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Spectra of Infinite Graphs Via Freeness with Amalgamation

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Spectra of Infinite Graphs Via Freeness with Amalgamation Spectra of Infinite Graphs via Freeness with Amalgamation Jorge Garza-Vargas Archit Kulkarni [email protected] [email protected] UC Berkeley UC Berkeley January 15, 2021 Abstract We use tools from free probability to study the spectra of Hermitian operators on infinite graphs. Special attention is devoted to universal covering trees of finite graphs. For operators on these graphs we derive a new variational formula for the spectral radius and provide new proofs of results due to Sunada and Aomoto using free probability. With the goal of extending the applicability of free probability techniques beyond universal covering trees, we introduce a new combinatorial product operation on graphs and show that, in the non-commutative probability context, it corresponds to the notion of freeness with amalgamation. We show that Cayley graphs of amalgamated free products of groups, as well as universal covering trees, can be constructed using our graph product. arXiv:1912.10137v4 [math.CO] 14 Jan 2021 1 Contents 1 Introduction 3 1.1 Results . .5 1.1.1 Universal Covers . .5 1.1.2 Amalgamated Graph Products . .8 1.2 Structure of the Paper . .9 1.3 Definitions and Conventions . 10 2 Motivation and related work 10 2.1 Density of States of Operators on Universal Covers . 10 2.2 Random Lifts . 12 2.3 Spectral Radii of Operators on Universal Covers . 13 2.4 Graph Products and Noncommutative Probability . 14 2.5 Random Walks on Groups and Free Probability . 15 3 Preliminaries on Free Probability 15 3.1 Free Probability . 16 3.2 Operator-valued Probability Spaces . 16 3.3 Freeness with Amalgamation . 17 3.4 The Amalgamated Free Product of Hilbert -modules . 18 B 3.5 The Operator-Valued R-Transform . 19 4 Band Structure of Universal Covers 20 4.1 Sunada’s Theorem via Asymptotic Freeness . 20 4.2 Spectral Splitting . 22 5 The Amalgamated Free Product for Graphs 25 5.1 Combinatorial Description . 25 5.2 Proof of Theorem 5.8..................................... 29 5.3 Amalgamated Free Product of Groups as Amalgamated Free Product of Graphs . 31 5.4 Implications for A ( ) .................................... 33 T G 6 Universal Covers: Algebraic Description of the Spectrum 35 6.1 Aomoto’s Equations via the R-transform . 35 6.2 Behavior of the Density of States at the Right Edge of the Spectrum . 36 6.3 Formula for the Spectral Radius . 38 7 Future Research 40 2 1 Introduction In the present work, by a graph we mean a locally finite undirected graph allowing loops and multi-edges, unless otherwise specified. Given a graph , we denote its vertex set by V( ) and its G G edge multiset by E( ). V( ); E( ) G G G The adjacency matrix, the graph Laplacian matrix, and transition matrices for symmetric random walks all fall under the umbrella of so-called “Jacobi matrices on graphs” [ABS20]. These are Hermitian operators A acting on = `2(V( )) defined by G H G X X (A f )(u) = bu f (u) + ae f (v) + 2 al f (u) (1) G e E( ) l E( ) e= u2;v G;v,u l=2 u;Gu f g f g 1 for real coefficients bu u V( ), and nonzero real coefficients ae e E( ) . (The factor 2 appears for loops f g 2 G f g 2 G because it will be convenient in our application to universal covers, but see Section 2.1 for further discussion involving half-loops.) Jacobi matrices on graphs will be the main operators considered in this work. We now recall the notion of spectral measure. Given a vertex u V( ) one can define the state 2 G τu : B( ) C as τu H ! def τu(X) = δu; Xδu ; X B( ); (2) h i 8 2 H 2 where δu ` (V( )) denotes the indicator function of the singleton u . Since A is self-adjoint, its δu 2 G f g G spectrum, which we denote by σ(A ), is contained in R. Moreover, by the functional calculus, τu σ( ) G · induces a probability measure µA ;u supported on σ(A ) with the property that µA ;u G G G Z k k x dµA ;u(x) = τu A R G G for all k Z 0. Throughout this work we refer to µA ;u as the spectral measure of A with respect to the 2 ≥ G G root u V( ). In terms of functional analysis, µA ;u is the composition of the usual projection-valued 2 G G spectral measure of A with τu. The spectra and spectralG measures of operators on infinite graphs have been extensively studied in the last several decades in different contexts. But despite significant progress in the area, current mathematical tools are still unable to answer simple questions. Since our goal is to provide a new perspective on problems of current interest, the content of this paper is a combination of new results, new proofs of existing results, and new tools for the analysis of infinite graphs, all through the lens of free probability. Our discussion includes the following families of graphs. Cayley Graphs. Let G be a finitely generated group and S G be a finite symmetric generating ⊂ set; i.e. we assume that if s S then s 1 S. We denote by Γ = Γ(G; S) the Cayley graph of Γ(G; S) 2 − 2 G with respect to S. Since S is symmetric, Γ is undirected. By definition of Γ, any weighting a : S R+ induces a weighting on the edges of Γ in the obvious way. The canonical scalar ! measure associated to AΓ is the spectral measure of Γ with respect to the root e G and the given 2 1 Note that we are deviating from the standard definition of Jacobi operator which requires the ae to be strictly positive. When the graph is a tree, gauge invariance allows to “turn” negative weights into positive without affecting the spectrum. However, in this work, allowing negative coefficients sometimes simplifies our exposition. 3 weighting. Both σ(AΓ) and µΓ;e have been studied thoroughly in the context of random walks on groups [Kes59, Woe86, Woe87, McL88, CS86]. However, several basic spectral questions about some natural Cayley graphs remain open [KFSH19]. The amalgamated free product of groups inspired the notions of free independence and freeness with amalgamation [Voi85], and since Voiculescu’s seminal work it became apparent that tools from free probability can provide important insights into the spectral theory of certain Cayley graphs. In this work we focus on a new connection, namely the role of freeness with amalgamation in the study of universal covering graphs. Universal Covering Graphs. Let be a connected graph with n vertices. One can form its universal G covering graph2 (also called universal covering tree or universal cover), which we denote by ( ). T G The universal covering graph comes with a covering map φ : ( ) . For simplicity, when is T G !G G clear from context, we will write instead of ( ). Recall that is an infinite tree if has at least T T G T G one cycle or loop and is itself when is a tree. G G Any Jacobi matrix A on can be lifted via φ to a Jacobi matrix A on , called a “periodic G G T T Jacobi matrix with period n”[ABS20] or a “pulled-back local operator” [AFH15]. We can then associate a canonical measure to A as follows. For any u V( ) we fix a representative u˜ V( ) 1 T 2 G 2 T in the fiber φ− (u), and then consider the respective state τu˜ as in (2). Now let τ T def 1 X τ = τu˜; T n u V( ) 2 G and denote by µA the induced scalar measure. We refer to µA as the density of states of A in µA accordance with theT physics and spectral theory literature. T T T It is a well known fact that is the limit, in the Benjamini-Schramm sense, of random lifts of T G (see Section 2.2). Hence, A can be regarded as a limit of random matrices. A bit of thought from the free probability perspectiveT shows that in fact A can be viewed as an operator-valued matrix with free entries (see Section4). This is the starting pointT of the present work. Previous results give an explicit description of σ(A ) and µ when has a particular structure T T G [Kes59, McK81, GM88, FTS94]. Others have made some progress in the case when is an arbitrary G graph [A+88, Aom91, SS92, Sun92, ABS20]. However, as we discuss in the last section, many fundamental questions remain open (also see[ABS20]). Amalgamated Free Product for Graphs. As mentioned above, since generic universal covering graphs are not Cayley graphs, the emergence of freeness with amalgmation in this context might seem somewhat fortuitous. Here, we clarify this connection by introducing a graph product that corresponds to the notion of freeness with amalgamation. Here is some context. Inspired by the combinatorial description of Cayley graphs of free products of groups, Quenell [Que94] introduced the free product of graphs. Only later it was understood by Accardi, Lenczewski, and Sałapata [ALS07] that this graph product is equivalent Voiculescu’s free product of Hilbert spaces [VDN92] and that free probability can be used to compute the spectra of graphs arising from Quenell’s graph product. We refer the reader to Section 2.4 for a detailed and more precise discussion. In the spirit of [Que94] and [ALS07], in this paper we define a combinatorial graph product and show that the machinery of freeness with amalgmation can be used to compute the spectra 2See Section 2.1 for a precise definition and a subtlety involving covers of loops.
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