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

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

Random-field Ising model • Standard Ising model: Domain Λ ⊂ ℤ 𝑒 . Boundary conditions 𝜐 outside Λ . Energy of configuration 𝜏: Λ → {−1,1} given by 𝐼 Λ,𝜐 𝜏 = −𝐾 𝜏 𝑣 𝜏 𝑤 − 𝐾 𝜏 𝑣 𝜐 𝑤 − ℎ 𝜏 𝑤 𝑣~𝑤, 𝑣~𝑤, 𝑤∈Λ 𝑣,𝑤∈Λ 𝑣∈Λ,𝑤∉Λ At temperature 𝑈 : • Prob 𝜏 ∝ exp − 1 𝑈 𝐼 Λ,𝜐 𝜏 • Random-field Ising model (RFIM): 𝐼 Λ,𝜐 𝜏 = −𝐾 𝜏 𝑣 𝜏 𝑤 − 𝐾 𝜏 𝑣 𝜐 𝑤 − 𝜁 𝜃 𝑤 𝜏 𝑤 𝑣~𝑤, 𝑣~𝑤, 𝑤∈Λ 𝑣,𝑤∈Λ 𝑣∈Λ,𝑤∉Λ with 𝜃 𝑤 a quenched random field. In this talk – 𝜃 𝑤 independent standard Gaussians. • 2

Long-range order The Ising model, a t 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 𝜃 𝑤 > 2𝑒 ⋅ 𝐾 𝜁 then necessarily sign 𝜏 𝑤 = sign 𝜃 𝑤 . At large 𝜁 , such vertices are likely to separate the origin from the • boundary of Λ(L) . Thus Λ 𝑀 ,+ 𝔽 < 𝜏 0 > 𝑈 𝑀→∞ 0 with convergence occurring exponentially fast in 𝑀 . • Recent more quantitative results by Camia-Jiang-Newman(18). 3

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 𝑒 𝐼 Λ 𝑀 ,+ + − 𝐼 Λ 𝑀 ,+ − ≈ J ⋅ 𝑀 𝑒−1 ± 𝜁 ⋅ 𝑀 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 ) 4

Rate of decay of boundary effect • How fast does the boundary effect decay in two dimensions? Λ 𝑀 ,+ ? How large is 𝔽 < 𝜏 0 > 𝑈 • Main result (power-law upper bound): In two dimensions, for any T ≥ 0, 𝐾, 𝜁 > 0, Λ 𝑀 ,+ ≤ 1 𝔽 < 𝜏 0 > 𝑈 𝑀 𝛿 for large L the obtained power 𝛿 is very small, behaving as 2 𝐾 for small 𝜁 𝛿 ≈ exp −𝑑 𝐾 𝜁 • Corollary (by FKG inequality): A similar power-law upper bound for correlations in the RFIM 1 log(log 𝑀 ) decay. • Improves Chatterjee (17) 5

Ideas of proof for 𝑈 = 0 Denote ground-state configuration by 𝜏 Λ,𝜐 . • Λ 𝑀 ,+ = ℙ 𝜏 0 Λ 𝑀 ,+ > 𝜏 0 Λ 𝑀 ,− Influence-percolation: 𝑄 𝑀 ≔ 𝔽 𝜏 0 • 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 Λ 3ℓ ,+ 𝜃 > 𝜏 𝑤 Λ 3ℓ ,− 𝜃 𝐸 ℓ 𝜃 : = 𝑤 ∈ Λ ℓ ∶ 𝜏 𝑤 ≥ ℓ 2 ⋅ 𝑄 Note: Using FKG inequality, 𝔽 𝐸 ℓ 𝜃 • 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: 𝜐 ℓ 𝜃 ≔ − ℰ +,+ + ℰ −,− − ℰ +,− − ℰ −,+ . • 6

Main steps Step 1 (upper bound): 𝔽[𝜐 ℓ 𝜃 ] ≤ 𝐷𝐾 ⋅ ℓ ⋅ 𝑄 ℓ−1 . • • Step 2 (exact expression): ∞ 𝔽[𝜐 ℓ 𝜃 ] = 2𝜁 𝔽[𝐸 ℓ 𝜃 𝑢 ] 𝑒𝑢 ℓ −∞ with 𝜃 𝑢 ≡ 𝜃 + t ℓ inside Λ ℓ , 𝜃 𝑢 ≡ 𝜃 outside Λ ℓ . increases by 𝑢 standard deviations in 𝜃 𝑢 Note: the sum 𝜃 𝑤 𝑤∈Λ ℓ • Put together, these imply the anti-concentration bound 𝔽(𝐸 ℓ ) < 1 𝐸 ℓ > 𝐷 ⋅ 𝐾 𝜁 ⋅ 𝑄 ℓ−1 ℙ ≥ ℙ 𝑂 0,1 2 𝑄 4ℓ 7

Variance bound • Anti-concentration bound: 𝔽(𝐸 ℓ ) < 1 𝐸 ℓ > 𝐷 ⋅ 𝐾 𝜁 ⋅ 𝑄 ℓ−1 ℙ ≥ ℙ 𝑂 0,1 2 𝑄 4ℓ Right-hand side is constant when 𝑄 ℓ approximately a power of ℓ . • • Step 3: This is contrasted with a variance bound: 2 . ℓ ≈ 1 If 𝑄 ℓ 𝜀 then Var 𝐸 ℓ ≤ 𝐷 ⋅ 𝜀 ⋅ 𝔽 𝐸 ℓ 𝔽(𝐸 ℓ ) < 1 𝐸 ℓ C hebyshev’s inequality implies that ℙ 2 < 𝐷 ⋅ 𝜀 • Contradiction arises if 𝜀 is too small. • 8

Step 1: surface tension upper bound Claim: 𝔽[𝜐 ℓ 𝜃 ] ≤ 𝐷𝐾 ⋅ ℓ ⋅ 𝑄 • ℓ−1 with 𝜐 ℓ 𝜃 ≔ − ℰ +,+ + ℰ −,− − ℰ +,− − ℰ −,+ . Proof: Let 𝜏 𝑡,𝑡′ be the ground state in Λ 3ℓ ∖ Λ ℓ subject to 𝑡, 𝑡′ • boundary conditions inside and outside. Then ℰ 𝑡,𝑡′ is its energy. +,− and 𝜏 −,+ : Form mixed configurations 𝜏 • 𝑡,𝑡′ ≡ 𝜏 𝑡,𝑡 on Λ 3ℓ ∖ Λ 2ℓ 𝜏 𝑡,𝑡′ ≡ 𝜏 𝑡 ′ ,𝑡 ′ on Λ 2ℓ ∖ Λ ℓ 𝜏 𝑡,𝑡′ for their energy with 𝑡, 𝑡′ boundary conditions. and write ℰ Of course, ℰ +,− ≤ ℰ +,− and ℰ +,− ≤ ℰ +,− by def. of ground state. • 𝜐 ℓ 𝜃 ≤ − ℰ +,+ + ℰ −,− − ℰ +,− − ℰ +,− Thus • The sole contribution to the right-hand side comes from the bonds of 𝜖Λ(2ℓ) where 𝜏 +,+ differs from 𝜏 −,− . Taking expectation over the random field finishes the proof. 9

Step 2: formula for surface tension 𝔽[𝐸 ℓ 𝜃 𝑢 ] 𝑒𝑢 Claim: 𝔽[𝜐 ℓ 𝜃 ] = 2𝜁 ∞ • ℓ −∞ with 𝜃 𝑢 ≡ 𝜃 + 𝑢 ℓ inside Λ ℓ , 𝜃 𝑢 ≡ 𝜃 outside Λ ℓ . Λ 3ℓ ,− 𝜃 𝑢 }| 𝐸 ℓ 𝜃 𝑢 : = |{𝑤 ∈ Λ ℓ ∶ 𝜏 𝑤 Λ 3ℓ ,+ (𝜃 𝑢 ) > 𝜏 𝑤 Proof: Let ℰ + , ℰ − be the ground-state energies in Λ 3ℓ with +,- • boundary conditions, respectively. Set 𝐻 𝜃 ≔ − ℰ + − ℰ − • • Then 𝜐 ℓ 𝜃 = − ℰ +,+ + ℰ −,− − ℰ +,− − ℰ −,+ = lim 𝑢→∞ 𝐻 𝜃 𝑢 − 𝐻(𝜃 −𝑢 ) • Now note that 𝜖𝐻 Λ 3ℓ ,− 𝜃 Λ 3ℓ ,+ (𝜃) > 𝜏 𝑤 (𝜃) = 2𝜁 ⋅ 1 𝜏 𝑤 𝜖𝜃 𝑤 10

Step 3: Variance upper bound 2 . ℓ ≈ 1 Claim: If 𝑄 ℓ 𝜀 then Var 𝐸 ℓ ≤ 𝐷 ⋅ 𝜀 ⋅ 𝔽 𝐸 ℓ • Λ 3ℓ ,+ 𝜃 > 𝜏 𝑤 Λ 3ℓ ,− 𝜃 Proof: Write 𝐹 𝑤 ≔ 𝜏 𝑤 . • Need to upper bound, for 𝑣, 𝑤 ∈ Λ ℓ , • Cov 1 𝐹 𝑣 , 1 𝐹 𝑤 = ℙ 𝐹 𝑣 ∩ 𝐹 𝑤 − ℙ(𝐹 𝑣 )ℙ(𝐹 𝑤 ) 4ℓ ≈ 4ℓ −𝜀 Use ℙ 𝐹 𝑣 ≥ 𝑄 • 2 ≈ dist(𝑣, 𝑤)/2 −2𝜀 ℙ 𝐹 𝑣 ∩ 𝐹 𝑤 ≤ 𝑄dist (𝑣,𝑤)/2 If 𝜀 is small and, say, dist 𝑣, 𝑤 ≥ ℓ/100 , get • 2𝜀 2𝜀 200 1 ≈ 𝑑 ⋅ 𝜀 ⋅ ℓ −2𝜀 Cov 1 𝐹 𝑣 , 1 𝐹 𝑤 ≤ − ℓ 4ℓ • Can sum such upper bounds to get required result. 11

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 𝑒 𝑑 = 4 (Imry- Ma 75, Aizenman-Wehr 89). Obtain a quantitative decay of correlations there . 12