Logarithmic and Riesz Equilibrium for Multiple Sources on the Sphere - PowerPoint PPT Presentation

OPCOP 2017, Peter Dragnev, IPFW Logarithmic and Riesz Equilibrium for Multiple Sources on the Sphere — the Exceptional Case . D. Dragnev ∗ P Indiana University-Purdue University Fort Wayne (IPFW) ∗ with J. Brauchart - TU Graz, E. Saff - Vanderbilt, R. Womersley - UNSW

OPCOP 2017, Peter Dragnev, IPFW Outline • OPC, OP and external field problems in C • OPC on S d - why minimize energy? • OPC separation - external field on S d • Multiple sources Log external fields on S 2 • Multiple sources ( d − 2 ) -external fields on S d

OPCOP 2017, Peter Dragnev, IPFW OPC, OP and external field problems in C Classical energy problems • Electrostatics - capacity, equilibrium measures; • Geometry - transfinite diameter (OPC); • Polynomials - Chebyshev constant (OP) • Classical theorem in potential theory External field problems • Characterization theorem of weighted equilibrium • Examples • Applications to orthogonal polynomials on the real line Constrained energy problems • Characterization theorem of constrained equilibrium • Examples • Applications to discrete orthogonal polynomials

OPCOP 2017, Peter Dragnev, IPFW Classical energy problem and equilibrium measure Electrostatics - capacity of a conductor cap ( E ) E - compact set in C , µ ∈ M ( E ) - probability measure on E ; Equilibrium occurs when potential ( logarithmic ) energy I ( µ ) is minimized. � � V E := inf { I ( µ ) := − log | x − y | d µ ( x ) d µ ( y ) } , cap ( E ) := exp ( − V E ) Remark : For Riesz energy we use Riesz kernel | x − y | − s instead. Equilibrium measure µ E If cap ( E ) > 0, there exists unique µ E : I ( µ E ) = V E . � Potential satisfies U µ E ( x ) = − log | x − y | d µ ( y ) = C on E . Examples • E = T , d µ E = d θ/ ( 2 π ) √ • E = [ − 1 , 1 ] , d µ E = dx /π 1 − x 2

OPCOP 2017, Peter Dragnev, IPFW Classical theorem in potential theory Geometry - transfinite diameter of a set δ ( E ) E ⊂ C - compact, Z n = { z 1 , z 2 , . . . , z n } ⊂ of E maximize Vandermond Fekete points (OPC)   2 / ( n ( n − 1 )) �   δ n ( E ) := max | z i − z j | , δ ( E ) := lim δ n ( E ) Z n ⊂ E 1 ≤ i < j ≤ n Approximation Theory - Chebyshev constant τ ( E ) E - compact set in C , T n ( x ) - monic polynomial of minimal uniform norm (OP for L 2 -norm); t n ( E ) := min {� x n − p n − 1 ( x ) � : p n − 1 ∈ P n − 1 } , τ ( E ) = lim t n 1 / n ( E ) Classical theorem (Fekete, Szegö) cap ( E ) = δ ( E ) = τ ( E )

OPCOP 2017, Peter Dragnev, IPFW External field problem - characterization theorem Electrostatics - add external field E - closed set in C , Q - lower semi-continuous on E (growth cond.); � V Q := inf { I Q ( µ ) := I ( µ ) + 2 Q ( x ) d µ ( x ) Theorem - Weighted equilibrium µ Q There exists unique µ Q : I Q ( µ Q ) = V Q . Potential satisfies: U µ Q ( x ) + Q ( x ) ≥ C q.e. on E U µ Q ( x ) + Q ( x ) ≤ C on supp ( µ Q ) . Applications • Orthogonal polynomials on real line • Approximation of functions by weighted polynomials • Integrable systems, Random matrices

OPCOP 2017, Peter Dragnev, IPFW Constrained energy problem Electrostatics - add external field and upper constraint Add constraint measure σ : σ ( E ) > 1 � V σ Q := inf { I Q ( µ ) := I ( µ ) + 2 Q ( x ) d µ ( x ) : µ ≤ σ Applications: Discrete orthogonal polynomials, random walks, numerical linear algebra methods, etc. Theorem (R ’96, Saff-D. ’97) - Constrained equilibrium λ σ Q There exists unique λ σ Q : I Q ( λ σ Q ) = V σ Q . Potential satisfies: U λ σ Q ( x ) + Q ( x ) ≥ C on supp ( σ − λ σ Q ) U λ σ Q ( x ) + Q ( x ) ≤ C on supp ( µ ) . Theorem (Saff-D. ’97) - Constrained vs. weighted equilibrium If Q ≡ 0, then σ − λ σ = ( � σ � − 1 ) µ Q for Q ( x ) = − U σ ( x ) / ( � σ � − 1 )

OPCOP 2017, Peter Dragnev, IPFW OPC on S 2 - why minimize energy? Electrostatics: Thomson Problem (1904) - (“plum pudding” model of an atom) Find the (most) stable (ground state) energy configuration ( code ) of N classical electrons (Coulomb law) constrained to move on the sphere S 2 . Generalized Thomson Problem ( 1 / r s potentials and log ( 1 / r ) ) A code C := { x 1 , . . . , x N } ⊂ S n − 1 that minimizes Riesz s -energy � � 1 1 E s ( C ) := | x j − x k | s , s > 0 , E log ( ω N ) := log | x j − x k | j � = k j � = k is called an optimal s -energy code .

OPCOP 2017, Peter Dragnev, IPFW OPC on S 2 - why minimize energy? Coding: Tammes Problem (1930) A Dutch botanist that studied modeling of the distribution of the orifices in pollen grain asked the following. Tammes Problem (Best-Packing, s = ∞ ) Place N points on the unit sphere so as to maximize the minimum distance between any pair of points. Definition Codes that maximize the minimum distance are called optimal (maximal) codes . Hence our choice of terms.

OPCOP 2017, Peter Dragnev, IPFW OPC on S 2 - why minimize energy? Nanotechnology: Fullerenes (1985) - (Buckyballs) Vaporizing graphite, Curl, Kroto, Smalley, Heath, and O’Brian discovered C 60 (Chemistry 1996 Nobel prize) Duality structure: 32 electrons and C 60 .

OPCOP 2017, Peter Dragnev, IPFW Other "Fullerenes" Under the lion paw Montreal biosphere

OPCOP 2017, Peter Dragnev, IPFW Computational "Fulerene" - Rob Womersley

OPCOP 2017, Peter Dragnev, IPFW Known OPC on S 2 Recall: Riesz Oprimal Configurations A configuration ω N := { x 1 , . . . , x N } ⊂ S d that minimizes Riesz s -energy � � 1 1 E s ( ω N ) := | x j − x k | s , s > 0 , E 0 ( ω N ) := log | x j − x k | j � = k j � = k is called an optimal s -energy configuration . OPC on S 2 • s = 0, Smale’s problem, logarithmic points (known for N = 1 − 6 , 12); • s = 1, Thomson Problem (known for N = 1 − 6 , 12) • s = − 1, Fejes-Toth Problem (known for N = 1 − 6 , 12) • s → ∞ , Tammes Problem (known for N = 1 − 12 , 13 , 14 , 24)

OPCOP 2017, Peter Dragnev, IPFW OPC separation on S d and external fields Separation Distance δ ( ω N ) := min j � = k | x j − x k | , ω N = { x 1 , . . . , x N } N ) ≍ N − 1 / d as N → ∞ , where ω ( s ) Expect: δ ( ω ( s ) optimal for S d N Definition N = 2 ⊂ S d is A sequence of N -point configurations { ω N } ∞ well-separated if there exists some c > 0 not depending on N s.t. δ ( ω N ) ≥ c N − 1 / d for all N .

OPCOP 2017, Peter Dragnev, IPFW OPC separation on S d and external fields Separation Results for Optimal Point Configurations on S d δ ( ω ( 0 ) N ) ≥ O ( N − 1 / 2 ) d = 2 , s = 0 R-S-Z (1995) δ ( ω ( s ) 0 < s < d − 2 N ) ≥ ? δ ( ω ( d − 1 ) ) ≥ O ( N − 1 / d ) s = d − 1 Dahlberg (1978) N δ ( ω ( s ) N ) ≥ O ( N − 1 / d ) d − 1 ≤ s < d K-S-S (2007) δ ( ω ( s ) N ) ≥ β s , d N − 1 / d d − 2 ≤ s < d D-S (2007) δ ( ω ( d ) N ) ≥ O (( N log N ) − 1 / d ) s = d K-S (1998) δ ( ω ( s ) N ) ≥ O ( N − 1 / d ) s > d K-S (1998) δ ( ω ( ∞ ) ) ≥ O ( N − 1 / d ) s = ∞ Conway-Sloane N Asymptotic Results (H-vdW (1951), Bo-H-S (2007))

OPCOP 2017, Peter Dragnev, IPFW Logarithmic Points on S 2 and external field Separation Results for Logarithmic Configurations on S 2 √ δ ( ω ( 0 ) N ) ≥ ( 3 / 5 ) / N R-S-Z (1995) √ δ ( ω ( 0 ) N ) ≥ ( 7 / 4 ) / N Dubickas (1997) √ δ ( ω ( 0 ) N ) ≥ 2 / N − 1 D. (2002)

OPCOP 2017, Peter Dragnev, IPFW Logarithmic Points on S 2 and external field Separation Results for Logarithmic Configurations on S 2 √ δ ( ω ( 0 ) N ) ≥ ( 3 / 5 ) / N R-S-Z (1995) √ δ ( ω ( 0 ) N ) ≥ ( 7 / 4 ) / N Dubickas (1997) √ δ ( ω ( 0 ) N ) ≥ 2 / N − 1 D. (2002) Proof. • R-S-Z, Dubickas: Stereographical projection with South Pole in ω N . • Dragnev: Stereographical projection with North Pole in ω N . This creates external field on projections of remaining N − 1 points { z k } . All weighted Fekete points are contained in support √ of continuous MEP , i.e. | z k | ≤ N − 2, which implies estimate.

OPCOP 2017, Peter Dragnev, IPFW OPC separation on S d and external fields Approach for S d • Fix a point of ω ( s ) and consider external field Q N it generates on N the remaining n = N − 1 points. • Study continuous energy problem for this external field Q N . • Discrete energy points for Q N are contained in CEP equilibrium support. Theorem (D-Saff 2007) OPC separation on S d for d − 2 ≤ s < d � � 1 / d ) ≥ K s , d 2 B ( d / 2 , 1 / 2 ) δ ( ω ( s , d ) N 1 / d , K s , d := , N B ( d / 2 , ( d − s ) / 2 ) where B ( x , y ) denotes the Beta function. In particular, � K d − 1 , d = 2 1 / d , K s , 2 = 2 1 − s / 2 . Remark: We need Principle of Domination, de la Valleè-Pousin type theorem, and Riesz balayage, hence the restriction on s .

OPCOP 2017, Peter Dragnev, IPFW Discrete MEP on S d for d − 2 ≤ s < d Q -optimal points Let Q be an external field . Find Q -optimal configuration of n points on S d , that solve   � �   n � 1 : x k ∈ S d min | x j − x k | s + Q ( x j ) + Q ( x k )   j � = k 2007 Separation: q = 1 / ( N − 2 ) , R = 1, q n = N − 1. Q ( x ) = | x − R p | s