Narrow Results

Kittel’s 1D model represents a natural DNA with two strands as a (molecular) zipper, which may be separated as the temperature is varied. We define multidimensional version of this model on a Cayley tree and study the set of Gibbs measures. We reduce description of Gibbs measures to solving of a nonlinear functional equation, with unknown functions (called boundary laws) defined on vertices of the Cayley tree. Each boundary law defines a Gibbs measure. We give a general formula of free energy depending on the boundary law. Moreover, we find some concrete boundary laws and corresponding Gibbs measures. Explicit critical temperature for occurrence of a phase transition (non-uniqueness of Gibbs measures) is obtained.

For the solid-on-solid (SOS) model with an external field and with spin values from the set of all integers on a Cayley tree, each (gradient) Gibbs measure corresponds to a boundary law (an infinite-dimensional vector function defined on vertices of the Cayley tree) satisfying a nonlinear functional equation. Recently some translation-invariant and height-periodic (non-normalizable) solutions to the equation are found. Here, our aim is to find non-height-periodic and non-normalizable boundary laws for the SOS model. By such a solution one can construct a non-probability Gibbs measure. We find explicitly several non-normalizable boundary laws. Moreover, we reduce the problem to solving of a nonlinear, second-order difference equation. We give analytic and numerical analyses of the difference equation.