  We consider a class of random loop models (including the random interchange
process) that are parametrised by a time parameter $\beta\geq 0$. Intuitively,
larger $\beta$ means more randomness. In particular, at $\beta=0$ we start with
loops of length 1 and as $\beta$ crosses a critical value $\beta_c$, infinite
loops start to occur almost surely. Our random loop models admit a natural
comparison to bond percolation with $p=1-e^{-\beta}$ on the same graph to
obtain a lower bound on $\beta_c$. For those graphs of diverging vertex degree
where $\beta_c$ and the critical parameter for percolation have been calculated
explicitly, that inequality has been found to be an equality. In contrast, we
show in this paper that for graphs of bounded degree the inequality is strict,
i.e. we show existence of an interval of values of $\beta$ where there are no
infinite loops, but infinite percolation clusters almost surely.
