In the non-amenable case the cheeger constant of $(\Gamma, S_k)$ goes to infinity with $k$ (with or without multiplicities) and this implies $p_c$ tends to $0$ by the standard exploration argument, see Theorem 2 of http://www.springerlink.com/content/t5l8401n32262112/fulltext.pdf

In the non-amenable case the cheeger constant of $(\Gamma, S_k)$ goes to infinity with $k$ (with or without multiplicities) and this implies $p_c$ tends to $0$ by the standard exploration argument, see Theorem 2 of Link