In this dissertation, we address a number of issues dealing with the edge spectrum of random Hecke operators. In particular, we focus on the spectral gap property of random $2d$-regular graphs, which can be thought of as random Hecke operators of form $z=\pi_1+\pi_1^{-1}+\cdots+\pi_d+\pi_d^{-1}$ with $\pi_i\in S_n$ under the permutation representation of symmetric group $S_n$. This dissertation is organized in essentially two parts.
In the first portion of the dissertation, we deal with the spectral gap property of random $2d$-regular graphs. Puder and Parzanchevski ([Puder11], [PP12]) developed a crucial theorem of cancellations between certain collections of closed walks in random $n$ lifts, which simplifies the work of counting the expected number of closed walks in random lifts. Our observation of the connection between generalized forms and core graphs enables us to adopt the cancellation theorem to estimate the expected number of closed walks under the permutation model. The resulting contribution from the cancellation theorem can be expanded to any order by using Friedman's expansion method [Fri91]. The freedom of choosing expansion order leads to an optimal estimation of the spectral gap. However, it is challenging to control high order terms from the expansion. We solve this key issue by separating the summation of irreducible walks into ``a good part'' and ``a bad part'', and showing the probability of the bad part occurring is small. With a lemma of complex random variables [Fri08], and Bartholdi identity [OS09], we provide an alternative proof of Friedman's strong Alon's conjecture $\la(G)\leq 2\sqrt{2d-1}+\ep$ for any $\ep>0$ with probability $1-\frac{c}{n^{\lceil\frac{\sqrt{d-1}-1}{2}\rceil}}$ in a simpler way.
There is a strong connection between random Hecke operators/random regular graphs and random matrices. It is conjectured that the edge spectra of random regular graphs can be modeled by certain Tracy-Widom distributions from random matrices. Due to the lack of a proper normalization factor, only indirect evidence [MNS08] is known. In Chapter 3, we consider a normalization factor obtained by matching the first four moments of random Hecke operators/random regular graphs with the corresponding moments of general $\beta$ ensembles. The validity of this normalization factor is supported by numerical analysis, where we are able to demonstrate that the edge spectra of both random Cayley graphs $\G(SL_2(\F_p),S_p)$ and the Fourier transform of random Hecke operators over $SU(2)$ at irreducible representations can be modeled by certain Tracy-Widom distributions.
