Percolation of worms

We introduce a new correlated percolation model on the d-dimensional lattice Zd called the random length worms model. Assume given a probability distribution on the set of positive integers (the length distribution) and v∈(0,∞) (the intensity parameter). From each site of Zd we start POI(v) independent simple random walks with this length distribution. We investigate the connectivity properties of the set Sv of sites visited by this cloud of random walks. It is easy to show that if the second moment of the length distribution is finite then Sv undergoes a percolation phase transition as v varies. Our main contribution is a sufficient condition on the length distribution which guarantees that Sv percolates for all v>0 if d≥5. E.g., if the probability mass function of the length distribution is m(ℓ)=c⋅ln(ln(ℓ))ɛ/(ℓ3ln(ℓ))1[ℓ≥ℓ0]for some ℓ0>ee and ɛ>0 then Sv percolates for all v>0. Note that the second moment of this length distribution is only “barely” infinite. In order to put our result in the context of earlier results about similar models (e.g., finitary random interlacements, loop percolation, Bernoulli hyper-edge percolation, Poisson Boolean model, ellipses percolation, etc.), we define a natural family of percolation models called the Poisson zoo and argue that the percolative behaviour of the random length worms model is quite close to being “extremal” in this family of models.