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High Dimensional Phenomenon in Convex Geometry and Spectral Theory of Random Graphs

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High Dimensional Phenomenon in Convex Geometry and Spectral Theory of Random Graphs High Dimensional Phenomenon in Convex Geometry and Spectral Theory of Random Graphs by Han Huang A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in the University of Michigan 2019 Doctoral Committee: Professor Mark Rudelson, Chair Professor Jinho Baik Professor Alexander Barvinok Professor Raj Rao Nadakuditi Han Huang [email protected] ORCID iD: 0000-0001-5694-7701 ©Han Huang 2019 TABLE OF CONTENTS List of Figures ..................................... iv Abstract ......................................... v Chapter 1 Introduction ..................................... 1 1.1 Geometric Functional Analysis.......................1 1.1.1 Overview of Results in Asymptotic Geometric Analysis.....3 1.2 Nodal Domains for graphs.........................9 1.3 Notations and Preliminaries......................... 12 1.3.1 Concentration on the Sphere.................... 14 1.3.2 Log-concave measures....................... 14 2 The Upper Bound in Dvoretzky theorem and Milman–Schechtman Theorem . 16 2.1 Proof of Theorem 2.3............................ 17 3 Barycenter and maximum volume ellipsoid .................... 21 3.1 Proof of Theorem 3.3............................ 22 3.1.1 Construction of Q ......................... 25 3.1.2 Proof of Theorem 3.3(1)...................... 26 3.1.3 Proof of Theorem 3.3 (2)...................... 30 3.2 The relation between the conjectures.................... 37 4 Approximation of Convex Bodies by Polytopes with few facets ......... 41 4.1 Proof of the main result........................... 44 4.1.1 Lower bound of facets....................... 45 4.1.2 Structure.............................. 46 4.1.3 Randomness............................ 48 4.1.4 The construction.......................... 49 4.2 Upper bound for small R .......................... 53 5 Ulam’s floating body ................................ 55 5.1 Introduction................................. 55 5.1.1 Metronoids............................. 55 5.1.2 Main results............................ 57 5.1.3 Some additional notation...................... 60 ii 5.2 Properties of Ulam floating bodies..................... 61 5.2.1 Basic properties.......................... 61 5.2.2 Ulam’s floating body problem................... 64 5.2.3 Connection to floating bodies.................... 66 5.2.4 Smoothness of Ulam floating bodies................ 68 5.3 Relation to p-affine surface area...................... 74 5.3.1 Preliminary results......................... 74 5.3.2 Proof of Theorem 5.3........................ 78 6 Size of nodal domains for G (n; p) graph ..................... 83 6.1 Introduction................................. 83 6.2 Tools..................................... 85 6.3 Bulk eigenvector.............................. 89 6.4 Edge Eigenvector.............................. 97 6.4.1 Outline of the proof........................ 99 6.4.2 A typical sample of M ...................... 101 6.4.3 Introduction of the shift...................... 106 > 6.4.4 Concentration of wi G (E) wj − dij + E ............. 115 6.4.5 Estimate of s (λ) .......................... 122 Bibliography ...................................... 128 iii LIST OF FIGURES FIGURES 1.1.1 Threshold Phenomenon of f (k) .........................4 1.2.1 Nodal Domains..................................9 3.1.1 B, B1, and B2 .................................. 24 4.1.1 Inner Product for shifted unit vectors...................... 47 5.1.1 H (δ; θ) is the hyperplane orthogonal to θ that cuts a set Cδ (θ) of volume δ from a convex body K. The point xθ is the barycenter of Cδ (θ). Then, Kδ ⊂ fx : hx; θi ≤ hyθ; θig while δ (K) ⊂ K \ fx : hx; θi ≤ hxθ; θig.... 56 + + 5.2.1 Hβ \ (K \ H (θ; δ)) .............................. 67 5.2.2 K±(") ....................................... 69 iv ABSTRACT In this thesis, we study high dimensional phenomena arising in convexity and probablistic combinatorics. The main object of the first part is high dimensional convex bodies. We study random almost spherical sections of a convex body, which is related to Dvoretzky’s theorem. We also investigate the mass distribution in a convex body with respect to its maximum volume ellipsoid. Furthemore, we study the approximation of convex bodies by polytopes with few facets. We also construct a special class of convex bodies which we use to define affine surface area. The second part of the thesis is devoted to the study of nodal domains of Erdös Rényi graphs. An Erdös Rényi graph G (n; p) is a random graph with n vertices where any two vertices are connected by an edge with probability p independently of other edges. Consider an eigenvector of the adjacency matrix of such random graph. A nodal domain corresponding to this eigenvector is a connected component of the set of vertices where the vector has a constant sign. It was proved by Dekel et. al. that with high probability, there are exactly two nodal domains for every non-leading eigenvector. We show that the sizes of these two nodal domains are almost exactly equal to each other. v CHAPTER 1 Introduction 1.1 Geometric Functional Analysis A convex body K ⊆ Rn is a convex, compact subset with non-empty interior. The study of convex bodies can be traced back to Euclid. In the past century, functional analysis provided a new perspective on the study of convex bodies. The connection between the two subjects is simple: If we embed a finite-dimensional normed space X to Rn, its unit ball becomes an origin-symmetric convex body in Rn. Moreover, for any origin-symmetric n convex body K ⊆ R , the gauge function kxkK := inf fr > 0 : x 2 rKg defines a norm. n The most natural examples of such convex bodies are the Euclidean ball B2 , the unit cube n n which is the l1-ball B1, and, correspondingly, the l1-ball B1 . In functional analysis, Banach-Mazur distance for two n-dimensional normed spaces X; Y is defined as −1 dBM (X; Y ) = inf kT k T : T 2 GL (X; Y ) ; where GL (X; Y ) is the collection of linear isomorphisms from X to Y . Notice that dBM is not a metric but its log is. However, it is more convenient to use dBM in the study of linear operators. Suppose we embedded both X and Y to Rn (we can choose any isomorphic embedding) and let BX and BY be the corresponding unit balls of X and Y respectively. Then, the Banach Mazur distance of X and Y can be expressed in terms of BX and BY : dBM (BX ;BY ) := inf fR ≥ 1 : 9T 2 GLn (R) such that TBY ⊆ BX ⊆ RT BY g ; where GLn (R) is the general linear group of degree n. This definition is geometric: We try to rotate, stretch, and compress BY so that it lies in BX . Meanwhile, we want to scale the transformed BY by the smallest possible factor R so that it contains BX . For n n p example, for Banach Mazur distance between the cube B1 and B2 is n. The ratio of n p n the outer and inner radius of B1 is exactly n. Also, for the crosspolytope B1 , we also 1 n n p have dBM (B1 ;B2 ) = n. Influenced by this, we often consider a convex body K and its affine linear images as one object. High dimensional measure concentration also plays an important role. To give a brief idea of what is this high dimensional phenomenon means, consider a unit sphere Sn−1. If n−1 p1 we intersect S with a centered strip with width about n , the remaining covers more pC than 99% of the volume of ball. In other words, a n neighbourhood of the equator basically covers almost everything volumetrically. This contradicts our intuition from small-dimensional world, but it demonstrates the measure concentration phenomenon. Relying on measure concentration on sphere, in 1971, Milman found a quantitative proof of Dvoretzky theorem. Roughly speaking, for any origin-symmetric convex body K ⊆ Rn, there exists a k large dimension k such that a random k-dimensional subspace F satisfies dBM K \ F; B2 ≤ 1 + " with high probability. Precisely, the theorem is stated as follows: Theorem 1.1. [24, Theorem 5.2.10] Let K ⊆ Rn be an origin-symmetric convex body. Let n−1 M be the average of kθkK over S (with respect to the normalized Haar measure on n−1 2 −1 M 2 S ) and b = maxθ2Sn−1 kθkK . For any " 2 (0; 1), if k ≤ C1" log (2=") n b , then 2 νn; k (fF 2 Grn; k : (1 − ") M < k·kK\F < (1 + ") Mg) ≥ 1 − exp C2" k ; where νn; k is the normalized Haar measure on Grassmannian Grn; k. For some convex bodies, k can be of order n. For instance, an n-dimensional crosspolytope is one of such examples. Also, C; C1;C2 etc, shall denote absolute constants that may change from line to line. Among all affine linear images of K, there are two classical choices: isotropic and John’s position (named after Fritz John). Isotropic position refers to the case where K is volumetrically balanced. An isotropic convex body K is a convex body of volume 1 with the following property: Suppose X is a random vector uniformly distributed on K, then, EX = 0 and for any α; β 2 Sn−1, EhX; αi2 = EhX; βi2 (the second moments in any direction are the same). John’s position focuses on the geometric shapes of convex bodies. A convex body K is n n in John’s position if K contains B2 and B2 has the maximum volume among all ellipsoids contained in K. In general, we will let EK ⊆ K be the maximum volume ellipsoid (John ellipsoid), so that it has the maximum volume among all ellipsoids that are contained in K. (For the existence and uniqueness of the maximum volume ellipsoid EK , we refer to [24, Proposition 2.1.6] ) A theorem of F. John shows that K is isotropic in a different sense if it is in John’s position: 2 n n Theorem 1.2. [24, Theorem 2.1.10] For a convex body K ⊆ R we have EK = B2 if and n m only if there exists a finite subset in the intersection of boundaries of K and B2 , fuigi=1 ⊆ n−1 m Pm Pm > @K \ S , and positive numbers fcigi=1 such that i=1 ciui = 0 and i=1 ciuiui = In.
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