Power-law exponent of the Bouchaud-M\'ezard model on regular random network

TL;DR

This paper derives an equation for the wealth distribution's power-law exponent in the Bouchaud-Mézard model on regular random networks, revealing it can be less than 2, contrasting with mean-field predictions, and confirms findings with simulations.

Contribution

The paper introduces a novel analytical approach to determine the wealth distribution exponent in the Bouchaud-Mézard model on regular random networks, showing it can be below 2.

Findings

Exponent can be less than 2.

Analytical results agree with numerical simulations.

Contrasts with mean-field analysis predictions.

Abstract

We study the Bouchaud-M\'ezard model on a regular random network. By assuming adiabaticity and independency, and utilizing the generalized central limit theorem and the Tauberian theorem, we derive an equation that determines the exponent of the probability distribution function of the wealth as $x\rightarrow \infty$. The analysis shows that the exponent can be smaller than 2, while a mean-field analysis always gives the exponent as being larger than 2. The results of our analysis are shown to be good agreement with those of the numerical simulations.