Random Matrices, Frobenius Eigenvalues, and Monodromy

✍ Scribed by Nicholas M. Katz, Peter Sarnak

• Publisher: AMS
• Year: 1999
• Tongue: English
• Leaves: 436
• Category: Library

✦ Table of Contents

RANDOM MATRICES, FROBENIUS EIGENVALUES, AND MONODROMY
Title Page
Copyright Page
Contents
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 NrnCOR 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 co-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 T 0 (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 N (x,y) version of the Weyl integration formula
5.2. The L N (x, y) rewriting of the Weyl integration formula
5.3. Estimates for L N (x, y)
5.4. The L N (x,y) determinants in terms of the sine ratios S N (x)
5.5. Case by case summary of explicit Weyl measure formulas via S N
5.6. Unified summary of explicit Weyl measure formulas via S N
5.7. Formulas for the expectation E(Z[n,F,G(N)])
5.8. Upper bound for E(Z[n,F,G(N)])
5.9. Interlude: The sin(πx)/πx kernel and its approximations
5.10. Large N limit of E(Z[n,F,G(N)]) via the sin(πx)/πz kernel
5.11. Upper bound for the variance
Chapter 6. Tail Estimates
6.0. Review: Operators of finite 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, C(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 ν(n, U(N))
6.12. Relation between the "n'th eigenvalue" measures ν(n, U(N)) and the expected value spacing measures μ(U(N), sep. k) on a fixed U(N)
6.13. Tail estimate for µ(U(N), sep. 0) and μ(univ, sep. 0)
6.14. Multi-eigenvalue location measures, static spacing measures and expected values of several variable spacing measures on U(N)
6.15. A failure of symmetry
6.16. Offset spacing measures and their relation to multi-eigenvalue location measures on U(N)
6.17. Interlude: "Tails" of measures on R r
6.18. Tails of offset spacing measures and tails of multi-eigenvalue location measures on U(N)
6.19. Moments of offset spacing measures and of multi-eigenvalue location measures on U(N)
6.20. Multi-eigenvalue location measures for the other G(N)
Chapter 7. Large N Limits and Fredholm Determinants
7.0. Generating series for the limit measures μ(univ, sep.'s a) in several variables: absolute continuity of these measures
7.1. Interlude: Proof of Theorem 1.7.6
7.2. Generating series in the case r = 1: relation to a Fredholm determinant
7.3. The Fredholm determinants E(T, s) and E±(T, s)
7.4. Interpretation of E(T, s) and E±(T, s) as large N scaling limits of E(N, T, s) and E±(N, T, s)
7.5. Large N limits of the measures ν(n, G(N)): the measures ν(n) and ν(±, n)
7.6. Relations among the measures μ n and the measures ν(n)
7.7. Recapitulation, and concordance with the formulas in [Mehta]
7.8. Supplement: Fredholm determinants and spectral determinants, with applications to E(T, s) and E±(T, s)
7.9. Interlude: Generalities on Fredholm determinants and spectral determinants
7.10. Application to E(T, s) and E+(T, s)
7.11. Appendix: Large N limits of multi-eigenvalue location measures and of static and offset spacing measures on U(N)
Chapter 8. Several Variables
8.0. Fredholm determinants in several variables and their measure-theoretic meaning (cf. [T-W])
8.1. Measure-theoretic application to the G(N)
8.2. Several variable Fredholm determinants for the sin(πx)/πx kernel and its ± variants
8.3. Large N scaling limits
8.4. Large N limits of multi-eigenvalue location measures attached to G(N)
8.5. Relation of the limit measure Off μ(univ, offsets c) with the limit measures ν(c)
Chapter 9. Equidistribution
9.0. Preliminaries
9.1. Interlude: zeta functions in families: how lisse pure F's arise in nature
9.2. A version of Deligne's equidistribution theorem
9.3. A uniform version of Theorem 9.2.6
9.4. Interlude: Pathologies around (9.3.7.1)
9.5. Interpretation of (9.3.7.2)
9.6. Return to a uniform version of Theorem 9.2.6
9.7. Another version of Deligne's equidistribution theorem
Chapter 10. Monodromy of Families of Curves
10.0. Explicit families of curves with big G geom
10.1. Examples in odd characteristic
10.2. Examples in characteristic two
10.3. Other examples in odd characteristic
10.4. Effective constants in our examples
10.5. Universal families of curves of genus g ≥ 2
10.6. The moduli space M g,3K for g ≥ 2
10.7. Naive and intrinsic measures on USp(2g)# attached to universal families of curves
10.8. Measures on USp(2g)# attached to universal families of hyperelliptic curves
Chapter 11. Monodromy of Some Other Families
11.0. Universal families of principally polarized ahelian varieties
11.1. Other "rational over the base field” ways of rigidifying curves and abelian varieties
11.2. Automorphisms of polarized abelian varieties
11.3. Naive and intrinsic measures on USp(2g)# attached to universal families of principally polarized abelian varieties
11.4. Monodromy of universal families of hypersurfaces
11.5. Projective automorphisms of hypersurfaces
11.6. First proof of 11.5.2
11.7. Second proof of 11.5.2
11.8. A properness result
11.9. Naive and intrinsic measures on USp(prim(n, d))# (if n is odd) or on O(prim(n, d))# (if n is even) attached to universal families of smooth hypersurfaces of degree d in P n+1
11.10. Monodromy of families of Kloosterman sums
Chapter 12. GUE Discrepancies in Various Families
12.0. A basic consequence of equidistribution: axiomatics
12.1. Application to GUE discrepancies
12.2. GUE discrepancies in universal families of curves
12.4. GUE discrepancies in universal families of hypersurfaces
12.5. GUE discrepancies in families of Kloosterman sums
Chapter 13. Distribution of Low-lying Frobenius Eigenvalues in Various Families
13.0. An elementary consequence of equidistribution
13.1. Review of the measures ν(c, G(N))
13.2. Equidistribution of low-lying eigenvalues in families of curves according to the measure ν(c, USp(2g))
13.3. Equidistribution of low-lying eigenvalues in families of abelian varieties according to the measure ν(c, USp(2g))
13.4. Equidistribution of low-lying eigenvalues in families of odd-dimensional hypersurfaces according to the measure ν(c, USp(prim(n, d)))
13.5. Equidistribution of low-lying eigenvalues of Kloosterman sums in evenly many variables according to the measure ν(c, USp(2n))
13.6. Equidistribution of low-lying eigenvalues of characteristic two Kloosterman sums in oddly many variables according to the measure ν(c, SO(2n + 1))
13.7. Equidistribution of low-lying eigenvalues in families of even-dimensional hypersurfaces according to the measures ν(c, SO(prim(n, d))) and ν(c, O_ (prim(n, d)))
13.8. Passage to the large N limit
Appendix: Densities
AD.0. Overview
AD.1. Basic definitions: W n (f, A, G(N)) and W n (f, G(N))
AD.2. Large N limits: the easy case
AD.3. Relations between eigenvalue location measures and densities: generalities
AD.4. Second construction of the large N limits of the eigenvalue location measures ν(c, G(N)) for G(N) one of U(N), SO(2N + 1), USp(2N), SO(2N), O_(2N + 2), O_(2N + 1)
AD.5. Large N limits for the groups U k (N): Widom's result
AD.6. Interlude: The quantities V r (φ, U k (N)) and V r (φ, U(N))
AD.7. Interlude: Integration formulas on U(N) and on U k (N)
AD.8. Return to the proof of Widom's theorem
AD.9. End of the proof of Theorem AD.5.2
AD.10. Large N limits of the eigenvalue location measures on the U k (N)
AD.11. Computation of the measures ν(c) via low-lying eigenvalues of Kloosterman sums in oddly many variables in odd characteristic
AD.12. A variant of the one-level scaling density
Appendix: Graphs
AG.0. How the graphs were drawn, and what they show
Figure 1. Densities of µ 0, ν̅(–, 1), and the unitary Wigner surmise
Figure 2. Density of ν(–, 1)
Figure 3. Density of ν(+, 1)
Figure 4. Density of ν(1)
References
Notation Index
Subject Index