Percolation on graphs of polynomial growth is local: analyticity, supercritical sharpness, isoperimetry
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911475140919296 |
|---|---|
| author | Martineau, Sébastien Panagiotis, Christoforos |
| author_facet | Martineau, Sébastien Panagiotis, Christoforos |
| contents | We investigate locality of the supercritical regime for Bernoulli percolation on transitive graphs with polynomial growth, by which we mean the following. Take a transitive graph of polynomial growth $\mathscr{G}$ satisfying $p_c(\mathscr{G})<1$ and take $p>p_c(\mathscr{G})$. Let $\mathscr{H}$ be another such graph and assume that $\mathscr{G}$ and $\mathscr{H}$ have the same ball of radius $r$ for $r$ large. We prove that various quantities regarding percolation of parameter close to $p$ on $\mathscr{H}$ can be well understood from $(\mathscr{G},p)$ alone.
This includes uniform versions of supercritical sharpness as well as the Kesten-Zhang bound on the probability of observing a large finite cluster: the constants involved can be chosen to depend only on $(\mathscr{G},p)$. We also prove that $θ_\mathscr{H}$ is an analytic function of $p$ in the whole supercritical regime and that, for a suitable $\varepsilon=\varepsilon(\mathscr{G},p)>0$, the analytic extension of $θ_\mathscr{H}$ to the $\varepsilon$-neighbourhood of $p$ in $\mathbb C$ is, uniformly, well approximated by the analytic extension of $θ_\mathscr{G}$. The proof relies on new results on the connectivity of minimal cutsets; in particular, we answer a question asked by Babson and Benjamini in 1999. We further discuss connections with the conjecture of non-percolation at criticality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01851 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Percolation on graphs of polynomial growth is local: analyticity, supercritical sharpness, isoperimetry Martineau, Sébastien Panagiotis, Christoforos Probability Combinatorics Group Theory We investigate locality of the supercritical regime for Bernoulli percolation on transitive graphs with polynomial growth, by which we mean the following. Take a transitive graph of polynomial growth $\mathscr{G}$ satisfying $p_c(\mathscr{G})<1$ and take $p>p_c(\mathscr{G})$. Let $\mathscr{H}$ be another such graph and assume that $\mathscr{G}$ and $\mathscr{H}$ have the same ball of radius $r$ for $r$ large. We prove that various quantities regarding percolation of parameter close to $p$ on $\mathscr{H}$ can be well understood from $(\mathscr{G},p)$ alone. This includes uniform versions of supercritical sharpness as well as the Kesten-Zhang bound on the probability of observing a large finite cluster: the constants involved can be chosen to depend only on $(\mathscr{G},p)$. We also prove that $θ_\mathscr{H}$ is an analytic function of $p$ in the whole supercritical regime and that, for a suitable $\varepsilon=\varepsilon(\mathscr{G},p)>0$, the analytic extension of $θ_\mathscr{H}$ to the $\varepsilon$-neighbourhood of $p$ in $\mathbb C$ is, uniformly, well approximated by the analytic extension of $θ_\mathscr{G}$. The proof relies on new results on the connectivity of minimal cutsets; in particular, we answer a question asked by Babson and Benjamini in 1999. We further discuss connections with the conjecture of non-percolation at criticality. |
| title | Percolation on graphs of polynomial growth is local: analyticity, supercritical sharpness, isoperimetry |
| topic | Probability Combinatorics Group Theory |
| url | https://arxiv.org/abs/2511.01851 |