We study a polymer model, which distribution depends on (quenched) random charges. Our main results are as follows. When the temperature is above a critical threshold, the distribution of the polymer converges (as the length of the polymer goes to infinity) to that of the random walk. Below the critical temperature, the maximum local time is of order the length of the chain. This transition is first order. In the low temperature regime, a large majority of the monomer lie on only four points, while the expectation of the end to end distance is bounded, uniformly in the length of the polymer.