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Markovian renormalisation for percolation in high-dimension: Semi-decidability of mean field behavior
Abstract
We develop a new approach to study Bernoulli percolation in dimensions d>6. The key idea is to approximate open paths at probability p'>p by a Markov chain of pointed p-open clusters. This allows us to transfer sharp information on the two point function from p to p'. This inductively gives good asymptotic estimates on the two point function as p↑ pc. As a main application, we show that having critical meanfield behavior is a semi-decidable problem. Along the way, we also prove sharp asymptotics for the susceptibility and the average radius of gyration, and prove a local central limit theorem for the slightly subcritical two-point function.
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BibTeX
@misc{blancrenaudie2026markovian,
title = {{Markovian renormalisation for percolation in high-dimension: Semi-decidability of mean field behavior}},
author = {Arthur Blanc-Renaudie},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20764},
eprint = {2609.20764},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05773590v1},
}
RIS
TY - UNPB
TI - Markovian renormalisation for percolation in high-dimension: Semi-decidability of mean field behavior
AU - Arthur Blanc-Renaudie
PY - 2026
DA - 2026/09/17
PB - HAL
DO - 10.48550/arxiv.2609.20764
UR - https://hal.science/hal-05773590v1
AB - We develop a new approach to study Bernoulli percolation in dimensions d>6. The key idea is to approximate open paths at probability p'>p by a Markov chain of pointed p-open clusters. This allows us to transfer sharp information on the two point function from p to p'. This inductively gives good asymptotic estimates on the two point function as p↑ p_c. As a main application, we show that having critical meanfield behavior is a semi-decidable problem. Along the way, we also prove sharp asymptotics for the susceptibility and the average radius of gyration, and prove a local central limit theorem for the slightly subcritical two-point function.
ER -