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In this paper we introduce a new combinatorial approach to analyze the traceof large powers of Wigner matrices. Our approach is motivated from the paper by\citet{sosh}. However the counting approach is different. We start withclassical word sentence approach similar to \citet{AZ05} and take themotivation from \citet{sinaisosh}, \citet{sosh} and\citet{peche2009universality} to encode the words to objects similar to Dyckpaths. To be precise the map takes a word to a Dyck path with some edgesremoved from it. Using this new counting we prove edge universality for largeWigner matrices with sub-Gaussian entries. One novelty of this approach isunlike \citet{sinaisosh}, \citet{sosh} and \citet{peche2009universality} we donot need to assume the entries of the matrices are symmetrically distributedaround $0$. The main technical contribution of this paper is two folded.Firstly we produce an encoding of the ``contributing words" (for definition onemight look at Section \ref{sec:word}) of the Wigner matrix which retrieves theedge universality. Hence this is the best one can do. We hope this method willbe applicable to many other scenarios in random matrices. Secondly in course ofthe paper we give a combinatorial description of the GOE Tracy Widom law. Theexplanation for GUE is very similar. This explanation might be important forthe models where exact calculations are not available but some combinatorialstructures are present.

A new combinatorial approach for edge universality of Wigner matrices