arXiv subjects

Daniil Fedotov

Publications and source records attributed to Daniil Fedotov.

5 recordsLinked to original sources

arXiv2026

Three faces of random walks in a hyperbolic domain: BKT, Lifshitz tails, and KPZ

We show that continuous random walks (diffusion) in the Poincaré upper half-plane $\mathbb{H}^2 = \{(x,y)| y>0\}$ provide a unified description of three seemingly unrelated phenomena: (i) the non-analytic divergence of the correlation length at the Berezinskii--Kosterlitz--Thouless (BKT) transition; (ii) the emergence of a Lifshitz tail (LT) in the 1D statistics of rare events; and (iii) the appearance of Kardar--Parisi--Zhang (KPZ) scaling in the fluctuations of stretched random walks constrained above an impermeable disc. We adapt the renormalization-group (RG) analysis originally developed for the Schrödinger equation with mixed boundary conditions in a 2D conformally invariant potential to the spectral problem in $\mathbb{H}^2$, and identify the conditions under which BKT-type behavior emerges. We show that the same RG equations can be interpreted in terms of the statistics of random walks in the large-deviation regime through an instanton approach, leading to Lifshitz tails (LT) in the associated spectral problem. We further derive KPZ-type scaling for the fluctuations of stretched random walks near the boundary of a disc in $\mathbb{R}^2$ and emphasize the similarity of this behavior with the boundary fluctuations of paths in $\mathbb{H}^2$ analyzed using a WKB approach.

cond-mat.stat-mech↗

arXiv2026

Non-algebraic first return probability of a stretched random walk near a convex boundary and its effect on adsorption

The $N$-step random walk, elongated in the vicinity of a disc (in 2D) or a sphere (in 3D) of radius $R$, demonstrates a non-algebraic stretched exponential decay $P_N\sim \exp\left(-{\rm const}\, N^{1/3}\right)$ for the first return probability $P_N$ in the double-scaling limit $N=\frac{L}{a}\gg 1, \frac{R}{a}\gg 1$ conditioned that $\frac{L}{R}=c={\rm const}$. Stretching means that the length of the walk, $L=Na$ (where $a$ is the unit step length) satisfies the condition $L = cR$, where $c > π$ and under "first return" we understand the radial first arrival to a boundary. Both analytic and numerical evidences of the non-algebraic behavior of $P_N$ are provided. Considering the model of a polymer loop stretched ("inflated") by external force, we show that non-algebraic behavior of $P_N$ affects the adsorption of a polymer at the boundary of a sticky disc in 2D, manifesting in a first order localization transition.

cond-mat.stat-mech↗

arXiv2024

KPZ-like scaling on a high-dimensional hypersphere

We consider the orientational diffusion controlled by the hyperspherical Laplacian, $\nabla^2_D$, on the surface of the $D$--dimensional hypersphere in the limit $D \to \infty$. We find that for stretched paths with lengths relatively short compared to the hypersphere's radius, the finite-size corrections in orientational correlations are controlled by the Kardar-Parisi-Zhang (KPZ) scaling exponent, $γ= 1/3$. In addition, we speculate about the topology of the orientational target space representing the surface of the hypersphere.

cond-mat.stat-mech↗

arXiv2024

Math behind everyday life: "black days", their manifestation as traffic jams, and beyond

In our daily lives, we encounter numerous independent events, each occurring with varying probabilities over time. This letter delves into the scientific background behind the inhomogeneous distribution of these events over time, often resulting in what we refer to as ``black days'', where multiple events seem to converge at once. In the first part of the work we performed an analysis involving $D$ independent periodic and random sequences of events. Using the Uniform Manifold Approximation and Projection (UMAP) technique, we observed a clustering of event sequences on a two-dimensional manifold ${\cal M}$ at a certain large $D$. We interpret this clustering as a signature of ``black days'', which bears a clear resemblance to traffic jams in vehicle flow. In the second part of the work we examined in detail clustering patterns of independently distributed $N$ points within the corners of a $D$-dimensional cube when $1\ll N<D$. Our findings revealed that a transition to a single-component cluster occurs at a critical dimensionality, $D_{cr}$, via a nearly third-order phase transition. Analyzing the spectral density, $ρ(λ)$, of the corresponding adjacency graph in the vicinity of the clustering transition we recover the singular ''Lifshitz tail'' behavior at the spectral boundary of $ρ(λ)$.

physics.soc-ph↗

arXiv2022

Approximate Quantum Algorithms as a Multiphoton Raman Excitation of a Quasicontinuum Edge

Many quantum algorithms can be seen as a transition from a well-defined initial quantum state of a complex quantum system, to an unknown target quantum state, corresponding to a certain eigenvalue either of the Hamiltonian or of a transition operator. Often such a target state corresponds to the minimum energy of a band of states. In this context, approximate quantum calculations imply transition not to the single, minimum energy, state but to a group of states close to the minimum. We consider dynamics and the result of two possible realization of such a process -- transition of population from a single initially populated isolated level to the quantum states at the edge of a band of levels. The first case deals with the time-independent Hamiltonian, while the other with a moving isolated level. We demonstrate that the energy width of the population energy distribution over the band is mainly dictated by the time-energy uncertainty principle, although the specific shape of the distribution depends on the particular setting. We consider the role of the statistics of the coupling matrix elements between the isolated level and the band levels. We have chosen the multiphoton Raman absorption by an ensemble of Rydberg atoms as the model for our analysis, although the results obtained can equally be applied to other quantum computing platforms.

quant-ph↗