Cycles and eigenvalues of sequentially growing random regular graphs (Q400565)

The authors consider random regular (multi-)graphs defined by the incidence matrices obtained as the sum of \(d\) i.i.d. random permutation matrices on \(n\) labels along with their transposes. They explore the joint asymptotic fluctuation for a coupling of all random regular graphs of various degrees constructed in such a way that each component permutation grows according to the Chinese restaurant process. Among other results, the authors prove that one can express the corresponding eigenvalue statistics in terms of a family of independent Yule processes with immigration. It is also observed that some GFF-like processes emerge when \(d\rightarrow \infty\).