# Emergence of order in random languages

## TL;DR

This paper investigates the behavior of large texts generated by weighted context-free grammars, revealing phase transitions and the necessity of replica symmetry breaking in the information-carrying phase.

## Contribution

It introduces a saddle-point analysis for large texts and demonstrates the need for replica symmetry breaking in the random language model.

## Key findings

- Large texts are governed by saddle-point equations.
- Replica symmetry breaking occurs in the information phase.
- The model is solved in the high-temperature disordered phase.

## Abstract

We consider languages generated by weighted context-free grammars. It is shown that the behaviour of large texts is controlled by saddle-point equations for an appropriate generating function. We then consider ensembles of grammars, in particular the Random Language Model of E. DeGiuli, Phys. Rev. Lett., 122, 128301, 2019. This model is solved in the replica-symmetric ansatz, which is valid in the high-temperature, disordered phase. It is shown that in the phase in which languages carry information, the replica symmetry must be broken.