A note on the finite collision property of random walks

We consider a wedge comb of ZZ and two independent random walks on it with continuous time parameter. They meet each other infinitely many times if the height of the comb increases at most of the order x1/5x1/5. Wedge combs with random heights are also discussed, as well as other related models. We prove that the infinite collision property is independent of the time parameter of the simple random walks when the graph is quasi-transitive and of subexponential growth.