ABSTRACT
In this chapter we shall put to use the results and concepts of the previous chapter to provide proofs of existence of the LSD of some real symmetric patterned matrices. These matrices are: Wigner, Toeplitz, Hankel, Reverse Circulant and the Symmetric Circulant. We shall see how certain types of words assume significance in the context of each matrix. It turns out that, for all of these matrices, p ( w ) = 1 $ p(w)=1 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429488436/725f3a92-cccc-4b3b-961a-84021a8d495e/content/inline-math2_1.tif"/> for all w in one common subclass of words, which we call the Catalan words. Moreover, for the Wigner matrix, p ( w ) = 0 $ p(w)=0 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429488436/725f3a92-cccc-4b3b-961a-84021a8d495e/content/inline-math2_2.tif"/> for all words outside this class and, for the Symmetric Circulant, p ( w ) = 1 $ p(w)=1 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429488436/725f3a92-cccc-4b3b-961a-84021a8d495e/content/inline-math2_3.tif"/> for all pair-matched words. The LSD is bounded for the Wigner matrix but is unbounded for all the other matrices. There are many interesting questions that arise from the main LSD results and associated words. The LSD do not depend on the specific nature of the input sequence but on the pattern. This feature is termed universality. We can define it formally as follows. First recall the three main assumptions from the previous chapter.
