A new combinatorial approach for Tracy-Widom law of Wigner matrices
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In this paper we introduce a new combinatorial approach to analyze the trace of large powers of Wigner matrices. Our approach is motivated from the paper by Soshnikov [36]. However the counting approach is different. We start with classical word sentence approach similar to Anderson and Zeitouni [1] and take the motivation from Sinai and Soshnikov [35], Soshnikov [36] and Péché [32] to encode the words to objects similar to Dyck paths. To be precise the map takes a word to a Dyck path with some edges removed from it. Using this new counting we prove edge universality for large Wigner matrices with sub-Gaussian entries. One novelty of this approach is unlike Sinai and Soshnikov [35], Soshnikov [36] and Péché [32] we do not need to assume the entries of the matrices are symmetrically distributed around 0. We hope this method will be applicable to many other scenarios in random matrices.
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