Proof of Proposition 6.2 and Lemma 6.3

We first show that the free energy is a function of the correlators, and then that the correlators only depend on{L^N_i (x<sup>`</sup>)}<sub>`>0,16i6d</sub> andτ_B^N. Finally, we deduce the largeN expansion of the free energy as well as its smoothness.

6.2.1. The free energy in terms of the correlators Recalling the definition of free energy, (6.5), and (6.4), we have

F_β^N,aV(A1, ..., Ad, B1, ..., Bm) := logI_β^N,aV

=

a

0

d

dulogI_β^N,uVdu

=N²

a

0

1

N^rTr^⊗rV dQ^N,uV_β du

=N^2−r

a

0

(W_1N^uV)^⊗r(V)du+r(r−1)N^2−r

a

0

W_2N^uV⊗(W_1N^uV)^r−2(V)du+RN, Z

Z Z

Z Z

whereRN has terms either with two cumulants of order 2, or a cumulant of order greater or equal to 3. By Lemma6.7(note that it applies uniformly inu∈[−a0, a0], for somea0

universally small), this latter term is at most of order 1/N, and is therefore negligible.

Moreover, using Corollary6.8and Lemmas6.12and 6.13, we find that F_β^N,aV(A₁, ..., A_d, B₁, ..., B_m) =N²

a

0

f₀^udu+N

a

0

f₁^udu+

a

0

f₂^udu+O 1

N

, with

f₀^u:= (τ₁₀^uV)^⊗r(V),

f₁^u:=rτ₁₁^uV⊗(τ₁₀^uV)^⊗(r−1)(V), f₂^u:=r(r−1)

(τ₁₁^uV)^⊗2+τ₂₀^uV

⊗(τ₁₀^uV)^⊗(r−2)(V)+rτ₁₂^uV⊗(τ₁₀^uV)^⊗(r−1)(V),

(6.29)

where we have used thatV is symmetric and such thatkVkξ₁,ζis finite forξ1big enough, so thatδξ₁,ζ(uV)<(1+max{2, r})⁻¹ providedu∈[−a0, a0] witha0 sufficiently small. In particular this implies that, fora0small enough and any 1<ξ4<ξ3<ξ2<ξ1,

δ_ξi,ζ(uV)<(1+max{2, r})⁻¹ for allu∈[−a₀, a₀], so that the previous lemmas apply. Hence, we deduce the following result.

Lemma 6.14. Let kVkξ₁,ζ₁ be finite for some ξ1 large enough and let ζ1>1. Then, there existsa0>0so that,fora∈[−a0, a0],and uniformly on Hermitian matrices{Ai}^d_i=1 and {Bi}16i6m whose operator norm is bounded by 1,we have

F_β^N,aV(A1, ..., Ad, B1, ..., Bm) =

2

X

l=0

N^2−lF_l^a+O 1

N

, with

F_l^a=

a

0

f_l^udu andf_l^u given by (6.29).

6.2.2. The correlators as functions of{L^N_i }^d_i=1 and τ_B^N Let us define the space

P:={Q(u1a1u⁻¹₁ , ..., udadu⁻¹_d , b1, ..., bk) :Q∈Chx1, ..., xd, b1, ..., bmi}.

As the functionsF_l^a only depend on the restriction toP ofτ₁₀^uV,τ₁₁^uV,τ₁₂^uV, andτ₂₀^uV for u∈[−a, a], we shall first prove that the latter only depend on

MA,B:=

1 N

N

X

j=1

(aⁱ_j)<sup>`</sup>:`>0 and 16i6d

∪{τ_B^N}.

Z Z Z

Z

• The restriction τ₀₁^aV|_P depends only on MA,B. We start by showing thatτ₀₁^aV can be defined inductively, as is the case when V=0, since it depends analytically on the potentialV in the following sense.

Lemma6.15. Let p∈L and V be a potential such that,for some ξ >1 and ζ>1, δ_ξ,ζ(V)< 1

1+max{2, r}. Then, for all a∈[−1,1],the solution τ₁₀^aV of

τ⊗τ(∂ip)+a(1+1β=1)τ(Di,τV p) = 0 (6.30) is uniquely defined. Moreover,we have the decomposition

τ₁₀^aV =X

n>0

aⁿτ_n^V,

with τ_n^V∈Lξ,ζ satisfying kτ_n^Vkξ,ζ6CnDⁿ, where {Cn}n>0 denote the Catalan numbers and D is a positive constant.

Proof. This result can be seen to be a consequence of the implicit function theorem.

However we will soon need additional information on theτ_n^V, and therefore give a proof

“by hand”.

By uniqueness of solutions, it is enough to show that there exists a solution of (6.30), or more precisely of (6.13), which is analytic ina. Let us therefore look for such a solution and writeτ^aV(p):=P

n>0aⁿτ_n^V(p). We then find thatτ^aV satisfies (6.13) if and only if τ_n^V(p)+ X

k∈{0,n}

τ_k^V⊗τ_n−k^V (Π ¯∆p)

=−

n−1

X

k=1

τ_k^V⊗τ_n−k^V (Π ¯∆p)

−X

hV, q1⊗...⊗qri

d

X

i=1 r

X

`=1

X

P

ik_i=n−1

Y

j6=`

τ_k^V_j(qj)

τ_k^V<sub>`</sub>(Diq`·DiD⁻¹p)

(6.31)

As ¯∆ splits monomialspinto simple tensorsq1⊗q2each of whose factors has degree strictly smaller than that ofp, we see that there exists a unique solution to this equation.

Moreover, we prove by induction that there exists finite constantD>0 such that, ifCn

denote the Catalan numbers, then

kτ_n^Vkξ,ζ6C_nDⁿ.

Indeed, for n=0, we simply have the law of free variables bounded by 1, so that the result is clear. Using the inductive hypothesis until n−1 to bound the right-hand side in (6.31), and (6.18) to bound the second term in the left-hand side of (6.31), we deduce that

(1−δξ,ζ(V))kτ_n^Vkξ,ζ6 8 ξ−1Dⁿ

n−1

X

k=1

CkC_n−k+Dⁿ⁻¹X

|hV, q1⊗...⊗qri|

× ^r

X

i=1

degqi

ζ^{Pri=1degA,B(qi)}ξ^Pri=1degUqi X

Pr

i=1ki6n−1 r

Y

i=1

Ck_i.

Using the fact thatPn

k=0CkC_n−k=Cn+164Cn, we find recursively that X

Pr

i=1k_i6n−1 r

Y

i=1

C_ki6C_n+r−164^r−1C_n.

Thus we can bound the last term by 4^r−1CnDⁿ⁻¹

|V|

_ξ, which implies that kτ_n^Vkξ,ζ6CnDⁿ

providedD is chosen sufficiently large. As Cn64ⁿ, this implies thatτ^aV=P

n>0aⁿτ_n^V is absolutely converging provided|a|<1/4D and it satisfies (6.30), so we get τ^aV=τ₀₁^aV as desired.

We finally show thatτ_n^V|_P only depends onMA,B. Again, we can argue by induction.

As already mentioned, this is clear whenn=0 asτ₀^V is the law of free variables. Also, if p∈P and deg(p)=0 thenpdepends only onb1, ..., bk, and thereforeτ_n^V(p) only depends onτ_B^N for alln>0. Thus, by the inductive hypothesis, we can assume that the result is true forτ_k^V(p) whenk6n−1 andp∈P, and forτ_n^V(p) whenp∈P and deg(p)6`.

To show that this property propagates we shall use the fact that (6.31) can be seen as an induction relation where all monomials belong to P. To this end, first note that {τ_n^V}n>0 are tracial, that is

τ_n^V(pq) =τ_n^V(qp) for allp, q∈ P.

Indeed this property is clear as it is satisfied byτ^aV, and{τ_n^V}n>0are derivatives ofτ^aV with respect toa.

Next, observe thatD⁻¹ keepsP stable. Moreover, if p=Q({uiaiu⁻¹_i }^m_i=1), where Q is a monomial, then

Dip= X

Q=q₁x_iq₂

(aiu⁻¹_i q2q1ui−u⁻¹_i q2q1uiai),

so that, up to cyclic symmetry,Dip·Diq∈P for eachiandq⊂P. (Here and in the sequel, cyclic symmetry is just the action of exchangingpqintoqp.) We also show that ¯∆ maps P into P ⊗P up to cyclic symmetry. Indeed, it follows from (6.12) that, forp∈P,

∆¯ip= X

p=p1uiaiu⁻¹_i p2

X

aiu⁻¹_i p2p1ui=aiu⁻¹_i q1uiaiu⁻¹_i q2ui

aiu⁻¹_i q1ui⊗aiu⁻¹_i q2ui

− X

a_iu⁻¹_i p₂p₁u_i=a_iu⁻¹_i q₁u_ia_iu⁻¹_i q₂u_i

(aiu⁻¹_i q1uiai⊗q2−ai⊗p2p1−p2p1⊗ai)

− X

u⁻¹_i p₂p₁u_ia_i=u⁻¹_i q₁u_ia_iu⁻¹_i q₂u_ia_i

q₁⊗aiu⁻¹_i q₂u_ia_i

+ X

u⁻¹_i p2p1uiai=u⁻¹_i q1uiaiu⁻¹_i q2uiai

u⁻¹_i q₁u_ia_i⊗u⁻¹_i q₂u_ia_i

,

so that, up to cyclic symmetry, ¯∆ip∈P ⊗P for alli∈{1, ..., m}andp∈P.

Hence, by induction we see thatτ_n^V restricted toP only depends on the restriction of{τ_k^V}k6n−1 toP, therefore only on the restriction of τ₀^V to P. Since we have already seen thatτ₀^V|P only depends onM_A,B, the conclusion follows.

• τ₁₁^aV depends only on M_A,B. A direct inspection shows that ˜∆ mapsP intoP up to cyclic symmetry. Indeed, ˜∆=P

i∆˜_i with

∆˜_ip= X

p=p1uiaiu⁻¹_i p2

X

aiu⁻¹_i p2p1ui=aiu⁻¹_i q1uiaiu⁻¹_i q2ui

u⁻¹_i q₂^∗u_ia²_iu⁻¹_i q₁u_i

− X

aiu⁻¹_i p2p1ui=aiu⁻¹_i q1uiaiu⁻¹_i q2ui

(u⁻¹_i q₂^∗uiaiu⁻¹_i q1uiai−u⁻¹_i p^∗₁p^∗₂uiai−aiu⁻¹_i p2p1ui)

− X

u⁻¹_i p₂p₁u_ia_i=u⁻¹_i q₁u_ia_iu⁻¹_i q₂u_ia_i

aiu⁻¹_i q₂^∗uiaiu⁻¹_i q1ui

+ X

u⁻¹_i p₂p₁u_ia_i=u⁻¹_i q₁u_ia_iu⁻¹_i q₂u_ia_i

a_iu⁻¹_i q₂^∗q₁u_ia_i

.

Moreover, the previous considerations showed that Ψ^aV_τ

10 mapsP intoP forasmall, and therefore

τ₁₁^aV(p) =1β=1τ₁₀^aV( ˜∆(Ψ^aV_τ10)⁻¹(p))

only depends on τ₁₀^aV|P. Since we just checked that the latter only depends on MA,B, this proves the result.

• τ₂₀^aV depends only on MA,B. By Lemma6.12,

τ₂₀^aV(Ψ^aV_τ10p, q) =−(1+1β=1)τ₁₀^aV(P^q_τaV

10

p),

and recalling thatτ₁₀^aV expands in a convergent series ina, we see that so doesτ₂₀^aV. We only need to check that the operators which appear in the equation defining τ₂₀^aV keep P stable. But we have already seen that both the operators ¯∆ andP^V keep P stable, and henceτ₂₀^aV(p, q) only depends onMA,B and it is in fact a convergent series in such elements.

• τ₁₂^aV depends only on MA,B. By Lemma6.13, τ₁₂^aV(Ψ^aV_τ

10p) =τ₁₁^aV( ˜∆p)−[τ₂₀^aV+τ₁₁^aV⊗τ₁₁^aV]( ¯∆p+S^aVβp),

from which we see thatτ₁₂^aV(p) is a convergent series in a(recall that we already proved thatτ₁₀^aV(p), τ₁₁^aV(p) andτ₂₀^aV(p) are convergent series ina). So the main point is to prove that, up to cyclic symmetry, ¯∆p+S^aVβp∈P ⊗P wheneverp∈P.

We already proved that this is the case for ¯∆p, so we focus onS^aVβp. We notice that it is the sum of two parts. One part is linear over tensors of two monomials appearing in the decomposition of aV, and as aV∈P^⊗r this part clearly belongs to P^⊗2. The other part is linear over tensors of one monomial appearing in the decomposition ofaV (which therefore belongs toP) andDip·Diqj withqj appearing in the decomposition of aV (which we have seen belongs to P up to cyclic symmetry). Hence also this second part satisfies the desired property, which concludes the proof.

6.2.3. Smoothness of the functionsF₂, F₁, and F₀

By Lemma6.14and the discussion in the previous subsection, we know that

F_β^N,aV =

2

X

l=0

N^2−lF_l^a(L^N₁, ..., L^N_d, τ_B^N)+O 1

N

,

where the functionalsF₀^a,F₁^aandF₂^adepend on{L^N_i }^d_i=1and onτ_B^N through the asymp-totic correlators{τ_1g^uV}²_g=0 andτ₂₀^uV. We finally prove that they are smooth functions of these measures.

Recall the notation introduced in (6.3). We show the following.

Lemma 6.16. There exists ξ0>1 large enough such that the following holds: let V have finite k · kξ,ζ norm for some ξ >ξ0 and ζ>1. Then there exists a0>0 such that, for all a∈[−a0, a0], F_l^a is Fr´echet differentiable ` times for all `∈N, and if νj= (ν₁^j, ..., ν_d^j, τj)∈P([−1,1])^d×T(B), we have

|D<sup>`</sup>F<sub>l</sub><sup>a</sup>(L<sup>N</sup><sub>1</sub> , ..., L<sup>N</sup><sub>d</sub>, τ<sub>B</sub><sup>N</sup>)[ν<sub>1</sub>, ..., ν<sub>`</sub>]|6C<sub>`</sub>|a| kν1kζ...kν`kζ.

Moreover,the derivativeDkF₀^a(L^N₁, ..., L^N_d, τ_B^N)=DF₀^a(L^N₁, ..., L^N_d , τ_B^N)[0, ...,0, δx,0, ...,0]

of F in the direction of the measure L^N_k is a function on the real line with finite k · kζ

norm for any k∈{1, ..., d}. As a consequence,it is of class C^∞ in an open neighborhood of [−1,1].

Proof. First, fix ξ₀ sufficiently large so that all previous results apply. By the previous section it is enough to show that {τ_1g^uV}²_g=0 and τ₂₀^uV depend smoothly on ({L^N_i }^d_i=1, τ_B^N), uniformly with respect to u∈[−a, a]. Indeed, by (6.29), F₀^a is the in-tegral of (τ₁₀^uV)^⊗r(V) over u∈[0, a]. We have seen in Lemma 6.8 that τ_AB^N 7!τ^uV,τ

N AB

01 is

` times Fr´echet differentiable. Moreover, we have also seen that, once restricted to P, it depends only on {L^N_i }^d_i=1 and τ_B^N, and not the full distribution τ_AB^N . As a conse-quence, the smoothness ofτ^uV,τ

N AB

01 as a function of τ_AB^N reduces to the smoothness as a function of the probability measures {L^N_i }^d_i=1 andτ_B^N. The fact that DF₀^a is C^∞ is a direct consequence of formulas (6.25) and (6.26). For instance, if we denote by Dk the derivative along L^N_k , and Π⁰_k is the projection onto the algebra generated by {ak}, for anyp∈Lξ,ζ∩P we have

D_kτ_0,1^V,τ1[p] =−Π⁰_k[ T_τΠ+τ1Π⁰+P^V_τΠ+τ^β

1Π⁰+ Q^V_τΠ+τ^β

1Π⁰](Id+Ξ^V

τ₀₁^V,τ1)⁻¹p∈ P, (6.32) where we use the fact (see Lemma6.2) that

[ T_τΠ+τ1Π0+P^V_τΠ+τ^β

1Π⁰+ Q^V_τΠ+τ^β

1Π⁰](Id+Ξ^V

τ₀₁^V,τ1)⁻¹(P)⊂ P

so that once we project it onA B we get only polynomials either in thea_i or in theb_i, and hence differentiating in the direction ofL^N_k we only keep those ina_k.

The same argument holds forF₁^u andF₀^u, since alsoτ₁₀^uV,τ₁₁^uV, andτ₂₀^uV are smooth and only depend on{L^N_i }^d_i=1 andτ_B^N.

7. Law of polynomials of random matrices

In document Universality in several-matrix models via approximate transport maps (Stránka 82-88)