arXiv subjects

Ulrik Thinggaard Hansen

Publications and source records attributed to Ulrik Thinggaard Hansen.

6 recordsLinked to original sources

arXiv2026

The End of the Double Random Current

The Ising correlation function can be expressed in terms of connectivity probabilities of a random graph known as the double random current. In 2016, Duminil-Copin asked whether this random graph only has one topological end. One-endedness implies that free and wired random-cluster measures are equal and that any translation-invariant Gibbs measure of the Ising model is a convex combination of the $+$ and $-$ Gibbs measures. In the general setting of transitive amenable graphs, these properties have been established by Raoufi, effectively circumventing the question of one-endedness. In this paper, we prove one-endedness of the double random current on any transitive, amenable one-ended graph and, relying on Raoufi's work, we answer the question of Duminil-Copin in the affirmative in this general setup. The results of this paper do not give a new proof of uniqueness of random-cluster measure or the characterisations of Gibbs states for the Ising model.

math.PR↗

arXiv2026

The Supercritical Loop O(1) and Random Current models: Uniqueness and Mixing

Much recent rigorous study of the classical ferromagnetic Ising model has been powered by its graphical representations, such as the random current and loop O(1) model (high temperature expansion). In this paper, we prove uniqueness of Gibbs measures and exponential ratio weak mixing for the loop O(1) and random current models corresponding to the supercritical Ising model on the hypercubic lattice $\Z^d$ in any dimension $d \geq 2$. The methods generalise to $q$-flow models and have natural applications for gradient measures of $\Z/q\mathbb{Z}$-gauge theories.

math.PR↗

arXiv2025

A General Coupling for Ising Models and Beyond

The couplings between the Ising model and its graphical representations, the random-cluster, random current and loop $\mathrm{O}(1)$ models, are put on common footing through a generalization of the Swendsen-Wang-Edwards-Sokal coupling. A new special case is that the single random current with parameter $2J$ on edges of agreement of XOR-Ising spins has the law of the double random current. The coupling also yields a general mechanism for constructing conditional Bernoulli percolation measures via uniform sampling, providing new perspectives on models such as the arboreal gas, random $d$-regular graphs, and self-avoiding walks. As an application of the coupling for the Loop-Cluster joint model, we prove that the regimes of exponential decay of the $q$-state Potts model, the random-cluster representation, and its $q$-flow loop representation coincide on the torus, generalizing the Ising result. By providing a new and equivalent definition of the plaquette random cluster model, we are able to generalize the loop-cluster coupling to lattice gauge theories.

math.PR↗

arXiv2024

Non-uniqueness of phase transitions for graphical representations of the Ising model on tree-like graphs

We consider the graphical representations of the Ising model on tree-like graphs. We construct a class of graphs on which the loop $\mathrm{O}(1)$ model and the single random current exhibit a non-unique phase transition with respect to the inverse temperature, highlighting the non-monotonicity of both models. It follows from the construction that there exist infinite graphs $\mathbb{G}\subseteq \mathbb{G}'$ such that the uniform even subgraph of $\mathbb{G}'$ percolates and the uniform even subgraph of $\mathbb{G}$ does not. We also show that on the wired $d$-regular tree, the phase transitions of the loop $\mathrm{O}(1)$, the single random current, and the random-cluster models are all unique and coincide.

math.PR↗

arXiv2023

The Uniform Even Subgraph and Its Connection to Phase Transitions of Graphical Representations of the Ising Model

The uniform even subgraph is intimately related to the Ising model, the random-cluster model, the random current model, and the loop $\mathrm{O}$(1) model. In this paper, we first prove that the uniform even subgraph of $Z^d$ percolates for $d \geq 2$ using its characterisation as the Haar measure on the group of even graphs. We then tighten the result by showing that the loop $\mathrm{O}$(1) model on $Z^d$ percolates for $d \geq 2$ for edge-weights $x$ lying in some interval $(1-\varepsilon,1]$. Finally, our main theorem is that the loop $\mathrm{O}$(1) model and random current models corresponding to a supercritical Ising model are always at least critical, in the sense that their two-point correlation functions decay at most polynomially and the expected cluster sizes are infinite.

math.PR↗

arXiv2023

Strict monotonicity, continuity and bounds on the Kertész line for the random-cluster model on $\mathbb{Z}^d$

Ising and Potts models can be studied using the Fortuin-Kasteleyn representation through the Edwards-Sokal coupling. This adapts to the setting where the models are exposed to an external field of strength $h>0$. In this representation, which is also known as the random-cluster model, the Kertész line separates the two regions of parameter space according to the existence of an infinite cluster in $\mathbb{Z}^d$. This signifies a geometric phase transition between the ordered and disordered phases even in cases where a thermodynamic phase transition does not occur. In this article, we prove strict monotonicity and continuity of the Kertész line. Furthermore, we give new rigorous bounds that are asymptotically correct in the limit $h \to 0$ complementing the bounds from the work of Ruiz and Wouts [J. Math. Phys. 49, 053303 (2008)], which were asymptotically correct for $h \to \infty$. Finally, using a cluster expansion, we investigate the continuity of the Kertész line phase transition.

math-ph↗