Infinitely many collisions between a recurrent simple random walk and arbitrary many transient random walks in a subballistic random environment

We consider d random walks (Sn(j))n∈N, 1≤j≤d, in the same random environment ω in Z, and a recurrent simple random walk (Zn)n∈N on Z. We assume that, conditionally on the environment ω, all the random walks are independent and start from even initial locations. Our assumption on the law of the environment is such that a single random walk in the environment ω is transient to the right but subballistic, with parameter 0<κ<1∕2. We show that – for every value of d – there are almost surely infinitely many times for which all these random walks, (Zn)n∈N and (Sn(j))n∈N, 1≤j≤d, are simultaneously at the same location, even though one of them is recurrent and the d others ones are transient.