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Register a new userOn spectra of representations and graphs. Erratum
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Unfortunately the proof of the main result of [1], Theorem 1, has a flaw. Namely, Lemma 13 used in the proof of Proposition 11 is correct only under an additional assumption that the operator $A$ is normal (adjoint for the one-sided shift operator in $l^2(mathbb N)$ provides a counterexample). Below we prove a version of Lemma 13 that does not require the normality assumption and apply it to prove Proposition 11. In addition, the same version of the lemma appears in paper [2] (as Lemma 3.1) where it is used in the proof of Theorem 1.6. We also explain here how to use the new version of Lemma 13 to correct the proof of Theorem 1.6 from [2].
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In this article we study the top of the spectrum of the normalized Laplace operator on infinite graphs. We introduce the dual Cheeger constant and show that it controls the top of the spectrum from above and below in a similar way as the Cheeger constant controls the bottom of the spectrum. Moreover, we show that the dual Cheeger constant at infinity can be used to characterize that the essential spectrum of the normalized Laplace operator shrinks to one point.
We consider the asymptotic behavior as $ntoinfty$ of the spectra of random matrices of the form [frac{1}{sqrt{n-1}}sum_{k=1}^{n-1}Z_{nk}rho_n ((k,k+1)),] where for each $n$ the random variables $Z_{nk}$ are i.i.d. standard Gaussian and the matrices $rho_n((k,k+1))$ are obtained by applying an irreducible unitary representation $rho_n$ of the symmetric group on ${1,2,...,n}$ to the transposition $(k,k+1)$ that interchanges $k$ and $k+1$ [thus, $rho_n((k,k+1))$ is both unitary and self-adjoint, with all eigenvalues either +1 or -1]. Irreducible representations of the symmetric group on ${1,2,...,n}$ are indexed by partitions $lambda_n$ of $n$. A consequence of the results we establish is that if $lambda_{n,1}gelambda_{n,2}ge...ge0$ is the partition of $n$ corresponding to $rho_n$, $mu_{n,1}gemu_{n,2}ge >...ge0$ is the corresponding conjugate partition of $n$ (i.e., the Young diagram of $mu_n$ is the transpose of the Young diagram of $lambda_n$), $lim_{ntoinfty}frac{lambda_{n,i}}{n}=p_i$ for each $ige1$, and $lim_{ntoinfty}frac{mu_{n,j}}{n}=q_j$ for each $jge1$, then the spectral measure of the resulting random matrix converges in distribution to a random probability measure that is Gaussian with random mean $theta Z$ and variance $1-theta^2$, where $theta$ is the constant $sum_ip_i^2-sum_jq_j^2$ and $Z$ is a standard Gaussian random variable.
We complete the proof of Proposition 5.3 of [GJR04].
Let $G$ be a connected undirected graph with $n$, $nge 3$, vertices and $m$ edges. Denote by $rho_1 ge rho_2 ge cdots > rho_n =0$ the normalized Laplacian eigenvalues of $G$. Upper and lower bounds of $rho_i$, $i=1,2,ldots , n-1$, are determined in terms of $n$ and general Randi c index, $R_{-1}$.
We investigate spectral properties of Kirchhoff Laplacians on radially symmetric antitrees. This class of metric graphs enjoys a rich group of symmetries, which enables us to obtain a decomposition of the corresponding Laplacian into the orthogonal sum of Sturm--Liouville operators. In contrast to the case of radially symmetric trees, the deficiency indices of the Laplacian defined on the minimal domain are at most one and they are equal to one exactly when the corresponding metric antitree has finite total volume. In this case, we provide an explicit description of all self-adjoint extensions including the Friedrichs extension. Furthermore, using the spectral theory of Krein strings, we perform a thorough spectral analysis of this model. In particular, we obtain discreteness and trace class criteria, criterion for the Kirchhoff Laplacian to be uniformly positive and provide spectral gap estimates. We show that the absolutely continuous spectrum is in a certain sense a rare event, however, we also present several classes of antitrees such that the absolutely continuous spectrum of the corresponding Laplacian is $[0,infty)$.
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