The Parisi theory of the Sherrington-Kirkpatrick model completely describes the geometry of the Gibbs sample in a sense that it predicts the limiting joint distribution of all scalar products, or overlaps, between i.i.d. replicas. One of the main predictions is that asymptotically the Gibbs measure concentrates on an ultrametric subset of all spin configurations. Another part of the theory are the Ghirlanda-Guerra identities which in various formulations have been proved rigorously. It is well known that together these two properties completely determine the joint distribution of the overlaps and for this reason they were always considered complementary. We show that in the case when overlaps take finitely many values the Ghirlanda-Guerra identities actually imply ultrametricity.

In the so called light bulb process of Rao, Rao and Zhang (2007), on days r =
1, . . . , n, out of n light bulbs, all initially off, exactly r bulbs, selected uniformly and
independent of the past, have their status changed from off to on or vice versa. With
X the number of bulbs on at the terminal time n, an even integer and = n/2, σ2 =
varX, we have
sup
∈R 􏰐
􏰐
P ( X −
σ ≤ z ) − P (Z ≤ z )
􏰐􏰐 ≤
n
2σ2 ∆0 + 1.64
n
σ3 +
2
σ
where Z is a
N (0, 1) random variable and
∆0
≤
1
2√n +
1
2n + e−
n/2
, for n
≥ 4,
yielding a bound of order O(n−1/2 ) as n
→ ∞.
The results are shown using a version of Steins method for bounded, monotone
size bias couplings. The argument for even n depends on the construction of a variable
X s on the same space as X which has the X size bias distribution, that is, which
satisfies
E[X g(X )] = E[g(X s )], for all bounded continuous g
and for which there exists a B
≥ 0, in this case, B = 2, such that X ≤ X
s
≤ X + B
almost surely. The argument for odd n is similar to that for n even, but one first
couples X closely to V , a symmetrized version of X, for which a size bias coupling of
V to V s can proceed as in the even case.