Adjacency matrices of sparse random regular graphs are long conjectured to lie within the universality class of random matrices. 
 However, there are few rigorously known results. We focus on fluctuations of linear eigenvalue statistics of a stochastic process of such adjacency 
 matrices growing in dimension. The idea is to compare with eigenvalues of minors of Wigner matrices whose fluctuation converges to the Gaussian Free 
 Field. We show that linear eigenvalue statistics can be described by a family of Yule processes with immigration. Certain key features of the Free 
 Field emerge as the degree tends to infinity. Based on joint work with Tobias Johnson.