Phase transitions for a class of gradient fields

We consider gradient fields on $${\mathbb {Z}}^d$$ Z d for potentials V that can be expressed as $$\begin{aligned} e^{-V(x)}=pe^{-\frac{qx^2}{2}}+(1-p)e^{-\frac{x^2}{2}}. \end{aligned}$$ e - V ( x ) = p e - q x 2 2 + ( 1 - p ) e - x 2 2 . This representation allows us to associate a random conductance type model to the gradient fields with zero tilt. We investigate this random conductance model and prove correlation inequalities, duality properties, and uniqueness of the Gibbs measure in certain regimes. We then show that there is a close relation between Gibbs measures of the random conductance model and gradient Gibbs measures with zero tilt for the potential V. Based on these results we can give a new proof for the non-uniqueness of ergodic zero-tilt gradient Gibbs measures in dimension 2. In contrast to the first proof of this result we rely on planar duality and do not use reflection positivity. Moreover, we show uniqueness of ergodic zero tilt gradient Gibbs measures for almost all values of p and q and, in dimension $$d\ge 4$$ d ≥ 4 , for q close to one or for $$p(1-p)$$ p ( 1 - p ) sufficiently small.

Generalized Fréchet Bounds for Cell Entries in Multidimensional Contingency Tables

We consider the lattice, $\mathcal{L}$, of all subsets of a multidimensional contingency table and establish the properties of monotonicity and supermodularity for the marginalization function, $n(\cdot)$, on $\mathcal{L}$. We derive from the supermodularity of $n(\cdot)$ some generalized Fr\'echet inequalities complementing and extending inequalities of Dobra and Fienberg. Further, we construct new monotonic and supermodular functions from $n(\cdot)$, and we remark on the connection between supermodularity and some correlation inequalities for probability distributions on lattices. We also apply an inequality of Ky Fan to derive a new approach to Fr\'echet inequalities for multidimensional contingency tables.