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The Phase Transition in Five Point Energy Minimization

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The Phase Transition in Five Point Energy Minimization The Phase Transition in Five Point Energy Minimization Richard Evan Schwartz September 26, 2018 arXiv:1610.03303v3 [math.OC] 21 Nov 2016 2 Contents I Introduction and Outline 9 1 Introduction 11 1.1 The Energy Minimization Problem . 11 1.2 MainResults ........................... 13 1.3 IdeasintheProof......................... 14 1.4 ComputerCode.......................... 16 2 Outline of the Proof 19 2.1 Under-Approximation . 19 2.2 DivideandConquer ....................... 20 2.3 Symmetrization .......................... 23 2.4 Endgame.............................. 25 2.5 ComputingtheConstant . 27 II Under-Approximation 29 3 Outline of the Forcing Lemma 31 3.1 AFourChapterPlan....................... 31 3.2 TheRegularTetrahedron. 31 3.3 TheGeneralApproach . 32 3.4 Case1 ............................... 33 3.5 Case2 ............................... 35 3.6 Case3 ............................... 36 3.7 Case4 ............................... 37 3.8 HaveFunPlayingAround . 37 3 4 CONTENTS 4 Polynomial Approximations 39 4.1 RationalIntervals......................... 39 4.2 IntervalPolynomials ....................... 40 4.3 ABoundonthePowerFunctions . 41 4.4 ApproximationofPowerCombos . 41 5 Positive Dominance 45 5.1 The Weak Positive Dominance Criterion . 45 5.2 The Positive Dominance Algorithm . 46 5.3 Discussion............................. 50 6ProofofTheForcingLemma 51 6.1 Positivity of the Coefficients . 51 6.2 UnderApproximation:Case1 . 53 6.3 UnderApproximation:Case2 . 55 6.4 Under-Approximation: Case3 . 57 6.5 UnderApproximation:Case4 . 58 III Divide and Conquer 59 7 The Configuration Space 61 7.1 NormalizedConfigurations . 61 7.2 CompactnessResults . .. .. 62 7.3 TheTBPConfigurations . 64 7.4 DyadicBlocks........................... 65 7.5 A Technical Result about Dyadic Boxes . 68 7.6 VeryNeartheTBP........................ 68 8 Spherical Geometry Estimates 71 8.1 Overview.............................. 71 8.2 SomeResultsaboutCircles . 72 8.3 TheHullApproximationLemma . 74 8.4 DotProductEstimates . 76 8.5 DisorderedBlocks......................... 77 8.6 IrrelevantBlocks ......................... 78 CONTENTS 5 9 The Energy Theorem 81 9.1 MainResult............................ 81 9.2 Generalizations and Consequences . 82 9.3 A Polynomial Inequality . 83 9.4 TheLocalEnergyLemma . 85 9.5 FromLocaltoGlobal....................... 87 10 The Main Algorithm 91 10.1GradingaBlock.......................... 91 10.2 The Divide and Conquer Algorithm . 91 10.3 Results of the Calculations . 92 10.4 Subdivision Recommendation . 93 10.5 IntervalArithmetic . 94 10.6 IntegerCalculations. 96 10.7AGildedApproach ........................ 97 10.8 APotentialSpeedUp. 97 10.9Debugging............................. 98 11 Local Analysis of the Hessian 99 11.1Outline............................... 99 11.2 LowerBoundsontheEigenvalues . 100 11.3 Taylor’sTheoremwithRemainder. 101 11.4 VariationoftheEigenvalues . .102 11.5 TheBiggestTerm. .. .. .104 11.6 TheRemainingTerms . .106 11.7 TheEndoftheProof. .107 IV Symmetrization 109 12 Preliminaries 111 12.1 MonotonicityLemma . .111 12.2 AThreeStepProcess. .112 12.3BaseandBows ..........................112 12.4 ProofStrategy. .. .. .114 6 CONTENTS 13 The Domains 117 13.1 BasicDescription . .. .. .117 13.2 AffineCubicalCoverings . .119 13.3 TheCoveringsforLemma1 . .120 13.4 TheCoveringsforLemma2 . .123 14 The First Symmetrization 125 14.1Inequality1 ............................125 14.2Inequality2 ............................127 14.3 ResultsaboutTriangles. .128 14.4Inequality3 ............................132 15 The Second Symmetrization 135 15.1Inequality1 ............................135 15.2Inequality2 ............................139 15.3 ImprovedMonotonicity. .140 15.4Inequality3 ............................141 15.5Discussion .............................143 V Endgame 145 16TheBabyEnergyTheorem 147 16.1TheMainResult .........................147 16.2 AGeneralEstimate. .149 16.3 AFormulafortheEnergy . .150 16.4 TheFirstEstimate . .151 16.5ATableofValues.........................152 16.6 TheSecondEstimate . .153 17 Divide and Conquer Again 155 17.1TheGoal .............................155 17.2 TheIdealCalculation. .156 17.3 Floating Point Calculations . 157 17.4 ConfrontingthePowerFunction . 159 17.5 The Rational Calculation . 161 17.6Discussion .............................162 CONTENTS 7 18 The Tiny Piece 163 18.1Overview..............................163 18.2 TheProofModuloDerivativeBounds . 164 18.3TheFirstBound .........................166 18.4 TheSecondBound . .. .. .169 19 The Final Battle 173 19.1Discussion .............................173 19.2 ProofOverview ..........................174 19.3 TheCriticalInterval . .175 19.4 TheSmallInterval . .177 19.5 TheLongInterval. .178 19.6TheEnd..............................180 8 CONTENTS Preface This monograph is a rigorous computer assisted analysis of the extent to which the triangular bi-pyramid (TBP) is the minimizer, amongst 5-point configurations on the sphere, with respect to a power law potential. Our main result settles a long-standing open question about this, but perhaps some people will object to both the length of the solution and to the computer- assisted nature. On the positive side, the proof divides neatly into 4 modular pieces, and there is a clear idea behind the attack at each step. −s Define the Riesz s-potential Rs(r) = sign(s) r . We prove there exists such that the TBP is the ...15.0480773927797 = ש a computable number The .(ש ,unique minimizer with respect to Rs if and only if s ( 2, 0) (0 is pronounced “shin”. Our result does∈ not− say what∪ happens ש Hebrew letter though we do prove that in the tiny ,ש for power law potentials much beyond ,the minimizer is a pyramid with square base. Thus [15+25/512 ,ש) interval we prove the existence of the phase transition that was conjectured to exist in the 1977 paper [MKS] by T. W. Melnyk, O. Knop, and W. R. Smith. This is still a draft of the monograph. The mathematics is all written down and the computer programs have all been run, but I am not sure if the programs are in their final form. While the programs are now pretty well documented and organized, I might still like to do more in this direction. Meanwhile, I am happy to answer any questions about the code. I thank Henry Cohn, John Hughes, Abhinav Kumar, Curtis McMullen, Ed Saff, and Sergei Tabachnikov, for discussions related to this monograph. I would specially like to thank Alexander Tumanov for his great idea about this problem, and Jill Pipher for her enthusiasm and encouragement. I thank my wife Brienne Brown for suggesting the name of the phase-transition constant. I thank my family for their forbearance while I worked on this project like Ahab going after the whale. I thank I.C.E.R.M. for facilitating this project. My interest in this prob- lem was rekindled during discussions about point configuration problems around the time of the I.C.E.R.M. Science Advisory Board meeting in Nov. 2015. I thank the National Science Foundation for their continued support, currently in the form of the Research Grant DMS-1204471. Finally, I thank the Simons Foundation for their support, in the form of a Simons Sabbatical Fellowship. Part I Introduction and Outline 9 Chapter 1 Introduction 1.1 The Energy Minimization Problem Let S2 denote the unit sphere in R3 and let P = p , ..., p be a finite list { 1 n} of distinct points on S2. Given some function f : (0, 2] R we can compute the total f-potential → (P )= f( p p ). (1.1) Ef k i − jk i<j X For fixed f and n, one can ask which configuration(s) minimize (P ). Ef For this problem, the energy functional f = Rs, where −s Rs(r) = sign(s)r , (1.2) is a natural one to consider. When s > 0, this is called the Riesz potential. When s< 0 this is called the Fejes-Toth potential. The case s = 1 is specially called the Coulomb potential or the electrostatic potential. This case of the energy minimization problem is known as Thomson’s problem. See [Th]. The case of s = 1, in which one tries to maximize the sum of the distances, is − known as Polya’s problem. There is a large literature on the energy minimization problem. See [F¨o] and [C] for some early local results. See [MKS] for a definitive numerical study on the minimizers of the Riesz potential for n relatively small. The website [CCD] has a compilation of experimental results which stretches all the way up to about n = 1000. The paper [SK] gives a nice survey of results, with an emphasis on the case when n is large. See also [RSZ]. The paper 11 12 CHAPTER 1. INTRODUCTION [BBCGKS] gives a survey of results, both theoretical and experimental, about highly symmetric configurations in higher dimensions. When n =2, 3 the problem is fairly trivial. In [KY] it is shown that when n =4, 6, 12, the most symmetric configurations – i.e. vertices of the relevant Platonic solids – are the unique minimizers for all Rs with s ( 2, ) 0 . See [A] for just the case n = 12 and see [Y] for an earler,∈ − partial∞ −{ result} in the case n = 4, 6. The result in [KY] is contained in the much more general and powerful result [CK, Theorem 1.2] concerning the so-called sharp configurations. The case n = 5 has been notoriously intractable. There is a general feeling that for a wide range of energy choices, and in particular for the Riesz potentials, the global minimizer is either the triangular bi-pyramid or else some pyramid with square base. The triangular bi-pyramid (TBP) is the configuration of 5 points with one point at the north pole, one at the south pole, and three arranged in an equilateral triangle around
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