JointworkwithV.JonesandD.Shlyakhtenko. Algebra,GeometryandPhysicsBonn/Berlinseminar Alice Guionnet Topologicalexpansions,Randommatricesandoperatoralgebras

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Topological expansions, Random matrices and operator algebras

AliceGuionnet

CNRS & ENS Lyon

Algebra, Geometry and Physics Bonn/Berlin seminar

Joint work with V. Jones and D. Shlyakhtenko.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

What is in common between

And

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

What is in common between

And

And

What is in common between

And

And

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Outline

Maps

Random Matrices and the enumeration of maps SD equations

Loop models Subfactors theory Transport

Topological expansions, Random matrices and operator algebras

Maps

Random Matrices and the enumeration of maps SD equations

Loop models Subfactors theory Transport

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

What is a map ?

A map is a connected graph which is properly embedded into a surface, that is embedded in such a way that its edges do not cross and the faces (obtained by cutting the surface along the edges of the graph) are homeomorphic to disks. The genus of a map is the genus of such a surface.

By Euler formula, 2−2g = #{vertices}

+#{faces} −#{edges}.

= 2 + 3−3

2 1 3

What is a map ?

Maps are connected graphs which are properly embedded into a surface, that is embedded in such a way that its edges do not cross and the faces (obtained by cutting the surface along the edges of the graph) are homeomorphic to disks. The genus of a map is the genus of such a surface.

By Euler formula, 2−2g = #{vertices}

+#{faces} −#{edges}.

= 2 + 1−3

1

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Enumeration of maps

Being given vertices with given valence, how many maps with genusg can we build ?

Recipe :

•Draw labeled vertices with labeled half-edges on a surface of genusg,

•Match the end points of these half-edges,

•Check the resulting map is properly embedded and could not be properly embedded on a surface with genus smaller than g,

•Count such matchings (which are the same only if matched labelled half-edges are the same).

8 1

2

4 3 5 6

8 7 1

2

3 4 5 6

7

Enumeration of maps

Being given vertices with given valence, how many maps with genusg can we build ?

Recipe :

•Draw labeled vertices with labeled half-edges on a surface of genusg,

•Match the end points of these half-edges,

•Check the resulting map is properly embedded and could not be properly embedded on a surface with genus smaller than g,

•Count such matchings (which are the same only if matched labelled half-edges are the same).

8 1

2

4 3 5 6

8 7 1

2

3 4 5 6

7

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Enumeration of maps

Being given vertices with given valence, how many maps with genusg can we build ?

Recipe :

•Draw vertices with labeled half-edges on a surface of genus g,

•Match the end points of these half-edges,

•Check the resulting map is properly embedded and could not be properly embedded on a surface with smaller genus,

•Count such matchings (which are the same only if matched labelled half-edges are the same).

5 6

8 7

1 2 3 4 5

6 7

8 1

2 3 4

Topological expansions, Random matrices and operator algebras

Maps

Random Matrices and the enumeration of maps SD equations

Loop models Subfactors theory Transport

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

The law of the GUE and the enumeration of maps

LetX^N be a matrix following theGaussian UnitaryEnsemble, that is aN×N Hermitian matrix with i.i.d centered complex Gaussian entries with covarianceN⁻¹, that is

dP(X^N) = 1

Z^N exp{−N

2Tr((X^N)²)}dX^N

Theorem (Harer-Zagier 86) For all p∈N

Z 1

NTr((X^N)^2p)dP(X^N) =X

g≥0

N^−2gM(2p;g).

equalsPN n=1

N n

(2p−1)!!2ⁿ⁻¹ p

n−1

.M(2p;g) denotes the number of maps with genus g build over a vertex of valence2p.

The law of the GUE and the enumeration of maps

LetX^N be a matrix following theGaussian UnitaryEnsemble, that is aN×N Hermitian matrix with i.i.d centered complex Gaussian entries with covarianceN⁻¹, that is

dP(X^N) = 1

Z^N exp{−N

2Tr((X^N)²)}dX^N

Theorem (Harer-Zagier 86) For all p∈N

Z 1

NTr((X^N)^2p)dP(X^N) =X

g≥0

N^−2gM(2p;g).

equalsPN n=1

N n

(2p−1)!!2ⁿ⁻¹ p

n−1

.M(2p;g) denotes the number of maps with genus g build over a vertex of valence2p.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Proof “ Feynman diagrams”

E[1

NTr((X^N)^p)] = 1 N

N

X

i(1),...,i(p)=1

E[X_i(1)i^N ₍₂₎X_i(2)i(3)^N · · ·X_i(p)i(1)^N ] Wick formula : If (G1,· · ·,G2n) is a centered Gaussian vector,

E[G1G2· · ·G2n] = X

1≤s1<s2..<sn≤2n ri>si

n

Y

j=1

E[Gs_jGr_j].

Example : IfG_i =G follows the standard Gaussian distribution E[G^p] = #{pair partitions of p points}

Proof “ Feynman diagrams”

E[Tr(X^N)^p] =

N

X

i(1),...,i(p)=1

E[X_i(1)i(2)^N X_i^N_(2)i(3)· · ·X_i^N_(p)i(1)]

E[X_i^N_(1)i(2)· · ·X_i^N_(p)i(1)] =

i(1) i(1) i(2)

i(2)

i(3)

i(3) i(4)

i(4) i(5) i(5) i(6)

i(6)

AsE[X_ij^NX_k^N<sub>`</sub>] =N<sup>−1</sup>1ij=`k, only matchings so that indices are constant along the boundary of the faces contribute.

E[Tr((X^N)^p)] = X graph 1 vertex

degree p

N^#faces^−p/2

= X

N^−2g+1M((x^p,1);g) by Euler formula

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Random matrices and the enumeration of maps

’t Hooft 74’ and Br´ezin-Itzykson-Parisi-Zuber 78’

Lett= (ti)1≤i≤n∈Rⁿ and setVt=Pn

i=1tixⁱ.Formally, 1

N²log Z

e^Ntr(Vt(X^N))dP(X^N)

= X

k1,..,kn∈N

X

g≥0

N^−2g

n

Y

j=1

(t_j)^kj

k_j! M((k_i)1≤i≤n;g) with

M((k_i)1≤i≤n;g) =]{maps of genus g with k_i vertices of degree i}

5 6

8 7

1 2 3 4 5

6 7

8 1

2 3 4

Enumeration of colored maps

Consider vertices with colored half-edges and enumerate maps build by matching half-edges of the same color.

Such vertices are in bijection with monomials:

toq(X1, . . . ,Xd) =Xi1Xi2· · ·Xip associate a “star of typeq” given by the vertex withp drawn on the plan so that the first half-edge has colori₁, the second colori₂ etc until the last which has colori_p. M((qi,ki)1≤i≤m,g) denotes the number of maps with genusg build onk_i stars of typeq_i,1≤i ≤m.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Random matrices and the enumeration of maps

’t Hooft (1974) and Br´ezin-Itzykson-Parisi-Zuber (1978) Let (q₁, . . . ,q_n) be monomials. Let t= (t_i)1≤i≤n∈Rⁿ and set Vt(X1, . . . ,Xm) =Pn

i=1tiqi(X1, . . . ,Xm).Formally, F_V^N_t = 1

N² log Z

e^Ntr(Vt(A1,· · ·,Am))dP^N(A1)· · ·dP^N(Am)

= X

k1,..,kn∈N

X

g≥0

N^−2g

n

Y

j=1

(tj)^kj

k_j! M((q_i,k_i)1≤i≤n,g) with

M((q_i,k_i)1≤i≤n,g) =]{maps of genusg with k_i vertices of type q_i} where maps are constructing by matching half-edges of the same color.

Example : The Ising model on random graphs

Takeq1(X1,X2) =X1X2,q2(X1,X2) =X₁⁴,q3(X1,X2) =X₂⁴ represented by

Then, 1 N²log

Z

e^{NTr(P3i=1tiqi(X1N,X2N))}dP(X₁^N)dP(X₂^N) is a generating function for the enumeration of the the Ising model on random

graphs. Solved by Mehta (1986).

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Random matrices, maps and tracial states

’t Hooft 74’ and Br´ezin-Itzykson-Parisi-Zuber 78’

Let (q1,· · · ,qn) be monomials,Vt=Pn

i=1t_iq_i and put

dPVt(X₁^N,· · ·,X_m^N) =e^{−N2FVNt+NTr}(^Vt(X1^N,···,X_m^N))dP(X₁^N)· · ·dP(X_m^N) Formally, for any monomialP

τ_t^N(P) :=

Z 1 NTr

P(X₁^N, . . . ,X_m^N)

dPVt(X₁^N, . . . ,X_m^N)

= ∂_sF_V^N

t+sP/N²|_s=0

= X

g≥0

N^−2g X

k1,..,kn∈N n

Y

j=1

(t_j)^kj

kj! M((P,1),(qi,ki)1≤i≤n;g)

τ_t^N is a tracial state :

τ_t^N(PP^∗)≥0, τ_t^N(1) = 1, τ_t^N(PQ) =τ_t^N(QP).

Random matrices, maps and tracial states

’t Hooft 74’ and Br´ezin-Itzykson-Parisi-Zuber 78’

Let (q1,· · · ,qn) be monomials,Vt=Pn

i=1t_iq_i and put

dPVt(X₁^N,· · ·,X_m^N) =e^{−N2FVNt+NTr}(^Vt(X1^N,···,X_m^N))dP(X₁^N)· · ·dP(X_m^N) Formally, for any monomialP

τ_t^N(P) :=

Z 1 NTr

P(X₁^N, . . . ,X_m^N)

dPVt(X₁^N, . . . ,X_m^N)

= ∂_sF_V^N

t+sP/N²|_s=0

= X

g≥0

N^−2g X

k1,..,kn∈N n

Y

j=1

(t_j)^kj

kj! M((P,1),(qi,ki)1≤i≤n;g) τ_t^N is a tracial state :

τ_t^N(PP^∗)≥0, τ_t^N(1) = 1, τ_t^N(PQ) =τ_t^N(QP).

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

What is a non-commutative law ?

What is a classical law onR^d? It is anon-negative linear map

Q :f ∈ C_b(R^d,R)→Q(f) = Z

f(x)dQ(x)∈R, Q(1) = 1

Anon-commutative law τ ofn self-adjoint variables is alinear map τ :P ∈ChX₁,· · ·,Xdi →τ(P)∈C

It should satisfy

• τ(PP^∗)≥0 for all P, (zXi1· · ·Xik)^∗ = ¯zXik· · ·Xi1.

• τ(1) = 1

• τ(PQ) =τ(QP) for all P,Q∈ChX₁,· · · ,X_di.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

What is a non-commutative law ?

What is a classical law onR^d? It is anon-negative linear map

Q :f ∈ C_b(R^d,R)→Q(f) = Z

f(x)dQ(x)∈R, Q(1) = 1

τ :P ∈ChX₁,· · ·,Xdi →τ(P)∈C It should satisfy

• τ(PP^∗)≥0 for all P, (zXi1· · ·Xik)^∗ = ¯zXik· · ·Xi1.

• τ(1) = 1

• τ(PQ) =τ(QP) for all P,Q∈ChX₁,· · · ,X_di.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

What is a non-commutative law ?

What is a classical law onR^d? It is anon-negative linear map

Q :f ∈ C_b(R^d,R)→Q(f) = Z

f(x)dQ(x)∈R, Q(1) = 1

Anon-commutative lawτ ofn self-adjoint variables is alinear map τ :P ∈ChX₁,· · ·,Xdi →τ(P)∈C

It should satisfy

• τ(PP^∗)≥0 for all P, (zXi1· · ·Xi_k)^∗ = ¯zXi_k· · ·Xi1.

• τ(1) = 1

• τ(PQ) =τ(QP) for allP,Q∈ChX₁,· · · ,X_di.

The law of free semicircle variables

TakeX₁^N,· · · ,X_d^N be independent GUE matrices, that is

P

dX₁^N,· · · ,dX_d^N

= 1

(Z^N)^d exp{−N 2Tr(

d

X

i=1

(X_i^N)²)}Y dX_i^N.

Theorem (Voiculescu(91))

For any polynomial P∈ChX₁,· · · ,Xdi

N→∞lim E[1

NTr(P(X₁^N,· · ·,X_d^N))] =σ(P) σ is the law of d free semicircle variables.

If P = Xi1Xi2· · ·Xik, σ(P) is the number of planar maps build over a star of type P.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

From formal to asymptotic topological expansions

Form∈N and (q1,· · · ,qn) monomials, Vt =Pn

i=1tiqi,M >2 dP^MVt(X₁^N,· · ·,X_m^N) = 1_kXN

i k≤M

Z_V^N,M e^NTr(^Vt(X₁^N,...,X_m^N))dP(X₁^N)· · ·dP(X_m^N) ForM >2, allK ∈N,t_i small enough so thatV_t=V_t^∗,for any monomialP

τ_t^N(P)= Z 1

NTr

P(X₁^N, . . . ,X_m^N)

dP^M_Vt(X₁^N,· · · ,X_m^N)

=

K

X

g=0

N^−2g X

k1,..,kn∈N n

Y

j=1

(tj)^kj

k_j! M((P,1),(qi,ki)1≤i≤n;g)+o(N^−2K)

-m= 1 : Ambj´orn et al. 95’, Albeverio, Pastur, Scherbina 01’, Ercolani-McLaughlin 03’

-m≥2 : G-Maurel-Segala 06’, G-Shlyakhtenko 09’, Dabrowski 18’ Jekel 19’

From formal to asymptotic topological expansions

Form∈N and (q1,· · · ,qn) monomials, Vt =Pn

i=1tiqi,M >2 dP^MVt(X₁^N,· · ·,X_m^N) = 1_kXN

i k≤M

Z_V^N,M e^NTr(^Vt(X₁^N,...,X_m^N))dP(X₁^N)· · ·dP(X_m^N) ForM >2, allK ∈N,t_i small enough so thatV_t=V_t^∗,for any monomialP

τ_t^N(P)= Z 1

NTr

P(X₁^N, . . . ,X_m^N)

dP^M_Vt(X₁^N,· · · ,X_m^N)

=

K

X

g=0

N^−2g X

k1,..,kn∈N n

Y

j=1

(tj)^kj

k_j! M((P,1),(qi,ki)1≤i≤n;g)+o(N^−2K) -m= 1 : Ambj´orn et al. 95’, Albeverio, Pastur, Scherbina 01’,

Ercolani-McLaughlin 03’

-m≥2 : G-Maurel-Segala 06’, G-Shlyakhtenko 09’, Dabrowski 18’

Jekel 19’

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Topological expansions, Random matrices and operator algebras

Maps

Random Matrices and the enumeration of maps SD equations

Loop models Subfactors theory Transport

Schwinger-Dyson equations

Both matrix integrals and map enumerations are related with a third mathematical objects : TheSchwinger-Dyson equations.

• They describe relations between moments, obtained thanks to integration by parts, for matrix integrals,

• They describe the induction relations for the enumeration of maps.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

First loop equation

LetV be a polynomial and set

dPV(X₁^N, . . . ,X_m^N) = (Z_V^N)⁻¹e^NTr(^V(X1^N,...,X_m^N))dP(X₁^N)· · ·dP(X_m^N) Then, for any polynomialP, anyi ∈ {1, . . . ,m}

Z 1

NTr⊗ 1

NTr(∂iP(X₁^N, . . . ,X_m^N))dPV(X₁^N, . . . ,X_m^N)

= Z 1

NTr((X_i −D_iV)P(X₁^N, . . . ,X_m^N))dPV(X₁^N, . . . ,X_m^N) where for any monomialq

∂_iq = X

q=q1Xiq2

q₁⊗q₂ D_iq= X

q=q1Xiq2

q₂q₁

Proof : Based onR

f⁰(x)e^−V(x)dx =R

f(x)V⁰(x)e^−V(x)dx.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

First order asymptotics

LetV be a polynomial and set

dPV(X₁^N, . . . ,X_m^N) = (Z_V^N)⁻¹e^NTr(^V(X1^N,...,X_m^N))dP(X₁^N)· · ·dP(X_m^N) AssumeV small (and add a cutoff if needed). The limit points τV

of

τ_XN(P) := 1

NTr(P(X₁^N, . . . ,X_d^N)) satisfy

(A) τ_V(X_iP) =τ_V ⊗τ_V(∂_iP) +τ_V(D_iVP) with∂_iq =P

q=q1Xiq2q₁⊗q₂, D_iq=P

q=q1Xiq2q₂q₁, (B) |τ_V(X_i1· · ·X_ik)| ≤4^k.

fork ≤ N. Hence (A) comes from the loop equation Z

τ_XN ⊗τ_XN(∂_iP)dPV = Z

τ_XN((X_i −D_iV)P)dPV

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

First order asymptotics

LetV be a polynomial and set

dPV(X₁^N, . . . ,X_m^N) = (Z_V^N)⁻¹e^NTr(^V(X1^N,...,X_m^N))dP(X₁^N)· · ·dP(X_m^N) AssumeV small (and add a cutoff if needed). The limit points τV

of

τ_XN(P) := 1

NTr(P(X₁^N, . . . ,X_d^N)) satisfy

(A) τ_V(X_iP) =τ_V ⊗τ_V(∂_iP) +τ_V(D_iVP) with∂_iq =P

q=q1Xiq2q₁⊗q₂, D_iq=P

q=q1Xiq2q₂q₁, (B) |τ_V(X_i1· · ·X_ik)| ≤4^k.

Proof : asPV is log-concave,τ_XN self-averages and satisfies (B) fork ≤√

N. Hence (A) comes from the loop equation Z

τ_XN ⊗τ_XN(∂_iP)dPV = Z

τ_XN((X_i −D_iV)P)dPV

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

First order asymptotics

IfV is small enough,there exists a unique solutionto (A) τ_V(X_iP) =τ_V ⊗τ_V(∂_iP) +τ_V(D_iVP)

⇔τ_V(X_iq) = X

q=q1Xiq2

τ_V(q₁)τ_V(q₂) +X

j

t_j X

qj=q^j₁Xiq₂^j

τ_V(q^j₂q₁^jq) (B) |τ_V(X_i1· · ·X_ik)| ≤4^k,

It is the generating function of planar maps τ_V(P) =X Yt_i^ki

k_i!M((P,1),(qi,ki); 0).

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

First order asymptotics

IfV is small enough,there exists a unique solutionto (A) τ_V(X_iP) =τ_V ⊗τ_V(∂_iP) +τ_V(D_iVP)

⇔τ_V(X_iq) = X

q=q1Xiq2

τ_V(q₁)τ_V(q₂) +X

j

t_j X

qj=q^j₁Xiq₂^j

τ_V(q^j₂q₁^jq)

(B) |τ_V(X_i1· · ·X_ik)| ≤4^k, Henceτ_XN converges to this solution.

It is the generating function of planar maps τ_V(P) =X Yt_i^ki

k_i!M((P,1),(qi,ki); 0).

First order asymptotics

IfV is small enough,there exists a unique solutionto (A) τ_V(X_iP) =τ_V ⊗τ_V(∂_iP) +τ_V(D_iVP)

⇔τ_V(X_iq) = X

q=q1Xiq2

τ_V(q₁)τ_V(q₂) +X

j

t_j X

qj=q^j₁Xiq₂^j

τ_V(q^j₂q₁^jq)

(B) |τ_V(X_i1· · ·X_ik)| ≤4^k, Henceτ_XN converges to this solution.

It is the generating function of planar maps τ_V(P) =X Yt_i^ki

k_i!M((P,1),(qi,ki); 0).

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Induction relations and non-commutative derivatives

Tutte’s surgery =Induction relations on maps.

Let M(p,n) be the number of planar maps with p vertices of degree 3 and one of degreen.

M(p,n)

= 3pM(p−1,n+ 1) +

n−2

X

k=0 p

X

`=0

C_p<sup>`</sup>M(`,k)M(p−`,n−k−2) M_t(xⁿ) =P

p≥0 t^p

p!M(p,n) satisfies the loop equation withV =x³ (A) Mt(xⁿ) =tMt(xⁿ⁻¹3x²) +Mt⊗Mt(∂x^p−1)

(B) |M_t(xⁿ)| ≤4ⁿ.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Induction relations and non-commutative derivatives

Tutte’s surgery =Induction relations on maps.

Let M(p,n) be the number of planar maps with p vertices of degree 3 and one of degreen.

M(p,n)

= 3pM(p−1,n+ 1) +

n−2

X

k=0 p

X

`=0

C_p<sup>`</sup>M(`,k)M(p−`,n−k−2)

p!

(A) Mt(xⁿ) =tMt(xⁿ⁻¹3x²) +Mt⊗Mt(∂x^p−1) (B) |M_t(xⁿ)| ≤4ⁿ.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Induction relations and non-commutative derivatives

Tutte’s surgery =Induction relations on maps.

Let M(p,n) be the number of planar maps with p vertices of degree 3 and one of degreen.

M(p,n)

= 3pM(p−1,n+ 1) +

n−2

X

k=0 p

X

`=0

C_p<sup>`</sup>M(`,k)M(p−`,n−k−2) M_t(xⁿ) =P

p≥0 t^p

p!M(p,n) satisfies the loop equation withV =x³ (A) Mt(xⁿ) =tMt(xⁿ⁻¹3x²) +Mt⊗Mt(∂x^p−1)

(B) |M_t(xⁿ)| ≤4ⁿ.

Topological expansions, Random matrices and operator algebras

Maps

Random Matrices and the enumeration of maps SD equations

Loop models Subfactors theory Transport

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Loop models

The Temperley-Lieb elements (TLE) are boxes with boundary points connected by non-intersecting strings, a shading and a marked boundary point.

*

LetS1, . . . ,Sn be (TLE) andβ1,· · · , βn be small real numbers.

The loop model is given, for any Temperley-Lieb elementS,by Tr_β,δ(S) =X

ni≥0

X Y

1≤i≤n

β_iⁿⁱ n_i!δ^]loops

where we sum over all planar maps with n_i ele- ments Si and one ele- mentS.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Main results

Theorem (G-Jones-Shlyakhtenko-Zinn Justin 10’ )

Let S₁, . . . ,S_n be Temperley-Lieb elements, β₁, . . . , β_n∈Rⁿ and consider the loop model

Tr_β,δ(S) =X

ni≥0

X Y

1≤i≤n

β_iⁿⁱ n_i!δ^]loops

Then,for δ∈I :={2 cos(^π_n)}_n≥3∪[2,∞[and β_i small enough Tr_β,δ is a limit of matrix models.

Theorem (G-Jones-Shlyakhtenko-Zinn Justin 10’ ) Forδ ∈I and a Temperley-Lieb element S of the form

there exists an explicit formula forTrβ,δ(S). Cf Bousquet-Melou–Bernardi, Borot, Duplantier, Eynard, Kostov, Staudacher ...

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Main results

Theorem (G-Jones-Shlyakhtenko-Zinn Justin 10’ )

Let S₁, . . . ,S_n be Temperley-Lieb elements, β₁, . . . , β_n∈Rⁿ and consider the loop model

Tr_β,δ(S) =X

ni≥0

X Y

1≤i≤n

β_iⁿⁱ n_i!δ^]loops

Then,for δ∈I :={2 cos(^π_n)}_n≥3∪[2,∞[and β_i small enough Tr_β,δ is a limit of matrix models.

For the Potts model, i.eS₁= ,S₂ =

Theorem (G-Jones-Shlyakhtenko-Zinn Justin 10’ ) Forδ ∈I and a Temperley-Lieb element S of the form

there exists an explicit formula forTrβ,δ(S).

Cf Bousquet-Melou–Bernardi, Borot, Duplantier, Eynard, Kostov, Staudacher ...

Random matrices and loop enumeration ; β = 0

Letδ=m∈N. For a (TLE) B, we denotep ∼^B ` if a string joins thepth boundary point with the`th boundary point in B, then we associate toB withk strings the polynomial

q_B(X) = X

ij=ip if ^j∼pB

1≤i`≤m

Xi1· · ·Xi_2k.

q_B(X) =

n

X

i,j,k=1

X_iX_jX_jX_iX_kX_k ⇔ Theorem

Ifν^N denotes the law of m independent GUE matrices,

N→∞lim Z 1

Ntr(qB(X))ν^N(dX) =X

m^]loops=Tr0(B)

where we sum over all planar maps that can be built on B.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Proof

By Voiculescu’s theorem, ifB = ,

N→∞lim Z 1

Ntr (q_B(X))ν^N(dX)

=

n

X

i,j,k=1 N→∞lim

Z 1

Ntr (X_iX_jX_jX_iX_kX_k)ν^N(dX)

=X

i,j,k

X

i j j k k

=X

n^]loops

because the indices have to be constant along loops.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Non integer fugacities, β = 0

Based on theconstruction of the planar algebra of a bipartite graph, Jones 99’.Recallp ∼^B j if a string joins thepth dot with the jth do in the TL element B

j

* p

.

q_B(X) = X

ij=ip if j∼p^B

X_i1· · ·X_i2k ⇒q_B^v(X) = X

ej=e_p^o ifj∼p^B

σ_B(w)X_e1· · ·X_e2k

adjacency matrix of Γ has eigenvalueδ with eigenvector (µ_v)v∈V

withµv ≥0 (∃ for anyδ∈ {2 cos(^π_n)}_n≥3∪[2,∞[)

•The sum runs over loops w =e₁· · ·e_2k in Γ which starts atv. v∈V₊ iff ∗is in a white region.

•σB(w) is a well chosen weight.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Non integer fugacities, β = 0

Based on theconstruction of the planar algebra of a bipartite graph, Jones 99’.Recallp ∼^B j if a string joins thepth dot with the jth do in the TL element B

j

* p

.

q_B(X) = X

ij=ip if j∼p^B

X_i1· · ·X_i2k ⇒q_B^v(X) = X

ej=e_p^o ifj∼p^B

σ_B(w)X_e1· · ·X_e2k

•e_i edges of a bipartite graph Γ = (V =V₊∪V−,E) so that the adjacency matrix of Γ has eigenvalueδ with eigenvector (µ_v)_v∈V withµv ≥0 (∃ for anyδ∈ {2 cos(^π_n)}_n≥3∪[2,∞[)

•The sum runs over loops w =e₁· · ·e_2k in Γ which starts atv. v∈V₊ iff ∗is in a white region.

•σB(w) is a well chosen weight.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Non integer fugacities, β = 0

Based on theconstruction of the planar algebra of a bipartite graph, Jones 99’.Recallp ∼^B j if a string joins thepth dot with the jth do in the TL element B

j

* p

.

q_B(X) = X

ij=ip if j∼p^B

X_i1· · ·X_i2k ⇒q_B^v(X) = X

ej=e_p^o ifj∼p^B

σ_B(w)X_e1· · ·X_e2k

•e_i edges of a bipartite graph Γ = (V =V₊∪V−,E) so that the adjacency matrix of Γ has eigenvalueδ with eigenvector (µ_v)_v∈V withµv ≥0 (∃ for anyδ∈ {2 cos(^π_n)}_n≥3∪[2,∞[)

•The sum runs over loops w =e₁· · ·e_2k in Γ which starts atv. v∈V₊ iff ∗is in a white region.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Non integer fugacities, β = 0

Based on theconstruction of the planar algebra of a bipartite graph, Jones 99’.Recallp ∼^B j if a string joins thepth dot with the jth do in the TL element B

j

* p

.

q_B(X) = X

ij=ip if j∼p^B

X_i1· · ·X_i2k ⇒q_B^v(X) = X

ej=e_p^o ifj∼p^B

σ_B(w)X_e1· · ·X_e2k

•e_i edges of a bipartite graph Γ = (V =V₊∪V−,E) so that the adjacency matrix of Γ has eigenvalueδ with eigenvector (µ_v)_v∈V withµv ≥0 (∃ for anyδ∈ {2 cos(^π_n)}_n≥3∪[2,∞[)

•The sum runs over loops w =e₁· · ·e_2k in Γ which starts atv. v∈V₊ iff ∗is in a white region.

•σB(w) is a well chosen weight.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Non integer fugacities,the matrix model, β = 0

Fore ∈E,e = (s(e),t(e)),X_e^M are independent (except X_e^o =X_e^∗) [Mµ_s(e)]×[Mµ_t(e)] matrices with i.i.d centered Gaussian entries with variance 1/(M√

µ_s(e)µ_t(e)).

Recall q_B^v(X^M) = X

w=e1···e2k∈LB s(e1)=v

σ_B(w)X_e^M₁ · · ·X_e^M

2k

Theorem (G-Jones-Shlyakhtenko 07’)

LetΓbe a bipartite graph as before. Let B be Temperley-Lieb element. For all v ∈V

M→∞lim E[ 1

Mµ_vtr(q^v_B(X^M))] =Tr_0,δ(B) =X δ^]loops where the sum runs above all planar maps built on B.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Non integer fugacities,the matrix model, β = 0

Fore ∈E,e = (s(e),t(e)),X_e^M are independent (except X_e^o =X_e^∗) [Mµ_s(e)]×[Mµ_t(e)] matrices with i.i.d centered Gaussian entries with variance 1/(M√

µ_s(e)µ_t(e)).

Recall q_B^v(X^M) = X

w=e1···e2k∈LB s(e1)=v

σ_B(w)X_e^M₁ · · ·X_e^M

2k

Theorem (G-Jones-Shlyakhtenko 07’)

LetΓbe a bipartite graph as before. Let B be Temperley-Lieb element. For all v ∈V

M→∞lim E[ 1

Mµ_vtr(q^v_B(X^M))] =Tr_0,δ(B) =X δ^]loops

where the sum runs above all planar maps built on B.

Based onP

e∈E:s(e)=vµ_t(e)=δµ_v.

Non integer fugacities, β 6= 0

LetBi be Temperley Lieb elements with ∗with colorσi ∈ {+,−}, 1≤i ≤p. Let Γ be a bipartite graph whose adjacency matrix has eigenvalueδ as before. Let ν^M be the law of the previous

independent rectangular Gaussian matrices and set dν_(B^M

i)i(Xe) = 1_kXek∞≤L

Z_B^N e^Mtr(Pp i=1βi

P

v∈Vσi µvq_Bi^v (X))

dν^M(Xe).

Theorem (G-Jones-Shlyakhtenko-Zinn Justin 10’) For any L>2, for β_i small enough real numbers, for any Temperley-Lieb element B with colorσ, any v ∈Vσ,

M→∞lim Z 1

Mµ_vtr(q_B^v(X))dν_(B^N

i)i(X) = X

ni≥0

Xδ^]loops

p

Y

i=1

β_iⁿⁱ n_i!

where we sum over the planar maps build on ni TL elements Bi

and one B. This isTr_β,δ(B).

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Topological expansions, Random matrices and operator algebras

Maps

Random Matrices and the enumeration of maps SD equations

Loop models Subfactors theory Transport

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Application to subfactors theory

Temperley-Lieb elements are boxes containing non-intersecting strings. We can endow this set with the multiplication :

and the trace given by τ(S) = X

R∈TL

δ^]loops in S.R

T T.S=

S

Theorem (G-Jones-Shlyakhtenko 07 ’ ,Popa 89’ and 93’ ) Takeδ ∈I :={2 cos(^π_n)}_n≥4∪]2,∞[

-τ is a tracial state, as a limit of matrix (or free var.) models.

-A tower of factors with index δ² can be built .

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Application to subfactors theory

Temperley-Lieb elements are boxes containing non-intersecting strings. We can endow this set with the multiplication :

and the trace given by τ(S) = X

R∈TL

δ^]loops in S.R

T T.S=

S

Theorem (G-Jones-Shlyakhtenko 07 ’ ,Popa 89’ and 93’ ) Takeδ ∈I :={2 cos(^π_n)}_n≥4∪]2,∞[

-τ is a tracial state, as a limit of matrix (or free var.) models.

-The corresponding von Neumann algebra is a factor.

-A tower of factors with index δ² can be built .

Application to subfactors theory

Temperley-Lieb elements are boxes containing non-intersecting strings. We can endow this set with the multiplication :

and the trace given by τ(S) = X

R∈TL

δ^]loops in S.R

T T.S=

S

Theorem (G-Jones-Shlyakhtenko 07 ’ ,Popa 89’ and 93’ ) Takeδ ∈I :={2 cos(^π_n)}_n≥4∪]2,∞[

-τ is a tracial state, as a limit of matrix (or free var.) models.

-The corresponding von Neumann algebra is a factor.

-A tower of factors with index δ² can be built .

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Topological expansions, Random matrices and operator algebras

Maps

Random Matrices and the enumeration of maps SD equations

Loop models Subfactors theory Transport

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Convergence of the empirical distribution of matrices

LetX^N = (X₁^N, . . . ,X_d^N) be a sequence ofN×N (random) Hermitian matrices and let ˆµ_N be its empirical distribution

ˆ

µ_N(P) = 1

NTr(P(X^N)) Assume that for any polynomialP

N→∞lim µˆ_N(P) = lim

N→∞

1

NTr(P(X^N)) =τ(P).(∗) Thenτ is a tracial state :

τ(PP^∗)≥0, τ(PQ) =τ(QP), τ(I) = 1.

sequence of matricesX^N such that (*) holds ?

Z. Ji, A. Natarajan,T. Vidick, J. Wright and H. Yuen (2020) : Answer is no(MIP*=RE). But a mystake in the proof was found and a patch posted.

Maps Random Matrices and the enumeration of maps SD equations Loop models Subfactors theory Transport

Convergence of the empirical distribution of matrices

LetX^N = (X₁^N, . . . ,X_d^N) be a sequence ofN×N (random) Hermitian matrices and let ˆµ_N be its empirical distribution

ˆ

µ_N(P) = 1

NTr(P(X^N)) Assume that for any polynomialP

N→∞lim µˆ_N(P) = lim

N→∞

1

NTr(P(X^N)) =τ(P).(∗) Thenτ is a tracial state :

τ(PP^∗)≥0, τ(PQ) =τ(QP), τ(I) = 1.

Connes Question:For any tracial stateτ can you find a sequence of matricesX^N such that (*) holds ?

Z. Ji, A. Natarajan,T. Vidick, J. Wright and H. Yuen (2020) : Answer is no(MIP*=RE). But a mystake in the proof was found and a patch posted.

Convergence of the empirical distribution of matrices

LetX^N = (X₁^N, . . . ,X_d^N) be a sequence ofN×N (random) Hermitian matrices and let ˆµ_N be its empirical distribution

ˆ

µ_N(P) = 1

NTr(P(X^N)) Assume that for any polynomialP

N→∞lim µˆ_N(P) = lim

N→∞

1

NTr(P(X^N)) =τ(P).(∗) Thenτ is a tracial state :

τ(PP^∗)≥0, τ(PQ) =τ(QP), τ(I) = 1.

Connes Question:For any tracial stateτ can you find a sequence of matricesX^N such that (*) holds ?

Z. Ji, A. Natarajan,T. Vidick, J. Wright and H. Yuen (2020) : Answer is no(MIP*=RE). But a mystake in the proof was found and a patch posted.