A new proof of the sharpness of the phase transition for Bernoulli percolation on $\mathbb Z^d$

We provide a new proof of the sharpness of the phase transition for nearest-neighbour Bernoulli percolation. More precisely, we show that

for $p < p_{c}$ , the probability that the origin is connected by an open path to distance $n$ decays exponentially fast in $n$ .
for $p > p_{c}$ , the probability that the origin belongs to an infinite cluster satisfies the mean-field lower bound $θ (p) \geq \frac{p - p_{c}}{p (1 - p_{c})}$ . This note presents the argument of \cite{DumTas15}, which is valid for long-range Bernoulli percolation (and for the Ising model) on arbitrary transitive graphs in the simpler framework of nearest-neighbour Bernoulli percolation on $Z^{d}$ .

We provide a new proof of the sharpness of the phase transition for nearest-neighbour Bernoulli percolation. More precisely, we show that

for $p < p_{c}$ , the probability that the origin is connected by an open path to distance $n$ decays exponentially fast in $n$ .
for $p > p_{c}$ , the probability that the origin belongs to an infinite cluster satisfies the mean-field lower bound $θ (p) \geq \frac{p - p_{c}}{p (1 - p_{c})}$ . This note presents the argument of \cite{DumTas15}, which is valid for long-range Bernoulli percolation (and for the Ising model) on arbitrary transitive graphs in the simpler framework of nearest-neighbour Bernoulli percolation on $Z^{d}$ .

A new proof of the sharpness of the phase transition for Bernoulli percolation on $Z^{d}$