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We study random one-Lipschitz integer functions $f$ on the vertices of afinite connected graph, sampled according to the weight $W(f) = \prod_{\langlev, w \rangle \in E} \mathbf{c}^{ \mathbb{I} \{ f(v) = f(w) \} }$ where$\mathbf{c} \geq 1$, and restricted by a boundary condition. For planar graphs,this is arguably the simplest ``2D random walk model'', and proving theconvergence of such models to the Gaussian free field (GFF) is a major openquestion. Our main result is that for subgraphs of the honeycomb lattice (andsome other cubic planar lattices), with flat boundary conditions and $1 \leq\mathbf{ c } \leq 2$, such functions exhibit logarithmic variations. This is inline with the GFF prediction and improves a non-quantitative delocalizationresult by P. Lammers. The proof goes via level-set percolation arguments,including a renormalization inequality and a dichotomy theorem for level-setloops. In another direction, we show that random Lipschitz functions havebounded variance whenever the wired FK-Ising model with $p=1-1/\mathbf{c}$percolates on the same lattice (corresponding to $\mathbf{c} > 2 + \sqrt{3}$ onthe honeycomb lattice). Via a simple coupling, this also implies, perhapssurprisingly, that random homomorphisms are localized on the rhombille lattice.

Logarithmic delocalization of random Lipschitz functions on honeycomb and other lattices