arXiv subjects

Moritz Dober

Publications and source records attributed to Moritz Dober.

4 recordsLinked to original sources

arXiv2026

Discontinuous transition in 2D Potts: II. Order-Order Interface convergence

The $q$-state Potts model is an archetypical model for various types of phase transitions. We consider it on the square grid and focus on the regime where it undergoes a discontinuous transition, that is $q>4$. At the transition point $T_c(q)$, there are exactly $q+1$ extremal Gibbs measures (pure phases): $q$ ordered (monochromatic) and one disordered (free). This work establishes for the first time the wetting phenomenon in a precise geometric form and in the entire regime of discontinuity $q>4$: at $T_c(q)$, between two ordered phases a disordered layer emerges and, in the diffusive scaling, its boundaries converge to a pair of Brownian motions conditioned not to intersect. This is starkly different from the subcritical ($T 4$.

math.PR↗

arXiv2025

Discontinuous transition in 2D Potts: I. Order-Disorder Interface convergence

We study a $q$-state Potts model on the square grid when $q>4$ at the point $T_c(q)$ of its (discontinous) transition. This model exhibits exactly $q+1$ extremal Gibbs measures: $q$ ordered (monochromatic) and one disordered (free). The current work deals with the Dobrushin order--disorder boundary conditions on a finite $N\times N$ box. Our main result is that this interface is a well-defined object, has $\sqrt{N}$ fluctuations, and converges to a Brownian bridge under diffusive scaling. The same holds also for the corresponding FK-percolation model for all $q>4$. Our proofs rely on a coupling between FK-percolation, the six-vertex model, and the random-cluster representation of an Ashkin--Teller model (ATRC), and on a detailed study of the latter. The coupling relates the interface in FK-percolation to a long subcritical cluster in the ATRC model. For this cluster we develop a ``renewal picture'' \`a la Ornstein-Zernike. This is based on fine mixing properties of the ATRC model that we establish using the link to the six-vertex model and its height function. Along the way, we derive various properties of the Ashkin-Teller model, such as Ornstein-Zernike asymptotics for its two-point function. In a companion work, we provide a detailed study of the Potts model under order-order Dobrushin conditions. We show emergence of a free layer of width $\sqrt{N}$ between the two ordered phases (wetting) and establish convergence of its boundaries to two Brownian bridges conditioned not to intersect.

math.PR↗

arXiv2024

On antiferromagnetic regimes in the Ashkin-Teller model

The Ashkin-Teller model can be represented by a pair $(\tau,\tau')$ of Ising spin configurations with coupling constants $J$ and $J'$ for each, and $U$ for their product. We study this representation on the integer lattice $\mathbb{Z}^d$ for $d\geq 2$. We confirm the presence of a partial antiferromagnetic phase in the isotropic case ($J=J'$) when $-U>0$ is sufficiently large and $J=J'>0$ is sufficiently small, by means of a graphical representation. In this phase, $\tau$ is disordered, admitting exponential decay of correlations, while the product $\tau\tau'$ is antiferromagnetically ordered, which is to say that correlations are bounded away from zero but alternate in sign. No correlation inequalities are available in this part of the phase diagram. In the planar case $d=2$, we construct a coupling with the six-vertex model and show, in analogy to the first result, that the corresponding height function is localised, although with antiferromagnetically ordered heights on one class of vertices of the graph. We then return to $d\geq 2$ and consider a part of the phase diagram where $U<0$ but where correlation inequalities still apply. Using the OSSS inequality, we proceed to establish a subcritical sharpness statement along suitable curves covering this part, circumventing the difficulty of the lack of general monotonicity properties in the parameters. We then address the isotropic case and provide indications of monotonicity.

math.PR↗

arXiv2023

Phase diagram of the Ashkin-Teller model

The Ashkin-Teller model is a pair of interacting Ising models and has two parameters: $J$ is a coupling constant in the Ising models and $U$ describes the strength of the interaction between them. In the ferromagnetic case $J,U>0$ on the square lattice, we establish a complete phase diagram conjectured in physics in 1970s (by Kadanoff and Wegner, Wu and Lin, Baxter and others): when $J<U$, the transitions for the Ising spins and their products occur at two distinct curves that are dual to each other; when $J\geq U$, both transitions occur at the self-dual curve. All transitions are shown to be sharp using the OSSS inequality. We use a finite-criterion argument and continuity to extend the result of Peled and the third author \cite{GlaPel19} from a self-dual point to its neighborhood. Our proofs go through the random-cluster representation of the Ashkin-Teller model introduced by Chayes-Machta and Pfister-Velenik and we rely on couplings to FK-percolation.

math.PR↗