Mathematics – Probability
Scientific paper
2011-12-16
Mathematics
Probability
Version 1.0, 22 pages, 3 figures
Scientific paper
One of the most studied random matrix ensembles is that of real symmetric matrices, where the limiting spectral measure converges to the semi-circle. Studies have also determined the limiting spectral measures for many structured ensembles, such as Toeplitz and circulant matrices. These systems have very different behavior; the limiting spectral measures for both have unbounded support. Given a structured ensemble, we introduce a parameter to continuously interpolate between these two behaviors. We fix a $p \in [1/2, 1]$ and study the ensemble of signed structured matrices by multiplying the $(i,j)$th and $(j,i)$th entries of a matrix by a randomly chosen $\epsilon_{ij} \in \{1, -1\}$, with ${\rm Prob}(\epsilon_{ij} = 1) = p$. For $p = 1/2$, we prove that the limiting spectral measure is the semi-circle. For all other $p$, for many structured ensembles (including the Toeplitz and circulant) we prove the measure has unbounded support, and converges to the original ensemble as $p \to 1$. The proofs are by Markov's Method of Moments. The analysis of the $2k^{\text{th}}$ moment for such distributions involves the pairings of $2k$ vertices on a circle. The contribution of each pairing in the signed case is weighted by a factor depending on $p$ and the number of vertices involved in at least one crossing. These numbers are of interest in their own right, appearing in problems in combinatorics and knot theory. The number of configurations with no vertices involved in a crossing is well-studied, and are the Catalan numbers. We discover and prove similar formulas for configurations with $4, 6, 8$ and 10 vertices in at least one crossing. For higher-order moments, we prove closed-form expressions for the expected value and variance for the number of vertices in at least one crossing. As the variance converges to 4, these results allow us to deduce properties of the limiting measure.
Beckwith Olivia
Miller Steven J.
Shen Karen
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