Author

Mostafa Mirabi

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Preprint Aug 2026

First-Order Laws for Random Geometric Graphs on the Torus

Let $G_D(n;r)$ be the random geometric graph generated by $n$ independent uniform points on the $D$-dimensional torus, with adjacency defined by torus $L^\infty$-distance at most $r$. We study first-order zero-one and convergence laws at fixed radius and in sparse regimes. At fixed radius, we determine the asymptotics...

Simi Haber, Mostafa Mirabi, S. Shelah · 0 citations

Preprint Jul 2026

Limit laws for component-pruned sparse random graphs and percolated tori

We prove an $\mathrm{MSO}_2$ zero-one law for a very sparse Erd\H{o}s-R\'enyi graph after pruning by component order. Let $p_n=c_n/n$, where $c_n\to0$, and delete every component of order less than $f(n)$, where $f(n)\to\infty$. If \[ f(n)\bigl(\log f(n)+\log(1/c_n)\bigr)=o(\log n), \] then the resulting graph satisfie...

Mostafa Mirabi, S. Shelah · 0 citations