pr.probability - Parity, Balls and Boxes

Start with a distribution $mu$ on [n], and drop m balls into these n+1 slots independently and according to the distribution &mu. That is, we have iid random variables x ₁ through x _m distributed according to &mu. Assume that m is close to the same size as n.

I call a collection of tuples (a(i), b (i)), with all a(i), b(i) between 1 and m, a partial matching if no number is repeated, and for all i, x _b(i) = x _a(i) + 1.

My (purposefully vague) question is: Is there, with high probability, a partial matching that includes many of the balls?

Of course, for some distributions this obviously doesn't happen. I would be happy if somebody had a 'standard argument' for this type of question in nice cases, and maybe we could get some understanding of what makes it run (and so in what cases it doesn't).

Some side comments:

We need that m be not much smaller than n, as otherwise most points will be fairly far apart.

For nice distributions, such as the uniform or binomial, one can do calculations much like the birthday paradox, and get some semi-plausible answers. I don't know if these are best possible, and would love to hear it if somebody out there has a nice clearly-tight argument for nice distributions. I was thinking of just asking about the uniform distribution, as I guess somebody out there must have a beautiful argument.