Brownian flights over a circle.

The stationary radial distribution, P(ρ), of a random walk with the diffusion coefficient D, which winds at the tangential velocity V around an impenetrable disk of radius R for R≫D/V converges to the distribution involving the Airy function. Typical trajectories are localized in the circular strip [R,R+δR^{1/3}], where δ is a constant which depends on the parameters D and V and is independent of R.

Anomalous one-dimensional fluctuations of a simple two-dimensional random walk in a large-deviation regime.

The following question is the subject of our work: could a two-dimensional (2D) random path pushed by some constraints to an improbable "large-deviation regime" possess extreme statistics with one-dimensional (1D) Kardar-Parisi-Zhang (KPZ) fluctuations? The answer is positive, though nonuniversal, since the fluctuations depend on the underlying geometry. We consider in detail two examples of 2D systems for which imposed external constraints force the underlying stationary stochastic process to stay in an atypical regime with anomalous statistics. The first example deals with the fluctuations of a stretched 2D random walk above a semicircle or a triangle. In the second example we consider a 2D biased random walk along a channel with forbidden voids of circular and triangular shapes. In both cases we are interested in the dependence of a typical span 〈d(t)〉∼t^{γ} of the trajectory of t steps above the top of the semicircle or the triangle. We show that γ=1/3, i.e., 〈d(t)〉 shares the KPZ statistics for the semicircle, while γ=0 for the triangle. We propose heuristic derivations of scaling exponents γ for different geometries, justify them by explicit analytic computations, and compare with numeric simulations. For practical purposes, our results demonstrate that the geometry of voids in a channel might have a crucial impact on the width of the boundary layer and, thus, on the heat transfer in the channel.

First-order transitions for n-vector models in two and more dimensions: rigorous proof.

We prove that various SO(n)-invariant n-vector models with interactions which have a deep and narrow enough minimum have a first-order transition in the temperature. The result holds in dimensions two or more and is independent of the nature of the low-temperature phase.