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Random matrices, differential operators and carousels matyd/BMC/slides/Valko - Random... · PDF file Random matrices, di erential operators and carousels Benedek Valko (University

Feb 18, 2020

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  • Random matrices, differential operators and carousels

    Benedek Valkó (University of Wisconsin – Madison)

    joint with B. Virág (Toronto)

    March 24, 2016

  • Basic question of RMT:

    What can we say about the spectrum of a large random matrix?

    -60 -40 -20 0 20 40 60

    5

    10

    15

    20

    25

    30

    35

    Ln

    b HLn - aL

    global local

    In this talk: local picture (point process limits)

  • Basic question of RMT:

    What can we say about the spectrum of a large random matrix?

    -60 -40 -20 0 20 40 60

    5

    10

    15

    20

    25

    30

    35

    Ln

    b HLn - aL

    global local

    In this talk: local picture (point process limits)

  • Basic question of RMT:

    What can we say about the spectrum of a large random matrix?

    -60 -40 -20 0 20 40 60

    5

    10

    15

    20

    25

    30

    35

    Ln

    b HLn - aL

    global local

    In this talk: local picture (point process limits)

  • A classical example: Gaussian Unitary Ensemble

    M = A+A ∗

    √ 2

    , A is n × n with iid complex std normal.

    Global picture: Wigner semicircle law -60 -40 -20 0 20 40 60 5

    10

    15

    20

    25

    30

    35

    Local picture: point process limit in the bulk and near the edge

    (Bulk: Dyson-Gaudin-Mehta, edge: Tracy-Widom)

    The limit processes are characterized by their joint intensity functions.

    Roughly: what is the probability of finding points near x1, . . . , xn

  • A classical example: Gaussian Unitary Ensemble

    M = A+A ∗

    √ 2

    , A is n × n with iid complex std normal.

    Global picture: Wigner semicircle law

    -60 -40 -20 0 20 40 60

    5

    10

    15

    20

    25

    30

    35

    Local picture: point process limit in the bulk and near the edge

    (Bulk: Dyson-Gaudin-Mehta, edge: Tracy-Widom)

    The limit processes are characterized by their joint intensity functions.

    Roughly: what is the probability of finding points near x1, . . . , xn

  • A classical example: Gaussian Unitary Ensemble

    M = A+A ∗

    √ 2

    , A is n × n with iid complex std normal.

    Global picture: Wigner semicircle law -60 -40 -20 0 20 40 60 5

    10

    15

    20

    25

    30

    35

    Local picture: point process limit in the bulk and near the edge

    (Bulk: Dyson-Gaudin-Mehta, edge: Tracy-Widom)

    The limit processes are characterized by their joint intensity functions.

    Roughly: what is the probability of finding points near x1, . . . , xn

  • A classical example: Gaussian Unitary Ensemble

    M = A+A ∗

    √ 2

    , A is n × n with iid complex std normal.

    Global picture: Wigner semicircle law -60 -40 -20 0 20 40 60 5

    10

    15

    20

    25

    30

    35

    Local picture: point process limit in the bulk and near the edge

    (Bulk: Dyson-Gaudin-Mehta, edge: Tracy-Widom)

    The limit processes are characterized by their joint intensity functions.

    Roughly: what is the probability of finding points near x1, . . . , xn

  • A classical example: Gaussian Unitary Ensemble

    M = A+A ∗

    √ 2

    , A is n × n with iid complex std normal.

    Global picture: Wigner semicircle law -60 -40 -20 0 20 40 60 5

    10

    15

    20

    25

    30

    35

    Local picture: point process limit in the bulk and near the edge

    (Bulk: Dyson-Gaudin-Mehta, edge: Tracy-Widom)

    The limit processes are characterized by their joint intensity functions.

    Roughly: what is the probability of finding points near x1, . . . , xn

  • Point process limit

    Ln

    b HLn - aL

    Finite n: spectrum of a random Hermitian matrix

    Limit point process: spectrum of ??

  • Point process limit

    Ln

    b HLn - aL

    Finite n: spectrum of a random Hermitian matrix

    Limit point process: spectrum of ??

  • Detour to number theory

    Riemann zeta function: ζ(s) = ∞∑ n=1

    1 ns , for Re s > 1.

    (Analytic continuation to C \ {1})

    Riemann hypothesis: the non-trivial zeros are on the line Re s = 12 .

    Dyson-Montgomery conjecture:

    After some scaling:

    non-trivial zeros of ζ( 1

    2 + i s) ∼ bulk limit process of GUE

    (Sine2 process)

    I Strong numerical evidence: Odlyzko

    I Certain weaker versions are proved (Montgomery, Rudnick-Sarnak)

  • Detour to number theory

    Riemann zeta function: ζ(s) = ∞∑ n=1

    1 ns , for Re s > 1.

    (Analytic continuation to C \ {1})

    Riemann hypothesis: the non-trivial zeros are on the line Re s = 12 .

    Dyson-Montgomery conjecture:

    After some scaling:

    non-trivial zeros of ζ( 1

    2 + i s) ∼ bulk limit process of GUE

    (Sine2 process)

    I Strong numerical evidence: Odlyzko

    I Certain weaker versions are proved (Montgomery, Rudnick-Sarnak)

  • Detour to number theory

    Riemann zeta function: ζ(s) = ∞∑ n=1

    1 ns , for Re s > 1.

    (Analytic continuation to C \ {1})

    Riemann hypothesis: the non-trivial zeros are on the line Re s = 12 .

    Dyson-Montgomery conjecture:

    After some scaling:

    non-trivial zeros of ζ( 1

    2 + i s) ∼ bulk limit process of GUE

    (Sine2 process)

    I Strong numerical evidence: Odlyzko

    I Certain weaker versions are proved (Montgomery, Rudnick-Sarnak)

  • Hilbert-Pólya conjecture: the Riemann hypotheses is true because

    non-trivial zeros of ζ( 1

    2 + i s)

    = ev’s of an unbounded self-adjoint operator

    A famous attempt to make this approach rigorous: de Branges

    (based on the theory of Hilbert spaces of entire functions)

    This approach would produce a self-adjoint differential operator with the appropriate spectrum.

  • Hilbert-Pólya conjecture: the Riemann hypotheses is true because

    non-trivial zeros of ζ( 1

    2 + i s)

    = ev’s of an unbounded self-adjoint operator

    A famous attempt to make this approach rigorous: de Branges

    (based on the theory of Hilbert spaces of entire functions)

    This approach would produce a self-adjoint differential operator with the appropriate spectrum.

  • Hilbert-Pólya conjecture: the Riemann hypotheses is true because

    non-trivial zeros of ζ( 1

    2 + i s)

    = ev’s of an unbounded self-adjoint operator

    A famous attempt to make this approach rigorous: de Branges

    (based on the theory of Hilbert spaces of entire functions)

    This approach would produce a self-adjoint differential operator with the appropriate spectrum.

  • Natural question:

    Is there a self-adjoint differential operator with a spectrum given by the bulk limit of GUE?

    Disclaimer: A positive answer would not get us closer to any of the conjectures or the Riemann hypothesis (unfortunately...)

    Borodin-Olshanski, Maples-Najnudel-Nikeghbali:

    ‘operator-like object’ with generalized eigenvalues distributed as Sine2

  • Natural question:

    Is there a self-adjoint differential operator with a spectrum given by the bulk limit of GUE?

    Disclaimer: A positive answer would not get us closer to any of the conjectures or the Riemann hypothesis (unfortunately...)

    Borodin-Olshanski, Maples-Najnudel-Nikeghbali:

    ‘operator-like object’ with generalized eigenvalues distributed as Sine2

  • Natural question:

    Is there a self-adjoint differential operator with a spectrum given by the bulk limit of GUE?

    Disclaimer: A positive answer would not get us closer to any of the conjectures or the Riemann hypothesis (unfortunately...)

    Borodin-Olshanski, Maples-Najnudel-Nikeghbali:

    ‘operator-like object’ with generalized eigenvalues distributed as Sine2

  • Starting point for deriving the Sine2 process:

    Joint eigenvalue density of GUE:

    1

    Zn

    ∏ i

  • Starting point for deriving the Sine2 process:

    Joint eigenvalue density of GUE:

    1

    Zn

    ∏ i

  • Starting point for deriving the Sine2 process:

    Joint eigenvalue density of GUE:

    1

    Zn

    ∏ i

  • β-ensemble: finite point process with joint density

    1

    Zn,f ,β

    ∏ i

  • Scaling limits - global picture

    Hermite β-ensemble semicircle law Laguerre β-ensemble Marchenko-Pastur law

    -2 2 1 2 3 4

    ↑ ↑ ↗ ↑ ↑ ↑ soft edge bulk s. e. hard edge bulk s. e.

  • Local limits

    Soft edge: Rider-Raḿırez-Virág (Hermite, Laguerre) Airyβ process

    Hard edge: Rider-Raḿırez (Laguerre) Besselβ,a processes

    Bulk: Killip-Stoiciu, V.-Virág (circular, Hermite) CβE and Sineβ processes

    Instead of joint intensities, the limit processes are described via their counting functions using coupled systems of SDEs.

    sign(λ) · (# of points in [0, λ])

  • Local limits

    Soft edge: Rider-Raḿırez-Virág (Hermite, Lague