Consider for example the Ising model on a square lattice. Fix zero magnetic field and plus boundary conditions.

Low temperature, one minus spin. With a Peierls argument one can prove that, given a vertex $v$, the probability of a minus spin on $v$, is bounded by $C\, e^{-c\beta}$ where $\beta$ is the inverse temperature and $C,c$ are suitable constants.

Low temperature, long minus spins path. Now, given a simple path $P=(v_0,v_1,\dots,v_l)$ with $l$ large, is it true that the probability of all minus spins on $P$ is bounded by something like $C^l\, e^{-c\beta\,l}$ (where the constants can be different from above)? The problem is that the spins are not independent, on the contrary the presence of a minus spin favours the presence of other neighbouring minus spins..

Consider for example the Ising model on a square lattice. Fix zero magnetic field and plus boundary conditions.

Low temperature, one minus spin. With a Peierls argument one can prove that, given a vertex $v$, the probability of having a minus spin on $v$, is bounded by $C\, e^{-c\beta}$ where $\beta$ is the inverse temperature and $C,c$ are suitable constants.

Low temperature, long path of minus spins. Now, given a simple path $P=(v_0,v_1,\dots,v_l)$ with $l$ large, is it true that the probability of having all minus spins on $P$ is bounded by something like $C^l\, e^{-c\beta\,l}$ (where the constants can be different from above)? The problem is that the spins are not independent, on the contrary the presence of a minus spin favours the presence of other neighbouring minus spins..