Thoughts on the Coulomb Plasma Yacin Ameur Centre for Mathematical - - PowerPoint PPT Presentation

Thoughts on the Coulomb Plasma

Yacin Ameur

Centre for Mathematical Sciences Lund University, Sweden Yacin.Ameur@maths.lth.se

OPCOP17: Castro Urdiales 2017

Particle systems

A system {ζi}n

1 ∈ C ("point charges”) in external field nQ.

Energy: Hn =

n

• j=k

log 1 |ζj − ζk| + n

n

• j=1

Q(ζj). Boltzmann–Gibbs law: dPn(ζ) = 1 Z β

n

e−βHn(ζ) d2nζ, ζ = (ζj)n

1.

(1)

• Assumptions. Q : C → R ∪ {+∞} is l.s.c., Cω-smooth, and

Q(ζ) >> log |ζ|, (ζ → ∞). A minimizer {ζj}n

1 of Hn is a Fekete-configuration.

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Frostman’s equilibrium measure

Q-energy of a Borel p.m. µ on C I(µ) :=

• log

1 |ζ − η| dµ(ζ)dµ(η) +

• Q dµ.

The equilibrium measure σ minimizes I(µ): µ p.m. Droplet S = S[Q] := supp σ. (2) Frostman: dσ(z) = χS(z) ∆Q(z) dA(z). Large deviation estimate: if {ζj}n

1 random sample, f continuous,

bounded, 1 nE(β)

n (f(ζ1) + . . . + f(ζn)) → σ(f).

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Example: Ginibre ensemble (β = 1)

Let Q(ζ) = |ζ|2. Then S = {|ζ| ≤ 1} and σ = χS dA. The process {ζi}n

1 can be interpreted as eigenvalues of an n × n-matrix

with i.i.d. centered complex Gaussian entries of variance 1/n.

• Figure : Left: Ginibre particles for β = 1. Right: boundary profiles for

β = 1, 2, 3, 4

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Sakai theory

Technical assumptions: Q is real-analytic in a nbh of S. ∆Q > 0 in a nbh of ∂S. Conclusions: Sc is an Unbounded Quadrature Domain (in wide sense of Shapiro). ∂S is a union of finitely many analytic curves. Possible singularities: cusps pointing out of S and double points.

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Droplets 1

0.5 0.5 1.0 1.5 1.0 0.5 0.5 1.0 1.0 0.8 0.6 0.4 0.2 1.0 0.5 0.5 1.0

Figure : The Deltoid is not admissible; it has three maximal 3/2 cusps. 5/2 cusp is OK.

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Droplets 2

Figure : Double point and 5/2 cusp under Laplacian growth.

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Global results

Linear statistics on (Cn, Pn) fluctn(f) =

n

• 1

f(ζj) − nσ(f), (f ∈ C∞

b (C)).

fluctn(f) converges in distribution to the normal N(ef, σ2

f ), where

ef = (1 β − 1 2)

• C

f · ∆(χS + LS), σ2

f =

• C

|∂f S|2, (L = log ∆Q). Here f S equals f in S and is harmonic and bounded in Sc. β = 1, ∂S C1-smooth, S connected, f C2-smooth. (MAH 2011) β > 0, f smooth, supported in the bulk. (BBNY 2016)

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Intensity functions

Let Nǫ(η) number of ζj which hit D(η, ǫ). 1-point function: Rn(η) = lim

ǫ→0

E(β)

n (Nǫ(η))

ǫ2 . 2-point function: Rn,2(η1, η2) = lim

ǫ→0

E(β)

n (Nǫ(η1) · Nǫ(η2))

ǫ4 . If β = 1, the process is determinantal, Rn,k(η1, . . . , ηk) = det

• Kn(ηi, ηj)

k

i,j=1 .

Here Kn is a "correlation kernel” = reprokernel for Wn := {q · e−nQ/2; degree(q) < n} ⊂ L2.

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Ward’s identity

Let {ζj}n

1 system. For smooth ψ define r.v.’s

Aψ = 1 2

n

• j=k

ψ(ζj) − ψ(ζk) ζj − ζk , Bψ = n

n

• 1

∂Q(ζj)ψ(ζj), Cψ =

n

• 1

∂ψ(ζj).

Theorem

For all ψ En(β · (Aψ − Bψ) + Cψ) = 0. This is an implicit relation between Rn and Rn,2. (Proof: reparametrization invariance of the partition function Zn :=

• Cn e−βHn dVn.)

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Microscopic scale

Fix p ∈ S. rn = rn(p) satisfies: n ·

• D(pn,rn)

∆Q(ζ) dA(ζ) = 1. Regular case: If ∆Q(p) > 0 then rn ∼ 1

• n∆Q(p)

. Vanishing equilibrium density to order k: If k is smallest s.t. ∆kQ(p) > 0 then rn ∼ (k[(k − 1)!]2 ∆kQ(p) )1/2k · n−1/2k.

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Rescaled system

zj = rn(ζj − p).

Figure : Left: a moving point pn approaching a cusp. Right: the profile of a translation invariant "candidate” for the micro-density at pn, β = 1.

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Rescaling at bulk singularities

Figure : These figures show the repelling effect of inserting a point charge close to a bulk singularity caused by vanishing equilibrium density.

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Free boundary vs hard edge

Figure : The hard edge is obtained by redefining Q = +∞ outside S. The intensity has been computed for β = 1.

Free boundary ↔ GFF with free BCs. Hard edge ↔ GFF with Neumann BCs. (Joint w/ H.-J. Tak.)

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Gaussian field with Dirichlet BCs in the disk

Figure : A field approximation Φn. The figure on the right shows the level curve Φn + h = 1/2 where h is harmonic measure for the upper half-circle. The level curve resembles an SLE4, in accordance with Sheffield-Schramm’s theorem.

Three relevant BCs: Dirichlet, Free, Neumann.

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Ward’s equation

Rn(z) = Rn(ζ) Rn,2(z, w) = Rn,2(ζ, η), z = rn(ζ − p), w = rn(η − p). Bn(z, w) = (Rn(z)Rn(w) − Rn,2(z, w))/Rn(z). Cn(z) := B(z,w)

z−w dA(w).

Ward’s equation: ¯ ∂Cn(z) = Rn(z) − 1 − 1 β ∆ log Rn(z) + o(1). If β = 1 then normal families show Rnk → R, Cnk → C = C, where R → C by analytic continuation. So ¯ ∂C = R − 1 − ∆ log R is an equation for the single function R.

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Translation invariance

To find a true micro-density, we need side-conditions in Ward’s equation. It is natural to assume translation invariance: R(z) = F(z + ¯ z) for some function F. The complete t.i. solution to Ward’s equation was given in AKM 14. The above might give a "physical proof” of universality, but for a mathematical proof we must rule out the possibility of non-t.i. solutions. For t.i. solutions, Ward’s equation can be written as a convolution equation and solved by Fourier analysis. For possibly non-t.i. solutions, we get a twisted convolution equation, known from Fourier analysis on the Heisenberg group. (Joint w/ J.-L. Romero.)

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Spacings

Fix 0 ∈ S and let zj = r −1

n ζj, j = 1, . . . , n.

Put Fn = {at least one particle falls in D}, ηn = P(β)

n (Fn).

Spacing at 0: s0 = min

zj∈D min k=j |zj − zk|,

{zj}n

1 ∈ Fn.

Repulsion theorem: if β > 1 then there is a constant c = c(n, β) > 0 so that P(β)

n ({s0 ≥ c · (ǫηn)

1 2(β−1) |Fn) ≥ 1 − m0ǫ,

0 < ǫ < 1, where m0 = 16c−2(ǫηn)−

1 β−1 .

Proof: (i) Estimate expected L2β-norm for weighted random Lagrange polynomials, (ii) Use Bernstein to estimate expected L2β norm of gradient, (iii) Morrey’s and Chebyshev’s inequalities give estimate for distance between zeros, with high probability.

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Crystallization

Corollary: if c < 1/(8√e) and n = n(c) large enough, then lim

β→∞ P(β) n ({s0 > c}) = 1.

Abrikosov conjecture: the right bound should be c < 21/23−1/4. Q: What patterns will emerge near a bulk singularity caused by vanishing equilibrium density?

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Hall effect

When β > 1 the intensity R has some "irregularities” near the boundary: We believe that a micro-density R is translation invariant and solves Ward: ¯ ∂C = R − 1 − 1

β∆ log R.

Q: Is there a critical "freezing temperature” 1/β0 after which crystallization takes place?

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Conical singularities

Fix c < 1. n-dependent potential Vn(ζ) = Q(ζ) + (c/n)G0(ζ) where G0 is Green’s function for S with pole at 0. Micro-scale: c +

• D(0,rn)

∆Q dA = 1. Rescale: zj = r −1

n ζj and let Rn be 1-point function of {ζj}n j=1.

Normal families: Rnk → R as distributions on C, locally uniformly on ˙ C. Microscopic potential: V0(z) = Q0(z) + 2c log |z| where Q0 is the dominant part of the Taylor expansion of Q about 0.

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Fock-Sobolev spaces

Let µ0 = e−V0dA. Fock-Sobolev space L2

a(µ0) consists of entire functions u s.t.

• |u|2e−V0 dA < ∞.

Let L0(z, w) Bergman kernel and R0(z) = L0(z, z) Bergman function. Theorem: In many situations, R = R0.

0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 2.5 3.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 2.0 2.5 3.0 5 10 15

Figure : The Bergman function R0 as a function of positive reals, for V0 = 2|z|2 − 2 log |z|, V0 = |z|2/2 + log |z|, and V0 = |z|4/2, respectively.

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Conical singularities

Micro-conformal metric ds2(z) = e−V0(z)|dz|2 = |z|−2ce−Q0(z) |dz|2. This metric has a conical singularity with total angle 2π(1 − c) at 0, i.e., the Gaussian curvature κ(z) = e2V0(z)∆V0(z) has a singularity there. This is related to CFT on Riemann surfaces of Kang-Makarov, Wiegmann et al.

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Orthogonal polynomials

Lee and Yang studied OP’s for Ginibre ensemble with log-singularity:

• 0.4
• 0.2

0.0 0.2 0.4 0.6

• 0.4
• 0.2

0.0 0.2 0.4

• 0.4
• 0.2

The zeros are located on a certain "potential theoretical skeleton”, likely to be universal.

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ESKERRIK!

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