Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0310232

Monotone properties of random geometric graphs have sharp thresholds

Ashish Goel·Sanatan Rai·Bhaskar Krishnamachari
View original source ↗Read source document ↗Export metadata ↓

Abstract

Random geometric graphs result from taking $n$ uniformly distributed points in the unit cube, $[0,1]^d$, and connecting two points if their Euclidean distance is at most $r$, for some prescribed $r$. We show that monotone properties for this class of graphs have sharp thresholds by reducing the problem to bounding the bottleneck matching on two sets of $n$ points distributed uniformly in $[0,1]^d$. We present upper bounds on the threshold width, and show that our bound is sharp for $d=1$ and at most a sublogarithmic factor away for $d\ge2$. Interestingly, the threshold width is much sharper for random geometric graphs than for Bernoulli random graphs. Further, a random geometric graph is shown to be a subgraph, with high probability, of another independently drawn random geometric graph with a slightly larger radius; this property is shown to have no analogue for Bernoulli random graphs.

Explore related subjects

math.PRcond-mat.stat-mechcs.DMmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ashish Goel, Sanatan Rai, Bhaskar Krishnamachari. 2006-02-24. Monotone properties of random geometric graphs have sharp thresholds. https://doi.org/10.1214/105051605000000575

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Line crossing problem for biased monotonic random walks in the plane

A $β$-biased monotonic random walk in the plane moves from $(x,y)$ to $(x+1,y)$ or $(x,y+1)$ with probabilities $1/(β+1)$ and $β/(β+1)$. We study $Φ_β(α,d)$, the probability that the walk started at the origin ever visits a lattice point strictly above the line $y=αx+d$, and $Ψ_β(α,d)$, the probability that it visits a lattice point \emph{on} the line after leaving the origin. For integer slopes $α=p$ we show $Φ_β(p,d)=ρ^{\lfloor d\rfloor+1}$, where $ρ$ is the smallest positive root of $y^{p+1}-(β+1)y+β=0$, give the Fuss--Catalan series for $ρ$, identify $ρ$ with the extinction probability of a Galton--Watson process, and compute the exact law and mean of the first crossing point. We prove that the return probability is $Ψ_β(p,0)=(p+1)(1-β/((β+1)ρ))$, which equals $(p+1)/(β+1)$ for $β\ge p$. For rational slopes $α=r/s$ we obtain $Φ_β$ in closed form from the $s$ large roots of the kernel $\tfracβ{β+1}z^{r+s}-z^r+\tfrac1{β+1}$, and $Ψ_β$ as a ratio of Green's functions. For arbitrary real $α$ we prove that the walk crosses every line $y=αx+d$ almost surely if and only if $β\geα$, and we establish the two-sided bound $e^{-θ(d+1)}\leΦ_β(α,d) β$.

math.PR↗
arXiv2026

Commutation relations for two-sided radial SLE

We study the commutation relation for 2-radial SLE in the unit disc starting from two distinct boundary points. We follow the framework introduced by Dubédat~[12]. Under an additional requirement of the interchangeability of the two curves, we classify all locally commuting 2-radial SLE$_κ$ for $κ\in (0,8)$: it is either two-sided radial SLE$_κ$ with spiral of constant spiraling rate or chordal SLE$_κ$ weighted by a power of the conformal radius of its complement. Namely, for fixed $κ$ and starting points, we have exactly two one-parameter families of locally commuting 2-radial SLE curves. We also discuss the semiclassical limit of the commutation relation as $κ\to 0$. In particular, we show that the limit for the second family with an appropriately chosen power of the conformal radius is a chord that minimizes a modified chordal Loewner energy, which is unique only when the endpoints are not antipodal.

math.PR↗
arXiv2026

Stochastic optimal transport for the Langevin dynamics and its zero-mass limit

We introduce a stochastic optimal transport for the Langevin dynamics with positive mass and study its zero--mass limit. The new aspect of this paper is that we only fix the initial and terminal probability distributions of the positions of particles under consideration, but not those of their velocities, with Heisenberg's uncertainty principle in mind. In the zero--mass limit, we show that the minimizers of our stochastic optimal transport is tight if and only if the initial momentum (=(mass)$\times$(velocity)) of a particle converges to zero. We also prove that the limit of minimizers of our stochastic optimal transport is a minimizer of a standard stochastic optimal transport for continuous semimartingales.

math.PR↗