They are not entirely independent. They are simply correlated to different links in the causality chain. The Muslims are there because it's a shitty area, not the other way around. If it were a good area, the houses would be too expensive for the latest wave of immigrants to move into.
It was the same way with the era of immigration in America. The first generation off the boat lived in the worst neighborhoods, and then the next generation rose to power and lorded it over the boats that came in on their watch. I imagine (though I don't know) that Hindus and Sikhs are more accepted as British than Muslims are, just because they have been around longer, and that they have long since moved into the better neighborhoods.
It was the same way with the era of immigration in America. The first generation off the boat lived in the worst neighborhoods, and then the next generation rose to power and lorded it over the boats that came in on their watch. I imagine (though I don't know) that Hindus and Sikhs are more accepted as British than Muslims are, just because they have been around longer, and that they have long since moved into the better neighborhoods.