Analytical solution of stochastic model of risk-spreading with global coupling

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.