On local behaviour of the phase separation line in the 2D Ising model

Summary. The aim of this note is to discuss some statistical properties of the phase separation line in the 2D low-temperature Ising model. We prove the functional central limit theorem for the probability distributions describing fluctuations of the phase boundary in the direction orthogonal to its orientation. The limiting Gaussian measure corresponds to a scaled Brownian bridge with direction dependent parameters. Up to the temperature factor, the variances of local increments of this limiting process are inversely proportional to the stiffness. Received: 6 June 1997/In revised form: 20 August 1997     

Centerpole sets for colorings of abelian groups

A subset C?G of a group G is called k-centerpole if for each k-coloring of G there is an infinite monochromatic subset G, which is symmetric with respect to a point c??C in the sense that S=cS ^?1 c. By c _k (G) we denote the smallest cardinality c _k (G) of a k-centerpole subset in G. We prove that c _k (G)=c _k (? ^m ) if G is an abelian group of free rank m??k. Also we prove that c ₁(? ⁿ⁺¹)=1, c ₂(? ⁿ⁺²)=3, c ₃(? ⁿ⁺³)=6, 8??c ₄(? ⁿ⁺⁴)??c ₄(?⁴)=12 for all n????, and ${\frac{1}{2}(k^{2}+3k-4)\le c_{k}(\mathbb{Z}^{n})\le2^{k}-1-\max_{s\le k-2}\binom {k-1}{s-1}}$ for all n??k??4.     

Random walk in mixed random environment without uniform ellipticity

We study a random walk in random environment on ?₊. The random environment is not homogeneous in law, but is a mixture of two kinds of site, one in asymptotically vanishing proportion. The two kinds of site are (i) points endowed with probabilities drawn from a symmetric distribution with heavy tails at 0 and 1, and (ii) “fast points” with a fixed systematic drift. Without these fast points, the model is related to the diffusion in heavy-tailed (“stable”) random potential studied by Schumacher and Singh; the fast points perturb that model. The two components compete to determine the behaviour of the random walk; we identify phase transitions in terms of the model parameters. We give conditions for recurrence and transience and prove almost sure bounds for the trajectories of the walk.     

Poly(2‐hydroxyethyl methacrylate‐co‐N,O‐dimethacryloylhydroxylamine) particles by dispersion polymerization

Poly(2‐hydroxyethyl methacrylate‐co‐N,O‐dimethacryloylhydroxylamine) particles were prepared by dispersion polymerization in toluene/2‐methylpropan‐1‐ol medium using cellulose acetate butyrate and dibenzoyl peroxide (BPO) as a steric stabilizer and initiator, respectively. The particle size was reduced with decreasing solvency of the reaction medium (more nuclei were generated) because the critical chain length of the precipitated oligomers decreased with an increasing toluene content, which is a poorer solvent for the polymer than 2‐methylpropan‐1‐ol. There is an optimum initiator concentration (2 wt % BPO relative to monomers) for producing low‐polydispersity particles under given conditions. Additionally, discrete spherical particles were obtained at a low monomer concentration and/or higher polymerization temperature. © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 40: 1625–1632, 2002     

Surface Tension and the Ornstein–Zernike Behaviour for the 2D Blume–Capel Model

We prove existence of the surface tension in the low temperature 2D Blume–Capel model and verify the Ornstein–Zernike asymptotics of the corresponding finite-volume interface partition function.     

Stochastic Model of Microtubule Dynamics

We introduce a continuous time stochastic process on strings made of two types of particle, whose dynamics mimics that of microtubules in a living cell. The long term behaviour of the system is described in terms of the velocity v of the string end. We show that v is an analytic function of its parameters and study its monotonicity properties. We give a complete characterisation of the phase diagram of the model and derive several criteria of the growth (\(v>0\)) and the shrinking (\(v<0\)) regimes of the dynamics.