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 { ζ i } n 1 ∈ C ("point charges”) in external field nQ . Energy: n n 1 � � H n = log | ζ j − ζ k | + n Q ( ζ j ) . j � = k j = 1 Boltzmann–Gibbs law: 1 e − β H n ( ζ ) d 2 n ζ, ζ = ( ζ j ) n d P n ( ζ ) = 1 . (1) Z β n Assumptions. Q : C → R ∪ { + ∞} is l.s.c., C ω -smooth, and Q ( ζ ) >> log | ζ | , ( ζ → ∞ ) . A minimizer { ζ j } n 1 of H n is a Fekete-configuration . OPCOP17: Castro Urdiales 2017 2 / Yacin Ameur (LU) Thoughts on the Coulomb Plasma 25
Frostman’s equilibrium measure Q-energy of a Borel p.m. µ on C 1 �� � I ( µ ) := log | ζ − η | d µ ( ζ ) d µ ( η ) + Q d µ. The equilibrium measure σ minimizes I ( µ ) : µ p.m. Droplet S = S [ Q ] := supp σ. (2) Frostman: d σ ( z ) = χ S ( z ) ∆ Q ( z ) dA ( z ) . Large deviation estimate: if { ζ j } n 1 random sample, f continuous, bounded, 1 n E ( β ) n ( f ( ζ 1 ) + . . . + f ( ζ n )) → σ ( f ) . OPCOP17: Castro Urdiales 2017 3 / Yacin Ameur (LU) Thoughts on the Coulomb Plasma 25
Example: Ginibre ensemble ( β = 1) Let Q ( ζ ) = | ζ | 2 . Then S = {| ζ | ≤ 1 } and σ = χ S dA . The process { ζ i } 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 OPCOP17: Castro Urdiales 2017 4 / Yacin Ameur (LU) Thoughts on the Coulomb Plasma 25
Sakai theory Technical assumptions: Q is real-analytic in a nbh of S . ∆ Q > 0 in a nbh of ∂ S . Conclusions: S c 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. OPCOP17: Castro Urdiales 2017 5 / Yacin Ameur (LU) Thoughts on the Coulomb Plasma 25
Droplets 1 1.0 1.0 0.5 0.5 � 0.5 0.5 1.0 1.5 � 1.0 � 0.8 � 0.6 � 0.4 � 0.2 � 0.5 � 0.5 � 1.0 � 1.0 Figure : The Deltoid is not admissible; it has three maximal 3/2 cusps. 5/2 cusp is OK. OPCOP17: Castro Urdiales 2017 6 / Yacin Ameur (LU) Thoughts on the Coulomb Plasma 25
Droplets 2 Figure : Double point and 5/2 cusp under Laplacian growth. OPCOP17: Castro Urdiales 2017 7 / Yacin Ameur (LU) Thoughts on the Coulomb Plasma 25
Global results Linear statistics on ( C n , P n ) n � ( f ∈ C ∞ fluct n ( f ) = f ( ζ j ) − n σ ( f ) , b ( C )) . 1 fluct n ( f ) converges in distribution to the normal N ( e f , σ 2 f ) , where e f = ( 1 β − 1 � � f · ∆( χ S + L S ) , σ 2 | ∂ f S | 2 , 2 ) f = ( L = log ∆ Q ) . C C Here f S equals f in S and is harmonic and bounded in S c . β = 1, ∂ S C 1 -smooth, S connected, f C 2 -smooth. (MAH 2011) β > 0, f smooth, supported in the bulk. (BBNY 2016) OPCOP17: Castro Urdiales 2017 8 / Yacin Ameur (LU) Thoughts on the Coulomb Plasma 25
Intensity functions Let N ǫ ( η ) number of ζ j which hit D ( η, ǫ ) . 1-point function: E ( β ) n ( N ǫ ( η )) R n ( η ) = lim . ǫ 2 ǫ → 0 2-point function: E ( β ) n ( N ǫ ( η 1 ) · N ǫ ( η 2 )) R n , 2 ( η 1 , η 2 ) = lim . ǫ 4 ǫ → 0 If β = 1, the process is determinantal , � k � R n , k ( η 1 , . . . , η k ) = det K n ( η i , η j ) i , j = 1 . Here K n is a "correlation kernel” = reprokernel for W n := { q · e − nQ / 2 ; degree ( q ) < n } ⊂ L 2 . OPCOP17: Castro Urdiales 2017 9 / Yacin Ameur (LU) Thoughts on the Coulomb Plasma 25
Ward’s identity Let { ζ j } n 1 system. For smooth ψ define r.v.’s n n n ψ ( ζ j ) − ψ ( ζ k ) A ψ = 1 � � � , B ψ = n ∂ Q ( ζ j ) ψ ( ζ j ) , C ψ = ∂ψ ( ζ j ) . 2 ζ j − ζ k j � = k 1 1 Theorem For all ψ E n ( β · ( A ψ − B ψ ) + C ψ ) = 0 . This is an implicit relation between R n and R n , 2 . (Proof: reparametrization invariance of the partition function C n e − β H n dV n . ) � Z n := OPCOP17: Castro Urdiales 2017 10 / Yacin Ameur (LU) Thoughts on the Coulomb Plasma 25
Microscopic scale Fix p ∈ S . r n = r n ( p ) satisfies: � n · ∆ Q ( ζ ) dA ( ζ ) = 1 . D ( p n , r n ) Regular case: If ∆ Q ( p ) > 0 then 1 r n ∼ . � n ∆ Q ( p ) Vanishing equilibrium density to order k : If k is smallest s.t. ∆ k Q ( p ) > 0 then r n ∼ ( k [( k − 1 )!] 2 ) 1 / 2 k · n − 1 / 2 k . ∆ k Q ( p ) OPCOP17: Castro Urdiales 2017 11 / Yacin Ameur (LU) Thoughts on the Coulomb Plasma 25
Rescaled system z j = r n ( ζ j − p ) . Figure : Left: a moving point p n approaching a cusp. Right: the profile of a translation invariant "candidate” for the micro-density at p n , β = 1. OPCOP17: Castro Urdiales 2017 12 / Yacin Ameur (LU) Thoughts on the Coulomb Plasma 25
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. OPCOP17: Castro Urdiales 2017 13 / Yacin Ameur (LU) Thoughts on the Coulomb Plasma 25
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.) OPCOP17: Castro Urdiales 2017 14 / Yacin Ameur (LU) Thoughts on the Coulomb Plasma 25
Gaussian field with Dirichlet BCs in the disk Figure : A field approximation Φ n . The figure on the right shows the level curve Φ n + h = 1 / 2 where h is harmonic measure for the upper half-circle. The level curve resembles an SLE 4 , in accordance with Sheffield-Schramm’s theorem. Three relevant BCs: Dirichlet, Free, Neumann. OPCOP17: Castro Urdiales 2017 15 / Yacin Ameur (LU) Thoughts on the Coulomb Plasma 25
Ward’s equation R n ( z ) = R n ( ζ ) R n , 2 ( z , w ) = R n , 2 ( ζ, η ) , z = r n ( ζ − p ) , w = r n ( η − p ) . B n ( z , w ) = ( R n ( z ) R n ( w ) − R n , 2 ( z , w )) / R n ( z ) . � B ( z , w ) C n ( z ) := z − w dA ( w ) . Ward’s equation: ∂ C n ( z ) = R n ( z ) − 1 − 1 ¯ β ∆ log R n ( z ) + o ( 1 ) . If β = 1 then normal families show R n k → R , C n k → C = C , where R → C by analytic continuation. So ¯ ∂ C = R − 1 − ∆ log R is an equation for the single function R . OPCOP17: Castro Urdiales 2017 16 / Yacin Ameur (LU) Thoughts on the Coulomb Plasma 25
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.) OPCOP17: Castro Urdiales 2017 17 / Yacin Ameur (LU) Thoughts on the Coulomb Plasma 25
Spacings Fix 0 ∈ S and let z j = r − 1 n ζ j , j = 1 , . . . , n . Put η n = P ( β ) F n = { at least one particle falls in D } , n ( F n ) . Spacing at 0: { z j } n s 0 = min z j ∈ D min k � = j | z j − z k | , 1 ∈ F n . Repulsion theorem: if β > 1 then there is a constant c = c ( n , β ) > 0 so that 1 P ( β ) 2 ( β − 1 ) | F n ) ≥ 1 − m 0 ǫ, n ( { s 0 ≥ c · ( ǫη n ) 0 < ǫ < 1 , 1 where m 0 = 16 c − 2 ( ǫη n ) − β − 1 . Proof: (i) Estimate expected L 2 β -norm for weighted random Lagrange polynomials, (ii) Use Bernstein to estimate expected L 2 β norm of gradient, (iii) Morrey’s and Chebyshev’s inequalities give estimate for distance between zeros, with high probability. OPCOP17: Castro Urdiales 2017 18 / Yacin Ameur (LU) Thoughts on the Coulomb Plasma 25
Crystallization Corollary: if c < 1 / ( 8 √ e ) and n = n ( c ) large enough, then β →∞ P ( β ) lim n ( { s 0 > c } ) = 1 . Abrikosov conjecture: the right bound should be c < 2 1 / 2 3 − 1 / 4 . Q: What patterns will emerge near a bulk singularity caused by vanishing equilibrium density? OPCOP17: Castro Urdiales 2017 19 / Yacin Ameur (LU) Thoughts on the Coulomb Plasma 25
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