Large scale numerical simulations of "ultrametric" long-range depinning

TL;DR

This paper demonstrates that using an ultrametric distance in large-scale simulations of long-range depinning does not alter the universality class, enabling more efficient computations and precise critical exponent determination.

Contribution

Introducing an ultrametric distance to reduce computational complexity in long-range depinning simulations without changing the universality class.

Findings

Critical exponents are unchanged by ultrametric distance.

Scaling functions remain consistent with Euclidean metric results.

Ultrametric approach enables larger scale simulations and precise measurements.

Abstract

The depinning of an elastic line interacting with a quenched disorder is studied for long range interactions, applicable to crack propagation or wetting. An ultrametric distance is introduced instead of the Euclidean distance, allowing for a drastic reduction of the numerical complexity of the problem. Based on large scale simulations, two to three orders of magnitude larger than previously considered, we obtain a very precise determination of critical exponents which are shown to be indistinguishable from their Euclidean metric counterparts. Moreover the scaling functions are shown to be unchanged. The choice of an ultrametric distance thus does not affect the universality class of the depinning transition and opens the way to an analytic real space renormalization group approach.