV,β µN ∈F| λ(N)∈[c, c+δ]

≤ −inf

µ∈FI_V^δ(c, µ) (22) lim inf

N

2

βN²lnP^NV,β µ_N ∈G| λ_(N)∈[c, c+δ]

≥ − inf

µ∈GI_V^δ(c, µ) (23) whereI_V^δ(c, µ) := inf_a∈[c,c+δ]IV(a, µ) satisfies

δ→0limI_V^δ(c, µ) =IV(c, µ). (24)

2. If c < b_V, set

J_V(c, µ) := I_V(c, µ)− inf

ν∈M₁^p(R⁺)

I_V(c, ν) (25)

= IV(c, µ)−J_V⁻(c), see Remark 2.14.

Then lim sup

N

2

βN²lnP^NV,β µN ∈F |λ_(N)∈[c−δ, c]

≤ − inf

µ∈FJ_V^δ(c, µ) (26) lim inf

N

2

βN²lnP^NV,β µN ∈G| λ(N)∈[c−δ, c]

≥ − inf

µ∈GJ_V^δ(c, µ) (27) whereJ_V^δ(c, µ) :== inf_{a∈[c−δ,c]}IV(a, µ)−J_V⁻(c)satisfies

lim

δ→0J_V^δ(c, µ) =JV(c, µ). (28) Remark 2.13 Let us notice that for fixed c,I_V(c,·)andJ_V(c,·)may be seen as conditional rate functions. From (15), we conclude that

I_V(c, µ) = 0 iff µ=τ_cµ_V. and from (25)

JV(c, µ) = 0 iff µ=µ_V(c−·).

Remark 2.14 Let us compute the projection onRof the rate function (16) i.e.

JV(c) := inf

µ∈M₁^p(R⁺)IV(c, µ). We have, by invariance

J(c) = inf

ν:∃µ∈M₁^p(R⁺):µ=τcν

−Σ(ν) + Z

V dν−cV .

Recall that the support ofµV is assumed to be[aV, bV]. Since then, eitherc≥bV

and then we can takeν =µV andJ(c) = 0, or c < bV and we get JV(c) = inf

ν:Suppν⊂(−∞,c)I^V(ν) :=J_V⁻(c). We recover Point 3. of Prop.2.5.

Remark 2.15 Let us now give some additional comments relative to the Gaus- sian case : V(x) = x²/2. In this case, κV(µ) is the mean of µ i.e. m(µ) = Rxdµ(x)andGV(µ)is its variance. Therefore,

I_V^DOS(µ) =−Σ(µ) +1

2Var(µ)−3 4,

and

dνV(x) =

√4x−x²

2π 1[0,4](x)dx .

As we have seen above, the rate function I_V^DOS is zero for probabilities of the formµ=τbµSC,b≥2 and thus is not a convex rate function. Notice that this particular functional is sermicontinuous not only fordW₁ but also for the weak topology. It is a consequence of the semicontinuity of Var. To prove this fact, use the representation

Var(µ) = 1

2E(X−Y)²

where X and Y are two real random variables independent and µ distributed, and then apply Fatou’s Lemma.

3. Proofs

In this section, we begin with the proofs of the easiest results and we end with the proof of the main result.

3.1. Proof of Corollary2.4 The set

KM ={µ∈M1(R), Z

|V(x)|dµ(x)≤M}

is compact for the weak topology and is used to prove the exponential tight- ness for µ^(N) in [AGZ10] p. 78. Actually KM is also a compact set for the q-Wasserstein distance forq < p(see the Appendix). It is then enough to apply Theorem 4.2.4 of [DZ98].

3.2. Proof of Proposition 2.6

Let f a bounded Lipschitz function with Lipschitz constant and uniform bound less than 1. Then,

| 1 N−1

N−1

X

1

f(λ_(N)−λ_(k))− 1 N

N

X

1

f(bV −λ_(k))|

= | 1

N−1

N−1

X

1

(f((λ_(N)−λ_(k))−f(bV −λ_(k))) + 1

N(N−1)

N−1

X

1

(f(b_V −λ_(k))−f(b_V −λ_(N)))|

≤ |λ_(N)−bV|+ 2 N

Therefored(µ_N, τ_bVµ^(N))≤ |λ(N)−b_V|+_N².

From the convergence ofµ^(N) toµ_V, andλ_(N) to b_V, we deduce thatµ_N con- verges toτ_bVµ_V.

3.3. Proof of Lemma 2.10

1) Notice that the uniqueness ofκV comes from the convexity ofV. 2) Letf(c) =R

V(c−x)dν(x). ThenκV(ν) is the solution of f⁰(c) =

Z

V⁰(c−x)dν(x) = 0.

Since the polynomial V⁰ is of degree p−1 and, for d_Wq, the functions ν 7→

Rx^kdν(x) are continuous fork≤p−1, this implies the continuity ofν7→κ_V(ν).

3) Denote bym_k(ν) thekth moment ofν. We can writeR

V(κ_V(ν)−x)dν(x) as

apmp(ν) +F(m1(ν), . . . , m_p−1(ν), κV(ν))

where F is a polynomial function and ap > 0. The function mp(ν) is lower semicontinuous as the supremum of the continuous functionsR

(|x|^p∧M)dν(x).

The functionsκV(ν) andmk(ν),k≤p−1 are continuous inνfordW_q. Therefore, GV is lower semicontinuous.

4) We refer to [AGZ10] for the same properties forIV, using e.g. for the positivity thatI_V^DOS(µ) =IV(τκ_V(µ)(µ)).

From [BAG97], −Σ(µ) is lower semicontinous for the topology of the weak convergence, and therefore is lower semicontinuous for the stronger topology W_q.

At last,G_V is lower semicontinuous from the 3).

5) First notice thatτ_bµ_V has a support inR⁺iffb≥b_V. From (19) and (15) we have then

I_V^DOS(τbµV) = inf

c IV(τcτbµV),

and this infimum is 0, reached atc=bsinceτ_b is an involution.

We could have also argued that, since the sequenceµ_N converges toτ_bVµ_V andI_V^DOSis the rate function in the LDP forµ_N, this insures thatI_V^DOS(τ_bVµ_V) = 0.

3.4. Proof of Theorem 2.9

It is enough to takec=κV(µ) in the lower bound (17) in Proposition2.7to obtain :

δ&0limlim inf

N→∞

2

βN²ln(P^NV,β(µN ∈BWp(µ, δ)))≥ −I_V^DOS(µ) which implies the lower bound for open sets.

For the upperbound, we takeF=Rin (18).

3.5. Proof of Proposition 2.7

Since the potential V is assumed to be a convex polynomial, it is lower bounded byVmin. ChangingV into V −Vmininduces a change ofcV intocV − Vmin, so in the above proofs we may and shall assumeV ≥0.

We denote by ¯QN the non normalized measure ¯Q^N_V,β =Z_V,β^N P^NV,β whereP^NV,β

is the distribution defined in (2). From [AGZ10, p.81] , we know that :

N→∞lim 2

βN²ln(Z_V,β^N ) =−cV .

Therefore, it is enough to prove the weak LDP for the measure ¯QN. The proof will consist in two parts : the lower bound and the upper bound.

3.5.1. Proof of the lower bound

We need an approximation lemma whose second statement is an easy conse- quence of Lemma 3.3 in [BAG97] (see also [AGZ10] p. 79). Indeed, the state- ment is given there for the distance of the weak convergence. Since the measure ν (and therefore its approximation) has compact support, the same is true for the Wasserstein distance.

Lemma 3.1 i) Letµ∈M₁^p(R⁺), for anyδ >0, there exists a supported com- pact probabilityν such that d_Wp(µ, ν)≤δ.

ii) Letν be probability on a compact set inR⁺, with no atoms. Let(x^i,N)the sequence of real numbers defined by

x1,N = inf{x|ν([0, x])≥ 1 N x^i+1,N = inf{x≥x^i,N|ν(]x^i+1,N, x])≥ 1

N, 1≤i≤N−2 Then,

x^1,N < x^N−2,N < . . . < x^N−1,N and for anyδ >0 andN large enough,

dW_p ν, 1 N−1

N−1

X

i=1

δ_xi,N

!

≤δ.

Proof of i)

For M > 0 and µ ∈ M₁^p(R⁺), we denote by µe^M the compactly supported probability defined bydµe^M = (µ([−M, M]))⁻¹1_{|x|≤M}dµ.

It is easy to see thatµe^M converges weakly toµasM tends to∞. Moreover, by dominated convergence theorem,

1 µ([−M, M])

Z

|x|^p1_{|x|≤M}dµ(x)→_M→∞

Z

|x|^pdµ(x).

This implies the convergence inW_p distance (see Proposition5.3, ii)).

To prove the lower bound (17), we will repeat almost verbatim the proof of [AGZ10] pp. 79-81, but follow step by step the rˆole played by λ_(N). We assume thatI(c, µ)<∞so thatµhas no atoms. We can also assume thatµis compactly supported, by consideringµe^M defined in Lemma3.1. One can check thatI(c,µe^M)→ I(c, µ).

Recall that

µN = 1 N−1

N−1

X

k=1

δλ_(N)−λ(k)

where theλ_(k)are the increasing sequence of eigenvalues.

Then, ifC= [c−δ⁰, c+δ⁰] andB :=B_Wp(µ, δ), Q¯N(λ_(N)∈C, µN ∈B)

= N!

Z

∆_N∩{_N−1¹ PN−1

k=1 δ_λN−λk∈B,λN∈C}

LbN(λ1, . . . λN))dλ1. . . dλN (29) where

∆N ={λ1< λ2< . . . < λN}, and

Lb_N(λ) = Y

i<j

|λ_i−λ_j|^βexp(−βN 2

N

X

i=1

V(λ_i)) (30)

= Y

i<j<N−1

|λ⁰_i−λ⁰_j|^β Y

i<N

|λ⁰_i|^βexp(−βN 2 (

N−1

X

i=1

V(λN −λ⁰_i) +V(λN))) := LN((λ⁰_i)i<N, λN)

where {λ⁰_i = λN −λi, i < N} is a decreasing family of positive numbers.

Since the densityL_N((λ⁰_i)_i<N, λ_N) is symmetric in (λ⁰_i), we can write : Q¯N(λ_(N)∈C, µN ∈B)

= N

Z

R^N−1+ ×C∩{_N−1¹ PN−1 k=1 δ_λ0

k∈B}

LN(λ⁰₁, . . . , λ⁰_N−1, λN))dλ⁰₁. . . dλ⁰_N−1dλN

From Lemma3.1, forN ≥Nδ,

(λ⁰_k)_k : |λ⁰_k−x^k,N| ≤ δ

2, ∀k < N

⊂ (

(λ⁰_k)_k: 1 N−1

N−1

X

k=1

δ_λ⁰

k∈B )

, and we can writeL_N as

LN = Y

i<j<N

|(λ⁰_i−x^i,N)−(λ⁰_j−x^j,N) +x^i,N−x^j,N|^β Y

i<N

|λ⁰_i|^β

× exp(−βN 2 (

N−1

X

i=1

V(λ_N −x^i,N −(λ⁰_i−x^i,N)) +V(λ_N −c+c)))

Sety_i=λ⁰_i−x^i,N,i < N andy_N =λ_N −c. Then, Q(µ¯ N ∈B)

≥ N Z

∆N(δ)

Y

i<j<N

|yi−yj+x^i,N−x^j,N|^β Y

i≤N−1

|yi+x^i,N|^β

exp(−βN 2 (

N−1

X

i=1

(V((yN +c)−(yi+x^i,N)) +V(yN+c)))Y

i≤N

dyi

where

∆⁰_N = {y1< y2< . . . < yN−1}

∆N(δ) = {(y1, . . . , yN) : (yi)i<N ∈[0, δ/2]^N−1, yN ∈[−δ, δ]} ∩∆⁰_N. Since on ∆⁰_N, the (yi) and the (x^i,N) form both increasing sequences, we have the lower bound:

|yi−y_j+x^i,N−x^j,N| ≥sup{|x^i,N−x^j,N|,|yi−y_j|}

and we use the same minoration as in [AGZ10] for the term

A := Y

i<j<N

|yi−yj+x^i,N −x^j,N|^β

≥ Y

i+1<j<N

|x^i,N−x^j,N|^β Y

i<N−1

|x^i,N −x^i+1,N|^β/2 Y

i<N−1

|yi−yi+1|^β/2.

For the second term, we use, since theyi andx^i,N are positive, Y

i<N−1

|yi+x^i,N|^β≥ Y

i<N−1

|yi|^β. We get:

Q(λ¯ _(N)∈C, µN ∈B)≥PN,1PN,2

where

P_N,1 = Nexp−βN 2

N−1

X

i=1

V(c−x^i,N) +V(c)

!

× Y

i+1<j<N

|x^i,N −x^j,N|^βY

|x^i,N−x^i+1,N|^β/2 (31) and

PN,2 = Z

∆N(δ)

Y

i<N−1

|yi−yi+1|^β/2|yi|^β

×exp−βN 2

N−1

X

i=1

V(c−x^i,N −yi+yN)−V(c−x^i,N)

×exp−βN

2 [V(y_N +c)−V(c)] Y

i≤N

dy_i.

Since we have assumed that µ is compactly supported, the {x^i,N,1 ≤ i ≤ N−1}are uniformly bounded and by continuity ofV,

δ→0limsup

N

sup

1≤i≤N

sup

|x|≤δ

|V(c−x^i,N +x)−V(c−x^i,N))|= 0, and

lim

δ→0 sup

|x|≤δ

|V(c+x)−V(c)|= 0.

Moreover, writingu₁=y₁, u_i+1=y_i+1−y_i, with δ⁰⁰= min(δ/2, δ⁰) Z

{(yi)i<N∈[0,^δ₂]^N−1}∩∆⁰_N∩{yN∈[−δ,δ]}

Y

i<N−1

|yi−yi+1|^β/2|yi|^β Y

i≤N

dyi

≥ Z

{(ui)_i<N∈[0,^δ_N⁰⁰]^N

u^β₁(u₂· · ·u_N−1)^3β/2u^β/2_N Y

i≤N

du_i

≥ C₁^−N δ⁰⁰

N ^C1N

for some constantC1, which yields lim

δ,δ⁰→0lim inf

N

2

βN²logPN,2≥0. On the other hand, from the choice of thex^i,N, we have

lim 1 N−1

N−1

X

i=1

V(c−x^i,N) = Z

V(c−x)dµ(x).

Finally the product in (31) can be managed exactly as in [AGZ10] p. 80. We conclude

lim

δ⁰&0lim

δ&0lim inf

N

2

βN²ln( ¯Q(λ_(N)∈C, µN ∈B)

≥ − Z Z

ln(y−x)dµ(x)dµ(y)− Z

V(c−x)dµ(x) which is the expected lower bound.

3.5.2. Proof of the upper bound

We start as in the proof of the lower bound with the representation (29).

Formula (30) can be rewritten as 2

β ln(bLN(λ)) = 2(N−1)² Z Z

x<y

ln(|x−y|)dµN(x)dµN(y) + 2(N−1)

Z

ln(|x|)dµN(x)

− (N−1)² Z

V(λN −x)dµN(x)−(N−1)V(λN)

−

N

X

i=1

V(λi) (32)

whereµN = _N¹₋₁PN−1

k=1 δλ_N−λk.

Under ¯Q_N, the λ_i are a.s. distinct so (µ_N)^⊗2({(x, y);x = y}) = _N¹₋₁ a.s..

Therefore, letM ∈R,

−2 Z Z

x<y

ln(|x−y|)dµN(x)dµN(y) =

−2 Z Z

x<y

ln(|x−y|)dµN(x)dµN(y) +M((µN)^⊗2({(x, y);x=y})− 1 N−1)

≥ Z Z

R²

(−ln(|x−y|)∧M)dµ_N(x)dµ_N(y)− M N−1 := −Σ^M(µ)− M

N−1. (33)

We have assumed V ≥ 0 so the second term in the third line of (32) is non positive. For the first term of the same line, notice that

−(N−1)² Z

V(λ_N −x)dµ_N(x) ≤ −(N−1)²inf

c∈F

Z

V(c−x)dµ_N(x).

We bound the term in the second line of (32) by (N−1)

Z

|x|dµN(x).

On the event{µN ∈B}, we have 2

βlogLbN(λ) ≤ −(N−1)² inf

ν∈B

−Σ^M(ν) + inf

c∈F

Z V dτcν

+(N−1) sup

ν∈B

Z

|x|dν(x) + (N−1)M

−β 2

N

X

i=1

V(λi)

Since

N!

Z

∆N

exp−β 2

N

X

i=1

V(λ_i) = Z

exp−β

2V(λ)dλ ^N

, and sup_ν∈BR

|x|dν(x)<∞, we obtain : lim sup

N

2

βN²ln( ¯Q(λ_(N)∈F, µN ∈B)≤ − inf

c∈F,ν∈B(−Σ^M(ν) + Z

V dτcν). We know that Σ^M is lower semicontinuous.

Assume thatF = [a,∞). SinceV is convex, the infimum inf_c≥aR

V(c−x)dν(x) is reached atc=κV(ν) or atc=a, so that

c≥ainf Z

V(c−x)dν(x) = Z

V(max(a, κV(ν))−x)dν(x) which is lower semicontinuous.

We obtain :

δ&0limlim sup

N

1

N²ln( ¯Q(µN ∈B, λ^(N)∈F))≤ −

−Σ^M(µ) + inf

c∈F

Z

V(c−x)dµ(x)

. (34) and since Σ^M grows to Σ as M goes to infinity, this yields the upper bound (18).

The same is true forF =]− ∞, a].

Now, takeF a non empty closed set. IfκV(µ)∈F, (34) is clearly true.

IfκV(µ)∈/F, thenκV(ν)∈/ F forν ∈B for smallδ.

Denote by a₋ = sup{x < κV(µ), x ∈ F} and a+ = inf{x > κV(µ), x ∈ F}.

Then,F ⊂]− ∞, a₋]∪[a+,∞[ and

δ&0limlim sup

N

1

N²ln( ¯Q(µN ∈B, λ^(N)∈F))≤

−

−Σ^M(µ) + Z

V(a−−x)dµ(x)∧ Z

V(a+−x)dµ(x)

. (35)

The last term in the above equation is inf_c∈FR

V(c−x)dµ(x).

3.6. Proof of Propostion 2.11 and Theorem 2.12

From Corollary2.8, we have lim sup 2

βN²lnP^NV,β(µN ∈F, λ_(N)∈[a, b])≤ − inf

µ∈F,c∈[a,b]IV(c, µ), (36) as soon asF is compact. To extend this property to closed sets, we follow the classical way and prove

Lemma 3.2 Let q < p. For any−∞< a < b <∞and M >0, there exists a compact setK_a,b,M ofM₁^q(R⁺) such that

lim sup

N

2

βN²ln(P^NV,β(λ_(N)∈[a, b], µ_N ∈/K_a,b,M))≤ −M . (37) Proof: Let

K_M :={µ∈M₁^p(R⁺) : Z

|V|dµ≤M}, which is compact ofM₁^q(R⁺) from Proposition5.3.

With our assumptions on the potentialV, there existsc1, c2>0 such that

|V(x)| ≤c₁x^p+c₂, Leta < b andC= sup{|a|,|b|}.

ForN ≥2, using the convexity ofx^p

{µN(V)≥M} ∩ {λ_(N)∈[a, b]} ⊂ {µ^(N)(V)≥M⁰} where

M =c₁(C^p2^p−1+ 2^p−1M⁰) +c₂.

It remains to use the exponential tightness for the ESDµ^(N), see [AGZ10], p. 77) where it is shown that :

lim sup

N

2

βN²ln(P^NV,β(µ^(N)∈/ KM¯))≤ −M

where ¯M is an affine function ofM.From Lemma3.2, (36) is satisfied forF a closed set ofM₁^q(R⁺).

3.6.1. Proof of Proposition2.11

By Proposition2.5, we know that for ∆ large enough, thenP^NV,β(λ_(N)∈∆)→ 1, so that

lim

N

2

βN²lnP^NV,β(λ_(N)∈∆) = 0, and then, for a closed setF,

lim sup

N

2

βN²lnP^NV,β(µ_N ∈F| λ_N ∈∆) = lim sup

N

2

βN²lnP^NV,β(µN ∈F, λN ∈∆)

≤ − inf

µ∈F,c∈∆I_V(c, µ), (38) from (36) for closed sets. Now, we use the easy bound

µ∈F,c∈∆inf IV(c, µ)≥ inf

µ∈F,c∈(−∞,∞)IV(c, µ) = inf

µ∈FI_V^DOS(µ).

3.6.2. Proof of Theorem2.12

We use Proposition 2.5 to estimate the probabilities of the conditioning events. On the one hand (36) for closed sets and (12) lead to (22) and on the other hand (36), (13) and Remark2.14lead to (26), using thatJ_V⁻is decreasing on [−∞, bV].

For the lower bounds, we use the lower bound coming from the LDP for both variables (Theorem2.71. or Corollary2.8) and (12) and (13), respectively.

It remains to prove (24) and (28), but it is straightforward since lim

δ→0 inf

a∈[c,c+δ]

Z

V(a−x)dµ(x) = lim

δ→0 inf

a∈[c−δ,c]

Z

V(a−x)dµ(x) = Z

V(c−x)dµ(x)

4. Fluctuations

We want to study the fluctuations ofµ_N around its limitν_V given in (14).

There are two contributions: the fluctuations of the largest eigenvalue and the fluctuations of the ESD. This yields a dichotomy according to the behavior of the test function. For the sake of simplicity, we choose a simple assumption on the test functionf which is far from optimal. ForV andβ we introduce a new assumption :

Assumption 4.1 V satisfies Assumption 2.2 andβ = 1,2,4, or V(x) =x²/2 andβ >0.

Proposition 4.2 Letf be a boundedC² function with two bounded derivatives.

1. If V andβ satisfy Assumption4.1 and ifνV(f⁰)6= 0, N^2/3(µ_N(f)−ν_V(f))⇒ν_V(f⁰)TW_β .

whereT Wβdenotes the Tracy-Widom distribution of indexβ(see [RRV11]

for a definition).

2. If V satisfies Assumption2.2 andβ >0 and ifνV(f⁰) = 0, N(µN(f)−νV(f))⇒ N(−f(0) +mV(f), σ²_V(f)) where

σ_V²(f) = 1 4β

∞

X

k=1

ka²_k, (39)

with

a_k= 2 π

Z π

0

f

b_V −a_V

2 (1−cosθ)

coskθ dθ , and

mV(f) = 2

β −1 Z

f(bV −t)dγV(t) (40) where γ_V is a signed measure on [a_V, b_V] given by formula (3.54) in [Joh98].

Let us notice, from Remark 3.5 in [Joh98], that in the Gaussian case,V(x) = x²/2, then

dγV(t) = 1

4δ₋₂+1 4δ2− 1

2π

√ dt 4−t². Proof: LetH > bV −aV. SetKH the random set defined by

KH ={(λ₍₁₎,· · · , λ_(N)) :∀i, −H ≤λ_(N)−λ_(i)≤H and −H ≤b−λ_(i)≤H}, and

M = max (1

2 sup

x∈[−H,H]

|f⁰⁰(x)|, sup

x∈[−H,H]

|f(x)|, sup

x∈[−H,H]

|xf⁰(x)|

) .

Setting

S_N(f) := (N−1)µ_N(f) =

N−1

X

1

f(λ_(N)−λ_(k)). we make a Taylor expansion off

f(λ_(N)−λ_(k)) = f(bV −λ_(k)) +εNf⁰(bV −λ_(k)) +rk,N(f) with

εN :=λ_(N)−bV

and

|rk,N(f)|1K_H ≤M ε²_N. Adding,

SN(f) =

N−1

X

1

f(bV −λ(k)) +εN N−1

X

1

f⁰(bV −λ(k)) +

N−1

X

1

rk,N

=

N

X

i=1

f(bV −λi) +εN N

X

i=1

f⁰(bV −λi) +RN(f) (41) where

RN(f) :=

N−1

X

1

rk,N(f) −f(−εN)−εNf⁰(−εN), satisfies

|RN(f)|1KH ≤M(N ε²_N+ 2). (42) Setting

∆_N(f) :=

N

X

i=1

f(b_V −λ_i)−N ν_V(f)

=

N

X

i=1

f(bV −λi)−N Z

f(bV −x)dµV(x), (43)

(41) gives

S_N(f)−N ν_V(f) =N ε_Nν_V(f⁰) + ∆_N(f) +ε_N∆_N(f⁰) +R_N(f). (44) The two sources of fluctuations are the convergences ofεN (rescaled) and ∆N(f).

On the one hand we know (Prop.2.5) that

εN →0 in probability, (45)

and the fluctuations are ruled by

N^2/3εN ⇒TWβ . (46)

(see [DG07] in the cases β = 1,2,4, and [RRV11] for the Gaussian case and β >0).

On the other hand, under our assumptions onV andf,

∆_N(f)⇒ N(m_V(f);σ_V²(f)), (47) where m_V(f) andσ²(f) are given by (40) and (39), respectively (see ([Joh98]

Theorem 2.4)).

1. IfνV(f⁰)6= 0, we set, for the sake of simplicity

N^−1/3[SN(f)−N ν(f)] =N^2/3εNνV(f⁰) +R⁰_N(f) We have then, if Φ(f) :=E[exp[iν(f⁰)TWβ]

Ee^iN−1/3[SN(f)−N ν(f)]−Φ(f) = Ee^{iN2/3εNν(f0)}−Φ(f)

+E

e^{iN2/3εNν(f0)}h

e^iR0N(f)−1i 1K_H

+E

e^{iN2/3εNν(f0)}h

e^iR0N(f)−1i 1K_H^c

. (48)

• The first term converges to zero, thanks to (46)

• The second term is bounded byE(|R⁰_N(f)∧2|1K_H) which tends to zero since onKH

|R⁰_N(f)| ≤N^−1/3|∆N(f)|+N^−1/3εN∆N(f⁰) +M N^2/3(εN)²+ 2M N^−1/3 and each of these terms tends to zero in probability, thanks to (47) and (45).

• The third one is bounded by 2P((K_H)^c) which tends to zero, since the extreme eigenvalues tend to the endpoints of the support.

This allows to conclude that

N^2/3(µN(f)−τb_VµV(f))⇒νV(f⁰)TWβ .

2. Ifν(f⁰) = 0, then,

SN(f)−N ν(f) = ∆N(f)−f(0) +εN∆N(f⁰) + ˜RN(f). (49) with

R˜N(f) :=

N−1

X

1

rk,N(f) −(f(−εN)−f(0))−εNf⁰(−εN).

From (47), ∆N(f)−f(0) converges in distribution toN(−f(0)+mV(f);σ_V²(f)).

MoreoverεN∆N(f⁰) tends to 0 in probability and ˜RN(f)1KN →0. The rest of the proof goes as before.

5. Appendix

We give some properties of the Wasserstein distancedW_p. Definition 5.1 Let p∈[1,∞[, and M₁^p(R) ={ν ∈M1(R),R

|x|^pdν(x)<∞}.

For two probabilitiesµandν inM₁^p(R), the Wasserstein distance of order pis defined by

dW_p(µ, ν) =

inf

π∈Π(µ,ν)

Z

R

|x−y|^pdπ(x, y) 1/p

(50) whereΠ(µ, ν)is the set of probabilities on R² with first marginalµand second marginalν.

Remark 5.2 For the Wasserstein distance of order 1, we have the duality for- mula

dW₁(µ, ν) = sup

kfkLip≤1

Z f dµ−

Z f dν

.

We now give a characterization of the convergence of probabilities in the topol- ogy induced bydW_p onM₁^p(R). We refer to [Vil09, Def. 6.8 and Theorem 6.9].

In the following, we denote by µn → µ the weak convergence of probabilities, i.e. against bounded continuous functions.

Proposition 5.3 Let (µn)n≥0 a sequence of probabilities in M₁^p(R) and µ ∈ M₁^p(R). The following assertions are equivalent :

i) dW_p(µn, µ)→0, ii) µn→µ and

Z

|x|^pdµn(x)→ Z

|x|^pdµ, iii) µ_n→µ andlim sup

n→∞

Z

|x|^pdµ_n(x)≤ Z

|x|^pdµ, iv) µn→µ and lim

R→∞lim sup

n→∞

Z

|x|≥R

|x|^pdµn(x) = 0,

v) For all continuous functionsf with|f(x)| ≤C(1 +|x|^p), one has Z

f(x)dµ_n(x)→ Z

f(x)dµ(x).

The condition in iv) is the condition of tightness, or relative compactness, in (M₁^p(R), d_Wp). In particular, it follows that, for anyM ∈R, the set

KM :={µ, Z

|x|^pdµ(x)≤M} is a compact set in (M₁^q(R), dW_q) for anyq < p.

Acknowledgments

We thank Gregory Schehr for the presentation of its model at the working group MEGA and Michel Ledoux for pointing out the semicontinuity in Remark 2.15.

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