Spectral properties of the Laplacian of Scale-Free Percolation models
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917099103846400 |
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| author | Hazra, Rajat Subhra Malhotra, Nandan |
| author_facet | Hazra, Rajat Subhra Malhotra, Nandan |
| contents | 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 $τ-1>0$, we connect any two points $i$ and $j$ on the torus with probability
$$p_{ij}= \frac{W_iW_j}{\|i-j\|^α} \wedge 1$$ for some parameter $α>0$.
We focus on the (centred) Laplacian operator of this random graph and study its empirical spectral distribution. We explicitly identify the limiting distribution when $α<1$ and $τ>3$, in terms of the spectral distribution of some non-commutative unbounded operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17552 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral properties of the Laplacian of Scale-Free Percolation models Hazra, Rajat Subhra Malhotra, Nandan Probability 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 $τ-1>0$, we connect any two points $i$ and $j$ on the torus with probability $$p_{ij}= \frac{W_iW_j}{\|i-j\|^α} \wedge 1$$ for some parameter $α>0$. We focus on the (centred) Laplacian operator of this random graph and study its empirical spectral distribution. We explicitly identify the limiting distribution when $α<1$ and $τ>3$, in terms of the spectral distribution of some non-commutative unbounded operators. |
| title | Spectral properties of the Laplacian of Scale-Free Percolation models |
| topic | Probability |
| url | https://arxiv.org/abs/2504.17552 |