Abstract
We give a mathematical introduction to percolation, and state the main questions. We introduce the various critical percolation thresholds, and argue that the most interesting behavior occurs close to it. We introduce critical exponents that describe the system at or close to the phase transition, and the role of the upper critical dimension. Finally, we explain basic techniques (in particular, the BK and FKG inequalities and Russo’s formula) and describe our aim of these lecture notes.
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Notes
- 1.
The careful reader may notice that we are anticipating here that \(p_{\mathrm {c}}=p_{\mathrm {T}}\).
- 2.
These events again do not depend on finitely many bonds, but they can be approximated by events depending on finitely many bonds, so that our conclusion remains to hold. See Exerc. 1.3.
- 3.
Grimmett [122] uses \(E \square F\) for the event in (1.3.7).
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Heydenreich, M., van der Hofstad, R. (2017). Introduction and Motivation. In: Progress in High-Dimensional Percolation and Random Graphs. CRM Short Courses. Springer, Cham. https://doi.org/10.1007/978-3-319-62473-0_1
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DOI: https://doi.org/10.1007/978-3-319-62473-0_1
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Online ISBN: 978-3-319-62473-0
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