Spectral properties of the Laplacian of Scale-Free Percolation models

We consider scale-free percolation on a discrete torus ${\mathbf{V}}_N$ of size $N$. Conditionally on an i.i.d. sequence of Pareto weights $(W_i{)}_{i\in {\mathbf{V}}_N}$ with tail exponent $\tau -1>0$, we connect any two points $i$ and $j$ on the torus with probability $$p_{ij}=\frac{W_i{W}_j}{\mathrm{\parallel }i-j{\mathrm{\parallel }}^{\alpha }}\wedge 1$$ for some parameter $\alpha >0$. We focus on the (centred) Laplacian operator of this random graph and study its empirical spectral distribution. We explicitly identify the limiting distribution when $\alpha <1$ and $\tau >3$, in terms of the spectral distribution of some non-commutative unbounded operators.