Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
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.
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.
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.
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.
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.
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.
Michael Anshelevich Free Brownian motion
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.
Michael Anshelevich Free Brownian motion
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.
Michael Anshelevich Free Brownian motion
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. . .
. . . . . . . . . . . . . . .
Michael Anshelevich Free Brownian motion
Definition II: Operators.
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. . .
. . . . . . . . . . . . . . .
Michael Anshelevich Free Brownian motion
Definition II: Operators.
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. . .
. . . . . . . . . . . . . . .
Michael Anshelevich Free Brownian motion
Definition II: Operators.
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. . .
. . . . . . . . . . . . . . .
Michael Anshelevich Free Brownian motion
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).
Michael Anshelevich Free Brownian motion
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).
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
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.
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
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.
Michael Anshelevich Free Brownian motion
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 .
Michael Anshelevich Free Brownian motion
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 .
Michael Anshelevich Free Brownian motion
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 .
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
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
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
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
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,
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.
Michael Anshelevich Free Brownian motion
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.
Michael Anshelevich Free Brownian motion
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.
Michael Anshelevich Free Brownian motion
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.
Michael Anshelevich Free Brownian motion
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.
Michael Anshelevich Free Brownian motion
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. . .
. . . . . . . . . . . . . . .
.
Michael Anshelevich Free Brownian motion
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. . .
. . . . . . . . . . . . . . .
.
Michael Anshelevich Free Brownian motion
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. . .
. . . . . . . . . . . . . . .
.
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
L(n, t) =1√n
[nt]∑i=1
(0i) ∈ C[S].
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
L(n, t) =1√n
[nt]∑i=1
(0i) ∈ C[S].
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
L(n, t) =1√n
[nt]∑i=1
(0i) ∈ C[S].
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
L(n, t) =1√n
[nt]∑i=1
(0i) ∈ C[S].
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
L(n, t) =1√n
[nt]∑i=1
(0i) ∈ C[S].
Michael Anshelevich Free Brownian motion
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.
Michael Anshelevich Free Brownian motion
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.
Michael Anshelevich Free Brownian motion
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.
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.
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