30 research outputs found
Multimodality of the Markov binomial distribution
We study the shape of the probability mass function of the Markov binomial
distribution, and give necessary and sufficient conditions for the probability
mass function to be unimodal, bimodal or trimodal. These are useful to analyze
the double-peaking results from a PDE reactive transport model from the
engineering literature. Moreover, we give a closed form expression for the
variance of the Markov binomial distribution, and expressions for the mean and
the variance conditioned on the state at time .Comment: 15 pages, 3 figure
Concentration Inequalities for Functions of Gibbs Fields with Application to Diffraction and Random Gibbs Measures
We derive useful general concentration inequalities for functions of Gibbs fields in the uniqueness regime. We also consider expectations of random Gibbs measures that depend on an additional disorder field, and prove concentration w.r.t. the disorder field. Both fields are assumed to be in the uniqueness regime, allowing in particular for non-independent disorder fields. The modification of the bounds compared to the case of an independent field can be expressed in terms of constants that resemble the Dobrushin contraction coefficient, and are explicitly computable.
On the basis of these inequalities, we obtain bounds on the deviation of a diffraction pattern created by random scatterers located on a general discrete point set in Euclidean space, restricted to a finite volume. Here we also allow for thermal dislocations of the scatterers around their equilibrium positions. Extending recent results for independent scatterers, we give a universal upper bound on the probability of a deviation of the random scattering measures applied to an observable from its mean. The bound is exponential in the number of scatterers with a rate that involves only the minimal distance between points in the point set.
A Finite-Volume Version of Aizenman-Higuchi Theorem for the 2d Ising Model
In the late 1970s, in two celebrated papers, Aizenman and Higuchi
independently established that all infinite-volume Gibbs measures of the
two-dimensional ferromagnetic nearest-neighbor Ising model are convex
combinations of the two pure phases. We present here a new approach to this
result, with a number of advantages: (i) We obtain an optimal finite-volume,
quantitative analogue (implying the classical claim); (ii) the scheme of our
proof seems more natural and provides a better picture of the underlying
phenomenon; (iii) this new approach might be applicable to systems for which
the classical method fails.Comment: A couple of typos corrected. To appear in Probab. Theory Relat.
Field
On the Gibbs states of the noncritical Potts model on Z^2
We prove that all Gibbs states of the q-state nearest neighbor Potts model on
Z^2 below the critical temperature are convex combinations of the q pure
phases; in particular, they are all translation invariant. To achieve this
goal, we consider such models in large finite boxes with arbitrary boundary
condition, and prove that the center of the box lies deeply inside a pure phase
with high probability. Our estimate of the finite-volume error term is of
essentially optimal order, which stems from the Brownian scaling of fluctuating
interfaces. The results hold at any supercritical value of the inverse
temperature.Comment: Minor typos corrected after proofreading. Final version, to appear in
Probab. Theory Relat. Field