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We address one open problem in a recent work due to Ding and Wirth, the firstversion of which was available in $2019$, relating to level-set percolation onmetric-graphs for the Gaussian free field in three dimensions, in which it wasshown that a percolation estimate that the authors employ for studyingconnectivity properties of different heights of the metric graph Gaussian freefield is bounded above poly-logarithmically. In three dimensions, in order toconstruct Kesten's incipient infinite cluster which was first seminallyintroduced for Bernoulli percolation in two dimensions, in $1986$, through theequality of two probabilistic quantities, we make use of a streamlined versionof the $1986$ argument due to Basu and Sapozhnikov, which was first madeavailable in $2016$, that introduces properties of crossing probabilities fordemonstrating that the IIC exists for Bernoulli percolation on an infiniteconnected, bounded degree graph. To make use of such arguments fordemonstrating the existence of the metric-graph GFF IIC in three dimensions, wealso address another open problem raised in a recent work, from October $2022$,due to Dubedat and Falconet, which expresses an open problem pertaining to theconstruction of an IIC-type limit for the Villain model.

Kesten's incipient infinite cluster for the three-dimensional,  metric-graph Gaussian free field, from critical level-set percolation, and  for the Villain model, from random cluster geometries and a Swendsen-Wang  type algorithm