Mixing Times of the Restricted Rook's Walk and a ...university-logo-udel Mixing Times of the...

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Mixing Times of the Restricted Rook’s Walkand a Generalized Curie-Weiss Model

Benjamin Savoie, Ana Wright, and Renjun ZhuUniversity of Michigan-Flint, Willamette University, and

University of California, Berkeley

Faculty Advisor: Peter T. Otto, Willamette University

July 29, 2016

As part of the Willamette Valley REU


Outline

BackgroundMarkov ChainsMixing TimeCouplingPath Coupling

Restricted Rook’s WalkFar / Near RestrictionCouplingsMixing Time Bounds / Long Term Behavior

Generalized Curie Weiss ModelEquilibrium Phase StrucureDynamic Phase StructureβM vs. βC


Motivation

7 0Z0Z0Z0Z6 Z0Z0Z0Z05 0Z0S0Z0Z4 Z0Z0Z0Z03 0Z0Z0Z0Z2 Z0Z0Z0Z01 0Z0Z0Z0Z0 Z0Z0Z0Z0

0 1 2 3 4 5 6 7

If the rook has made one move, what do we know about itsstarting position?If the rook has made two moves, what do we know about itsstarting position?


Markov Chains

7 0Z0Z0Z0Z6 Z0Z0Z0Z05 0Z0S0Z0Z4 Z0Z0Z0Z03 0Z0Z0Z0Z2 Z0Z0Z0Z01 0Z0Z0Z0Z0 Z0Z0Z0Z0

0 1 2 3 4 5 6 7

Markov chain: a sequence of random variables/vectorsX1,X2, . . . such that

P[Xt+1 = xt+1|Xt = xt ,Xt−1 = xt−1, . . . ,X0 = x0] = P[Xt+1 = xt+1|Xt = xt ]

Where x0, . . . , xt are states at time t , and Ω is the state space.


Markov chains

Transition probability:

p(x , y) = P(Xt+1 = y |Xt = x)

Transition matrix:P = [p(x , y)]

Distribution at time t : P t (x , ·) = P(Xt = ·|X0 = x)This is the x-th row of the transition matrix.


Convergence Theorem

For irreducible and aperiodic Markov chains

P t (x , ·) =⇒ π as t →∞where π is the unique stationary distribution of the chain, i.e.πP = π.

1 RZ0 Z0

0 1

5 0Z0Z0Z4 Z0Z0Z03 0Z0Z0Z2 Z0Z0Z01 RZ0Z0Z0 Z0Z0Z0

0 1 2 3 4 5


Mixing time

Total variation distance

‖µ− ν‖TV := maxA⊂Ω|µ(A)− ν(A)| =

12

∑x∈Ω

|µ(x)− ν(x)|

Distance from stationarity

d(t) := maxx∈Ω‖P t (x , ·)− π‖TV

Mixing time: a measure of the convergence rate of thechain to its stationary distribution.

tmix(ε) := mint : d(t) ≤ ε


Bounding the Mixing Time

A coupling of two distributions µ and ν is a pair of randomvariables (X ,Y ) on a common probability space with marginalsµ and ν.

7 0Z0Z0Z0Z6 Z0Z0Z0Z05 0Z0Z0Z0Z4 Z0Z0Z0Z03 0Z0s0Z0Z2 Z0Z0Z0Z01 RZ0Z0Z0Z0 Z0Z0Z0Z0

0 1 2 3 4 5 6 7


Coupling Markov Chains

Coupling Inequality [LPW]:

‖µ− ν‖TV = ‖P t (x , ·)− P t (y , ·)‖TV ≤ P(Xt 6= Yt )

Proof: For any A ⊂ Ω,

µ(A)−ν(A) = P(Xt ∈ A)−P(Yt ∈ A) ≤ P(Xt ∈ A,Yt /∈ A) ≤ P(Xt 6= Yt )

So,

d(t) = maxx∈Ω‖P t (x , ·)− π‖TV ≤ max

x ,y‖P t (x , ·)− P t (y , ·)‖TV

≤ maxx ,y

P(Xt 6= Yt )


Coupling of Markov Chains

With a metric ρ on our state space in conjunction with Markov’sinequality and results about d(t) give us:

d(t) ≤ maxx ,y

P(Xt 6= Yt ) = maxx ,y

P[ρ(Xt ,Yt ) ≥ 1] ≤maxx ,y

E [ρ(Xt ,Yt )]

1

Mean coupling distance: maxx ,y

E [ρ(Xt ,Yt )]



Want to show contraction after one step:

E [ρ(Xt ,Yt )|xt−1, yt−1] = (1− α)ρ(xt−1, yt−1) ≤ e−αρ(xt−1, yt−1)

with 0 < α < 1

Iteration gives:

E [ρ(Xt ,Yt )] ≤ e−αtE [ρ(X0,Y0)]



Sinced(t) ≤ max

x ,yE [ρ(Xt ,Yt )]

and:d(tmix ) ≤ ε

then:maxx ,y

E [ρ(X0,Y0)]e−αt ≤ ε

thus we have the Mixing Time Theorem:

tmix (ε) ≤ 1α

log

maxx ,y

E [ρ(X0,Y0)]

ε


Path Coupling

If neighboring pairs contract, then a pair of chains will contractfrom any two states distance r apart.

Xt = x0, x1, ..., xr = Yt

7 0Z0Z0Z0Z6 Z0Z0Z0Z05 0Z0Z0Z0Z4 Z0Z0Z0Z03 0Z0s0Z0Z2 Z0Z0Z0Z01 RZ0Z0Z0Z0 Z0Z0Z0Z0

0 1 2 3 4 5 6 7

E [ρ(Xt ,Yt )] ≤r∑

i=1E [ρ(Xt ,i ,Xt ,i−1)]



Overall goal: Find a ’good’ coupling; i.e a coupling thatcontracts with each time step. (α > 0)

E [ρ(Xt ,Yt )|xt−1, yt−1] ≤ e−αρ(xt−1, yt−1) = e−α


”Good” Coupling

1 A ’good’ coupling rule encourages two rooks to meet asfast as possible.

2 One approach we made was to require that after one step,ρ(Xt ,Yt ) = 0 or 1, so that we could guarantee the rookswould contract.

3 It is possible to have bigger ρ(Xt ,Yt ) for coupling, and onaverage they might sill contract.


Literature Review: Unrestricted

7 0Z0Z0Z0Z6 Z0Z0Z0s05 0Z0Z0Z0Z4 Z0Z0Z0Z03 0Z0Z0Z0Z2 Z0Z0Z0Z01 0S0Z0Z0Z0 Z0Z0Z0Z0

0 1 2 3 4 5 6 7

Theorem [MORS]

Mixing time bound: tmix (ε) ≤⌈

log( dε

)

log d(n−1)(d−1)(n−1)+1

⌉≤⌈

d(n−1)n−2 log(d

ε )⌉


Definition: Far Restricted Rook’s Walk

Legal Moves: K = 1,2,3 = bn2c − r

9 0Z0Z0Z0Z0Z8 Z0Z0Z0Z0Z07 0Z0Z0Z0Z0Z6 Z0Z0Z0Z0Z05 0Z0Z0Z0Z0Z4 Z0Z0Z0Z0Z03 0Z0Z0s0Z0Z2 Z0Z0Z0Z0Z01 0Z0Z0Z0Z0Z0 Z0Z0Z0Z0Z0

0 1 2 3 4 5 6 7 8 9

10Z0Z0Z0Z0Z0Z9 0Z0Z0Z0Z0Z08 Z0Z0Z0Z0Z0Z7 0Z0Z0Z0Z0Z06 Z0Z0Z0Z0Z0Z5 0Z0Z0Z0Z0Z04 Z0Z0Z0Z0Z0Z3 0Z0Z0s0Z0Z02 Z0Z0Z0Z0Z0Z1 0Z0Z0Z0Z0Z00 Z0Z0Z0Z0Z0Z

0 1 2 3 4 5 6 7 8 9 10

n = 10, r = 2, n = 11, r = 2


Method: Far Restriction

Coupling Rule 1: Move to Common Accessible set, so they willmatch in 1 dimension.

9 0Z0Z0Z0Z0Z8 Z0Z0Z0Z0Z07 0Z0S0Z0Z0Z6 Z0Z0Z0Z0Z05 0Z0Z0Z0Z0Z4 Z0Z0Z0Z0Z03 0Z0Z0ZrZ0Z2 Z0Z0Z0Z0Z01 0Z0Z0Z0Z0Z0 Z0Z0Z0Z0Z0

0 1 2 3 4 5 6 7 8 9

9 0Z0Z0Z0Z0Z8 Z0Z0Z0Z0Z07 0Z0ZRZ0Z0Z6 Z0Z0Z0Z0Z05 0Z0Z0Z0Z0Z4 Z0Z0Z0Z0Z03 0Z0ZrZ0Z0Z2 Z0Z0Z0Z0Z01 0Z0Z0Z0Z0Z0 Z0Z0Z0Z0Z0

0 1 2 3 4 5 6 7 8 9

n = 10, r = 2, so K = 1,2,3



Coupling Rule 2: Move to circle set (accessible for one, notthe other). ”First Match First”, etc.

9 0Z0Z0Z0Z0Z8 Z0Z0Z0Z0Z07 0Z0S0Z0Z0Z6 Z0Z0Z0Z0Z05 0Z0Z0Z0Z0Z4 Z0Z0Z0Z0Z03 0Z0Z0ZrZ0Z2 Z0Z0Z0Z0Z01 0Z0Z0Z0Z0Z0 Z0Z0Z0Z0Z0

0 1 2 3 4 5 6 7 8 9

9 0Z0Z0Z0Z0Z8 Z0Z0Z0Z0Z07 RZ0Z0Z0Z0Z6 Z0Z0Z0Z0Z05 0Z0Z0Z0Z0Z4 Z0Z0Z0Z0Z03 0Z0Z0Z0s0Z2 Z0Z0Z0Z0Z01 0Z0Z0Z0Z0Z0 Z0Z0Z0Z0Z0

0 1 2 3 4 5 6 7 8 9

n = 10, r = 2, so K = 1,2,3



Coupling Rule 3: Swap.

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n = 10, r = 2, so K = 1,2,3



Coupling Rule 4: Move to Common Accessible square, sothey will match.

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0 1 2 3 4 5 6 7 8 9

n = 10, r = 2, so K = 1,2,3


Far Restriction: NOT Contract

What if n = 8 and r = 2 ???=⇒ρ(Xt ,Yt ) increases

E [ρ(Xt ,Yt )|ρ(xt−1, yt−1) = 1] = 28 · 2 + 4+1

8 · 1 + 18 · 0 = 9

8 > 1

7 0Z0Z0Z0Z6 S0s0Z0Z05 0Z0Z0Z0Z4 Z0Z0Z0Z03 0Z0Z0Z0Z2 Z0Z0Z0Z01 0Z0Z0Z0Z0 Z0Z0Z0Z0

0 1 2 3 4 5 6 7

Here, we have ρ(Xt ,Yt ) = 2 in 2 cases with K = 1,2.


Far Restriction: Condition

Claim: For even n, if n ≥ 6r − 2, then ρ(Xt ,Yt ) = 1,0 for allneighboring pairs.

Idea: Prevent increase in ρ(Xt ,Yt ). The number of inaccessiblesquares is n − 2k − 1 = 2r − 1. This needs to be reached byother rook. So:

n − 2k − 1 ≤ n2 − r

n ≥ 6r − 2

0 S0ZrZ0Z0Z00 1 2 3 4 5 6 7 8 9

For r = 2 Example, 8 = n < 6 · 2− 2 = 10, condition fails!!

0 S0s0Z0Z00 1 2 3 4 5 6 7


Result: Far Restriction Condition

Theorem [OSWZ’16]

For any far restricted rook’s walk on an nd board with legalmoves K = 1,2, ..., bn

2c − r, if n ≥ 6r − 2 for even n, andn ≥ 6r + 1 for odd n, then

1 diam(Ω) = 2d ;2 ρ(Xt ,Yt ) = 1,0 for all neighboring pairs ρ(Xt−1,Yt−1).


Analysis: Mean Coupling Distance

Condition: n ≥ 6r − 2.

Eeven[ρ(Xt+1,Yt+1) | ρ(xt , yt ) = 1] = (d−1d ) · 1 + (2r−1)+1

2d( n2−r)

· 1 +[1− d−1

d − (2r−1)+12d( n

2−r)

]·0 = 1−

[1d −

rd( n

2−r)

]= 1−

[ n2−2r

d( n2−r)

]≤ e−α

120Z0Z0Z0Z0Z0Z11Z0Z0Z0Z0Z0Z0100Z0Z0Z0Z0Z0Z9Z0Z0Z0Z0Z0Z080Z0Z0Z0Z0Z0Z7S0Z0s0Z0Z0Z060Z0Z0Z0Z0Z0Z5Z0Z0Z0Z0Z0Z040Z0Z0Z0Z0Z0Z3Z0Z0Z0Z0Z0Z020Z0Z0Z0Z0Z0Z1Z0Z0Z0Z0Z0Z0

1 2 3 4 5 6 7 8 9 10 11 12


Result: Mixing Bound (Far Restriction)

Contraction bound:

E[ρ(Xt + 1,Yt + 1)|ρ(xt , yt ) = 1] ≤ e−α·diam(Ω) = 2d

Mixing bound for Restricted Rook’s Walk:

Theorem [OSWZ’16]

Mixing Time Bound: tmix (ε) ≤⌈

d(n−2r)n−4r log

(2dε

)⌉, for even n.


Result: Odd n (Far Restriction)

Condition: n ≥ 6r + 1

Eodd [ρ(Xt+1,Yt+1) | ρ(xt , yt ) = 1] =

(d−1d ) · 1 + (2r)+1

2d( n−12 −r)

· 1 + (1− d−1d − 2r+1

2d( n−12 )−r

) · 0 =

1−[

1d −

2r+1d(n−2r−1)

]= 1−

[n−4r−2

d(n−2r−1)

]≤ e−α

Theorem [OSWZ’16]


d(n−2r−1)n−4r−2 log

(2dε

)⌉, for odd n.


Far Restriction: Mixing Time Bound Behavior

limn→∞

Eeven/odd [ρ(Xt ,Yt )|ρ(xt−1, yt−1) = 1] =d − 1

d= 1− 1

d

CorollaryFor both even and odd n,

limn→∞

tmix (ε) ≤⌈

d log(

2dε

)⌉

CorollaryMixing Time Bound tmix (ε) strictly increases with r .


Definition: Near Restriction

For the near restriction, K = r + 1, r + 2, ...,⌊n

2

⌋, where r is

the restriction.

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Example: n = 12, r = 2, so K = 3,4,5,6


Method: Near Restriction

Coupling Rule: ”First to First, No Swap!!!” Circle set maintainsthe same distance apart, ρ(Xt ,Yt ) = 1.

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0 1 2 3 4 5 6 7 8 9 10 11

n = 12, r = 2, so K = 3,4,5,6,


Result: ρ(Xt ,Yt) Theorem (Near Restriction)

Theorem [OSWZ’16]

For any neighboring pair of near restricted rook’s walk on an nd

board with legal movesK = r + 1, r + 2, ..., bn

2c, ρ(Xt ,Yt ) = 0,1.

0 S0Z0ZrZ0Z0Z0Z0 1 2 3 4 5 6 7 8 9 10 11 12

n = 13, r = 2, K = 3,4,5,6Common Accessible set: ρ(Xt ,Yt ) = 0Rest: ρ(Xt ,Yt ) = 1, i.e. 3→ 11,4→ 12, etc. 5 apart.


Result: Mixing Time Bound (Near Restriction)

Condition n ≥ 4r + 4 for even n, and n ≥ 4r + 3 for odd n;

then:Eeven/odd [ρ(Xt ,Yt ) | ρ(xt−1, yt−1) = 1] =

(d−1d ) · 1 + ( 2r+1

d(n−2r−1) ) · 1 + (1− d−1d − 2r+1

d(n−2r−1) ) · 0 =

1−[

1d −

2r+1d(n−2r−1)

]= 1− n−4r−2

d(n−2r−1) ≤ e−α

Theorem [OSWZ’16]



(2dε

)⌉


Near Restriction: Mixing Time Bound Behavior

limn→∞

Eeven/odd [ρ(Xt ,Yt )|ρ(xt−1, yt−1 = 1] =d − 1

d= 1− 1

d

Corollary

limn→∞

tmix (ε) ≤⌈d log

(2dε

)⌉Corollarytmix (ε) strictly increases with r


Different Restriction

Corollary

For both the far and near restricted rook’s walk, if r = nf , then

limn→∞ tmix (ε) ≤ d(f−2)f−4 log

(2dε

)Corollary

For a near or far restricted rook’s walk, if r = o(n), i.e. r = n1p ,

then limn→∞ tmix (ε) ≤⌈d · log

(2dε )⌉


Summary: Unrestricted Rook’s Walk

1 No Restriction: r = 0tmix (ε) ≤

⌈d(n−1)

n−2 log(2dε )⌉

2 Far Restriction: r < n6

tmix (ε) ≤⌈

d(n−2r)n−4r log

(2dε

)⌉, for even n.

tmix (ε) ≤⌈


(2dε

)⌉, for odd n.

3 Near Restriction: r < n4

tmix (ε) ≤⌈


(2dε

)⌉Remark: As n→∞, all mixting time bound converge to thebound of

⌈d · log(2d

ε )⌉.


Statistical Mechanics

”In statistical mechanics, one derives macroscopicproperties of a substance from a probability distributionthat describes the complicated interactions among theindividual constituent particles.” [1]


Curie Weiss Model

In the Curie Weiss model, there are n particles, each withspin +1 or −1. A state is a complete configurationω ∈ −1,1n that describes the spin at each particle.Sn(ω) is the sum of all the spins of ω.

Microscopic quantity: Spin at each particle.

Macroscopic quantity: The mean spin / magnetizationSn(ω)

n .


Curie Weiss Model

A particular configuration ω where Sn(ω)n = − 1

15 .


Gibbs Ensemble

The stationary distribution is given by the Gibbs Ensemble:

Pn,β(ω) =1

Zn(β)enβg( Sn(ω)

n )n∏

i=1

ρ(ωi)

The partition function Zn(β) =∑ω∈Ω

enβg( Sn(ω)n )

n∏i=1

ρ(ωi)

normalizes the probabilities and β = 1T .

g(Sn(ω)n ) defines the interaction between particles.


Our Generalized Curie-Weiss Model

In the classical CW model, g(x) = x2.

We instead have:

g(x) =α1

4!x4 +

α2

2!x2

α1 > 0 and α2 > 0 represent interaction strengths.

The Gibbs Ensemble then becomes:

Pn,β(ω) =1

Zn(β)enβ α1

4!

(Sn(ω)

n

)4+α22!

(Sn(ω)

n

)2 n∏i=1

ρ(ωi)


Phase Transition Structure

In the Curie Weiss model, we see how changing thetemperature (1/β) affects the equilibrium structure anddynamic structure.

Equilibrium Structure: How many global minimizers ofthe free energy function Gβ(z)?

Dynamic Structure: Does the Glauber dynamics Markovchain mix rapidly or slowly? fβ(z)


Equilibrium Phase Transition Structure

Gβ(Sn/n) for second order continuous phase transition.

Gβ(Sn/n) for first order discontinuous phase transition.


Glauber Dynamics

Glauber dynamics defines a Markov chain that is guaranteedto converge to a given stationary distribution (Gibbs Ensemble).


Glauber Dynamics

Glauber dynamics have local update probabilities.

p±1(ω, k) =enβ(

α14!

( S(ω,k)±1n )4+

α22!

( S(ω,k)±1n )2)

enβ(α14!

( S(ω,k)+1n )4+

α22!

( S(ω,k)+1n )2) + enβ(

α14!

( S(ω,k)−1n )4+

α22!

( S(ω,k)−1n )2)


Greedy Coupling

Let Xt = σ and Yt = τ where σ and τ are any two differentconfigurations on −1,1n. With a common source ofrandomness, U ∈ (0,1), define the greedy coupling:

X (k) =

−1 if 0 ≤ U ≤ p−1(σ, k)

+1 if p−1(σ, k) ≤ U ≤ 1

Y (k) =

−1 if 0 ≤ U ≤ p−1(τ, k)

+1 if p−1(τ, k) ≤ U ≤ 1

Define the metric ρ(s, t) = 12Σn

i=1|si − ti |.


Aggregate Path Coupling

With standard path coupling, contraction is requiredbetween all neighboring configurations.

With aggregate path coupling, contraction is only requiredbetween a configuration close to the equilibrium and anyother configuration.


Mean Coupling Distance

Let σ be a configuration close to the origin and τ be any otherconfiguration that is neighboring σ. Then the mean couplingdistance is:

Eσ,τ [ρ(X ,Y )] ≤ 1−(

1n− 1

2

)[fβ

(Sn(σ)

n

)− fβ

(Sn(τ)

n

)]+O

(1n2

)

fβ(

Sn(σ)n

)− fβ

(Sn(τ)

n

)(

Sn(σ)n

)−(

Sn(τ)n

) ≤ α′

Eσ,τ [ρ(X ,Y )] ≤ 1+α′ − 1

n+O

(1n2

)


Continuous Phase Transition (α1 < 2α2)

-1.0 -0.5 0.5 1.0

Sn (ω)

n

-1.0

-0.5

0.5

1.0

0 < β < βC



-1.0 -0.5 0.5 1.0

Sn (ω)

n

-1.0

-0.5

0.5

1.0

β = βC



-1.0 -0.5 0.5 1.0

Sn (ω)

n

-1.0

-0.5

0.5

1.0

β > βC


Disontinuous Phase Transition (α1 > 2α2)

-1.0 -0.5 0.5 1.0

Sn (ω)

n

-1.0

-0.5

0.5

1.0

0 < β < βM



-1.0 -0.5 0.5 1.0

Sn (ω)

n

-1.0

-0.5

0.5

1.0

β = βM



-1.0 -0.5 0.5 1.0

Sn (ω)

n

-1.0

-0.5

0.5

1.0

βM < β < βC



-1.0 -0.5 0.5 1.0

Sn (ω)

n

-1.0

-0.5

0.5

1.0

β = βC


Curie-Weiss Conclusion

Main Results:βM = sup

β>0α′β < 1

βC = supβ>0Gβ(0) = G′β(0) = 0,G′′β(0) ≥ 0

For the second order continuous phase transition,βM = βC .

For the first order discontinuous phase transition, βM < βC .


Acknowledgments

Willamette Valley REU Consortium for MathematicsResearchNational Science Foundation for funding the REUDr. Peter Otto, faculty mentor


References

S. Kim,Mixing time of a Rook’s Walk.(2012)

D. Levin, Y. Peres, E. Wilmer,Markov Chains and Mixing Times.American Mathematical Society, USA (2009)

C. Mcleman, P. Otto, J. Rahmani , M. SutterMixing Times For The Rook’s Walk Via Path Coupling(2014)

R. S. Ellis,Entropy, Large Deviations and Statistical Mechanics2006

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