Dynamical Freezing in a Spin Glass with Logarithmic Correlations - - PowerPoint PPT Presentation

Setup Motivation Results Proof Idea

Dynamical Freezing in a Spin Glass with Logarithmic Correlations

Based on joint work with A. Cortines, J. Gold and A. Svejda Weizmann Institute of Science, 18/1/2018

Setup Motivation Results Proof Idea

Outline

Setup Motivation Results Proof Idea

Setup Motivation Results Proof Idea

Setup

Potential: hN ≡

• hN(x) : x ∈ VN
• , VN = (0, N)2 ∩ Z2, where

hN ∼ Discrete Gaussian Free Field on VN (with 0 b.c.). = ⇒ hN ∼ N(0, GN), where GN(x, y) = Ex ∞

n=0 1{Sn=y , ∀k≤n :Sk∈VN}

• .
• GN is the Green Function on VN:
• Cov
• hN(X), hN(y)
• = −g log y−x

N

+ O(1).

• E
• hN(x) − hN(y)

2 = 2g log y − x + O(1).

(In the bulk, g = 2/π.)

= ⇒ Logarithmic correlations.

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Setup - Cont’d

Dynamics: XN =

• XN(t) : t ≥ 0
• , where:

XN is (conditionally on hN) a Markov jump process on VN with transition rates: qN(x, y) :=

• eβ[a hN(y) − (1−a) hN(x)]

if x∼y ,

• therwise .

x ∼ y stands for nearest neighbors in the torus. β ≥ 0 is the interaction strength parameter (inverse temperature). a ∈ [0, 1] is a symmetry parameter. Question Long and short time behavior of XN for large β?

Setup Motivation Results Proof Idea

Outline

Setup Motivation Results Proof Idea

Setup Motivation Results Proof Idea

Motivation: Glauber Dynamics for a Spin Glass

Easy to check that for all a and β: eβhN(x)qN(x, y) = eβhN(y)qN(y, x) = eβa[hN(x)+hN(y)] = ⇒ XN is reversible with respect to

• MN,β =

MN,β MN,β(VN) , MN,β =

• x∈VN

eβhN(x)δx .

• MN,β is a spin-glass Gibbs distribution with energy states:
• − hN(x) : x ∈ VN
• .

= ⇒ XN is its Glauber dynamics.

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Limiting Gibbs Distribution

There exists βc ∈ (0, ∞) such that Mβ is asymptotically discrete if and

• nly if β > βc.

Equivalently, iff β > βc, for large N most of the mass of MN,β is concentrated on (essentially) finitely many vertices (energy levels). “Known”: MN,β(N·) = ⇒ Mβ as N → ∞, where (formally):

• Mβ(dx) =

eβh(x)dx

• V eβh(y)dy ,

h ∼ continous GFF on V .

• Mβ is a normalized version of the Liouville Quantum Gravity Measure

with respect to the continuous Gaussian free field on V := [0, 1]2 at parameter β. Only really proved for β > βc (Zindy-Arguin, Biskup, L.).

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Limiting Gibbs Distribution - Supercritical case

What is actually proved is MN,β = ⇒ Mβ as N → ∞, where

• MN,β(dx) =

     N2√gβ(log N)−3√gβ/4MN,β(dx) β > βc , MN,β(dx)

• EeβhN(x)−1 ,

β < βc , (log N)MN,β(dx)

• EeβhN(x)−1 ,

β = βc . Mβ is the (“real”) LQGM on V. Indeed a.s. discrete iff β > βc. When β > βc we have Mβ =

Mβ |Mβ| and Mβ = k τkδξk where

• (ξk, τk) : k ≥ 1
• atoms of χβ ∼ PPP
• Z(dx) ⊗ κβ|Z|t−1−βc/βdt
• .

! Z = |Z| Z is some random measure on V, κβ ∈ (0, ∞). Equivalently Mβ =

k≥1

τkδξk , where ( τk : k ≥ 1) are Poisson Dirichlet with parameter β/βc and ξk ∼ Z are independently i.i.d.

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Glassy Phase

The regime β > βc is called the glassy (or frozen) phase. Conjectured to be a universal feature of (mean-field) spin-glass-type distributions at low temperature. “Static” features: Few and isolated dominant energy levels. Poisson Dirichlet statistics in the limit. Overlap distribution supported on {0, 1} (1 RSB). · · · ”Dynamical” features - Dynamical Freezing: Metastability. Tunneling. Aging.

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Dynamical Freezing in other Spin-Glass models

Previously studied models: The Random Energy Model. Underlying graph is Hn = {−1, +1}n with N = 2n. hN(x) are i.i.d. centered Gaussians with variance n. The Bouchaud Trap Model. Underlying graph is VN (or Zd for d ≥ 2). ehN(x) are i.i.d. stable random variables with index 1. The p-spin model. Underlying graph is Hn (or the sphere Sn−1 in Rn). hN(x) is Gaussian, with EhN(x)hN(y) = n

• 1

nx, y

• p, p ≥ 2.

(In our case correlations are logarithmic).

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Outline

Setup Motivation Results Proof Idea

Setup Motivation Results Proof Idea

In-equlibirium timescales

From now on we take a = 0 (random hopping times dynamics). Let τN = τN(β) := N2√gβ(log N)1−3√gβ/4. ! Evolution rate of XN in equilibrium. Let V = V∪{∞} with ∞ at finite distance from V = [0, 1]2. Theorem (Cortines, Gold, L. ’17) Let β > βc. There exists a process Yβ ≡

• Yβ(t) : t ≥ 0
• taking values

in V such that for any t > 0,

• 1

N XN(τNt) : t ∈ [0, t]

• =

⇒

• Yβ(t) : t ∈ [0, t]
• as N → ∞ ,

as random functions in L1 [0, t] → V

• .

Q: What is Yβ?

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The limiting process

Ingredient 1 - The K process: Fix (τk)k≥1 be a deterministic decreasing sequence of “trap depths”. Let

• Ak = (Ak(u) : u ≥ 0)
• k≥1 be indep. rate 1 Poisson processes.

The jumps of Ak are denoted by σk = (σk(j) : j = 0, . . . ). Let

• Ek(j) : k, j = 1, . . . ) be i.i.d Exp(1) (indep of Ak-s).

The clock process is T : [0, ∞) → [0, ∞), where: T(u) :=

∞

• k=1

τk

Ak(u)

• j=1

Ek(j) , The K-process is K : [0, ∞) → N := N∪{∞}, where: K(t) :=

• k

if t ∈

• T
• σk(j)−
• , T
• σk(j)
• for some k, j ,

∞

• therwise.

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The limiting process

Ingredient 2 - The trapping landscape: Recall that

• (ξk, τk) : k = 1, . . .
• enumerate the atoms of

χβ ∼ PPP

• Z(dz) ⊗ κβ|Z|t−1−βc/βdt
• ,

(Limiting Gibbs dist. is Mβ =

k τkδξk / k τk for β > βc).

Order (ξk, τk)k≥1 so that τ1 > τ2 > . . . . Think of (ξk, τk) ∈ V × R+ are position and depth of k-th deepest trap. Construction of Yβ: Draw the trapping landscape:

• (ξk, τk) : k ≥ 1
• .

Draw a K-process K =

• K(t) : t ≥ 0
• with trap depths (τk)k≥1.

The limiting process Yβ : [0, ∞) → V is Yβ(t) :=

• ξK(t)

if K(t) = ∞ , ∞ if K(t) = ∞ .

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Properties of the Limiting Process

The K-process was introduced in this context by Fontes and Mathieu (2008) who studied dynamics for BTM on the complete graph. Named after Kolmogorov (1951) who studied a variant of K(t). The evolution of Yβ can be described as follows: When at trap k the process spends ∼ Exp(τ −1

k

) time at position ξk. Then jumps to ∞. At ∞ the process spends zero time (that is the exit time is a.s. 0). Then jumps to a new trap.

= ⇒ ∞ is called an unstable state..

Traps are chosen such that following a visit to ∞:

• The hitting time of any finite subset of traps is finite a.s.
• The entrance distribution to any such subset is uniform.

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Properties of the Limiting Process - Cont’d

The set of times when the process is at ∞ has Leb. measure zero. = ⇒ ∞ is called a fictitious state.. On a (compactifed version of) N ∪ {∞} the process K is strongly Markovian and has a version with càdlàg paths. Yβ can be seen as super-critical Liouville Brownian motion. Formally, Yβ(t) = X

• T −1(t)
• ,

T(u) = u

s=0

eβhX(s)ds , where (X(t) : t ≥ 0) is a SBM on V and h is the CGFF on V. This has been constructed recently for β ≤ βc.

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The Asymptotic Picture

For large N, the process XN:

• Spends most (≍ τN) time trapped in clusters of diameter O(1) of

dominant or meta-stable energy states of MN,β.

• Spends negligible (≪ τN) time transitioning or tunneling between

such clusters.

• Next cluster to visit is chosen (effectively) uniformly.

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Pre-equilibrium Timescales

What about pre-equilibrium timescales, that is when t ≪ τN(β)? Theorem (Cortines, Svejda, L. ’18+) Let β > βc (and a = 0). There exists 0 < γ0 < γ1 < ∞ and θ0 > 0 such that lim

N→∞ r→∞

P

• XN
• (1 + θ)t
• − XN
• t
• < r
• = Aslβc/β
• 1

1+θ

• .

uniformly in t ∈

• τN/(log N)γ1 , τN/(log N)γ0

and θ ∈ (0, θ0]. Aslκ is the CDF of the generalized arcsine distribution (on [0, 1]): Aslκ(u) = u sin κπ π vκ−1(1 − v)−κdv , Aslκ(0) = 0 , Aslκ(1) = 1 . θ0, γ0 can be chosen arbitrarily large, resp. small. True also conditionally on hN (in probability).

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Asymptotic Picture

Let N be very large and take any t ∈

• τN/(log N)γ1 , τN/(log N)γ0

.

• For any θ > 0, sampled at time t and time (1 + θ)t, the process

XN did not move by more than O(1) w.p. ≈ Aslβc/β(

1 1+θ)> 0.

• In particular, this probability is close to 1 if θ is close to 0.
• This is true for any choice of t in the above range of times.

= ⇒ After t time, the process gets trapped for time proportional to t. = ⇒ XN slows down or ages over time.

• Compare with in-equilibrium timescales: If t ≫ τN, then this

probability is the same as the probability of choosing the same atom when sampling twice from Mβ and does not depend on θ.

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Outline

Setup Motivation Results Proof Idea

Setup Motivation Results Proof Idea

Thank You!