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Mixing in Product Spaces
Elchanan Mossel
Elchanan Mossel Mixing in Product Spaces
Poincare Recurrence Theorem
Theorem (Poincare, 1890)
Let f : X → X be a measure preserving transformation. LetE ⊂ X measurable. Then
P[x ∈ E : f n(x) /∈ E , n > N(x)] = 0
One of the first results in Ergodic Theory.
Long term mixing.
This talk is about short term mixing.
Elchanan Mossel Mixing in Product Spaces
Poincare Recurrence Theorem
Theorem (Poincare, 1890)
Let f : X → X be a measure preserving transformation. LetE ⊂ X measurable. Then
P[x ∈ E : f n(x) /∈ E , n > N(x)] = 0
One of the first results in Ergodic Theory.
Long term mixing.
This talk is about short term mixing.
Elchanan Mossel Mixing in Product Spaces
Finite Markov Chains
As a first example consider a Finite Markov chain.
Let M be a k × k doubly stochastic symmetric matrix.
Pick X 0 uniformly at random from 1, . . . , k.
Given X i = a, let X i+1 = b with probability Ma,b.
Theorem (Long Term Mixing for Markov Chains)
Suppose that other than 1, all eigenvalues λi of M satisfy|λi | ≤ λ < 1. Then for any two sets A,B ⊂ [k], it holds that∣∣∣P[X 0 ∈ A,X t ∈ B]− P[A]P[B]
∣∣∣ ≤ λt
Elchanan Mossel Mixing in Product Spaces
Finite Markov Chains
As a first example consider a Finite Markov chain.
Let M be a k × k doubly stochastic symmetric matrix.
Pick X 0 uniformly at random from 1, . . . , k.
Given X i = a, let X i+1 = b with probability Ma,b.
Theorem (Long Term Mixing for Markov Chains)
Suppose that other than 1, all eigenvalues λi of M satisfy|λi | ≤ λ < 1. Then for any two sets A,B ⊂ [k], it holds that∣∣∣P[X 0 ∈ A,X t ∈ B]− P[A]P[B]
∣∣∣ ≤ λt
Elchanan Mossel Mixing in Product Spaces
Short Term Mixing for Markov Chains
Theorem∣∣∣P[X 0 ∈ A,X 1 ∈ B]− P[A]P[B]∣∣∣ is upper bounded by
λ√
P[A](1− P[A])P[B](1− P[B])
Shows: mixing in one step for large sets.
Proof: 1A = P[A]1 + f , 1B = P[B]1 + g , where f , g ⊥ 1
P[X 0 ∈ A,X 1 ∈ B] =1k
(P[A]1 + f )tM(P[B]1 + g)
= P[A]P[B] +1kf tMg ,
1k|f tMg | ≤ λ‖f ‖2‖g‖2 = λ
√P[A](1− P[A])P[B](1− P[B])
Also called Expander Mixing Lemma.Used a lot in computer science, e.g. in (de)randomization.
Elchanan Mossel Mixing in Product Spaces
Short Term Mixing for Markov Chains
Theorem∣∣∣P[X 0 ∈ A,X 1 ∈ B]− P[A]P[B]∣∣∣ is upper bounded by
λ√
P[A](1− P[A])P[B](1− P[B])
Shows: mixing in one step for large sets.Proof: 1A = P[A]1 + f , 1B = P[B]1 + g , where f , g ⊥ 1
P[X 0 ∈ A,X 1 ∈ B] =1k
(P[A]1 + f )tM(P[B]1 + g)
= P[A]P[B] +1kf tMg ,
1k|f tMg | ≤ λ‖f ‖2‖g‖2 = λ
√P[A](1− P[A])P[B](1− P[B])
Also called Expander Mixing Lemma.Used a lot in computer science, e.g. in (de)randomization.
Elchanan Mossel Mixing in Product Spaces
Short Term Mixing for Markov Chains
Theorem∣∣∣P[X 0 ∈ A,X 1 ∈ B]− P[A]P[B]∣∣∣ is upper bounded by
λ√
P[A](1− P[A])P[B](1− P[B])
Shows: mixing in one step for large sets.Proof: 1A = P[A]1 + f , 1B = P[B]1 + g , where f , g ⊥ 1
P[X 0 ∈ A,X 1 ∈ B] =1k
(P[A]1 + f )tM(P[B]1 + g)
= P[A]P[B] +1kf tMg ,
1k|f tMg | ≤ λ‖f ‖2‖g‖2 = λ
√P[A](1− P[A])P[B](1− P[B])
Also called Expander Mixing Lemma.Used a lot in computer science, e.g. in (de)randomization.
Elchanan Mossel Mixing in Product Spaces
Short Term Mixing for Markov Chains
Theorem∣∣∣P[X 0 ∈ A,X 1 ∈ B]− P[A]P[B]∣∣∣ is upper bounded by
λ√
P[A](1− P[A])P[B](1− P[B])
Shows: mixing in one step for large sets.Proof: 1A = P[A]1 + f , 1B = P[B]1 + g , where f , g ⊥ 1
P[X 0 ∈ A,X 1 ∈ B] =1k
(P[A]1 + f )tM(P[B]1 + g)
= P[A]P[B] +1kf tMg ,
1k|f tMg | ≤ λ‖f ‖2‖g‖2 = λ
√P[A](1− P[A])P[B](1− P[B])
Also called Expander Mixing Lemma.Used a lot in computer science, e.g. in (de)randomization.
Elchanan Mossel Mixing in Product Spaces
The tensor property
Consider (Y1,Z1), . . . , (Yn,Zn) which are drawnindependently from the distribution of (X 0,X 1).
Equivalently, the transition matrix from Y = (Y1, . . . ,Yn) toZ = (Z1, . . . ,Zn) is M⊗n.
Thm =⇒ that for any sets A,B ⊂ [k]n:∣∣∣P[Y ∈ A,Z ∈ B]−P[A]P[B]∣∣∣ ≤ λ√P[A](1− P[A])P[B](1− P[B])
Follows immediately from tensorization of the spectrum.
Elchanan Mossel Mixing in Product Spaces
The tensor property
Consider (Y1,Z1), . . . , (Yn,Zn) which are drawnindependently from the distribution of (X 0,X 1).
Equivalently, the transition matrix from Y = (Y1, . . . ,Yn) toZ = (Z1, . . . ,Zn) is M⊗n.
Thm =⇒ that for any sets A,B ⊂ [k]n:∣∣∣P[Y ∈ A,Z ∈ B]−P[A]P[B]∣∣∣ ≤ λ√P[A](1− P[A])P[B](1− P[B])
Follows immediately from tensorization of the spectrum.
Elchanan Mossel Mixing in Product Spaces
The tensor property
Consider (Y1,Z1), . . . , (Yn,Zn) which are drawnindependently from the distribution of (X 0,X 1).
Equivalently, the transition matrix from Y = (Y1, . . . ,Yn) toZ = (Z1, . . . ,Zn) is M⊗n.
Thm =⇒ that for any sets A,B ⊂ [k]n:∣∣∣P[Y ∈ A,Z ∈ B]−P[A]P[B]∣∣∣ ≤ λ√P[A](1− P[A])P[B](1− P[B])
Follows immediately from tensorization of the spectrum.
Elchanan Mossel Mixing in Product Spaces
Log Sobolev inequalities
Entropy, Log Sobolev and hyper-contraction
A similar story could be told using more sophisticated analytictools. Easier to work with Markov semi-groups Tt = e−tL.
Entropy, Dirchelet Form
Ent(f ) = E(f log f )− Ef · logEf
E(f , g) = E(fLg) = E(gLf ) = E(g , f ) = − ddtEfTtg
∣∣∣t=0
.
Definition of Log-Sob
p-logSob(C) ⇐⇒ ∀f ,Ent(f p) ≤ Cp2
4(p−1)E(f p−1, f ) (p 6= 0, 1)
1-logSob(C) ⇐⇒ ∀f ,Ent(f ) ≤ C4 E(f , log f )
0-logSob(C) ⇐⇒ ∀f ,Var(log f ) ≤ −C2 E(f , 1/f )
Elchanan Mossel Mixing in Product Spaces
Log Sobolev inequalities
Entropy, Log Sobolev and hyper-contraction
A similar story could be told using more sophisticated analytictools. Easier to work with Markov semi-groups Tt = e−tL.
Entropy, Dirchelet Form
Ent(f ) = E(f log f )− Ef · logEf
E(f , g) = E(fLg) = E(gLf ) = E(g , f ) = − ddtEfTtg
∣∣∣t=0
.
Definition of Log-Sob
p-logSob(C) ⇐⇒ ∀f ,Ent(f p) ≤ Cp2
4(p−1)E(f p−1, f ) (p 6= 0, 1)
1-logSob(C) ⇐⇒ ∀f ,Ent(f ) ≤ C4 E(f , log f )
0-logSob(C) ⇐⇒ ∀f ,Var(log f ) ≤ −C2 E(f , 1/f )
Elchanan Mossel Mixing in Product Spaces
Log Sobolev inequalities
Entropy, Log Sobolev and hyper-contraction
A similar story could be told using more sophisticated analytictools. Easier to work with Markov semi-groups Tt = e−tL.
Entropy, Dirchelet Form
Ent(f ) = E(f log f )− Ef · logEf
E(f , g) = E(fLg) = E(gLf ) = E(g , f ) = − ddtEfTtg
∣∣∣t=0
.
Definition of Log-Sob
p-logSob(C) ⇐⇒ ∀f ,Ent(f p) ≤ Cp2
4(p−1)E(f p−1, f ) (p 6= 0, 1)
1-logSob(C) ⇐⇒ ∀f ,Ent(f ) ≤ C4 E(f , log f )
0-logSob(C) ⇐⇒ ∀f ,Var(log f ) ≤ −C2 E(f , 1/f )
Elchanan Mossel Mixing in Product Spaces
Log Sob. Inequalities and Hyper-Contraction
Hyper-Contraction (Gross, Nelson 1960 ... )
r -logSob with constant C implies
‖Tt f ‖p ≤ ‖f ‖q, t ≥ C
4log
p − 1q − 1
, 1 < p < q < r or r ′ < q < p
=⇒ |E[g(X0)f (Xt)]| = |E [gTt f | ≤ ‖g‖p′‖Tf ‖p ≤ ‖g‖p′‖f ‖qIf f = 1A and g = 1B , get:
P[X0 ∈ A,Xt ∈ B] ≤ ‖1A‖q‖1B‖p′ = P[A]1/qP[B]1/p′,
Now optimize over norms to get a better bound than CS.
Elchanan Mossel Mixing in Product Spaces
Reverse-Hyper-Contraction
Log-Sobolev and Rev. Hyper-Contraction(M-Oleszkiewicz-Sen-13)
Let Tt = e−tL be a general Markov semi-group satisfying
2-Logsob with constant C or
1-Logsob inequality with constant C .
Then for all q < p < 1, all positive f , g and all t ≥ C4 log 1−q1−p it
holds that
‖Tt f ‖q ≥ ‖f ‖p =⇒
E[g(X0)f (Xt)] = E [gTt f ] ≥ ‖g‖q′‖f ‖p
Elchanan Mossel Mixing in Product Spaces
Short-Time Implications
Theorem (M-Oleszkiewicz-Sen-13 ; Short-Time Implications)
Let Tt = e−tL, where L satisfy 1 or 2-LogSob inequality withconstant C . Let A,B ⊂ Ωn with P[A] ≥ ε and P[B] ≥ ε. Then:
P[X (0) ∈ A,X (t) ∈ B] ≥ ε2
1−e−2t/C
Comments
1. Works for small sets too.2. Tensorizes.3. Some examples where it is (almost) tight.4. Uses in social choice analysis, queuing theory.
Elchanan Mossel Mixing in Product Spaces
Short-Time Implications
Theorem (M-Oleszkiewicz-Sen-13 ; Short-Time Implications)
Let Tt = e−tL, where L satisfy 1 or 2-LogSob inequality withconstant C . Let A,B ⊂ Ωn with P[A] ≥ ε and P[B] ≥ ε. Then:
P[X (0) ∈ A,X (t) ∈ B] ≥ ε2
1−e−2t/C
Comments
1. Works for small sets too.2. Tensorizes.3. Some examples where it is (almost) tight.4. Uses in social choice analysis, queuing theory.
Elchanan Mossel Mixing in Product Spaces
Comment: typical application MCMC
Long Time Behavior
Log Sobolev inequalities play a major role in analyzing long termmixing of Markov chains, in particular in analysis of mixing times(Diaconis, Saloff-Coste etc.)
Long Time Behavior
The ε-total variation mixing time of a finite Markov chain isbounded by:
1λ
(log(1/π∗) + log(1/ε))
1C
(log log(1/π∗) + log(1/ε))
for a continuous time Markov chain with spectral gap λ and2-LogSob C .
Elchanan Mossel Mixing in Product Spaces
Comment: typical application MCMC
Long Time Behavior
Log Sobolev inequalities play a major role in analyzing long termmixing of Markov chains, in particular in analysis of mixing times(Diaconis, Saloff-Coste etc.)
Long Time Behavior
The ε-total variation mixing time of a finite Markov chain isbounded by:
1λ
(log(1/π∗) + log(1/ε))
1C
(log log(1/π∗) + log(1/ε))
for a continuous time Markov chain with spectral gap λ and2-LogSob C .
Elchanan Mossel Mixing in Product Spaces
What are these lectures about?
High Dimensional Phenomena
High dimensional mixing: mixing of product processes on productspaces Ωn with n large.
Tight bounds
For which processes, given measures a and b can we findprecise upper/lower bounds for
sup(P[X0 ∈ A,Xt ∈ B] : P[A] = a,P[B] = b
)Interested in product space/processes of dimension n andanswers as n→∞.
Most important examples / techniques from probability /analysis.
Elchanan Mossel Mixing in Product Spaces
What are these lectures about?
Mulit-step prcoesses
How to bound P[X0 ∈ A0,X1 ∈ A1, . . . ,Xk ∈ Ak ] forprocesses X0, . . . ,Xk?
Interested in product space/processes of dimension n andanswers as n→∞.
Most important examples / techniques from additivecombinatorics.
Elchanan Mossel Mixing in Product Spaces
What are these lectures about?
And more
Theory that does both?
Applications?
Elchanan Mossel Mixing in Product Spaces
Today: tight bounds
Borell’s result.
Open Problem: The Boolean cube.
The state of affairs - partition into 3 parts or more.
Elchanan Mossel Mixing in Product Spaces
Two Examples: Gaussian, Boolean
Correlated pairs (M-O’Donnell-Regev-Steif-Sudakov-05):
Let x , y ∈ −1, 1n be e−t correlated:
x is chosen uniformly and y is Tt correlated version.
i.e. E[xiyi ] = e−t for all i independently
Let A,B ⊂ −1, 1n1/2 with P[A] ≥ ε and P[B] ≥ ε
Then: P[x ∈ A, y ∈ B] ≥ ε2
1−e−t
Easy to prove when A = B ...
Gaussian Version
Let x , y ∈ Rn two Gaussian vectors:
x ∼ N(0, 1), y ∼ N(0, 1),E [xiyj ] = e−tδi ,j
Let A,B ⊂ Rn with P[A] ≥ ε and P[B] ≥ ε
Then: P[x ∈ A, y ∈ B] ≥ ε2
1−e−t
Elchanan Mossel Mixing in Product Spaces
Two Examples: Gaussian, Boolean
Correlated pairs (M-O’Donnell-Regev-Steif-Sudakov-05):
Let x , y ∈ −1, 1n be e−t correlated:
x is chosen uniformly and y is Tt correlated version.
i.e. E[xiyi ] = e−t for all i independently
Let A,B ⊂ −1, 1n1/2 with P[A] ≥ ε and P[B] ≥ ε
Then: P[x ∈ A, y ∈ B] ≥ ε2
1−e−t
Easy to prove when A = B ...
Gaussian Version
Let x , y ∈ Rn two Gaussian vectors:
x ∼ N(0, 1), y ∼ N(0, 1),E [xiyj ] = e−tδi ,j
Let A,B ⊂ Rn with P[A] ≥ ε and P[B] ≥ ε
Then: P[x ∈ A, y ∈ B] ≥ ε2
1−e−t
Elchanan Mossel Mixing in Product Spaces
Borell’s Result and Open Problems
Borell (85): In Gaussian case the maximum and minimum ofP[x ∈ A, y ∈ B] as a function of P[A] and P[B] is obtainedfor parallel half-spaces.
Do not know what is the optimum in −1, 1n. In particular:
Open Problem:
limn→∞
min(P[X ∈ A,Y ∈ B] : A,B ⊂ −1, 1n,P[A] = P[B] = 1/4)
and similarly for max.
Partition to 3 or more parts even in Gaussian space.
Elchanan Mossel Mixing in Product Spaces
Borell’s Result and Open Problems
Borell (85): In Gaussian case the maximum and minimum ofP[x ∈ A, y ∈ B] as a function of P[A] and P[B] is obtainedfor parallel half-spaces.
Do not know what is the optimum in −1, 1n. In particular:
Open Problem:
limn→∞
min(P[X ∈ A,Y ∈ B] : A,B ⊂ −1, 1n,P[A] = P[B] = 1/4)
and similarly for max.
Partition to 3 or more parts even in Gaussian space.
Elchanan Mossel Mixing in Product Spaces
Borell’s Result and Open Problems
Borell (85): In Gaussian case the maximum and minimum ofP[x ∈ A, y ∈ B] as a function of P[A] and P[B] is obtainedfor parallel half-spaces.
Do not know what is the optimum in −1, 1n. In particular:
Open Problem:
limn→∞
min(P[X ∈ A,Y ∈ B] : A,B ⊂ −1, 1n,P[A] = P[B] = 1/4)
and similarly for max.
Partition to 3 or more parts even in Gaussian space.
Elchanan Mossel Mixing in Product Spaces
Borell’s Result and Open Problems
Borell (85): In Gaussian case the maximum and minimum ofP[x ∈ A, y ∈ B] as a function of P[A] and P[B] is obtainedfor parallel half-spaces.
Do not know what is the optimum in −1, 1n. In particular:
Open Problem:
limn→∞
min(P[X ∈ A,Y ∈ B] : A,B ⊂ −1, 1n,P[A] = P[B] = 1/4)
and similarly for max.
Partition to 3 or more parts even in Gaussian space.
Elchanan Mossel Mixing in Product Spaces
If there is time before the break ...
A cute proof of a special case of Borell’s result.
Connections to social choice Theory.
Elchanan Mossel Mixing in Product Spaces
Simple Example 1
Cosmic coin problem(M-O’Donnell-05):
x ∈ −1, 1n uniform.
(y i )m1 conditionally independent given x .
Each pair (x , y i ) is ρ-correlated.
Problem: What is the largest P[y1 ∈ A, . . . ym ∈ A] can be?
Elchanan Mossel Mixing in Product Spaces
Simple Example 2
(y i ,j)1≤i<j≤m is an exchangeable collection of vectors in−1, 1n.
If |I ∩ J| = 1 then yI , yJ are −1/3 correlated.
Otherwise independent.
Why?
If n voters rank alternatives uniformly at random, the pairwisepreferences between alternatives will be given by the collectiony .
Elchanan Mossel Mixing in Product Spaces
Full support finite Ω using hyper-contraction
Thm: More General Reverse Hypercontractivity Theorem(M-Oleszkiewicz-Sen-13)
Let a the measure Ψ over a finite Ωk satisfyminx1,...,xk∈Ω Pr[X1 = x , . . . ,Xk = xk ] = α > 0 and have equalmarginals.
Consider the distribution Ψn and let A1, . . . ,Ak ⊆ Ωn, µ(Ai ) ≥ µ.Then:
Pr[X1 ∈ A1, . . .Xk ∈ Ak ] ≥ µO( 1α) ,
where (X1(i), . . . ,Xk(i)) are i.i.d. according to Ψ.
Note
This is a key tool of analyzing the examples above as well as manyothers.
Elchanan Mossel Mixing in Product Spaces
Full support finite Ω using hyper-contraction
Thm: More General Reverse Hypercontractivity Theorem(M-Oleszkiewicz-Sen-13)
Let a the measure Ψ over a finite Ωk satisfyminx1,...,xk∈Ω Pr[X1 = x , . . . ,Xk = xk ] = α > 0 and have equalmarginals.Consider the distribution Ψn and let A1, . . . ,Ak ⊆ Ωn, µ(Ai ) ≥ µ.Then:
Pr[X1 ∈ A1, . . .Xk ∈ Ak ] ≥ µO( 1α) ,
where (X1(i), . . . ,Xk(i)) are i.i.d. according to Ψ.
Note
This is a key tool of analyzing the examples above as well as manyothers.
Elchanan Mossel Mixing in Product Spaces
Notation
X X 1 X 2 . . . X i. . . X n
X (1) X(1)1 X
(1)2
· · · X(1)i
· · · X(1)n
X (2) X(2)1 X
(2)2
· · · X(2)i
· · · X(2)n
......
......
...
X (j) X(j)1 X
(j)2
· · · X(j)i
· · · X(j)n
......
......
...
X (`) X(`)1 X
(`)2
· · · X(`)i
· · · X(`)n
Tuples X i arei.i.d. according to P. Themarginals of P are πj .
Vectors X (j) aredistributedaccording toπj := πnj .
Distributedaccording toP := Pn.
α(P) := minx∈ΩP(x , x , . . . , x)
ρ(P) : See Definition ??
X(j)i ∈ Ω
X (j) ∈ Ω := Ωn
X i ∈ Ω := Ω`
X ∈ Ω := Ωn·`
S ⊆ Ω
Figure: Naming of the random variables in the general case. The columnsX i are distributed i .i .d according to P. Each X
(j)i is distributed
according to πj . The overall distribution of X is P.
Elchanan Mossel Mixing in Product Spaces
Lower Bounds
We are mostly interested in two types of lower bounds:
Set hitting: Lower bounds on
P[X 1 ∈ A1, . . . ,Xk ∈ Ak ]
in terms of P[A1], . . . ,P[Ak ]
Same set hitting: Lower bounds on
P[X 1 ∈ A, . . . ,X k ∈ A]
in terms of P[A].
Set hitting will require something ... - e.g.X 1 = X 2 = . . . = X k .
Elchanan Mossel Mixing in Product Spaces
Gaussian Bounds
Borell (85) k = 2 - parallel half-spaces are optimal (alsoIsaksson-Mossel, Neeman)
By a Reverse Brascamp-Lieb inq. (Ledoux,Chen-Dafnis-Paouris 14-15) for A, . . .C ⊂ Rn:
P[U ∈ A, . . . ,Z ∈ C ] ≥ (P[A] · · ·P[C ])1/(1−ρ2),
where ρ is the second eigenvalue of Σ.
Doesn’t require independence of coordinates
Elchanan Mossel Mixing in Product Spaces
Full Support Case
Thm: More General Reverse Hypercontractivity Theorem(M-Oleszkiewicz-Sen-13)
Let a the measure Ψ over a finite Ωk satisfyminx1,...,xk∈Ω Pr[X1 = x , . . . ,Xk = xk ] = α > 0 and have equalmarginals.
Then:
Pr[X1 ∈ A1, . . .Xk ∈ Ak ] ≥ µO( 1α) ,
where (X1(i), . . . ,Xk(i)) are i.i.d. according to Ψ.
Elchanan Mossel Mixing in Product Spaces
Full Support Case
Thm: More General Reverse Hypercontractivity Theorem(M-Oleszkiewicz-Sen-13)
Let a the measure Ψ over a finite Ωk satisfyminx1,...,xk∈Ω Pr[X1 = x , . . . ,Xk = xk ] = α > 0 and have equalmarginals.Then:
Pr[X1 ∈ A1, . . .Xk ∈ Ak ] ≥ µO( 1α) ,
where (X1(i), . . . ,Xk(i)) are i.i.d. according to Ψ.
Elchanan Mossel Mixing in Product Spaces
Non full support?
What if the support of Ω is not full?
Do we care?
Maybe: This is what additive combinatorics is all about.
In particular: finite cominatorics in finite field models(Green-04 ... ).
Many other applications in combinatorics and computerscience.
Elchanan Mossel Mixing in Product Spaces
Non full support?
What if the support of Ω is not full?
Do we care?
Maybe: This is what additive combinatorics is all about.
In particular: finite cominatorics in finite field models(Green-04 ... ).
Many other applications in combinatorics and computerscience.
Elchanan Mossel Mixing in Product Spaces
Additive combinatorics perspective
Example:
Theorem (Finite Field Roth Theorem)
Y ,R be chosen uniformly at random at F n3 .
Then for every µ > 0 there exists c(µ) > 0,N(µ) such that ifn ≥ N(µ) and
A ⊂ F n3 satisfies P[A] ≥ µ, then:
P[Y ∈ A,Y + R ∈ A,Y + 2R ∈ A] ≥ c(µ).
Why is this true?
Elchanan Mossel Mixing in Product Spaces
Additive combinatorics perspective
Example:
Theorem (Finite Field Roth Theorem)
Y ,R be chosen uniformly at random at F n3 .
Then for every µ > 0 there exists c(µ) > 0,N(µ) such that ifn ≥ N(µ) and
A ⊂ F n3 satisfies P[A] ≥ µ, then:
P[Y ∈ A,Y + R ∈ A,Y + 2R ∈ A] ≥ c(µ).
Why is this true?
Elchanan Mossel Mixing in Product Spaces
Fourier Obstructions
Theorem (Finite Field Roth Theorem - Analysis)
Let Y ,R be chosen uniformly at random at F n3 . Let A,B,C ⊂ F n
3then
|P[Y ∈ A,Y + R ∈ B,Y + 2R ∈ C ]− P[A]P[B]P[C ]| ≤ ‖A‖∞
Only obstruction to uniformity is linear structure If A = B = C ,high Fourier coefficient =⇒ can restrict to linear subspace withhigher denisty Density increase arguments ...
Elchanan Mossel Mixing in Product Spaces
Fourier Obstructions
Theorem (Finite Field Roth Theorem - Analysis)
Let Y ,R be chosen uniformly at random at F n3 . Let A,B,C ⊂ F n
3then
|P[Y ∈ A,Y + R ∈ B,Y + 2R ∈ C ]− P[A]P[B]P[C ]| ≤ ‖A‖∞
Only obstruction to uniformity is linear structure
If A = B = C ,high Fourier coefficient =⇒ can restrict to linear subspace withhigher denisty Density increase arguments ...
Elchanan Mossel Mixing in Product Spaces
Fourier Obstructions
Theorem (Finite Field Roth Theorem - Analysis)
Let Y ,R be chosen uniformly at random at F n3 . Let A,B,C ⊂ F n
3then
|P[Y ∈ A,Y + R ∈ B,Y + 2R ∈ C ]− P[A]P[B]P[C ]| ≤ ‖A‖∞
Only obstruction to uniformity is linear structure If A = B = C ,high Fourier coefficient =⇒ can restrict to linear subspace withhigher denisty
Density increase arguments ...
Elchanan Mossel Mixing in Product Spaces
Fourier Obstructions
Theorem (Finite Field Roth Theorem - Analysis)
Let Y ,R be chosen uniformly at random at F n3 . Let A,B,C ⊂ F n
3then
|P[Y ∈ A,Y + R ∈ B,Y + 2R ∈ C ]− P[A]P[B]P[C ]| ≤ ‖A‖∞
Only obstruction to uniformity is linear structure If A = B = C ,high Fourier coefficient =⇒ can restrict to linear subspace withhigher denisty Density increase arguments ...
Elchanan Mossel Mixing in Product Spaces
Higher Order Arithmetic Obstructions
Furstenberg-Weiss (80s): For longer arithmetic progressions,obstructions other than Fourier.
Gowers: Obstructions can be identified using the Gowersnorms.
Again - use obstruction to your benefit.
Thm: (Gowers 08; Rodel and Skokan 04,06):
If q is prime and ` ≤ q thenfor every µ > 0 there exists c(µ) > 0,N(µ) such that ifn ≥ N(µ) andA ⊂ F n
q satisfies P[A] ≥ µ, then:
P[Y ∈ A,Y + R ∈ A, . . . ,Y + (`− 1)R ∈ A] ≥ c(µ),
where A,R ∈ F nq are chosen uniformly at random.
Question: Is the additive structure necessary?
Elchanan Mossel Mixing in Product Spaces
Higher Order Arithmetic Obstructions
Furstenberg-Weiss (80s): For longer arithmetic progressions,obstructions other than Fourier.
Gowers: Obstructions can be identified using the Gowersnorms.
Again - use obstruction to your benefit.
Thm: (Gowers 08; Rodel and Skokan 04,06):
If q is prime and ` ≤ q thenfor every µ > 0 there exists c(µ) > 0,N(µ) such that ifn ≥ N(µ) andA ⊂ F n
q satisfies P[A] ≥ µ, then:
P[Y ∈ A,Y + R ∈ A, . . . ,Y + (`− 1)R ∈ A] ≥ c(µ),
where A,R ∈ F nq are chosen uniformly at random.
Question: Is the additive structure necessary?
Elchanan Mossel Mixing in Product Spaces
Obstruction to Chaos
Consider the support of Ω as a graph G with vertex V = allatoms with non-zero weight and edges between any twoatoms that differ in one coordinate.
We say that ρ < 1 if the graph G is connected.
More formally:
Definition
ρ(P,S ,T ) := supCov [f (X (S)), g(X (T ))]
∣∣∣ f : Ω(S) → R, g : Ω(T ) → R,
Var [f (X (S))] = Var [g(X (T ))] = 1.
The correlation of P is ρ(P) := maxj∈[`] ρ (P, j, [`] \ j).
Elchanan Mossel Mixing in Product Spaces
The quest for a unifying theory
Is there one theory that explains both the noisy examples and theadditive theory?
Elchanan Mossel Mixing in Product Spaces
Example
Let X be uniform in F n3 .
Let Yi = Xi or Xi + 1 with probability 1/2 independently foreach coordinate.
Theorem =⇒ P[X ∈ A,Y ∈ a] ≥ c(P[A]).
Motivation from understanding “parallel repetition”.
Does not follow from hyper-contraction nor does it followfrom additive techniques ...
Elchanan Mossel Mixing in Product Spaces
A General Result
Theorem (+ Hazla, Holenstein)
Suppose (X ,Y ) is distributed in a finite Ω2 such that:
α = minaP[X = Y = a] > 0.
P[X = a] = P[Y = a] for all a.
Then for any set A ⊂ Ωn with PX⊗n [A] = PY⊗n [A] ≥ µ it holdsthat
P[X ∈ A,Y ∈ A] ≥ c(α, µ) > 0
Our c is pretty bad:
c = 1/ exp(exp(exp(1/(µ)D))), D = D(α)
Related to the fact that the proof is interesting:1 Lose in “Regularity Lemma” type arguments.2 Lose in “Invariance” transforming the problem to a Gaussian
problem.
Elchanan Mossel Mixing in Product Spaces
A General Result
Theorem (+ Hazla, Holenstein)
Suppose (X ,Y ) is distributed in a finite Ω2 such that:
α = minaP[X = Y = a] > 0.
P[X = a] = P[Y = a] for all a.
Then for any set A ⊂ Ωn with PX⊗n [A] = PY⊗n [A] ≥ µ it holdsthat
P[X ∈ A,Y ∈ A] ≥ c(α, µ) > 0
Our c is pretty bad:
c = 1/ exp(exp(exp(1/(µ)D))), D = D(α)
Related to the fact that the proof is interesting:1 Lose in “Regularity Lemma” type arguments.2 Lose in “Invariance” transforming the problem to a Gaussian
problem.
Elchanan Mossel Mixing in Product Spaces
A Markov Chain Theorem and a general process theorem
Theorem[+Hazla, Holenstein]
Xi ,Yi ,Zi , . . . ,Wi be a Markov chain over Ω withminx∈Ω Pr[Xi = Yi = Zi = . . .Wi = x ] = β > 0 and uniformmarginals.Let A ⊆ Ωn, µ(A) = µ > 0.
Pr[X ∈ A ∧ Y ∈ A ∧ Z ∈ A, . . . ,∧W ∈ A] ≥ f (µ, β) > 0 .
Theorem[+Hazla, Holenstein]
Xi ,Yi ,Zi , . . . ,Wi be distributed over Ωk withminx∈Ω Pr[Xi = Yi = Zi = . . .Wi = x ] = β > 0 and uniformmarginals. Suppose further that ρ(Xi ,Yi , . . . ,Wi ) < 1. LetA ⊆ Ωn, µ(A) = µ > 0.
Pr[X ∈ A ∧ Y ∈ A ∧ Z ∈ A, . . . ,∧W ∈ A] ≥ f (µ, β) > 0 .
Elchanan Mossel Mixing in Product Spaces
The condition ρ < 1
Weaker than full support.
Does not hold in arithmetic setups.
ρ = 1 iff the support of Ψ is connected with respect tochanging one coordinate at a time.
Example: (x , y) ∈ F 23 where y = x , x + 1 has ρ < 1 but notfull support.
Elchanan Mossel Mixing in Product Spaces
The condition ρ < 1
Weaker than full support.
Does not hold in arithmetic setups.
ρ = 1 iff the support of Ψ is connected with respect tochanging one coordinate at a time.
Example: (x , y) ∈ F 23 where y = x , x + 1 has ρ < 1 but notfull support.
Elchanan Mossel Mixing in Product Spaces
The condition ρ < 1
Weaker than full support.
Does not hold in arithmetic setups.
ρ = 1 iff the support of Ψ is connected with respect tochanging one coordinate at a time.
Example: (x , y) ∈ F 23 where y = x , x + 1 has ρ < 1 but notfull support.
Elchanan Mossel Mixing in Product Spaces
The condition ρ < 1
Weaker than full support.
Does not hold in arithmetic setups.
ρ = 1 iff the support of Ψ is connected with respect tochanging one coordinate at a time.
Example: (x , y) ∈ F 23 where y = x , x + 1 has ρ < 1 but notfull support.
Elchanan Mossel Mixing in Product Spaces
Open Problems
Still searching for unified theory.
Concrete Example: Suppose Ψ is uniform over
(0, 0, 0), (1, 1, 1), (2, 2, 2), (0, 1, 2), (1, 2, 0), (2, 0, 1)
ρ = 1 but not arithmetic.
Do not understand.
Elchanan Mossel Mixing in Product Spaces
Questions??
Thank you!
Elchanan Mossel Mixing in Product Spaces
Questions??
Thank you!
Elchanan Mossel Mixing in Product Spaces