  We study a reversible continuous-time Markov dynamics on lozenge tilings of
the plane, introduced by Luby et al. Single updates consist in concatenations
of $n$ elementary lozenge rotations at adjacent vertices. The dynamics can also
be seen as a reversible stochastic interface evolution. When the update rate is
chosen proportional to $1/n$, the dynamics is known to enjoy especially nice
features: a certain Hamming distance between configurations contracts with time
on average and the relaxation time of the Markov chain is diffusive, growing
like the square of the diameter of the system. Here, we present another
remarkable feature of this dynamics, namely we derive, in the diffusive time
scale, a fully explicit hydrodynamic limit equation for the height function (in
the form of a non-linear parabolic PDE). While this equation cannot be written
as a gradient flow w.r.t. a surface energy functional, it has nice analytic
properties, for instance it contracts the $\mathbb L^2$ distance between
solutions. The mobility coefficient $\mu$ in the equation has non-trivial but
explicit dependence on the interface slope and, interestingly, is directly
related to the system's surface free energy. The derivation of the hydrodynamic
limit is not fully rigorous, in that it relies on an unproven assumption of
local equilibrium.
