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

## 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

## 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

## 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

## 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

## 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^{−2g}M(2p;g).

equalsPN n=1

N n

(2p−1)!!2^{n−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^{−2g}M(2p;g).

equalsPN n=1

N n

(2p−1)!!2^{n−1}
p

n−1

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

## 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_{j}Gr_{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}^{N}X_{k}^{N}_{`}] =N^{−1}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}^{+1}M((x^{p},1);g) by Euler formula

## Random matrices and the enumeration of maps

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

Lett= (ti)1≤i≤n∈R^{n} and setVt=Pn

i=1tix^{i}.Formally,
1

N^{2}log
Z

e^{N}tr(Vt(X^{N}))dP(X^{N})

= X

k1,..,kn∈N

X

g≥0

N^{−2g}

n

Y

j=1

(t_{j})^{k}^{j}

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 colori_{1}, the second colori_{2} 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.

## Random matrices and the enumeration of maps

’t Hooft (1974) and Br´ezin-Itzykson-Parisi-Zuber (1978)
Let (q_{1}, . . . ,q_{n}) be monomials. Let t= (t_{i})1≤i≤n∈R^{n} and set
Vt(X1, . . . ,Xm) =Pn

i=1tiqi(X1, . . . ,Xm).Formally,
F_{V}^{N}_{t} = 1

N^{2} log
Z

e^{N}tr(Vt(A1,· · ·,Am))dP^{N}(A1)· · ·dP^{N}(Am)

= X

k1,..,kn∈N

X

g≥0

N^{−2g}

n

Y

j=1

(tj)^{k}^{j}

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) =X_{1}^{4},q3(X1,X2) =X_{2}^{4}
represented by

Then,
1
N^{2}log

Z

e^{NTr(}^{P}^{3}^{i}^{=1}^{t}^{i}^{q}^{i}^{(X}^{1}^{N}^{,X}^{2}^{N}^{))}dP(X_{1}^{N})dP(X_{2}^{N})
is a generating function for the enumeration of the
the Ising model on random

graphs. Solved by Mehta (1986).

## 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_{i}q_{i} and put

dPVt(X_{1}^{N},· · ·,X_{m}^{N}) =e^{−N}^{2}^{F}^{V}^{N}^{t}^{+NTr}(^{V}^{t}^{(X}1^{N},···,X_{m}^{N}))dP(X_{1}^{N})· · ·dP(X_{m}^{N})
Formally, for any monomialP

τ_{t}^{N}(P) :=

Z 1 NTr

P(X_{1}^{N}, . . . ,X_{m}^{N})

dPVt(X_{1}^{N}, . . . ,X_{m}^{N})

= ∂_{s}F_{V}^{N}

t+sP/N^{2}|_{s=0}

= X

g≥0

N^{−2g} X

k1,..,kn∈N n

Y

j=1

(t_{j})^{k}^{j}

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_{i}q_{i} and put

dPVt(X_{1}^{N},· · ·,X_{m}^{N}) =e^{−N}^{2}^{F}^{V}^{N}^{t}^{+NTr}(^{V}^{t}^{(X}1^{N},···,X_{m}^{N}))dP(X_{1}^{N})· · ·dP(X_{m}^{N})
Formally, for any monomialP

τ_{t}^{N}(P) :=

Z 1 NTr

P(X_{1}^{N}, . . . ,X_{m}^{N})

dPVt(X_{1}^{N}, . . . ,X_{m}^{N})

= ∂_{s}F_{V}^{N}

t+sP/N^{2}|_{s=0}

= X

g≥0

N^{−2g} X

k1,..,kn∈N n

Y

j=1

(t_{j})^{k}^{j}

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).

## 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 ∈ChX_{1},· · ·,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∈ChX_{1},· · · ,X_{d}i.

## 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 ∈ChX_{1},· · ·,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∈ChX_{1},· · · ,X_{d}i.

## 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 ∈ChX_{1},· · ·,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∈ChX_{1},· · · ,X_{d}i.

## The law of free semicircle variables

TakeX_{1}^{N},· · · ,X_{d}^{N} be independent GUE matrices, that is

P

dX_{1}^{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∈ChX_{1},· · · ,Xdi

N→∞lim E[1

NTr(P(X_{1}^{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.

## From formal to asymptotic topological expansions

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

i=1tiqi,M >2
dP^{M}Vt(X_{1}^{N},· · ·,X_{m}^{N}) = 1_{kX}N

i k≤M

Z_{V}^{N,M} e^{NTr}(^{V}t(X_{1}^{N},...,X_{m}^{N}))dP(X_{1}^{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(X_{1}^{N}, . . . ,X_{m}^{N})

dP^{M}_{V}_{t}(X_{1}^{N},· · · ,X_{m}^{N})

=

K

X

g=0

N^{−2g} X

k1,..,kn∈N n

Y

j=1

(tj)^{k}^{j}

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^{M}Vt(X_{1}^{N},· · ·,X_{m}^{N}) = 1_{kX}N

i k≤M

Z_{V}^{N,M} e^{NTr}(^{V}t(X_{1}^{N},...,X_{m}^{N}))dP(X_{1}^{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(X_{1}^{N}, . . . ,X_{m}^{N})

dP^{M}_{V}_{t}(X_{1}^{N},· · · ,X_{m}^{N})

=

K

X

g=0

N^{−2g} X

k1,..,kn∈N n

Y

j=1

(tj)^{k}^{j}

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’

## 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.

## First loop equation

LetV be a polynomial and set

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

Z 1

NTr⊗ 1

NTr(∂iP(X_{1}^{N}, . . . ,X_{m}^{N}))dPV(X_{1}^{N}, . . . ,X_{m}^{N})

= Z 1

NTr((X_{i} −D_{i}V)P(X_{1}^{N}, . . . ,X_{m}^{N}))dPV(X_{1}^{N}, . . . ,X_{m}^{N})
where for any monomialq

∂_{i}q = X

q=q1Xiq2

q_{1}⊗q_{2} D_{i}q= X

q=q1Xiq2

q_{2}q_{1}

Proof : Based onR

f^{0}(x)e^{−V}^{(x)}dx =R

f(x)V^{0}(x)e^{−V}^{(x)}dx.

## First order asymptotics

LetV be a polynomial and set

dPV(X_{1}^{N}, . . . ,X_{m}^{N}) = (Z_{V}^{N})^{−1}e^{NTr}(^{V}^{(X}1^{N},...,X_{m}^{N}))dP(X_{1}^{N})· · ·dP(X_{m}^{N})
AssumeV small (and add a cutoff if needed). The limit points τV

of

τ_{X}N(P) := 1

NTr(P(X_{1}^{N}, . . . ,X_{d}^{N}))
satisfy

(A) τ_{V}(X_{i}P) =τ_{V} ⊗τ_{V}(∂_{i}P) +τ_{V}(D_{i}VP)
with∂_{i}q =P

q=q1Xiq2q_{1}⊗q_{2}, D_{i}q=P

q=q1Xiq2q_{2}q_{1},
(B) |τ_{V}(X_{i}_{1}· · ·X_{i}_{k})| ≤4^{k}.

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

τ_{X}N ⊗τ_{X}N(∂_{i}P)dPV =
Z

τ_{X}N((X_{i} −D_{i}V)P)dPV

## First order asymptotics

LetV be a polynomial and set

dPV(X_{1}^{N}, . . . ,X_{m}^{N}) = (Z_{V}^{N})^{−1}e^{NTr}(^{V}^{(X}1^{N},...,X_{m}^{N}))dP(X_{1}^{N})· · ·dP(X_{m}^{N})
AssumeV small (and add a cutoff if needed). The limit points τV

of

τ_{X}N(P) := 1

NTr(P(X_{1}^{N}, . . . ,X_{d}^{N}))
satisfy

(A) τ_{V}(X_{i}P) =τ_{V} ⊗τ_{V}(∂_{i}P) +τ_{V}(D_{i}VP)
with∂_{i}q =P

q=q1Xiq2q_{1}⊗q_{2}, D_{i}q=P

q=q1Xiq2q_{2}q_{1},
(B) |τ_{V}(X_{i}_{1}· · ·X_{i}_{k})| ≤4^{k}.

Proof : asPV is log-concave,τ_{X}N self-averages and satisfies (B)
fork ≤√

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

τ_{X}N ⊗τ_{X}N(∂_{i}P)dPV =
Z

τ_{X}N((X_{i} −D_{i}V)P)dPV

## First order asymptotics

IfV is small enough,there exists a unique solutionto
(A) τ_{V}(X_{i}P) =τ_{V} ⊗τ_{V}(∂_{i}P) +τ_{V}(D_{i}VP)

⇔τ_{V}(X_{i}q) = X

q=q1Xiq2

τ_{V}(q_{1})τ_{V}(q_{2}) +X

j

t_{j} X

qj=q^{j}_{1}Xiq_{2}^{j}

τ_{V}(q^{j}_{2}q_{1}^{j}q)
(B) |τ_{V}(X_{i}_{1}· · ·X_{i}_{k})| ≤4^{k},

It is the generating function of planar maps
τ_{V}(P) =X Yt_{i}^{k}^{i}

k_{i}!M((P,1),(qi,ki); 0).

## First order asymptotics

IfV is small enough,there exists a unique solutionto
(A) τ_{V}(X_{i}P) =τ_{V} ⊗τ_{V}(∂_{i}P) +τ_{V}(D_{i}VP)

⇔τ_{V}(X_{i}q) = X

q=q1Xiq2

τ_{V}(q_{1})τ_{V}(q_{2}) +X

j

t_{j} X

qj=q^{j}_{1}Xiq_{2}^{j}

τ_{V}(q^{j}_{2}q_{1}^{j}q)

(B) |τ_{V}(X_{i}_{1}· · ·X_{i}_{k})| ≤4^{k},
Henceτ_{X}N converges to this solution.

It is the generating function of planar maps
τ_{V}(P) =X Yt_{i}^{k}^{i}

k_{i}!M((P,1),(qi,ki); 0).

## First order asymptotics

IfV is small enough,there exists a unique solutionto
(A) τ_{V}(X_{i}P) =τ_{V} ⊗τ_{V}(∂_{i}P) +τ_{V}(D_{i}VP)

⇔τ_{V}(X_{i}q) = X

q=q1Xiq2

τ_{V}(q_{1})τ_{V}(q_{2}) +X

j

t_{j} X

qj=q^{j}_{1}Xiq_{2}^{j}

τ_{V}(q^{j}_{2}q_{1}^{j}q)

(B) |τ_{V}(X_{i}_{1}· · ·X_{i}_{k})| ≤4^{k},
Henceτ_{X}N converges to this solution.

It is the generating function of planar maps
τ_{V}(P) =X Yt_{i}^{k}^{i}

k_{i}!M((P,1),(qi,ki); 0).

## 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}^{`}M(`,k)M(p−`,n−k−2)
M_{t}(x^{n}) =P

p≥0 t^{p}

p!M(p,n) satisfies the loop equation withV =x^{3}
(A) Mt(x^{n}) =tMt(x^{n−1}3x^{2}) +Mt⊗Mt(∂x^{p−1})

(B) |M_{t}(x^{n})| ≤4^{n}.

## 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}^{`}M(`,k)M(p−`,n−k−2)

p!

(A) Mt(x^{n}) =tMt(x^{n−1}3x^{2}) +Mt⊗Mt(∂x^{p−1})
(B) |M_{t}(x^{n})| ≤4^{n}.

## 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}^{`}M(`,k)M(p−`,n−k−2)
M_{t}(x^{n}) =P

p≥0 t^{p}

p!M(p,n) satisfies the loop equation withV =x^{3}
(A) Mt(x^{n}) =tMt(x^{n−1}3x^{2}) +Mt⊗Mt(∂x^{p−1})

(B) |M_{t}(x^{n})| ≤4^{n}.

## Topological expansions, Random matrices and operator algebras

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_{i}^{n}^{i}
n_{i}!δ^{]loops}

where we sum over all
planar maps with n_{i} ele-
ments Si and one ele-
mentS.

## Main results

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

Let S_{1}, . . . ,S_{n} be Temperley-Lieb elements, β_{1}, . . . , β_{n}∈R^{n} and
consider the loop model

Tr_{β,δ}(S) =X

ni≥0

X Y

1≤i≤n

β_{i}^{n}^{i}
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 ...

## Main results

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

Let S_{1}, . . . ,S_{n} be Temperley-Lieb elements, β_{1}, . . . , β_{n}∈R^{n} and
consider the loop model

Tr_{β,δ}(S) =X

ni≥0

X Y

1≤i≤n

β_{i}^{n}^{i}
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.eS_{1}= ,S_{2} =

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}^{∼p}^{B}

1≤i`≤m

Xi1· · ·Xi_{2k}.

q_{B}(X) =

n

X

i,j,k=1

X_{i}X_{j}X_{j}X_{i}X_{k}X_{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.

## 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_{i}X_{j}X_{j}X_{i}X_{k}X_{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.

## 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_{i}_{1}· · ·X_{i}_{2k} ⇒q_{B}^{v}(X) = X

ej=e_{p}^{o} ifj∼p^{B}

σ_{B}(w)X_{e}_{1}· · ·X_{e}_{2k}

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 =e_{1}· · ·e_{2k} in Γ which starts atv.
v∈V_{+} iff ∗is in a white region.

•σB(w) is a well chosen weight.

## 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_{i}_{1}· · ·X_{i}_{2k} ⇒q_{B}^{v}(X) = X

ej=e_{p}^{o} ifj∼p^{B}

σ_{B}(w)X_{e}_{1}· · ·X_{e}_{2k}

•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 =e_{1}· · ·e_{2k} in Γ which starts atv.
v∈V_{+} iff ∗is in a white region.

•σB(w) is a well chosen weight.

## 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_{i}_{1}· · ·X_{i}_{2k} ⇒q_{B}^{v}(X) = X

ej=e_{p}^{o} ifj∼p^{B}

σ_{B}(w)X_{e}_{1}· · ·X_{e}_{2k}

•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 =e_{1}· · ·e_{2k} in Γ which starts atv.
v∈V_{+} iff ∗is in a white region.

## Non integer fugacities, β = 0

^{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_{i}_{1}· · ·X_{i}_{2k} ⇒q_{B}^{v}(X) = X

ej=e_{p}^{o} ifj∼p^{B}

σ_{B}(w)X_{e}_{1}· · ·X_{e}_{2k}

•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,∞[)

_{1}· · ·e_{2k} in Γ which starts atv.
v∈V_{+} iff ∗is in a white region.

•σB(w) is a well chosen weight.

## 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}^{M}_{1} · · ·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_{v}tr(q^{v}_{B}(X^{M}))] =Tr_{0,δ}(B) =X
δ^{]loops}
where the sum runs above all planar maps built on B.

## 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}^{M}_{1} · · ·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_{v}tr(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_{kX}_{e}_{k}_{∞}_{≤L}

Z_{B}^{N} e^{M}tr(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_{v}tr(q_{B}^{v}(X))dν_{(B}^{N}

i)i(X) = X

ni≥0

Xδ^{]loops}

p

Y

i=1

β_{i}^{n}^{i}
n_{i}!

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

and one B. This isTr_{β,δ}(B).

## Topological expansions, Random matrices and operator algebras

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 δ^{2} 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 δ^{2} 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 δ^{2} can be built .

## Topological expansions, Random matrices and operator algebras

Maps

Random Matrices and the enumeration of maps SD equations

Loop models Subfactors theory Transport

## Convergence of the empirical distribution of matrices

LetX^{N} = (X_{1}^{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.

## Convergence of the empirical distribution of matrices

LetX^{N} = (X_{1}^{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} = (X_{1}^{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.