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
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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
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Motivation
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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?
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Markov Chains
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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.
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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.
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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
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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
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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.
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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 )
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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 )
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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 )
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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 )]
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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 )]
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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 )]
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Coupling of Markov Chains
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)]
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Coupling of Markov Chains
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
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Path Coupling
If neighboring pairs contract, then a pair of chains will contractfrom any two states distance r apart.
Xt = x0, x1, ..., xr = Yt
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E [ρ(Xt ,Yt )] ≤r∑
i=1E [ρ(Xt ,i ,Xt ,i−1)]
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Coupling of Markov Chains
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
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”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.
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Literature Review: Unrestricted
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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 )⌉
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Definition: Far Restricted Rook’s Walk
Legal Moves: K = 1,2,3 = bn2c − r
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n = 10, r = 2, n = 11, r = 2
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Method: Far Restriction
Coupling Rule 1: Move to Common Accessible set, so they willmatch in 1 dimension.
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n = 10, r = 2, so K = 1,2,3
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Method: Far Restriction
Coupling Rule 1: Move to Common Accessible set, so they willmatch in 1 dimension.
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n = 10, r = 2, so K = 1,2,3
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Method: Far Restriction
Coupling Rule 2: Move to circle set (accessible for one, notthe other). ”First Match First”, etc.
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n = 10, r = 2, so K = 1,2,3
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Method: Far Restriction
Coupling Rule 2: Move to circle set (accessible for one, notthe other). ”First Match First”, etc.
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n = 10, r = 2, so K = 1,2,3
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Method: Far Restriction
Coupling Rule 3: Swap.
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n = 10, r = 2, so K = 1,2,3
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Method: Far Restriction
Coupling Rule 3: Swap.
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n = 10, r = 2, so K = 1,2,3
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Method: Far Restriction
Coupling Rule 4: Move to Common Accessible square, sothey will match.
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n = 10, r = 2, so K = 1,2,3
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Method: Far Restriction
Coupling Rule 4: Move to Common Accessible square, sothey will match.
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n = 10, r = 2, so K = 1,2,3
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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
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Here, we have ρ(Xt ,Yt ) = 2 in 2 cases with K = 1,2.
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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
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For r = 2 Example, 8 = n < 6 · 2− 2 = 10, condition fails!!
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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).
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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
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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.
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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.
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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]
Mixing Time Bound: tmix (ε) ≤⌈
d(n−2r−1)n−4r−2 log
(2dε
)⌉, for odd n.
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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 .
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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
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Method: Near Restriction
Coupling Rule: ”First to First, No Swap!!!” Circle set maintainsthe same distance apart, ρ(Xt ,Yt ) = 1.
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n = 12, r = 2, so K = 3,4,5,6,
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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.
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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]
Mixing Time Bound: tmix (ε) ≤⌈
d(n−2r−1)n−4r−2 log
(2dε
)⌉
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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
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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 )⌉
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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) ≤⌈
d(n−2r−1)n−4r−2 log
(2dε
)⌉, for odd n.
3 Near Restriction: r < n4
tmix (ε) ≤⌈
d(n−2r−1)n−4r−2 log
(2dε
)⌉Remark: As n→∞, all mixting time bound converge to thebound of
⌈d · log(2d
ε )⌉.
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Statistical Mechanics
”In statistical mechanics, one derives macroscopicproperties of a substance from a probability distributionthat describes the complicated interactions among theindividual constituent particles.” [1]
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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 .
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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 .
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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 .
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Curie Weiss Model
A particular configuration ω where Sn(ω)n = − 1
15 .
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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.
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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.
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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.
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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.
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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)
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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)
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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)
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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)
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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)
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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)
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Equilibrium Phase Transition Structure
Gβ(Sn/n) for second order continuous phase transition.
Gβ(Sn/n) for first order discontinuous phase transition.
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Glauber Dynamics
Glauber dynamics defines a Markov chain that is guaranteedto converge to a given stationary distribution (Gibbs Ensemble).
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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)
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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 |.
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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 |.
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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.
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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.
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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
)
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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
)
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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
)
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Phase Transition Structure
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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
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Continuous Phase Transition (α1 < 2α2)
-1.0 -0.5 0.5 1.0
Sn (ω)
n
-1.0
-0.5
0.5
1.0
β = βC
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Continuous Phase Transition (α1 < 2α2)
-1.0 -0.5 0.5 1.0
Sn (ω)
n
-1.0
-0.5
0.5
1.0
β > βC
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Phase Transition Structure
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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
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Disontinuous Phase Transition (α1 > 2α2)
-1.0 -0.5 0.5 1.0
Sn (ω)
n
-1.0
-0.5
0.5
1.0
β = βM
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Disontinuous Phase Transition (α1 > 2α2)
-1.0 -0.5 0.5 1.0
Sn (ω)
n
-1.0
-0.5
0.5
1.0
βM < β < βC
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Disontinuous Phase Transition (α1 > 2α2)
-1.0 -0.5 0.5 1.0
Sn (ω)
n
-1.0
-0.5
0.5
1.0
β = βC
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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 .
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Acknowledgments
Willamette Valley REU Consortium for MathematicsResearchNational Science Foundation for funding the REUDr. Peter Otto, faculty mentor
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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