arXiv subjects
Publications and source records attributed to Ritvik Ramanan Radhakrishnan.
We prove that for supercritical percolation on every infinite transitive graph, the probability that the origin belongs to a finite cluster of size at least $n$ decays exponentially in $Φ(n)$, where $Φ$ is the isoperimetric function of the graph.
We discuss a general method to prove quantitative improvements on correlation inequalities and apply it to arm estimates for Bernoulli bond percolation on the square lattice. Our first result is that the two-arm exponent is strictly larger than twice the one-arm exponent and can be seen as a quantitative improvement on the Harris-FKG inequality. This answers a question of Garban and Steif, which was motivated by the study of exceptional times in dynamical percolation. Our second result is that the monochromatic arm exponents are strictly larger than their polychromatic versions, and can be seen as a quantitative improvement on Reimer's main lemma. This second result is not new and was already proved by Beffara and Nolin using a different argument.
We study Voronoi percolation on a large class of $d$-dimensional Riemannian manifolds, which includes the hyperbolic spaces $\mathbb{H}^d$, $d\geq 2$. We prove that as the intensity $λ$ of the underlying Poisson point process tends to infinity, both critical parameters $p_c(M,λ)$ and $p_u(M,λ)$ converge to the Euclidean critical parameter $p_c(\mathbb{R}^d)$. This extends a recent result of Hansen & Müller in the special case $M=\mathbb{H}^2$ to a general class of manifolds of arbitrary dimension. A crucial step in our proof, which may be of independent interest, is to show that if $M$ is simply connected and one-ended, then embedded graphs induced by a general class of tessellations on $M$ have connected minimal cutsets. In particular, this result applies to $\varepsilon$-nets, allowing us to implement a "fine-graining" argument.