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 $$p_{ij}= \frac{W_iW_j}{\|i-j\|^{\alpha}} \wedge 1$$ 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.