The following question is the subject of our work: could a two-dimensional random path pushed by some constraints to an improbable "large deviation regime", possess extreme statistics with one-dimensional Kardar-Parisi-Zhang (KPZ) fluctuations?The answer is positive, though non-universal, since the fluctuations depend on the underlying geometry. We consider in details 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.

Using scaling arguments and extensive numerical simulations, we study dynamics of a tracer particle in a corrugated channel represented by a periodic sequence of broad chambers and narrow funnel-like bottlenecks...

The desire to create nanometer-scale switching devices has motivated an active search for bistate macromolecular systems allowing for sharp conformational transitions in response to stimuli. Using full-atomic simulations, we found particular oligomers of thermosensitive polymers, themselves only a few nanometers in size, that possessed conformational bistability and reacted to power loads as nonlinear mechanical systems, termed “catastrophe machines”. We established the bifurcation and hysteresis effects, spontaneous vibrations, and stochastic resonance for these oligomers. It is important to note that the spontaneous vibrations and stochastic resonance were activated by thermal fluctuations. Because of such mechanic-like characteristics, short oligomers are a promising platform for the design of nanodevices and molecular machines.

We consider a particular example of interplay between statistical models related to CFT on one hand, and to the spectral properties of ODE, known as ODE/IS correspondence, on the other hand. We focus at the representation of wave functions of Schrödinger operators in terms of spectral properties of associated transfer matrices on "super trees" (the trees whose vertex degree changes with the distance from the root point). Such trees with varying branchings encode the structure of the Fock space of the model. We discuss basic spectral properties of "averaged random matrix ensembles" in terms of Hermite polynomials for the transfer matrix of super trees. At small "branching velocities" we have related the problem of paths counting on super trees to the statistics of area-weighted one-dimensional Dyck paths. We also discuss the connection of the spectral statistics of random walks on super trees with the Kardar-Parisi-Zhang scaling.

We consider statistics of a planar ideal polymer loop of length L in a large deviation regime, when a gyration radius, R g, is slightly less than the radius of a fully inflated ring, L 2 π . Specifically, we study analytically and via off-lattice Monte-Carlo simulations relative fluctuations of chain monomers in an ensemble of Brownian loops. We have shown that these fluctuations in the regime with fixed large gyration radius are Gaussian with the critical exponent γ = 1 2 . However, if we insert inside the inflated loop the impenetrable disc of radius R d = R g, the fluctuations become non-Gaussian with the critical exponent γ = 1 3 typical for the Kardar–Parisi–Zhang universality class.