R. Cerf

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math.PR2026

$θ (p_{c}, Z^{d}) = 0 ?$

Raphaël Cerf

We discuss the famous conjecture { $θ (p_{c}, Z^{d}) = 0$ } for the Bernoulli site percolation model on $Z^{d}$ with $d \geq 3$ . We prove that, for $d \geq 3$ , in the Bernoulli percol…

math.PR2026

The case of equality in BK

RaphaÃ«l Cerf, Pierre Tesio

We characterize the pairs of increasing events $A, B$ for which there is equality in the BK inequality. Namely, we show that $P (A \circ B) = P (A) P (B)$ if and only if all the configurat…

math.PR2024

The pivotal set of a Boolean function

RaphaÃ«l Cerf

We define the pivotal set of a Boolean function and we prove a fundamental inequality on its expected size, when the inputs are independent random coins of parameter~ $p$ . We give t…

θ(pc​,Zd)=0?

The case of equality in BK

The pivotal set of a Boolean function

$θ (p_{c}, Z^{d}) = 0 ?$