Logarithmic delocalization of random Lipschitz functions on honeycomb and other lattices

TL;DR

This paper proves that random Lipschitz functions on honeycomb and similar lattices exhibit logarithmic fluctuations, aligning with Gaussian free field behavior, and explores their delocalization and localization properties.

Contribution

It establishes the logarithmic delocalization of random Lipschitz functions on honeycomb lattices for certain parameters, advancing understanding of their fluctuation behavior.

Findings

Logarithmic variance of functions on honeycomb lattices for c between 1 and 2.

Delocalization results consistent with Gaussian free field predictions.

Bounded variance and localization for certain percolation regimes.

Abstract

We study random one-Lipschitz integer functions $f$ on the vertices of a finite connected graph, sampled according to the weight $W(f) = \prod_{\langle v, w \rangle \in E} \mathbf{c}^{ \mathbb{I} \{ f(v) = f(w) \} }$ 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 the convergence of such models to the Gaussian free field (GFF) is a major open question. Our main result is that for subgraphs of the honeycomb lattice (and some other cubic planar lattices), with flat boundary conditions and $1 \leq \mathbf{ c } \leq 2$, such functions exhibit logarithmic variations. This is in line with the GFF prediction and improves a non-quantitative delocalization result by P. Lammers. The proof goes via level-set percolation arguments, including a renormalization inequality and a dichotomy theorem for level-set loops. In another direction, we show that random Lipschitz functions have bounded variance whenever the wired FK-Ising model with $p=1-1/\mathbf{c}$ percolates on the same lattice (corresponding to $\mathbf{c} > 2 + \sqrt{3}$ on the honeycomb lattice). Via a simple coupling, this also implies, perhaps surprisingly, that random homomorphisms are localized on the rhombille lattice.