We start by surveying known properties of first passage percolation (FPP) geodesics on a Gromov-hyperbolic group $G$. Due to a coalescence phenomenon established in earlier work of the authors, a random tree $T(\xi,\omega)$ consisting of infinite random geodesics to a point $\xi$ on the Gromov boundary $\partial G$ emerges naturally. These trees have a rich random geometry that we explore further in this paper.
We show that, for Weil--Petersson random closed hyperbolic surfaces of large genus, the normalized weighted count of prime geodesics with norm in the interval $(X,X+H]$ is asymptotically Gaussian, provided $H \leq X$ and $H/\log X\to\infty$ as $X\to\infty$. In particular, it applies to intervals which are not necessari...
We prove locality of the critical probability for Bernoulli site percolation on infinite, connected, locally finite, vertex-transitive graphs, under the usual assumption that the critical probabilities stay uniformly below one. The proof develops site-percolation forms of the two-ghost inequality, sharp-threshold and s...
For Bernoulli bond percolation on every regular tree of degree at least three, we prove that infinite clusters appear strictly before absolutely continuous spectrum. Throughout an explicit supercritical interval, the adjacency operator of every open cluster almost surely has no absolutely continuous component. Together...
In this article, we prove the Defect Conjecture for Cayley graphs of abelian groups, dihedral groups, and groups of the form $\mathbb{Z}/p\mathbb{Z} \rtimes \mathbb{Z}/(p-1)\mathbb{Z}$ where $p\geq 3$ is a prime. We also compute the Iwasawa invariants of the Bowen--Franks groups associated with these graphs in several...
Debanjana Kundu, Katharina Müller, A. Newton et al.· 0 citations
We prove that every compact connected Riemannian isometric filling $M$ of a circle of length $2\pi$ satisfies $\operatorname{Area}(M) \geq \frac{14\zeta(3)}{\pi} \approx 5.35677$, regardless of orientability or topological types. Our new approach uses the odd Fourier coefficients of the distance functions from boundary...
We show that the percolation components on a square grid can be $5$-colored as a factor of iid so that neighboring components get different colors. Above the critical probability, we observe that these regions can be 4-colored. Along the way, we extend the classical correspondence between fiid process and measurable la...
Justin Hsu, Daniel Sium, R. Thornton· 0 citations
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