Wulff construction in statistical mechanics and in combinatorics

We present the geometric solutions to some variational problems of statistical mechanics and combinatorics. Together with the Wulff construction, which predicts the shape of the crystals, we discuss the construction which exhibits the shape of a typical Young diagram and of a typical skyscraper.Comment: A revie

Rotating states in driven clock- and XY-models

We consider 3D active plane rotators, where the interaction between the spins is of XY-type and where each spin is driven to rotate. For the clock-model, when the spins take N\gg1 possible values, we conjecture that there are two low-temperature regimes. At very low temperatures and for small enough drift the phase diagram is a small perturbation of the equilibrium case. At larger temperatures the massless modes appear and the spins start to rotate synchronously for arbitrary small drift. For the driven XY-model we prove that there is essentially a unique translation-invariant and stationary distribution despite the fact that the dynamics is not ergodic

Crystals in the Void

We study the problem of the crystal formation in the vacuum

Poisson Hypothesis for Information Networks II. Cases of Violations and Phase Transitions

We present examples of queuing networks that never come to equilibrium. That is achieved by constructing Non-linear Markov Processes, which are non-ergodic, and possess eternal transience property

Random Lattices and Random Sphere Packings: Typical Properties

We review results about the density of typical lattices in $R^n.$ They state that such density is of the order of $2^{-n}.$ We then obtain similar results for random packings in $R^n$: after taking suitably a fraction $\nu$ of a typical random packing $\sigma$, the resulting packing $\tau$ has density $C(\nu) 2^{-n},$ with a reasonable $C(\nu).$ We obtain estimates on $C(\nu).

Ising model fog drip: the first two droplets

We present here a simple model describing coexistence of solid and vapour phases. The two phases are separated by an interface. We show that when the concentration of supersaturated vapour reaches the dew-point, the droplet of solid is created spontaneously on the interface, adding to it a monolayer of a visible size

Self-averaging property of queuing systems

We establish the averaging property for a queuing process with one server, M(t)/GI/1. It is a new relation between the output flow rate and the input flow rate, crucial in the study of the Poisson Hypothesis. Its implications include the statement that the output flow always possesses more regularity than the input flow.Comment: 18 pages, one typo remove

A manifold of pure Gibbs states of the Ising model on a Cayley tree

We study the Ising model on a Cayley tree. A wide class of new Gibbs states is exhibited