Extremality of Disordered Phase of λ-Model on Cayley Trees

In this paper, we consider the λ-model for an arbitrary-order Cayley tree that has a disordered phase. Such a phase corresponds to a splitting Gibbs measure with free boundary conditions. In communication theory, such a measure appears naturally, and its extremality is related to the solvability of the non-reconstruction problem. In general, the disordered phase is not extreme; hence, it is natural to find a condition for their extremality. In the present paper, we present certain conditions for the extremality of the disordered phase of the λ-model.

Equilibrium measures of the natural extension of -shifts

We give a necessary and sufficient condition on $\beta $ of the natural extension of a $\beta $ -shift, so that any equilibrium measure for a function of bounded total oscillations is a weak Gibbs measure.

Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity I: measures

In this two-paper series, we prove the invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree nonlinearity. The main novelty is the singularity of the Gibbs measure with respect to the Gaussian free field. The singularity has several consequences in both measure-theoretic and dynamical aspects of our argument. In this paper, we construct and study the Gibbs measure. Our approach is based on earlier work of Barashkov and Gubinelli for the$$\Phi ^4_3$$Φ34-model. Most importantly, our truncated Gibbs measures are tailored towards the dynamical aspects in the second part of the series. In addition, we develop new tools dealing with the non-locality of the Hartree interaction. We also determine the exact threshold between singularity and absolute continuity of the Gibbs measure depending on the regularity of the interaction potential.

Thermodynamics of DNA–RNA renaturation

We consider a new model which consists of a DNA together with a RNA. Here, we assume that DNA is from a mammal or bird but RNA comes from a virus. To study thermodynamic properties of this model, we use methods of statistical mechanics, namely, the theory of Gibbs measures. We use these measures to describe phases (states) of the DNA–RNA system. Using a Markov chain (corresponding to Gibbs measure) we give conditions (on temperature) of DNA–RNA renaturation.

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 $$\sqrt{\log N}$$ 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.

The Free-Fermion Eight-Vertex Model: Couplings, Bipartite Dimers and Z-Invariance

We study the eight-vertex model at its free-fermion point. We express a new “switching” symmetry of the model in several forms: partition functions, order-disorder variables, couplings, Kasteleyn matrices. This symmetry can be used to relate free-fermion 8V-models to free-fermion 6V-models, or bipartite dimers. We also define new solution of the Yang–Baxter equations in a “checkerboard” setting, and a corresponding Z-invariant model. Using the bipartite dimers of Boutillier et al. (Probab Theory Relat Fields 174:235–305, 2019), we give exact local formulas for edge correlations in the Z-invariant free-fermion 8V-model on lozenge graphs, and we deduce the construction of an ergodic Gibbs measure.