Ostap Hryniv

Fluctuations of shapes of large areas under paths of random walks

SummaryWe discuss statistical properties of random walks conditioned by fixing a large area under their paths. We prove the functional central limit theorem (invariance principle) for these conditional distributions. The limiting Gaussian measure coincides with the conditional probability distribution of certain timenonhomogeneous Gaussian random process obtained by an integral transformation of the white noise. From the point of view of statistical mechanics the studied problem is the problem of describing the fluctuations of the phase boundary in the one-dimensional SOS-model.

Universality of critical behaviour in a class of recurrent random walks

Abstract.Let X0=0, X1, X2,.. be an aperiodic random walk generated by a sequence ξ1, ξ2,... of i.i.d. integer-valued random variables with common distribution p(·) having zero mean and finite variance. For anN-step trajectory and a monotone convex functionV: withV(0)=0, define Further, let be the set of all non-negative paths compatible with the boundary conditionsX0=a, XN=b. We discuss asymptotic properties of under the probability distribution N→∞ and λ→0, Za,bN,+,λ being the corresponding normalization. If V(·) grows not faster than polynomially at infinity, define H(λ) to be the unique solution to the equation Our main result reads that as λ→0, the typical height of X[α, N] scales as H(λ) and the correlations along decay exponentially on the scale H(λ)2. Using a suitable blocking argument, we show that the distribution tails of the rescaled height decay exponentially with critical exponent 3/2. In the particular case of linear potential V(·), the characteristic length H(λ) is proportional to λ-1/3 as λ→0.

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.

THE OPINION GAME: STOCK PRICE EVOLUTION FROM MICROSCOPIC MARKET MODELING

We propose a class of Markovian agent based models for the time evolution of a share price in an interactive market. The models rely on a microscopic description of a market of buyers and sellers who change their opinion about the stock value in a stochastic way. The actual price is determined in realistic way by matching (clearing) offers until no further transactions can be performed. Some analytic results for simple special cases are presented. We also propose basic interaction mechanisms and show in simulations that these already reproduce certain particular features of prices in real stock markets.

Excursions and path functionals for stochastic processes with asymptotically zero drifts.

We study discrete-time stochastic processes (Xt) on [0,∞) with asymptotically zero mean drifts. Specifically, we consider the critical (Lamperti-type) situation in which the mean drift at x is about c/x. Our focus is the recurrent case (when c is not too large). We give sharp asymptotics for various functionals associated with the process and its excursions, including results on maxima and return times. These results include improvements on existing results in the literature in several respects, and also include new results on excursion sums and additive functionals of the form ∑s≤tXsα, α>0. We make minimal moments assumptions on the increments of the process. Recently there has been renewed interest in Lamperti-type process in the context of random polymers and interfaces, particularly nearest-neighbour random walks on the integers; some of our results are new even in that setting. We give applications of our results to processes on the whole of R and to a class of multidimensional ‘centrally biased’ random walks on Rd; we also apply our results to the simple harmonic urn, allowing us to sharpen existing results and to verify a conjecture of Crane et al.

Some rigorous results on semiflexible polymers, I: Free and confined polymers

We introduce a class of models of semiflexible polymers. The latter are characterized by a strong rigidity, the correlation length associated with the gradient-gradient correlations, called the persistence length, being of the same order as the polymer length. We determine the macroscopic scaling limit, from which we deduce bounds on the free energy of a polymer confined inside a narrow tube.

Long-time behaviour in a model of microtubule growth

We study a continuous-time stochastic process on strings made of two types of particle, whose dynamics mimic the behaviour of microtubules in a living cell; namely, the strings evolve via a competition between (local) growth/shrinking as well as (global) hydrolysis processes. We give a complete characterization of the phase diagram of the model, and derive several criteria of the transient and recurrent regimes for the underlying stochastic process.

Random Walk in Mixed Random Environment without Uniform Ellipticity

We study a random walk in random environment on ℤ+. 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.