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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.05748 |
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| _version_ | 1866917191892336640 |
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| author | Adhikari, Kartick Kumar, Kiran Saha, Koushik |
| author_facet | Adhikari, Kartick Kumar, Kiran Saha, Koushik |
| contents | In this paper, we consider the multi-parameter random simplicial complex model, which generalizes the Linial-Meshulam model and random clique complexes by allowing simplices of different dimensions to be included with distinct probabilities. For $n,d \in \mathbb{N}$ and $\mathbf{p}=(p_1,p_2,\ldots, p_d)$ such that $p_i \in (0,1]$ for all $1 \leq i \leq d$, the multi-parameter random simplicial complex $Y_d(n,\mathbf{p})$ is constructed inductively. Starting with $n$ vertices, edges (1-cells) are included independently with probability $p_1$, yielding the Erdős-Rényi graph $\mathcal{G}(n,p_1)$, which forms the $1$-skeleton. Conditional on the $(k-1)$-skeleton, each possible $k$-cell is included independently with probability $p_k$, for $2 \leq k \leq d$.
We study the signed and unsigned adjacency matrices of $d$-dimensional multi-parameter random simplicial complexes $Y_d(n,\mathbf{p}),$ under the assumptions $\min_{i=1,\ldots d-1}\liminf p_i >0$ and $np_d \rightarrow \infty$ with $p_d=o(1)$. In general, these matrices have random dimensions and exhibit dependency among its entries. We prove that the empirical spectral distributions of both matrices converge weakly to the semicircle law in probability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05748 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Semicircle law for multi-parameter random simplicial complexes Adhikari, Kartick Kumar, Kiran Saha, Koushik Probability 60B20, 60B10, 05E45, 05C80 In this paper, we consider the multi-parameter random simplicial complex model, which generalizes the Linial-Meshulam model and random clique complexes by allowing simplices of different dimensions to be included with distinct probabilities. For $n,d \in \mathbb{N}$ and $\mathbf{p}=(p_1,p_2,\ldots, p_d)$ such that $p_i \in (0,1]$ for all $1 \leq i \leq d$, the multi-parameter random simplicial complex $Y_d(n,\mathbf{p})$ is constructed inductively. Starting with $n$ vertices, edges (1-cells) are included independently with probability $p_1$, yielding the Erdős-Rényi graph $\mathcal{G}(n,p_1)$, which forms the $1$-skeleton. Conditional on the $(k-1)$-skeleton, each possible $k$-cell is included independently with probability $p_k$, for $2 \leq k \leq d$. We study the signed and unsigned adjacency matrices of $d$-dimensional multi-parameter random simplicial complexes $Y_d(n,\mathbf{p}),$ under the assumptions $\min_{i=1,\ldots d-1}\liminf p_i >0$ and $np_d \rightarrow \infty$ with $p_d=o(1)$. In general, these matrices have random dimensions and exhibit dependency among its entries. We prove that the empirical spectral distributions of both matrices converge weakly to the semicircle law in probability. |
| title | Semicircle law for multi-parameter random simplicial complexes |
| topic | Probability 60B20, 60B10, 05E45, 05C80 |
| url | https://arxiv.org/abs/2601.05748 |