目录
Introduction
Chapter 1.Statements of the Main Results
1.0.Measures attached to spacings of eigenvalues
1.1.Expected values of spacing measures
1.2.Existence, universality and discrepancy theorems for limits of expected values of spacing measures: the three main theorems
1.3.Interlude: A functorial property of Haar measure on compact groups
1.4.Application: Slight economies in proving Theorems 1.2.3 and 1.2.6
1.5.Application: An extension of Theorem 1.2.6
1.6.Corollaries of Theorem 1.5.3
1.7.Another generalization of Theorem 1.2.6
1.8.Appendix: Continuity properties of "the i'th eigenvalue" as a function on U(N)
Chapter 2.Reformulation of the Main Results
2.0."Naive" versions of the spacing measures
2.1.Existence, universality and discrepancy theorems for limits of expected values of naive spacing measures: the main theorems bis
2.2.Deduction of Theorems 1.2.1, 1.2.3 and 1.2.6 from their bis versions
2.3.The combinatorics of spacings of finitely many points on a line: first discussion
2.4.The combinatorics of spacings of finitely many points on a line: second discussion
2.5.The combinatorics of spacings of finitely many points on a line: third discussion: variations on Sep(a)and Clump(a)
2.6.The combinatorics of spacings of finitely many points of a line: fourth discussion: another variation on Clump(a)
2.7.Relation to naive spacing measures on G(N): Int, Cor and TCor
2.8.Expected value measures via INT and COR and TCOR
2.9.The axiomatics of proving Theorem 2.1.3
2.10.Large N COR limits and formulas for limit measures
2.11.Appendix: Direct image properties of the spacing measures
Chapter 3.Reduction Steps in Proving the Main Theorems
3.0.The axiomatics of proving Theorems 2.1.3 and 2.1.5
3.1.A mild generalization of Theorem 2.1.5: the φ-version
3.2.M-grid discrepancy, L cutoff and dependence on the choice of coordinates
3.3.A weak form of Theorem 3.1.6
3.4.Conclusion of the axiomatic proof of Theorem 3.1.6
3.5.Making explicit the constants
Chapter 4.Test Functions
4.0.The classes T(n) and To(n) of test functions
4.1.The random variable Z[n, F, G(N)]on G(N) attached to a function F in T(n)
4.2.Estimates for the expectation E(Z[n, F, G(N)]) and variance Var(Z[n, F, G(N)])of Z[n, F, G(N)]on G(N)
Chapter 5.Haar Measure
5.0.The Weyl integration formula for the various G(N)
5.1.The K(x, y)version of the Weyl integration formula
5.2.The L(r, y) rewriting of the Weyl integration formula
5.3.Estimates for Ly(x, y)
5.4.The Lv(x, y) determinants in terms of the sine ratios Sv(x)
5.5.Case by case summary of explicit Weyl measure formulas via Sv
5.6.Unified summary of explicit Weyl measure formulas via Sy
5.7.Formulas for the expectation B(Z[n, F, G(N)])
5.8.Upper bound for E(Z[n, F, G(N)])
5.9.Interlude: The sin(rx)/rx kernel and its approximations
5.10.Large N limit of E(ZIn, F, G(N)]) via the sin(rx)/rx kernel
5.11.Upper bound for the variance
Chapter 6.Tail Estimates
6.0.Review: Operators of inite rank and their (reversed)characteristic polynomials
6.1.Integral operators of finite rank: a basic compatibility between spectral and Fredholm determinants
6.2.An integration formula
6.3.Integrals of determinants over G(N)as Fredholm determinants
6.4.A new special case: O_(2N +1)
6.5.Interlude: A determinant-trace inequality
6.6.First application of the determinant-trace inequality
6.7.Application: Estimates for the numbers eigen(n, s, G(N))
6.8.Some curious identities among various eigen(n, s, G(N))
6.9.Normalized "n'th eigenvalue" measures attached to G(N)
6.10.Interlude: Sharper upper bounds for eigen(0, s, SO(2N), for eigen(0, s, O_(2N +1)), and for eigen(0, s, U(N))
6.11.A more symmetric construction of the "n'th eigenvalue" measures v(n, U(N))
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