# The fractional p-Laplacian emerging from homogenization of the random conductance model with degenerate ergodic weights and unbounded-range jumps

## TL;DR

This paper demonstrates that in a random conductance model with long-range jumps and ergodic weights, the homogenized limit of the discrete p-Laplace operator converges to a fractional p-Laplace operator, linking discrete models to fractional PDEs.

## Contribution

It establishes the emergence of a fractional p-Laplace operator as the homogenized limit in a complex random conductance setting with long-range interactions.

## Key findings

- Homogenized limit is a fractional p-Laplace operator.
- Under certain moment conditions, spectral homogenization to fractional Laplace is achieved.
- The approach uses a variational formulation and ergodic assumptions.

## Abstract

We study a general class of discrete $p$-Laplace operators in the random conductance model with long-range jumps and ergodic weights. Using a variational formulation of the problem, we show that under the assumption of bounded first moments and a suitable lower moment condition on the weights, the homogenized limit operator is a fractional $p$-Laplace operator. Under strengthened lower moment conditions, we can apply our insights also to the spectral homogenization of the discrete Laplace operator to the continuous fractional Laplace operator.