Random-Field Curie-Weiss-Potts Model: From average to pointwise - - PowerPoint PPT Presentation

• 10. February 2011

Martin Slowik

Random-Field Curie-Weiss-Potts Model:

From average to pointwise estimates of metastable times

Warwick Statistical Mechanics Seminar 1 (21)

Metastability in stochastic dynamics

Metastability: A common phenomenon

The paradigm. Metastability is a phenomenon related to the dynamics of first order phase transitions. Changing the parameters of a system quickly across the line of a first order phase transition, the behaviour of the system reveals the existence of multiple, well separated time-scales:

⊲ Short time-scales. The system reaches quickly a quasi-equilibrium and remains for

a long time in a confined subset of the phase space, called the metastable state.

⊲ Larger time-scales. Rapid transitions between metastable states occur which are

influenced by random fluctuations. The goal. Understanding of quantitative aspects of dynamical phase transitions:

⊲ expected time of a transition from a metastable to a stable state, ⊲ distribution of the exit time from a metastable state,

Warwick Statistical Mechanics Seminar 2 (21)

Metastability in stochastic dynamics

Stochastic spin models

We are interested in studying the stochastic dynamics of (disordered) spin systems, i.e. Markov chain with

⊲ State space

SΛ = SΛ, where Λ ⊂ Zd and S finite set,

⊲ Hamiltonian

HΛ : SΛ → R,

⊲ Gibbs measure

µΛ,β(σ) = Z−1

Λ,β exp

` − βHΛ(σ) ´ ,

⊲ Order parameter e.g. ̺Λ(σ) =

1 |Λ|

P

i∈Λ δσi,

⊲ Transition rates

pΛ,β(σ, η) reversible with respect to µΛ,β and ”local”, i.e. essentially single site flips only.

Warwick Statistical Mechanics Seminar 3 (21)

Metastability in stochastic dynamics

Well understood situations

Low temperature limit. If β ↑ ∞ and the state space SN is finite, then

⊲ metastable states correspond to local minima of HN, ⊲ exit from metastable states occur through minimal saddle points of HN connecting

• ne minimum to deeper ones, only a few path are relevant,

⊲ the mean exit time of a metastable state is proportional to

exp ` β(HN(saddle) − HN(min)) ´ . Mean-field models. If HN(σ) = E(̺N(σ)) for some mesoscopic variable ̺N and ̺N(σ(t)) is itself a Markov chain, then

⊲ an exact reduction to a low-dimensional model is possible; nearest-neighbor

random walk in the free energy landscape.

Warwick Statistical Mechanics Seminar 4 (21)

Properties of the random field CWP model

The random field Curie–Weiss–Potts Model and the dynamics

Random Hamiltonian. HN(σ) = − 1 N

N

X

i,j=1

δ(σi, σj) −

N

X

i=1 q

X

r=1

hr

i δ(σi, r),

σ ∈ SN ≡ {1, . . . , q}N {hi}i∈N are (bounded) i.i.d. random variables taking values in Rq. Gibbs measure. µN(σ) = Z−1

N

exp ` −βHN(σ) ´ q−N Equilibrium properties.

⊲ J.M. Amaro de Matos, A.E. Patrick, V.A. Zagrebnov (JSP

, 1992), C. Külske (JSP , 1997, 1998)

⊲ G. Iacobelli, C. Külske (JSP

, 2010)

Glauber dynamics. Consider a discrete-time Markov chain {σ(t)}t∈N0 on SN reversible w.r.t. µN with Metropolis transition probabilities pN(σ, η) = 1 N(q − 1) exp ` −β ˆ HN(η) − HN(σ) ˜

+

´ if σ, η ∈ SN differ on a single coordinate, and zero else.

Warwick Statistical Mechanics Seminar 5 (21)

Properties of the random field CWP model

Free energy landscape

Macroscopic variable. ̺N : SN → Γ

N ⊂ Rq,

̺N(σ) =

1 N N

P

i=1

δσi Induced measure. QN = µN ◦ ̺−1

N

• n the set Γ

N

Using sharp large deviation estimates, one gets in the limit when N ↑ ∞ ZN QN(x) = exp ` −Nβ FN(x) ´ ` 1 + O

N(1)

´ (2πN)

q−1 2

q˛ ˛ det ˆ ∇2UN(t∗) ˜˛ ˛ , where FN(x) := −x2 + 1

β IN(x) and IN(x) is the Legendre–Fenchel transform of

UN(t) = 1 N XN

i=1 ln

Xq

r=1

1 q exp ` β hi

r + tr

´ . Critical points. Solutions of z∗

l

= 1 N XN

i=1 ln

exp ` β (2z∗

l + hi l)

´ Pq

r=1 exp

` β (2z∗

r + hi r)

´, l = 1, . . . , q. Moreover, at any critical point z∗ one gets an explicit expression for FN(z∗).

Warwick Statistical Mechanics Seminar 6 (21)

Properties of the random field CWP model

Main question

Let m∗ be a local minimum and consider the set of ’deeper’ local minima M = ˘ m | FN(m) ≤ FN(m∗) ¯ . We will be interested in the metastable exit time τS[M] = inf ˘ t > 0 ˛ ˛ σ(t) ∈ S[M], σ(0) ∈ S[m∗] ¯ , where S[I] := {σ ∈ SN | ̺N(σ) ∈ I}. Question: What can we say about Eσ ˆ τS[M] ˜ for all σ ∈ S[m∗]? Question: What can we say about the distribution of τS[M]?

Warwick Statistical Mechanics Seminar 7 (21)

Properties of the random field CWP model

Coarse graining

The entropic problem can be solved by passing on to Mesoscopic variables. ̺n : SN → Γn ⊂ Rn·q, ̺n(σ) =

n

P

k=1

ek ⊗

1 N

P

i∈Λk

δσi

⊲ {Hk}n

k=1 is a partition of support of the distribution of the random field.

⊲ Λk =

˘ i ∈ {1, . . . , N} | hi ∈ Hk ¯ is a random partition of {1, . . . , N}.

Induced measure. Qn = µN ◦ (̺n)−1

• n the set Γn

This allows to rewrite the Hamiltonian as HN(σ) = −NE ` ̺n(σ) ´ − Xn

k=1

X

i∈Λk˜

hi, eσ

⊲ ˜

hi = hi − ¯ hk for i ∈ Λk and ‚ ‚˜ hi‚ ‚ ≤ c/n ≡ ε(n)

• Strategy. Approximate the original dynamics by effective mesoscopic dynamics on Γn

which are reversible w.r.t. Qn with rN(x, y) = 1 Qn(x) X

σ∈Sn[x]

µN(σ) X

η∈Sn[y]

pN(σ, η).

Warwick Statistical Mechanics Seminar 8 (21)

Potential theoretic approach

Potential theory

Consider two disjoint sets A, B ⊂ SN and let LN = PN − I be the discrete generator. Equilibrium potential. Solution of the equation 8 > < > : ` LNhA,B ´ (σ) = 0, σ ∈ SN \ ` A ∪ B ´ hA,B(σ) = 1, σ ∈ A hA,B(σ) = 0, σ ∈ B Equilibrium measure. eA,B(σ) = − ` LNhA,B ´ (σ) Capacity. cap(A, B) = P

σ∈B µN(σ) eA,B(σ)

Dirichlet form. E(h) = 1 2 X

σ,η∈SN

µN(σ) pN(σ, η) ` h(σ) − h(η) ´2 First hitting probability. P

σ

ˆ τA < τB ˜ = 8 > < > : hA,B(σ), σ ∈ A ∪ B 1 − eA,B(σ), σ ∈ A eB,A(σ), σ ∈ B

Warwick Statistical Mechanics Seminar 9 (21)

Potential theoretic approach

Probabilistic interpretation

Mean hitting time. EνA,B ˆ τB ˜ = 1 cap(A, B) X

σ∈SN

µN(σ) hA,B(σ), where νA,B is a measure on A νA,B(σ) = µN(σ) eA,B(σ) cap(A, B) = µN(σ) P

σ

ˆ τB < τA ˜ P

η∈A µN(σ) P η

ˆ τB < τA ˜. The full beauty. To obtain sharp estimates for the mean hitting time, we need:

⊲ precise control on capacities. ⊲ some rough bounds on the equilibrium potential.

△

!

To obtain pointwise estimates it would be much simpler, if we were aware of a reasonable quantitative version of an elliptic Harnack inequality for such processes.

Warwick Statistical Mechanics Seminar 10 (21)

Potential theoretic approach

Probabilistic interpretation

Mean hitting time. EνA,B ˆ τB ˜ = 1 cap(A, B) X

σ∈SN

µN(σ) hA,B(σ), where νA,B is a measure on A νA,B(σ) = µN(σ) eA,B(σ) cap(A, B) = µN(σ) P

σ

ˆ τB < τA ˜ P

η∈A µN(σ) P η

ˆ τB < τA ˜. The full beauty. To obtain sharp estimates for the mean hitting time, we need:

⊲ precise control on capacities. ⊲ some rough bounds on the equilibrium potential.

△

!

To obtain pointwise estimates it would be much simpler, if we were aware of a reasonable quantitative version of an elliptic Harnack inequality for such processes.

Warwick Statistical Mechanics Seminar 10 (21)

Potential theoretic approach

Computation of capacities

Variational principles for capacities offers two convenient options for upper and lower bounds: Dirichlet principle. cap(A, B) = inf

h∈HA,B

1 2 X

σ,η

µ(σ) p(σ, η) ` h(σ) − h(η) ´2 HA,B is the space of functions with boundary constraints; minimizer harmonic function Berman-Konsowa principle. cap(A, B) = sup

f∈UA,B

Ef " X

(σ,η)∈X

f(σ, η) µ(σ) p(σ, η) #−1 UA,B is the space of unit flows; maximizer harmonic flow. Ef denotes the law of a directed Markov chain with transition probabilities proportional to the flow.

Warwick Statistical Mechanics Seminar 11 (21)

Metastable exit times in the random-field CWP model

Averaged result

Theorem 1. Let m∗ be a local minimum of FN and let z∗ be the critical point of index 1 separating m∗ from M. Then, Ph-a.s., Eν ˆ τS[M] ˜ = C(β, q, m∗, z∗) N eβN(FN (z∗)−FN(m∗)) ` 1 + O

N(1)

´ where ν is a probability measure on S[m∗] and C(β, q, m∗, z∗) is the prefactor (explicit formulae). Previous and related work

⊲ F. den Hollander and P

. dai Pra (JSP , 1996) large deviations, logarithmic asymptotics

⊲ P

. Mathieu and P . Picco (JSP , 1998) Bernoulli distribution, up to polynomial errors

⊲ A. Bovier, M. Eckhoff, V. Gayrard and M. Klein (PTRL, 2001) discrete distribution, up to a

multiplicative constant

⊲ A. Bianchi, A. Bovier and D. Ioffe (EJP

, 2008) bounded continuous distribution, precise prefactor

Warwick Statistical Mechanics Seminar 12 (21)

Elements of the proof

The program

The key step in the proof of the upper and lower bound on capacities is to

• 1. find a function which is almost harmonic in a small neighborhood of the relevant

saddle point z∗. Two scale construction:

• 2. Construct a mesoscopic unit flow on variables x from the approximate harmonic
• function. This yields a good lower bound in the mesoscopic Dirichlet form.
• 3. Construct a subordinate microscopic unit flow for each mesoscopic path.
• 4. Use that the magnetic field is almost constant in any block Λk to show strong

concentration properties along microscopic paths. This yields a lower bound that differs from the upper bound only by a factor 1 + O(1/n).

Warwick Statistical Mechanics Seminar 13 (21)

Elements of the proof

Approximate harmonic function

The key step in the proof of the upper and lower bound is to find a function which is almost harmonic in a small neighborhood of the relevant saddle point. Consider the following two parameter family of functions g : Γn → [0, 1] g(x) := f(v, x − z∗) for a suitable vector v ∈ Rn·q and f: R → [0, 1] f(s) := r βN|ˆ γ1| 2π

s

Z

−∞

e− 1

2 βN|ˆ

γ1|u2du.

The parameter v and ˆ γ1 are chosen in such a way that g is as harmonic as possible. This gives an upper bound for the macroscopic capacity which will turn out to be the correct answer when N ↑ ∞.

Warwick Statistical Mechanics Seminar 14 (21)

Elements of the proof

Renewal equations

Final step in control of metastable exit times: X

σ

µN(σ) hS[m∗],S[M](σ) ∼ QN ˆ m∗ − x ≤ δ ˜ This requires to show that

⊲ hS[m∗],S[M](σ) ∼ 1 for all σ in a neighborhood of S[m∗], ⊲ hS[m∗],S[M](σ) ≤ e−βN(FN (z∗)−FN(̺N (σ))−δ)

if FN(̺N(σ)) ≤ FN(m∗). Strategy.

• 1. Use averaged renewal equations: Let A, B, X ⊂ SN be mutually disjoint. Then

P

νX,A∪B

ˆ τA < τB ˜ = P

µX

ˆ τA < τB∪X ˜ P

µX

ˆ τA∪B < τX ˜ ≤ cap(X, A) cap(X, B).

• 2. Construct a coupling of two Markov chains started in σ, η ∈ Sn[x] and show that

min

σ∈X P σ

ˆ τA∪B < τX ˜ ≥ e−ε(n)N max

σ∈X P σ

ˆ τA∪B < τX ˜ , X = Sn[x].

Warwick Statistical Mechanics Seminar 15 (21)

From average to pointwise estimates

From average to pointwise estimates

Questions.

⊲ Does the metastable time really depend on the last exit biased distribution ν? ⊲ Under which conditions can we deduce pointwise estimates?

Heuristic. The time spent in the starting well before reaching S[M] is much larger then the mixing time of the dynamics conditioned to stay in the well: Eσ ˆ τS[M] ˜ ∼ Eη ˆ τS[M] ˜ ∀ σ, η ∈ S[m∗]. After the system is mixed, the return times to S[m∗] are i.i.d. random variables, and the number of returns to S[m∗] is geometric. Provided that the mixing time is small enough respect to Eν[τS[M]] , the metastable time is expected to be exponential distributed.

Warwick Statistical Mechanics Seminar 16 (21)

From average to pointwise estimates

Coupling I

Properties of PN. Let σ, η ∈ SN such that ̺n(σ) = ̺n(η).

⊲ if σi = ηi then

pN(σ, σi,r) = pN(η, ηi,r).

⊲ if σi = ηj for i, j ∈ Λk then

pn(η, ηj,r) pN(σ, σi,r) ≤ eε(n), Lemma 2. Suppose for µ, ν ∈ M1(S) there exists δ ∈ (0, 1) s.th. δµ(r) ≤ ν(r), ∀ r ∈ S. Then, there exists an optimal coupling (X, Y ) of µ and ν with the additional property that there exists a r.v. U ∼ Ber(δ) independent of X s.th. P ˆ Y = s ˛ ˛ U = 1, X = r ˜ = δ(r, s).

Warwick Statistical Mechanics Seminar 17 (21)

From average to pointwise estimates

Coupling II

Let σ, η ∈ S[m∗] and assume that until time t, ̺n(σ(t)) = ̺n(η(t)). η(t) σ(t)

Warwick Statistical Mechanics Seminar 18 (21)

From average to pointwise estimates

Coupling II

Let σ, η ∈ S[m∗] and assume that until time t, ̺n(σ(t)) = ̺n(η(t)). η(t) σ(t) i i Choose a lattice site i uniform at random.

Warwick Statistical Mechanics Seminar 18 (21)

From average to pointwise estimates

Coupling II

Let σ, η ∈ S[m∗] and assume that until time t, ̺n(σ(t)) = ̺n(η(t)). η(t + 1) σ(t + 1) If ηi(t) = σi(t), then ηi(t + 1) = σi(t + 1) with probability one.

Warwick Statistical Mechanics Seminar 18 (21)

From average to pointwise estimates

Coupling II

Let σ, η ∈ S[m∗] and assume that until time t, ̺n(σ(t)) = ̺n(η(t)). η(t) σ(t) i i If ηi(t) = σi(t),

Warwick Statistical Mechanics Seminar 18 (21)

From average to pointwise estimates

Coupling II

Let σ, η ∈ S[m∗] and assume that until time t, ̺n(σ(t)) = ̺n(η(t)). η(t) σ(t) i j If ηi(t) = σi(t), choose a lattice site j uniform at random from ˘ j ∈ Λk ˛ ˛ ηi(t) = σi(t), ηj(t) = σj(t) and ηi(t) = σj(t) ¯ and toss a coin U with probability of heads equal e−ε(n).

Warwick Statistical Mechanics Seminar 18 (21)

From average to pointwise estimates

Coupling II

Let σ, η ∈ S[m∗] and assume that until time t, ̺n(σ(t)) = ̺n(η(t)). η(t + 1) σ(t + 1) If U = 1, then ̺n` η(t + 1) ´ = ̺n` σ(t + 1) ´ .

Warwick Statistical Mechanics Seminar 18 (21)

From average to pointwise estimates

Coupling II

Let σ, η ∈ S[m∗] and assume that until time t, ̺n(σ(t)) = ̺n(η(t)). η(t + 1) σ(t + 1)

• Consequence. As long as the coin tosses come up heads,

dH ` η(t), σ(t) ´ is non-increasing.

Warwick Statistical Mechanics Seminar 18 (21)

From average to pointwise estimates

Cycle decomposition

Question: How to proceed if U = 0?

⊲ Stop the coupling and let the chain {σ(t)} evolves until it returns to Sn[̺n(η)]. Let

s1 denote the corresponding stopping time.

⊲ Make a second attempt to couple {σ(t)} starting from σ(τs1) and an independent

copy of {η(t)} starting in η.

⊲ Do this iteratively until either both chains merged or {σ(t)} hits S[M ∗].

Warwick Statistical Mechanics Seminar 19 (21)

Metastable exit times in the random-field CWP model

Pointwise results

Let m∗ and M be the mesoscopic minima in Γn corresponding to m∗ and M. Theorem 2. For n large enough, Eσ ˆ τSn[M ] ˜ = Eη ˆ τSn[M ] ˜ ` 1 + O

N(1)

´ for all σ, η ∈ Sn[m∗]. Theorem 3. For n large enough and all t > 0 P

σ

ˆ τSn[M ]/ Eσ ˆ τSn[M ] ˜ > t ˜ → e−t, as N → ∞ for all σ, η ∈ Sn[m∗]. Previous and related work

⊲ D.A. Levin, M. Luczak, Y. Peres (PTRF, 2010) without random field, coupling construction ⊲ A. Bianchi, A. Bovier and D. Ioffe (accepted Ann. Prob.) continuous distribution, coupling

construction for Ising spins

Warwick Statistical Mechanics Seminar 20 (21)

Summary and outlook

Conclusions

What has been done so far.

⊲ Sharp estimates on metastable exit times in a model without symmetry when

entropy is relevant.

⊲ Description of distribution of metastable exit times. ⊲ Averaged version of renewal equations for harmonic functions. ⊲ Construction of a coupling when the underlying single spin space is finite.

Future challenges.

⊲ Control of the small eigenvalues of the generator! ⊲ Hopfield model with infinitely many patterns.

Warwick Statistical Mechanics Seminar 21 (21)