# Localization of soft modes at the depinning transition

## TL;DR

This paper investigates the properties of soft modes at the depinning transition, revealing universal spectral features and their relation to localization phenomena, with implications for understanding avalanche instabilities.

## Contribution

It characterizes the soft modes at the depinning transition, compares them with Anderson models, and links localization length to the Larkin length, highlighting a new localization transition in a fully connected model.

## Key findings

- Density of states shows a universal linear tail at the spectrum edge.
- Eigenvectors of the dynamical matrix are similar to those in Anderson models.
- The ground state soft mode is localized and related to the Larkin length.

## Abstract

We characterize the soft modes of the dynamical matrix at the depinning transition, and compare it with the properties of the Anderson model (and long--range generalizations). The density of states at the edge of the spectrum displays a universal linear tail, different from the Lifshitz tails. The eigenvectors are instead very similar in the two matrix ensembles. We focus on the ground state (soft mode), which represents the epicenter of avalanche instabilities. We expect it to be localized in all finite dimensions, and make a clear connection between its localization length and the Larkin length of the depinning model. In the fully connected model, the weak--strong pinning transition coincides with a peculiar localization transition of the ground state.