## Tuesday, March 20, 2007

### Elementary Maurey's theorem + ode to interaction

Maurey's theorem (rather, one of them) states that if P is the Haar (i.e., normalized counting) measure on S_n (the group of all permutations on n elements) and d is the normalized Hamming metric on this group, then
P(f-Ef > t) <= exp(-nt^2 / 8),
where f:S_n -> R is 1-Lipschitz w.r.t. d. See Schechtman's paper for a proof; it is done via the notion of metric space length.

In our conversation at Berkeley, Jim Pitman suggested a simple way to parametrize uniformly random permutations as an independent process. At each step i, 1<=i<=n, you uniformly pick an integer between 1 and i as the location in which to insert the next element. Thus we can recover Maurey's theorem in a simpler way, and with better constants, via McDiarmid's inequality:
P(f-Ef > t) <= exp(-2nt^2),
where P and f are as above. At first I couldn't believe it would be this simple, but Gideon tells me that I'm right.

The greater moral of this story is that I should talk to people more. I can't overstate the value of this blog as a research tool -- look no further than the high quality discussions here, here, here, and here (and this isn't counting the email exchanges that some posts generate). But you can't count on the experts to always stumble on your blog post -- nothing beats talking to one in person!