[Paper Review] A central limit theorem for the determinant of a Wigner matrix

Terence Tao, Van Vu|arXiv (Cornell University)|Nov 27, 2011

Random Matrices and Applications31 references3 citations

TL;DR

This paper establishes a central limit theorem for the log-determinant of Wigner matrices under four-moment matching with either the Gaussian Unitary Ensemble (GUE) or Gaussian Orthogonal Ensemble (GOE). It shows that the log-determinant converges in distribution to a normal law with mean and variance depending on the ensemble: $\mathcal{N}(\log\sqrt{n!} - \frac{1}{2}\log n, \frac{1}{2}\log n)$ for GUE and $\mathcal{N}(\log\sqrt{n!} - \frac{1}{4}\log n, \frac{1}{4}\log n)$ for GOE, under matching of the first four moments off-diagonal and second moment on-diagonal.

ABSTRACT

We establish a central limit theorem for the log-determinant $\log|\det(M_n)|$ of a Wigner matrix $M_n$, under the assumption of four matching moments with either the GUE or GOE ensemble. More specifically, we show that this log-determinant is asymptotically distributed like $N(\log \sqrt{n!} - 1/2 \log n, 1/2 \log n)_\R$ when one matches moments with GUE, and $N(\log \sqrt{n!} - 1/4 \log n, 1/4 \log n)_\R$ when one matches moments with GOE.

Motivation & Objective

• To establish a central limit theorem for the log-determinant of Wigner matrices under moment matching conditions.
• To determine the asymptotic distribution of $\log|\det M_n|$ when $M_n$ matches GUE or GOE to fourth order off-diagonal and second order on-diagonal.
• To extend previous results on iid random matrices to the Hermitian case, where row independence no longer holds.
• To resolve the limiting distribution of the log-determinant in the absence of full independence, using resolvent swapping and moment matching techniques.

Proposed method

• Uses resolvent swapping to compare the log-determinant of a general Wigner matrix to that of a GUE or GOE matrix.
• Applies moment matching up to fourth order for off-diagonal entries and second order for diagonal entries to control the difference in distribution.
• Employs combinatorial analysis of permutations via cycle types to compute the second moment of the determinant's logarithm.
• Uses generating function and double-counting arguments to bound sums over permutations with specific cycle structures, particularly $2$-cycles.
• Relies on Stirling's approximation and asymptotic enumeration of permutations with fixed cycle counts to derive the limiting variance.
• Establishes convergence in distribution by controlling the cumulants and using the Lindeberg-Feller type argument for weak convergence.

Experimental results

Research questions

• RQ1 What is the limiting distribution of $\log|\det M_n|$ for a Wigner matrix $M_n$ that matches GUE to fourth order?
• RQ2 How does the asymptotic mean and variance of the log-determinant depend on the underlying ensemble (GUE vs. GOE)?
• RQ3 Can the central limit theorem for the log-determinant be extended from iid matrices to Wigner matrices with dependent entries?
• RQ4 What role do the fourth moments of the entries play in determining the limiting distribution of the log-determinant?
• RQ5 To what extent can the GUE/GOE assumptions be relaxed while preserving the same limiting behavior?

Key findings

• The log-determinant $\log|\det M_n|$ converges in distribution to a normal law with mean $\log\sqrt{n!} - \frac{1}{2}\log n$ and variance $\frac{1}{2}\log n$ when $M_n$ matches GUE to fourth order.
• For GOE matching, the limiting distribution is $\mathcal{N}(\log\sqrt{n!} - \frac{1}{4}\log n, \frac{1}{4}\log n)$, reflecting the different fourth moment structure.
• The central limit theorem holds under the minimal condition that the matrix matches GUE or GOE to fourth order off-diagonal and second order on-diagonal.
• The second moment of $|\det M_n|$ is asymptotically $n!$, consistent with known identities for Gaussian matrices.
• The proof relies on controlling the contribution of permutations with $m$ 2-cycles, showing that their total weight is $O(n \cdot n!)$ and $\gg n \cdot n!$ for $m \leq n/4$, respectively.
• The result extends to non-Gaussian Wigner matrices as long as the first four moments match those of GUE or GOE.