In this paper, we present a new combinatorial approach for analyzing the traces of great powers of the Wigner matrix. Our approach is based on the \citet{sosh} paper. However, the counting method is different. Starting with a classical word-sentence approach similar to \citet{AZ05}, and taking motives from \citet{sinaisosh}, \citet{sosh}, and \citet{peche2009universality}, we put words into Dyck path-like objects. encode the To be precise, the map goes to the Dyck path with some edges removed from the words. Using this new count, we prove the edge universality for large Wigner matrices with sub-Gaussian entries. One novelty of this approach is that unlike \citet{sinaisosh}, \citet{sosh}, and \citet{peche2009universality}, it assumes that the matrix entries are symmetrically distributed around $0$. you don’t have to. There are two main technical contributions of this paper. First, we create an encoding of the “contributing words” (see section \ref{sec:word} for definitions) of the Wigner matrix that captures edge universality. So this is the best you can do. This method can be applied to many other scenarios for random matrices. Explanations may be important for models where exact calculations are made.There are some combinatorial structures that are not available.
