The effect of small quenched noise on connectivity properties of random interlacements.

*(English)*Zbl 1347.60132The authors study the effect of a small quenched noise on the connectivity properties of random interlacements (at level \(u\)) and vacant sets. In particular, given \(\varepsilon > 0\), each vertex of the random interlacement, considered as occupied, is allowed to become vacant with probability \(1- \varepsilon\), as well as each vacant vertex is allowed to become occupied with probability \(\varepsilon\), independently of the randomness of the interlacement and independently for different vertices. They prove that for any \(d \geq 3\) and \(u > 0\), the perturbed random interlacement percolates for sufficiently small values of the noise parameter \(\varepsilon\) almost surely, by proving the stronger statement that the Bernoulli percolation on the random interlacement graph has a non-trivial phase transition in wide enough slabs. Moreover, they show that any electric network with i.i.d. positive resistances on the interlacement graph is transient, strengthening their previous result on the transience of random interlacements. Furthermore, they show that, when the noise parameter is small enough, the vacant set at level \(u\) undergoes a non-trivial phase transition in \(u\) as in the non-random case (\(\varepsilon = 0\)), giving explicit upper and lower bounds for the critical threshold when \(\varepsilon\) tends to zero.

Reviewer: Michele Gianfelice (Rende)

##### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82B43 | Percolation |