Phase transition in a stochastic prime number generator

TL;DR

This paper presents a stochastic prime number generator that exhibits a phase transition between a prime-reducing active phase and a frozen low-prime-density phase, supported by simulations and analytical modeling.

Contribution

It introduces a novel stochastic algorithm for prime generation and characterizes its phase transition using numerical and analytical methods.

Findings

Identifies a continuous phase transition in the algorithm's dynamics.

Shows critical slowing down near the transition point.

Provides an analytical approximation that collapses simulation data.

Abstract

We introduce a stochastic algorithm that acts as a prime number generator. The dynamics of such algorithm gives rise to a continuous phase transition which separates a phase where the algorithm is able to reduce a whole set of integers into primes and a phase where the system reaches a frozen state with low prime density. We present both numerical simulations and an analytical approach in terms of an annealed approximation, by means of which the data are collapsed. A critical slowing down phenomenon is also outlined.