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Aspects of Coulomb gases
Aspects of Coulomb gases
Djalil CHAFAÏ
CEREMADE / Université Paris-Dauphine / PSL
Random matrices
Oberwolfach
December 11, 2019
MFO 2019 � 1/25
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Aspects of Coulomb gases
Electrostatics
Outline
Electrostatics
Gases
Dynamics for planar case
Conditioning
Jellium
MFO 2019 � 2/25
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Aspects of Coulomb gases
Electrostatics
Coulomb kernel in mathematical physics
■ Coulomb kernel in Rd , d ≥ 1,
x ∈Rd 7→ g(x)=
log
1|x | if d = 2
1
(d −2)|x |d−2 if not.
■ Fundamental solution of Poisson's equation
∆g =−cd δ0 where cd = |Sd−1| = 2πd/2
Γ(d/2).
■ Repulsion for charges of same sign, singular when d ≥ 2
■ Riesz kernel |x |−s , if s = d −α then fractional Laplacian ∆α
MFO 2019 � 3/25
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Aspects of Coulomb gases
Electrostatics
Coulomb kernel in mathematical physics
■ Coulomb kernel in Rd , d ≥ 1,
x ∈Rd 7→ g(x)=
log
1|x | if d = 2
1
(d −2)|x |d−2 if not.
■ Fundamental solution of Poisson's equation
∆g =−cd δ0 where cd = |Sd−1| = 2πd/2
Γ(d/2).
■ Repulsion for charges of same sign, singular when d ≥ 2
■ Riesz kernel |x |−s , if s = d −α then fractional Laplacian ∆α
MFO 2019 � 3/25
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Aspects of Coulomb gases
Electrostatics
Coulomb kernel in mathematical physics
■ Coulomb kernel in Rd , d ≥ 1,
x ∈Rd 7→ g(x)=
log
1|x | if d = 2
1
(d −2)|x |d−2 if not.
■ Fundamental solution of Poisson's equation
∆g =−cd δ0 where cd = |Sd−1| = 2πd/2
Γ(d/2).
■ Repulsion for charges of same sign, singular when d ≥ 2
■ Riesz kernel |x |−s , if s = d −α then fractional Laplacian ∆α
MFO 2019 � 3/25
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Aspects of Coulomb gases
Electrostatics
Coulomb kernel in mathematical physics
■ Coulomb kernel in Rd , d ≥ 1,
x ∈Rd 7→ g(x)=
log
1|x | if d = 2
1
(d −2)|x |d−2 if not.
■ Fundamental solution of Poisson's equation
∆g =−cd δ0 where cd = |Sd−1| = 2πd/2
Γ(d/2).
■ Repulsion for charges of same sign, singular when d ≥ 2
■ Riesz kernel |x |−s , if s = d −α then fractional Laplacian ∆α
MFO 2019 � 3/25
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Aspects of Coulomb gases
Electrostatics
From now on d ≥ 2
|x |
g(x)
10
1
−1
d = 3d = 2d = 1
MFO 2019 � 4/25
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Aspects of Coulomb gases
Electrostatics
Coulomb potential and Coulomb energy■ Coulomb potential of a probability measure µ at point x
Uµ(x)=∫g(x −y)dµ(y)= (g ∗µ)(x)
■ Inversion formula (g =−cd∆−1)
∆g = cdδ0 ⇒ −∆Uµ = cdµ.
■ Coulomb energy of probability measure µ
E (µ)= 12
Ïg(x −y)µ(dx)µ(dy)
■ Integration by parts and �carré du champ�, η=µ−ν,
E (η)= 12
∫Uηdη=− 1
2cd
∫Uη∆Uηdx = 1
2cd
∫|∇Uη|2dx .
MFO 2019 � 5/25
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Aspects of Coulomb gases
Electrostatics
Coulomb potential and Coulomb energy■ Coulomb potential of a probability measure µ at point x
Uµ(x)=∫g(x −y)dµ(y)= (g ∗µ)(x)
■ Inversion formula (g =−cd∆−1)
∆g = cdδ0 ⇒ −∆Uµ = cdµ.
■ Coulomb energy of probability measure µ
E (µ)= 12
Ïg(x −y)µ(dx)µ(dy)
■ Integration by parts and �carré du champ�, η=µ−ν,
E (η)= 12
∫Uηdη=− 1
2cd
∫Uη∆Uηdx = 1
2cd
∫|∇Uη|2dx .
MFO 2019 � 5/25
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Aspects of Coulomb gases
Electrostatics
Coulomb potential and Coulomb energy■ Coulomb potential of a probability measure µ at point x
Uµ(x)=∫g(x −y)dµ(y)= (g ∗µ)(x)
■ Inversion formula (g =−cd∆−1)
∆g = cdδ0 ⇒ −∆Uµ = cdµ.
■ Coulomb energy of probability measure µ
E (µ)= 12
Ïg(x −y)µ(dx)µ(dy)
■ Integration by parts and �carré du champ�, η=µ−ν,
E (η)= 12
∫Uηdη=− 1
2cd
∫Uη∆Uηdx = 1
2cd
∫|∇Uη|2dx .
MFO 2019 � 5/25
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Aspects of Coulomb gases
Electrostatics
Coulomb potential and Coulomb energy■ Coulomb potential of a probability measure µ at point x
Uµ(x)=∫g(x −y)dµ(y)= (g ∗µ)(x)
■ Inversion formula (g =−cd∆−1)
∆g = cdδ0 ⇒ −∆Uµ = cdµ.
■ Coulomb energy of probability measure µ
E (µ)= 12
Ïg(x −y)µ(dx)µ(dy)
■ Integration by parts and �carré du champ�, η=µ−ν,
E (η)= 12
∫Uηdη=− 1
2cd
∫Uη∆Uηdx = 1
2cd
∫|∇Uη|2dx .
MFO 2019 � 5/25
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Aspects of Coulomb gases
Electrostatics
Con�nement and equilibrium measure
■ External con�ning potential V :Rd → (−∞,+∞]
lim|x |→∞
(V (x)− log |x |1d=2)>−∞.
■ Coulomb energy with external (con�ning) potential
EV (µ)= E (µ)+∫Vdµ.
■ EV strictly convex, lower semi-continuous, compact level sets
■ Equilibrium probability measure (electrostatics)
µV = arg minP (Rd )
EV
■ supp(µV ) is compact if lim|x |→∞(V (x)− log |x |1d=2)=+∞
MFO 2019 � 6/25
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Aspects of Coulomb gases
Electrostatics
Con�nement and equilibrium measure
■ External con�ning potential V :Rd → (−∞,+∞]
lim|x |→∞
(V (x)− log |x |1d=2)>−∞.
■ Coulomb energy with external (con�ning) potential
EV (µ)= E (µ)+∫Vdµ.
■ EV strictly convex, lower semi-continuous, compact level sets
■ Equilibrium probability measure (electrostatics)
µV = arg minP (Rd )
EV
■ supp(µV ) is compact if lim|x |→∞(V (x)− log |x |1d=2)=+∞
MFO 2019 � 6/25
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Aspects of Coulomb gases
Electrostatics
Con�nement and equilibrium measure
■ External con�ning potential V :Rd → (−∞,+∞]
lim|x |→∞
(V (x)− log |x |1d=2)>−∞.
■ Coulomb energy with external (con�ning) potential
EV (µ)= E (µ)+∫Vdµ.
■ EV strictly convex, lower semi-continuous, compact level sets
■ Equilibrium probability measure (electrostatics)
µV = arg minP (Rd )
EV
■ supp(µV ) is compact if lim|x |→∞(V (x)− log |x |1d=2)=+∞
MFO 2019 � 6/25
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Aspects of Coulomb gases
Electrostatics
Con�nement and equilibrium measure
■ External con�ning potential V :Rd → (−∞,+∞]
lim|x |→∞
(V (x)− log |x |1d=2)>−∞.
■ Coulomb energy with external (con�ning) potential
EV (µ)= E (µ)+∫Vdµ.
■ EV strictly convex, lower semi-continuous, compact level sets
■ Equilibrium probability measure (electrostatics)
µV = arg minP (Rd )
EV
■ supp(µV ) is compact if lim|x |→∞(V (x)− log |x |1d=2)=+∞
MFO 2019 � 6/25
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Aspects of Coulomb gases
Electrostatics
Con�nement and equilibrium measure
■ External con�ning potential V :Rd → (−∞,+∞]
lim|x |→∞
(V (x)− log |x |1d=2)>−∞.
■ Coulomb energy with external (con�ning) potential
EV (µ)= E (µ)+∫Vdµ.
■ EV strictly convex, lower semi-continuous, compact level sets
■ Equilibrium probability measure (electrostatics)
µV = arg minP (Rd )
EV
■ supp(µV ) is compact if lim|x |→∞(V (x)− log |x |1d=2)=+∞MFO 2019 � 6/25
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Aspects of Coulomb gases
Electrostatics
Convexity and Bochner positivity
■ Convexity/Positivity for probability measures µ and ν
tEV (µ)+ (1− t)EV (ν)−EV (tµ+ (1− t)ν)
t(1− t)
= E (µ−ν)= 12cd
∫Rd
|∇Uµ−ν|2dx .
■ Euler�Lagrange: if cV = E (µV )−∫VdµV then q.e.
UµV +V
{= cV on supp(µV )
≥ cV outside
MFO 2019 � 7/25
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Aspects of Coulomb gases
Electrostatics
Convexity and Bochner positivity
■ Convexity/Positivity for probability measures µ and ν
tEV (µ)+ (1− t)EV (ν)−EV (tµ+ (1− t)ν)
t(1− t)
= E (µ−ν)= 12cd
∫Rd
|∇Uµ−ν|2dx .
■ Euler�Lagrange: if cV = E (µV )−∫VdµV then q.e.
UµV +V
{= cV on supp(µV )
≥ cV outside
MFO 2019 � 7/25
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Aspects of Coulomb gases
Electrostatics
Examples of equilibrium measures
Dimension d Potential V Equilibrium measure µV
≥ 1 ∞1|·|>r Uniform on sphere of radius r
≥ 1 <∞ and C 2 c−1d ∆V on interior of support
≥ 1 12 |·|2 Uniform on unit ball
(Ginibre) 2 12 |·|2 Uniform on unit disc
(Spherical) 2 12 log(1+|·|2) Heavy-tailed 1
π(1+|·|2)2
(CUE) 2 ∞1([a,b]×{0})c Arcsine s 7→ 1s∈[a,b]
πp
(s−a)(b−s)
(GUE) 2
|·|22 1R×{0} +∞1(R×{0})c Semicircle s 7→
p4−s22π 1s∈[−2,2]
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Aspects of Coulomb gases
Electrostatics
Examples of equilibrium measures
Dimension d Potential V Equilibrium measure µV
≥ 1 ∞1|·|>r Uniform on sphere of radius r
≥ 1 <∞ and C 2 c−1d ∆V on interior of support
≥ 1 12 |·|2 Uniform on unit ball
(Ginibre) 2 12 |·|2 Uniform on unit disc
(Spherical) 2 12 log(1+|·|2) Heavy-tailed 1
π(1+|·|2)2
(CUE) 2 ∞1([a,b]×{0})c Arcsine s 7→ 1s∈[a,b]
πp
(s−a)(b−s)(GUE) 2 |·|2
2 1R×{0} +∞1(R×{0})c Semicircle s 7→p4−s22π 1s∈[−2,2]
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Aspects of Coulomb gases
Gases
Outline
Electrostatics
Gases
Dynamics for planar case
Conditioning
Jellium
MFO 2019 � 9/25
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Aspects of Coulomb gases
Gases
Coulomb gas or one-component plasma■ Particles subject to con�nement and singular pair repulsion
βEn(x1, . . . ,xn)=βn2(1n
n∑i=1
V (xi )+1n2
∑i<j
g(xi −xj))
=βn2E 6=V(µx1,...,xn)
■ Empirical measure µx1,...,xn =1n
n∑i=1
δxi and
E 6=V(µ)=
∫Vdµ+ 1
2
Ï6=g(u−v)dµ(u)dµ(v)
■ Boltzmann�Gibbs measure when e−nβ(V−log(1+|·|)1d=2) ∈ L1(dx)
dPn(x1, . . . ,xn)= e−βEn(x1,...,xn)
Zndx1 · · ·dxn
MFO 2019 � 10/25
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Aspects of Coulomb gases
Gases
Coulomb gas or one-component plasma■ Particles subject to con�nement and singular pair repulsion
βEn(x1, . . . ,xn)=βn2(1n
n∑i=1
V (xi )+1n2
∑i<j
g(xi −xj))
=βn2E 6=V(µx1,...,xn)
■ Empirical measure µx1,...,xn =1n
n∑i=1
δxi and
E 6=V(µ)=
∫Vdµ+ 1
2
Ï6=g(u−v)dµ(u)dµ(v)
■ Boltzmann�Gibbs measure when e−nβ(V−log(1+|·|)1d=2) ∈ L1(dx)
dPn(x1, . . . ,xn)= e−βEn(x1,...,xn)
Zndx1 · · ·dxn
MFO 2019 � 10/25
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Aspects of Coulomb gases
Gases
Coulomb gas or one-component plasma■ Particles subject to con�nement and singular pair repulsion
βEn(x1, . . . ,xn)=βn2(1n
n∑i=1
V (xi )+1n2
∑i<j
g(xi −xj))
=βn2E 6=V(µx1,...,xn)
■ Empirical measure µx1,...,xn =1n
n∑i=1
δxi and
E 6=V(µ)=
∫Vdµ+ 1
2
Ï6=g(u−v)dµ(u)dµ(v)
■ Boltzmann�Gibbs measure when e−nβ(V−log(1+|·|)1d=2) ∈ L1(dx)
dPn(x1, . . . ,xn)= e−βEn(x1,...,xn)
Zndx1 · · ·dxn
MFO 2019 � 10/25
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Aspects of Coulomb gases
Gases
Examples: exactly solvable RMT and Coulomb gases■ d = 2 gives
e−∑i V (xi )−
∑i<j g(xi−xj ) = e−
∑i V (xi )
∏i<j
|xi −xj |
■ d = 2 and β= 2 give determinantal structure
Pn,k(dx1, . . . ,dxk)= det(KV ,n(xi ,xj ))1≤i ,j≤k
■ M ∈Mn,n(C), M ∝ e−Trace(MM∗)
Ensemble Random Matrix Potential V (d =β= 2)
Ginibre M 12 |·|2
(GUE) Hermite 1p2(M +M∗) 1
2 |·|21R×{0} +∞1(R×{0})c
(LUE) Laguerre MM∗ 12 |·|1R+×{0} +∞1(R+×{0})c
Spherical M1M2−1 1
2n+1n log(1+|·|2)
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Aspects of Coulomb gases
Gases
Examples: exactly solvable RMT and Coulomb gases■ d = 2 gives
e−∑i V (xi )−
∑i<j g(xi−xj ) = e−
∑i V (xi )
∏i<j
|xi −xj |
■ d = 2 and β= 2 give determinantal structure
Pn,k(dx1, . . . ,dxk)= det(KV ,n(xi ,xj ))1≤i ,j≤k
■ M ∈Mn,n(C), M ∝ e−Trace(MM∗)
Ensemble Random Matrix Potential V (d =β= 2)
Ginibre M 12 |·|2
(GUE) Hermite 1p2(M +M∗) 1
2 |·|21R×{0} +∞1(R×{0})c
(LUE) Laguerre MM∗ 12 |·|1R+×{0} +∞1(R+×{0})c
Spherical M1M2−1 1
2n+1n log(1+|·|2)
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Aspects of Coulomb gases
Gases
Examples: exactly solvable RMT and Coulomb gases■ d = 2 gives
e−∑i V (xi )−
∑i<j g(xi−xj ) = e−
∑i V (xi )
∏i<j
|xi −xj |
■ d = 2 and β= 2 give determinantal structure
Pn,k(dx1, . . . ,dxk)= det(KV ,n(xi ,xj ))1≤i ,j≤k
■ M ∈Mn,n(C), M ∝ e−Trace(MM∗)
Ensemble Random Matrix Potential V (d =β= 2)
Ginibre M 12 |·|2
(GUE) Hermite 1p2(M +M∗) 1
2 |·|21R×{0} +∞1(R×{0})c
(LUE) Laguerre MM∗ 12 |·|1R+×{0} +∞1(R+×{0})c
Spherical M1M2−1 1
2n+1n log(1+|·|2)
MFO 2019 � 11/25
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Aspects of Coulomb gases
Gases
Laplace method point of view
■ Laplace point of view
dPn(x1, . . . ,xn)= e−βn2E 6=
V (µx1 ,...,xn )
Zndx1 · · ·dxn.
■ Large Deviation Principle (also works if β=βn with nβn →∞)
Pn(µx1,...,xn ∈B) ≈n→∞ e−βn
2 infB(EV−EV (µV ))
■ Law of Large Numbers : if X ∼ Pn then almost surely
µX1,...,Xn−→n→∞µV = argminEV
. . . , Voiculescu, Ben Arous �Guionnet, Hiai�Petz,. . .
. . . , Serfaty et al, C.�Gozlan�Zitt, Berman, García-Zelada,. . .
MFO 2019 � 12/25
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Aspects of Coulomb gases
Gases
Laplace method point of view
■ Laplace point of view
dPn(x1, . . . ,xn)= e−βn2E 6=
V (µx1 ,...,xn )
Zndx1 · · ·dxn.
■ Large Deviation Principle (also works if β=βn with nβn →∞)
Pn(µx1,...,xn ∈B) ≈n→∞ e−βn
2 infB(EV−EV (µV ))
■ Law of Large Numbers : if X ∼ Pn then almost surely
µX1,...,Xn−→n→∞µV = argminEV
. . . , Voiculescu, Ben Arous �Guionnet, Hiai�Petz,. . .
. . . , Serfaty et al, C.�Gozlan�Zitt, Berman, García-Zelada,. . .
MFO 2019 � 12/25
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Aspects of Coulomb gases
Gases
Laplace method point of view
■ Laplace point of view
dPn(x1, . . . ,xn)= e−βn2E 6=
V (µx1 ,...,xn )
Zndx1 · · ·dxn.
■ Large Deviation Principle (also works if β=βn with nβn →∞)
Pn(µx1,...,xn ∈B) ≈n→∞ e−βn
2 infB(EV−EV (µV ))
■ Law of Large Numbers : if X ∼ Pn then almost surely
µX1,...,Xn−→n→∞µV = argminEV
. . . , Voiculescu, Ben Arous �Guionnet, Hiai�Petz,. . .
. . . , Serfaty et al, C.�Gozlan�Zitt, Berman, García-Zelada,. . .
MFO 2019 � 12/25
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Aspects of Coulomb gases
Gases
Asymptotic analysis of �uctuations
■ Gaussian structure: Pn ≈ e−n2
2 ⟨−βcd∆−1µ,µ⟩−n2β⟨V ,µ⟩
■ Asymptotics: Central Limit Theorem with Gaussian Free Field
n∑i=1
f (Xi )−E(· · ·) law−→n→∞ N
(0,
1βcd
∫Rd
|∇f |2dx +·· ·)
. . . , Johansson, . . . , Rider�Virag, . . . , Serfaty et al, . . .
■ Quantitative: concentration of measure inequalities
P(dist(µX1,...,Xn
,µV )≥ r)
≤ e−aβn2r2
+1d=2( β4n logn)+bβn2−2/d+c(β)n
. . . , Guionnet�Zeitouni, Rougerie�Serfaty, Hardy�C.�Maïda, Berman, . . .
MFO 2019 � 13/25
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Aspects of Coulomb gases
Gases
Asymptotic analysis of �uctuations
■ Gaussian structure: Pn ≈ e−n2
2 ⟨−βcd∆−1µ,µ⟩−n2β⟨V ,µ⟩
■ Asymptotics: Central Limit Theorem with Gaussian Free Field
n∑i=1
f (Xi )−E(· · ·) law−→n→∞ N
(0,
1βcd
∫Rd
|∇f |2dx +·· ·)
. . . , Johansson, . . . , Rider�Virag, . . . , Serfaty et al, . . .
■ Quantitative: concentration of measure inequalities
P(dist(µX1,...,Xn
,µV )≥ r)
≤ e−aβn2r2+1d=2( β4n logn)+bβn2−2/d+c(β)n
. . . , Guionnet�Zeitouni, Rougerie�Serfaty, Hardy�C.�Maïda, Berman, . . .
MFO 2019 � 13/25
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Aspects of Coulomb gases
Gases
Asymptotic analysis of �uctuations
■ Gaussian structure: Pn ≈ e−n2
2 ⟨−βcd∆−1µ,µ⟩−n2β⟨V ,µ⟩
■ Asymptotics: Central Limit Theorem with Gaussian Free Field
n∑i=1
f (Xi )−E(· · ·) law−→n→∞ N
(0,
1βcd
∫Rd
|∇f |2dx +·· ·)
. . . , Johansson, . . . , Rider�Virag, . . . , Serfaty et al, . . .
■ Quantitative: concentration of measure inequalities
P(dist(µX1,...,Xn
,µV )≥ r)
≤ e−aβn2r2+1d=2( β4n logn)+bβn2−2/d+c(β)n
. . . , Guionnet�Zeitouni, Rougerie�Serfaty, Hardy�C.�Maïda, Berman, . . .
MFO 2019 � 13/25
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Aspects of Coulomb gases
Dynamics for planar case
Outline
Electrostatics
Gases
Dynamics for planar case
Conditioning
Jellium
MFO 2019 � 14/25
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Aspects of Coulomb gases
Dynamics for planar case
Langevin dynamics■ Overdamped Langevin dynamics on (Rd )n: Xt
law−→t→∞ Pn
dXt =√2α
βdBt −α∇En(Xt)dt, L =α(β−1∆−∇En ·∇)
Dyson, Bru, Lassalle, Rogers�Shi, Guionnet et al,. . . , Erd®s�Yau et al. . .
Bolley�C.�Fontbona, C.�Lehec, Akkeman�Byun, . . .
■ Mean-�eld McKean�Vlasov limit: if σ= limn→∞ αnβn
then (?)
limn→∞µXt
= νt where ∂tνt =σ∆νt +∇· ((∇V +∇g ∗νt)νt).
■ Underdamped Langevin or kinetic Dyson�Ornstein�Uhlenbeck
dXt =αYtdt, dYt =−α∇En(Xt)dt+√2γα
βdBt −γαYtdt
Good for numerical simulation via Hamiltonian Monte Carlo
MFO 2019 � 15/25
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Aspects of Coulomb gases
Dynamics for planar case
Langevin dynamics■ Overdamped Langevin dynamics on (Rd )n: Xt
law−→t→∞ Pn
dXt =√2α
βdBt −α∇En(Xt)dt, L =α(β−1∆−∇En ·∇)
■ Mean-�eld McKean�Vlasov limit: if σ= limn→∞ αnβn
then (?)
limn→∞µXt
= νt where ∂tνt =σ∆νt +∇· ((∇V +∇g ∗νt)νt).
. . . , Carrillo�McCann-Villani, . . . , Guionnet et al, . . . , Jabin et al, . . .
■ Underdamped Langevin or kinetic Dyson�Ornstein�Uhlenbeck
dXt =αYtdt, dYt =−α∇En(Xt)dt+√2γα
βdBt −γαYtdt
Good for numerical simulation via Hamiltonian Monte Carlo
MFO 2019 � 15/25
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Aspects of Coulomb gases
Dynamics for planar case
Langevin dynamics■ Overdamped Langevin dynamics on (Rd )n: Xt
law−→t→∞ Pn
dXt =√2α
βdBt −α∇En(Xt)dt, L =α(β−1∆−∇En ·∇)
■ Mean-�eld McKean�Vlasov limit: if σ= limn→∞ αnβn
then (?)
limn→∞µXt
= νt where ∂tνt =σ∆νt +∇· ((∇V +∇g ∗νt)νt).
■ Underdamped Langevin or kinetic Dyson�Ornstein�Uhlenbeck
dXt =αYtdt, dYt =−α∇En(Xt)dt+√2γα
βdBt −γαYtdt
Good for numerical simulation via Hamiltonian Monte CarloFerré�C., Lu�Mattingly, Dolbeault et al
MFO 2019 � 15/25
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Aspects of Coulomb gases
Dynamics for planar case
Example of kinetic Dyson HMC in 1D
MFO 2019 � 16/25
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Aspects of Coulomb gases
Dynamics for planar case
Exact computations for β Ginibre
■ Exact computation for d = 2, V = 12 |·|2, and any β> 0
■ β ∈ 2N Laughlin wave function fractional quantum Hall e�ect
■ Xt,1+·· ·+Xt,n is an Ornstein�Uhlenbeck process
■ |Xt,1|2+·· ·+ |Xt,n|2 is a Cox�Ingersoll�Ross process
■ As a consequence if Xn ∼ Pn then
X1+·· ·+Xn ∼ N(0, I2
β
)|X1|2+·· ·+ |Xn|2 ∼ Gamma
(n+βn(n−1)
4 ,βn2
)■ Lack of useful tridiagonal model? Special eigenfunctions!
MFO 2019 � 17/25
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Aspects of Coulomb gases
Dynamics for planar case
Exact computations for β Ginibre
■ Exact computation for d = 2, V = 12 |·|2, and any β> 0
■ β ∈ 2N Laughlin wave function fractional quantum Hall e�ect
■ Xt,1+·· ·+Xt,n is an Ornstein�Uhlenbeck process
■ |Xt,1|2+·· ·+ |Xt,n|2 is a Cox�Ingersoll�Ross process
■ As a consequence if Xn ∼ Pn then
X1+·· ·+Xn ∼ N(0, I2
β
)|X1|2+·· ·+ |Xn|2 ∼ Gamma
(n+βn(n−1)
4 ,βn2
)■ Lack of useful tridiagonal model? Special eigenfunctions!
Bolley�C.�Fontbona, C.-Lehec
MFO 2019 � 17/25
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Aspects of Coulomb gases
Dynamics for planar case
Exact computations for β Ginibre
■ Exact computation for d = 2, V = 12 |·|2, and any β> 0
■ β ∈ 2N Laughlin wave function fractional quantum Hall e�ect
■ Xt,1+·· ·+Xt,n is an Ornstein�Uhlenbeck process
■ |Xt,1|2+·· ·+ |Xt,n|2 is a Cox�Ingersoll�Ross process
■ As a consequence if Xn ∼ Pn then
X1+·· ·+Xn ∼ N(0, I2
β
)|X1|2+·· ·+ |Xn|2 ∼ Gamma
(n+βn(n−1)
4 ,βn2
)■ Lack of useful tridiagonal model? Special eigenfunctions!
Bolley�C.�Fontbona, C.-Lehec
MFO 2019 � 17/25
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Aspects of Coulomb gases
Dynamics for planar case
Exact computations for β Ginibre
■ Exact computation for d = 2, V = 12 |·|2, and any β> 0
■ β ∈ 2N Laughlin wave function fractional quantum Hall e�ect
■ Xt,1+·· ·+Xt,n is an Ornstein�Uhlenbeck process
■ |Xt,1|2+·· ·+ |Xt,n|2 is a Cox�Ingersoll�Ross process
■ As a consequence if Xn ∼ Pn then
X1+·· ·+Xn ∼ N(0, I2
β
)|X1|2+·· ·+ |Xn|2 ∼ Gamma
(n+βn(n−1)
4 ,βn2
)■ Lack of useful tridiagonal model? Special eigenfunctions!
Bolley�C.�Fontbona, C.-Lehec
MFO 2019 � 17/25
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Aspects of Coulomb gases
Dynamics for planar case
Exact computations for β Ginibre
■ Exact computation for d = 2, V = 12 |·|2, and any β> 0
■ β ∈ 2N Laughlin wave function fractional quantum Hall e�ect
■ Xt,1+·· ·+Xt,n is an Ornstein�Uhlenbeck process
■ |Xt,1|2+·· ·+ |Xt,n|2 is a Cox�Ingersoll�Ross process
■ As a consequence if Xn ∼ Pn then
X1+·· ·+Xn ∼ N(0, I2
β
)|X1|2+·· ·+ |Xn|2 ∼ Gamma
(n+βn(n−1)
4 ,βn2
)
■ Lack of useful tridiagonal model? Special eigenfunctions!Bolley�C.�Fontbona, C.-Lehec
MFO 2019 � 17/25
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Aspects of Coulomb gases
Dynamics for planar case
Exact computations for β Ginibre
■ Exact computation for d = 2, V = 12 |·|2, and any β> 0
■ β ∈ 2N Laughlin wave function fractional quantum Hall e�ect
■ Xt,1+·· ·+Xt,n is an Ornstein�Uhlenbeck process
■ |Xt,1|2+·· ·+ |Xt,n|2 is a Cox�Ingersoll�Ross process
■ As a consequence if Xn ∼ Pn then
X1+·· ·+Xn ∼ N(0, I2
β
)|X1|2+·· ·+ |Xn|2 ∼ Gamma
(n+βn(n−1)
4 ,βn2
)■ Lack of useful tridiagonal model? Special eigenfunctions!
Bolley�C.�Fontbona, C.-Lehec
MFO 2019 � 17/25
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Aspects of Coulomb gases
Conditioning
Outline
Electrostatics
Gases
Dynamics for planar case
Conditioning
Jellium
MFO 2019 � 18/25
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Aspects of Coulomb gases
Conditioning
Conditioned Coulomb gas
■ X ∼ Pn and Y ∼ Law(X |ϕ(X1)+·· ·+ϕ(Xn)= 0
)
■ If ϕ is regular enough then almost surely
limn→∞
1n
n∑i=1
δYi=µV+αϕ = argminEV+αϕ.
■ Exactly solvable when V (x)= c |x |2 and ϕ(x)= ax +b. Shift!
■ Gibbs conditioning principle for non-⊗ singular Gibbs measures
■ Maing technical di�culty: regularity sets of EV
■ Quadratic conditioning gives perturbation of g instead of V
■ Numerical simulation: constrained HMC via kinetic Langevin
MFO 2019 � 19/25
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Aspects of Coulomb gases
Conditioning
Conditioned Coulomb gas
■ X ∼ Pn and Y ∼ Law(X |ϕ(X1)+·· ·+ϕ(Xn)= 0
)■ If ϕ is regular enough then almost surely
limn→∞
1n
n∑i=1
δYi=µV+αϕ = argminEV+αϕ.
■ Exactly solvable when V (x)= c |x |2 and ϕ(x)= ax +b. Shift!
■ Gibbs conditioning principle for non-⊗ singular Gibbs measures
■ Maing technical di�culty: regularity sets of EV
■ Quadratic conditioning gives perturbation of g instead of V
■ Numerical simulation: constrained HMC via kinetic Langevin
MFO 2019 � 19/25
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Aspects of Coulomb gases
Conditioning
Conditioned Coulomb gas
■ X ∼ Pn and Y ∼ Law(X |ϕ(X1)+·· ·+ϕ(Xn)= 0
)■ If ϕ is regular enough then almost surely
limn→∞
1n
n∑i=1
δYi=µV+αϕ = argminEV+αϕ.
■ Exactly solvable when V (x)= c |x |2 and ϕ(x)= ax +b. Shift!
■ Gibbs conditioning principle for non-⊗ singular Gibbs measures
■ Maing technical di�culty: regularity sets of EV
■ Quadratic conditioning gives perturbation of g instead of V
■ Numerical simulation: constrained HMC via kinetic Langevin
MFO 2019 � 19/25
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Aspects of Coulomb gases
Conditioning
Conditioned Coulomb gas
■ X ∼ Pn and Y ∼ Law(X |ϕ(X1)+·· ·+ϕ(Xn)= 0
)■ If ϕ is regular enough then almost surely
limn→∞
1n
n∑i=1
δYi=µV+αϕ = argminEV+αϕ.
■ Exactly solvable when V (x)= c |x |2 and ϕ(x)= ax +b. Shift!
■ Gibbs conditioning principle for non-⊗ singular Gibbs measures
■ Maing technical di�culty: regularity sets of EV
■ Quadratic conditioning gives perturbation of g instead of V
■ Numerical simulation: constrained HMC via kinetic Langevin
MFO 2019 � 19/25
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Aspects of Coulomb gases
Conditioning
Conditioned Coulomb gas
■ X ∼ Pn and Y ∼ Law(X |ϕ(X1)+·· ·+ϕ(Xn)= 0
)■ If ϕ is regular enough then almost surely
limn→∞
1n
n∑i=1
δYi=µV+αϕ = argminEV+αϕ.
■ Exactly solvable when V (x)= c |x |2 and ϕ(x)= ax +b. Shift!
■ Gibbs conditioning principle for non-⊗ singular Gibbs measures
■ Maing technical di�culty: regularity sets of EV
■ Quadratic conditioning gives perturbation of g instead of V
■ Numerical simulation: constrained HMC via kinetic Langevin
MFO 2019 � 19/25
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Aspects of Coulomb gases
Conditioning
Conditioned Coulomb gas
■ X ∼ Pn and Y ∼ Law(X |ϕ(X1)+·· ·+ϕ(Xn)= 0
)■ If ϕ is regular enough then almost surely
limn→∞
1n
n∑i=1
δYi=µV+αϕ = argminEV+αϕ.
■ Exactly solvable when V (x)= c |x |2 and ϕ(x)= ax +b. Shift!
■ Gibbs conditioning principle for non-⊗ singular Gibbs measures
■ Maing technical di�culty: regularity sets of EV
■ Quadratic conditioning gives perturbation of g instead of V
■ Numerical simulation: constrained HMC via kinetic Langevin
MFO 2019 � 19/25
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Aspects of Coulomb gases
Conditioning
Conditioned Coulomb gas
■ X ∼ Pn and Y ∼ Law(X |ϕ(X1)+·· ·+ϕ(Xn)= 0
)■ If ϕ is regular enough then almost surely
limn→∞
1n
n∑i=1
δYi=µV+αϕ = argminEV+αϕ.
■ Exactly solvable when V (x)= c |x |2 and ϕ(x)= ax +b. Shift!
■ Gibbs conditioning principle for non-⊗ singular Gibbs measures
■ Maing technical di�culty: regularity sets of EV
■ Quadratic conditioning gives perturbation of g instead of V
■ Numerical simulation: constrained HMC via kinetic Langevin
MFO 2019 � 19/25
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Aspects of Coulomb gases
Conditioning
Conditioned Coulomb gas
■ X ∼ Pn and Y ∼ Law(X |ϕ(X1)+·· ·+ϕ(Xn)= 0
)■ If ϕ is regular enough then almost surely
limn→∞
1n
n∑i=1
δYi=µV+αϕ = argminEV+αϕ.
■ Exactly solvable when V (x)= c |x |2 and ϕ(x)= ax +b. Shift!
■ Gibbs conditioning principle for non-⊗ singular Gibbs measures
■ Maing technical di�culty: regularity sets of EV
■ Quadratic conditioning gives perturbation of g instead of V
■ Numerical simulation: constrained HMC via kinetic LangevinFerré�C.�Stoltz
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Aspects of Coulomb gases
Conditioning
Conditioned Coulomb gases � HMC/Julia
V (x)= |x |2MFO 2019 � 20/25
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Aspects of Coulomb gases
Conditioning
Conditioned Coulomb gases � HMC/Julia
V (x)= |x |2, ϕ(x)= a ·x +b
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Aspects of Coulomb gases
Conditioning
Conditioned Coulomb gases � HMC/Julia
V (x)= |x |4MFO 2019 � 20/25
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Aspects of Coulomb gases
Conditioning
Conditioned Coulomb gases � HMC/Julia
V (x)= |x |4, ϕ(x)= ax1+b
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Aspects of Coulomb gases
Conditioning
Conditioned Coulomb gases � HMC/Julia
V (x)= |x |2, ϕ(x)= c −b(cos(ax1)+cos(ax2))
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Aspects of Coulomb gases
Conditioning
Conditioned Coulomb gases � HMC/Julia
V (x)= |x |2, ϕ(x)= c −b(cos(ax1)+cos(ax2))
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Aspects of Coulomb gases
Jellium
Outline
Electrostatics
Gases
Dynamics for planar case
Conditioning
Jellium
MFO 2019 � 21/25
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Aspects of Coulomb gases
Jellium
A bit of chronology
Random matrices and Coulomb gases■ 1920 : Fisher, Wishart, . . .
■ 1930 : Bartlett, Wigner (jellium), . . .
■ 1940 : Goldstine�von Neumann, . . .
■ 1950 : Anderson, Wigner (matrix), . . .
■ 1960 : Mehta, Dyson, Ginibre, Marchenko�Pastur, . . .
■ . . .
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Aspects of Coulomb gases
Jellium
A bit of chronology
Random matrices and Coulomb gases■ 1920 : Fisher, Wishart, . . .
■ 1930 : Bartlett, Wigner (jellium), . . .
■ 1940 : Goldstine�von Neumann, . . .
■ 1950 : Anderson, Wigner (matrix), . . .
■ 1960 : Mehta, Dyson, Ginibre, Marchenko�Pastur, . . .
■ . . .
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Aspects of Coulomb gases
Jellium
A bit of chronology
Random matrices and Coulomb gases■ 1920 : Fisher, Wishart, . . .
■ 1930 : Bartlett, Wigner (jellium), . . .
■ 1940 : Goldstine�von Neumann, . . .
■ 1950 : Anderson, Wigner (matrix), . . .
■ 1960 : Mehta, Dyson, Ginibre, Marchenko�Pastur, . . .
■ . . .
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Aspects of Coulomb gases
Jellium
A bit of chronology
Random matrices and Coulomb gases■ 1920 : Fisher, Wishart, . . .
■ 1930 : Bartlett, Wigner (jellium), . . .
■ 1940 : Goldstine�von Neumann, . . .
■ 1950 : Anderson, Wigner (matrix), . . .
■ 1960 : Mehta, Dyson, Ginibre, Marchenko�Pastur, . . .
■ . . .
MFO 2019 � 22/25
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Aspects of Coulomb gases
Jellium
A bit of chronology
Random matrices and Coulomb gases■ 1920 : Fisher, Wishart, . . .
■ 1930 : Bartlett, Wigner (jellium), . . .
■ 1940 : Goldstine�von Neumann, . . .
■ 1950 : Anderson, Wigner (matrix), . . .
■ 1960 : Mehta, Dyson, Ginibre, Marchenko�Pastur, . . .
■ . . .
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Aspects of Coulomb gases
Jellium
A bit of chronology
Random matrices and Coulomb gases■ 1920 : Fisher, Wishart, . . .
■ 1930 : Bartlett, Wigner (jellium), . . .
■ 1940 : Goldstine�von Neumann, . . .
■ 1950 : Anderson, Wigner (matrix), . . .
■ 1960 : Mehta, Dyson, Ginibre, Marchenko�Pastur, . . .
■ . . .
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Aspects of Coulomb gases
Jellium
Wigner jellium and Coulomb gas■ Eugene P. Wigner 1938 : Electrons in a piece S ⊂Rd of metal
■ Simpli�cation of Hartree�Fock quantum model
■ Background of opposite charges on supp(ρ)⊂ S
EJelliumn (x1, . . . ,xn)=
∑i<j
g(xi −xj)−αn∑i=1
Uρ(xi )+α2c
■ Charge neutral if α= n, Boltzmann�Gibbs measure e−βEJelliumn
Z Jelliumn
■ Jellium is a Coulomb gas Pn with con�nement potential
V ={−α
n Uρ on S
+∞ outsideand µV = α
nρ
■ Coulomb gas is a Jellium with ρ = nαcd
∆V on S =Rd
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Aspects of Coulomb gases
Jellium
Wigner jellium and Coulomb gas■ Eugene P. Wigner 1938 : Electrons in a piece S ⊂Rd of metal
■ Simpli�cation of Hartree�Fock quantum model
■ Background of opposite charges on supp(ρ)⊂ S
EJelliumn (x1, . . . ,xn)=
∑i<j
g(xi −xj)−αn∑i=1
Uρ(xi )+α2c
■ Charge neutral if α= n, Boltzmann�Gibbs measure e−βEJelliumn
Z Jelliumn
■ Jellium is a Coulomb gas Pn with con�nement potential
V ={−α
n Uρ on S
+∞ outsideand µV = α
nρ
■ Coulomb gas is a Jellium with ρ = nαcd
∆V on S =Rd
MFO 2019 � 23/25
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Aspects of Coulomb gases
Jellium
Wigner jellium and Coulomb gas■ Eugene P. Wigner 1938 : Electrons in a piece S ⊂Rd of metal
■ Simpli�cation of Hartree�Fock quantum model
■ Background of opposite charges on supp(ρ)⊂ S
EJelliumn (x1, . . . ,xn)=
∑i<j
g(xi −xj)−αn∑i=1
Uρ(xi )+α2c
■ Charge neutral if α= n, Boltzmann�Gibbs measure e−βEJelliumn
Z Jelliumn
■ Jellium is a Coulomb gas Pn with con�nement potential
V ={−α
n Uρ on S
+∞ outsideand µV = α
nρ
■ Coulomb gas is a Jellium with ρ = nαcd
∆V on S =Rd
MFO 2019 � 23/25
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Aspects of Coulomb gases
Jellium
Wigner jellium and Coulomb gas■ Eugene P. Wigner 1938 : Electrons in a piece S ⊂Rd of metal
■ Simpli�cation of Hartree�Fock quantum model
■ Background of opposite charges on supp(ρ)⊂ S
EJelliumn (x1, . . . ,xn)=
∑i<j
g(xi −xj)−αn∑i=1
Uρ(xi )+α2c
■ Charge neutral if α= n, Boltzmann�Gibbs measure e−βEJelliumn
Z Jelliumn
■ Jellium is a Coulomb gas Pn with con�nement potential
V ={−α
n Uρ on S
+∞ outsideand µV = α
nρ
■ Coulomb gas is a Jellium with ρ = nαcd
∆V on S =Rd
MFO 2019 � 23/25
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Aspects of Coulomb gases
Jellium
Wigner jellium and Coulomb gas■ Eugene P. Wigner 1938 : Electrons in a piece S ⊂Rd of metal
■ Simpli�cation of Hartree�Fock quantum model
■ Background of opposite charges on supp(ρ)⊂ S
EJelliumn (x1, . . . ,xn)=
∑i<j
g(xi −xj)−αn∑i=1
Uρ(xi )+α2c
■ Charge neutral if α= n, Boltzmann�Gibbs measure e−βEJelliumn
Z Jelliumn
■ Jellium is a Coulomb gas Pn with con�nement potential
V ={−α
n Uρ on S
+∞ outsideand µV = α
nρ
■ Coulomb gas is a Jellium with ρ = nαcd
∆V on S =Rd
MFO 2019 � 23/25
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Aspects of Coulomb gases
Jellium
Wigner jellium and Coulomb gas■ Eugene P. Wigner 1938 : Electrons in a piece S ⊂Rd of metal
■ Simpli�cation of Hartree�Fock quantum model
■ Background of opposite charges on supp(ρ)⊂ S
EJelliumn (x1, . . . ,xn)=
∑i<j
g(xi −xj)−αn∑i=1
Uρ(xi )+α2c
■ Charge neutral if α= n, Boltzmann�Gibbs measure e−βEJelliumn
Z Jelliumn
■ Jellium is a Coulomb gas Pn with con�nement potential
V ={−α
n Uρ on S
+∞ outsideand µV = α
nρ
■ Coulomb gas is a Jellium with ρ = nαcd
∆V on S =RdMFO 2019 � 23/25
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Aspects of Coulomb gases
Jellium
Two dimensional jellium with uniform background■ d = 2 and ρ = Uniform(D(0,R))
V (x)=−αnUρ(x)
= α
n
( |x |22R
−1+ logR)1|x |≤R + α
nlog |x |1|x |>R .
■ Z Jelliumn <∞ i� α−n> 2
β−1
■ Impossible: charge neutral (α= n) with determinantal (β= 2)
■ If nβn →∞ and αnn →λ≥ 1, then µV = Uniform(D(0,R/
pλ))
■ Transition for edge �uctuations when β= 2 and αn ∼λn
Gumbel if λ> 1 and Heavy-tailed if λ= 1.
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Aspects of Coulomb gases
Jellium
Two dimensional jellium with uniform background■ d = 2 and ρ = Uniform(D(0,R))
V (x)=−αnUρ(x)
= α
n
( |x |22R
−1+ logR)1|x |≤R + α
nlog |x |1|x |>R .
■ Z Jelliumn <∞ i� α−n> 2
β−1
■ Impossible: charge neutral (α= n) with determinantal (β= 2)
■ If nβn →∞ and αnn →λ≥ 1, then µV = Uniform(D(0,R/
pλ))
■ Transition for edge �uctuations when β= 2 and αn ∼λn
Gumbel if λ> 1 and Heavy-tailed if λ= 1.
MFO 2019 � 24/25
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Aspects of Coulomb gases
Jellium
Two dimensional jellium with uniform background■ d = 2 and ρ = Uniform(D(0,R))
V (x)=−αnUρ(x)
= α
n
( |x |22R
−1+ logR)1|x |≤R + α
nlog |x |1|x |>R .
■ Z Jelliumn <∞ i� α−n> 2
β−1
■ Impossible: charge neutral (α= n) with determinantal (β= 2)
■ If nβn →∞ and αnn →λ≥ 1, then µV = Uniform(D(0,R/
pλ))
■ Transition for edge �uctuations when β= 2 and αn ∼λn
Gumbel if λ> 1 and Heavy-tailed if λ= 1.
MFO 2019 � 24/25
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Aspects of Coulomb gases
Jellium
Two dimensional jellium with uniform background■ d = 2 and ρ = Uniform(D(0,R))
V (x)=−αnUρ(x)
= α
n
( |x |22R
−1+ logR)1|x |≤R + α
nlog |x |1|x |>R .
■ Z Jelliumn <∞ i� α−n> 2
β−1
■ Impossible: charge neutral (α= n) with determinantal (β= 2)
■ If nβn →∞ and αnn →λ≥ 1, then µV = Uniform(D(0,R/
pλ))
■ Transition for edge �uctuations when β= 2 and αn ∼λn
Gumbel if λ> 1 and Heavy-tailed if λ= 1.
MFO 2019 � 24/25
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Aspects of Coulomb gases
Jellium
Two dimensional jellium with uniform background■ d = 2 and ρ = Uniform(D(0,R))
V (x)=−αnUρ(x)
= α
n
( |x |22R
−1+ logR)1|x |≤R + α
nlog |x |1|x |>R .
■ Z Jelliumn <∞ i� α−n> 2
β−1
■ Impossible: charge neutral (α= n) with determinantal (β= 2)
■ If nβn →∞ and αnn →λ≥ 1, then µV = Uniform(D(0,R/
pλ))
■ Transition for edge �uctuations when β= 2 and αn ∼λn
Gumbel if λ> 1 and Heavy-tailed if λ= 1.
García-Zelada�C.�Jung. More: Butez�García-Zelada
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Aspects of Coulomb gases
Jellium
Thank you very much for your attention!
MFO 2019 � 25/25