Glauber Dynamics for Ising Model on the
Complete Graph
David A. Levin, U. Oregon
Malwina Luczak, London School of Economics
Yuval Peres, Berkeley and Microsoft Research
March 2008
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Ising model
Let Gn = (Vn , En ) be a graph with n = |Vn | < ∞ vertices.
The nearest-neighbor Ising model on Gn is the probability distribution
on {−1, 1}Vn given by



 X
σ(u)σ(v) ,
µ(σ) = Z(β)−1 exp β
(u,v)∈En
where σ ∈ {−1, 1}Vn .
The interaction strength β is a parameter which has physical
interpretation as 1/T, where T = temperature.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Three regimes
High temperature (β < βc ):
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Three regimes
low temperature (β > βc ),
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Three regimes
critical temperature (β = βc ),
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Glauber dynamics
The (single-site) Glauber dynamics for µ is a Markov chain (Xt )
having µ as its stationary distribution.
Transitions are made from state σ as follows:
1 a vertex v is chosen uniformly at random from V .
n
2 The new state σ0 agrees with σ everywhere except possibly at v,
where σ0 (v) = 1 with probability
eβS(σ,v)
eβS(σ,v) + e−βS(σ,v)
where
S(σ, v) :=
X
σ(w).
w:w∼v
Note the probability above equals the µ-conditional probability of a
positive spin at v, given that that all spins agree with σ at vertices
different from v.
Note that the Glauber dynamics is reversible with respect to the Gibbs
measure µ.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Glauber dynamics
The (single-site) Glauber dynamics for µ is a Markov chain (Xt )
having µ as its stationary distribution.
Transitions are made from state σ as follows:
1 a vertex v is chosen uniformly at random from V .
n
2 The new state σ0 agrees with σ everywhere except possibly at v,
where σ0 (v) = 1 with probability
eβS(σ,v)
eβS(σ,v) + e−βS(σ,v)
where
S(σ, v) :=
X
σ(w).
w:w∼v
Note the probability above equals the µ-conditional probability of a
positive spin at v, given that that all spins agree with σ at vertices
different from v.
Note that the Glauber dynamics is reversible with respect to the Gibbs
measure µ.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Glauber dynamics
The (single-site) Glauber dynamics for µ is a Markov chain (Xt )
having µ as its stationary distribution.
Transitions are made from state σ as follows:
1 a vertex v is chosen uniformly at random from V .
n
2 The new state σ0 agrees with σ everywhere except possibly at v,
where σ0 (v) = 1 with probability
eβS(σ,v)
eβS(σ,v) + e−βS(σ,v)
where
S(σ, v) :=
X
σ(w).
w:w∼v
Note the probability above equals the µ-conditional probability of a
positive spin at v, given that that all spins agree with σ at vertices
different from v.
Note that the Glauber dynamics is reversible with respect to the Gibbs
measure µ.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Mixing time
For a Markov chain (Xt ) on state-space Ω and with stationary
distribution π, write
d(t) = max kP{Xt ∈ · | X0 = x} − πkTV
x∈Ω
Define
tmix () := min{t ≥ 0 : d(t) ≤ }.
Consider now a sequence of Markov chains, (Xtn ), and write dn (t) and
n ().
tmix
A sequence of Markov chains has a cutoff if
n ()
tmix
n (1 − )
tmix
→1
Levin, Luczak, Peres
as
n → ∞.
Glauber Dynamics for Ising Model on the Complete Graph
Cutoff more precisely
The Glauber dynamics is said to exhibit a cut-off at {tn } with window
{wn } if wn = o(tn ) and
lim lim inf dn (tn − γwn ) = 1,
γ→∞ n→∞
lim lim sup dn (tn + γwn ) = 0.
γ→∞
n→∞
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
A conjecture
For the Glauber dynamics on graph sequences with bounded degree,
n = Ω(n log n).
tmix
(T. Hayes and A. Sinclair)
Conjecture (due to Y. Peres): If the Glauber dynamics for a
n = O(n log n), then there is a
sequence of transitive graphs satisfies tmix
cut-off.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Mean field case
Take Gn = Kn , the complete
graph on the n vertices: Vn = {1, . . . , n},
n
and En contains all 2 possible edges.
The total interaction strength should be O(1), so replace β by β/n.
The probability of updating to a +1 is then
eβ(S−σ(v))/n
eβ(S−σ(v))/n + e−β(S−σ(v))/n
where S is the total magnetization
S=
n
X
σ(i).
i=1
The statistic S is almost sufficient for determining the updating
probability.
Now, in this case St = S(Xt ) is a Markov chain in its own right; (St )
will be key to analysis of the dynamics.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Mean field case
Take Gn = Kn , the complete
graph on the n vertices: Vn = {1, . . . , n},
n
and En contains all 2 possible edges.
The total interaction strength should be O(1), so replace β by β/n.
The probability of updating to a +1 is then
eβ(S−σ(v))/n
eβ(S−σ(v))/n + e−β(S−σ(v))/n
where S is the total magnetization
S=
n
X
σ(i).
i=1
The statistic S is almost sufficient for determining the updating
probability.
Now, in this case St = S(Xt ) is a Markov chain in its own right; (St )
will be key to analysis of the dynamics.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Mean field case
Take Gn = Kn , the complete
graph on the n vertices: Vn = {1, . . . , n},
n
and En contains all 2 possible edges.
The total interaction strength should be O(1), so replace β by β/n.
The probability of updating to a +1 is then
eβ(S−σ(v))/n
eβ(S−σ(v))/n + e−β(S−σ(v))/n
where S is the total magnetization
S=
n
X
σ(i).
i=1
The statistic S is almost sufficient for determining the updating
probability.
Now, in this case St = S(Xt ) is a Markov chain in its own right; (St )
will be key to analysis of the dynamics.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Mean field has tmix = O(n log n)
A consequence (see, e.g., Aizenman and Holley (1987)) of the
Dobrushin-Shlosman uniqueness criterion: For the Glauber dynamics
on Kn , if β < 1, then
tmix = O(n log n).
(See also Bubley and Dyer (1998).)
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
High temperature mean-field
Theorem (L.-L.-P.)
Let (Xtn ) be the Glauber dynamics for the Ising model on Kn . If β < 1,
n log n
then tmix () = (1 + o(1)) 2(1−β)
and there is a cut-off.
In fact, we show that there is window of size O(n) centered about
tn =
1
n log n.
2(1 − β)
That is,
lim sup dn (tn + γn) → 0
as
γ → ∞.
n
and
lim inf dn (tn + γn) → 1
n
Levin, Luczak, Peres
as
γ → −∞.
Glauber Dynamics for Ising Model on the Complete Graph
Critical temperature mean-field
Theorem (L.-L.-P.)
Let (Xtn ) be the Glauber dynamics for the Ising model on Kn . If β = 1,
then there are constants c1 and c2 so that
c1 n3/2 ≤ tmix ≤ c2 n3/2 .
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Low temperature mean-field
If β > 1, then
n
tmix
> c1 ec2 n .
This can be established using Cheeger constant – there is a bottleneck
going between states with positive magnetization and states with
negative magnetization.
Our results show that once this barrier to mixing is removed, the
mixing time is reduced to n log n.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Low temperature mean-field
If β > 1, then
n
tmix
> c1 ec2 n .
This can be established using Cheeger constant – there is a bottleneck
going between states with positive magnetization and states with
negative magnetization.
Our results show that once this barrier to mixing is removed, the
mixing time is reduced to n log n.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Low temperature mean-field
If β > 1, then
n
tmix
> c1 ec2 n .
This can be established using Cheeger constant – there is a bottleneck
going between states with positive magnetization and states with
negative magnetization.
Our results show that once this barrier to mixing is removed, the
mixing time is reduced to n log n.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Lower bound via Cheeger constant
Let
X
Q(A, Ac ) =
π(x)P(x, y).
x∈A,y∈Ac
If
ΦA =
Q(A, Ac )
,
π(A)
then for π(A) ≤ 1/2,
tmix ≥
Levin, Luczak, Peres
1
.
4ΦA
Glauber Dynamics for Ising Model on the Complete Graph
The bottleneck
Let Ak be those configurations with k spins equal to +1.Then
µ(Abαnc ) =
1 −n[φ(α)+o(1)]
e
.
Z(β)
The function φ changes shape at β = 1:
0.2
0.4
0.6
0.8
0.695
0.69
0.685
0.68
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
The bottleneck
If Ω+ are configurations with strictly positive magnetization,
Q(Ω+ , (Ω+ )c ) exp n[φ(1/2) + o(1)]
≤
.
π(Abn/2c )
exp n[φ(α0 ) + o(1)]
If β > 1, there is α0 so that φ(α0 ) > φ(1/2) and then
φS ≤ c1 e−c2 n .
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Truncated dynamics for low temperature mean-field
If the bottleneck at zero magnetization is removed by truncating the
dynamics at zero magnetization, then the chain converges fast:
Theorem (L.-L.-P.)
Let β > 1. Let (Xt ) be the Glauber dynamics on Kn , restricted to the
set of configurations with non-negative magnetization. Then
n = O(n log n).
tmix
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Proof idea for low temperature
Use coupling: Show that for arbitrary starting states, can run together
two copies of the chain so that the chains meet with high probability
in O(n log n) steps.
First show that the magnetizations will agree after O(n log n)
steps, when chains are run independently. (Hard part – involves
hitting time calculations.)
After magnetizations agree, couple the chains as below.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Proof idea for low temperature
Use coupling: Show that for arbitrary starting states, can run together
two copies of the chain so that the chains meet with high probability
in O(n log n) steps.
First show that the magnetizations will agree after O(n log n)
steps, when chains are run independently. (Hard part – involves
hitting time calculations.)
After magnetizations agree, couple the chains as below.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Proof idea for low temperature
Use coupling: Show that for arbitrary starting states, can run together
two copies of the chain so that the chains meet with high probability
in O(n log n) steps.
First show that the magnetizations will agree after O(n log n)
steps, when chains are run independently. (Hard part – involves
hitting time calculations.)
After magnetizations agree, couple the chains as below.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
A coupling (any temperature)
Write (Xt ) and (X̃t ) for the two chains. We assume that S(Xt ) = S(X̃t ).
Let J be the vertex selected for updating in Xt , and let s ∈ {−1, 1} be
the spin used to update Xt (J).
The X̃-chain will also be updated with the spin s at a site J̃ which has
X̃t (J̃) = Xt (J), although it will not always be that J = J̃.
If Xt (J) = X̃t (J), then update both chains at J.
If Xt (J) , X̃t (J), then pick J̃ uniformly at random from
{i : X̃t (i) , Xt (i) and X̃t (i) = Xt (J)}.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
A coupling (any temperature)
Write (Xt ) and (X̃t ) for the two chains. We assume that S(Xt ) = S(X̃t ).
Let J be the vertex selected for updating in Xt , and let s ∈ {−1, 1} be
the spin used to update Xt (J).
The X̃-chain will also be updated with the spin s at a site J̃ which has
X̃t (J̃) = Xt (J), although it will not always be that J = J̃.
If Xt (J) = X̃t (J), then update both chains at J.
If Xt (J) , X̃t (J), then pick J̃ uniformly at random from
{i : X̃t (i) , Xt (i) and X̃t (i) = Xt (J)}.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
A coupling (any temperature)
Write (Xt ) and (X̃t ) for the two chains. We assume that S(Xt ) = S(X̃t ).
Let J be the vertex selected for updating in Xt , and let s ∈ {−1, 1} be
the spin used to update Xt (J).
The X̃-chain will also be updated with the spin s at a site J̃ which has
X̃t (J̃) = Xt (J), although it will not always be that J = J̃.
If Xt (J) = X̃t (J), then update both chains at J.
If Xt (J) , X̃t (J), then pick J̃ uniformly at random from
{i : X̃t (i) , Xt (i) and X̃t (i) = Xt (J)}.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
A coupling (any temperature)
Write (Xt ) and (X̃t ) for the two chains. We assume that S(Xt ) = S(X̃t ).
Let J be the vertex selected for updating in Xt , and let s ∈ {−1, 1} be
the spin used to update Xt (J).
The X̃-chain will also be updated with the spin s at a site J̃ which has
X̃t (J̃) = Xt (J), although it will not always be that J = J̃.
If Xt (J) = X̃t (J), then update both chains at J.
If Xt (J) , X̃t (J), then pick J̃ uniformly at random from
{i : X̃t (i) , Xt (i) and X̃t (i) = Xt (J)}.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
A coupling (any temperature)
Write (Xt ) and (X̃t ) for the two chains. We assume that S(Xt ) = S(X̃t ).
Let J be the vertex selected for updating in Xt , and let s ∈ {−1, 1} be
the spin used to update Xt (J).
The X̃-chain will also be updated with the spin s at a site J̃ which has
X̃t (J̃) = Xt (J), although it will not always be that J = J̃.
If Xt (J) = X̃t (J), then update both chains at J.
If Xt (J) , X̃t (J), then pick J̃ uniformly at random from
{i : X̃t (i) , Xt (i) and X̃t (i) = Xt (J)}.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
A coupling, continued
If Dt is the number of sites where Xt and X̃t disagree, then when
Dt ≥ 0,
c1 Dt .
E[Dt+1 | Ft ] ≤ 1 −
n
It takes O(n log n) steps to drive this expectation down to .
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
A coupling, continued
If Dt is the number of sites where Xt and X̃t disagree, then when
Dt ≥ 0,
c1 Dt .
E[Dt+1 | Ft ] ≤ 1 −
n
It takes O(n log n) steps to drive this expectation down to .
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Low temperature in more detail
How exactly do we restrict the dynamics? Whenever in a state with
non-negative magnetization and a move is proposed to a state η with
negative magnetization, we move to −η instead.
Show there is a coupling (Xt+ , X̃t+ ) of restricted dynamics started from
σ, σ̃ such that
lim sup Pσ,σ̃ (τmag > cn log n) → 0
as c → ∞.
n→∞
Here τmag is the first time t such that St+ = S̃t+ .
By monotonicity, it is enough to consider σ = 0 and σ̃ = 1.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Low temperature in more detail
How exactly do we restrict the dynamics? Whenever in a state with
non-negative magnetization and a move is proposed to a state η with
negative magnetization, we move to −η instead.
Show there is a coupling (Xt+ , X̃t+ ) of restricted dynamics started from
σ, σ̃ such that
lim sup Pσ,σ̃ (τmag > cn log n) → 0
as c → ∞.
n→∞
Here τmag is the first time t such that St+ = S̃t+ .
By monotonicity, it is enough to consider σ = 0 and σ̃ = 1.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Low temperature in more detail
How exactly do we restrict the dynamics? Whenever in a state with
non-negative magnetization and a move is proposed to a state η with
negative magnetization, we move to −η instead.
Show there is a coupling (Xt+ , X̃t+ ) of restricted dynamics started from
σ, σ̃ such that
lim sup Pσ,σ̃ (τmag > cn log n) → 0
as c → ∞.
n→∞
Here τmag is the first time t such that St+ = S̃t+ .
By monotonicity, it is enough to consider σ = 0 and σ̃ = 1.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Low temperature in more detail
How exactly do we restrict the dynamics? Whenever in a state with
non-negative magnetization and a move is proposed to a state η with
negative magnetization, we move to −η instead.
Show there is a coupling (Xt+ , X̃t+ ) of restricted dynamics started from
σ, σ̃ such that
lim sup Pσ,σ̃ (τmag > cn log n) → 0
as c → ∞.
n→∞
Here τmag is the first time t such that St+ = S̃t+ .
By monotonicity, it is enough to consider σ = 0 and σ̃ = 1.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Low temperature in more detail
How exactly do we restrict the dynamics? Whenever in a state with
non-negative magnetization and a move is proposed to a state η with
negative magnetization, we move to −η instead.
Show there is a coupling (Xt+ , X̃t+ ) of restricted dynamics started from
σ, σ̃ such that
lim sup Pσ,σ̃ (τmag > cn log n) → 0
as c → ∞.
n→∞
Here τmag is the first time t such that St+ = S̃t+ .
By monotonicity, it is enough to consider σ = 0 and σ̃ = 1.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Low temperature continued
We couple both these, StB and StT with an equilibrium copy, St , and we
try to get them to “cross over”.
For this, we need to consider hitting times:
τ1 = min{t ≥ 0 : StT ≤ s∗ + c1 n−1/2 }
τ2 = min{t ≥ 0 : StB ≥ s∗ + c2 n−1/2 },
where s∗ is the unique positive solution of tanh(βs) = s.
These hitting times can be analysed, as nS(Xt+ )/2 is a birth-and-death
chain.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Low temperature continued
We couple both these, StB and StT with an equilibrium copy, St , and we
try to get them to “cross over”.
For this, we need to consider hitting times:
τ1 = min{t ≥ 0 : StT ≤ s∗ + c1 n−1/2 }
τ2 = min{t ≥ 0 : StB ≥ s∗ + c2 n−1/2 },
where s∗ is the unique positive solution of tanh(βs) = s.
These hitting times can be analysed, as nS(Xt+ )/2 is a birth-and-death
chain.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Low temperature continued
We couple both these, StB and StT with an equilibrium copy, St , and we
try to get them to “cross over”.
For this, we need to consider hitting times:
τ1 = min{t ≥ 0 : StT ≤ s∗ + c1 n−1/2 }
τ2 = min{t ≥ 0 : StB ≥ s∗ + c2 n−1/2 },
where s∗ is the unique positive solution of tanh(βs) = s.
These hitting times can be analysed, as nS(Xt+ )/2 is a birth-and-death
chain.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Birth-and-death chain hitting times
Let (Zt ) be a birth-and-death chain on {0, . . . , N} with transition
probabilities
pk = P(Zt+1 − Zt = +1 | Zt = k),
k = 0, . . . , N − 1,
qk = P(Zt+1 − Zt = −1 | Zt = k),
k = 1, . . . , N,
rk = P(Zt+1 − Zt = 0 | Zt = k),
k = 0, . . . , N.
Let π = (π(k)) be the stationary measure.
Let Zt(`) be a restriction of Zt to {0, . . . , `}. Note that π(`) (k) ∼ π(k) for
k = 0, . . . , `.
Then for ` = 0, 1, . . . , N − 1,
1
π(`) (`)
= 1 + q` E`−1 (τ` ).
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Birth-and-death chain hitting times
Let (Zt ) be a birth-and-death chain on {0, . . . , N} with transition
probabilities
pk = P(Zt+1 − Zt = +1 | Zt = k),
k = 0, . . . , N − 1,
qk = P(Zt+1 − Zt = −1 | Zt = k),
k = 1, . . . , N,
rk = P(Zt+1 − Zt = 0 | Zt = k),
k = 0, . . . , N.
Let π = (π(k)) be the stationary measure.
Let Zt(`) be a restriction of Zt to {0, . . . , `}. Note that π(`) (k) ∼ π(k) for
k = 0, . . . , `.
Then for ` = 0, 1, . . . , N − 1,
1
π(`) (`)
= 1 + q` E`−1 (τ` ).
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Birth-and-death chain hitting times
Let (Zt ) be a birth-and-death chain on {0, . . . , N} with transition
probabilities
pk = P(Zt+1 − Zt = +1 | Zt = k),
k = 0, . . . , N − 1,
qk = P(Zt+1 − Zt = −1 | Zt = k),
k = 1, . . . , N,
rk = P(Zt+1 − Zt = 0 | Zt = k),
k = 0, . . . , N.
Let π = (π(k)) be the stationary measure.
Let Zt(`) be a restriction of Zt to {0, . . . , `}. Note that π(`) (k) ∼ π(k) for
k = 0, . . . , `.
Then for ` = 0, 1, . . . , N − 1,
1
π(`) (`)
= 1 + q` E`−1 (τ` ).
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Hitting times for magnetization
Let `∗ = bns∗ c. Then
√
√
E`−1 [τ` ] ≤
n(1 + O(n−1/2 )), 1 ≤ ` ≤ C n
√
C1 n
E`−1 [τ` ] ≤
, C n ≤ ` ≤ `∗ /2
`
√
E`−1 [τ` ] ≤ C2 n`∗ − `, `∗ /2 ≤ ` ≤ `∗ − C n
√
√
√
E`−1 [τ` ] ≤
n(1 + O(n−1/2 )), `∗ − C n ≤ ` ≤ `∗ + C n
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Magnetization chain: key equation
If St =
Pn
i=1 Xt (i),
then for St ≥ 0,
"
#
St
E[St+1 − St | Ft ] ≈ −
− tanh(βSt /n) .
n
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
β<1
When β < 1, using the inequality tanh(x) ≤ x for x ≥ 0 shows that for
St ≥ 0,
!
1−β
E[St+1 | Ft ] ≤ St 1 −
n
√
Need [2(1 − β)]−1 n log n steps to drive E[St ] to n.
Additional O(n) steps needed for magnetization to hit zero. (Compare
with simple random walk.)
Can couple two versions of the chain so that the magnetizations agree
by the time the magnetization of the top chain hits zero.
Once magnetizations agree, use a two-dimensional process to bound
time until full configurations agree. Takes an additional O(n) steps.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
β<1
When β < 1, using the inequality tanh(x) ≤ x for x ≥ 0 shows that for
St ≥ 0,
!
1−β
E[St+1 | Ft ] ≤ St 1 −
n
√
Need [2(1 − β)]−1 n log n steps to drive E[St ] to n.
Additional O(n) steps needed for magnetization to hit zero. (Compare
with simple random walk.)
Can couple two versions of the chain so that the magnetizations agree
by the time the magnetization of the top chain hits zero.
Once magnetizations agree, use a two-dimensional process to bound
time until full configurations agree. Takes an additional O(n) steps.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
β<1
When β < 1, using the inequality tanh(x) ≤ x for x ≥ 0 shows that for
St ≥ 0,
!
1−β
E[St+1 | Ft ] ≤ St 1 −
n
√
Need [2(1 − β)]−1 n log n steps to drive E[St ] to n.
Additional O(n) steps needed for magnetization to hit zero. (Compare
with simple random walk.)
Can couple two versions of the chain so that the magnetizations agree
by the time the magnetization of the top chain hits zero.
Once magnetizations agree, use a two-dimensional process to bound
time until full configurations agree. Takes an additional O(n) steps.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
β<1
When β < 1, using the inequality tanh(x) ≤ x for x ≥ 0 shows that for
St ≥ 0,
!
1−β
E[St+1 | Ft ] ≤ St 1 −
n
√
Need [2(1 − β)]−1 n log n steps to drive E[St ] to n.
Additional O(n) steps needed for magnetization to hit zero. (Compare
with simple random walk.)
Can couple two versions of the chain so that the magnetizations agree
by the time the magnetization of the top chain hits zero.
Once magnetizations agree, use a two-dimensional process to bound
time until full configurations agree. Takes an additional O(n) steps.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
β<1
When β < 1, using the inequality tanh(x) ≤ x for x ≥ 0 shows that for
St ≥ 0,
!
1−β
E[St+1 | Ft ] ≤ St 1 −
n
√
Need [2(1 − β)]−1 n log n steps to drive E[St ] to n.
Additional O(n) steps needed for magnetization to hit zero. (Compare
with simple random walk.)
Can couple two versions of the chain so that the magnetizations agree
by the time the magnetization of the top chain hits zero.
Once magnetizations agree, use a two-dimensional process to bound
time until full configurations agree. Takes an additional O(n) steps.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Variance estimates: Markov chain lemma
Let (Zt ) be a Markov chain taking values in R and with transition
matrix P. We will write Pz and Ez for its probability measure and
expectation, respectively, when Z0 = z. Suppose that there is some
0 < ρ < 1 such that for all pairs of starting states (z, z̃),
| Ez [Zt ] − Ez̃ [Zt ] | ≤ ρt |z − z̃|.
Then vt := supz0 varz0 (Zt ) satisfies
vt ≤ v1 min{t, (1 − ρ2 )−1 }.
This lemma enables us to show that var(St ) = O(n) for β < 1 and
var(St ) = O(t) for β = 1.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Variance estimates: Markov chain lemma
Let (Zt ) be a Markov chain taking values in R and with transition
matrix P. We will write Pz and Ez for its probability measure and
expectation, respectively, when Z0 = z. Suppose that there is some
0 < ρ < 1 such that for all pairs of starting states (z, z̃),
| Ez [Zt ] − Ez̃ [Zt ] | ≤ ρt |z − z̃|.
Then vt := supz0 varz0 (Zt ) satisfies
vt ≤ v1 min{t, (1 − ρ2 )−1 }.
This lemma enables us to show that var(St ) = O(n) for β < 1 and
var(St ) = O(t) for β = 1.
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Conjecture
Consider the d-dimensional torus (Z/nZ)d . Let βc be the critical
temperature.
Conjectures:
1
2
3
For β < βc , there is a cut-off.
For β = βc , the mixing time is polynomial in n.
Stronger: tmix = O(|Vn |3/2 ) for d > dc .
For β > βc , if the dynamics are truncated, the mixing time is
polynomial in n.
Stronger: for d > dc , the mixing time is O(|Vn | log |Vn |).
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Conjecture
Consider the d-dimensional torus (Z/nZ)d . Let βc be the critical
temperature.
Conjectures:
1
2
3
For β < βc , there is a cut-off.
For β = βc , the mixing time is polynomial in n.
Stronger: tmix = O(|Vn |3/2 ) for d > dc .
For β > βc , if the dynamics are truncated, the mixing time is
polynomial in n.
Stronger: for d > dc , the mixing time is O(|Vn | log |Vn |).
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph
Conjecture
Consider the d-dimensional torus (Z/nZ)d . Let βc be the critical
temperature.
Conjectures:
1
2
3
For β < βc , there is a cut-off.
For β = βc , the mixing time is polynomial in n.
Stronger: tmix = O(|Vn |3/2 ) for d > dc .
For β > βc , if the dynamics are truncated, the mixing time is
polynomial in n.
Stronger: for d > dc , the mixing time is O(|Vn | log |Vn |).
Levin, Luczak, Peres
Glauber Dynamics for Ising Model on the Complete Graph