Main Problem Combinatorial Objects of Study Main Results Summary. . . On the Limiting Distribution of Eigenvalues of Large Random d -Regular Graphs with Weighted Edges M. C. Khoury S. J. Miller Joint Mathematics Meetings 2012 (Boston, MA) M. C. Khoury, S. J. Miller Randomly Weighted Regular Graphs
Main Problem Spectra of Random Regular Graphs Combinatorial Objects of Study Weighting the Edges Main Results Eigendistributions Summary. . . Spectra of Large d -Regular Graphs Let { G i } be an infinite sequence of d -regular graphs such that the number of cycles of a given length is growing slowly relative to the number of vertices. (The technical condition.) For each G i consider its spectral measure ν G i , the uniform measure on the eigenvalues of its adjacency matrix. Theorem (McKay) The limit ν d = lim i →∞ ν G i exists and depends only on d. If we normalize so that the support of ν ′ d is [ − 1 , 1 ] , then lim d →∞ ν ′ d is the semicircle measure. M. C. Khoury, S. J. Miller Randomly Weighted Regular Graphs
Main Problem Spectra of Random Regular Graphs Combinatorial Objects of Study Weighting the Edges Main Results Eigendistributions Summary. . . Weighting the Edges Now we begin with a probability distribution µ with finite moments. Fix d and µ . Take a sequence { G i } of d -regular graphs satisfying the same technical condition. For each graph, assign each edge a weight (independently, using distribution µ ) and form the uniform probability measure on the modified adjacency matrix. Average over all possible weights to get a spectral measure ν G i ,µ . The limiting spectral measure T d ( µ ) = ν d ,µ = lim i →∞ ν G i ,µ depends only on d and µ . We are interested in the relationship between µ and T d ( µ ) . M. C. Khoury, S. J. Miller Randomly Weighted Regular Graphs
Main Problem Spectra of Random Regular Graphs Combinatorial Objects of Study Weighting the Edges Main Results Eigendistributions Summary. . . Fixed Points A natural question is whether there are fixed points in any sense. Recall the notion of rescaling a probability measure on R . S λ ( µ ) is the measure defined by S λ ( µ )( I ) = µ ( λ I ) for any interval I . For all µ, d , λ , S λ ( T d ( µ ) = T d ( S λ ( µ )) . Can we describe all the “eigendistributions”? That is, for which µ do we have T d ( µ ) = S λ ( µ ) for some λ ? M. C. Khoury, S. J. Miller Randomly Weighted Regular Graphs
Main Problem Spectra of Random Regular Graphs Combinatorial Objects of Study Weighting the Edges Main Results Eigendistributions Summary. . . Eigenmoments We work at the level of moments of distributions. S λ multiplies the k th moment of µ by a factor of λ − k . Write σ k for the k th moment of µ and ˜ σ k for the k th moment of T d ( µ ) . σ k = λ − k σ k . We seek sequences { σ k } so that ˜ σ 2 = d σ 2 , so λ = d − 1 / 2 . It turns out that ˜ σ k = d k / 2 σ k . We seek sequences { σ k } so that ˜ Without loss of generality, we can rescale so that σ 2 = 1 4 . M. C. Khoury, S. J. Miller Randomly Weighted Regular Graphs
Main Problem Key Ideas Combinatorial Objects of Study Closed Acyclic Path Patterns Main Results Moment Relations Summary. . . Key Ideas The sum of the k th powers of the eigenvalues of A is the trace of A k . Thus ˜ σ k corresponds to an average diagonal element of A k , where A is the weighted adjacency matrix. Nonzero contributions to a given diagonal entry of A k come from closed paths starting and ending at a particular vertex in the graph. The size of the contribution of any particular path depends on the weights assigned, so on average the moments σ k will appear. The technical condition means that paths including cycles are negligible. M. C. Khoury, S. J. Miller Randomly Weighted Regular Graphs
Main Problem Key Ideas Combinatorial Objects of Study Closed Acyclic Path Patterns Main Results Moment Relations Summary. . . Closed Acyclic Path Patterns Definition The set P 2 k of closed acyclic path patterns of length 2k is defined as follows. For k > 0, P 2 k contains all (equivalence classes of) strings π of 2 k symbols with the following properties. In the substring of symbols between any two consecutive 1 instances of the same symbol, every symbol appears an even number of times. Every symbol appears an even number of times. 2 Two c.a.p.p. are the same if they differ only by a relabelling of the symbols. M. C. Khoury, S. J. Miller Randomly Weighted Regular Graphs
Main Problem Key Ideas Combinatorial Objects of Study Closed Acyclic Path Patterns Main Results Moment Relations Summary. . . Closed Acyclic Path Patterns aabccbaa ∈ P 8 , but aabccbcc �∈ P 8 aabccbaa = bbdppdbb = ααβγγβαα Because we always count up to this equivalence, each P k is a finite set. P 2 k parametrizes all possible types of closed paths of length 2 k starting (and ending) at a given vertex in a large tree (or locally treelike graph) with plenty of edges at each vertex. M. C. Khoury, S. J. Miller Randomly Weighted Regular Graphs
Main Problem Key Ideas Combinatorial Objects of Study Closed Acyclic Path Patterns Main Results Moment Relations Summary. . . Closed Acyclic Path Patterns: Small Examples P 2 = { aa } P 4 = { aaaa , aabb , abba } P 6 = { aaaaaa , aaaabb , aabbaa , aabbbb , aabbcc , abbaaa , abbacc , aaabba , aabccb , abbbba , abbcca , abccba } M. C. Khoury, S. J. Miller Randomly Weighted Regular Graphs
Main Problem Key Ideas Combinatorial Objects of Study Closed Acyclic Path Patterns Main Results Moment Relations Summary. . . M. C. Khoury, S. J. Miller Randomly Weighted Regular Graphs
Main Problem Key Ideas Combinatorial Objects of Study Closed Acyclic Path Patterns Main Results Moment Relations Summary. . . Moment Relations What we end up with is a sum of the following form. � σ 2 k = ˜ m ( π ) w σ ( π ) π ∈ P 2 k m ( π ) is a polynomial in d corresponding to how many way the pattern can be realized in a d -ary tree. w σ ( π ) is product of moments σ i corresponding to how many times each symbol appears in the pattern. In particular: The odd moments vanish. For even k , ˜ σ k depends only on σ 2 , σ 4 , . . . , σ k . M. C. Khoury, S. J. Miller Randomly Weighted Regular Graphs
Main Problem Key Ideas Combinatorial Objects of Study Closed Acyclic Path Patterns Main Results Moment Relations Summary. . . Moment Relations: Examples ˜ σ 2 = d σ 2 σ 4 = d σ 4 + 2 d ( d − 1 ) σ 2 ˜ 2 σ 6 = d σ 6 + 6 d ( d − 1 ) σ 4 σ 2 +[ 3 d ( d − 1 ) 2 + 2 d ( d − 1 )( d − 2 )] σ 3 ˜ 2 8 d ( d − 1 ) σ 6 σ 2 + 6 d ( d − 1 ) σ 2 σ 8 = d σ 8 ˜ + 4 [ 16 d ( d − 1 ) 2 + 12 d ( d − 1 )( d − 2 )] σ 4 σ 2 + 2 [ 4 d ( d − 1 ) 3 + 8 d ( d − 1 ) 2 ( d − 2 ) + · · · + · · · + 2 d ( d − 1 )( d − 2 )( d − 3 )] σ 4 2 M. C. Khoury, S. J. Miller Randomly Weighted Regular Graphs
Main Problem Unique Existence of Eigendistributions Combinatorial Objects of Study Limiting Moments of Eigendistributions Main Results Improving the Error Estimate Summary. . . Theorem 1 Theorem (Unique Existence) For each d ≥ 2 , there exists a unique sequence of eigenmoments σ ⋆ k = σ ⋆ k ( d ) satisfying σ ⋆ 2 = 1 / 4 . Furthermore, σ ⋆ k ( d ) is a rational function of d. σ ⋆ k = 0 identically for odd k . The proof is straightforward by induction using the moment relations. M. C. Khoury, S. J. Miller Randomly Weighted Regular Graphs
Main Problem Unique Existence of Eigendistributions Combinatorial Objects of Study Limiting Moments of Eigendistributions Main Results Improving the Error Estimate Summary. . . Proof Sketch Substitute σ k = σ ⋆ σ k = d k / 2 σ ⋆ k and ˜ k into the moment relation. d k / 2 σ ⋆ k = � π ∈ P k m ( π ) w σ ⋆ ( π ) σ ⋆ k appears only once on the right hand side, for the path that uses only one edge. ( d k / 2 − d ) σ ⋆ k = � k m ( π ) w σ ⋆ ( π ) , where only strictly π ∈ P ′ smaller moments now appear on the right hand side. M. C. Khoury, S. J. Miller Randomly Weighted Regular Graphs
Main Problem Unique Existence of Eigendistributions Combinatorial Objects of Study Limiting Moments of Eigendistributions Main Results Improving the Error Estimate Summary. . . Small Examples We begin with σ ⋆ 2 = 1 / 4 and solve recursively. d 2 σ ⋆ 4 = d σ ⋆ 4 + 2 d ( d − 1 )( σ ⋆ 2 ) 2 This gives σ ⋆ 4 = 1 / 8. d 3 σ ⋆ 6 = 2 +[ 3 d ( d − 1 ) 2 + 2 d ( d − 1 )( d − 2 )]( σ ⋆ d σ ⋆ 6 + 6 d ( d − 1 ) σ ⋆ 4 σ ⋆ 2 ) 3 This gives σ ⋆ 6 = 5 / 64. Note the appearance of the moments of the semicircle distribution and the absence of d . M. C. Khoury, S. J. Miller Randomly Weighted Regular Graphs
Main Problem Unique Existence of Eigendistributions Combinatorial Objects of Study Limiting Moments of Eigendistributions Main Results Improving the Error Estimate Summary. . . Small Examples We begin with σ ⋆ 2 = 1 / 4 and solve recursively. d 2 σ ⋆ 4 = d σ ⋆ 4 + 2 d ( d − 1 )( σ ⋆ 2 ) 2 This gives σ ⋆ 4 = 1 / 8. d 3 σ ⋆ 6 = 2 +[ 3 d ( d − 1 ) 2 + 2 d ( d − 1 )( d − 2 )]( σ ⋆ d σ ⋆ 6 + 6 d ( d − 1 ) σ ⋆ 4 σ ⋆ 2 ) 3 This gives σ ⋆ 6 = 5 / 64. Note the appearance of the moments of the semicircle distribution and the absence of d . 7 1 However, σ ⋆ 8 = 128 + 128 ( d 2 + d + 1 ) . M. C. Khoury, S. J. Miller Randomly Weighted Regular Graphs
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