Free Brownian motion - Texas A&M University

Free Brownian motion

Michael Anshelevich

Texas A&M University

October 8, 2009

Michael Anshelevich Free Brownian motion

Definition I: Combinatorics.

Catalan numbers

ck =1

k + 1

(2kk

): 1, 1, 2, 5, 14, . . .

Count many things (Stanley Exercise 6.19(a-nnn) +109).

Count lattice paths: , , ,

, , , ,

How many with k steps?

m2k+1 = 0.

m2k = ck = Catalan number.



Catalan numbers

ck =1

k + 1

(2kk

): 1, 1, 2, 5, 14, . . .



, , , ,


m2k+1 = 0.




Catalan numbers

ck =1

k + 1

(2kk

): 1, 1, 2, 5, 14, . . .


Count lattice paths: ,

, ,

, , , ,


m2k+1 = 0.




Catalan numbers

ck =1

k + 1

(2kk

): 1, 1, 2, 5, 14, . . .



, , , ,


m2k+1 = 0.




Catalan numbers

ck =1

k + 1

(2kk

): 1, 1, 2, 5, 14, . . .



, , , ,


m2k+1 = 0.




Catalan numbers

ck =1

k + 1

(2kk

): 1, 1, 2, 5, 14, . . .



, , , ,


m2k+1 = 0.



Numbers→combinatorialstructures→ measures.

Interpret {mk} as moments. More precisely,{mk t

k/2}

.

Want a measure µt on R such that

mk tk/2 =

∫ ∞−∞

xk dµt(x)

Combinatorics⇒ µt =1

2πt

√4t− x2 dx = semicircle laws.


Numbers→combinatorialstructures→ measures.

Interpret {mk} as moments. More precisely,{mk t

k/2}

.

Want a measure µt on R such that

mk tk/2 =

∫ ∞−∞

xk dµt(x)

Combinatorics⇒ µt =1

2πt

√4t− x2 dx = semicircle laws.


Definition II: Operators.

a+ = right shift.

2 3 4 5 61

a− = left shift.

a−a+ = Id, a+a− 6= Id

Matrices

a+ ∼

0 0 0 0

. . .

1 0 0 0. . .

0 1 0 0. . .

0 0 1 0. . .

. . . . . . . . . . . . . . .

, a− ∼

0 1 0 0

. . .

0 0 1 0. . .

0 0 0 1. . .

0 0 0 0. . .

. . . . . . . . . . . . . . .



a+ = right shift.

0 2 3 4 5 61

a− = left shift.

a−a+ = Id, a+a− 6= Id

Matrices

a+ ∼

0 0 0 0

. . .

1 0 0 0. . .

0 1 0 0. . .

0 0 1 0. . .

. . . . . . . . . . . . . . .

, a− ∼

0 1 0 0

. . .

0 0 1 0. . .

0 0 0 1. . .

0 0 0 0. . .

. . . . . . . . . . . . . . .



a+ = right shift.

0 2 3 4 5 61

a− = left shift.

a−a+ = Id, a+a− 6= Id

Matrices

a+ ∼

0 0 0 0

. . .

1 0 0 0. . .

0 1 0 0. . .

0 0 1 0. . .

. . . . . . . . . . . . . . .

, a− ∼

0 1 0 0

. . .

0 0 1 0. . .

0 0 0 1. . .

0 0 0 0. . .

. . . . . . . . . . . . . . .



a+ = right shift.

0 2 3 4 5 61

a− = left shift.

a−a+ = Id, a+a− 6= Id

Matrices

a+ ∼

0 0 0 0

. . .

1 0 0 0. . .

0 1 0 0. . .

0 0 1 0. . .

. . . . . . . . . . . . . . .

, a− ∼

0 1 0 0

. . .

0 0 1 0. . .

0 0 0 1. . .

0 0 0 0. . .

. . . . . . . . . . . . . . .


Operators.

X ∼ a+ + a− ∼

0 1 0 0

. . .

1 0 1 0. . .

0 1 0 1. . .

0 0 1 0. . .

. . . . . . . . . . . . . . .

.

Symmetric matrix; in fact a self-adjoint operator.

Tri-diagonal (orthogonal polynomials).


Operators.

X ∼ a+ + a− ∼

0 1 0 0

. . .

1 0 1 0. . .

0 1 0 1. . .

0 0 1 0. . .

. . . . . . . . . . . . . . .

.

Symmetric matrix; in fact a self-adjoint operator.

Tri-diagonal (orthogonal polynomials).


Operators.

Can realize (more complicated) operators{a+(t), a−(t) : t ≥ 0

}with

a−(s)a+(t) = min(s, t) Id

andX(t) = a+(t) + a−(t).

Each X(t) = self-adjoint operator.

Proposition. ⟨X(t)ke1, e1

⟩= mk t

k/2,

so X(t) ∼ µt, X(t) has distribution µt.


Operators.

Can realize (more complicated) operators{a+(t), a−(t) : t ≥ 0

}with

a−(s)a+(t) = min(s, t) Id

andX(t) = a+(t) + a−(t).

Each X(t) = self-adjoint operator.

Proposition. ⟨X(t)ke1, e1

⟩= mk t

k/2,

so X(t) ∼ µt, X(t) has distribution µt.


Operators.

Why?

⟨X(t)4e1, e1

⟩=⟨(a+ + a−)(a+ + a−)(a+ + a−)(a+ + a−)e1, e1

⟩Using a−(t)e1 = 0 and a−(t)a+(t) = t Id, only left with

−−++ = t2 ,

−+−+ = t2 .


Operators.

Why?

⟨X(t)4e1, e1

⟩=⟨(a+ + a−)(a+ + a−)(a+ + a−)(a+ + a−)e1, e1

⟩

Using a−(t)e1 = 0 and a−(t)a+(t) = t Id, only left with

−−++ = t2 ,

−+−+ = t2 .


Operators.

Why?

⟨X(t)4e1, e1

⟩=⟨(a+ + a−)(a+ + a−)(a+ + a−)(a+ + a−)e1, e1

⟩Using a−(t)e1 = 0 and a−(t)a+(t) = t Id, only left with

−−++ = t2 ,

−+−+ = t2 .


Free Brownian motion.

{Xt} not just individual operators with these distributions.

{X(t) : t ≥ 0} form a process.

{X(t)} = free Brownian motion.

Each X(t) ∼ µt. Increments

X(t1)−X(t0), X(t2)−X(t1), . . . , X(tk)−X(tk−1)

freely independent.

Xt0 · · · · · · · · · · · ·Xt1 · · · · · ·Xt2 · · · · · · · · · · · · · · · · · ·Xt3 .

Have other processes, other types of increments.







X(t1)−X(t0), X(t2)−X(t1), . . . , X(tk)−X(tk−1)

freely independent.

Xt0 · · · · · · · · · · · ·Xt1 · · · · · ·Xt2 · · · · · · · · · · · · · · · · · ·Xt3 .







Each X(t) ∼ µt.

Increments

X(t1)−X(t0), X(t2)−X(t1), . . . , X(tk)−X(tk−1)

freely independent.

Xt0 · · · · · · · · · · · ·Xt1 · · · · · ·Xt2 · · · · · · · · · · · · · · · · · ·Xt3 .








X(t1)−X(t0), X(t2)−X(t1), . . . , X(tk)−X(tk−1)

freely independent.

Xt0 · · · · · · · · · · · ·Xt1 · · · · · ·Xt2 · · · · · · · · · · · · · · · · · ·Xt3 .








X(t1)−X(t0), X(t2)−X(t1), . . . , X(tk)−X(tk−1)

freely independent.

Xt0 · · · · · · · · · · · ·Xt1 · · · · · ·Xt2 · · · · · · · · · · · · · · · · · ·Xt3 .



Definition III: Random matrices.

Mn(t) = n× n symmetric random matrix,

Mn(t) =

1√nB2t

1√nBt

1√nBt ...

1√nBt

1√nB2t

1√nBt ...

1√nBt

1√nBt

1√nB2t ...

......

.... . .

.Bt = (usual) Brownian motion.

1n

Tr(Mn(t)k) = (random) number.


Definition III: Random matrices.

Mn(t) = n× n symmetric random matrix,

Mn(t) =

1√nB2t

1√nBt

1√nBt ...

1√nBt

1√nB2t

1√nBt ...

1√nBt

1√nBt

1√nB2t ...

......

.... . .

.Bt = (usual) Brownian motion.

1n

Tr(Mn(t)k) = (random) number.


Random matrices.

Proposition.As the size of the matrix n→∞,

1n

Tr(Mn(t1)Mn(t2) . . .Mn(tk)

)−→ 〈X(t1)X(t2) . . . X(tk)e1, e1〉.

In particular, 1n Tr

(Mn(t)k

)→ mnt

k/2.

{Mn(t) : t ≥ 0} = asymptotically free Brownian motion.


Random matrices.


1n


)−→ 〈X(t1)X(t2) . . . X(tk)e1, e1〉.


(Mn(t)k

)→ mnt

k/2.



Random matrices.


1n


)−→ 〈X(t1)X(t2) . . . X(tk)e1, e1〉.


(Mn(t)k

)→ mnt

k/2.



Random matrices.

Proof I. (Wigner 1958, L. Arnold 1967)

1n

Tr(Mn(t1)Mn(t2) . . .Mn(tn)

)=∞∑k=0

1nk/2

paths.

Proof II. (Trotter 1984)

Mntridiagonalization−→

1√nN 1√

nχn 0

. . .

1√nχn

1√nN 1√

nχn

. . .

0 1√nχn

1√nN

. . .. . . . . . . . . . . .

n→∞−→

0 1 0

. . .

1 0 1. . .

0 1 0. . .

. . . . . . . . . . . . . . .

.


Random matrices.


1n


)=∞∑k=0

1nk/2

paths.



1√nN 1√

nχn 0

. . .

1√nχn

1√nN 1√

nχn

. . .

0 1√nχn

1√nN

. . .. . . . . . . . . . . .

n→∞−→

0 1 0

. . .

1 0 1. . .

0 1 0. . .

. . . . . . . . . . . . . . .

.


Random matrices.


1n


)=∞∑k=0

1nk/2

paths.



1√nN 1√

nχn 0

. . .

1√nχn

1√nN 1√

nχn

. . .

0 1√nχn

1√nN

. . .. . . . . . . . . . . .

n→∞−→

0 1 0

. . .

1 0 1. . .

0 1 0. . .

. . . . . . . . . . . . . . .

.


Definition IV: Permutations.

S = infinite symmetric group.

C[S] = its group algebra= formal linear combinations of permutations.

ϕ[w] = constant term= coefficient of the identity permutation in w.

(0a) transposition.

Denote

L(n, t) =1√n

[nt]∑i=1

(0i) ∈ C[S].






(0a) transposition.

Denote

L(n, t) =1√n

[nt]∑i=1

(0i) ∈ C[S].






(0a) transposition.

Denote

L(n, t) =1√n

[nt]∑i=1

(0i) ∈ C[S].






(0a) transposition.

Denote

L(n, t) =1√n

[nt]∑i=1

(0i) ∈ C[S].






(0a) transposition.

Denote

L(n, t) =1√n

[nt]∑i=1

(0i) ∈ C[S].


Permutations.

Proposition.As n→∞,

ϕ[L(n, t1)L(n, t2) . . . L(n, tk)

]−→ 〈X(t1)X(t2) . . . X(tk)e1, e1〉.

In particularϕ[L(n, t)k

]→ mk t

k/2.


Permutations.

Why?

L(n, t) =1√n

[nt]∑i=1

(0i) no e

so ϕ [L(n, t)] = 0.

L(n, t)2=1n

[nt]∑i1,i2=1

(0i1)(0i2)

=1n

∑i1 6=i2

(0i1i2) + [nt]e

≈ . . .+ te

so ϕ[L(n, t)2

]= t. Etc.


Permutations.

Why?

L(n, t) =1√n

[nt]∑i=1

(0i) no e

so ϕ [L(n, t)] = 0.

L(n, t)2=1n

[nt]∑i1,i2=1

(0i1)(0i2)

=1n

∑i1 6=i2

(0i1i2) + [nt]e

≈ . . .+ te

so ϕ[L(n, t)2

]= t. Etc.


Free Probability Theory.

Combinatorics.Operator representations.Random matrix theory.Group algebras (symmetric and free).

Other approaches:

Orthogonal polynomials.Asymptotic representation theory (Young diagrams).Operator algebras applications.Complex analysis techniques.



Combinatorics.Operator representations.Random matrix theory.Group algebras (symmetric and free).

Other approaches:

Orthogonal polynomials.Asymptotic representation theory (Young diagrams).Operator algebras applications.Complex analysis techniques.



Upshot:

Do not need to know all of this.

Can enter the field by knowing one of these.

Helps to learn the rest as time goes on.


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