Dynamics of a planar Coulomb gas F . Bolley, D. Chafa , J. - - PowerPoint PPT Presentation

Dynamics of a planar Coulomb gas

Dynamics of a planar Coulomb gas

F . Bolley, D. Chafa¨ ı, J. Fontbona

Jussieu, Dauphine, Santiago

Optimal Point Configurations and Orthogonal Polynomials April 19–22, 2017 Castro Urdiales, Cantabria, Spain

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Outline

Poincar´ e for diffusions Dyson Process Ginibre process

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Diffusions

Markov process pXtqtě0 Stochastic Differential Equation dXt “ ? 2dBt ´ ∇HpXtqdt

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Diffusions

Markov process pXtqtě0 Stochastic Differential Equation dXt “ ? 2dBt ´ ∇HpXtqdt Energy H : x P Rd ÞÑ Hpxq P R with e´H P L1pdxq

3/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Diffusions

Markov process pXtqtě0 Stochastic Differential Equation dXt “ ? 2dBt ´ ∇HpXtqdt Energy H : x P Rd ÞÑ Hpxq P R with e´H P L1pdxq Non-explosion: if ∇2H ě c P R

3/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Diffusions

Markov process pXtqtě0 Stochastic Differential Equation dXt “ ? 2dBt ´ ∇HpXtqdt Energy H : x P Rd ÞÑ Hpxq P R with e´H P L1pdxq Non-explosion: if ∇2H ě c P R Reversible (and thus invariant) Boltzmann-Gibbs measure µpdxq “ e´Hpxq Z dx X0 „ µ ñ pX0, Xtq d “ pXt, X0q @t ě 0

3/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Diffusions

Markov process pXtqtě0 Stochastic Differential Equation dXt “ ? 2αdBt ´ α∇HpXtqdt Energy H : x P Rd ÞÑ Hpxq P R with e´H P L1pdxq Non-explosion: if ∇2H ě c P R Reversible (and thus invariant) Boltzmann-Gibbs measure µpdxq “ e´Hpxq Z dx X0 „ µ ñ pX0, Xtq d “ pXt, X0q @t ě 0

3/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Diffusions

Markov process pXtqtě0 Stochastic Differential Equation dXt “ c 2α β dBt ´ α∇HpXtqdt Energy H : x P Rd ÞÑ Hpxq P R with e´H P L1pdxq Non-explosion: if ∇2H ě c P R Reversible (and thus invariant) Boltzmann-Gibbs measure µβpdxq “ e´βHpxq Zβ dx X0 „ µ ñ pX0, Xtq d “ pXt, X0q @t ě 0

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Inifinitesimal generator

Conditional laws: Stp¨qpxq “ LawpXt | X0 “ xq

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Inifinitesimal generator

Conditional laws: Stp¨qpxq “ LawpXt | X0 “ xq Markov semigroup: Stpfqpxq “ EpfpXtq | X0 “ xq S0 “ Identity, St ˝ St1 “ St`t1

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Inifinitesimal generator

Conditional laws: Stp¨qpxq “ LawpXt | X0 “ xq Markov semigroup: Stpfqpxq “ EpfpXtq | X0 “ xq S0 “ Identity, St ˝ St1 “ St`t1 Infinitesimal generator G “ ∆ ´ ∇H ¨ ∇ d dt ˇ ˇ ˇ ˇ

t“0

Stpfqpxq “ ∆fpxq ´ x∇Hpxq, ∇fpxqy “ Gpfqpxq

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Inifinitesimal generator

Conditional laws: Stp¨qpxq “ LawpXt | X0 “ xq Markov semigroup: Stpfqpxq “ EpfpXtq | X0 “ xq S0 “ Identity, St ˝ St1 “ St`t1 Infinitesimal generator G “ ∆ ´ ∇H ¨ ∇ d dt ˇ ˇ ˇ ˇ

t“0

Stpfqpxq “ ∆fpxq ´ x∇Hpxq, ∇fpxqy “ Gpfqpxq The operators G and St “ etG are symmetric in L2pµq

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Inifinitesimal generator

Conditional laws: Stp¨qpxq “ LawpXt | X0 “ xq Markov semigroup: Stpfqpxq “ EpfpXtq | X0 “ xq S0 “ Identity, St ˝ St1 “ St`t1 Infinitesimal generator G “ ∆ ´ ∇H ¨ ∇ d dt ˇ ˇ ˇ ˇ

t“0

Stpfqpxq “ ∆fpxq ´ x∇Hpxq, ∇fpxqy “ Gpfqpxq The operators G and St “ etG are symmetric in L2pµq Fokker-Planck equation if ft “ dµt

dµ with µt “ LawpXtq then

ft “ Stpf0q and Btft “ Gft

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Exactly solvable model: Ornstein-Uhlenbeck process

Gaussian model: Hpxq “ 1

2|x|2, dXt “

? 2dBt ´ Xtdt, µ “ Np0, Idq, Gfpxq “ ∆fpxq ´ xx, ∇fpxqy

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Exactly solvable model: Ornstein-Uhlenbeck process

Gaussian model: Hpxq “ 1

2|x|2, dXt “

? 2dBt ´ Xtdt, µ “ Np0, Idq, Gfpxq “ ∆fpxq ´ xx, ∇fpxqy Mehler formula: Stp¨qpxq “ LawpXt | X0 “ xq “ Npxe´t, 1 ´ e´2tq

5/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Exactly solvable model: Ornstein-Uhlenbeck process

Gaussian model: Hpxq “ 1

2|x|2, dXt “

? 2dBt ´ Xtdt, µ “ Np0, Idq, Gfpxq “ ∆fpxq ´ xx, ∇fpxqy Mehler formula: Stp¨qpxq “ LawpXt | X0 “ xq “ Npxe´t, 1 ´ e´2tq Hermite polynomials:

5/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Exactly solvable model: Ornstein-Uhlenbeck process

Gaussian model: Hpxq “ 1

2|x|2, dXt “

? 2dBt ´ Xtdt, µ “ Np0, Idq, Gfpxq “ ∆fpxq ´ xx, ∇fpxqy Mehler formula: Stp¨qpxq “ LawpXt | X0 “ xq “ Npxe´t, 1 ´ e´2tq Hermite polynomials:

§ L2pµq “ k8

n“0vectpPnq

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Exactly solvable model: Ornstein-Uhlenbeck process

Gaussian model: Hpxq “ 1

2|x|2, dXt “

? 2dBt ´ Xtdt, µ “ Np0, Idq, Gfpxq “ ∆fpxq ´ xx, ∇fpxqy Mehler formula: Stp¨qpxq “ LawpXt | X0 “ xq “ Npxe´t, 1 ´ e´2tq Hermite polynomials:

§ L2pµq “ k8

n“0vectpPnq

§ GPn “ ´nPn and StpPnq “ e´ntPn

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Exactly solvable model: Ornstein-Uhlenbeck process

Gaussian model: Hpxq “ 1

2|x|2, dXt “

? 2dBt ´ Xtdt, µ “ Np0, Idq, Gfpxq “ ∆fpxq ´ xx, ∇fpxqy Mehler formula: Stp¨qpxq “ LawpXt | X0 “ xq “ Npxe´t, 1 ´ e´2tq Hermite polynomials:

§ L2pµq “ k8

n“0vectpPnq

§ GPn “ ´nPn and StpPnq “ e´ntPn § G “ ´ ř8

n“0 nΠPn and St “ ř8 n“0 e´ntΠPn

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Exactly solvable model: Ornstein-Uhlenbeck process

Gaussian model: Hpxq “ 1

2|x|2, dXt “

? 2dBt ´ Xtdt, µ “ Np0, Idq, Gfpxq “ ∆fpxq ´ xx, ∇fpxqy Mehler formula: Stp¨qpxq “ LawpXt | X0 “ xq “ Npxe´t, 1 ´ e´2tq Hermite polynomials:

§ L2pµq “ k8

n“0vectpPnq

§ GPn “ ´nPn and StpPnq “ e´ntPn § G “ ´ ř8

n“0 nΠPn and St “ ř8 n“0 e´ntΠPn

Exponential decay (spectral gap):

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Exactly solvable model: Ornstein-Uhlenbeck process

Gaussian model: Hpxq “ 1

2|x|2, dXt “

? 2dBt ´ Xtdt, µ “ Np0, Idq, Gfpxq “ ∆fpxq ´ xx, ∇fpxqy Mehler formula: Stp¨qpxq “ LawpXt | X0 “ xq “ Npxe´t, 1 ´ e´2tq Hermite polynomials:

§ L2pµq “ k8

n“0vectpPnq

§ GPn “ ´nPn and StpPnq “ e´ntPn § G “ ´ ř8

n“0 nΠPn and St “ ř8 n“0 e´ntΠPn

Exponential decay (spectral gap):

§ }ft ´ 1}L2pµq ď e´t}f0 ´ 1}L2pµq

5/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Exactly solvable model: Ornstein-Uhlenbeck process

Gaussian model: Hpxq “ 1

2|x|2, dXt “

? 2dBt ´ Xtdt, µ “ Np0, Idq, Gfpxq “ ∆fpxq ´ xx, ∇fpxqy Mehler formula: Stp¨qpxq “ LawpXt | X0 “ xq “ Npxe´t, 1 ´ e´2tq Hermite polynomials:

§ L2pµq “ k8

n“0vectpPnq

§ GPn “ ´nPn and StpPnq “ e´ntPn § G “ ´ ř8

n“0 nΠPn and St “ ř8 n“0 e´ntΠPn

Exponential decay (spectral gap):

§ }ft ´ 1}L2pµq ď e´t}f0 ´ 1}L2pµq § VarµpStfq ď e´tVarµpfq

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Exactly solvable model: Ornstein-Uhlenbeck process

Gaussian model: Hpxq “ 1

2|x|2, dXt “

? 2dBt ´ Xtdt, µ “ Np0, Idq, Gfpxq “ ∆fpxq ´ xx, ∇fpxqy Mehler formula: Stp¨qpxq “ LawpXt | X0 “ xq “ Npxe´t, 1 ´ e´2tq Hermite polynomials:

§ L2pµq “ k8

n“0vectpPnq

§ GPn “ ´nPn and StpPnq “ e´ntPn § G “ ´ ř8

n“0 nΠPn and St “ ř8 n“0 e´ntΠPn

Exponential decay (spectral gap):

§ }ft ´ 1}L2pµq ď e´t}f0 ´ 1}L2pµq § VarµpStfq ď e´tVarµpfq

Poincar´ e inequality: Varµpfq ď ´EµpfGfq “ Eµp|∇f|2q

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Comparison to Gaussianity via convexity

Theorem (Brascamp–Lieb 1976)

If ∇2H ą 0 on Rd then for any smooth f : Rd Ñ R, Varµpfq ď Eµxp∇2Hq´1∇f, ∇fy

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Comparison to Gaussianity via convexity

Theorem (Brascamp–Lieb 1976)

If ∇2H ą 0 on Rd then for any smooth f : Rd Ñ R, Varµpfq ď Eµxp∇2Hq´1∇f, ∇fy Proof by induction on dimension d

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Comparison to Gaussianity via convexity

Theorem (Brascamp–Lieb 1976)

If ∇2H ą 0 on Rd then for any smooth f : Rd Ñ R, Varµpfq ď Eµxp∇2Hq´1∇f, ∇fy Proof by induction on dimension d Ornstein–Uhlenbeck: ∇2H “ Id

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Comparison to Gaussianity via convexity

Theorem (Brascamp–Lieb 1976)

If ∇2H ą 0 on Rd then for any smooth f : Rd Ñ R, Varµpfq ď Eµxp∇2Hq´1∇f, ∇fy Proof by induction on dimension d Ornstein–Uhlenbeck: ∇2H “ Id Convexity: ∇2H ě ρ ą 0 gives Poincar´ e with 1{ρ

6/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Comparison to Gaussianity via convexity

Theorem (Brascamp–Lieb 1976)

If ∇2H ą 0 on Rd then for any smooth f : Rd Ñ R, Varµpfq ď Eµxp∇2Hq´1∇f, ∇fy Proof by induction on dimension d Ornstein–Uhlenbeck: ∇2H “ Id Convexity: ∇2H ě ρ ą 0 gives Poincar´ e with 1{ρ H convex means that µpdxq “ e´Hdx is log-concave

6/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Comparison to Gaussianity via convexity

Theorem (Brascamp–Lieb 1976)

If ∇2H ą 0 on Rd then for any smooth f : Rd Ñ R, Varµpfq ď Eµxp∇2Hq´1∇f, ∇fy Proof by induction on dimension d Ornstein–Uhlenbeck: ∇2H “ Id Convexity: ∇2H ě ρ ą 0 gives Poincar´ e with 1{ρ H convex means that µpdxq “ e´Hdx is log-concave Jensen divergence: Varµpfq “ EµΦpfq ´ ΦpEµfq, Φpuq “ u2

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Comparison to Gaussianity via convexity

Theorem (Bakry– ´ Emery 1984)

If ∇2H ě ρ ą 0 then for any convex Φ : I Ñ R with pu, vq ÞÑ Φ2puqv2 convex and any smooth f : Rd Ñ I EµΦpfq ´ ΦpEµfq ď 1 ρEµpΦ2pfq|∇f|2q

7/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Comparison to Gaussianity via convexity

Theorem (Bakry– ´ Emery 1984)

If ∇2H ě ρ ą 0 then for any convex Φ : I Ñ R with pu, vq ÞÑ Φ2puqv2 convex and any smooth f : Rd Ñ I EµΦpfq ´ ΦpEµfq ď 1 ρEµpΦ2pfq|∇f|2q Proof by semigroup interpolation St´upΦpSupfqqq

7/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Comparison to Gaussianity via convexity

Theorem (Bakry– ´ Emery 1984)

If ∇2H ě ρ ą 0 then for any convex Φ : I Ñ R with pu, vq ÞÑ Φ2puqv2 convex and any smooth f : Rd Ñ I EµΦpfq ´ ΦpEµfq ď 1 ρEµpΦ2pfq|∇f|2q Proof by semigroup interpolation St´upΦpSupfqqq Ornstein–Uhlenbeck: ∇2H “ Id

7/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Comparison to Gaussianity via convexity

Theorem (Bakry– ´ Emery 1984)

If ∇2H ě ρ ą 0 then for any convex Φ : I Ñ R with pu, vq ÞÑ Φ2puqv2 convex and any smooth f : Rd Ñ I EµΦpfq ´ ΦpEµfq ď 1 ρEµpΦ2pfq|∇f|2q Proof by semigroup interpolation St´upΦpSupfqqq Ornstein–Uhlenbeck: ∇2H “ Id Poincar´ e: I “ R, Φpuq “ u2

7/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Comparison to Gaussianity via convexity

Theorem (Bakry– ´ Emery 1984)

If ∇2H ě ρ ą 0 then for any convex Φ : I Ñ R with pu, vq ÞÑ Φ2puqv2 convex and any smooth f : Rd Ñ I EµΦpfq ´ ΦpEµfq ď 1 ρEµpΦ2pfq|∇f|2q Proof by semigroup interpolation St´upΦpSupfqqq Ornstein–Uhlenbeck: ∇2H “ Id Poincar´ e: I “ R, Φpuq “ u2 Beckner: I “ R`, Φpuq “ up, 1 ă p ď 2

7/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Comparison to Gaussianity via convexity

Theorem (Bakry– ´ Emery 1984)

If ∇2H ě ρ ą 0 then for any convex Φ : I Ñ R with pu, vq ÞÑ Φ2puqv2 convex and any smooth f : Rd Ñ I EµΦpfq ´ ΦpEµfq ď 1 ρEµpΦ2pfq|∇f|2q Proof by semigroup interpolation St´upΦpSupfqqq Ornstein–Uhlenbeck: ∇2H “ Id Poincar´ e: I “ R, Φpuq “ u2 Beckner: I “ R`, Φpuq “ up, 1 ă p ď 2 Logarithmic Sobolev: I “ R`, Φpuq “ u logpuq

7/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Comparison to Gaussianity via convexity

Theorem (Caffarelli 2000)

If ∇2H ě ρ ą 0 then µ is the image of Np0, Idq by a Lipschitz function F : Rd Ñ Rd with }F}Lip ď

1 ?ρ.

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Comparison to Gaussianity via convexity

Theorem (Caffarelli 2000)

If ∇2H ě ρ ą 0 then µ is the image of Np0, Idq by a Lipschitz function F : Rd Ñ Rd with }F}Lip ď

1 ?ρ.

Proof by Monge-Amp` ere equation: f “ detpDFqg

8/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Comparison to Gaussianity via convexity

Theorem (Caffarelli 2000)

If ∇2H ě ρ ą 0 then µ is the image of Np0, Idq by a Lipschitz function F : Rd Ñ Rd with }F}Lip ď

1 ?ρ.

Proof by Monge-Amp` ere equation: f “ detpDFqg Gives Poincar´ e from the Gaussian by transportation: Varµpfq “ VarNp0,IdqpfpFqq ď ENp0,Idqp|∇fpFq|2q ď 1 ρEµp|∇f|2q

8/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

Comparison to Gaussianity via convexity

Theorem (Caffarelli 2000)

If ∇2H ě ρ ą 0 then µ is the image of Np0, Idq by a Lipschitz function F : Rd Ñ Rd with }F}Lip ď

1 ?ρ.

Proof by Monge-Amp` ere equation: f “ detpDFqg Gives Poincar´ e from the Gaussian by transportation: Varµpfq “ VarNp0,IdqpfpFqq ď ENp0,Idqp|∇fpFq|2q ď 1 ρEµp|∇f|2q Gives also any Φ-Sobolev inequality from the Gaussian!

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Dynamics of a planar Coulomb gas Poincar´ e for diffusions

KLS conjecture

Conjecture (Kannan–Lov´ asz–Simonovits 1995)

If ∇2H ě 0 with Eµpxixjq “ δij then µ satisfies to a Poincar´ e inequality with a universal constant (independent of d and H). . . .

9/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

KLS conjecture

Conjecture (Kannan–Lov´ asz–Simonovits 1995)

If ∇2H ě 0 with Eµpxixjq “ δij then µ satisfies to a Poincar´ e inequality with a universal constant (independent of d and H). . . . Bobkov 1999: true with d1{2

9/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

KLS conjecture

Conjecture (Kannan–Lov´ asz–Simonovits 1995)

If ∇2H ě 0 with Eµpxixjq “ δij then µ satisfies to a Poincar´ e inequality with a universal constant (independent of d and H). . . . Bobkov 1999: true with d1{2 . . . , Bourgain, . . . , Klartag, . . . , Eldan, . . .

9/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions

KLS conjecture

Conjecture (Kannan–Lov´ asz–Simonovits 1995)

If ∇2H ě 0 with Eµpxixjq “ δij then µ satisfies to a Poincar´ e inequality with a universal constant (independent of d and H). . . . Bobkov 1999: true with d1{2 . . . , Bourgain, . . . , Klartag, . . . , Eldan, . . . Lee & Vempala 2016: true with d1{4

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Dynamics of a planar Coulomb gas Dyson Process

Outline

Poincar´ e for diffusions Dyson Process Ginibre process

10/20

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Hermitian Random Matrices

HermNˆN ” RN`2 N2´N

2

“ RN2 “ Rd

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Dynamics of a planar Coulomb gas Dyson Process

Gaussian Hermitian Random Matrices

HermNˆN ” RN`2 N2´N

2

“ RN2 “ Rd Boltzmann–Gibbs measure µpdMq “ e´ N

2 TrpM2q

Z dM

11/20

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Hermitian Random Matrices

HermNˆN ” RN`2 N2´N

2

“ RN2 “ Rd Boltzmann–Gibbs measure µpdMq “ e´ N

2 TrpM2q

Z dM Stochastic Differential Equation ` a la Ornstein–Uhlenbeck dMt “ ? 2dBt ´ NMtdt.

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Dynamics of a planar Coulomb gas Dyson Process

Gaussian Hermitian Random Matrices

HermNˆN ” RN`2 N2´N

2

“ RN2 “ Rd Boltzmann–Gibbs measure µpdMq “ e´ N

2 TrpM2q

Z dM Stochastic Differential Equation ` a la Ornstein–Uhlenbeck dMt “ a 2αNdBt ´ αNNMtdt.

11/20

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Hermitian Random Matrices

HermNˆN ” RN`2 N2´N

2

“ RN2 “ Rd Boltzmann–Gibbs measure µpdMq “ e´ N

2 TrpM2q

Z dM Stochastic Differential Equation ` a la Ornstein–Uhlenbeck dMt “ c 2 N dBt ´ Mtdt.

11/20

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Hermitian Random Matrices

HermNˆN ” RN`2 N2´N

2

“ RN2 “ Rd Boltzmann–Gibbs measure µpdMq “ e´ N

2 TrpM2q

Z dM Stochastic Differential Equation ` a la Ornstein–Uhlenbeck dMt “ c 2 N dBt ´ Mtdt. Change of variable: if specpMq “ tx1, . . . , xNu, M “ UDU˚ with D “ diagpx1, . . . , xNq

11/20

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Hermitian Random Matrices

HermNˆN ” RN`2 N2´N

2

“ RN2 “ Rd Boltzmann–Gibbs measure µpdMq “ e´ N

2 TrpM2q

Z dM Stochastic Differential Equation ` a la Ornstein–Uhlenbeck dMt “ c 2 N dBt ´ Mtdt. Change of variable: if specpMq “ tx1, . . . , xNu, M “ UDU˚ with D “ diagpx1, . . . , xNq Stochastic process of spectrum?

11/20

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Unitary Ensemble and Dyson Process

State space D “ tpx1, . . . , xNq P RN : x1 ă ¨ ¨ ¨ ă xNu

12/20

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Unitary Ensemble and Dyson Process

State space D “ tpx1, . . . , xNq P RN : x1 ă ¨ ¨ ¨ ă xNu Boltzmann–Gibbs measure via change of variable µpdxq “ e´ N

2

řN

i“1 x2 i ś

iăjpxj ´ xiq2

Z dx

12/20

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Unitary Ensemble and Dyson Process

State space D “ tpx1, . . . , xNq P RN : x1 ă ¨ ¨ ¨ ă xNu Boltzmann–Gibbs measure via change of variable µpdxq “ e

´ N

2

řN

i“1 x2 i ´2 ř iăj log 1 xj ´xi

Z dx

12/20

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Unitary Ensemble and Dyson Process

State space D “ tpx1, . . . , xNq P RN : x1 ă ¨ ¨ ¨ ă xNu Boltzmann–Gibbs measure via change of variable µpdxq “ e

´ N

2

řN

i“1 x2 i ´2 ř iăj log 1 xj ´xi

Z dx Dyson Ornstein–Uhlenbeck process via Itˆ

• formula

dXt,i “ c 2 N dBt,i´ ˆ Xt,i ` 2 N ÿ

iăj

1 Xt,j ´ Xt,i ˙ dt

12/20

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Unitary Ensemble and Dyson Process

State space D “ tpx1, . . . , xNq P RN : x1 ă ¨ ¨ ¨ ă xNu Boltzmann–Gibbs measure via change of variable µpdxq “ e

´ N

2

řN

i“1 x2 i ´2 ř iăj log 1 xj ´xi

Z dx Dyson Ornstein–Uhlenbeck process via Itˆ

• formula

dXt,i “ c 2 N dBt,i´ ˆ Xt,i ` 2 N ÿ

iăj

1 Xt,j ´ Xt,i ˙ dt Non-explosion and solution: . . . , Rogers–Shi, . . . , AGZ, . . .

12/20

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Unitary Ensemble and Dyson Process

State space D “ tpx1, . . . , xNq P RN : x1 ă ¨ ¨ ¨ ă xNu Boltzmann–Gibbs measure via change of variable µpdxq “ e

´ N

2

řN

i“1 x2 i ´2 ř iăj log 1 xj ´xi

Z dx Dyson Ornstein–Uhlenbeck process via Itˆ

• formula

dXt,i “ c 2 N dBt,i´ ˆ Xt,i ` 2 N ÿ

iăj

1 Xt,j ´ Xt,i ˙ dt Non-explosion and solution: . . . , Rogers–Shi, . . . , AGZ, . . . Poincar´ e and log-Sobolev: . . . , Erd˝

• s–Yau et al, . . .

12/20

Dynamics of a planar Coulomb gas Dyson Process

Gaussian Unitary Ensemble and Dyson Process

State space D “ tpx1, . . . , xNq P RN : x1 ă ¨ ¨ ¨ ă xNu Boltzmann–Gibbs measure via change of variable µpdxq “ e

´ N

2

řN

i“1 x2 i ´2 ř iăj log 1 xj ´xi

Z dx Dyson Ornstein–Uhlenbeck process via Itˆ

• formula

dXt,i “ c 2 N dBt,i´ ˆ Xt,i ` 2 N ÿ

iăj

1 Xt,j ´ Xt,i ˙ dt Non-explosion and solution: . . . , Rogers–Shi, . . . , AGZ, . . . Poincar´ e and log-Sobolev: . . . , Erd˝

• s–Yau et al, . . .

Erd˝

• s–Yau: book Dynamical approach to RMT

12/20

Dynamics of a planar Coulomb gas Dyson Process

Optimal Poincar´ e constant (mind the gap!)

Boltzmann–Gibbs measure µpdxq “ e´Hpxq Z dx with Hpxq “ N 2

N

ÿ

i“1

x2

i `2

ÿ

iăj

log 1 xj ´ xi

13/20

Dynamics of a planar Coulomb gas Dyson Process

Optimal Poincar´ e constant (mind the gap!)

Boltzmann–Gibbs measure µpdxq “ e´Hpxq Z dx with Hpxq “ N 2

N

ÿ

i“1

x2

i `2

ÿ

iăj

log 1 xj ´ xi Log-concavity ∇2Hpxq ě N.

13/20

Dynamics of a planar Coulomb gas Dyson Process

Optimal Poincar´ e constant (mind the gap!)

Boltzmann–Gibbs measure µpdxq “ e´Hpxq Z dx with Hpxq “ N 2

N

ÿ

i“1

x2

i `2

ÿ

iăj

log 1 xj ´ xi Log-concavity ∇2Hpxq ě N. Brascamp–Lieb or Bakry– ´ Emery or Caffarelli Varµpfq ď 1 N Eµp|∇f|2q.

13/20

Dynamics of a planar Coulomb gas Dyson Process

Optimal Poincar´ e constant (mind the gap!)

Boltzmann–Gibbs measure µpdxq “ e´Hpxq Z dx with Hpxq “ N 2

N

ÿ

i“1

x2

i `2

ÿ

iăj

log 1 xj ´ xi Log-concavity ∇2Hpxq ě N. Brascamp–Lieb or Bakry– ´ Emery or Caffarelli Varµpfq ď 1 N Eµp|∇f|2q. Equality achieved for fpxq “ x1 ` ¨ ¨ ¨ ` xN (compute traces)

13/20

Dynamics of a planar Coulomb gas Dyson Process

Optimal Poincar´ e constant (mind the gap!)

Boltzmann–Gibbs measure µpdxq “ e´Hpxq Z dx with Hpxq “ N 2

N

ÿ

i“1

x2

i `2

ÿ

iăj

log 1 xj ´ xi Log-concavity ∇2Hpxq ě N. Brascamp–Lieb or Bakry– ´ Emery or Caffarelli Varµpfq ď 1 N Eµp|∇f|2q. Equality achieved for fpxq “ x1 ` ¨ ¨ ¨ ` xN (compute traces) Lipschitz deformation of Gaussian (Hoffman–Wielandt)

13/20

Dynamics of a planar Coulomb gas Ginibre process

Outline

Poincar´ e for diffusions Dyson Process Ginibre process

14/20

Dynamics of a planar Coulomb gas Ginibre process

Ginibre process

Boltzmann–Gibbs measure on MatNˆNpCq µpMq “ e´NTrpMM˚q Z dM

15/20

Dynamics of a planar Coulomb gas Ginibre process

Ginibre process

Boltzmann–Gibbs measure on MatNˆNpCq µpMq “ e´NTrpMM˚q Z dM Schur unitary decomposition: if tx1, . . . , xNu “ specpMq, M “ UTU˚ with T “ D`N and D “ diagpx1, . . . , xNq.

15/20

Dynamics of a planar Coulomb gas Ginibre process

Ginibre process

Boltzmann–Gibbs measure on MatNˆNpCq µpMq “ e´NTrpMM˚q Z dM Schur unitary decomposition: if tx1, . . . , xNu “ specpMq, M “ UTU˚ with T “ D`N and D “ diagpx1, . . . , xNq. Lack of normality is generic: µptN “ 0uq “ 0

15/20

Dynamics of a planar Coulomb gas Ginibre process

Ginibre process

Boltzmann–Gibbs measure on MatNˆNpCq µpMq “ e´NTrpMM˚q Z dM Schur unitary decomposition: if tx1, . . . , xNu “ specpMq, M “ UTU˚ with T “ D`N and D “ diagpx1, . . . , xNq. Lack of normality is generic: µptN “ 0uq “ 0 Process on spectrum melts N and D (Ñ G. Dubach PhD)

15/20

Dynamics of a planar Coulomb gas Ginibre process

Ginibre process

Boltzmann–Gibbs measure on MatNˆNpCq µpMq “ e´NTrpMM˚q Z dM Schur unitary decomposition: if tx1, . . . , xNu “ specpMq, M “ UTU˚ with T “ D`N and D “ diagpx1, . . . , xNq. Lack of normality is generic: µptN “ 0uq “ 0 Process on spectrum melts N and D (Ñ G. Dubach PhD) How about direct analogue of Dyson O.–U. process on x?

15/20

Dynamics of a planar Coulomb gas Ginibre process

State space D “ CNz Yi‰j tpx1, . . . , xNq P CN : xi “ xju

16/20

Dynamics of a planar Coulomb gas Ginibre process

State space D “ CNz Yi‰j tpx1, . . . , xNq P CN : xi “ xju Boltzmann–Gibbs measure via change of variable µpdxq “ e´N řN

i“1 |xi|2

Z ź

iăj

|xi ´ xj|2dx

16/20

Dynamics of a planar Coulomb gas Ginibre process

State space D “ CNz Yi‰j tpx1, . . . , xNq P CN : xi “ xju Boltzmann–Gibbs measure via change of variable µpdxq “ e

´N řN

i“1 |xi|2´2 ř iăj log 1 |xi ´xj |

Z dx

16/20

Dynamics of a planar Coulomb gas Ginibre process

State space D “ CNz Yi‰j tpx1, . . . , xNq P CN : xi “ xju Boltzmann–Gibbs measure via change of variable µpdxq “ e

´N řN

i“1 |xi|2´2 ř iăj log 1 |xi ´xj |

Z dx Ginibre process on CN “ pR2qN dXt,i “ c 2 N dBt,i ´ 2Xt,idt ´ 2 N ÿ

j‰i

Xt,i ´ Xt,j ˇ ˇXt,i ´ Xt,j ˇ ˇ2 dt.

16/20

Dynamics of a planar Coulomb gas Ginibre process

State space D “ CNz Yi‰j tpx1, . . . , xNq P CN : xi “ xju Boltzmann–Gibbs measure via change of variable µpdxq “ e

´βN ˆ

1 N

řN

i“1 |xi|2` 1 N2

ř

iăj log 1 |xi ´xj |

˙

Z dx Ginibre process on CN “ pR2qN dXt,i “ c 2αN βN dBt,i ´ 2αN N Xt,idt ´ 2αN N ÿ

j‰i

Xt,i ´ Xt,j ˇ ˇXt,i ´ Xt,j ˇ ˇ2 dt.

16/20

Dynamics of a planar Coulomb gas Ginibre process

State space D “ CNz Yi‰j tpx1, . . . , xNq P CN : xi “ xju Boltzmann–Gibbs measure via change of variable µpdxq “ e

´βN ˆ

1 N

řN

i“1 |xi|2` 1 N2

ř

iăj log 1 |xi ´xj |

˙

Z dx Ginibre process on CN “ pR2qN dXt,i “ c 2αN βN dBt,i ´ 2αN N Xt,idt ´ 2αN N ÿ

j‰i

Xt,i ´ Xt,j ˇ ˇXt,i ´ Xt,j ˇ ˇ2 dt. RMT: βN “ N2

16/20

Dynamics of a planar Coulomb gas Ginibre process

State space D “ CNz Yi‰j tpx1, . . . , xNq P CN : xi “ xju Boltzmann–Gibbs measure via change of variable µpdxq “ e

´βN ˆ

1 N

řN

i“1 |xi|2` 1 N2

ř

iăj log 1 |xi ´xj |

˙

Z dx Ginibre process on CN “ pR2qN dXt,i “ c 2αN βN dBt,i ´ 2αN N Xt,idt ´ 2αN N ÿ

j‰i

Xt,i ´ Xt,j ˇ ˇXt,i ´ Xt,j ˇ ˇ2 dt. RMT: βN “ N2 No convexity / Brascamp–Lieb / Bakry– ´ Emery / Caffarelli

16/20

Dynamics of a planar Coulomb gas Ginibre process

State space D “ CNz Yi‰j tpx1, . . . , xNq P CN : xi “ xju Boltzmann–Gibbs measure via change of variable µpdxq “ e

´βN ˆ

1 N

řN

i“1 |xi|2` 1 N2

ř

iăj log 1 |xi ´xj |

˙

Z dx Ginibre process on CN “ pR2qN dXt,i “ c 2αN βN dBt,i ´ 2αN N Xt,idt ´ 2αN N ÿ

j‰i

Xt,i ´ Xt,j ˇ ˇXt,i ´ Xt,j ˇ ˇ2 dt. RMT: βN “ N2 No convexity / Brascamp–Lieb / Bakry– ´ Emery / Caffarelli No Hoffman–Wielandt for non-normal matrices

16/20

Dynamics of a planar Coulomb gas Ginibre process

First results

Theorem (Absence of explosion)

For any X0 “ x P D, PpTBD “ `8q “ 1

17/20

Dynamics of a planar Coulomb gas Ginibre process

First results

Theorem (Absence of explosion)

For any X0 “ x P D, PpTBD “ `8q “ 1

Theorem (Poincar´ e inequality)

For any N, the law µN satisfies a Poincar´ e inequality.

17/20

Dynamics of a planar Coulomb gas Ginibre process

First results

Theorem (Absence of explosion)

For any X0 “ x P D, PpTBD “ `8q “ 1

Theorem (Poincar´ e inequality)

For any N, the law µN satisfies a Poincar´ e inequality.

Theorem (Second moment dynamics)

For any x P D and t ě 0, we have E ˆ|Xt|2 N | X0 “ x ˙ “ |x| N e´

4αN N t`

ˆ1 2 ` N βN ´ 1 2N ˙´ 1 ´ e´

4αN N t¯

.

17/20

Dynamics of a planar Coulomb gas Ginibre process

Uniform Poincar´ e for the one particle marginal

Theorem (Uniform Poincar´ e for one-particle)

If βN “ N2 then the one-particle marginal of µ is log-concave and satisfies a Poincar´ e inequality with a constant uniform in N.

18/20

Dynamics of a planar Coulomb gas Ginibre process

Uniform Poincar´ e for the one particle marginal

Theorem (Uniform Poincar´ e for one-particle)

If βN “ N2 then the one-particle marginal of µ is log-concave and satisfies a Poincar´ e inequality with a constant uniform in N. If βN “ N2 then one particle marginal of µ has density z P C ÞÑ ϕpzq “ e´N|z|2 π

N´1

ÿ

ℓ“0

Nℓ|z|2ℓ ℓ! .

18/20

Dynamics of a planar Coulomb gas Ginibre process

Uniform Poincar´ e for the one particle marginal

Theorem (Uniform Poincar´ e for one-particle)

If βN “ N2 then the one-particle marginal of µ is log-concave and satisfies a Poincar´ e inequality with a constant uniform in N. If βN “ N2 then one particle marginal of µ has density z P C ÞÑ ϕpzq “ e´N|z|2 π

N´1

ÿ

ℓ“0

Nℓ|z|2ℓ ℓ! . Circular law lim

NÑ8 sup zPK

ˇ ˇ ˇ ˇϕpzq ´ 1t|z|ď1u π ˇ ˇ ˇ ˇ “ 0.

18/20

Dynamics of a planar Coulomb gas Ginibre process

Uniform Poincar´ e for the one particle marginal

Theorem (Uniform Poincar´ e for one-particle)

If βN “ N2 then the one-particle marginal of µ is log-concave and satisfies a Poincar´ e inequality with a constant uniform in N. If βN “ N2 then one particle marginal of µ has density z P C ÞÑ ϕpzq “ e´N|z|2 π

N´1

ÿ

ℓ“0

Nℓ|z|2ℓ ℓ! . Circular law lim

NÑ8 sup zPK

ˇ ˇ ˇ ˇϕpzq ´ 1t|z|ď1u π ˇ ˇ ˇ ˇ “ 0. The function z ÞÑ log řN´1

ℓ“0 |z|2ℓ ℓ!

is concave!

18/20

Dynamics of a planar Coulomb gas Ginibre process

Uniform Poincar´ e for the one particle marginal

Theorem (Uniform Poincar´ e for one-particle)

If βN “ N2 then the one-particle marginal of µ is log-concave and satisfies a Poincar´ e inequality with a constant uniform in N. If βN “ N2 then one particle marginal of µ has density z P C ÞÑ ϕpzq “ e´N|z|2 π

N´1

ÿ

ℓ“0

Nℓ|z|2ℓ ℓ! . Circular law lim

NÑ8 sup zPK

ˇ ˇ ˇ ˇϕpzq ´ 1t|z|ď1u π ˇ ˇ ˇ ˇ “ 0. The function z ÞÑ log řN´1

ℓ“0 |z|2ℓ ℓ!

is concave! Second moment of ϕ bounded in N (then Bobkov theorem)

18/20

Dynamics of a planar Coulomb gas Ginibre process

McKean–Vlasov mean-field limit

dXt,i “ c 2αN βN dBt,i ´ 2αN N Xt,idt ´ 2αN N ÿ

j‰i

Xt,i ´ Xt,j ˇ ˇXt,i ´ Xt,j ˇ ˇ2 dt. νN,t “ 1 N

N

ÿ

i“1

δX i

t 19/20

Dynamics of a planar Coulomb gas Ginibre process

McKean–Vlasov mean-field limit

Theorem (Limiting McKean–Vlasov equation)

If σ “ limNÑ8

αN βN P r0, 8q then limNÑ8 νN,t “ νt with

d dt ż

R2fpxq νtpdxq “ σ

ż ∆fpxq νtpdxq ´ 2 ż

R2x ¨ ∇fpxq νtpdxq

` ż

R4

px ´ yq ¨ p∇fpxq ´ ∇fpyqq |x ´ y|2 νtpdxqνtpdyq

• r more compactly: Btνt “ σ∆νt ` ∇ ¨ pp∇V ` ∇W ˚ νtqνtq.

19/20

Dynamics of a planar Coulomb gas Ginibre process

McKean–Vlasov mean-field limit

Theorem (Limiting McKean–Vlasov equation)

If σ “ limNÑ8

αN βN P r0, 8q then limNÑ8 νN,t “ νt with

d dt ż

R2fpxq νtpdxq “ σ

ż ∆fpxq νtpdxq ´ 2 ż

R2x ¨ ∇fpxq νtpdxq

` ż

R4

px ´ yq ¨ p∇fpxq ´ ∇fpyqq |x ´ y|2 νtpdxqνtpdyq

• r more compactly: Btνt “ σ∆νt ` ∇ ¨ pp∇V ` ∇W ˚ νtqνtq.

Semi-linear Partial Differential Equation (Sznitman)

19/20

Dynamics of a planar Coulomb gas Ginibre process

McKean–Vlasov mean-field limit

Theorem (Limiting McKean–Vlasov equation)

If σ “ limNÑ8

αN βN P r0, 8q then limNÑ8 νN,t “ νt with

d dt ż

R2fpxq νtpdxq “ σ

ż ∆fpxq νtpdxq ´ 2 ż

R2x ¨ ∇fpxq νtpdxq

` ż

R4

px ´ yq ¨ p∇fpxq ´ ∇fpyqq |x ´ y|2 νtpdxqνtpdyq

• r more compactly: Btνt “ σ∆νt ` ∇ ¨ pp∇V ` ∇W ˚ νtqνtq.

Semi-linear Partial Differential Equation (Sznitman) Regimes: pαN, βNq “ pN, N2q and pαN, βNq “ pN, Nq

19/20

Dynamics of a planar Coulomb gas Ginibre process

That’s all folks! Thank you for your attention.

20/20