Uniform Lipschitz functions on the triangular lattice have logarithmic variations

Uniform integer-valued Lipschitz functions on a domain of size $N$ of the triangular lattice are shown to have variations of order $\sqrt{\log N}$. The level lines of such functions form a loop $O(2)$ model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for the loop O(2) model is constructed as a thermodynamic limit and is shown to be unique. It contains only finite loops and has properties indicative of scale-invariance: macroscopic loops appearing at every scale. The existence of the infinite-volume measure carries over to height functions pinned at the origin; the uniqueness of the Gibbs measure does not. The proof is based on a representation of the loop $O(2)$ model via a pair of spin configurations that are shown to satisfy the FKG inequality. We prove RSW-type estimates for a certain connectivity notion in the aforementioned spin model.Comment: Compared to v1: Theorem 1.3 (uniqueness of the Gibbs measure) is added; proof of Theorem 3.2 (delocalization) is significantly shortened; more details added in Section 4 (proof of the dichotomy theorem

On the probability that self-avoiding walk ends at a given point

We prove two results on the delocalization of the endpoint of a uniform self-avoiding walk on Z^d for d>1. We show that the probability that a walk of length n ends at a point x tends to 0 as n tends to infinity, uniformly in x. Also, for any fixed x in Z^d, this probability decreases faster than n^{-1/4 + epsilon} for any epsilon >0. When |x|= 1, we thus obtain a bound on the probability that self-avoiding walk is a polygon.Comment: 31 pages, 8 figures. Referee corrections implemented; removed section 5.

Delocalisation and continuity in 2D: loop O(2), six-vertex, and random-cluster models

We prove the existence of macroscopic loops in the loop O(2) model with $\frac12\leq x^2\leq 1$ or, equivalently, delocalisation of the associated integer-valued Lipschitz function on the triangular lattice. This settles one side of the conjecture of Fan, Domany, and Nienhuis (1970s-80s) that $x^2 = \frac12$ is the critical point. We also prove delocalisation in the six-vertex model with $0<a,\,b\leq c\leq a+b$. This yields a new proof of continuity of the phase transition in the random-cluster and Potts models in two dimensions for $1\leq q\leq 4$ relying neither on integrability tools (parafermionic observables, Bethe Ansatz), nor on the Russo-Seymour-Welsh theory. Our approach goes through a novel FKG property required for the non-coexistence theorem of Zhang and Sheffield, which is used to prove delocalisation all the way up to the critical point. We also use the $\mathbb T$-circuit argument in the case of the six-vertex model. Finally, we extend an existing renormalisation inequality in order to quantify the delocalisation as being logarithmic, in the regimes $\frac12\leq x^2\leq 1$ and $a=b\leq c\leq a+b$. This is consistent with the conjecture that the scaling limit is the Gaussian free field.Comment: 50 pages, 10 figure

A minimal model of quantized conductance in interacting ballistic quantum wires

We review what we consider to be the minimal model of quantized conductance in a finite interacting quantum wire. Our approach utilizes the simplicity of the equation of motion description to both deal with general spatially dependent interactions and finite wire geometry. We emphasize the role of two different kinds of boundary conditions, one associated with local "chemical" equilibrium in the sense of Landauer, the other associated with screening in the proximity of the Fermi liquid metallic leads. The relation of our analysis to other approaches to this problem is clarified. We then use our formalism to derive a Drude type expression for the low frequency AC-conductance of the finite wire with general interaction profile.Comment: 6 pages, 2 figures; extended discussion, references adde