Browsing by Author "Hirviniemi, Olli"

In this thesis we examine the properties of Wigner matrices. We will give proofs for two fundamental limit theorems of random Hermitian matrices. One of them is Wigner's semicircular law which states that the distribution of the eigenvalues approaches the Wigner's semicircular distribution when the size of the matrix increases. The other, Bai-Yin theorem, tells that the operator norm of such n × n matrix is almost surely (2+o(1))\sqrt{n}. In Chapter 1 we begin by introducing Wigner matrices and deduce the proper normalizing factor. We will also consider what does it mean for the distribution of the eigenvalues to converge by demanding that the empirical spectral distribution should converge almost surely in weak topology. In Chapter 2 we introduce the Stieltjes transform, a useful tool for finding the limit measure. We prove some basic properties and prove that the weak convergence of measures is equivalent to the convergence of their Stieltjes transforms. In Chapter 3 we prove the Wigner's semicircular law. The proof is based on using the Stieltjes transform, and contains several steps. First we find the pointwise expectation of the Stieltjes transform by deriving a polynomial equation for it. While deriving this equation we use several techniques and theorems from both probability theory and linear algebra, covered in the appropriate Appendix. After deducing that the expectation converges, we see quite straightforwardly that the pointwise limit is in fact almost sure. From this we deduce that the empirical spectral distribution of the Wigner matrix converges to the Wigner's semicircular distribution. We begin Chapter 4 by concluding the lower bound in Bai-Yin theorem directly by the semicircular law. For upper bound, we can split the matrix into three parts and use triangle inequality. The diagonal part is easily seen to grow slower than \sqrt{n}. The part with smaller elements is estimated using even moments and combinatorics to see that it has the wanted upper bound. Finally the part containing all large elements is almost surely sparse, and therefore its operator norm grows slower than \sqrt{n}.