2023/09/26 by Karrila, Alex M. · 1 citation
#60D05 #60K35 #82B20 #82B27 #82B41 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.2309.14895
We study random one-Lipschitz integer functions f on the vertices of a finite connected graph, sampled according to the weight W(f) = ∏⟨ v, w ⟩ ∈ E c^ \mathbbI \ f(v) = f(w) \ where c ≥ 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 ≤ c ≤ 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/c percolates on the same lattice (corresponding to c > 2 + √(3) on the honeycomb lattice). Via a simple coupling, this also implies, perhaps surprisingly, that random homomorphisms are localized on the rhombille lattice.