Semicircular law with a few independent entries in a random matrix

Abstract: It is well known in random matrix literature that the limiting spectral distribution of a Wigner matrix is the semi circular law while the limiting spectral distributions of other patterned matrices like Toeplitz, Hankel, symmetric circulant and reverse circulant matrices have unbounded supports. One fundamental difference between the Wigner matrices and the other matrices mentioned above is that the Wigner matrices have $O(n<sup>2)$ independent random variables while the others have $O(n)$ independent random variables. In this paper, we show that this is not true in general. In particular, we form matrices with $O(n)$ independent random variables whose empirical spectral distributions are arbitrary close to the semi circular law.

• The paper constructs a symmetric random matrix using exactly n independent variables and proves that its empirical spectral distribution converges in probability to the standard semicircular law on [-2,2].
• The authors use moment bounds, matched-partition estimates, and permutation-induced weak dependence to verify sufficient conditions for semicircular convergence, including conditional convergence for almost every permutation pattern.
• The result shows that repetition geometry, rather than the number of independent entries, determines whether a patterned matrix has bounded semicircular or unbounded limiting spectral behavior.

The paper addresses a long-standing heuristic in the theory of patterned random matrices: that matrices built from only $O(n)$ independent random variables, such as Toeplitz, Hankel, symmetric circulant, and reverse circulant ensembles, necessarily have limiting spectral distributions (LSDs) with unbounded support, in contrast to Wigner matrices whose $O(n^2)$ independent entries yield the semicircular law on $[-2,2]$. The authors show this dichotomy is not a consequence of the number of independent variables per se. They construct an explicit family of symmetric random matrices with only $n$ independent inputs whose empirical spectral distributions (ESDs) converge to the standard semicircular law (2608.17648).

The model

Let $\xi_1,\ldots,\xi_n$ be independent, centered, unit-variance random variables with uniformly bounded moments, i.e., $M_r := \sup_i E|\xi_i|^r < \infty$ for each fixed $r$. Let $\Phi_1,\ldots,\Phi_n$ be i.i.d. uniform random permutations of $[n]$, and set $Y_{ij} := \xi_{\Phi_i(j)}$. The matrix of interest is the symmetrization

$O(n^2)$0

Each row $O(n^2)$1 draws its entries from a fresh random permutation of the same pool $O(n^2)$2, so the entire ensemble depends on only $O(n^2)$3 independent random variables. The permutation structure induces weak dependence: $O(n^2)$4 when $O(n^2)$5 and zero otherwise. After symmetrization, off-diagonal variances are $O(n^2)$6 and diagonal variances are $O(n^2)$7, giving $O(n^2)$8, which fixes the correct scaling.

Main result

Theorem. If all moments of $O(n^2)$9 are finite, then the ESD of $[-2,2]$0 converges weakly in probability to the standard semicircular law with density $[-2,2]$1 on $[-2,2]$2.

Equivalently, $[-2,2]$3 in probability under the bounded-Lipschitz metric. The corollary that carries the conceptual weight of the paper is a conditional LSD statement: for every $[-2,2]$4 there exist sets $[-2,2]$5 with $[-2,2]$6 such that, conditioned on any realization $[-2,2]$7 of the permutations, the conditional probability that the ESD deviates from the semicircle by more than $[-2,2]$8 is at most $[-2,2]$9. In other words, for almost all fixed choices of permutations — deterministic patterns of entry repetition — the resulting matrix still has a semicircular LSD. This directly answers the question posed by Arup Bose: it is not the count of independent variables but the structure of the repetitions (diagonal-parallel lines in Toeplitz matrices, cross-diagonal lines in Hankel matrices) that inflates the moments and produces unbounded support. Destroying that structure via random permutations restores Wigner-type behavior while retaining only $n$0 randomness.

Proof technique

The proof reduces the problem, via Theorem 5 of Hochstättler et al., to verifying two moment conditions on distinct index sets: mixed moments of order $n$1 satisfy $n$2, and $n$3. The key observation is that a product of the $n$4's has nonzero expectation only if the multiset of indices $n$5 is matched (each value appearing with multiplicity at least two). A union bound over the at most $n$6 matched partitions, combined with the fact that equality events among $n$7 values carry probability $n$8 per constraint, yields the $n$9 decay; the counting hinges on the bound $\xi_1,\ldots,\xi_n$0, where $\xi_1,\ldots,\xi_n$1 counts blocks containing no $\xi_1,\ldots,\xi_n$2-indices. The second condition follows from Hölder's inequality together with $\xi_1,\ldots,\xi_n$3, giving convergence at rate $\xi_1,\ldots,\xi_n$4. The conditional corollary is then a direct Markov inequality argument.

A simulation with $\xi_1,\ldots,\xi_n$5 and Gaussian $\xi_1,\ldots,\xi_n$6 shows the eigenvalue histogram matching the semicircular density closely, consistent with the theorem.

Limitations and open questions

Several points are left open. The result is convergence in probability of the ESD, not almost sure convergence, and no rate of convergence in $\xi_1,\ldots,\xi_n$7 is established beyond what the proof implicitly suggests. The moment method verifies the two sufficient conditions but does not provide fluctuation results, local statistics, or operator norm bounds for $\xi_1,\ldots,\xi_n$8. It also remains open whether analogous constructions can reproduce other classical LSDs (e.g., Marchenko–Pastur-type behavior) using $\xi_1,\ldots,\xi_n$9 independent variables, and whether the requirement of finite moments of all orders can be relaxed to a Lindeberg-type condition as in the Wigner setting. Finally, the paper does not quantify how "structured" a deterministic pattern must be before the LSD departs from the semicircle; the conditional corollary asserts typicality over permutations but gives no characterization of the exceptional set $M_r := \sup_i E|\xi_i|^r < \infty$0.

Conclusion

This paper refutes the folk belief that $M_r := \sup_i E|\xi_i|^r < \infty$1 independent entries force unbounded-limiting-support behavior in patterned random matrices. By randomly permuting a single pool of $M_r := \sup_i E|\xi_i|^r < \infty$2 variables across rows, the authors obtain a symmetric ensemble with exactly $M_r := \sup_i E|\xi_i|^r < \infty$3 independent inputs whose ESD converges in probability to the semicircular law, and which does so conditionally on almost every realization of the permutation pattern. The result cleanly isolates repetition geometry, rather than independence count, as the determinant of the LSD in patterned ensembles.