Local Universality of Repulsive Particle Systems and Random Matrices - - PowerPoint PPT Presentation

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Local Universality of Repulsive Particle Systems and Random Matrices - - PowerPoint PPT Presentation

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Local Universality of Repulsive Particle Systems and Random Matrices Friedrich Gtze joint with M.Venker, A.Naumov and A.Tikhomirov Bielefeld University www.math.uni-bielefeld.de/ goetze Workshop Random Matrices and their

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Local Universality of Repulsive Particle Systems and Random Matrices

Friedrich Götze

joint with M.Venker, A.Naumov and A.Tikhomirov

Bielefeld University www.math.uni-bielefeld.de/∼goetze

Workshop ”Random Matrices and their Applications” Télécom Paristech October 9, 2012

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 1 / 25

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Topics

Local Correlation Statistics for Repulsive Systems: GUE-Limits ( G.- M.Venker)

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 2 / 25

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Topics

Local Correlation Statistics for Repulsive Systems: GUE-Limits ( G.- M.Venker) Local Correlation Statistics for Repulsive Systems: β-Ensemble Universality (M.Venker)

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 2 / 25

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Topics

Local Correlation Statistics for Repulsive Systems: GUE-Limits ( G.- M.Venker) Local Correlation Statistics for Repulsive Systems: β-Ensemble Universality (M.Venker) Wigner’s Semicircular Law for Martingale Ensembles (G.- A.Naumov and A.Tikhomirov )

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 2 / 25

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Repulsive Particle Systems and GUE Empirical evidence for ”repulsive particle systems” in I R

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 3 / 25

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Repulsive Particle Systems and GUE Empirical evidence for ”repulsive particle systems” in I R exhibiting GUE-spacing statistics?

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 3 / 25

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Repulsive Particle Systems and GUE Empirical evidence for ”repulsive particle systems” in I R exhibiting GUE-spacing statistics? Distances between parked cars. (Abul-Magd (2005))

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 3 / 25

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Repulsive Particle Systems and GUE Empirical evidence for ”repulsive particle systems” in I R exhibiting GUE-spacing statistics? Distances between parked cars. (Abul-Magd (2005)) Bus stop waiting times in certain cities. (Krbalek, Seba (2000) , Beik,Borodin,Deift,Suidan (2006))

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 3 / 25

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Repulsive Particle Systems and GUE Empirical evidence for ”repulsive particle systems” in I R exhibiting GUE-spacing statistics? Distances between parked cars. (Abul-Magd (2005)) Bus stop waiting times in certain cities. (Krbalek, Seba (2000) , Beik,Borodin,Deift,Suidan (2006)) Spacings of zeros of the Riemann Zeta function on the critical line.

  • H. Montgomery, J. Keating, N. Snaith
  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 3 / 25

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Repulsive Particle Systems and GUE Empirical evidence for ”repulsive particle systems” in I R exhibiting GUE-spacing statistics? Distances between parked cars. (Abul-Magd (2005)) Bus stop waiting times in certain cities. (Krbalek, Seba (2000) , Beik,Borodin,Deift,Suidan (2006)) Spacings of zeros of the Riemann Zeta function on the critical line.

  • H. Montgomery, J. Keating, N. Snaith

All these particle systems show the phenomenon of repulsion.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 3 / 25

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Class of Models for Repulsive Particle Systems Suggestive to study repulsion models with relation to matrix ensembles:

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 4 / 25

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Class of Models for Repulsive Particle Systems Suggestive to study repulsion models with relation to matrix ensembles: For a ”smooth” even function ϕ > 0 define a density of particles xj, 1 ≤ j ≤ N

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 4 / 25

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Class of Models for Repulsive Particle Systems Suggestive to study repulsion models with relation to matrix ensembles: For a ”smooth” even function ϕ > 0 define a density of particles xj, 1 ≤ j ≤ N w.r.t. a potential Q by

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 4 / 25

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Class of Models for Repulsive Particle Systems Suggestive to study repulsion models with relation to matrix ensembles: For a ”smooth” even function ϕ > 0 define a density of particles xj, 1 ≤ j ≤ N w.r.t. a potential Q by 1 ZN,Q

  • j<k

ϕ(xk − xj)e−N N

j=1 Q(xj).

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 4 / 25

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Class of Models for Repulsive Particle Systems Suggestive to study repulsion models with relation to matrix ensembles: For a ”smooth” even function ϕ > 0 define a density of particles xj, 1 ≤ j ≤ N w.r.t. a potential Q by 1 ZN,Q

  • j<k

ϕ(xk − xj)e−N N

j=1 Q(xj).

Writing: ϕ(t) := t2e−h(t), the non unitary invariant deformation of PN,Q(x) := 1 ZN,Q exp

  • 2
  • j<k

log |xk − xj| − N

N

  • j=1

Q(xj)

  • ,

is

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 4 / 25

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Class of Models for Repulsive Particle Systems Suggestive to study repulsion models with relation to matrix ensembles: For a ”smooth” even function ϕ > 0 define a density of particles xj, 1 ≤ j ≤ N w.r.t. a potential Q by 1 ZN,Q

  • j<k

ϕ(xk − xj)e−N N

j=1 Q(xj).

Writing: ϕ(t) := t2e−h(t), the non unitary invariant deformation of PN,Q(x) := 1 ZN,Q exp

  • 2
  • j<k

log |xk − xj| − N

N

  • j=1

Q(xj)

  • ,

is Ph

N,Q(x)

:= 1 Z h

N,Q

exp

  • 2
  • j<k

log |xk − xj| −

  • j<k

h (xk − xj) − N

N

  • j=1

Q(xj)

  • .
  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 4 / 25

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Assumptions

Assumptions on potential Q:

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 5 / 25

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Assumptions

Assumptions on potential Q: symmetric around zero

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 5 / 25

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Assumptions

Assumptions on potential Q: symmetric around zero

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 5 / 25

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Assumptions

Assumptions on potential Q: symmetric around zero real analytic

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 5 / 25

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Assumptions

Assumptions on potential Q: symmetric around zero real analytic

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 5 / 25

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Assumptions

Assumptions on potential Q: symmetric around zero real analytic strongly convex: mint∈R Q′′(t) > 0

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 5 / 25

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Assumptions

Assumptions on potential Q: symmetric around zero real analytic strongly convex: mint∈R Q′′(t) > 0

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 5 / 25

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Assumptions

Assumptions on potential Q: symmetric around zero real analytic strongly convex: mint∈R Q′′(t) > 0 Assumptions on h:

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 5 / 25

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Assumptions

Assumptions on potential Q: symmetric around zero real analytic strongly convex: mint∈R Q′′(t) > 0 Assumptions on h: symmetric around zero

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 5 / 25

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Assumptions

Assumptions on potential Q: symmetric around zero real analytic strongly convex: mint∈R Q′′(t) > 0 Assumptions on h: symmetric around zero

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 5 / 25

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Assumptions

Assumptions on potential Q: symmetric around zero real analytic strongly convex: mint∈R Q′′(t) > 0 Assumptions on h: symmetric around zero Schwartz function

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 5 / 25

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Assumptions

Assumptions on potential Q: symmetric around zero real analytic strongly convex: mint∈R Q′′(t) > 0 Assumptions on h: symmetric around zero Schwartz function

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 5 / 25

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Assumptions

Assumptions on potential Q: symmetric around zero real analytic strongly convex: mint∈R Q′′(t) > 0 Assumptions on h: symmetric around zero Schwartz function real analytic

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 5 / 25

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Global Marginal Distributions ρh,k

N,Q(x1, . . . , xk) :=

  • RN−k Ph

N,Q(x)dxk+1 . . . dxN :

the k-th correlation function of Ph

N,Q.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 6 / 25

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Global Marginal Distributions ρh,k

N,Q(x1, . . . , xk) :=

  • RN−k Ph

N,Q(x)dxk+1 . . . dxN :

the k-th correlation function of Ph

N,Q.

Thm (G.-Venker ’12) For all h above exist αh > 0 s.th. for all Q above with mint∈R Q′′(t) > αh,

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 6 / 25

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Global Marginal Distributions ρh,k

N,Q(x1, . . . , xk) :=

  • RN−k Ph

N,Q(x)dxk+1 . . . dxN :

the k-th correlation function of Ph

N,Q.

Thm (G.-Venker ’12) For all h above exist αh > 0 s.th. for all Q above with mint∈R Q′′(t) > αh, there exists µh

Q, p-measure with compact support s.th.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 6 / 25

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Global Marginal Distributions ρh,k

N,Q(x1, . . . , xk) :=

  • RN−k Ph

N,Q(x)dxk+1 . . . dxN :

the k-th correlation function of Ph

N,Q.

Thm (G.-Venker ’12) For all h above exist αh > 0 s.th. for all Q above with mint∈R Q′′(t) > αh, there exists µh

Q, p-measure with compact support s.th.

(k-th correlation measure of Ph

N,Q) ⇒

  • µh

Q

⊗k as N → ∞,

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 6 / 25

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Global Marginal Distributions ρh,k

N,Q(x1, . . . , xk) :=

  • RN−k Ph

N,Q(x)dxk+1 . . . dxN :

the k-th correlation function of Ph

N,Q.

Thm (G.-Venker ’12) For all h above exist αh > 0 s.th. for all Q above with mint∈R Q′′(t) > αh, there exists µh

Q, p-measure with compact support s.th.

(k-th correlation measure of Ph

N,Q) ⇒

  • µh

Q

⊗k as N → ∞, i.e. for g ∈ Cb(Rk)

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 6 / 25

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Global Marginal Distributions ρh,k

N,Q(x1, . . . , xk) :=

  • RN−k Ph

N,Q(x)dxk+1 . . . dxN :

the k-th correlation function of Ph

N,Q.

Thm (G.-Venker ’12) For all h above exist αh > 0 s.th. for all Q above with mint∈R Q′′(t) > αh, there exists µh

Q, p-measure with compact support s.th.

(k-th correlation measure of Ph

N,Q) ⇒

  • µh

Q

⊗k as N → ∞, i.e. for g ∈ Cb(Rk) lim

N→∞

  • Rk g(t)ρh,k

N,Q(t)dkt =

  • Rk g(t)(µh

Q)⊗k(dt).

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Global Marginal Distributions ρh,k

N,Q(x1, . . . , xk) :=

  • RN−k Ph

N,Q(x)dxk+1 . . . dxN :

the k-th correlation function of Ph

N,Q.

Thm (G.-Venker ’12) For all h above exist αh > 0 s.th. for all Q above with mint∈R Q′′(t) > αh, there exists µh

Q, p-measure with compact support s.th.

(k-th correlation measure of Ph

N,Q) ⇒

  • µh

Q

⊗k as N → ∞, i.e. for g ∈ Cb(Rk) lim

N→∞

  • Rk g(t)ρh,k

N,Q(t)dkt =

  • Rk g(t)(µh

Q)⊗k(dt).

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 6 / 25

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Local Correlations in the Bulk Thm (G.-Venker 2012, arxiv:1205.0671) Above assumptions on Q, h and αh > 0:

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 7 / 25

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Local Correlations in the Bulk Thm (G.-Venker 2012, arxiv:1205.0671) Above assumptions on Q, h and αh > 0: For k ≥ 1 and a ∈ supp(µh

Q)◦

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 7 / 25

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Local Correlations in the Bulk Thm (G.-Venker 2012, arxiv:1205.0671) Above assumptions on Q, h and αh > 0: For k ≥ 1 and a ∈ supp(µh

Q)◦

density µh

Q(a) > 0, uniformly on compacts in t1, . . . , tk

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 7 / 25

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Local Correlations in the Bulk Thm (G.-Venker 2012, arxiv:1205.0671) Above assumptions on Q, h and αh > 0: For k ≥ 1 and a ∈ supp(µh

Q)◦

density µh

Q(a) > 0, uniformly on compacts in t1, . . . , tk

lim

N→∞

1 µh

Q(a)k ρh,k N,Q

  • a +

t1 Nµh

Q(a), . . . , a +

tk Nµh

Q(a)

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 7 / 25

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Local Correlations in the Bulk Thm (G.-Venker 2012, arxiv:1205.0671) Above assumptions on Q, h and αh > 0: For k ≥ 1 and a ∈ supp(µh

Q)◦

density µh

Q(a) > 0, uniformly on compacts in t1, . . . , tk

lim

N→∞

1 µh

Q(a)k ρh,k N,Q

  • a +

t1 Nµh

Q(a), . . . , a +

tk Nµh

Q(a)

  • = det

sin (π(ti − tj)) π(ti − tj)

  • 1≤i,j≤k

.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 7 / 25

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Extensions to β-Ensembles Let ϕ be an even, smooth, nonnegative function with ϕ(0) = 0 and ϕ(t) > 0 for t = 0.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 8 / 25

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Extensions to β-Ensembles Let ϕ be an even, smooth, nonnegative function with ϕ(0) = 0 and ϕ(t) > 0 for t = 0. Assume that for some β > 0 and c > 0

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 8 / 25

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Extensions to β-Ensembles Let ϕ be an even, smooth, nonnegative function with ϕ(0) = 0 and ϕ(t) > 0 for t = 0. Assume that for some β > 0 and c > 0 lim

ε→0

ϕ(ε) |ε|β = c.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 8 / 25

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Extensions to β-Ensembles Let ϕ be an even, smooth, nonnegative function with ϕ(0) = 0 and ϕ(t) > 0 for t = 0. Assume that for some β > 0 and c > 0 lim

ε→0

ϕ(ε) |ε|β = c. Let Q be a strictly convex function of sufficient growth at infinity and define Pϕ,β

N,Q as the probability measure on RN with density

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 8 / 25

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Extensions to β-Ensembles Let ϕ be an even, smooth, nonnegative function with ϕ(0) = 0 and ϕ(t) > 0 for t = 0. Assume that for some β > 0 and c > 0 lim

ε→0

ϕ(ε) |ε|β = c. Let Q be a strictly convex function of sufficient growth at infinity and define Pϕ,β

N,Q as the probability measure on RN with density

Pϕ,β

N,Q(x) :=

1 Z ϕ,β

N,Q

  • i<j

ϕ(xi − xj)e−N N

j=1 Q(xj)dx.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 8 / 25

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Extensions to β-Ensembles Let ϕ be an even, smooth, nonnegative function with ϕ(0) = 0 and ϕ(t) > 0 for t = 0. Assume that for some β > 0 and c > 0 lim

ε→0

ϕ(ε) |ε|β = c. Let Q be a strictly convex function of sufficient growth at infinity and define Pϕ,β

N,Q as the probability measure on RN with density

Pϕ,β

N,Q(x) :=

1 Z ϕ,β

N,Q

  • i<j

ϕ(xi − xj)e−N N

j=1 Q(xj)dx.

We conjecture that Pϕ,β

N,Q has the same bulk local k-correlation, say ρk β, as the

Gaussian-β ensemble Pβ

N(x) := 1

Z β

N

  • j<k

|xk − xj|β e−N N

j=1 x2 j .

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 8 / 25

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β-Ensembles

Theorem (Venker 2012)

Write ϕ(x) := |x|β exp{h}, h real analytic and even Schwartz function, αh ≥ 0 s. th. for all real analytic, strongly convex and even Q with αQ > αh:

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 9 / 25

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β-Ensembles

Theorem (Venker 2012)

Write ϕ(x) := |x|β exp{h}, h real analytic and even Schwartz function, αh ≥ 0 s. th. for all real analytic, strongly convex and even Q with αQ > αh:The correlation measure ρh,1

N,Q,β converges weakly to a compactly

supported p.m. µh

Q,β.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 9 / 25

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β-Ensembles

Theorem (Venker 2012)

Write ϕ(x) := |x|β exp{h}, h real analytic and even Schwartz function, αh ≥ 0 s. th. for all real analytic, strongly convex and even Q with αQ > αh:The correlation measure ρh,1

N,Q,β converges weakly to a compactly

supported p.m. µh

Q,β.

Conditions on Q as above, there is µQ,β of compact support, semicircular for Q(x) = x2

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 9 / 25

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β-Ensembles

Theorem (Venker 2012)

Write ϕ(x) := |x|β exp{h}, h real analytic and even Schwartz function, αh ≥ 0 s. th. for all real analytic, strongly convex and even Q with αQ > αh:The correlation measure ρh,1

N,Q,β converges weakly to a compactly

supported p.m. µh

Q,β.

Conditions on Q as above, there is µQ,β of compact support, semicircular for Q(x) = x2 and a scaled deformed correlation function

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 9 / 25

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β-Ensembles

Theorem (Venker 2012)

Write ϕ(x) := |x|β exp{h}, h real analytic and even Schwartz function, αh ≥ 0 s. th. for all real analytic, strongly convex and even Q with αQ > αh:The correlation measure ρh,1

N,Q,β converges weakly to a compactly

supported p.m. µh

Q,β.

Conditions on Q as above, there is µQ,β of compact support, semicircular for Q(x) = x2 and a scaled deformed correlation function 1 µh

Q,β(a)k ρh,k N,Q,β

  • a +

t1 Nµh

Q,β(a), . . . , a +

tk Nµh

Q,β(a)

  • ,

(1) where a ∈ supp(µh

Q,β(a))◦.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 9 / 25

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SLIDE 53

β-Ensembles

Theorem (Venker 2012)

Write ϕ(x) := |x|β exp{h}, h real analytic and even Schwartz function, αh ≥ 0 s. th. for all real analytic, strongly convex and even Q with αQ > αh:The correlation measure ρh,1

N,Q,β converges weakly to a compactly

supported p.m. µh

Q,β.

Conditions on Q as above, there is µQ,β of compact support, semicircular for Q(x) = x2 and a scaled deformed correlation function 1 µh

Q,β(a)k ρh,k N,Q,β

  • a +

t1 Nµh

Q,β(a), . . . , a +

tk Nµh

Q,β(a)

  • ,

(1) where a ∈ supp(µh

Q,β(a))◦. For h = 0 and Q = x2,

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 9 / 25

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SLIDE 54

β-Ensembles

Theorem (Venker 2012)

Write ϕ(x) := |x|β exp{h}, h real analytic and even Schwartz function, αh ≥ 0 s. th. for all real analytic, strongly convex and even Q with αQ > αh:The correlation measure ρh,1

N,Q,β converges weakly to a compactly

supported p.m. µh

Q,β.

Conditions on Q as above, there is µQ,β of compact support, semicircular for Q(x) = x2 and a scaled deformed correlation function 1 µh

Q,β(a)k ρh,k N,Q,β

  • a +

t1 Nµh

Q,β(a), . . . , a +

tk Nµh

Q,β(a)

  • ,

(1) where a ∈ supp(µh

Q,β(a))◦. For h = 0 and Q = x2, the limit N → ∞ exists for

Q(x) = G(x) := x2 and h = 0 by Valko-Virag (09).

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 9 / 25

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Universality of Deformed (Averaged) Local Correlations of β-Ensembles

Compare local correlations of PN,hM

η with those of the Gaussian β-Ensemble

PN,G,β:

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 10 / 25

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SLIDE 56

Universality of Deformed (Averaged) Local Correlations of β-Ensembles

Compare local correlations of PN,hM

η with those of the Gaussian β-Ensemble

PN,G,β:

Theorem (Venker 2012, arxiv 1209.317)

h and Q as above. Let 0 < ξ ≤ 1/2 and sN := N−1+ξ.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 10 / 25

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SLIDE 57

Universality of Deformed (Averaged) Local Correlations of β-Ensembles

Compare local correlations of PN,hM

η with those of the Gaussian β-Ensemble

PN,G,β:

Theorem (Venker 2012, arxiv 1209.317)

h and Q as above. Let 0 < ξ ≤ 1/2 and sN := N−1+ξ. For k = 1, 2, . . . , any a ∈ supp(µh

Q,β)◦, any a′ supp(µG,β)◦, any smooth

function f : Rk − → R with compact support

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 10 / 25

slide-58
SLIDE 58

Universality of Deformed (Averaged) Local Correlations of β-Ensembles

Compare local correlations of PN,hM

η with those of the Gaussian β-Ensemble

PN,G,β:

Theorem (Venker 2012, arxiv 1209.317)

h and Q as above. Let 0 < ξ ≤ 1/2 and sN := N−1+ξ. For k = 1, 2, . . . , any a ∈ supp(µh

Q,β)◦, any a′ supp(µG,β)◦, any smooth

function f : Rk − → R with compact support lim

N→∞

  • dtkf(t)

a+sN

a−sN

1 µh

Q,β(a)k ρh,k N,Q,β

  • u +

t1 Nµh

Q,β(a), . . . , u +

tk Nµh

Q,β(a)

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 10 / 25

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SLIDE 59

Universality of Deformed (Averaged) Local Correlations of β-Ensembles

Compare local correlations of PN,hM

η with those of the Gaussian β-Ensemble

PN,G,β:

Theorem (Venker 2012, arxiv 1209.317)

h and Q as above. Let 0 < ξ ≤ 1/2 and sN := N−1+ξ. For k = 1, 2, . . . , any a ∈ supp(µh

Q,β)◦, any a′ supp(µG,β)◦, any smooth

function f : Rk − → R with compact support lim

N→∞

  • dtkf(t)

a+sN

a−sN

1 µh

Q,β(a)k ρh,k N,Q,β

  • u +

t1 Nµh

Q,β(a), . . . , u +

tk Nµh

Q,β(a)

  • −

a′+sN

a′−sN

1 µG,β(a′)k ρk

N,G,β

  • u +

t1 NµG,β(a′), . . . , u + tk NµG,β(a′) du 2sN = 0.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 10 / 25

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SLIDE 60

Universality of Deformed (Averaged) Local Correlations of β-Ensembles

Compare local correlations of PN,hM

η with those of the Gaussian β-Ensemble

PN,G,β:

Theorem (Venker 2012, arxiv 1209.317)

h and Q as above. Let 0 < ξ ≤ 1/2 and sN := N−1+ξ. For k = 1, 2, . . . , any a ∈ supp(µh

Q,β)◦, any a′ supp(µG,β)◦, any smooth

function f : Rk − → R with compact support lim

N→∞

  • dtkf(t)

a+sN

a−sN

1 µh

Q,β(a)k ρh,k N,Q,β

  • u +

t1 Nµh

Q,β(a), . . . , u +

tk Nµh

Q,β(a)

  • −

a′+sN

a′−sN

1 µG,β(a′)k ρk

N,G,β

  • u +

t1 NµG,β(a′), . . . , u + tk NµG,β(a′) du 2sN = 0. Local correlation limits using relaxation flow methods of Bourgade, Erd˝

  • s,
  • B. Schlein, and H.-T. Yau (2011,2012) instead of potential theory.
  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 10 / 25

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SLIDE 61

Proof: Simple Example h(x) := −x2 and γ > 0 Pγ

N,α(x)

:= Z −1

N,α,γPGUE N,α (x) exp{γ

  • i<j

(xi − xj)2},

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 11 / 25

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SLIDE 62

Proof: Simple Example h(x) := −x2 and γ > 0 Pγ

N,α(x)

:= Z −1

N,α,γPGUE N,α (x) exp{γ

  • i<j

(xi − xj)2}, PGUE

N,α (x)

:= 1 ZN,α

  • 1≤i<j≤N
  • xi − xj
  • 2 exp{−αN
  • j

x2

j },

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 11 / 25

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SLIDE 63

Proof: Simple Example h(x) := −x2 and γ > 0 Pγ

N,α(x)

:= Z −1

N,α,γPGUE N,α (x) exp{γ

  • i<j

(xi − xj)2}, PGUE

N,α (x)

:= 1 ZN,α

  • 1≤i<j≤N
  • xi − xj
  • 2 exp{−αN
  • j

x2

j },

exp{γ

  • i<j

(xi − xj)2} = exp{−γN M2(x)} exp{γM1(x)2},

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 11 / 25

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SLIDE 64

Proof: Simple Example h(x) := −x2 and γ > 0 Pγ

N,α(x)

:= Z −1

N,α,γPGUE N,α (x) exp{γ

  • i<j

(xi − xj)2}, PGUE

N,α (x)

:= 1 ZN,α

  • 1≤i<j≤N
  • xi − xj
  • 2 exp{−αN
  • j

x2

j },

exp{γ

  • i<j

(xi − xj)2} = exp{−γN M2(x)} exp{γM1(x)2}, where

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 11 / 25

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SLIDE 65

Proof: Simple Example h(x) := −x2 and γ > 0 Pγ

N,α(x)

:= Z −1

N,α,γPGUE N,α (x) exp{γ

  • i<j

(xi − xj)2}, PGUE

N,α (x)

:= 1 ZN,α

  • 1≤i<j≤N
  • xi − xj
  • 2 exp{−αN
  • j

x2

j },

exp{γ

  • i<j

(xi − xj)2} = exp{−γN M2(x)} exp{γM1(x)2}, where Mp(x) :=

N

  • j=1

xp

j ,

p = 1, 2,

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 11 / 25

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SLIDE 66

Proof: Simple Example h(x) := −x2 and γ > 0 Pγ

N,α(x)

:= Z −1

N,α,γPGUE N,α (x) exp{γ

  • i<j

(xi − xj)2}, PGUE

N,α (x)

:= 1 ZN,α

  • 1≤i<j≤N
  • xi − xj
  • 2 exp{−αN
  • j

x2

j },

exp{γ

  • i<j

(xi − xj)2} = exp{−γN M2(x)} exp{γM1(x)2}, where Mp(x) :=

N

  • j=1

xp

j ,

p = 1, 2, exp{γM1(x)2} = c

  • R

exp{ε√γM1(x)} exp{−ε2/4}dε,

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 11 / 25

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SLIDE 67

Orthogonal Polynomials Pγ

N,α(x)

= c′

  • R

ZN,ε ZN PN,ε(x) exp{−ε2/4}dε,

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 12 / 25

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SLIDE 68

Orthogonal Polynomials Pγ

N,α(x)

= c′

  • R

ZN,ε ZN PN,ε(x) exp{−ε2/4}dε, PN,ε(x) := ∆(x)2 ZN,ε exp{−

N

  • j=1

(N (α + γ)x2

j + √γεxj)}

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 12 / 25

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SLIDE 69

Orthogonal Polynomials Pγ

N,α(x)

= c′

  • R

ZN,ε ZN PN,ε(x) exp{−ε2/4}dε, PN,ε(x) := ∆(x)2 ZN,ε exp{−

N

  • j=1

(N (α + γ)x2

j + √γεxj)}

ZN,ε/ZN =

  • 1 + γ

α exp

  • γε2

4(α + γ)

  • .
  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 12 / 25

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SLIDE 70

Orthogonal Polynomials Pγ

N,α(x)

= c′

  • R

ZN,ε ZN PN,ε(x) exp{−ε2/4}dε, PN,ε(x) := ∆(x)2 ZN,ε exp{−

N

  • j=1

(N (α + γ)x2

j + √γεxj)}

ZN,ε/ZN =

  • 1 + γ

α exp

  • γε2

4(α + γ)

  • .

Orthogonal polynomials w.r.t. the kernel exp{−N(α + γ)t2 + ε√γt} are shifted Hermite polynomials.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 12 / 25

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SLIDE 71

Orthogonal Polynomials Pγ

N,α(x)

= c′

  • R

ZN,ε ZN PN,ε(x) exp{−ε2/4}dε, PN,ε(x) := ∆(x)2 ZN,ε exp{−

N

  • j=1

(N (α + γ)x2

j + √γεxj)}

ZN,ε/ZN =

  • 1 + γ

α exp

  • γε2

4(α + γ)

  • .

Orthogonal polynomials w.r.t. the kernel exp{−N(α + γ)t2 + ε√γt} are shifted Hermite polynomials. The ensemble Pε

N is determinantal with kernel:

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 12 / 25

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SLIDE 72

Orthogonal Polynomials Pγ

N,α(x)

= c′

  • R

ZN,ε ZN PN,ε(x) exp{−ε2/4}dε, PN,ε(x) := ∆(x)2 ZN,ε exp{−

N

  • j=1

(N (α + γ)x2

j + √γεxj)}

ZN,ε/ZN =

  • 1 + γ

α exp

  • γε2

4(α + γ)

  • .

Orthogonal polynomials w.r.t. the kernel exp{−N(α + γ)t2 + ε√γt} are shifted Hermite polynomials. The ensemble Pε

N is determinantal with kernel:

K ∗

N(t, s) = exp{ω′2 ε2

4N }KN(t − ω′ε

2N , s − ω′ε 2N ),

ω′ := √γ α + γ

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 12 / 25

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SLIDE 73

Orthogonal Polynomials Pγ

N,α(x)

= c′

  • R

ZN,ε ZN PN,ε(x) exp{−ε2/4}dε, PN,ε(x) := ∆(x)2 ZN,ε exp{−

N

  • j=1

(N (α + γ)x2

j + √γεxj)}

ZN,ε/ZN =

  • 1 + γ

α exp

  • γε2

4(α + γ)

  • .

Orthogonal polynomials w.r.t. the kernel exp{−N(α + γ)t2 + ε√γt} are shifted Hermite polynomials. The ensemble Pε

N is determinantal with kernel:

K ∗

N(t, s) = exp{ω′2 ε2

4N }KN(t − ω′ε

2N , s − ω′ε 2N ),

ω′ := √γ α + γ where KN is the kernel of rescaled GUEω.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 12 / 25

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SLIDE 74

Universality ρε,k

N :

k-th correlation function of Pε

N,

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 13 / 25

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SLIDE 75

Universality ρε,k

N :

k-th correlation function of Pε

N,

σ Wigner density on [−ω, ω], ω := (α + γ)−1/2 ,

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 13 / 25

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SLIDE 76

Universality ρε,k

N :

k-th correlation function of Pε

N,

σ Wigner density on [−ω, ω], ω := (α + γ)−1/2 , for all ε ∈ R:

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 13 / 25

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SLIDE 77

Universality ρε,k

N :

k-th correlation function of Pε

N,

σ Wigner density on [−ω, ω], ω := (α + γ)−1/2 , for all ε ∈ R: ρ1,ε

N

= ⇒ σ and

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 13 / 25

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SLIDE 78

Universality ρε,k

N :

k-th correlation function of Pε

N,

σ Wigner density on [−ω, ω], ω := (α + γ)−1/2 , for all ε ∈ R: ρ1,ε

N

= ⇒ σ and lim

N→∞

1 σ(a)k ρε,k

N

  • a +

t1 Nσ(a), . . . , a + tk Nσ(a)

  • = det

sin(π(ti − tj) π(ti − tj)

  • 1≤i,j≤k

,

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 13 / 25

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SLIDE 79

Universality ρε,k

N :

k-th correlation function of Pε

N,

σ Wigner density on [−ω, ω], ω := (α + γ)−1/2 , for all ε ∈ R: ρ1,ε

N

= ⇒ σ and lim

N→∞

1 σ(a)k ρε,k

N

  • a +

t1 Nσ(a), . . . , a + tk Nσ(a)

  • = det

sin(π(ti − tj) π(ti − tj)

  • 1≤i,j≤k

, locally uniformly in t1, . . . tk, and a in compact subsets of (−ω, ω).

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 13 / 25

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SLIDE 80

Universality ρε,k

N :

k-th correlation function of Pε

N,

σ Wigner density on [−ω, ω], ω := (α + γ)−1/2 , for all ε ∈ R: ρ1,ε

N

= ⇒ σ and lim

N→∞

1 σ(a)k ρε,k

N

  • a +

t1 Nσ(a), . . . , a + tk Nσ(a)

  • = det

sin(π(ti − tj) π(ti − tj)

  • 1≤i,j≤k

, locally uniformly in t1, . . . tk, and a in compact subsets of (−ω, ω). Thm (Venker ’11) ρk,γ

N,α, kth correlation function of Pγ N,α:

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 13 / 25

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SLIDE 81

Universality ρε,k

N :

k-th correlation function of Pε

N,

σ Wigner density on [−ω, ω], ω := (α + γ)−1/2 , for all ε ∈ R: ρ1,ε

N

= ⇒ σ and lim

N→∞

1 σ(a)k ρε,k

N

  • a +

t1 Nσ(a), . . . , a + tk Nσ(a)

  • = det

sin(π(ti − tj) π(ti − tj)

  • 1≤i,j≤k

, locally uniformly in t1, . . . tk, and a in compact subsets of (−ω, ω). Thm (Venker ’11) ρk,γ

N,α, kth correlation function of Pγ N,α:

lim

N→∞

1 σ(a)k ργ,k

N,α

  • a +

t1 Nσ(a), . . . , a + tk Nσ(a)

  • =

det

  • sin(π(ti−tj)

π(ti−tj)

  • 1≤i,j≤k
  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 13 / 25

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SLIDE 82

Sketch of Proof: Recentering Hoeffding type decomposition of interaction

  • i<j h (xi − xj) =

i<j ˜

h (xi − xj) + N

i

  • h(xi − s)dµh

Q(s) + const.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 14 / 25

slide-83
SLIDE 83

Sketch of Proof: Recentering Hoeffding type decomposition of interaction

  • i<j h (xi − xj) =

i<j ˜

h (xi − xj) + N

i

  • h(xi − s)dµh

Q(s) + const.

into centered fluctuation (w.r.t to µh

Q) and additional potential h ∗ µh Q

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 14 / 25

slide-84
SLIDE 84

Sketch of Proof: Recentering Hoeffding type decomposition of interaction

  • i<j h (xi − xj) =

i<j ˜

h (xi − xj) + N

i

  • h(xi − s)dµh

Q(s) + const.

into centered fluctuation (w.r.t to µh

Q) and additional potential h ∗ µh Q

Ensemble Ph

N,Q:

Ph

N,Q(x)

:= 1 Z h

N,Q

∆(x)2 exp{−N

N

  • j=1

Q(xj)} exp{−

  • j<k

h (xk − xj)}

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 14 / 25

slide-85
SLIDE 85

Sketch of Proof: Recentering Hoeffding type decomposition of interaction

  • i<j h (xi − xj) =

i<j ˜

h (xi − xj) + N

i

  • h(xi − s)dµh

Q(s) + const.

into centered fluctuation (w.r.t to µh

Q) and additional potential h ∗ µh Q

Ensemble Ph

N,Q:

Ph

N,Q(x)

:= 1 Z h

N,Q

∆(x)2 exp{−N

N

  • j=1

Q(xj)} exp{−

  • j<k

h (xk − xj)} = 1 ¯ Z h

N,Q

∆(x)2 exp{−N

N

  • j=1

V(xj)} exp{−

  • j<k

˜ h (xk − xj)},

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 14 / 25

slide-86
SLIDE 86

Sketch of Proof: Recentering Hoeffding type decomposition of interaction

  • i<j h (xi − xj) =

i<j ˜

h (xi − xj) + N

i

  • h(xi − s)dµh

Q(s) + const.

into centered fluctuation (w.r.t to µh

Q) and additional potential h ∗ µh Q

Ensemble Ph

N,Q:

Ph

N,Q(x)

:= 1 Z h

N,Q

∆(x)2 exp{−N

N

  • j=1

Q(xj)} exp{−

  • j<k

h (xk − xj)} = 1 ¯ Z h

N,Q

∆(x)2 exp{−N

N

  • j=1

V(xj)} exp{−

  • j<k

˜ h (xk − xj)}, V(t) := Q(t) +

  • h(t − s)dµh

Q(s)

Claim: Correlation-Fct. of Ph

N,Q equivalent to PN,V as N → ∞, where

PN,V(x) := 1 ZN,V ∆(x)2 exp{−N

N

  • j=1

V(xj)}

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 14 / 25

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SLIDE 87

Equilibrium Measure For ν ∈ M1(R) (Q, h as above) consider potential

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 15 / 25

slide-88
SLIDE 88

Equilibrium Measure For ν ∈ M1(R) (Q, h as above) consider potential Vν,Q(t) := Q(t) +

  • h(t − s)dν(s).
  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 15 / 25

slide-89
SLIDE 89

Equilibrium Measure For ν ∈ M1(R) (Q, h as above) consider potential Vν,Q(t) := Q(t) +

  • h(t − s)dν(s).

Equilibrium measure for potential V (like Vν,Q above) is the unique solution, say µ = T(V),

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 15 / 25

slide-90
SLIDE 90

Equilibrium Measure For ν ∈ M1(R) (Q, h as above) consider potential Vν,Q(t) := Q(t) +

  • h(t − s)dν(s).

Equilibrium measure for potential V (like Vν,Q above) is the unique solution, say µ = T(V), to the minimization problem min

µ∈M1(R)

  • V(t)dµ(t) +

log |t − s|−1 dµ(t)dµ(s).

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 15 / 25

slide-91
SLIDE 91

Equilibrium Measure For ν ∈ M1(R) (Q, h as above) consider potential Vν,Q(t) := Q(t) +

  • h(t − s)dν(s).

Equilibrium measure for potential V (like Vν,Q above) is the unique solution, say µ = T(V), to the minimization problem min

µ∈M1(R)

  • V(t)dµ(t) +

log |t − s|−1 dµ(t)dµ(s). By Schauder let µ = µh

Q be a fixed point of

ν → T(Vν,Q), i.e.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 15 / 25

slide-92
SLIDE 92

Equilibrium Measure For ν ∈ M1(R) (Q, h as above) consider potential Vν,Q(t) := Q(t) +

  • h(t − s)dν(s).

Equilibrium measure for potential V (like Vν,Q above) is the unique solution, say µ = T(V), to the minimization problem min

µ∈M1(R)

  • V(t)dµ(t) +

log |t − s|−1 dµ(t)dµ(s). By Schauder let µ = µh

Q be a fixed point of

ν → T(Vν,Q), i.e. Selfconsistency: µ = T(Vµ,Q),

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 15 / 25

slide-93
SLIDE 93

Equilibrium Measure For ν ∈ M1(R) (Q, h as above) consider potential Vν,Q(t) := Q(t) +

  • h(t − s)dν(s).

Equilibrium measure for potential V (like Vν,Q above) is the unique solution, say µ = T(V), to the minimization problem min

µ∈M1(R)

  • V(t)dµ(t) +

log |t − s|−1 dµ(t)dµ(s). By Schauder let µ = µh

Q be a fixed point of

ν → T(Vν,Q), i.e. Selfconsistency: µ = T(Vµ,Q), with continuous density µh

Q(x) and compact support.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 15 / 25

slide-94
SLIDE 94

Fourier Representation of ˜ h Fourier representation,

  • h real,

µ = µh

Q,

−

  • l=k

˜ h (xl − xk) = −

  • h(t)
  • j

ei xjt − ei xj·µ

  • 2

dt,

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 16 / 25

slide-95
SLIDE 95

Fourier Representation of ˜ h Fourier representation,

  • h real,

µ = µh

Q,

−

  • l=k

˜ h (xl − xk) = −

  • h(t)
  • j

ei xjt − ei xj·µ

  • 2

dt, Let e.g. S(t) =

j sin(t xj).

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 16 / 25

slide-96
SLIDE 96

Fourier Representation of ˜ h Fourier representation,

  • h real,

µ = µh

Q,

−

  • l=k

˜ h (xl − xk) = −

  • h(t)
  • j

ei xjt − ei xj·µ

  • 2

dt, Let e.g. S(t) =

j sin(t xj).

If g(t) = − h ≥ 0

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 16 / 25

slide-97
SLIDE 97

Fourier Representation of ˜ h Fourier representation,

  • h real,

µ = µh

Q,

−

  • l=k

˜ h (xl − xk) = −

  • h(t)
  • j

ei xjt − ei xj·µ

  • 2

dt, Let e.g. S(t) =

j sin(t xj).

If g(t) = − h ≥ 0 exp 1 2 ∞ g(t)S(t)2dt

  • = E exp{

∞ g1/2(t)S(t)dBt} =: Eω exp{

  • j

f(xj, ω)},

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 16 / 25

slide-98
SLIDE 98

Fourier Representation of ˜ h Fourier representation,

  • h real,

µ = µh

Q,

−

  • l=k

˜ h (xl − xk) = −

  • h(t)
  • j

ei xjt − ei xj·µ

  • 2

dt, Let e.g. S(t) =

j sin(t xj).

If g(t) = − h ≥ 0 exp 1 2 ∞ g(t)S(t)2dt

  • = E exp{

∞ g1/2(t)S(t)dBt} =: Eω exp{

  • j

f(xj, ω)}, and one may linearize −

l=k ˜

h (xl − xk) .

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 16 / 25

slide-99
SLIDE 99

Fourier Representation of ˜ h Fourier representation,

  • h real,

µ = µh

Q,

−

  • l=k

˜ h (xl − xk) = −

  • h(t)
  • j

ei xjt − ei xj·µ

  • 2

dt, Let e.g. S(t) =

j sin(t xj).

If g(t) = − h ≥ 0 exp 1 2 ∞ g(t)S(t)2dt

  • = E exp{

∞ g1/2(t)S(t)dBt} =: Eω exp{

  • j

f(xj, ω)}, and one may linearize −

l=k ˜

h (xl − xk) . Need real f: extend limit results to g(t) = − h(t) < 0:

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 16 / 25

slide-100
SLIDE 100

Fourier Representation of ˜ h Fourier representation,

  • h real,

µ = µh

Q,

−

  • l=k

˜ h (xl − xk) = −

  • h(t)
  • j

ei xjt − ei xj·µ

  • 2

dt, Let e.g. S(t) =

j sin(t xj).

If g(t) = − h ≥ 0 exp 1 2 ∞ g(t)S(t)2dt

  • = E exp{

∞ g1/2(t)S(t)dBt} =: Eω exp{

  • j

f(xj, ω)}, and one may linearize −

l=k ˜

h (xl − xk) . Need real f: extend limit results to g(t) = − h(t) < 0: to family: gz := g+ + zg− ≥ 0, z ≥ 0

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 16 / 25

slide-101
SLIDE 101

Fourier Representation of ˜ h Fourier representation,

  • h real,

µ = µh

Q,

−

  • l=k

˜ h (xl − xk) = −

  • h(t)
  • j

ei xjt − ei xj·µ

  • 2

dt, Let e.g. S(t) =

j sin(t xj).

If g(t) = − h ≥ 0 exp 1 2 ∞ g(t)S(t)2dt

  • = E exp{

∞ g1/2(t)S(t)dBt} =: Eω exp{

  • j

f(xj, ω)}, and one may linearize −

l=k ˜

h (xl − xk) . Need real f: extend limit results to g(t) = − h(t) < 0: to family: gz := g+ + zg− ≥ 0, z ≥ 0 Analytical extension of limits to Re (z) < 0: get g−1 = − h

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 16 / 25

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SLIDE 102

Concentration Inequalities

For equivalence use concentration inequalities: For f Lipschitz, some cf, C > 0,

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 17 / 25

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SLIDE 103

Concentration Inequalities

For equivalence use concentration inequalities: For f Lipschitz, some cf, C > 0, PN,Q  

  • N
  • j=1

f(xj) − EN,Q

N

  • j=1

f(xj)

  • > η

  ≤ Ce−cf η2.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 17 / 25

slide-104
SLIDE 104

Concentration Inequalities

For equivalence use concentration inequalities: For f Lipschitz, some cf, C > 0, PN,Q  

  • N
  • j=1

f(xj) − EN,Q

N

  • j=1

f(xj)

  • > η

  ≤ Ce−cf η2. Truncation of xj to range [−L, L]: s.th. cf is bounded a.s. for f(xj) = f(xj, ω) Gaussian process in Ck

b[−L, L].

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 17 / 25

slide-105
SLIDE 105

Concentration Inequalities

For equivalence use concentration inequalities: For f Lipschitz, some cf, C > 0, PN,Q  

  • N
  • j=1

f(xj) − EN,Q

N

  • j=1

f(xj)

  • > η

  ≤ Ce−cf η2. Truncation of xj to range [−L, L]: s.th. cf is bounded a.s. for f(xj) = f(xj, ω) Gaussian process in Ck

b[−L, L].

Approximate Ph

N,Q by a mixture of asymptotic unitary invariant

ensembles w.r.t. the Wiener measure.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 17 / 25

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SLIDE 106

Concentration Inequalities

For equivalence use concentration inequalities: For f Lipschitz, some cf, C > 0, PN,Q  

  • N
  • j=1

f(xj) − EN,Q

N

  • j=1

f(xj)

  • > η

  ≤ Ce−cf η2. Truncation of xj to range [−L, L]: s.th. cf is bounded a.s. for f(xj) = f(xj, ω) Gaussian process in Ck

b[−L, L].

Approximate Ph

N,Q by a mixture of asymptotic unitary invariant

ensembles w.r.t. the Wiener measure. Show PN,V,f =

1 ZN,V,f

  • i<j|xi − xj|2 exp{− N

j=1

  • N V(xj) + f(xj, ω)
  • }

= ⇒

  • µh

Q

⊗k (µh

Q equilibrium measure of V):

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 17 / 25

slide-107
SLIDE 107

Concentration Inequalities

For equivalence use concentration inequalities: For f Lipschitz, some cf, C > 0, PN,Q  

  • N
  • j=1

f(xj) − EN,Q

N

  • j=1

f(xj)

  • > η

  ≤ Ce−cf η2. Truncation of xj to range [−L, L]: s.th. cf is bounded a.s. for f(xj) = f(xj, ω) Gaussian process in Ck

b[−L, L].

Approximate Ph

N,Q by a mixture of asymptotic unitary invariant

ensembles w.r.t. the Wiener measure. Show PN,V,f =

1 ZN,V,f

  • i<j|xi − xj|2 exp{− N

j=1

  • N V(xj) + f(xj, ω)
  • }

= ⇒

  • µh

Q

⊗k (µh

Q equilibrium measure of V):

(Johansson ’98):

Independence of f in the global limits.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 17 / 25

slide-108
SLIDE 108

Concentration Inequalities

For equivalence use concentration inequalities: For f Lipschitz, some cf, C > 0, PN,Q  

  • N
  • j=1

f(xj) − EN,Q

N

  • j=1

f(xj)

  • > η

  ≤ Ce−cf η2. Truncation of xj to range [−L, L]: s.th. cf is bounded a.s. for f(xj) = f(xj, ω) Gaussian process in Ck

b[−L, L].

Approximate Ph

N,Q by a mixture of asymptotic unitary invariant

ensembles w.r.t. the Wiener measure. Show PN,V,f =

1 ZN,V,f

  • i<j|xi − xj|2 exp{− N

j=1

  • N V(xj) + f(xj, ω)
  • }

= ⇒

  • µh

Q

⊗k (µh

Q equilibrium measure of V):

(Johansson ’98):

Independence of f in the global limits.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 17 / 25

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SLIDE 109

Asymptotic Approximations

Compare EN,Q N

j=1 f(xj) with N

  • f dµQ:
  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 18 / 25

slide-110
SLIDE 110

Asymptotic Approximations

Compare EN,Q N

j=1 f(xj) with N

  • f dµQ:
  • fdρN,Q =
  • f dµQ + O

1 N2

  • for real analytic Q, f.

(Kriecherbauer, Shcherbina ’10).

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 18 / 25

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SLIDE 111

Asymptotic Approximations

Compare EN,Q N

j=1 f(xj) with N

  • f dµQ:
  • fdρN,Q =
  • f dµQ + O

1 N2

  • for real analytic Q, f.

(Kriecherbauer, Shcherbina ’10).

Uniform convergence of ρN,Q,f to the density of µQ

  • n compacts in supp(µQ)◦. (Totik ’00).
  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 18 / 25

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SLIDE 112

Asymptotic Approximations

Compare EN,Q N

j=1 f(xj) with N

  • f dµQ:
  • fdρN,Q =
  • f dµQ + O

1 N2

  • for real analytic Q, f.

(Kriecherbauer, Shcherbina ’10).

Uniform convergence of ρN,Q,f to the density of µQ

  • n compacts in supp(µQ)◦. (Totik ’00).

Local bulk universality of correlation functions of ensembles of the form PN,Q,f. (Levin, Lubinsky ’08).

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 18 / 25

slide-113
SLIDE 113

Asymptotic Approximations

Compare EN,Q N

j=1 f(xj) with N

  • f dµQ:
  • fdρN,Q =
  • f dµQ + O

1 N2

  • for real analytic Q, f.

(Kriecherbauer, Shcherbina ’10).

Uniform convergence of ρN,Q,f to the density of µQ

  • n compacts in supp(µQ)◦. (Totik ’00).

Local bulk universality of correlation functions of ensembles of the form PN,Q,f. (Levin, Lubinsky ’08). For β = 2 M. Venker used relaxation flow techniques of Bourgade, Erd˝

  • s, B. Schlein, and H.-T. Yau (2011,2012).
  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 18 / 25

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SLIDE 114

Semicircle Law Let Xjk, 1 ≤ j ≤ k < ∞ triangular array, s.th. E Xjk = 0 and E X 2

jk = σ2 jk,

Xjk = Xkj for 1 ≤ j < k < ∞.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 19 / 25

slide-115
SLIDE 115

Semicircle Law Let Xjk, 1 ≤ j ≤ k < ∞ triangular array, s.th. E Xjk = 0 and E X 2

jk = σ2 jk,

Xjk = Xkj for 1 ≤ j < k < ∞. Let Xn := {Xjk}n

j,k=1.

λ1 ≤ . . . ≤ λn eigenvalues of n−1/2Xn with spectral distr.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 19 / 25

slide-116
SLIDE 116

Semicircle Law Let Xjk, 1 ≤ j ≤ k < ∞ triangular array, s.th. E Xjk = 0 and E X 2

jk = σ2 jk,

Xjk = Xkj for 1 ≤ j < k < ∞. Let Xn := {Xjk}n

j,k=1.

λ1 ≤ . . . ≤ λn eigenvalues of n−1/2Xn with spectral distr. FXn(x) = 1 n

n

  • i=1

1(λi ≤ x),

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 19 / 25

slide-117
SLIDE 117

Semicircle Law Let Xjk, 1 ≤ j ≤ k < ∞ triangular array, s.th. E Xjk = 0 and E X 2

jk = σ2 jk,

Xjk = Xkj for 1 ≤ j < k < ∞. Let Xn := {Xjk}n

j,k=1.

λ1 ≤ . . . ≤ λn eigenvalues of n−1/2Xn with spectral distr. FXn(x) = 1 n

n

  • i=1

1(λi ≤ x), Let F Xn(x) := E FXn(x) and G(x) d.f. of standard semicircle law.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 19 / 25

slide-118
SLIDE 118

Semicircle Law Let Xjk, 1 ≤ j ≤ k < ∞ triangular array, s.th. E Xjk = 0 and E X 2

jk = σ2 jk,

Xjk = Xkj for 1 ≤ j < k < ∞. Let Xn := {Xjk}n

j,k=1.

λ1 ≤ . . . ≤ λn eigenvalues of n−1/2Xn with spectral distr. FXn(x) = 1 n

n

  • i=1

1(λi ≤ x), Let F Xn(x) := E FXn(x) and G(x) d.f. of standard semicircle law. σ-algebras F(i,j) := σ{Xkl : 1 ≤ k ≤ l ≤ n, (k, l) = (i, j)}, 1 ≤ i ≤ j ≤ n.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 19 / 25

slide-119
SLIDE 119

Semicircle Law for Martingale Ensembles For any τ > 0 matricial Lindeberg:

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 20 / 25

slide-120
SLIDE 120

Semicircle Law for Martingale Ensembles For any τ > 0 matricial Lindeberg: Ln(τ) := 1 n2

n

  • i,j=1

E |Xij|2 1(|Xij| ≥ τ √ n) → 0 as n → ∞. (1) In addition:

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 20 / 25

slide-121
SLIDE 121

Semicircle Law for Martingale Ensembles For any τ > 0 matricial Lindeberg: Ln(τ) := 1 n2

n

  • i,j=1

E |Xij|2 1(|Xij| ≥ τ √ n) → 0 as n → ∞. (1) In addition: E(Xij|F(i,j)) = 0; and (2

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 20 / 25

slide-122
SLIDE 122

Semicircle Law for Martingale Ensembles For any τ > 0 matricial Lindeberg: Ln(τ) := 1 n2

n

  • i,j=1

E |Xij|2 1(|Xij| ≥ τ √ n) → 0 as n → ∞. (1) In addition: E(Xij|F(i,j)) = 0; and (2 1 n2

n

  • i,j=1

E | E(X 2

ij |F(i,j)) − σ2 ij |

→ 0 as n → ∞; (3)

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 20 / 25

slide-123
SLIDE 123

Semicircle Law for Martingale Ensembles For any τ > 0 matricial Lindeberg: Ln(τ) := 1 n2

n

  • i,j=1

E |Xij|2 1(|Xij| ≥ τ √ n) → 0 as n → ∞. (1) In addition: E(Xij|F(i,j)) = 0; and (2 1 n2

n

  • i,j=1

E | E(X 2

ij |F(i,j)) − σ2 ij |

→ 0 as n → ∞; (3) For all 1 ≤ i ≤ n: average column-variances B2

i := 1

n

n

  • j=1

σ2

ij .

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 20 / 25

slide-124
SLIDE 124

Martingale Wigner Law

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 21 / 25

slide-125
SLIDE 125

Martingale Wigner Law

1 n

n

i=1 |B2 i − 1| → 0

as n → ∞;

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 21 / 25

slide-126
SLIDE 126

Martingale Wigner Law

1 n

n

i=1 |B2 i − 1| → 0

as n → ∞; (4) max1≤i≤n Bi ≤ C < ∞, absolute constant. (5)

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 21 / 25

slide-127
SLIDE 127

Martingale Wigner Law

1 n

n

i=1 |B2 i − 1| → 0

as n → ∞; (4) max1≤i≤n Bi ≤ C < ∞, absolute constant. (5)

Theorem (G.-Naumov-Tikhomirov 2012)

Let Xn satisfy conditions (1)–(5). Then sup

x

|F Xn(x) − G(x)| → 0 as n → ∞.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 21 / 25

slide-128
SLIDE 128

Martingale Wigner Law

1 n

n

i=1 |B2 i − 1| → 0

as n → ∞; (4) max1≤i≤n Bi ≤ C < ∞, absolute constant. (5)

Theorem (G.-Naumov-Tikhomirov 2012)

Let Xn satisfy conditions (1)–(5). Then sup

x

|F Xn(x) − G(x)| → 0 as n → ∞. Previous results with σij = const.:

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 21 / 25

slide-129
SLIDE 129

Martingale Wigner Law

1 n

n

i=1 |B2 i − 1| → 0

as n → ∞; (4) max1≤i≤n Bi ≤ C < ∞, absolute constant. (5)

Theorem (G.-Naumov-Tikhomirov 2012)

Let Xn satisfy conditions (1)–(5). Then sup

x

|F Xn(x) − G(x)| → 0 as n → ∞. Previous results with σij = const.:

Wigner (1958), Arnold (1971), Pastur (1973), G.-Tikhomirov (2006).

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 21 / 25

slide-130
SLIDE 130

Martingale Wigner Law

1 n

n

i=1 |B2 i − 1| → 0

as n → ∞; (4) max1≤i≤n Bi ≤ C < ∞, absolute constant. (5)

Theorem (G.-Naumov-Tikhomirov 2012)

Let Xn satisfy conditions (1)–(5). Then sup

x

|F Xn(x) − G(x)| → 0 as n → ∞. Previous results with σij = const.:

Wigner (1958), Arnold (1971), Pastur (1973), G.-Tikhomirov (2006).

extensions σi,j:

Erd˝

  • s, Yau and Yin (2010) used maxi,j σ2

ij .

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 21 / 25

slide-131
SLIDE 131

Martingale Wigner Law

1 n

n

i=1 |B2 i − 1| → 0

as n → ∞; (4) max1≤i≤n Bi ≤ C < ∞, absolute constant. (5)

Theorem (G.-Naumov-Tikhomirov 2012)

Let Xn satisfy conditions (1)–(5). Then sup

x

|F Xn(x) − G(x)| → 0 as n → ∞. Previous results with σij = const.:

Wigner (1958), Arnold (1971), Pastur (1973), G.-Tikhomirov (2006).

extensions σi,j:

Erd˝

  • s, Yau and Yin (2010) used maxi,j σ2

ij .

Note that (4)-(5) is implied by

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 21 / 25

slide-132
SLIDE 132

Martingale Wigner Law

1 n

n

i=1 |B2 i − 1| → 0

as n → ∞; (4) max1≤i≤n Bi ≤ C < ∞, absolute constant. (5)

Theorem (G.-Naumov-Tikhomirov 2012)

Let Xn satisfy conditions (1)–(5). Then sup

x

|F Xn(x) − G(x)| → 0 as n → ∞. Previous results with σij = const.:

Wigner (1958), Arnold (1971), Pastur (1973), G.-Tikhomirov (2006).

extensions σi,j:

Erd˝

  • s, Yau and Yin (2010) used maxi,j σ2

ij .

Note that (4)-(5) is implied by max

1≤i≤n

  • B2

i − 1

  • → 0

as n → ∞.

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 21 / 25

slide-133
SLIDE 133

Martingale Wigner Law

1 n

n

i=1 |B2 i − 1| → 0

as n → ∞; (4) max1≤i≤n Bi ≤ C < ∞, absolute constant. (5)

Theorem (G.-Naumov-Tikhomirov 2012)

Let Xn satisfy conditions (1)–(5). Then sup

x

|F Xn(x) − G(x)| → 0 as n → ∞. Previous results with σij = const.:

Wigner (1958), Arnold (1971), Pastur (1973), G.-Tikhomirov (2006).

extensions σi,j:

Erd˝

  • s, Yau and Yin (2010) used maxi,j σ2

ij .

Note that (4)-(5) is implied by max

1≤i≤n

  • B2

i − 1

  • → 0

as n → ∞. Marcenko-Pastur laws for martingale ensembles: G.-Tikhomirov (2004/6), Adamczak (2011)

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 21 / 25

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SLIDE 134

Counterexamples I

Xn =   A B BT D   , n = 2m, m = 500 even A: m × m symmetric N(0, 1), B: m × m: i.i.d. N(0, 1). D: m × m: N(0, 1) diagonal matrix

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 22 / 25

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SLIDE 135

Counterexamples I

Xn =   A B BT D   , n = 2m, m = 500 even A: m × m symmetric N(0, 1), B: m × m: i.i.d. N(0, 1). D: m × m: N(0, 1) diagonal matrix (4) does not hold: Simulated density of F Xn:

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 22 / 25

slide-136
SLIDE 136

Counterexamples I

Xn =   A B BT D   , n = 2m, m = 500 even A: m × m symmetric N(0, 1), B: m × m: i.i.d. N(0, 1). D: m × m: N(0, 1) diagonal matrix (4) does not hold: Simulated density of F Xn:

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 22 / 25

slide-137
SLIDE 137

Counterexamples II

Xn =   A B BT D   , n = 4000, m = 1000 even A: 1000 × 1000 symmetric N(0, 10) B: 1000 × 3000: i.i.d. N(0, 1). D: 3000 × 3000: N(0, 10)

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 23 / 25

slide-138
SLIDE 138

Counterexamples II

Xn =   A B BT D   , n = 4000, m = 1000 even A: 1000 × 1000 symmetric N(0, 10) B: 1000 × 3000: i.i.d. N(0, 1). D: 3000 × 3000: N(0, 10) (4) does not hold: Simulated density of F Xn:

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 23 / 25

slide-139
SLIDE 139

Counterexamples II

Xn =   A B BT D   , n = 4000, m = 1000 even A: 1000 × 1000 symmetric N(0, 10) B: 1000 × 3000: i.i.d. N(0, 1). D: 3000 × 3000: N(0, 10) (4) does not hold: Simulated density of F Xn:

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 23 / 25

slide-140
SLIDE 140

Counterexamples II

Xn =   A B BT D   , n = 4000, m = 1000 even A: 1000 × 1000 symmetric N(0, 10) B: 1000 × 3000: i.i.d. N(0, 1). D: 3000 × 3000: N(0, 10) (4) does not hold: Simulated density of F Xn: Can be proved via asymptotic freeness of blocks or Lenczewski (arxiv 2012).

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 23 / 25

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SLIDE 141

Steps of Proof Lindeberg-type universality: Replacing Xij by Gaussian Yij using Stieltjes-transforms and conditional moments Graph summation using moment methods for non identical Gaussian entries

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 24 / 25

slide-142
SLIDE 142

Thank You!

  • F. Götze (Bielefeld)

Local/Global Universality October 9, 2012 25 / 25