arXiv subjects

Georgii Zakharov

Publications and source records attributed to Georgii Zakharov.

6 recordsLinked to original sources

arXiv2026

Matching Random Point Sets with Minimum Distortion

How well can two independent random samples from a metric measure space be matched while preserving pairwise distances? Concretely, given two independent copies, each of $n$ points sampled from the space, what is their typical metric distortion as finite $n$-point metric spaces? We prove that the space $[0,1]$ equipped with the uniform measure already exhibits a rich behavior. We show that with high probability the distortion of the two independent samples is $n^{2/3}$ up to polylogarithmic factors. The lower bound reflects two competing obstructions. A bijection that maps points close to their original positions or their reflections in $[0,1]$ must expand the smallest gaps, in both directions. On the other hand, we show that large displacement from both the identity and reflection itself forces large distortion. Balancing these obstructions yields the lower-bound exponent $2/3$. For the upper bound, we construct a partial matching of the points that maps each gap between consecutive points of length $d\lesssim n^{-4/3}$ to a pair at distance $\widetilde{O}(dn^{1/3})$, with analogous control in the reverse direction. We then use a delicate multiscale oscillation construction to extend this partial matching to a low-distortion bijection.

math.PR↗

arXiv2026

Enumerating Complexity Revisited

We study the relation between deterministic and randomized $\textit{enumerating}$ Kolmogorov complexity of sets $S\subseteq \mathbb{N}$. Let $D(S)$ be the length of a shortest input on which a fixed optimal deterministic machine enumerates $S$, and let $ R(S) $ be the probability that a fixed optimal randomized machine enumerates $S$. It is easy to see that $-\log R(S) \le D(S)+O(\log D(S))$ for every $S$; the converse direction is much harder. Solovay proved $D(S)\le -3\log R(S) +O(\log\log\frac{1}{ R(S) })$ for all $S$, while for finite sets the sharper bound with the constant $2$ is known. We show that the case of infinite sets carries no extra cost: $\textit{any}$ inequality of the form $D(F)\le -α\log R(F)+O(\log\log\frac{1}{R(F)})$ with a computable $α$, valid for all finite $F$, automatically extends to all sets. More generally, we prove a single dichotomy from which such transfer results follow for an arbitrary computable non-decreasing bound, and which also shows that every c.e. set $S$ has a finite subset $F$ such that $F$ has at least as large deterministic complexity as $S$ but randomized one approximately the same or lower. Our result makes qualitative progress on a problem studied by Solovay in 1977 by showing that the search for the optimal constant can be reduced to a purely combinatorial question about a game played with finite sets. In particular, our result yields $D(S)\le-2\log R(S) +O(\log\log\frac{1}{ R(S) })$ for every $S$, improving Solovay's constant $3$ to $2$.

cs.CC↗

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↗