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Parametric RMT , discrete symmetries, and cross-correlations between L -functions. Igor Smolyarenko Cavendish Laboratory. Collaborators: B. D. Simons, B. Conrey. July 12, 2004.

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Parametric rmt discrete symmetries and cross correlations between l functions

Parametric RMT, discrete symmetries, and cross-correlations between L-functions

Igor Smolyarenko

Cavendish Laboratory

Collaborators: B. D. Simons, B. Conrey

July 12, 2004


Parametric rmt discrete symmetries and cross correlations between l functions

“…the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.” (S. Banach)

  • Pair correlations of zeta zeros: GUE and beyond

  • Analogy with dynamical systems

  • Cross-correlations between different chaotic spectra

  • Cross-correlations between zeros of different(Dirichlet)

  • L-functions

  • Analogy: Dynamical systems with discrete symmetries

  • Conclusions: conjectures and fantasies


Pair correlations of zeros
Pair correlations of zeros theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.”

  • Montgomery ‘73:

universal GUE behavior

(

)

As T→ 1

Data: M. Rubinstein

How much does the universal GUE formula tell us about

the (conjectured) underlying “Riemann operator”?

Q:

Not much, really… However,…

A:


Beyond gue aim is nothing but the movement is everything
Beyond GUE: theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.” “…aim… is nothing , but the movement is everything"

Non-universal (lower order in

) features

of the pair correlation function contain a lot of information

  • Berry’86-’91; Keating ’93; Bogomolny, Keating ’96; Berry, Keating ’98-’99:

and similarly for any Dirichlet L-function with

How can this information be extracted?


Poles and zeros
Poles and zeros theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.”

  • The pole of zeta at → 1

What about the rest of the structure of (1+i)?

  • Low-lying critical (+ trivial) zeros turn out to be connected

    to the classical analogue of “Riemann dynamics”

Discussion of the poles and zeros;

the meaning of leading vs. subleading terms


Parametric rmt discrete symmetries and cross correlations between l functions

Number theory theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.” vs. chaotic dynamics

Classical spectral

determinant

Andreev, Altshuler, Agam

via supersymmetric

nonlinear -model

Quantum mechanics of

classically chaotic systems:

spectral determinants

and their derivatives

Statistics

of (E)

regularized modes of

(Perron-Frobenius spectrum)

via periodic orbit

theory

Berry, Bogomolny, Keating

Dynamic

zeta-function

Periodic orbits

Prime numbers

Dictionary:

Statistics of zeros

Number theory:

zeros of (1/2+i)and L(1/2+i, )

(1+i)


Generic chaotic dynamical systems periodic orbits and perron frobenius modes
Generic chaotic dynamical systems: theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.” periodic orbits and Perron-Frobenius modes

  • Number theory: zeros, arithmetic information, but the underlying

    operators are not known

  • Chaotic dynamics: operator (Hamiltonian) is known,

    but not the statistics of periodic orbits

Correlation functions for chaotic spectra (under simplifying assumptions):

(Bogomolny, Keating, ’96)

Cf.:

Z(i) – analogue of the -function on the Re s =1 line

(1-i) becomes a complementary source of information about “Riemann dynamics”


Parametric rmt discrete symmetries and cross correlations between l functions

What else can be learned? theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.”

  • In Random Matrix Theory and in theory of dynamical systems

    information can be extracted from parametric correlations

  • Simplest: H→H+V(X)

X

Spectrum ofH´=H+V

Spectrum ofH

  • If spectrum of H exhibits GUE

    (or GOE, etc.) statistics, spectra of

    H and H´ togetherexhibit “descendant”

    parametric statistics

Under certain conditions

on V (it has to be small

either in magnitude or

in rank):

Inverse problem: given two chaotic spectra,

parametric correlations can be used to extract

information about V=H-H


Can pairs of l functions be viewed as related chaotic spectra
Can pairs of theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.” L-functionsbe viewed as related chaotic spectra?

Bogomolny, Leboeuf, ’94; Rudnick and Sarnak, ’98:

No cross-correlations to the leading order in

Using Rubinstein’s data on zeros of Dirichlet L-functions:

Cross-correlation function between L(s,8)and L(s,-8):

R11()

1.2

1.0

0.8


Examples of parametric spectral statistics
Examples of parametric spectral statistics theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.”

(*)

R11(x≈0.2)

R2

-- norm of V

Beyond the leading Parametric GUE terms:

Perron-Frobenius

modes

Analogue of the diagonal contribution

(*) Simons, Altshuler, ‘93


Cross correlations between l function zeros analytical results
Cross-correlations between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.” L-function zeros:analytical results

Diagonal contribution:

Off-diagonal contribution:

Convergent product

over primes

Being computed

L(1-i) is regular at 1 – consistent with the absence

of a leading term


Dynamical systems with discrete symmetries
Dynamical systems with discrete symmetries theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.”

Consider the simplest possible discrete group

If H is invariant under G:

then

Spectrum can be split into two parts, corresponding to

symmetric

and antisymmetric

eigenfunctions


Discrete symmetries beyond parametric gue
Discrete symmetries: Beyond Parametric GUE theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.”

Consider two irreducible representations 1 and 2 of G

Define P1 and P2 – projection operators onto subspaces which

transform according to 1 and 2

The cross-correlation between the spectra of P1HP1 and P2HP2

are given by the analog of the dynamical zeta-function formed

by projecting Perron-Frobenius operator onto subspace of the

phase space which transforms according to

!!


Parametric rmt discrete symmetries and cross correlations between l functions

Number theory theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.” vs. chaotic dynamics II:

Cross-correlations

Classical spectral

determinant

via supersymmetric

nonlinear -model

Quantum mechanics of

classically chaotic systems:

spectral determinants

and their derivatives

Correlations

between

1(E) and 2(E+)

regularized modes of

via periodic orbit

theory

“Dynamic

L-function”

Periodic orbits

Prime numbers

Cross-correlations of zeros

Number theory:

zeros of L(1/2+i,1)and L(1/2+i, 2)

L(1-i,12)


The incomplete to do list
The (incomplete?) “to do” list theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.”

0. Finish the calculation and compare to numerical data

  • Find the correspondence between

and the eigenvalues of

information on analogues of ?

  • Generalize to L-functions of degree > 1