Free Brownian motion
Michael Anshelevich
Texas A&M University
October 8, 2009
Michael Anshelevich
Free Brownian motion
Definition I: Combinatorics.
Catalan numbers
2k
1
:
ck =
k+1 k
Michael Anshelevich
1, 1, 2, 5, 14, . . .
Free Brownian motion
Definition I: Combinatorics.
Catalan numbers
2k
1
:
ck =
k+1 k
1, 1, 2, 5, 14, . . .
Count many things (Stanley Exercise 6.19(a-nnn) +109).
Michael Anshelevich
Free Brownian motion
Definition I: Combinatorics.
Catalan numbers
2k
1
:
ck =
k+1 k
1, 1, 2, 5, 14, . . .
Count many things (Stanley Exercise 6.19(a-nnn) +109).
Count lattice paths:
,
Michael Anshelevich
Free Brownian motion
Definition I: Combinatorics.
Catalan numbers
2k
1
:
ck =
k+1 k
1, 1, 2, 5, 14, . . .
Count many things (Stanley Exercise 6.19(a-nnn) +109).
Count lattice paths:
,
Michael Anshelevich
,
Free Brownian motion
,
Definition I: Combinatorics.
Catalan numbers
2k
1
:
ck =
k+1 k
1, 1, 2, 5, 14, . . .
Count many things (Stanley Exercise 6.19(a-nnn) +109).
Count lattice paths:
,
,
,
,
Michael Anshelevich
,
,
Free Brownian motion
,
Definition I: Combinatorics.
Catalan numbers
2k
1
:
ck =
k+1 k
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.
Michael Anshelevich
Free Brownian motion
,
combinatorial
Numbers → structures → measures.
Interpret {mk } as moments. More precisely, mk tk/2 .
Want a measure µt on R such that
Z ∞
k/2
mk t
=
xk dµt (x)
−∞
Michael Anshelevich
Free Brownian motion
combinatorial
Numbers → structures → measures.
Interpret {mk } as moments. More precisely, mk tk/2 .
Want a measure µt on R such that
Z ∞
k/2
mk t
=
xk dµt (x)
−∞
1 √
Combinatorics ⇒ µt =
4t − x2 dx = semicircle laws.
2πt
Michael Anshelevich
Free Brownian motion
Definition II: Operators.
a+ = right shift.
1
2
3
4
Michael Anshelevich
5
6
Free Brownian motion
Definition II: Operators.
a+ = right shift.
0
1
2
3
4
5
6
a− = left shift.
Michael Anshelevich
Free Brownian motion
Definition II: Operators.
a+ = right shift.
0
1
2
3
4
5
6
a− = left shift.
a− a+ = Id,
a+ a− 6= Id
Michael Anshelevich
Free Brownian motion
Definition II: Operators.
a+ = right shift.
0
1
2
3
4
6
5
a− = left shift.
a− a+ = Id,
a+ a− 6= Id
Matrices

0



+
a ∼



0
0
0
..
.

. 
1 0 0 0 .. 

..  ,
.
0 1 0 0
.. 
.
0 0 1 0
.. .. .. .. ..
. . . . .
Michael Anshelevich

0



−
a ∼



1
0
0
..
.

.. 
.

.. 
.
0 0 0 1
.. 
.
0 0 0 0
.. .. .. .. ..
. . . . .
0
0
Free Brownian motion
1
0
Operators.

0



X ∼ a+ + a− ∼ 



Michael Anshelevich
1
0
0
..
.

.. 
.

..  .

.
0 1 0 1
.. 
.
0 0 1 0
.. .. .. .. ..
. . . . .
1
0
1
0
Free Brownian motion
Operators.

0



X ∼ a+ + a− ∼ 



1
0
0
..
.

.. 
.

..  .

.
0 1 0 1
.. 
.
0 0 1 0
.. .. .. .. ..
. . . . .
1
0
1
0
Symmetric matrix; in fact a self-adjoint operator.
Tri-diagonal (orthogonal polynomials).
Michael Anshelevich
Free Brownian motion
Operators.
Can realize (more complicated) operators
+
a (t), a− (t) : t ≥ 0
with
a− (s)a+ (t) = min(s, t) Id
and
X(t) = a+ (t) + a− (t).
Each X(t) = self-adjoint operator.
Michael Anshelevich
Free Brownian motion
Operators.
Can realize (more complicated) operators
+
a (t), a− (t) : t ≥ 0
with
a− (s)a+ (t) = min(s, t) Id
and
X(t) = a+ (t) + a− (t).
Each X(t) = self-adjoint operator.
Proposition.
D
E
X(t)k e1 , e1 = mk tk/2 ,
so X(t) ∼ µt , X(t) has distribution µt .
Michael Anshelevich
Free Brownian motion
Operators.
Why?
Michael Anshelevich
Free Brownian motion
Operators.
Why?
X(t)4 e1 , e1 = (a+ + a− )(a+ + a− )(a+ + a− )(a+ + a− )e1 , e1
Michael Anshelevich
Free Brownian motion
Operators.
Why?
X(t)4 e1 , 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
.
Michael Anshelevich
Free Brownian motion
Free Brownian motion.
{Xt } not just individual operators with these distributions.
{X(t) : t ≥ 0} form a process.
Michael Anshelevich
Free Brownian motion
Free Brownian motion.
{Xt } not just individual operators with these distributions.
{X(t) : t ≥ 0} form a process.
{X(t)} = free Brownian motion.
Michael Anshelevich
Free Brownian motion
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 .
Michael Anshelevich
Free Brownian motion
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 .
Michael Anshelevich
Free Brownian motion
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.
Michael Anshelevich
Free Brownian motion
Definition III: Random matrices.
Mn (t) = n × n symmetric random matrix,
 √1

Mn (t) = 

√1 Bt √1 Bt
B
n 2t
n
n
1
√ Bt √1 B2t √1 Bt
n
n
n
√1 Bt √1 Bt √1 B2t
n
n
n
..
.
..
.
..
.
...
...
...
..
.
Bt = (usual) Brownian motion.
Michael Anshelevich
Free Brownian motion


.

Definition III: Random matrices.
Mn (t) = n × n symmetric random matrix,
 √1

Mn (t) = 

√1 Bt √1 Bt
B
n 2t
n
n
1
√ Bt √1 B2t √1 Bt
n
n
n
√1 Bt √1 Bt √1 B2t
n
n
n
..
.
..
.
..
.
...
...
...
..


.

.
Bt = (usual) Brownian motion.
1
Tr(Mn (t)k ) = (random) number.
n
Michael Anshelevich
Free Brownian motion
Random matrices.
Proposition.
As the size of the matrix n → ∞,
1 Tr Mn (t1 )Mn (t2 ) . . . Mn (tk ) −→ hX(t1 )X(t2 ) . . . X(tk )e1 , e1 i.
n
Michael Anshelevich
Free Brownian motion
Random matrices.
Proposition.
As the size of the matrix n → ∞,
1 Tr Mn (t1 )Mn (t2 ) . . . Mn (tk ) −→ hX(t1 )X(t2 ) . . . X(tk )e1 , e1 i.
n
In particular, n1 Tr Mn (t)k → mn tk/2 .
Michael Anshelevich
Free Brownian motion
Random matrices.
Proposition.
As the size of the matrix n → ∞,
1 Tr Mn (t1 )Mn (t2 ) . . . Mn (tk ) −→ hX(t1 )X(t2 ) . . . X(tk )e1 , e1 i.
n
In particular, n1 Tr Mn (t)k → mn tk/2 .
{Mn (t) : t ≥ 0} = asymptotically free Brownian motion.
Michael Anshelevich
Free Brownian motion
Random matrices.
Proof I. (Wigner 1958, L. Arnold 1967)
∞
X
1
1 Tr Mn (t1 )Mn (t2 ) . . . Mn (tn ) =
paths.
k/2
n
n
k=0
Michael Anshelevich
Free Brownian motion
Random matrices.
Proof I. (Wigner 1958, L. Arnold 1967)
∞
X
1
1 Tr Mn (t1 )Mn (t2 ) . . . Mn (tn ) =
paths.
k/2
n
n
k=0
Proof II. (Trotter 1984)

√1 N
n
√1 χn
n
0


tridiagonalization  √1 χn √1 N √1 χn
 n
n
n
Mn
−→

 0 √1 χ √1 N

n n
n
..
..
..
.
.
.
Michael Anshelevich
..
.


.. 
.

.. 
.

..
.
Free Brownian motion
Random matrices.
Proof I. (Wigner 1958, L. Arnold 1967)
∞
X
1
1 Tr Mn (t1 )Mn (t2 ) . . . Mn (tn ) =
paths.
k/2
n
n
k=0
Proof II. (Trotter 1984)

√1 N
n
√1 χn
n
0


tridiagonalization  √1 χn √1 N √1 χn
 n
n
n
Mn
−→

 0 √1 χ √1 N

n n
n
..
..
..
.
.
.
Michael Anshelevich
..
.



 0 1 0
.. 

.  n→∞ 
1 0 1
−→ 


.. 
 0 1 0
.
.. .. ..
. . .
..
.
Free Brownian motion
..
.
..
.
..
.
.. ..
. .



.


Definition IV: Permutations.
S = infinite symmetric group.
Michael Anshelevich
Free Brownian motion
Definition IV: Permutations.
S = infinite symmetric group.
C[S] = its group algebra
= formal linear combinations of permutations.
Michael Anshelevich
Free Brownian motion
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.
Michael Anshelevich
Free Brownian motion
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.
Michael Anshelevich
Free Brownian motion
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
[nt]
1 X
L(n, t) = √
(0i) ∈ C[S].
n
i=1
Michael Anshelevich
Free Brownian motion
Permutations.
Proposition.
As n → ∞,
h
i
ϕ L(n, t1 )L(n, t2 ) . . . L(n, tk ) −→ hX(t1 )X(t2 ) . . . X(tk )e1 , e1 i.
In particular
i
h
ϕ L(n, t)k → mk tk/2 .
Michael Anshelevich
Free Brownian motion
Permutations.
Why?
[nt]
1 X
L(n, t) = √
(0i)
n
no e
i=1
so ϕ [L(n, t)] = 0.
Michael Anshelevich
Free Brownian motion
Permutations.
Why?
[nt]
1 X
L(n, t) = √
(0i)
n
no e
i=1
so ϕ [L(n, t)] = 0.
[nt]
1 X
(0i1 )(0i2 )
n
i1 ,i2 =1


X
1
(0i1 i2 ) + [nt]e ≈ . . . + te
= 
n
L(n, t)2 =
i1 6=i2
so ϕ L(n, t)2 = t. Etc.
Michael Anshelevich
Free Brownian motion
Free Probability Theory.
Combinatorics.
Operator representations.
Random matrix theory.
Group algebras (symmetric and free).
Michael Anshelevich
Free Brownian motion
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.
Michael Anshelevich
Free Brownian motion
Free Probability Theory.
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.
Michael Anshelevich
Free Brownian motion