Random matrices, Frobenius eigenvalues, and monodromy:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
1999
|
| Schriftenreihe: | American Mathematical Society Colloquium publications
volume 45 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XI, 419 Seiten |
| ISBN: | 0821810170 |
Internformat
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| 100 | 1 | |a Katz, Nicholas M. |d 1943- |e Verfasser |0 (DE-588)141265558 |4 aut | |
| 245 | 1 | 0 | |a Random matrices, Frobenius eigenvalues, and monodromy |c Nicholas M. Katz ; Peter Sarnak |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c 1999 | |
| 264 | 4 | |c © 1999 | |
| 300 | |a XI, 419 Seiten | ||
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| 650 | 7 | |a L-functies |2 gtt | |
| 650 | 4 | |a Matrices aléatoires | |
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Datensatz im Suchindex
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| adam_text |
Contents
Introduction 1
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 combinatorics 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 ^ 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
vii
viii CONTENTS
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 7o(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 Km{x, y) version of the Weyl integration formula 109
5.2. The L]v(x,y) rewriting of the Weyl integration formula 116
5.3. Estimates for Ln (x, y) 117
5.4. The LN(x,y) determinants in terms of the sine ratios Sn(x) 118
5.5. Case by case summary of explicit Weyl measure formulas via Sn 120
5.6. Unified summary of explicit Weyl measure formulas via Sn 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(?ra;)/7ra; kernel and its approximations 124
5.10. Large N limit of E{Z[n, F, G(N)]) via the sin(7ra)/7ra: 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 v{n, U(N)) 169
6.12. Relation between the "n'th eigenvalue" measures u(n,U(N))
and the expected value spacing measures n(U(N), sep. k)
on a fixed U(N) 170
6.13. Tail estimate for fi(U(N), sep. 0) and /i(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
CONTENTS ix
6.17. Interlude: "Tails" of measures on Rr 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 /u(univ, sep.'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±(T, s) 211
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) 212
7.5. Large N limits of the measures u(n,G(N)): the measures v(n)
andu(±,n) 215
7.6. Relations among the measures fin and the measures u(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±(T,s) 229
7.9. Interlude: Generalities on Fredholm determinants and spectral
determinants 232
7.10. Application to E(T,s) and E±(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 sm(nx)/irx kernel
and its ± 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 /x(univ, offsets c) with the limit
measures v(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
x CONTENTS
Chapter 10. Monodromy of Families of Curves 293
10.0. Explicit families of curves with big Ggeom 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 2 304
10.6. The moduli space M9,3k for g 2 307
10.7. Naive and intrinsic measures on USp(2g)# attached to universal
families of curves 315
10.8. Measures on USp(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 USp(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 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 Pâ„¢+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 i/(c, G(N)) 363
13.2. Equidistribution of low lying eigenvalues in families of curves
according to the measure v(c, USp(2g)) 364
13.3. Equidistribution of low lying eigenvalues in families of abelian
varieties according to the measure i/(c, USp{2g)) 365
13.4. Equidistribution of low lying eigenvalues in families of odd
dimensional hypersurfaces according to the measure
i/(c, USp(pnm(n, d))) 366
CONTENTS xi
13.5. Equidistribution of low lying eigenvalues of Kloosterman sums in
evenly many variables according to the measure v(c, USp{2n)) 367
13.6. Equidistribution of low lying eigenvalues of characteristic two
Kloosterman sums in oddly many variables according to the
measure u(c, S0(2n + 1)) 367
13.7. Equidistribution of low lying eigenvalues in families of even
dimensional hypersurfaces according to the measures
v(c, SO(prim(n, d))) and i/(c, O_ (prim(n, d))) 368
13.8. Passage to the large N limit 369
Appendix: Densities 373
AD.O. Overview 373
AD.l. Basic definitions: Wn(f, A, G(N)) and Wn(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 TV limits of the eigenvalue
location measures i/(c,G(N)) for G(N) one of U(N),
SO(2N + 1), USp(2N), SO(2N), O_(27V + 2), O_(27V + 1) 381
AD.5. Large N limits for the groups Uk(N): Widom's result 385
AD.6. Interlude: The quantities Vr( p, Uk(N)) and Vr{y, U(N)) 386
AD.7. Interlude: Integration formulas on U(N) and on Uk(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 iV limits of the eigenvalue location measures on the
Uk(N) 401
AD.11. Computation of the measures u(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
Appendix: Graphs 411
AG.0. How the graphs were drawn, and what they show 411
Figure 1 413
Figure 2 414
Figure 3 415
Figure 4 416
References 417 |
| any_adam_object | 1 |
| author | Katz, Nicholas M. 1943- Sarnak, Peter 1953- |
| author_GND | (DE-588)141265558 (DE-588)102958365X |
| author_facet | Katz, Nicholas M. 1943- Sarnak, Peter 1953- |
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| ctrlnum | (OCoLC)39122834 (DE-599)BVBBV012429054 |
| dewey-full | 512.9434 515/.56 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra 515 - Analysis |
| dewey-raw | 512.9434 515/.56 |
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| dewey-sort | 3512.9434 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV012429054 |
| illustrated | Not Illustrated |
| indexdate | 2025-03-03T13:02:17Z |
| institution | BVB |
| isbn | 0821810170 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008434382 |
| oclc_num | 39122834 |
| open_access_boolean | |
| owner | DE-739 DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-20 DE-11 DE-703 DE-83 |
| owner_facet | DE-739 DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-20 DE-11 DE-703 DE-83 |
| physical | XI, 419 Seiten |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | American Mathematical Society Colloquium publications |
| series2 | American Mathematical Society Colloquium publications |
| spelling | Katz, Nicholas M. 1943- Verfasser (DE-588)141265558 aut Random matrices, Frobenius eigenvalues, and monodromy Nicholas M. Katz ; Peter Sarnak Providence, Rhode Island American Mathematical Society 1999 © 1999 XI, 419 Seiten txt rdacontent n rdamedia nc rdacarrier American Mathematical Society Colloquium publications volume 45 Fonctions L Fonctions l ram Fonctions zeta ram Fonctions zêta Groupes de monodromie Groupes de monodromie ram L-functies gtt Matrices aléatoires Matrices aléatoires ram Matrices gtt Monodromie gtt Théorèmes des limites (Théorie des probabilités) ram Théorèmes limites (Théorie des probabilités) Zeta-functies gtt Functions, Zeta L-functions Limit theorems (Probability theory) Monodromy groups Random matrices Stochastische Matrix (DE-588)4057624-3 gnd rswk-swf Monodromiegruppe (DE-588)4194644-3 gnd rswk-swf Zetafunktion (DE-588)4190764-4 gnd rswk-swf Grenzwertsatz (DE-588)4158163-5 gnd rswk-swf L-Funktion (DE-588)4137026-0 gnd rswk-swf Zetafunktion (DE-588)4190764-4 s L-Funktion (DE-588)4137026-0 s Stochastische Matrix (DE-588)4057624-3 s Grenzwertsatz (DE-588)4158163-5 s Monodromiegruppe (DE-588)4194644-3 s DE-604 Sarnak, Peter 1953- Verfasser (DE-588)102958365X aut Elektronische Reproduktion 978-1-4704-3191-4 American Mathematical Society Colloquium publications volume 45 (DE-604)BV035417609 45 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008434382&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Katz, Nicholas M. 1943- Sarnak, Peter 1953- Random matrices, Frobenius eigenvalues, and monodromy American Mathematical Society Colloquium publications Fonctions L Fonctions l ram Fonctions zeta ram Fonctions zêta Groupes de monodromie Groupes de monodromie ram L-functies gtt Matrices aléatoires Matrices aléatoires ram Matrices gtt Monodromie gtt Théorèmes des limites (Théorie des probabilités) ram Théorèmes limites (Théorie des probabilités) Zeta-functies gtt Functions, Zeta L-functions Limit theorems (Probability theory) Monodromy groups Random matrices Stochastische Matrix (DE-588)4057624-3 gnd Monodromiegruppe (DE-588)4194644-3 gnd Zetafunktion (DE-588)4190764-4 gnd Grenzwertsatz (DE-588)4158163-5 gnd L-Funktion (DE-588)4137026-0 gnd |
| subject_GND | (DE-588)4057624-3 (DE-588)4194644-3 (DE-588)4190764-4 (DE-588)4158163-5 (DE-588)4137026-0 |
| title | Random matrices, Frobenius eigenvalues, and monodromy |
| title_auth | Random matrices, Frobenius eigenvalues, and monodromy |
| title_exact_search | Random matrices, Frobenius eigenvalues, and monodromy |
| title_full | Random matrices, Frobenius eigenvalues, and monodromy Nicholas M. Katz ; Peter Sarnak |
| title_fullStr | Random matrices, Frobenius eigenvalues, and monodromy Nicholas M. Katz ; Peter Sarnak |
| title_full_unstemmed | Random matrices, Frobenius eigenvalues, and monodromy Nicholas M. Katz ; Peter Sarnak |
| title_short | Random matrices, Frobenius eigenvalues, and monodromy |
| title_sort | random matrices frobenius eigenvalues and monodromy |
| topic | Fonctions L Fonctions l ram Fonctions zeta ram Fonctions zêta Groupes de monodromie Groupes de monodromie ram L-functies gtt Matrices aléatoires Matrices aléatoires ram Matrices gtt Monodromie gtt Théorèmes des limites (Théorie des probabilités) ram Théorèmes limites (Théorie des probabilités) Zeta-functies gtt Functions, Zeta L-functions Limit theorems (Probability theory) Monodromy groups Random matrices Stochastische Matrix (DE-588)4057624-3 gnd Monodromiegruppe (DE-588)4194644-3 gnd Zetafunktion (DE-588)4190764-4 gnd Grenzwertsatz (DE-588)4158163-5 gnd L-Funktion (DE-588)4137026-0 gnd |
| topic_facet | Fonctions L Fonctions l Fonctions zeta Fonctions zêta Groupes de monodromie L-functies Matrices aléatoires Matrices Monodromie Théorèmes des limites (Théorie des probabilités) Théorèmes limites (Théorie des probabilités) Zeta-functies Functions, Zeta L-functions Limit theorems (Probability theory) Monodromy groups Random matrices Stochastische Matrix Monodromiegruppe Zetafunktion Grenzwertsatz L-Funktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008434382&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035417609 |
| work_keys_str_mv | AT katznicholasm randommatricesfrobeniuseigenvaluesandmonodromy AT sarnakpeter randommatricesfrobeniuseigenvaluesandmonodromy |