Statistics of low energy excitations for the directed polymer in a $1+d$ random medium ($d=1,2,3$)

TL;DR

This study numerically analyzes the statistics of low energy excitations in directed polymers within 1, 2, and 3-dimensional random media, revealing scaling behaviors and singularities related to energy fluctuations.

Contribution

It provides a detailed numerical characterization of excitation densities and their scaling functions, highlighting differences between bulk and boundary excitations in directed polymers.

Findings

Both bulk and boundary excitation densities follow a specific scaling form with system size.

The scaling functions exhibit distinct behaviors, with boundary excitations showing a non-monotonic pattern.

The decay exponents suggest a potential relation between excitation density decay and energy fluctuation exponent.

Abstract

We consider a directed polymer of length $L$ in a random medium of space dimension $d=1,2,3$. The statistics of low energy excitations as a function of their size $l$ is numerically evaluated. These excitations can be divided into bulk and boundary excitations, with respective densities $\rho^{bulk}_L(E=0,l)$ and $\rho^{boundary}_L(E=0,l)$. We find that both densities follow the scaling behavior $\rho^{bulk,boundary}_L(E=0,l) = L^{-1-\theta_d} R^{bulk,boundary}(x=l/L)$, where $\theta_d$ is the exponent governing the energy fluctuations at zero temperature (with the well-known exact value $\theta_1=1/3$ in one dimension). In the limit $x=l/L \to 0$, both scaling functions $R^{bulk}(x)$ and $R^{boundary}(x)$ behave as $R^{bulk,boundary}(x) \sim x^{-1-\theta_d}$, leading to the droplet power law $\rho^{bulk,boundary}_L(E=0,l)\sim l^{-1-\theta_d} $ in the regime $1 \ll l \ll L$. Beyond their common singularity near $x \to 0$, the two scaling functions $R^{bulk,boundary}(x)$ are very different : whereas $R^{bulk}(x)$ decays monotonically for $0<x<1$, the function $R^{boundary}(x)$ first decays for $0<x<x_{min}$, then grows for $x_{min}<x<1$, and finally presents a power law singularity $R^{boundary}(x)\sim (1-x)^{-\sigma_d}$ near $x \to 1$. The density of excitations of length $l=L$ accordingly decays as $\rho^{boundary}_L(E=0,l=L)\sim L^{- \lambda_d} $ where $\lambda_d=1+\theta_d-\sigma_d$. We obtain $\lambda_1 \simeq 0.67$, $\lambda_2 \simeq 0.53$ and $\lambda_3 \simeq 0.39$, suggesting the possible relation $\lambda_d= 2 \theta_d$.