Singularity of sparse random matrices: simple proofs

Consider a random $n\times n$ zero-one matrix with ‘sparsity’ p, sampled according to one of the following two models: either every entry is independently taken to be one with probability p (the ‘Bernoulli’ model) or each row is independently uniformly sampled from the set of all length-n zero-one vectors with exactly pn ones (the ‘combinatorial’ model). We give simple proofs of the (essentially best-possible) fact that in both models, if $\min(p,1-p)\geq (1+\varepsilon)\log n/n$ for any constant $\varepsilon>0$ , then our random matrix is nonsingular with probability $1-o(1)$ . In the Bernoulli model, this fact was already well known, but in the combinatorial model this resolves a conjecture of Aigner-Horev and Person.

Fluctuations of extreme eigenvalues of sparse Erdős–Rényi graphs

We consider a class of sparse random matrices which includes the adjacency matrix of the Erdős–Rényi graph $${{\mathcal {G}}}(N,p)$$ G ( N , p ) . We show that if $$N^{\varepsilon } \leqslant Np \leqslant N^{1/3-\varepsilon }$$ N ε ⩽ N p ⩽ N 1 / 3 - ε then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. This extends the result (Huang et al. in Ann Prob 48:916–962, 2020) on the fluctuations of the extreme eigenvalues from $$Np \geqslant N^{2/9 + \varepsilon }$$ N p ⩾ N 2 / 9 + ε down to the optimal scale $$Np \geqslant N^{\varepsilon }$$ N p ⩾ N ε . The main technical achievement of our proof is a rigidity bound of accuracy $$N^{-1/2-\varepsilon } (Np)^{-1/2}$$ N - 1 / 2 - ε ( N p ) - 1 / 2 for the extreme eigenvalues, which avoids the $$(Np)^{-1}$$ ( N p ) - 1 -expansions from Erdős et al. (Ann Prob 41:2279–2375, 2013), Huang et al. (2020) and Lee and Schnelli (Prob Theor Rel Fields 171:543–616, 2018). Our result is the last missing piece, added to Erdős et al. (Commun Math Phys 314:587–640, 2012), He (Bulk eigenvalue fluctuations of sparse random matrices. arXiv:1904.07140), Huang et al. (2020) and Lee and Schnelli (2018), of a complete description of the eigenvalue fluctuations of sparse random matrices for $$Np \geqslant N^{\varepsilon }$$ N p ⩾ N ε .

Stability of synchronous states in sparse neuronal networks

The stability of synchronous states is analyzed in the context of two populations of inhibitory and excitatory neurons, characterized by two different pulse-widths. The problem is reduced to that of determining the eigenvalues of a suitable class of sparse random matrices, randomness being a consequence of the network structure. A detailed analysis, which includes also the study of finite-amplitude perturbations, is performed in the limit of narrow pulses, finding that the overall stability depends crucially on the relative pulse-width. This has implications for the overall property of the asynchronous (balanced) regime.

Stability of synchronous states in sparse neuronal networks

The stability of synchronous states is analysed in the context of two populations of inhibitory and excitatory neurons, characterized by different pulse-widths. The problem is reduced to that of determining the eigenvalues of a suitable class of sparse random matrices, randomness being a consequence of the network structure. A detailed analysis, which includes also the study of finite-amplitude perturbations, is performed in the limit of narrow pulses, finding that the stability depends crucially on the relative pulse-width. This has implications for the overall property of the asynchronous (balanced) regime.