Universality results

In this section we explain how Corollaries2.6,2.7and2.9follow from Theorem2.5.

Proof of Corollary 2.6. Given ϑ>0, we define the set

G_ϑ:={λˆ∈R^dN:|λ^{`</sup><sub>i</sub>−γ<sub>i/N</sub><sup>`} |6N^ϑ−2/3min{i, N+1−i}^1/3for alliand`}. (5.1) As proved in [29] in the special case of the Gaussian ensembles and then generalized in [13, Theorem 2.4] to potentialsWk satisfying much weaker conditions than the ones assumed here, the following rigidity estimate holds: for all ϑ>0 there exist ¯c>0 and C <∞ such that, for allN>0,

Pe_β^N,0 R^N\Gϑ

6Ce ^−Nc¯. (5.2)

Also, due to the fact thatµ⁰_k has a density which is strictly positive inside its support [a⁰_k, b⁰_k] except at the two boundary points where it goes to zero as a square root (see Lemma3.2), we deduce that

m N > 1

C

γ^k_(i+m)/N

γ_i/N^k

minp

s−a⁰_k,p

b⁰_k−s ds,

from which it follows easily that

|γ^k_(i+m)/N−γ_i/N^k |6 C N^2/3min

m^2/3, m

min{i, N+1−i}^1/3

. (5.3)

Since

|λ^k_i+m−λ^k_i|6|λ^k_i−γ_i/N^k |+|λ^k_i+m−γ_(i+m)/N^k |+|γ_(i+m)/N^k −γ^k_i/N|, (5.4) using (5.2) and (5.3) and recalling that by assumptionmN, we deduce that

|N(λ^k_i

k+j−λ^k_i

k)|6C(N^ϑ+m) for all ˆλ∈Gϑ,ik∈[N ε, N(1−ε)], j= 1, ..., m, (5.5) and

|N^2/3(λ^k_j−a⁰_k)|6C(N^ϑ+m^2/3) for all ˆλ∈G_ϑ,j= 1, ..., m. (5.6) Z

Now, given a bounded function χ:R^dN!R, applying (2.8) to 1

2

1+ χ kχk∞

withk=0 andη=ϑ, we deduce that

χT^NdP_β^N,0− χ dP_β^N,aV

6CN^ϑ−1kχk_∞. (5.7)

Recall that the mapT^N is given byX₁^N,whereX_t^N is the flow of the vector fieldY^N_t that has the very special form (4.13) (see Proposition4.13). In particular, since the functions y⁰_k,t,y¹_k,t,ζk`,t(·, y) are uniformly Lipschitz, we see that

|( ˙X_t^N)^k_i−( ˙X_t^N)^k_j|6L|(X_t^N)^k_i−(X_t^N)^k_j| for alli, j= 1, ..., N andk= 1, ..., d.

Hence, sinceX₁^N=T^N andX₀^N=Id, Gr¨onwall’s inequality yields

e^−L(λ^k_i−λ^k_j)6(T^N)^k_i(ˆλ)−(T^N)^k_j(ˆλ)6e^L(λ^k_i−λ^k_j) for allλ^k_i >λ^k_j. (5.8) We now remark that the lawPe_β^N,aV is obtained as the image of the law ofλ^k=(λ^k₁, ..., λ^k_N), 16k6dunderP_β^N,aV under the map

R:b R^dN!R^dN, R(λb ¹, ..., λ^k, ..., λ^d) := (R(λ¹), ...,R(λ^k), ...,R(λ^d)), (5.9) whereR:R^N!R^N is defined as

[R(x1, ..., xN)]i:= min

#J=imax

j∈J xj for alli= 1, ..., N. (5.10) Hence, due to (5.8), it follows thatT^N andRb commute, namely

RbT^N=T^NR.b (5.11)

We now consider a test functionχof the form

χ(ˆλ) =f((N(λ^k_ik+1−λ^k_ik), ..., N(λ^k_ik+m−λ^k_ik))^d_k=1). (5.12) Then

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 = χRbdP_β^N,aV, and it follows by (5.7) and (5.11) that

χ dPe_β^N,aV− χT^NRbdP_β^N,0

6CN^ϑ−1kfk∞.

Z Z

Z

Z Z

Z

LetX0,t,X1,t,andX2,tbe as in Proposition4.13, and note the following fact: whenever

SinceX_0,t<sup>`</sup> is a smooth diffeomorphism which sends the quantiles ofµ<sup>∗</sup><sub>`,0</sub>onto the quantiles ofµ^∗_`,t, we deduce that and by the same argument as the one used in the proof of Proposition4.13to show (4.39) and (4.40) we get

(this follows by the same proof as the one of (5.8), compare also with [5, Equation (5.2)]).

In addition,

and hence, by the definition ofGϑ, Hence, we have proved that(¹)

For the second statement we choose

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

(¹) This estimate, as well as the one at the edge that we shall prove below, should be compared with the one obtained in [5, Theorem 1.5]. While the estimates here are considerably stronger than the ones in [5, Theorem 1.5] (this follows from the fact that we have better bounds on our approximate transport maps), as a small “loss” we now haveN^ϑ−1instead of a term (logN)³/N. The reason for this small difference comes from the fact that we decided to apply (2.8) to deduce (5.7). It is worth noticing that the argument in§4combined with [5, Lemma 2.2] proves that also the stronger bound

holds. However, since in general (2.8) is much more powerful than the estimate above (as it allows to deal with functions that grow polynomially with respect to the dimension) and the improvement between (logN)³/NandN^ϑ−1 is minimal, we have decided not to state also this second estimate.

Z

and we note thatT₀^k(a⁰_k)=a^aV_k . Then, due to (4.39) and (5.13), we get using the rigidity estimate (5.6), we may replace

N^2/3(T₀^k(λ^k₁)−T₀^k(a⁰_k)) by (T₀^k)⁰(a⁰_k)N^2/3(λ^k₁−a⁰_k) which proves the second statement by choosingϑ6θ.

Proof of Corollary 2.7. We first note that the proof of Corollary 2.6 could be re-peated verbatim in the context of [5] to show that [5, Theorem 1.5] holds with the same estimates as we obtained here. Hence, by combining this result with Corollary 2.6, we have

whereγ_ik/N satisfies µsc((−∞, γ_ik/N))=ik/N. Note that the transport relations (2.10) and (2.11) imply thatT₀^kS₀^k(γ_ik/N)=γ_i^k

k/N,a, whereγ_i^k

k/N,a satisfies µ^aV_k ((−∞, γ^k_i

k/N,a)) =ik

N, and hence (again by (2.10) and (2.11))

(T₀^kS₀^k)⁰(γ_ik/N) = %sc(γ_ik/N)

%^aV_k (γ_i^k

k/N,a).

Finally, since |σ_k−i_k/N|6C/N and σ_k∈(0,1), arguing as we did for proving (5.3), we deduce that|γ_ik/N−γ_σk|6C/Ne , so up to another small error we may replace

%_sc(γ_ik/N)

%^aV_k (γ_i^k

k/N,a) by %sc(γσ_k)

%^aV_k (γ_σk,k).

This concludes the proof of of the first statement, while the second one is just a conse-quence of Corollary2.6(2) and [5, Theorem 1.5 (2)].

Proof of Corollary 2.8. As is clear by looking at the proof of Corollaries2.6and2.7, the fact of dealing at the same time with the eigenvalues of different matrices does not complicate the proof. For this reason, since the proof of Corollary 2.8 is already very involved, to make the argument more transparent we shall prove the result when the test function is of the form

−

Rk(E)+N^−ζR⁰_k(E)

R_k(E)−N^−ζR⁰_k(E)

X

i_jdistinct

f N(λ^k_i

1−E), ..., Ne (λ^k_i

m−E)e dEe

for someE∈(−2,2), the proof in the general case being completely analogous and just notationally heavier.

To simplify the notation, we set gEe(ˆλ) := X

i16=...6=im

f(N(λ^k_i1−E), ..., N(λe ^k_im−E)),e

A_k:=

−

R_k(E)+N^−ζR⁰_k(E)

R_k(E)−N^−ζR⁰_k(E)

g

EedEe

dP_β^N,aV.

It follows by (2.8) withη=θthat

|log(1+Ak)−log(1+A1,k)|6CN^θ−1, (5.15) Z

Z Z

where

A1,k:=

−

R_k(E)+N^−ζR⁰_k(E)

Rk(E)−N^−ζR⁰_k(E)

gEe(T^N)^kdEe

dP_β^N,0

=

−

R_k(E)+N^−ζR⁰_k(E)

Rk(E)−N^−ζR⁰_k(E)

X

i16=...6=im

f(N((T^N)^k_i1(ˆλ)−E), ..., N((Te ^N)^k_im(ˆλ)−E))e dEe

dP_β^N,0.

Define the quantilesγ_i/N^k ∈(S_k⁰(−2), S_k⁰(2)) as in Corollary2.6, and given ϑ>0 small (to be fixed later) we consider the setGϑ defined in (5.1).

Since the integrandg

Ee(T^N)^k is pointwise bounded bykfk∞N^m, it follows by (5.2) that

A1,k=A2,k+O(e^−Nc), (5.16)

where

A2,k:=

G_ϑ

−

R_k(E)+N^−ζR⁰_k(E)

R_k(E)−N^−ζR⁰_k(E)

gEe(T^N)^kdEe

dP_β^N,0. Observe that if ˆλ∈Gϑ then, by definition,

|λ^k_i−λ^k_j|>|γ^k_i/N−γ_j/N^k |−N^−2/3+ϑmin{i, N+1−i}^−1/3−N^−2/3+ϑmin{j, N+1−j}^−1/3. Hence, sinceγ_(i+1)/N^k −γ_i/N^k >c0N^−2/3min{i, N+1−i}^−1/3 for alli, we deduce that

|λ^k_i−λ^k_j|>N^ϑ−1 provided|i−j|>C₀N^ϑ, which, combined with (5.8) yields, for ˆλ∈Gϑ,

|(T^N)^k_i(ˆλ)−(T^N)^k_j(ˆλ)|>e^−LN^ϑ−1 provided|i−j|>C₀N^ϑ. (5.17) We now notice that, sincef is compactly supported, the quantity

f(N((T^N)^k_i

1(ˆλ)−E), ..., Ne ((T^N)^k_i

m(ˆλ)−E))e can be non-zero only if

|(T^N)^k_ij(ˆλ)−E|e 6C1

N for allj= 1, ..., m.

Therefore, if ¯i∈{1, ..., N}is an index (depending on ˆλandE) such thate

|(T^N)^k¯i(ˆλ)−E|e 6C1

N, Z Z

Z Z

Z Z

then (5.17) yields

|(T^N)^k_i(ˆλ)−E|e 6C1

N =⇒ |i−¯i|6C₀N^ϑ. This proves that, for any ˆλ∈Gϑ, there exists a set of indices

Jˆλ,Ee⊂ {(i1, ..., im)∈ {1, ..., N}^m:i16=...6=im} such that #Jλ,ˆEe6CN^mϑand

A2,k=

Gϑ

−

Rk(E)+N^−ζR⁰_k(E)

Rk(E)−N^−ζR⁰_k(E)

ˆ

gEe(T^N)^kdEe

dP_β^N,0, where

ˆ

gEe(ˆλ) := X

(i1,...,im)∈J_λ,fˆE

f(N(λ^k_i1−E), ..., N(λe ^k_im−E))e

satisfies|ˆg_Tk

0(E)e |6Ckfk_∞N^mϑ.

We now perform the change of variableEe7!T₀^k(E), which givese

Rk(E)+N^−ζR⁰_k(E)

R_k(E)−N^−ζR⁰_k(E)

ˆ g

Ee(T^N)^kdEe=

(T₀^k)⁻¹[Rk(E)+N^−ζR⁰_k(E)]

(T₀^k)⁻¹[R_k(E)−N^−ζR_k⁰(E)]

ˆ g_Tk

0(E)e (T^N)^k(T₀^k)⁰(E)e dE.e Recalling thatRk=T₀^kS₀^k and that these maps are all smooth diffeomorphisms ofR, we see that for

Ee∈[(T₀^k)⁻¹[R_k(E)−N^−ζR⁰_k(E)],(T₀^k)⁻¹[R_k(E)+N^−ζR⁰_k(E)]]

it holds

|(T₀^k)⁰(E)−(Te ₀^k)⁰S₀^k(E)|6CN^−ζ, R⁰_k(E) = [(T₀^k)⁰S₀^k(E)] (S₀^k)⁰(E), and

(T₀^k)⁻¹[Rk(E)±N^−ζR⁰_k(E)] =S₀^k(E)±N^−ζ(S₀^k)⁰(E)+O(N^−2ζ).

Hence, since|ˆg_Tk

0(E)e |6CN^mϑ,

−

(T₀^k)⁻¹[R_k(E)+N^−ζR⁰_k(E)]

(T₀^k)⁻¹[R_k(E)−N^−ζR⁰_k(E)]

ˆ g_Tk

0(E)e (T^N)^k(T₀^k)⁰(E)e dEe

=−

S^k₀(E)−N^−ζ(S^k₀)⁰(E)

S₀^k(E)−N^−ζ(S₀^k)⁰(E)

ˆ g_Tk

0(E)e (T^N)^kdEe+O(N^mϑ−ζ),

Z Z

Z Z

Z

Z

which proves that

Due to Theorem2.5, we can write ˆ arguing as we did for (5.13), we get

|∂EeX_1,^k_ˆλ(E)|e 6CN^ϑ, |(X_1,1^N )^k_i(ˆλ)−X_1,^k_λˆ(E)|e 6CN^ϑ|λ^k_i−E|e for all ˆλ∈Gϑ. (5.20)

Z Z

Z

Z Z

Z

In addition, by the same reasoning,

maxi,k ∂1zk`,t(X<sub>0,t</sub><sup>k</sup> (λ<sup>k</sup><sub>i</sub>), y)dM<sub>X</sub><sup>N</sup>`

0,t(y) =O(N^ϑ) for all ˆλ∈Gϑ, and the argument used to prove (4.39) (see in particular (4.50)) yields

max

i,k |(X_2,1^N )^k_i|6CN^2ϑ for all ˆλ∈G_ϑ. Hence, since #Jλ,ˆEe6CN^mϑwe immediately deduce that

O 1

N _Gϑ

X

(i₁,...,i_m)∈Jλ,fˆE

|(X_2,1^N )^k_ij|dP_β^N,0

=O N^(m+2)ϑ−1

. (5.21)

Now, to get rid of the term X^k

1,λˆ(E) insidee h

Ee we take advantage of (5.20) and the average with respect toE: more precisely, we consider the change of variablee

Ee7−!Φˆλ(E) := (Te ₀^k)⁻¹

T₀^k(E)+e 1 NX^k

1,ˆλ(E)e

so that

−

S₀^k(E)−N^−ζ(S₀^k)⁰(E)

S^k₀(E)−N^−ζ(S^k₀)⁰(E)

h

EedEe

=−

S₀^k(E)−N^−ζ(S₀^k)⁰(E)

S^k₀(E)−N^−ζ(S^k₀)⁰(E)

X

(i1,...,im)∈J_λ,fˆE

f(N(T₀^k(λ^k_i

1)−T₀^k(E))+[(Xe _1,1^N )^k_i

1(ˆλ)−X^k

1,λˆ(E)e , ..., N(T₀^k(λ^k_im)−T₀^k(E))+[(Xe _1,1^N )^k_im(ˆλ)−X_1,^k_λˆ(E)])∂e

EeΦλˆ(E)e dE.e Therefore, since∂

EeΦ_λˆ(E)=1+O Ne ^ϑ−1

(due to (5.20)),|h

Ee|6CN^mϑ, and the interval [S₀^k(E)−N^−ζ(S₀^k)⁰(E), S₀^k(E)−N^−ζ(S₀^k)⁰(E)] has length of orderN^−ζ, we deduce that

−

S₀^k(E)−N^−ζ(S₀^k)⁰(E)

S^k₀(E)−N^−ζ(S^k₀)⁰(E)

h

EedEe

=−

S₀^k(E)−N^−ζ(S₀^k)⁰(E)

S^k₀(E)−N^−ζ(S^k₀)⁰(E)

X

(i1,...,im)∈J_λ,fˆE

f(N(T₀^k(λ^k_i1)−T₀^k(E))+[(Xe _1,1^N )^k_i1(ˆλ)−X_1,^k_λˆ(E)],e ..., N(T₀^k(λ^k_im)−T₀^k(E))+[(Xe _1,1^N )^k_im(ˆλ)−X_1,^k_ˆλ(E)])e dEe+O(N^ζN^mϑN^ϑ−1).

(5.22) We now observe that, since T₀^k:R!R is a diffeomorphism with (T₀^k)⁰>e^−L>0 (see (5.14)), it follows by (5.20) that

|(X_1,1^N )^k_i1(ˆλ)−X_1,^k_λˆ(E)|e 6CN^ϑ|T₀^k(λ^k_i)−T₀^k(E)|.e Z

Z

Z Z

Z Z

Therefore, sincef is compactly supported, we see that the expression f(N(T₀^k(λ^k_i1)−T₀^k(E))+[(Xe _1,1^N )^k_i1(ˆλ)−X_1,^k_λˆ(E)],e

..., N(T₀^k(λ^k_i

m)−T₀^k(E))+[(Xe _1,1^N )^k_i

m(ˆλ)−X^k

1,λˆ(E)])e is non-zero only if

|T₀^k(λ^k_ij)−T₀^k(E)|e 6C1

N for allj= 1, ..., m.

In particular, using again that (T₀^k)⁰>e^−L>0, this implies that|λ^k_ij−E|e 6C/N. Thus

|T₀^k(λ^k_ij)−T₀^k(E)−(Te ₀^k)⁰(E)[λ^k_ij−E]|e =O 1

N²

and

N^ϑ|T₀^k(λ^k_ij)−T₀^k(E)|e =O N^ϑ−1 , and we get

f(N(T₀^k(λ^k_i

1)−T₀^k(E)e

+[(X_1,1^N )^k_i

1(ˆλ)−X^k

1,λˆ(E)],e

..., N(T₀^k(λ^k_im)−T₀^k(E))+[(Xe _1,1^N )^k_im(ˆλ)−X_1,^k_λˆ(E)])e

=f((T₀^k)⁰(E)N(λ^k_ij−E), ...,e (T₀^k)⁰(E)N(λ^k_ij−E))+O(k∇fe k_∞N^ϑ−1).

Combining this estimate with (5.22) and the fact that #Jλ,ˆEe6CN^mϑwe conclude that

−

S^k₀(E)−N^−ζ(S₀^k)⁰(E)

S^k₀(E)−N^−ζ(S^k₀)⁰(E)

hEedEe= ¯gE+O N^{(m+1)ϑ+ζ−1} , where

¯ gE(ˆλ)

:=−

S₀^k(E)−N^−ζ(S₀^k)⁰(E)

S^k₀(E)−N^−ζ(S^k₀)⁰(E)

X

(i₁,...,i_m)∈Jλ,fˆE

f((T₀^k)⁰(E)N(λ^k_ij−E), ...,e (T₀^k)⁰(E)N(λ^k_ij−E))e dE.e

Also, by the argument above it follows that we can add back into the sum all the in-dices outsideJλ,ˆEe (since, up to infinitesimal errors, the function above vanishes on such indices), and therefore

−

S^k₀(E)−N^−ζ(S^k₀)⁰(E)

S₀^k(E)−N^−ζ(S^k₀)⁰(E)

h

EedEe= ¯¯g_E+O(N^{(m+1)ϑ+ζ−1}), Z

Z

Z

with Combining this bound with (5.15), (5.16), (5.18), (5.19), and (5.21), we conclude that

|log(1+Ak)−log(1+ ¯A¯k)|6C(N^mϑ−ζ+N^(m+2)ϑ−1+N^{(m+1)ϑ+ζ−1}), (5.23) Combining this estimate with (5.23), we get

|log(1+Ak)−log(1+ ˆAk)|6C(N^mϑ−ζ+N^(m+2)ϑ−1+N^{(m+1)ϑ+ζ−1}).

Choosingϑsmall enough so that (m+2)ϑ<θ, this gives

|log(1+Ak)−log(1+ ˆAk)|6C(N^θ+ζ−1+N^θ−1/2+N^θ−ζ)6C(N^θ+ζ−1+N^θ−ζ), and since ˆAk is uniformly bounded inN (see for instance [65]) and the right-hand side is infinitesimal (recall thatθ<min{ζ,1−ζ}), we conclude that

|Ak−Aˆk|6C(N^θ+ζ−1+N^θ−ζ).

Recalling the definition ofA_k and ˆA_k, this proves that

which corresponds to our statement when f depends only on the eigenvalues of one matrix. As explained at the beginning of the proof, the very same argument presented above extends also to the general case.

Z

Z Z

Z Z

Z Z

Z

Proof of Corollary 2.9. We begin by noticing that the proof of Theorem 2.5 could be repeated verbatim in the context of [5] to show that [5, Theorem 1.4] holds with the same estimates as we obtained here.

To prove the gap estimates, it is enough to show that the approximate transport maps do not change gaps in the bulk uniformly (away from the edges). Due to Theo-rem2.5and [5, Theorem 1.4], we have the expansions

(T^N)^k_i(ˆλ) =T₀^k(λ^k_i)+ 1

N(X_1,1^N )^k_i(ˆλ)+ 1

N²(X_2,1^N )^k_i(ˆλ), (S_k^N)i(λ^k) =S₀^k(λ^k_i)+ 1

N(Sk,1)i(λ^k)+ 1

N²(Sk,2)i(λ^k),

where (S_k,1)_i and (S_k,2)_i satisfy the same estimates as (X₁^N)^k_i and (X₂^N)^k_i. Hence, by the formulas above we deduce that

(T^N)^k_i S₁^N(λ¹), ..., S_d^N(λ^d)

=T₀^kS₀^k(λ^k_i)+ 1

N[(T₀^k)⁰S₀^k(λ^k_i)](Sk,1)i(λ^k) +1

N(X_1,1^N )^k_i

S₀¹(λ¹₁)+ 1

N(S1,1)1(λ¹), ..., S₀^d(λ^d_N)+ 1

N(SN,d)N(λ^d)

+Ei,

(5.24)

where the errorEi satisfies (due to the bounds in Theorem2.5and [5, Theorem 1.4]) s

X

i

kEik²_L2(P_GVE,β^N )=O

(logN)² N^3/2

(5.25)

Also, by using again Theorem2.5 and [5, Theorem 1.4], with probability greater than 1−e^−c(logN)2 and uniformly with respect toi∈{1, ..., N}, we have

|[(T₀^k)⁰S₀^k(λ^k_i+1)](Sk,1)i+1(λ^k)−[(T₀^k)⁰S₀^k(λ^k_i)](Sk,1)i(λ^k)|

6C(logN)N^1/(σ−15)|λ^k_i+1−λ^k_i|, and

|(X_1,1^N )^k_i+1−(X_1,1^N )^k_i|

(S₀¹)^⊗N+ 1

NS_1,1, ...,(S₀^d)^⊗N+ 1 NS_d,1

(ˆλ) 6C(logN)N^1/(σ−15)

|S₀^k(λ^k_i+1)−S₀^k(λ^k_i)|+1

N|(Sk,1)i+1(λ^k)−(Sk,1)i(λ^k)|

6ClogN N^1/(σ−15)|λ^k_i+1−λ^k_i|, while

T₀^kS₀^k(λ^k_i+1)−T₀^kS₀^k(λ^k_i) = (T₀^kS₀^k)⁰(λ^k_i)[λ^k_i+1−λ^k_i]+O(|λ^k_i+1−λ^k_i|²).

Recalling that, with probability greater than 1−e^−N¯c, |λ^k_i+1−λ^k_i|6CN^θ−1 when the {λ^k_i}^N_i=1 are ordered andi∈[εN,(1−ε)N] (see (5.2) and (5.4)), we conclude that, with probability greater than 1−e^−c(logN)2, and uniformly with respect to i∈[εN,(1−ε)N], we have

[(T^N)^k_i+1−(T^N)^k_i](S₁^N(λ¹), ..., S_d^N(λ^d))

= (T₀^kS₀^k)⁰(λ^k_i)[λ^k_i+1−λ^k_i]+O

(logN)N^1/(σ−15) N^2−θ

. Combining this estimate with (5.25) and noticing that

N^4/3

(logN)N^2/(σ−15)

N^2−θ +(logN)² N^3/2

!0 as N!∞,

providedθ<¹₆ (recall that by assumptionσ>36; see Hypothesis2.1), the two statements follow from the fact thatT^N(S₁^N, ... S^N_d ):R^dN!R^dN is an approximate transport map from (P_GVE,β^N )^⊗d to P_β^N,aV and that the results are true underP_GVE,β^N due to [6, Theo-rem 1.3 and Corollary 1.5].

In document Universality in several-matrix models via approximate transport maps (Stránka 50-63)