arXiv subjects
Publications and source records attributed to Pranav Chinmay.
Consider critical Bernoulli percolation on $\mathbb{Z}^d$ for $d$ large; let $y_0, \dots, y_{k-1}$ be $k$ distinct points in $\mathbb{R}^d$. We prove that the probability that $\{\lfloor n y_i\rfloor\}_{i=0}^{k-1}$ all lie in the same open cluster, rescaled by an appropriate power of $n$, converges as $n \to \infty$ to an explicit constant. This confirms a conjecture of Aizenman and Newman.
We identify the asymptotic distribution of the chemical distance and other natural metrics (including the resistance) in high-dimensional critical percolation. When rescaled by the square of the Euclidean distance, each of these metrics converges in distribution to a multiple of the hitting time $T$ of a Brownian motion to hit $\mathbf{e}_1$ conditional on $\{T < \infty\}$. We extend these results to the near-critical regime, where the limiting distribution is now the analogue of $T$ for a killed Brownian motion. These results are intended to be the foundation for an understanding of the metric space structure of high-dimensional clusters. They follow from a general theorem we find interesting in its own right, a ``law of large numbers'' for local functions summed along the backbone of a long open connection. A mixing result for open clusters \cite{CCHS} in the form of a robust convergence to the incipient infinite cluster measure plays a key role in the proofs.
We give a new construction of the incipient infinite cluster (IIC) associated with high-dimensional percolation in a broad setting and under minimal assumptions. Our arguments differ substantially from earlier constructions of the IIC; we do not directly use the machinery of the lace expansion or similar diagrammatic expansions. We show that the IIC may be constructed by conditioning on the cluster of a vertex being infinite in the supercritical regime $p > p_c$ and then taking $p \searrow p_c$. Furthermore, at criticality, we show that the IIC may be constructed by conditioning on a connection to an arbitrary distant set $V$, generalizing previous constructions where one conditions on a connection to a single distant vertex or the boundary of a large box. The input to our proof are the asymptotics for the two-point function obtained by Hara, van der Hofstad, and Slade. Our construction thus applies in all dimensions for which those asymptotics are known, rather than an unspecified high dimension considered in previous works. The results in this paper will be instrumental in upcoming work related to structural properties and scaling limits of various objects involving high-dimensional percolation clusters at and near criticality.