The Wulff droplet arises by conditioning a spin system in a dominant
phase to have an excess of signs of opposite type. These gather
together to form a droplet, with a macroscopic Wulff profile, a
solution to an isoperimetric problem.
I will discuss recent work proving that the phase boundary that
delimits the signs of opposite type has a characteristic scale, both
at the level of exponents and their logarithmic corrections.
This behaviour is expected to be shared by a broad class of stochastic
interface models in the Kardar-Parisi-Zhang class. Universal
distributions such as Tracy-Widom arise in this class, for example, as
the maximum behaviour of repulsive particle systems. time permitting,
I will explain how probabilistic resampling ideas employed in spin
systems may help to develop a qualitative understanding of the random
mechanisms at work in the KPZ class.