The Singular Values of the GOE

As a unifying framework for examining several properties that nominally involve eigenvalues, we present a particular structure of the singular values of the Gaussian orthogonal ensemble (GOE): the even-location singular values are distributed as the positive eigenvalues of a Gaussian ensemble with chiral unitary symmetry (anti-GUE), while the odd-location singular values, conditioned on the even-location ones, can be algebraically transformed into a set of independent $\chi$-distributed random variables. We discuss three applications of this structure: first, there is a pair of bidiagonal square matrices, whose singular values are jointly distributed as the even- and odd-location ones of the GOE; second, the magnitude of the determinant of the GOE is distributed as a product of simple independent random variables; third, on symmetric intervals, the gap probabilities of the GOE can be expressed in terms of the Laguerre unitary ensemble (LUE). We work specifically with matrices of finite order, but by passing to a large matrix limit, we also obtain new insight into asymptotic properties such as the central limit theorem of the determinant or the gap probabilities in the bulk-scaling limit. The analysis in this paper avoids much of the technical machinery (e.g. Pfaffians, skew-orthogonal polynomials, martingales, Meijer $G$-function, etc.) that was previously used to analyze some of the applications.Comment: Introduction extended, typos corrected, reference added. 31 pages, 1 figur

The singular values of the GUE (less is more)

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 quarter-circle 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 x n GUE as a product n independent random variables to gain new insight into its asymptotic log-normality. 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.National Science Foundation (U.S.) (Grant DMS–1035400)National Science Foundation (U.S.) (Grant DMS–1016125

The combinatorics of the Jack parameter and the genus series for topological maps

Informally, a rooted map is a topologically pointed embedding of a graph in a surface. This thesis examines two problems in the enumerative theory of rooted maps. The b-Conjecture, due to Goulden and Jackson, predicts that structural similarities between the generating series for rooted orientable maps with respect to vertex-degree sequence, face-degree sequence, and number of edges, and the corresponding generating series for rooted locally orientable maps, can be explained by a unified enumerative theory. Both series specialize M(x,y,z;b), a series defined algebraically in terms of Jack symmetric functions, and the unified theory should be based on the existence of an appropriate integer valued invariant of rooted maps with respect to which M(x,y,z;b) is the generating series for locally orientable maps. The conjectured invariant should take the value zero when evaluated on orientable maps, and should take positive values when evaluated on non-orientable maps, but since it must also depend on rooting, it cannot be directly related to genus. A new family of candidate invariants, η, is described recursively in terms of root-edge deletion. Both the generating series for rooted maps with respect to η and an appropriate specialization of M satisfy the same differential equation with a unique solution. This shows that η gives the appropriate enumerative theory when vertex degrees are ignored, which is precisely the setting required by Goulden, Harer, and Jackson for an application to algebraic geometry. A functional equation satisfied by M and the existence of a bijection between rooted maps on the torus and a restricted set of rooted maps on the Klein bottle show that η has additional structural properties that are required of the conjectured invariant. The q-Conjecture, due to Jackson and Visentin, posits a natural combinatorial explanation, for a functional relationship between a generating series for rooted orientable maps and the corresponding generating series for 4-regular rooted orientable maps. The explanation should take the form of a bijection, ϕ, between appropriately decorated rooted orientable maps and 4-regular rooted orientable maps, and its restriction to undecorated maps is expected to be related to the medial construction. Previous attempts to identify ϕ have suffered from the fact that the existing derivations of the functional relationship involve inherently non-combinatorial steps, but the techniques used to analyze η suggest the possibility of a new derivation of the relationship that may be more suitable to combinatorial analysis. An examination of automorphisms that must be induced by ϕ gives evidence for a refinement of the functional relationship, and this leads to a more combinatorially refined conjecture. The refined conjecture is then reformulated algebraically so that its predictions can be tested numerically

Social comparisons in oligopsony

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Pengiuns

Children's non-fiction book about penguins. Several types of penguins are described with their eating habits, natural habitats, physical desriptions, and family structure