On the algebraic numbers computable by some generalized Ehrenfest urns

TL;DR

This paper investigates the algebraic numbers that can be approximated by a class of stochastic urn models inspired by distributed computing, revealing that only certain algebraic numbers are attainable through these processes.

Contribution

It characterizes which algebraic numbers can be obtained via generalized Ehrenfest urns, showing that not all algebraic numbers are computable by such stochastic protocols.

Findings

Proves convergence of black ball proportion to algebraic numbers

Identifies limitations on algebraic numbers achievable by urn models

Demonstrates that some algebraic numbers cannot be computed by these protocols

Abstract

This article deals with some stochastic population protocols, motivated by theoretical aspects of distributed computing. We modelize the problem by a large urn of black and white balls from which at every time unit a fixed number of balls are drawn and their colors are changed according to the number of black balls among them. When the time and the number of balls both tend to infinity the proportion of black balls converges to an algebraic number. We prove that, surprisingly enough, not every algebraic number can be "computed" this way.