arXiv subjects

Georgii Zakharov

Publications and source records attributed to Georgii Zakharov.

5 recordsLinked to original sources

arXiv2026

Sharp threshold for reconstructing points on the line

For a set of $n$ points $V \subseteq \mathbb{R}$ let $G(V, p)$ be the random graph on $V$ where each possible edge is present independently with probability $p$. We call a subset $U \subseteq V$ {\emph {reconstructible}} if every injection $φ:V\to \mathbb{R}$ that preserves the distances along the edges of $G(V, p)$ also preserves all pairwise distances in $U$. How large is the size $\mathsf{R}$ of a largest reconstructible subset? Girão, Illingworth, Michel, Powierski and Scott conjectured that the answer is linear whp when $p = (1+\varepsilon)/n$ for every $\varepsilon > 0$. In this paper, we show that for every $\varepsilon>0$ whp there exists a reconstructible subset $U$ of the largest component $\mathcal{C}$ of the 2-core satisfying $|U| = |V(\mathcal{C})|(1-o(1))$, proving a stronger form of the conjecture. The bound is asymptotically best possible, since for $V \subseteq \mathbb{R}$ linearly independent over $\mathbb{Q}$ it is straightforward to verify that $\mathsf{R} \leq \max(2, |V(\mathcal{C})|)$. Furthermore, we extend these results to every $\varepsilon:= \varepsilon(n)$ satisfying $\varepsilon = ω(1/\ln n)$.

math.CO↗

arXiv2025

On the threshold for triangulations inside convex polygons

Start with a large convex polygon and add all other edges inside independently with probability $p$. At what critical threshold $p_c$ do triangulations of the polygon begin to appear? The first author and Gravner asked this question, and observed that $p_c=Θ(1)$, using the relationship with the Catalan numbers and a coupling with oriented site percolation on ${\mathbb Z}^2$. More recently, Archer, Hartarsky, the first author, Olesker-Taylor, Schapira and Valesin proved that $1/4<p_c<p_c^o$, where $1/4$ is the Catalan exponential growth rate and $p_c^o$ is the critical threshold for oriented percolation. The upper bound is strict, but non-quantitative, and follows by a renormalization argument. We show that $p_c<1/2$ using a simple ear clipping algorithm, which can be analyzed using the gambler's ruin problem. This bound is closer to the truth (perhaps near $0.4$) and shows that most configurations of edges inside large convex polygons contain triangulations.

math.PR↗

arXiv2025

Sums along the edges of bounded degree graphs

Let $G$ be a graph on $n$ vertices and $(H,+)$ be an abelian group. What is the minimum size ${\sf S}_H(G)$ of the set of all sums $A(u)+A(v)$ over all injections $A:V(G)\to H$? In 2012, the first author, Angel, the second author, and Lubetzky proved that, for expander graphs and $H=\mathbb{Z}$, this minimum is at least $Ω(\log n)$, and this bound is tight -- there exists a regular expander $G$ with ${\sf S}_{\mathbb{Z}}(G)=O(\log n)$. We prove that, for every constant $d\geq 3$, the random $d$-regular graph $\mathcal{G}_{n,d}$ has significantly larger sum-sets: with high probability, for every abelian group $H$, ${\sf S}_H(\mathcal{G}_{n,d})=Ω(n^{1-2/d})$. In particular, this proves that, for every $\varepsilon>0$, there exists a regular graph with $O(n)$ edges and with sum-sets of size at least $n^{1-\varepsilon}$, for all abelian groups. The bound ${\sf S}_H(\mathcal{G}_{n,d})=Ω(n^{1-2/d})$ is tight up to a polylogarithmic factor: We show that, for every $3\leq d\leq \ln n/ \ln \ln n$, there exists an abelian group $H$ such that, for every graph $G$ on $n$ vertices with maximum degree at most $d$, ${\sf S}_H(G) \leq n^{1-2/d}(\log n)^{O(1)}$. We also prove that, for $d\gg\ln^2 n$, with high probability, for every abelian group $H$, ${\sf S}_H(\mathcal{G}_{n,d})=n(1-o(1))$ and determine the second-order term, up to a polylogarithmic factor.

math.CO↗

arXiv2025

Majority dynamics on finite trees

For an arbitrary finite tree $T$, we find the exact value of the wort-case stabilisation time of majority dynamics on $T$. We also prove that for a perfect rooted cubic tree $T$ with diameter $D$ and uniformly random initial opinions, the dynamics stabilises in time $τ\in(D/4,D/3)$ with high probability.

math.CO↗

arXiv2023

Enumerating Complexity Revisited

Consider a subset of positive integers $S$. In this paper, we reduce the upper bound on the length of a minimum program that enumerates $S$ in terms of the probability of $S$ being enumerated by a random program. So far, the best-known upper bound was given by Solovay. Solovay proved that the minimum length of a program enumerating $S$ is bounded by $3$ times minus binary logarithm of the probability that a random program enumerates $S$. Later, Vereshchagin showed that the constant can be improved from $3$ to $2$ for finite sets. By improving the method proposed by Solovay, we demonstrate that any bound for finite sets implies the same bound for infinite sets, modulo logarithmic factors. Thus, the constant can be replaced by $2$ for every set $S$ due to the result of Vereshchagin.

cs.CC↗