TL;DR: For a reasonable definition of the set of "sparse, $d \times d$, real matrices" and a (probability) measure on it, will any such matrix almost surely have $d$ distinct eigenvalues?
Background: Let $$M := \{ A \in \mathbb{R}^{d \times d}| A \text{ has $d$ distinct eigenvalues}\}.$$ According to this paper, the set $\mathbb{R}^{d\times d} \setminus M$ is a negligible set with respect to the Lebesgue measure (statement directly after Proposition 2.1). They also claim that this holds with respect to probability measures associated with common random matrix ensembles such as Ginibre, orthogonal Gaussian, Wishart, etc. While I don't see how that follows from the cited sources, I'm willing to believe that statement.
My question is whether any sparse matrix $A \in \mathbb{R}^{d\times d}$ still almost surely has $d$ distinct eigenvalues?
This leaves open two choices: What is the set of "sparse matrices" I care about and what measure should I consider. I welcome suggestions for both, but my initial thoughts are:
- Let $$S := \{A \in \mathbb{R}^{d \times d} | A \text{ has at most $k$ non-zero entries}\}$$ with the measure induced by the Lebesgue measure on $\mathbb{R}^{d \times d}$. Is $S \setminus M$ negligible under this measure?
- The most common sparse random matrix ensemble I could find is the Erdös-Reyni matrix, see e.g., here, but it allows only for binary entries and requires symmetry. I'd like to consider an ensemble where each row has on average $p\cdot n$ non-zero entries for $p \ll 1$ and all non-zero entries are iid Gaussians (or uniform on, e.g., [-1,1]). Have those been studied? If so, under the induced probability measure, is $S \setminus M$ negligible?