# Numerical calculation of overlap distribution of $(2+1)$-dimensional directed polymer in random media

## TL;DR

This study numerically analyzes the overlap distribution of (2+1)-dimensional directed polymers in Gaussian random media, revealing no replica symmetry breaking and contrasting diffusive behaviors of polymer end displacements.

## Contribution

It provides the first numerical calculation of the overlap distribution in (2+1)D directed polymers, demonstrating the absence of replica symmetry breaking.

## Key findings

- No replica symmetry breaking observed.
- Superdiffusive behavior of individual polymer ends.
- Subdiffusive behavior of relative distance between polymer ends.

## Abstract

We investigate $(2+1)$-dimensional discretized directed polymers in Gaussian random media. By numerically calculating the probability distribution function of overlap between two independent and identical systems on a common random potential, we show that there is no replica symmetry breaking. We also show that while the mean-squared displacement of one polymer end exhibits superdiffusive behavior, the mean-squared relative distance of two polymer ends exhibits subdiffusive behavior.