Schramm's Problem 2.3: Gaussian free field and SLE$_4$ limits of uniform real Lipschitz heights on the triangular lattice

Principal manuscript: on approximations of smooth simply connected domains, the centered uniform real Lipschitz height converges as a random distribution to a multiple of the Dirichlet GFF, and at one tuned amplitude $\lambda$ the zero-height interface converges in uniform curve distance to chordal SLE$_4$; the variance-boundary relation is given through an implicit coefficient, so constants are not explicit. Companion: the field part of Problem 2.2 (uniform odd integer heights, boundary $\pm1$) with an absolutely convergent normalization formula; the Problem 2.2 interface is not proved. Companion: GFF limits for zero-boundary integer Lipschitz heights weighted by $x\in[1/\sqrt2,1]$ (loop $O(2)$), part of a conjecture discussed by Glazman and Lammers; larger $x$ is not covered.

Principal manuscript: on approximations of smooth simply connected domains, the centered uniform real Lipschitz height converges as a random distribution to a multiple of the Dirichlet GFF, and at one tuned amplitude $\lambda$ the zero-height interface converges in uniform curve distance to chordal SLE$_4$; the variance-boundary relation is given through an implicit coefficient, so constants are not explicit. Companion: the field part of Problem 2.2 (uniform odd integer heights, boundary $\pm1$) with an absolutely convergent normalization formula; the Problem 2.2 interface is not proved. Companion: GFF limits for zero-boundary integer Lipschitz heights weighted by $x\in[1/\sqrt2,1]$ (loop $O(2)$), part of a conjecture discussed by Glazman and Lammers; larger $x$ is not covered.