arXiv subjects

Karol Kajetan Kozlowski

Publications and source records attributed to Karol Kajetan Kozlowski.

3 recordsLinked to original sources

arXiv2026

Rotational invariance in critical planar lattice models

We prove that the large-scale properties of a number of two-dimensional lattice models are rotationally invariant. More precisely, we prove that the random-cluster model on the square lattice with cluster-weight $1\le q\le 4$ exhibits rotational invariance at large scales. This covers the case of Bernoulli percolation on the square lattice as an important example. We deduce that the correlations of the critical Potts models with $q\in\{2,3,4\}$ colours are rotationally invariant at large scales. Our result is instrumental in proving the convergence of the six-vertex model to the Gaussian Free Field in a separate paper.

math.PR↗

arXiv2026

Gaussian free field convergence of the six-vertex model with $-1\leqΔ\leq-\frac12$

We study the isotropic six-vertex model on $\mathbb{Z}^2$ with spectral parameter $Δ\in[-1,-1/2]$, that is, with weights $\mathbf{a}=\mathbf{b}=1$ and $\mathbf{c}\in[\sqrt{3},2]$. We show that the associated height function converges, in the scaling limit, to a properly scaled full-plane Gaussian free field. The result extends to anisotropic weights $\mathbf{a}\neq\mathbf{b}$ upon using a suitable embedding of the lattice.

math-ph↗

arXiv2022

On the six-vertex model's free energy

In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime $Δ<1$. As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches $1/2$. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when $a=b=1$ and $c\ge1$, and the rotational invariance of the six-vertex model and the Fortuin-Kasteleyn percolation.

math-ph↗