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

TL;DR

This paper provides an analytical solution to a stochastic risk-spreading model with global coupling, revealing that optimal dispersion follows Zipf's law and that ensemble averages converge due to finite sample sizes.

Contribution

It offers the first analytical results for a stochastic risk-spreading model with global coupling, extending beyond well-mixed cases.

Findings

Optimal dispersion follows Zipf's law.

Arithmetic and geometric averages of growth rates converge.

Analytical solution extends understanding of risk-spreading models.

Abstract

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.