[PPT] - A power-law upper bound on the correlations in the two-dimensional PowerPoint Presentation

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A power-law upper bound on the correlations in the two-dimensional random-field Ising model

Ron Peled, Tel Aviv University

Based on a joint work with Michael Aizenman, Princeton

ICMP 2018, Montreal

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Random-field Ising model

Standard Ising model:

Domain Λ ⊂ ℤ𝑒. Boundary conditions 𝜐 outside Λ. Energy of configuration 𝜏: Λ → {−1,1} given by 𝐼Λ,𝜐 𝜏 = −𝐾 𝜏𝑣𝜏𝑤 −

𝑣~𝑤, 𝑣,𝑤∈Λ

𝐾 𝜏𝑣𝜐𝑤

𝑣~𝑤, 𝑣∈Λ,𝑤∉Λ

− ℎ 𝜏𝑤

𝑤∈Λ

At temperature 𝑈:

Prob 𝜏 ∝ exp − 1 𝑈 𝐼Λ,𝜐 𝜏

Random-field Ising model (RFIM):

𝐼Λ,𝜐 𝜏 = −𝐾 𝜏𝑣𝜏𝑤 −

𝑣~𝑤, 𝑣,𝑤∈Λ

𝐾 𝜏𝑣𝜐𝑤

𝑣~𝑤, 𝑣∈Λ,𝑤∉Λ

− 𝜁 𝜃𝑤𝜏𝑤

𝑤∈Λ

with 𝜃𝑤 a quenched random field.

In this talk – 𝜃𝑤 independent standard Gaussians.

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Long-range order

The Ising model, at ℎ = 0, exhibits long-range order at low

temperatures.

Is this the case also for the random-field Ising model?
No, when 𝜁 is large! (strong disorder regime)
Proof for 𝑈 = 0:

If 𝜃𝑤 > 2𝑒 ⋅ 𝐾

𝜁 then necessarily sign 𝜏𝑤 = sign 𝜃𝑤 .

At large 𝜁, such vertices are likely to separate the origin from the

boundary of Λ(L). Thus 𝔽 < 𝜏0 >𝑈

Λ 𝑀 ,+ 𝑀→∞ 0

with convergence occurring exponentially fast in 𝑀.

Recent more quantitative results by Camia-Jiang-Newman(18).

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Imry-Ma phenomenon

Imry-Ma (75) considered small 𝜁 (weak disorder) and argued that:
Long-range order occurs in dimensions 𝑒 ≥ 3.
No long-range order in two dimensions:

Unique Gibbs state for all 𝑈 ≥ 0. Even for arbitrarily weak disorder!

An essence of the argument:

With plus bounday conditions, is the plus configuration favored over the minus configuration? Energy difference is 𝐼Λ 𝑀 ,+ + − 𝐼Λ 𝑀 ,+ − ≈ J ⋅ 𝑀𝑒−1 ± 𝜁 ⋅ 𝑀

𝑒 2

Boundary wins when 𝑒 ≥ 3. Random field wins, due to random fluctuations, when 𝑒 = 2.

Proofs. d ≥ 3: Imbrie (𝑈 = 0, 85), Bricmont-Kupiainen (88)

𝑒 = 2: Aizenman-Wehr (89) (quantum: Aizenman-Greenblatt-Lebowitz 09)

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Rate of decay of boundary effect

How fast does the boundary effect decay in two dimensions?

How large is 𝔽 < 𝜏0 >𝑈

Λ 𝑀 ,+ ?

Main result (power-law upper bound):

In two dimensions, for any T ≥ 0, 𝐾, 𝜁 > 0, 𝔽 < 𝜏0 >𝑈

Λ 𝑀 ,+ ≤ 1

𝑀𝛿 for large L the obtained power 𝛿 is very small, behaving as 𝛿 ≈ exp −𝑑

𝐾 𝜁 2

for small 𝜁

𝐾

Corollary (by FKG inequality):

A similar power-law upper bound for correlations in the RFIM

Improves Chatterjee (17)

1 log(log 𝑀 ) decay.

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Ideas of proof for 𝑈 = 0

Denote ground-state configuration by 𝜏Λ,𝜐.
Influence-percolation: 𝑄

𝑀 ≔ 𝔽 𝜏0 Λ 𝑀 ,+ = ℙ 𝜏0 Λ 𝑀 ,+ > 𝜏0 Λ 𝑀 ,−

Main result: Power-law upper bound on 𝑄

𝑀.

First observable: The number of sites in Λ ℓ influenced by

boundary conditions on Λ 3ℓ 𝐸ℓ 𝜃 : = 𝑤 ∈ Λ ℓ ∶ 𝜏𝑤

Λ 3ℓ ,+ 𝜃 > 𝜏𝑤 Λ 3ℓ ,− 𝜃

Note: Using FKG inequality, 𝔽 𝐸ℓ 𝜃

≥ ℓ2 ⋅ 𝑄

4ℓ.

Second observable: Work in annulus Λ 3ℓ ∖ Λ(ℓ) with + or −

boundary conditions inside and outside.

Ground-state energies ℰ+,+, ℰ−,−, ℰ+,−, ℰ−,+. Functions of field 𝜃.
Surface tension: 𝜐ℓ 𝜃 ≔ − ℰ+,+ + ℰ−,− − ℰ+,− − ℰ−,+ .

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Main steps

Step 1 (upper bound): 𝔽[𝜐ℓ 𝜃 ] ≤ 𝐷𝐾 ⋅ ℓ ⋅ 𝑄

ℓ−1.

Step 2 (exact expression):

𝔽[𝜐ℓ 𝜃 ] = 2𝜁 ℓ 𝔽[𝐸ℓ 𝜃𝑢 ] 𝑒𝑢

∞ −∞

with 𝜃𝑢 ≡ 𝜃 + t

ℓ inside Λ ℓ ,

𝜃𝑢 ≡ 𝜃 outside Λ ℓ . Note: the sum 𝜃𝑤

𝑤∈Λ ℓ

increases by 𝑢 standard deviations in 𝜃𝑢

Put together, these imply the anti-concentration bound

ℙ 𝐸ℓ 𝔽(𝐸ℓ) < 1 2 ≥ ℙ 𝑂 0,1 > 𝐷 ⋅ 𝐾 𝜁 ⋅ 𝑄

ℓ−1

𝑄

4ℓ

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Variance bound

Anti-concentration bound:

ℙ 𝐸ℓ 𝔽(𝐸ℓ) < 1 2 ≥ ℙ 𝑂 0,1 > 𝐷 ⋅ 𝐾 𝜁 ⋅ 𝑄

ℓ−1

𝑄

4ℓ

Right-hand side is constant when 𝑄

ℓ approximately a power of ℓ.

Step 3: This is contrasted with a variance bound:

If 𝑄

ℓ ≈ 1 ℓ𝜀 then Var 𝐸ℓ ≤ 𝐷 ⋅ 𝜀 ⋅ 𝔽 𝐸ℓ 2.

Chebyshev’s inequality implies that ℙ

𝐸ℓ 𝔽(𝐸ℓ) < 1 2 < 𝐷 ⋅ 𝜀

Contradiction arises if 𝜀 is too small.

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Step 1: surface tension upper bound

Claim: 𝔽[𝜐ℓ 𝜃 ] ≤ 𝐷𝐾 ⋅ ℓ ⋅ 𝑄

ℓ−1

with 𝜐ℓ 𝜃 ≔ − ℰ+,+ + ℰ−,− − ℰ+,− − ℰ−,+ .

Proof: Let 𝜏𝑡,𝑡′ be the ground state in Λ 3ℓ ∖ Λ ℓ subject to 𝑡, 𝑡′

boundary conditions inside and outside. Then ℰ𝑡,𝑡′ is its energy.

Form mixed configurations 𝜏

+,− and 𝜏 −,+: 𝜏 𝑡,𝑡′ ≡ 𝜏𝑡,𝑡 on Λ 3ℓ ∖ Λ 2ℓ 𝜏 𝑡,𝑡′ ≡ 𝜏𝑡′,𝑡′ on Λ 2ℓ ∖ Λ ℓ and write ℰ 𝑡,𝑡′ for their energy with 𝑡, 𝑡′ boundary conditions.

Of course, ℰ+,− ≤ ℰ

+,− and ℰ+,− ≤ ℰ +,− by def. of ground state. Thus 𝜐ℓ 𝜃 ≤ − ℰ+,+ + ℰ−,− − ℰ +,− − ℰ +,−

The sole contribution to the right-hand side comes from the bonds
f 𝜖Λ(2ℓ) where 𝜏+,+ differs from 𝜏−,−.

Taking expectation over the random field finishes the proof.

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Step 2: formula for surface tension

Claim: 𝔽[𝜐ℓ 𝜃 ] = 2𝜁

ℓ

𝔽[𝐸ℓ 𝜃𝑢 ] 𝑒𝑢

∞ −∞

with 𝜃𝑢 ≡ 𝜃 + 𝑢

ℓ inside Λ ℓ ,

𝜃𝑢 ≡ 𝜃 outside Λ ℓ . 𝐸ℓ 𝜃𝑢 : = |{𝑤 ∈ Λ ℓ ∶ 𝜏𝑤

Λ 3ℓ ,+(𝜃𝑢) > 𝜏𝑤 Λ 3ℓ ,− 𝜃𝑢 }|

Proof: Let ℰ+, ℰ− be the ground-state energies in Λ 3ℓ with +,-

boundary conditions, respectively.

Set 𝐻 𝜃 ≔ − ℰ+ − ℰ−
Then

𝜐ℓ 𝜃 = − ℰ+,+ + ℰ−,− − ℰ+,− − ℰ−,+ = lim

𝑢→∞ 𝐻 𝜃𝑢 − 𝐻(𝜃−𝑢)

Now note that

𝜖𝐻 𝜖𝜃𝑤 (𝜃) = 2𝜁 ⋅ 1 𝜏𝑤

Λ 3ℓ ,+(𝜃) > 𝜏𝑤 Λ 3ℓ ,− 𝜃

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Step 3: Variance upper bound

Claim: If 𝑄

ℓ ≈ 1 ℓ𝜀 then Var 𝐸ℓ ≤ 𝐷 ⋅ 𝜀 ⋅ 𝔽 𝐸ℓ 2.

Proof: Write 𝐹𝑤 ≔ 𝜏𝑤

Λ 3ℓ ,+ 𝜃 > 𝜏𝑤 Λ 3ℓ ,− 𝜃

.

Need to upper bound, for 𝑣, 𝑤 ∈ Λ ℓ ,

Cov 1 𝐹𝑣 , 1 𝐹𝑤 = ℙ 𝐹𝑣 ∩ 𝐹𝑤 − ℙ(𝐹𝑣)ℙ(𝐹𝑤)

Use ℙ 𝐹𝑣 ≥ 𝑄

4ℓ ≈ 4ℓ −𝜀

ℙ 𝐹𝑣 ∩ 𝐹𝑤 ≤ 𝑄dist(𝑣,𝑤)/2

2

≈ dist(𝑣, 𝑤)/2 −2𝜀

If 𝜀 is small and, say, dist 𝑣, 𝑤 ≥ ℓ/100, get

Cov 1 𝐹𝑣 , 1 𝐹𝑤 ≤ 200 ℓ

2𝜀

− 1 4ℓ

2𝜀

≈ 𝑑 ⋅ 𝜀 ⋅ ℓ−2𝜀

Can sum such upper bounds to get required result.

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Open questions

Is there a Kosterlitz-Thouless-type transition from exponential to

power-law decay of correlations as the random field becomes weaker? Mechanism which would imply power-law bound: If the influence percolation behaves like Mandelbrot percolation. (connectivity of Mandelbrot percolation – Chayes-Chayes-Durrett)

For systems with continuous symmetry, such as the random-field XY

model, the critical dimension for long-range order is 𝑒𝑑 = 4 (Imry- Ma 75, Aizenman-Wehr 89). Obtain a quantitative decay of correlations there.

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