Random matrices, Frobenius eigenvalues, and monodromy / Nicholas M. Katz, Peter Sarnak.

Format:

Book

Author/Creator:

Katz, Nicholas M., 1943-

Contributor:

Sarnak, Peter.

Series:

Colloquium publications (American Mathematical Society) ; v. 45.

Colloquium publications / American Mathematical Society, 0065-9258 ; v. 45

Language:

English

Subjects (All):

Functions, Zeta.

L-functions.

Random matrices.

Limit theorems (Probability theory).

Monodromy groups.

Physical Description:

xi, 419 pages : illustrations ; 26 cm.

Place of Publication:

Providence, RI : American Mathematical Society, [1999]

Contents:

Chapter 1. Statements of the Main Results 17

1.0. Measures attached to spacings of eigenvalues 17

1.1. Expected values of spacing measures 23

1.2. Existence, universality and discrepancy theorems for limits of expected values of spacing measures: the three main theorems 24

1.3. Interlude: A functorial property of Haar measure on compact groups 25

1.4. Application: Slight economies in proving Theorems 1.2.3 and 1.2.6 25

1.5. Application: An extension of Theorem 1.2.6 26

1.6. Corollaries of Theorem 1.5.3 28

1.7. Another generalization of Theorem 1.2.6 30

1.8. Appendix Continuity properties of "the i'th eigenvalue" as a function on U(N) 32

Chapter 2. Reformulation of the Main Results 35

2.0. "Naive" versions of the spacing measures 35

2.1. Existence, universality and discrepancy theorems for limits of expected values of naive spacing measures: the main theorems bis 37

2.2. Deduction of Theorems 1.2.1, 1.2.3 and 1.2.6 from their bis versions 38

2.3. The combinatorics of spacings of finitely many points on a line: first discussion 42

2.4. The combinatorics of spacings of finitely many points on a line: second discussion 45

2.5. The combinatorics of spacings of finitely many points on a line: third discussion: variations on Sep(a) and Clump(a) 49

2.6. The combinatiorics of spacings of finitely many points of a line: fourth discussion: another variation on Clump(a) 54

2.7. Relation to naive spacing measures on G(N): Int, Cor and TCor 54

2.8. Expected value measures via INT and COR and TCOR 57

2.9. The axiomatics of proving Theorem 2.1.3 58

2.10. Large N COR limits and formulas for limit measures 63

2.11. Appendix Direct image properties of the spacing measures 65

Chapter 3. Reduction Steps in Proving the Main Theorems 73

3.0. The axiomatics of proving Theorems 2.1.3 and 2.1.5 73

3.1. A mild generalization of Theorem 2.1.5: the [open phi]-version 74

3.2. M-grid discrepancy, L cutoff and dependence on the choice of coordinates 77

3.3. A weak form of Theorem 3.1.6 89

3.4. Conclusion of the axiomatic proof of Theorem 3.1.6 90

3.5. Making explicit the constants 98

Chapter 4. Test Functions 101

4.0. The classes T(n) and T[subscript 0](n) of test functions 101

4.1. The random variable Z[n, F, G(N)] on G(N) attached to a function F in T(n) 103

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) 104

Chapter 5. Haar Measure 107

5.0. The Weyl integration formula for the various G(N) 107

5.1. The K[subscript N](x, y) version of the Weyl integration formula 109

5.2. The L[subscript N](x, y) rewriting of the Weyl integration formula 116

5.3. Estimates for L[subscript N](x, y) 117

5.4. The L[subscript N](x,y) determinants in terms of the sine ratios S[subscript N](x) 118

5.5. Case by case summary of explicit Weyl measure formulas via S[subscript N] 120

5.6. Unified summary of explicit Weyl measure formulas via S[subscript N] 121

5.7. Formulas for the expectation E(Z[n, F, G(N)]) 122

5.8. Upper bound for E(Z[n, F, G(N)]) 123

5.9. Interlude: The sin([pi]x)/[pi]x kernel and its approximations 124

5.10. Large N limit of E(Z[n, F, G(N)]) via the sin ([pi]x)/[pi]x kernel 127

5.11. Upper bound for the variance 133

Chapter 6. Tail Estimates 141

6.0. Review: Operators of finite rank and their (reversed) characteristic polynomials 141

6.1. Integral operators of finite rank: a basic compatibility between spectral and Fredholm determinants 141

6.2. An integration formula 143

6.3. Integrals of determinants over G(N) as Fredholm determinants 145

6.4. A new special case: O_(2N + 1) 151

6.5. Interlude: A determinant-trace inequality 154

6.6. First application of the determinant-trace inequality 156

6.7. Application: Estimates for the numbers eigen(n, s, G(N)) 159

6.8. Some curious identities among various eigen(n, s, G(N)) 162

6.9. Normalized "n'th eigenvalue" measures attached to G(N) 163

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)) 166

6.11. A more symmetric construction of the "n'th eigenvalue" measures [nu](n, U(N)) 169

6.12. Relation between the "n'th eigenvalue" measures [nu](n, U(N)) and the expected value spacing measures [mu](U(N), sep. k) on a fixed U(N) 170

6.13. Tail estimate for [mu](U(N), sep. 0) and [mu](univ, sep. 0) 174

6.14. Multi-eigenvalue location measures, static spacing measures and expected values of several variable spacing measures on U(N) 175

6.15. A failure of symmetry 183

6.16. Offset spacing measures and their relation to multi-eigenvalue location measures on U(N) 185

6.17. Interlude: "Tails" of measures on R[superscript r] 189

6.18. Tails of offset spacing measures and tails of multi-eigenvalue location measures on U(N) 192

6.19. Moments of offset spacing measures and of multi-eigenvalue location measures on U(N) 194

6.20. Multi-eigenvalue location measures for the other G(N) 195

Chapter 7. Large N Limits and Fredholm Determinants 197

7.0. Generating series for the limit measures [mu](univ, spe.'s a) in several variables: absolute continuity of these measures 197

7.1. Interlude: Proof of Theorem 1.7.6 205

7.2. Generating series in the case r = 1: relation to a Fredholm determinant 208

7.3. The Fredholm determinants E(T, s) and E[plus or minus](T, s) 211

7.4. Interpretation of E(T, s) and E[plus or minus](T, s) as large N scaling limits of E(N, T, s) and E[plus or minus](N, T, s) 212

7.5. Large N limits of the measures [nu](n, G(N)): the measures [nu](n) and [nu]([plus or minus], n) 215

7.6. Relations among the measures [mu][subscript n] and the measures [nu](n) 225

7.7. Recapitulation, and concordance with the formulas in [Mehta] 228

7.8. Supplement: Fredholm determinants and spectral determinants, with applications to E(T, s) and E[plus or minus](T, s) 229

7.9. Interlude: Generalities on Fredholm determinants and spectral determinants 232

7.10. Application to E(T, s) and E[plus or minus](T, s) 235

7.11. Appendix Large N limits of multi-eigenvalue location measures and of static and offset spacing measures on U(N) 235

Chapter 8. Several Variables 245

8.0. Fredholm determinants in several variables and their measure-theoretic meaning (cf.

[T-W]) 245

8.1. Measure-theoretic application to the G(N) 248

8.2. Several variable Fredholm determinants for the sin ([pi]x)/[pi]x kernel and its [plus or minus] variants 249

8.3. Large N scaling limits 251

8.4. Large N limits of multi-eigenvalue location measures attached to G(N) 257

8.5. Relation of the limit measure Off [mu](univ, offsets c) with the limit measures [nu](c) 263

Chapter 9. Equidistribution 267

9.0. Preliminaries 267

9.1. Interlude: zeta functions in families: how lisse pure F's arise in nature 270

9.2. A version of Deligne's equidistribution theorem 275

9.3. A uniform version of Theorem 9.2.6 279

9.4. Interlude: Pathologies around (9.3.7.1) 280

9.5. Interpretation of (9.3.7.2) 283

9.6. Return to a uniform version of Theorem 9.2.6 283

9.7. Another version of Deligne's equidistribution theorem 287

Chapter 10. Monodromy of Families of Curves 293

10.0. Explicit families of curves with big G[subscript geom] 293

10.1. Examples in odd characteristic 293

10.2. Examples in characteristic two 301

10.3. Other examples in odd characteristic 302

10.4. Effective constants in our examples 303

10.5. Universal families of curves of genus g [greater than or equal] 2 304

10.6. The moduli space M[subscript g,3K] for g [greater than or equal] 2 307

10.7. Naive and intrinsic measures on U Sp(2g) # attached to universal families of curves 315

10.8. Measures on U Sp(2g) # attached to universal families of hyperelliptic curves 320

Chapter 11. Monodromy of Some Other Families 323

11.0. Universal families of principally polarized abelian varieties 323

11.1. Other "rational over the base field" ways of rigidifying curves and abelian varieties 324

11.2. Automorphisms of polarized abelian varieties 327

11.3. Naive and intrinsic measures on U Sp(2g) # attached to universal families of principally polarized abelian varieties 328

11.4. Monodromy of universal families of hypersurfaces 331

11.5. Projective automorphisms of hypersurfaces 335

11.6. First proof of 11.5.2 335

11.7. Second proof of 11.5.2 337

11.8. A properness result 342

11.9. Naive and intrinsic measures on U Sp(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[superscript n+1] 346

11.10. Monodromy of families of Kloosterman sums 347

Chapter 12. GUE Discrepancies in Various Families 351

12.0. A basic consequence of equidistribution: axiomatics 351

12.1. Application to GUE discrepancies 352

12.2. GUE discrepancies in universal families of curves 353

12.3. GUE discrepancies in universal families of abelian varieties 355

12.4. GUE discrepancies in universal families of hypersurfaces 356

12.5. GUE discrepancies in families of Kloosterman sums 358

Chapter 13. Distribution of Low-lying Frobenius Eigenvalues in Various Families 361

13.0. An elementary consequence of equidistribution 361

13.1. Review of the measures [nu](c, G(N)) 363

13.2. Equidistribution of low-lying eigenvalues in families of curves according to the measure [nu](c, U Sp(2g)) 364

13.3. Equidistribution of low-lying eigenvalues in families of abelian varieties according to the measure [nu](c, U Sp(2g)) 365

13.4. Equidistribution of low-lying eigenvalues in families of odd-dimensional hypersurfaces according to the measure [nu](c, U Sp(prim(n,d))) 366

13.5. Equidistribution of low-lying eigenvalues of Kloosterman sums in evenly many variables according to the measure [nu](c, U Sp(2n)) 367

13.6. Equidistribution of low-lying eigenvalues of characteristic two Kloosterman sums in oddly many variables according to the measure [nu](c, SO(2n+1)) 367

13.7. Equidistribution of low-lying eigenvalues in families of even-dimensional hypersurfaces according to the measures [nu](c, SO(prim(n,d))) and [nu](c,O_(prim(n,d))) 368

13.8. Passage to the large N limit 369

Appendix Densities 373

AD.1. Basic definitions: W[subscript n](f, A, G(N)) and W[subscript n](f, G(N)) 373

AD.2. Large N limits: the easy case 374

AD.3. Relations between eigenvalue location measures and densities: generalities 378

AD.4. Second construction of the large N limits of the eigenvalue location measures [nu](c, G(N)) for G(N) one of U(N), SO(2N + 1), U Sp(2N), SO(2N), O_(2N + 2), O_(2N + 1) 381

AD.5. Large N limits for the groups U[subscript k](N): Widom's result 385

AD.6. Interlude: The quantities V[subscript r]([open phi], U[subscript k](N)) and V[subscript r]([open phi], U (N)) 386

AD.7. Interlude: Integration formulas on U (N) and on U[subscript k](N) 390

AD.8. Return to the proof of Widom's theorem 392

AD.9. End of the proof of Theorem AD.5.2 399

AD.10. Large N limits of the eigenvalue location measures on the U[subscript k](N) 401

AD.11. Computation of the measures v(c) via low-lying eigenvalues of Kloosterman sums in oddly many variables in odd characteristic 403

AD.12. A variant of the one-level scaling density 405

AG.0. How the graphs were drawn, and what they show 411.

Notes:

Includes bibliographical references.

ISBN:

0821810170

OCLC:

39122834