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

Crystals in the Void

We study the problem of the crystal formation in the vacuum

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

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

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

Poisson Hypothesis for information networks (A study in non-linear Markov processes)

In this paper we prove the Poisson Hypothesis for the limiting behavior of the large queueing systems in some simple ("mean-field") cases. We show in particular that the corresponding dynamical systems, defined by the non-linear Markov processes, have a line of fixed points which are global attractors. To do this we derive the corresponding non-linear integral equation and we explore its self-averaging properties. Our derivation relies on a solution of a combinatorial problem of rode placements.Comment: 70 page

Dobrushin Interfaces via Reflection Positivity

We study the interfaces separating different phases of 3D systems by means of the Reflection Positivity method. We treat discrete non-linear sigma-models, which exhibit power-law decay of correlations at low temperatures, and we prove the rigidity property of the interface. Our method is applicable to the Ising and Potts models, where it simplifies the derivation of some known results. The method also works for large-entropy systems of continuous spins.Comment: 48 pages, 4 figures; updated for publication (to appear in CMP

Stationary States of the Generalized Jackson Networks

We consider Jackson Networks on general countable graphs and with arbitrary service times. We find natural sufficient conditions for existence and uniqueness of stationary distributions. They generalise these obtained earlier by Kelbert, Kontsevich and Rybko.Comment: 18 pages, minor change