Author

I. Benjamini

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Preprint Sep 2026

Very sharp distance and range transitions for random walk bridges on Ramanujan graphs

For vertex-transitive Ramanujan graphs with logarithmic girth, a simple random walk bridge of length of order $\log N$, where $N$ is the size of the graph, has a maximum distance that changes from order $\sqrt{\log N}$ to order $\log N$ in a bounded critical window. We prove this by separating bridges whose lifts to th...

I. Benjamini · 0 citations

Open access Mar 2023

The Number of Tiles of $\mathbb{Z}^d$

For fixed $d\ge 1$, let $t_{n,d}$ be the number of subsets of$[n]^d$ that tile $\mathbb{Z}^d$ by translations. We prove that\[t_{n,d}=(3^{1/3})^{n^d\pm o(n^d)}.\]

I. Benjamini, G. Kozma, Elad Tzalik · Electronic Journal of Combin... · 0 citations

Preprint Aug 2026

The Uniform Gromov Hausdorff Gap Problem for Approximating Spheres by Finite Homogeneous Spaces

Let $S^n$ be the unit round sphere with its intrinsic angular metric, normalized so that $\operatorname{diam}S^n=\pi$. For finite homogeneous metric spaces $X$, put \[ \delta_n=\inf_X d_{GH}(X,S^n). \] The main open problem is whether $\inf_{n\ge2}\delta_n>0$. Gelander's theorem gives $\delta_n>0$ in each fixed dimensi...

I. Benjamini · 0 citations