Sharpness of the phase transition
The Duminil-Copin–Tassion proof of exponential decay below p_c: the quantity phi_p(S), Russo's formula and pivotal edges, exploration and last-exit decompositions, the differential inequality, the mean-field bound and subcritical decay.
- The Duminil-Copin–Tassion quantity φ_p(S) and the threshold p̃_c
- Bernoulli percolation in the inverse-temperature parametrization
- Russo's formula for percolation
- Exponential decay below the critical point
- Sharpness of the phase transition, assuming its two analytic inputs
- Mean-field lower bound and critical points for Bernoulli percolation
- Duminil-Copin–Tassion differential inequality
- Sharpness of the phase transition for Bernoulli bond percolation on ℤ^d
- Aizenman–Barsky inequality
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