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A combinatorial relationship between Eulerian maps and hypermaps in orientable surfaces
, 1997
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The quadrangulation conjecture for orientable surfaces
 J. COMBIN. THEORY SER. B
, 2001
"... ... The Quadrangulation Conjecture concerns the problem of finding a natural bijection of this type. Tutte's medial construction is a solution in the special case g = 0 of planar maps. We give a construction of a bijection e\Xi which both extends Tutte's medial construction to nonplanar m ..."
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... The Quadrangulation Conjecture concerns the problem of finding a natural bijection of this type. Tutte's medial construction is a solution in the special case g = 0 of planar maps. We give a construction of a bijection e\Xi which both extends Tutte's medial construction to nonplanar maps and preserves the parameter n of the Quadrangulation Conjecture. (The parameter g is not generally preserved, except
THE SINGULAR VALUES OF THE GUE (LESS IS MORE)
, 2014
"... Some properties that nominally involve the eigenvalues of Gaussian Unitary Ensemble (GUE) can instead be phrased in terms of singular values. By discarding the signs of the eigenvalues, we gain access to a surprising decomposition: the singular values of the GUE are distributed as the union of the ..."
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Some properties that nominally involve the eigenvalues of Gaussian Unitary Ensemble (GUE) can instead be phrased in terms of singular values. By discarding the signs of the eigenvalues, we gain access to a surprising decomposition: the singular values of the GUE are distributed as the union of the singular values of two independent ensembles of Laguerre type. This independence is remarkable given the well known phenomenon of eigenvalue repulsion. The structure of this decomposition reveals that several existing observations about large n limits of the GUE are in fact manifestations of phenomena that are already present for finite random matrices. We relate the semicircle law to the quartercircle law by connecting Hermite polynomials to generalized Laguerre polynomials with parameter ±1/2. Similarly, we write the absolute value of the determinant of the n×n GUE as a product n independent random variables to gain new insight into its asymptotic lognormality. The decomposition also provides a description of the distribution of the smallest singular value of the GUE, which in turn permits the study of the leading order behavior of the condition number of GUE matrices. The study is motivated by questions involving the enumeration of orientable maps, and is related to questions involving powers of complex Ginibre matrices. The inescapable conclusion of this work is that the singular values of the GUE play an unpredictably important role that had gone unnoticed for decades even though, in hindsight, so many clues had been around.