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.

Self-isolation or borders closing: What prevents the spread of the epidemic better?

Pandemic propagation of COVID-19 motivated us to discuss the impact of the human network clustering on epidemic spreading. Today, there are two clustering mechanisms which prevent of uncontrolled disease propagation in a connected network: an "internal" clustering, which mimics self-isolation (SI) in local naturally arranged communities, and an "external" clustering, which looks like a sharp frontiers closing (FC) between cities and countries, and which does not care about the natural connections of network agents. SI networks are "evolutionarily grown" under the condition of maximization of small cliques in the entire network, while FC networks are instantly created.

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.

Finite plateau in spectral gap of polychromatic constrained random networks.

We consider critical behavior in the ensemble of polychromatic Erdős-Rényi networks and regular random graphs, where network vertices are painted in different colors. The links can be randomly removed and added to the network subject to the condition of the vertex degree conservation. In these constrained graphs we run the Metropolis procedure, which favors the connected unicolor triads of nodes.