TY - JOUR

T1 - Approximate Domain Markov Property for Rigid Ising Interfaces

AU - Gheissari, Reza

AU - Lubetzky, Eyal

N1 - Funding Information:
R.G. thanks the Miller Institute for Basic Research in Science for its support. E.L. was supported in part by NSF grant DMS-1812095.
Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2023/5

Y1 - 2023/5

N2 - Consider the Ising model on a centered box of side length n in Zd with ∓ -boundary conditions that are minus in the upper half-space and plus in the lower half-space. Dobrushin famously showed that in dimensions d≥ 3 , at low-temperatures the Ising interface (dual-surface separating the plus/minus phases) is rigid, i.e., it has O(1) height fluctuations. Recently, the authors decomposed these oscillations into pillars and identified their typical shape, leading to a law of large numbers and tightness of their maximum. Suppose we condition on a height-h level curve of the interface, bounding a set S⊂ Zd-1 , along with the entire interface outside the cylinder S× Z : what does the interface in S× Z look like? Many models of random surfaces (e.g., SOS and DGFF) fundamentally satisfy the domain Markov property, whereby their heights on S only depend on the heights on Sc through the heights on ∂S . The Ising interface importantly does not satisfy this property; the law of the interface depends on the full spin configuration outside S× Z . Here we establish an approximate domain Markov property inside the level curves of the Ising interface. We first extend Dobrushin’s result to this setting, showing the interface in S× Z is rigid about height h, with exponential tails on its height oscillations. Then we show that the typical tall pillars in S× Z are uniformly absolutely continuous with respect to tall pillars of the unconditional Ising interface. Using this we identify the law of large numbers, tightness, and sharp Gumbel tail bounds on the maximum oscillations in S× Z about height h, showing that these only depend on the conditioning through the cardinality of S.

AB - Consider the Ising model on a centered box of side length n in Zd with ∓ -boundary conditions that are minus in the upper half-space and plus in the lower half-space. Dobrushin famously showed that in dimensions d≥ 3 , at low-temperatures the Ising interface (dual-surface separating the plus/minus phases) is rigid, i.e., it has O(1) height fluctuations. Recently, the authors decomposed these oscillations into pillars and identified their typical shape, leading to a law of large numbers and tightness of their maximum. Suppose we condition on a height-h level curve of the interface, bounding a set S⊂ Zd-1 , along with the entire interface outside the cylinder S× Z : what does the interface in S× Z look like? Many models of random surfaces (e.g., SOS and DGFF) fundamentally satisfy the domain Markov property, whereby their heights on S only depend on the heights on Sc through the heights on ∂S . The Ising interface importantly does not satisfy this property; the law of the interface depends on the full spin configuration outside S× Z . Here we establish an approximate domain Markov property inside the level curves of the Ising interface. We first extend Dobrushin’s result to this setting, showing the interface in S× Z is rigid about height h, with exponential tails on its height oscillations. Then we show that the typical tall pillars in S× Z are uniformly absolutely continuous with respect to tall pillars of the unconditional Ising interface. Using this we identify the law of large numbers, tightness, and sharp Gumbel tail bounds on the maximum oscillations in S× Z about height h, showing that these only depend on the conditioning through the cardinality of S.

KW - Interface

KW - Ising model

KW - Low temperature

KW - Markov property

KW - Rigidity

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U2 - 10.1007/s10955-023-03101-x

DO - 10.1007/s10955-023-03101-x

M3 - Article

AN - SCOPUS:85159003623

SN - 0022-4715

VL - 190

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 5

M1 - 99

ER -