  We study a stochastic matrix model to understand the mechanics of
risk-spreading (or bet-hedging) by dispersion. Such model has been mostly dealt
numerically except for well-mixed case, so far. Here, we present an analytical
result, which shows that optimal dispersion leads to Zipf's law. Moreover, we
found that the arithmetic ensemble average of the total growth rate converges
to the geometric one, because the sample size is finite.
