Searcharxiv⌕ Search

arXiv subjects

Roman Kotecký

Publications and source records attributed to Roman Kotecký.

10 recordsLinked to original sources
arXiv2026

The Widom-Rowlinson model: Mesoscopic fluctuations for the critical droplet

We study the critical droplet for a close-to-equilibrium Widom-Rowlinson model of interacting particles, represented by disks of radius $1$, in the two-dimensional plane at low temperature. The critical droplet is the set of macroscopic states that correspond to saddle points for the passage from a low-density supersaturated vapour to a stable high-density liquid. We analyse the mesoscopic fluctuations of the surface of the critical droplet, which turns out to be the set of particle configurations that are close to a disk of a certain deterministic radius. Our results represent the first detailed rigorous analysis of the surface fluctuations of a continuum interacting particle system exhibiting condensation and, as such, constitute a fundamental step in the study of phase separation from the perspective of stochastic geometry. At the same time, our results serve as a basis for the study of a non-equilibrium version of the Widom-Rowlinson model, to be analysed elsewhere, where they lead to a correction term in the Arrhenius formula for the average vapour-liquid crossover time.

math-ph↗
arXiv2024

Cauchy-Born Rule from Microscopic Models with Non-convex Potentials

We study gradient field models on an integer lattice with non-convex interactions. These models emerge in distinct branches of physics and mathematics under various names. In particular, as zero-mass lattice (Euclidean) quantum field theory, models of random interfaces, and as mass-string models of nonlinear elasticity.Our attention is mostly devoted to the latter with random vector valued fields as displacements for atoms of crystal structures,where our aim is to prove the strict convexity of the free energy as a function of affine deformations for low enough temperatures and small enough deformations. This claim can be interpreted as a form of verification of the Cauchy-Born rule at small non-vanishing temperatures for a class of these models. We also show that the scaling limit of the Laplace transform of the corresponding Gibbs measure (under a proper rescaling) corresponds to the Gaussian gradient field with a particular covariance. The proofs are based on a multi-scale (renormalisation group analysis) techniques needed in view of strong correlations of studied gradient fields. To cover sufficiently wide class of models, we extend these techniques from the standard case with rotationally symmetric nearest neighbour interaction to a more general situation with finite range interactions without any symmetry. Our presentation is entirely self-contained covering the details of the needed renormalisation group methods.

math-ph↗
arXiv2018

Staggered long-range order for diluted quantum spin models

We study an annealed site diluted quantum XY model with spin $S\in \frac{1}{2}\mathbb{N}$. We find regions of the parameter space where, in spite of being a priori favourable for a densely occupied state, phases with staggered occupancy occur at low temperatures.

cond-mat.stat-mech↗
arXiv2016

Strict Convexity of the Surface Tension for Non-convex Potentials

We study gradient models on the lattice $\mathbb{Z}^d$ with non-convex interactions. These Gibbs fields (lattice models with continuous spin) emerge in various branches of physics and mathematics. In quantum field theory they appear as massless field theories. Even though our motivation stems from considering vector valued fields as displacements for atoms of crystal structures and the study of the Cauchy-Born rule for these models, our attention here is mostly devoted to interfaces, with the gradient field as an \emph{effective} interface interaction. In this case we prove the strict convexity of the surface tension (interface free energy) for low temperatures and sufficiently small interface tilts using muli-scale (renormalisation group analysis) techniques following the approach of Brydges and coworkers \cite{B07}. This is a complement to the study of the high temperature regime in \cite{CDM09} and it is an extension of Funaki and Spohn's result \cite{FS97} valid for strictly convex interactions.

math-ph↗
arXiv2013

Entropy-driven phase transition in low-temperature antiferromagnetic Potts models

We prove the existence of long-range order at sufficiently low temperatures, including zero temperature, for the three-state Potts antiferromagnet on a class of quasi-transitive plane quadrangulations, including the diced lattice. More precisely, we show the existence of (at least) three infinite-volume Gibbs measures, which exhibit spontaneous magnetization in the sense that vertices in one sublattice have a higher probability to be in one state than in either of the other two states. For the special case of the diced lattice, we give a good rigorous lower bound on this probability, based on computer-assisted calculations that are not available for the other lattices.

math-ph↗
arXiv2013

Two-dimensional Potts antiferromagnets with a phase transition at arbitrarily large q

We exhibit infinite families of two-dimensional lattices (some of which are triangulations or quadrangulations of the plane) on which the q-state Potts antiferromagnet has a finite-temperature phase transition at arbitrarily large values of q. This unexpected result is proven rigorously by using a Peierls argument to measure the entropic advantage of sublattice long-range order. Additional numerical data are obtained using transfer matrices, Monte Carlo simulation, and a high-precision graph-theoretic method.

cond-mat.stat-mech↗
arXiv2012

Nonlinear elastic free energies and gradient Young-Gibbs measures

We investigate, in a fairly general setting, the limit of large volume equilibrium Gibbs measures for elasticity type Hamiltonians with clamped boundary conditions. The existence of a quasiconvex free energy, forming the large deviations rate functional, is shown using a new interpolation lemma for partition functions. The local behaviour of the Gibbs measures can be parametrized by Young measures on the space of gradient Gibbs measures. In view of unboundedness of the state space, the crucial tool here is an exponential tightness estimate that holds for a vast class of potentials and the construction of suitable compact sets of gradient Gibbs measures.

math-ph↗
arXiv2012

Finite range decomposition for families of gradient Gaussian measures

Let a family of gradient Gaussian vector fields on $ \mathbb{Z}^d $ be given. We show the existence of a uniform finite range decomposition of the corresponding covariance operators, that is, the covariance operator can be written as a sum of covariance operators whose kernels are supported within cubes of diameters $ \sim L^k $. In addition we prove natural regularity for the subcovariance operators and we obtain regularity bounds as we vary within the given family of gradient Gaussian measures.

math-ph↗
arXiv1994

Pathological Behavior of Renormalization-Group Maps at High Fields and Above the Transition Temperature

We show that decimation transformations applied to high-$q$ Potts models result in non-Gibbsian measures even for temperatures higher than the transition temperature. We also show that majority transformations applied to the Ising model in a very strong field at low temperatures produce non-Gibbsian measures. This shows that pathological behavior of renormalization-group transformations is even more widespread than previous examples already suggested.

hep-lat↗