Resumen de Subcritical percolation with a line of defects

S. Friedli, D. Ioffe, Yvan Velenik

• We consider the Bernoulli bond percolation process Pp,p′ on the nearest-neighbor edges of Zd, which are open independently with probability p < pc, except for those lying on the first coordinate axis, for which this probability is p′. Define ξp,p′:=−limn→∞n−1logPp,p′(0↔ne1) and ξp:=ξp,p. We show that there exists p′c=p′c(p,d) such that ξp,p' = ξp if p′ < p′c and ξp,p′ < ξp if p′ > p′c. Moreover, p′c(p,2)=p′c(p,3)=p, and p′c(p,d)>p for d≥4. We also analyze the behavior of ξp−ξp,p′ as p′↓p′c in dimensions d=2,3. Finally, we prove that when p′>p′c, the following purely exponential asymptotics holds:

  Pp,p′(0↔ne1)=ψde−ξp,p′n(1+o(1)) for some constant ψd=ψd(p,p′), uniformly for large values of n. This work gives the first results on the rigorous analysis of pinning-type problems, that go beyond the effective models and don’t rely on exact computations.