Statement of the results

We are interested in the joint law of the eigenvalues underP^N,Vβ . We shall in fact consider a slightly more general model, where the interaction potential may not be linear in the trace, but rather some tensor power of the trace. This is necessary to deal with the law of a polynomial in several matrices. Hence, we consider the probability measure

dP^N,V_β (X1, ..., Xd) := 1

Z_β^N,Ve^{N2−rTr⊗rV(X1,...,Xd,B1,...,Bm)}

d

Y

k=1

dR^N,W_β,M^k(Xk) with

dR^N,W_β,M(X) := 1

Z_β,M^N,We^−NTr(W(X))1_kXk∞6MdX,

where1_Edenotes the indicator function of a setE, andZ_β^N,V andZ_β,M^N,W are normalizing constants. Here,

• β=2 (resp.β=1) corresponds to integration over the Hermitian (resp. symmetric) setH^N_β ofN×N matrices with complex (resp. real) entries. In particular

dX= Q

16j6`6NdX`j, ifβ= 1,

Q

16j6`6NdRe(X`j)Q

16j<`6NdIm(X`j), ifβ= 2.

• Tr denotes the trace overN×N matrices, that is, TrA=PN j=1A_jj.

• W_k:R!Rare uniformly convex functions, that is W_k⁰⁰(x)>c0>0 for allx∈R,

and given a functionW:R!Rand aN×N Hermitian matrixX, we defineW(X) as W(X) :=U W(D)U^∗,

where U is a unitary matrix which diagonalizes X as X=U DU^∗, and W(D) is the diagonal matrix with entries (W(D₁₁), ..., W(D_{N N})).

• B1, ..., Bmare Hermitian (resp. symmetric) matrices ifβ=2 (resp.β=1).

• Chx1, ..., xd, b1, ..., bmi^⊗r denotes the space ofrth tensor products of polynomials in d non-commutative variables with complex (resp. real) coefficients whenβ=2 (resp.

β=1). Forp∈Chx1, ..., xd, b1, ..., bmi^⊗r we denote by p=X

hp, q1⊗q2...⊗qriq1⊗q2...⊗qr

its decomposition on the monomial basis, and letp^∗denote its adjoint given by p^∗:=X

hp, q1⊗q2...⊗qriq^∗₁⊗q^∗₂...⊗q_r^∗, where∗ denotes the involution given by

(Yi₁... Yi_{`</sub>)<sup>∗</sup>=Yi<sub>`}... Yi₁ for alli1, ..., i`∈ {1, ..., d+m}, where{Y_i=X_i}^d_i=1and{Y_j+d=B_j}^m_j=1. We takeV to belong to the closure of

Chx1, ..., xd, b1, ..., bmi^⊗r for the norm given, forξ >1 andζ>1, by

kpkξ,ζ:=X

|hp, q1⊗q2...⊗qri|ξ^{Pri=1degX(qi)}ζ^{Pri=1degB(qi)} (2.1) where deg_X(q) (resp. deg_B(q)) denotes the number of letters {Xi}^d_i=1 (resp. {Bi}^m_i=1) contained in q. If ponly depends on the Xi (resp. the Bi), its norm does not depend onζ (resp.ξ) and we simply denote it bykpkξ (resp.kpkζ). We also assume thatV is self-adjoint, that isV(X1, ..., Xd, B1, ..., Bm)^∗=V(X1, ..., Xd, B1, ..., Bm).

• We usek · k∞ to denote the spectral radius norm.

Performing the change of variables Xk7!UkD(λ^k)U_k^∗, with Uk being unitary and D(λ^k) being the diagonal matrix with entries λ^k:=(λ^k₁, ..., λ^k_N), we find that the joint law of the eigenvalues is given by

dP_β^N,V(λ¹, ..., λ^d) = 1

Ze_β^N,VI_β^N,V(λ¹, ..., λ^d)

d

Y

k=1

dR_β,M^N,Wk(λ^k), (2.2) where

I_β^N,V(λ¹, ..., λ^d) := e^{N2−rTr⊗rV(U1D(λ1)U1∗,...,UdD(λd)Ud∗,B1,...,Bm)}dU1... dUd, dU being the Haar measure on the unitary group whenβ=2 (resp. the orthogonal group whenβ=1),Ze_β^N,V>0 is a normalization constant, andR_β,M^N,W is the probability measure onR^N given by

dR^N,W_β,M(λ) := 1 Z_β,M^N,W

Y

i<j

|λi−λj|^βe^{−NPNi=1W(λi)}

N

Y

i=1

1_|λi|6Mdλi, λ= (λ1, ..., λN). (2.3) Z

As we shall prove in§3, if Wk are uniformly convex and V is sufficiently small, for all k∈{1, ..., d}the empirical measureL^N_k of the eigenvalues ofXk converges to a compactly supported probability measureµ^V_k. In particular, if the cut-offM is chosen sufficiently large so that [−M, M]csupp(µ⁰_k), forV sufficiently small [−M, M]csupp(µ^V_k) and the limiting measures µ^V_k will be independent of M. Hence, we shall assume that M is a universally large constant (i.e., the largeness depends only on the potentialsWk). More precisely, throughout the whole paper we will suppose that the following holds.

Hypothesis 2.1. Assume that:

• Wk:R!Ris uniformly convex for any k∈{1, ..., d}, that is,W_k⁰⁰(x)>c0>0 for all x∈R. Moreover,W_k∈C^σ(R) for someσ>36.

• M >1 is a large universal constant.

• V is self-adjoint and kVkM ξ,ζ<∞ for some ξ large enough (the largeness being universal, see Lemma6.16) andζ>1.

• The spectral radius of each of the Hermitian matricesB₁, ..., B_mis bounded by 1.

Remark 2.2. The convexity assumption on the potentialsWk could be relaxed. In-deed, the main reasons for this assumption are:

– To ensure that the equilibrium measures, obtained as limits of the empirical mea-sure of the eigenvalues, enjoy the properties described in§3.

– To guarantee that the operatorΞtappearing in Proposition4.4is invertible.

– To prove the concentration inequalities in §4.5.

– To have rigidity estimates on the eigenvalues, needed in the universality proofs in§5.

As shown in the papers [12], [11], [5], the properties above hold under weaker as-sumptions on the Wk’s. However, because the proofs of our results are already very delicate, we decided to introduce the convexity assumptions in order to avoid additional technicality that would obscure the main ideas in the paper.

In order to be able to apply the approximate transport strategy introduced in [5], a key result we will prove is the following large dimension expansion ofI_β^N,V.

Theorem 2.3. Under Hypothesis 2.1,there exists a₀>0such that,for a∈[−a₀, a₀], I_β^N,aV(λ¹, ..., λ^k) =

1+O

1 N

e^{P2l=0N2−lFla(LN1,...,LNd,τBN)}, (2.4) where L^N_k are the spectral measures

L^N_k := 1 N

N

X

i=1

δ_λk i,

O(1/N) depends only on M, τ_B^N denotes the non-commutative distribution of the Bi

given by the collection of complex numbers τ_B^N(p) := 1

NTr(p(B1, ..., Bm)), p∈Chb1, ..., bmi, (2.5) and {F_l^a(µ1, ..., µd, τ)}²_l=0 are smooth functions of (µ1, ..., µd, τ) for the weak topology generated on the space of probability measures P([−M,+M])by

kµkζM:= max

k>1(M ζ)^−k|µ(x^k)|

and the norm sup_kpkζ61|τ(p)|on linear forms τ on Chb1, ..., bmi.

This result is proved in §6. We notice that it was already partially proved in [37]

in the unitary case. However, only the case where r=1 was considered there, and the expansion was shown to hold only in terms of the joint non-commutative distribution of the diagonal matrices{D(λ^k)}^d_k=1 rather than the spectral measure of each of them.

From the latter expansion of the density ofP_β^N,aV we can deduce the convergence of the spectral measures by standard large deviation techniques.

Corollary 2.4. Assume that, for any polynomial p∈Chb1, ..., bmi,

Nlim!∞τ_B^N(p) =τB(p). (2.6) Then, under Hypothesis 2.1, there exists a0>0 such that, for a∈[−a0, a0],the empirical measures {L^N_k}^d_k=1 converge almost surely under P_β^N,aV towards probability measures {µ^aV_k }^d_k=1 on the real line.

In the caser=1 this result is already a consequence of [35] and [18]. The existence and study of the equilibrium measures is performed in§3.

Starting from the representation of the density given in Theorem 2.3 (see §4), we are able to prove the following existence results on approximate transport maps.

Theorem 2.5. Under Hypothesis 2.1with ζ >1, suppose additionally that τ_B^N(p) =τ_B⁰(p)+ 1

Nτ_B¹(p)+ 1

N²τ_B²(p)+O 1

N³

, (2.7)

where the error is uniform on balls for k · kζ. Then there exists a constant α>0 such that, provided |a|6α, we can construct a map

T^N= ((T^N)¹₁, ...,(T^N)¹_N, ...,(T^N)^d₁, ...,(T^N)^d_N):R^dN−!R^dN

satisfying the following property: Let χ:R^dN!R⁺be a non-negative measurable function such that kχk_∞6N^k for some k>0. Then, for any η >0,we have

log

1+ χT^NdP_β^N,0

−log

1+ χ dP_β^N,aV

6C_k,ηN^η−1 (2.8) for some constant Cη,k independent of N. Also,with ˆλ:=(λ¹₁, ..., λ^d_N),T^N has the form

(T^N)^k_i(ˆλ) =T₀^k(λ^k_i)+ 1

N(T₁^N)^k_i(ˆλ) for all i= 1, ..., N and k= 1, ..., d,

where T₀^k:R!R and T₁^N:R^dN!R^dN are of class C^σ−3 and satisfy uniform (in N) regularity estimates. More precisely, we have the decomposition

T₁^N=X_1,1^N + 1 NX_2,1^N , where

max

16k6d 16i6N

k(X_1,1^N )^k_ik_L4(P_β^N,0)6ClogN and max

16k6d 16i6N

k(X_2,1^N )^k_ik_L2(P_β^N,0)6C(logN)²,

for some constant C >0 independent of N. In addition, with P_β^N,0-probability greater than 1−e^−c(logN)2,

max

i,k |(X_1,1^N )^k_i|6C(logN)N^1/(σ−14), max

i,k |(X_2,1^N )^k_i|6C(logN)²N^2/(σ−15), max

16i,i⁰6N|(X_1,1^N )^k_i(ˆλ)−(X_1,1^N )^k_i0(ˆλ)|6C(logN)N^1/(σ−15)|λ^k_i−λ^k_i0| for allk= 1..., d,

16maxi,i⁰6N|(X_2,1^N )^k_i(ˆλ)−(X_2,1^N )^k_i0(ˆλ)|6C(logN)²N^2/(σ−17)|λ^k_i−λ^k_i0| for allk= 1, ..., d, max

16i,j6N|∂_λ`

j(X_1,1^N )^k_i|(ˆλ)6C(logN)N^1/(σ−15) for allk, `= 1, ..., d.

As explained in§5, the existence of an approximate transport map satisfying regular-ity properties as above allows us to show universalregular-ity properties for the local fluctuations of the spectrum. For instance, we can prove the following result.

Corollary 2.6. Under the hypotheses of Theorem 2.5 the following holds: Let T₀^k be as in Theorem 2.5 and denote by Pe_β^N,aV the distribution of the increasingly ordered eigenvalues ({λ^k_i}^N_i=1)^d_k=1 under the law P_β^N,aV. Also, let µ⁰_k and µ^aV_k be as in Corol-lary 2.4,andαas in Theorem2.5. Then,for any θ∈ 0,¹₆

there exists a constant C >0,b independent of N,such that the following two facts hold true provided |a|6α:

(1) Let {ik}^d_k=1⊂[εN,(1−ε)N] for some ε>0. Then, choosing γ_i^k

k/N∈Rsuch that µ⁰_k((−∞, γ_i^k_k/N)) =ik

N,

Z Z

if m6N^2/3−θ then,for any bounded Lipschitz function f:R^dm!R,

f((N(λ^k_i

k+1−λ^k_ik), ..., N(λ^k_i

k+m−λ^k_ik))^d_k=1)dPe_β^N,aV

− f(((T₀^k)⁰(γ_i^k

k/N)N(λ^k_i

k+1−λ^k_i

k), ...,(T₀^k)⁰(γ^k_i

k/N)N(λ^k_i

k+m−λ^k_i

k))^d_k=1)dPe_β^N,0

6CNb ^θ−1kfk_∞+Cmb ^3/2N^θ−1k∇fk_∞.

(2) Let a⁰_k (resp.a^aV_k )denote the smallest point in the support of µ⁰_k(resp.µ^aV_k ),so that supp(µ⁰_k)⊂[a⁰_k,∞) (resp.supp(µ^aV_k )⊂[a^aV_k ,∞)). If m6N^4/7 then,for any bounded Lipschitz function f:R^dm!R,

f((N^2/3(λ^k₁−a^aV_k ), ..., N^2/3(λ^k_m−a^aV_k ))^d_k=1)dPe_β^N,aV

− f(((T₀^k)⁰(a⁰_k)N^2/3(λ^k₁−a⁰_k), ...,(T₀^k)⁰(a⁰_k)N^2/3(λ^k_m−a⁰_k))^d_k=1)dPe_β^N,0

6CNb ^θ−1kfk_∞+C(mb ^1/2N^θ−1/3+m^7/6N^−2/3)k∇fk_∞. The same bound holds around the largest point in the support of µ^aV_k .

Similar results could be derived with functions of both statistics in the bulk and at the edge. Let us remark that for a=0 the eigenvalues of the different matrices are uncorrelated andP_β^N,0 becomes a product:

dP_β^N,0=

d

Y

k=1

dR^N,W_β,M^k.

Universality under the latterβ-models was already proved in [13], [14], [56], [5]. Also, by the results in [5] we can find approximate transport mapsS_k^N:R^N!R^N from the law P_GVE,β^N (this is the law of GUE matrices when β=2 and GOE matrices whenβ=1) to R^N,W_β,M^k for any k=1, ..., d. Hence (S₁^N, ..., S^N_d ):R^dN!R^dN is an approximate transport from (P_GVE,β^N )^⊗d (i.e., the law of d independent GUE matrices when β=2 and GOE matrices whenβ=1) toP_β^N,0, and this allows us to deduce that the local statistics are in the same universality class as GUE (resp. GOE) matrices.

More precisely, as already observed in [5], the leading orders in the transport can be restated in terms of the equilibrium densities: denoting by

%sc(x) := 1 2π

p(4−x²)+ (2.9)

the density of the semi-circle distribution and by%⁰_k the density ofµ⁰_k, then the leading-order term ofS_k^N is given by (S^k₀)^⊗N, where S₀^k:R!Ris the monotone transport from

%_scdxto%⁰_kdxthat can be found solving the ordinary differential equation (ODE) (S₀^k)⁰(x) = %sc

%⁰_k(S₀^k)(x), S₀^k(−2) =a⁰_k. (2.10) Z

Z

Z Z

Also, the transportT₀^k:R!Rappearing in Corollary2.6solves (T₀^k)⁰(x) = %⁰_k

%^aV_k (T₀^k)(x), T₀^k(a⁰_k) =a^aV_k . (2.11) Set

c^aV_k := lim

x!−2⁺

%_sc

%^aV_k (T₀^kS₀^k)(x). (2.12) Due to these observations, we can easily prove the following result.

Corollary 2.7. Let m∈N. Under the hypotheses of Theorem 2.5, the following holds: Denote by Pe_β^N,aV (resp. (Pe_GVE,β^N )^⊗d)the distribution of the increasingly ordered eigenvalues ({λ^k_i}^N_i=1)^d_k=1 under the law P_β^N,aV (resp.(P_GVE,β^N )^⊗d). Also, let αbe as in Theorem 2.5. Then,for any θ∈ 0,¹₆

and C0>0 there exists a constant C >0,b indepen-dent of N, such that the following two facts hold true provided |a|6α:

(1) Given {σk}^d_k=1⊂(0,1), let γσ_k∈R be such that µsc((−∞, γσ_k))=σk, and γσ_k,k

such that µ^aV_k ((−∞, γσ_k,k))=σk. Then, if |ik/N−σk|6C0/N and m6N^2/3−θ, for any bounded Lipschitz function f:R^dm!Rwe have

f((N(λ^k_i

k+1−λ^k_ik), ..., N(λ^k_i

k+m−λ^k_ik))^d_k=1)dPe_β^N,aV

− f

%sc(γσ_k)

%^aV_k (γσ_k,k)N(λ^k_ik+1−λ^k_ik), ..., %sc(γσ_k)

%^aV_k (γσ_k,k)N(λ^k_ik+m−λ^k_ik) ^d

k=1

d(Pe_GVE,β^N )^⊗d

6CNb ^θ−1kfk∞+Cmb ^3/2N^θ−1k∇fk∞.

(2) Let c^aV_k be as in (2.12). If m6N^4/7 then, for any bounded Lipschitz function f:R^m!R, we have

f((N^2/3(λ^k₁−a^aV_k ), ..., N^2/3(λ^k_m−a^aV_k ))^d_k=1)dPe_β^N,aV

− f(c^aV_k N^2/3(λ^k₁+2), ..., c^aV_k N^2/3(λ^k_m+2))^d_k=1)d(Pe_GVE,β^N )^⊗d

6CNb ^θ−1kfk∞+C(mb ^1/2N^θ−1/3+m^7/6N^−2/3)k∇fk∞. The same bound holds around the largest point in the support of µ^aV_k .

While the previous results deal only with bounded test function, in the next theo-rem we take full advantage of the estimate (2.8) to show averaged energy universality in our multi-matrix setting. Note that, to show this result, we need to consider as test functions averages (with respect to E) of m-points correlation functions of the form P

i₁6=...6=i_mf(N(λ^k_i

1−E), ..., N(λ^k_i

m−E)), where E belongs to the bulk of the spectrum.

Z

Z

Z

Z

In particular, these test functions have L^∞ norm of size N^m. Actually, as in Corol-laries 2.6 and 2.7, we can deal with test functions depending at the same time on the eigenvalues of the different matrices.

Here and in the following, we use⁻_I to denote the averaged integral over an interval I⊂R, namely⁻_I=(1/|I|) _I.

Corollary2.8. Fix m∈Nand ζ∈(0,1), and let αbe as in Theorem 2.5. Also,let T₀^kand S₀^k be as in (2.11)and (2.10),and defineR_k:=T₀^kS₀^k. Then,given{E_k}_16k6d⊂ (−2,2), θ∈(0,min{ζ,1−ζ}), and a non-negative Lipschitz function f:R^dm!R⁺ with compact support,there exists a constant C >b 0,independent of N,such that the following holds true provided |a|6α:

−

R1(E1)+N^−ζR⁰₁(E1)

R₁(E₁)−N^−ζR⁰₁(E₁)

dEe₁...−

Rd(Ed)+N^−ζR⁰_d(Ed)

R_d(E_d)−N^−ζR⁰_d(E_d)

dEe_d

× X

i_k,16=...6=i_k,m

f((N(λ^k_ik,1−Eek), ..., N(λ^k_ik,m−Eek))^d_k=1)

dP_β^N,aV

−

−

E₁+N^−ζ

E₁−N^−ζ

dEe1...−

E_d+N^−ζ

E_d−N^−ζ

dEed

× X

ik,16=...6=ik,m

f((R⁰_k(Ek)N(λ^k_ik,1−Eek), ..., R⁰_k(Ek)N(λ^k_ik,m−Eek))^d_k=1)

dP_GVE^N

6C(Nb ^θ+ζ−1+N^θ−ζ).

It is worth mentioning that, in the single-matrix case, Bourgade, Erd˝os, Yau, and Yin [15] have recently been able to remove the average with respect to E and prove the Wigner–Dyson–Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. We believe that combining their techniques with ours one should be able to remove the average with respect toE in the previous theorem. However, this would go beyond the scope of this paper and we shall not investigate this here.

Another consequence of our transportation approach is the universality of other observables, such as the minimum spacing in the bulk. The next result is restricted to the caseβ=2 since we rely on [6, Theorem 1.4] which is proved in the caseβ=2 and is currently unknown forβ=1.

Corollary2.9. Letβ=2,fix k∈{1, ..., d},letIkbe a compact subset of (−a^aV_k , b^aV_k ) with non-empty interior, and denote the renormalized gaps by

∆^k_i := λ^k_i+1−λ^k_i

(T₀^kS^k₀)⁰(γi/N), λ^k_i ∈Ik, Z

Z Z

Z Z

Z

Z

Z Z

where γ_i/N∈Ris such that µsc((−∞, γ_i/N))=i/N. Also, denote by Pe_β,k^N,aV the distribu-tion of the increasingly ordered eigenvalues {λ^k_i}^N_i=1 under P_β,k^N,aV, the law of the eigen-values of the k-th matrix under P_β^N,aV. Then,under the hypotheses of Theorem 2.5,the following statements hold:

• (Smallest gaps) Let t˜_N,k¹ <˜t_N,k² ...<t˜_N,k^p denote the p smallest renormalized spac-ings ∆^k_i of the eigenvalues of the k-th matrix lying in I, and set

˜ τ_N,k^p :=

1 144π² _(Tk

0S₀^k)⁻¹(I)

(4−x²)²dx 1/3

t˜_N,k^p .

Then, as N!∞, N^4/3˜τ_N,k^p converges in law towards τ^p whose density is given by 3

(p−1)!x^3p−1e^−x3dx.

• (Largest gaps)Let `<sup>1</sup><sub>N,k</sub>(I)>`²_N,k(I)>... be the largest gaps of the form ∆^k_i with λ^k_i∈Ik. Let {rN}_N∈Nbe a family of positive integers such that

logrN

logN !0 as N!∞.

Then, as N!∞,

√ N

32 logN`^r_N,k^N !1 in L^q(Pe_β,k^N,aV) for any q <∞.

All the above corollaries are proved in §5.

As an important application of our results, we consider the law of the eigenvalues of a self-adjoint polynomials in several GUE or GOE matrices. Indeed, ifεis sufficiently small andX1, ..., Xdare independent GUE or GOE matrices, a change of variable formula shows that the law of the eigenvalues of thedrandom matrices given by

Yi=Xi+ε Pi(X1, ..., Xd), 16i6d,

follows a distribution of the form P_β^N,aV with r=2 and V a convergent series, see §7.

Hence we have the following result.

Corollary 2.10. Let P1, ..., Pd∈Chx1, ..., xd, b1, ..., bmibe self-adjoint polynomials.

There exists ε₀>0 such that the following holds: Let X_i be independent GUE or GOE matrices and set

Y_i:=X_i+ε P_i(X₁, ..., X_d).

Z

Then, for ε∈[−ε0, ε0],the eigenvalues of the matrices {Yi}^d_i=1 fluctuate in the bulk or at the edge as when ε=0,up to rescaling. The same result holds for

Y_i=X_i+ε P_i(X₁, ..., X_d, B₁, ..., B_m)

provided τ_B^N satisfies (2.7). Namely, in both models,the law Pe_β^N,εP of the ordered eigen-values of the matrices Y_k satisfies the same conclusions as Pe_β^N,aV in Corollaries 2.7 and 2.9.

Remark 2.11. Recall that, as already stated at the beginning of§2, when β=1 the matricesBi, as well as the coefficients ofP, are assumed to be real. In particular, in the statement above, ifXi are GOE then the matrices Yi must be orthogonal. The reason for this is that we need the map (X₁, ..., X_d)7!(Y₁, ..., Y_d) to be an isomorphism close to identity at least for uniformly bounded matrices. Our result should generalize to mixed polynomials in GOE and GUE which satisfy this property, but it does not include the case of the perturbation of a GOE matrix by a small GUE matrix which is Hermitian but not orthogonal.

Acknowledgments. A. F. was partially supported by NSF Grant DMS-1262411 and NSF Grant DMS-1361122. A. G. was partially supported by the Simons Foundation and by NSF Grant DMS-1307704. The authors would like to thank an anonymous referee for his challenging questions.

In document Universality in several-matrix models via approximate transport maps (Stránka 7-16)