arXiv subjects

Vyacheslav Koval

Publications and source records attributed to Vyacheslav Koval.

4 recordsLinked to original sources

arXiv2026

Peripheral Traps and Lower Bounds on Mixing Times for Random Walks on Sparse Heavy-Tailed Random Intersection Graphs

This paper analyzes mixing time lower bounds for random walks on sparse, heavy-tailed Random Intersection Graphs. In sparse feature regimes, heavy-tailed feature distributions lead to the formation of peripheral trap -- chains of overlapping low-weight feature cliques attached to high-weight hub nodes within the graph's giant component. By modeling escape trajectories from these traps as continuous limit hitting times for reflected Brownian motion, the analysis demonstrates that random walks experience logarithmic squared delays. Consequently, the mixing time is bounded below by $Ω(\log^2 n)$, and the local total variation distance exhibits non-concentrated decay, formally preventing a sharp cutoff phenomenon.

math.PR↗

arXiv2026

Meeting and coalescence times for random walks in the largest component of the Erdős-Rényi random graph

We prove that the stationary and worst-case expected meeting times of two independent continuous-time random walks on the largest component of the Erdős-Rényi random graph $G(n,p)$ have order $n$ throughout the strictly supercritical, the slightly supercritical and the critical regimes. Using these bounds along with a fine-tuned combination of comparison inequalities due to Oliveira (2012) and Kanade-Mallmann-Trenn-Sauerwald (KMS, 2023), we deduce that expected coalescence time and full voter-model consensus also have order $n$ throughout these three regimes.

math.PR↗

arXiv2026

Hausdorff dimension of reflected Bedford--McMullen carpets

We study Bedford--McMullen type carpets whose selected grid rectangles may be reflected in one or both coordinates. The organizing principle is that the Hausdorff dimension is controlled by the entropy of the weak-coordinate projection. When this weak projection is separated, we obtain an explicit McMullen-type formula. This yields stability under arbitrary horizontal reflections and row-compatible weak reflections, and it gives several computable mixed-sign classes, including signed row-branch systems, interval-window systems and finite-block separated systems. We also explain why fully arbitrary weak-coordinate reflection patterns lead instead to a projection-entropy problem.

math.DS↗

arXiv2010

Long-range percolation on the hierarchical lattice

We study long-range percolation on the hierarchical lattice of order $N$, where any edge of length $k$ is present with probability $p_k=1-\exp(-β^{-k} α)$, independently of all other edges. For fixed $β$, we show that the critical value $α_c(β)$ is non-trivial if and only if $N < β< N^2$. Furthermore, we show uniqueness of the infinite component and continuity of the percolation probability and of $α_c(β)$ as a function of $β$. This means that the phase diagram of this model is well understood.

math.PR↗