We study stochastic dynamics of an inverted pendulum subject to a random force in the horizontal direction. Considered at the entire time axis, the problem admits a unique solution which always remains in the upper half plane. We develop a new technique for treating statistical properties of this unique non-falling trajectory. In our approach based on the supersymmetric formalism of Parisi and Sourlas, statistics of the non-falling trajectory is expressed in terms of the zero mode of a corresponding transfer-matrix Hamiltonian. The emerging mathematical structure is similar to that of the Fokker-Planck equation, but it is rather written for the «square root» of the distribution function.
We derive the specific boundary conditions that correspond to the non-falling trajectory. Our results for the distribution function of the angle and its velocity at the non-falling trajectory are in perfect agreement with direct numerical simulations of the stochastic pendulum equation. In the limit of very strong noise, an exact analytical solution is obtained.

I plan to talk about explicit formulas for solutions of the Painlevé equations and their q-difference analogues. The formulas for generic solutions are written in terms of the Nekrasov partition functions or (according to the AGT correspondence) in terms of conformal blocks. Technically, these equations are reduced to bilinear relations on the partition functions. I plan to talk about the geometric meaning of these bilinear relations.

In this article, we consider the phenomenon of complete coincidence of the key properties of pairs of Calabi-Yau manifolds realized as hypersurfaces in two different weighted projective spaces. More precisely, the first manifold in such a pair is realized as a hypersurface in a weighted projective space, and the second as a hypersurface in the orbifold of another weighted projective space. The two manifolds in each pair have the same Hodge numbers and special K\"ahler geometry on the complex structure moduli space and are associated with the same $N=2$ gauge linear sigma model. We give the explanation of this interesting coincidence using the Batyrev's correspondence between Calabi-Yau manifolds and the reflexive polyhedra.

We develop a theory of quantum T=0 phase transition (q-SMT) between metal and superconducting ground states in a two-dimensional metal with frozen-in spatial fluctuations of the Cooper attraction constant. When strength of these fluctuations exceeds some critical magnitude, usual mean-field-like scenario of the q-SMT breaks down due to spontaneous formation of local droplets of superconducting phase. The density of these droplets grows exponentially with the increase of average attraction constant. Interaction between the droplet's order parameters is due to proximity effect via normal metal and scales with distance as inverse power of distance, with power exponent slightly larger than 2. We treat this interaction by means of strong-disorder real-space renormalization group and find the RG flow formally similar to the Berezinski-Kosterlitz-Thouless RG for 2D XY model. Line of fixed points of this RG corresponds to a Griffiths phase of a metal with large fractal clusters of strongly coupled superconducting islands. Superconducting side of the transition is characterized by a non-monotonic variation of physical properties with logarithm of the temperature T, which results of a very weak T-dependence in a broad temperature range.

One of the most exciting phenomena observed in crystalline disordered membranes, including a suspended graphene, is rippling, i.e. a formation of static flexural deformations. Despite an active research, it still remains unclear whether the rippled phase exists in the thermodynamic limit, or it is destroyed by thermal fluctuations. We demonstrate that a sufficiently strong short-range disorder stabilizes ripples, whereas in the case of a weak disorder the thermal flexural fluctuations dominate in the thermodynamic limit. The phase diagram of the disordered suspended graphene contains two separatrices: the crumpling transition line dividing the flat and crumpled phases and the rippling transition line demarking the rippled and clean phases. At the intersection of the separatrices there is the unstable, multicritical point which splits up all four phases. Most remarkably, rippled and clean flat phases are described by a single stable fixed point which belongs to the rippling transition line.

In the talk, I will discuss the examples of integrable differential-difference equations of order 2 with respect to the discrete variable constructed in [1]. Over the past few years, some particular classification results have been obtained for such type equations, but their general description remains a very difficult open problem. The considered examples are related with continuous symmetries of discrete equations on a triangular lattice. I demonstrate that their linear combination can be written as a scalar chain of order 2, under restriction at one of the lattice axes. Although this construction is rather special, helps to significantly expand the list of known examples. [1] V.E. Adler. Integrable seven-point discrete equations and second-order evolution chains. Theor. Math. Phys. 195:2 (2018) 513-528.

It is studied long-time dynamics of solitons in the frame of the super compact Zakharov equation for unidirectional waves. It is shown that after multiple collisions of breathers (solitons), only one soliton having initially a larger number of particles survives. Besides, it was found numerically bi-solutions solutions in the framework of both super compact and fully nonlinear equations.

We investigate the bursts of intensity in a telecommunication communication line that arise as a result of the action of chromatic dispersion during the propagation of a signal with quasi-random information encoded in it. As encodings, we consider the model format of sequential pulses of a Gaussian shape and the OFDM (orthogonal frequency division multiplexing) format used in practice. The communication line is assumed to be long, so that the pulses corresponding to the bits of information become broadened due to the dispersion so much that a large number of such pulses are overlapped at one point. The initial phase pulse multipliers can take a fixed set of values, constituting the encoding alphabet. Bursts of intensity are the result of the fact that the incident phases of the pulses, acquired as they propagate along the transmission line, may (partially) compensate for the phases of the symbols of the alphabet. As a result, constructive interference of the pulse fields occurs. Such events can be attributed as random. With a fixed distance travelled by the signal, there is a maximum possible value for the amplitude of such bursts. We investigate bursts of amplitude close to the limiting for the two indicated formats: we establish their amplitude and profile. Further, we carry out numerical modelling and take into account the influence of weak nonlinearity. Finally, we examine how the capacity of the communication channel decreases if it uses primarily sequences of characters that result in strong bursts of amplitude at a fixed distance from the beginning of the line.

Поставлен вопрос о влиянии конечности шага на оценку вероятности пересечения окружности при случайном блуждании на плоскости. Численно выявлена зависимость точности оценки от величины шага. Предложен аналитический вид зависимости. Предложен эффективный алгоритм изменения величины шага.
[1] Olga Klimenkova, Anton Menshutin, Lev N. Shchur, "Influence of the random walk finite step on the first-passage probability", Physics and beyond (CSP2017), 9-12 Oct., 2017, Moscow
[2] Olga Klimenkova, Anton Yu. Menshutin, Lev N. Shchur, "Variable-step-length algorithms for a random walk: hitting probability and computation performance", Computer Phys. Commun., 241, 28-32 (2019)