Approximate domain Markov property for rigid Ising interfaces

Consider the Ising model on a centered box of side length $n$ in $\mathbb Z^d$ with $\mp$-boundary conditions that are minus in the upper half-space and plus in the lower half-space. Dobrushin famously showed that in dimensions $d\ge 3$, 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 $S \subset \mathbb Z^{d-1}$, along with the entire interface outside the cylinder $S\times \mathbb Z$: what does the interface in $S\times \mathbb Z$ 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 $S^c$ through the heights on $\partial S$. The Ising interface importantly does not satisfy this property; the law of the interface depends on the full spin configuration outside $S\times \mathbb Z$. 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 $S\times \mathbb Z$ is rigid about height $h$, with exponential tails on its height oscillations. Then we show that the typical tall pillars in $S\times \mathbb Z$ 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 Gumbel tail bounds on the maximum oscillations in $S\times \mathbb Z$ about height $h$, showing that these only depend on the conditioning through the cardinality of $S$.