On Extracting Common Random Bits From Correlated Sources on Large Alphabets

Suppose Alice and Bob receive strings X=(X
₁
,...,X
_n
) and Y=(Y
₁
,...,Y
_n
) each uniformly random in [s]
ⁿ
, but so that X and Y are correlated. For each symbol i, we have that Y
_i
=X
_i
with probability 1-ε and otherwise Y
_i
is chosen independently and uniformly from [s]. Alice and Bob wish to use their respective strings to extract a uniformly chosen common sequence from [s]
^k
, but without communicating. How well can they do? The trivial strategy of outputting the first k symbols yields an agreement probability of (1-ε+ε/s)
^k
. In a recent work by Bogdanov and Mossel, it was shown that in the binary case where s=2 and k=k(ε) is large enough then it is possible to extract k bits with a better agreement probability rate. In particular, it is possible to achieve agreement probability (kε)
^-1/2
·2
^{-kε/(2(1-ε/2))}
using a random construction based on Hamming balls, and this is optimal up to lower order terms. In this paper, we consider the same problem over larger alphabet sizes s and we show that the agreement probability rate changes dramatically as the alphabet grows. In particular, we show no strategy can achieve agreement probability better than (1-ε)
^k
(1+δ(s))
^k
where δ(s)→ 0 as s→∞. We also show that Hamming ball-based constructions have much lower agreement probability rate than the trivial algorithm as s→∞. Our proofs and results are intimately related to subtle properties of hypercontractive inequalities.